on the emission of gamma rays by young pulsars a …dn649cn8855/kw... · 2011. 9. 22. · abstract...

ON THE EMISSION OF GAMMA RAYS BY YOUNG PULSARS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Kyle Padia Watters

November 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/dn649cn8855

© 2011 by Kyle Padia Watters. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/dn649cn8855

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Roger Romani, Primary Adviser


Chao-Lin Kuo


Peter Michelson

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

The γ-ray emission from young rotation-powered pulsars has been studied in detail.

The antenna patterns of emission are calculated for a variety of geometric emission

models. These antenna patterns vary with magnetic inclination and spindown power;

correspondingly, libraries of emission patterns are calculated across the phase space of

possible parameter combinations. These libraries allow one to explore the possible γ-

ray light curves under a given model, for a given set of input parameters. A method of

quantitative comparison was developed to allow the determination of a goodness-of-fit

between an observed pulsar light curve and all of the possible simulated light curves

for a given model. This method can localize the region of input parameter space that

produces a model’s best fit to an observed light curve. This can then be compared

to external constraints on input parameters, often supplied by observations at other

wavelengths (i.e., radio polarization and x-ray morphology). Finally, a simulation is

run to reproduce the Galactic young pulsar population as faithfully as possible. Onto

this population we apply the antenna patterns of various models in order to study the

properties of the young γ-ray pulsar population as a whole, constraining properties

of the objects and evaluating the ability of the geometric γ-ray models to reproduce

the population-wide distributions of observables.

v

To Laura

vii

Acknowledgments

No man is an island, and no thesis has but truly one author. So it is for me and for

this work. First and foremost, I must thank my advisor, Prof. Roger W. Romani. He

has been a truly outstanding guide and teacher throughout this process. His insights

have been invaluable, his drive motivating, and his enthusiasm contagious. Those

close to me can attest to the numerous times that I have emerged from his office filled

with a renewed motivation and inspired by the excitement of a new puzzle to solve.

I must thank my parents, who have been through this endeavor as they have ever

been - loving, supportive, prayerful, hilarious, giving, guiding, and loud. They are

more than I could ever ask for, and I remain confident that they are in fact the best

parents in the world.

My years at Stanford have been a great adventure, and the people here have made

it one that I will always look back on fondly. The Physics Department graduate stu-

dent entering class of 2006 is an awesome group of people. My fellow students helped

me through graduate school in so many ways: scores of late-night study sessions en-

sured that my problem sets actually got finished, parties and hijinks kept spirits high

in between said study sessions, and close camaraderie helped me realize that I wasn’t

the only one who thought this stuff was hard. For this and so much more I will always

remember Phil, Darin, Christian, David, Dan, Johan, Doug, and Conrad. Adam Van

Etten has been classmate, officemate, roommate, and friend, and has been a stalwart

of weekend adventures, intramural sports teams, and my random queries regarding

puslar nebulae. Jason Pelc would remind us that there is no written qual and no TA

requirements in Applied Physics. Ewen was cordial every time he had to explain that

ix

he did not study physics but that he chose to associate with us nonetheless.

During my time at Stanford I encountered yearly milestones, quarterly forms,

committees, reimbursements, seminar organization meetings, and a countless number

of free-food-offering events (without which I almost surely would have succumbed to

malnutrition some time around winter of second year). Successful navigation of all

this was made possible by the wonderful staff of the physics department and KIPAC.

In the front office, Maria, Violet, and Elva always greeted me with a smile and solved

any problem I could bring them. Similarly for Martha and Ziba up at SLAC. And

Christine was always just down the hall, and kept us well supplied with pizza.

I owe a great debt of gratitude to Peter Michelson for his support which allowed

me to work with Fermi and to travel to Australia to work at the Parkes Observatory,

a fantastic opportunity.

I owe it all to Steve Healey, whose singular insights into pulsar astrophysics were

largely stolen by me. At least, that’s what the acknowledgments said when I received,

from Steve, the Stanford University thesis template files. For those files, and myriad

other tips, tricks, and guides to survival as a graduate student, I am grateful to Steve,

my de facto graduate student mentor.

Finally, to Laura: my girlfriend at the outset of this adventure, my patient fiancee

in the middle, my amazing wife at the culmination. I love her more than I can express.

She makes me feel like there is nothing that we cannot do. I am so excited to be with

her for the rest of the way.

x

Contents

Abstract v

Acknowledgments ix

1 Introduction 1

1.1 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Magnetosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 The Fermi Gamma-ray Space Telescope . . . . . . . . . . . . . . . . . 3

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Compiling an Atlas of Model Light Curves 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Radio Emission . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Pulse Profile Characterization . . . . . . . . . . . . . . . . . . 11

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Determining Individual Pulsar Geometries . . . . . . . . . . . 19

2.3.2 The Polar Cap Model . . . . . . . . . . . . . . . . . . . . . . . 20

xi

2.3.3 Observed Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Light Curve Atlases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Towards a Quantitative Method of Comparison 32

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Modeling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Magnetic Field Structures . . . . . . . . . . . . . . . . . . . . 37

3.2.2 Current-induced B Perturbations . . . . . . . . . . . . . . . . 40

3.2.3 Gap Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.4 Computational Grid . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.5 Lightcurve Fitting . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Fit Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.1 External Angle Constraints . . . . . . . . . . . . . . . . . . . 49

3.3.2 Testing Field Perturbations . . . . . . . . . . . . . . . . . . . 50

3.3.3 Other EGRET Pulsars . . . . . . . . . . . . . . . . . . . . . . 53

3.3.4 Luminosities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5 Light Curve Atlases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Simulating the Galactic Pulsar Population 67

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 Simulating the Galactic Young Pulsar Population . . . . . . . . . . . 69

4.2.1 Galactic Structure . . . . . . . . . . . . . . . . . . . . . . . . 71

xii

4.2.2 Birth Spin Distribution and Pulsar Radio Emission . . . . . . 73

4.2.3 γ-ray Emission . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.4 Pulsar Samples and Detection Sensitivity . . . . . . . . . . . . 81

4.3 γ-ray Pulse Morphology . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.1 Peak Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.2 Peak Locations . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3.3 Bridge and Off-pulse Emission . . . . . . . . . . . . . . . . . . 89

4.4 γ-ray Evolution with E . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.5 Possible Amendments to the γ-ray Model . . . . . . . . . . . . . . . . 98

4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 Conclusions 108

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2.1 Magnetospheric Currents . . . . . . . . . . . . . . . . . . . . . 110

5.2.2 Gap Widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2.3 Magnetic Alignment . . . . . . . . . . . . . . . . . . . . . . . 111

5.3 Millisecond Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

xiii

List of Tables

2.1 Geometry and Beaming of Bright EGRET Pulsars . . . . . . . . . . . 23

3.1 PFF Model Angles and Efficiencies . . . . . . . . . . . . . . . . . . . 59

4.1 Model Probabilities from Peak Number Comparison . . . . . . . . . . 85

4.2 Model Probabilities from Pulse Morphology Comparisons . . . . . . . 86

xiv

List of Figures

2.1 Sample model antenna patterns . . . . . . . . . . . . . . . . . . . . . 9

2.2 Comparative atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 TPC atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 OG atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 High resolution atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 ∆ vs. δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 ∆ vs. δ with high altitude radio emission . . . . . . . . . . . . . . . . 25

2.8 Pulsar γ-ray luminosities and efficiencies . . . . . . . . . . . . . . . . 27

2.9 TPC light curve atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.10 OG light curve atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 Dipole field geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Polar cap shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Effect of currents on antenna patterns . . . . . . . . . . . . . . . . . . 43

3.4 Vela pulsar parameter fitting . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Vela pulsar model light curves . . . . . . . . . . . . . . . . . . . . . . 52

3.6 Vela pulsar parameter fitting with currents . . . . . . . . . . . . . . . 52

xv

3.7 Parameter fitting for PSR J1952+3252 . . . . . . . . . . . . . . . . . 54

3.8 Parameter fitting for PSR J1709-4429 . . . . . . . . . . . . . . . . . . 55

3.9 PSR J1709-4429 model light curves . . . . . . . . . . . . . . . . . . . 56

3.10 Parameter fitting for PSR J1057-5226 . . . . . . . . . . . . . . . . . . 57

3.11 Parameter fitting for Geminga . . . . . . . . . . . . . . . . . . . . . . 58

3.12 Geminga model light curves . . . . . . . . . . . . . . . . . . . . . . . 60

3.13 Pulsar γ-ray luminosities . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.14 Updated TPC light curve atlas . . . . . . . . . . . . . . . . . . . . . 64

3.15 Updated OG light curve atlas . . . . . . . . . . . . . . . . . . . . . . 65

3.16 fΩ-∆ overlay atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 The local Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 Comparing radio luminosity functions . . . . . . . . . . . . . . . . . . 76

4.3 Comparing initial spin period distributions . . . . . . . . . . . . . . . 79

4.4 Number of peaks as a function of model and spin-down energy . . . . 84

4.5 ∆ vs. δ as a function of spin-down energy . . . . . . . . . . . . . . . 87

4.6 Bridge and off-pulse emission . . . . . . . . . . . . . . . . . . . . . . 90

4.7 γ-ray detections with E . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.8 γ-ray detections in E and galactic latitude . . . . . . . . . . . . . . . 95

4.9 γ-ray detections in E and distance from Earth . . . . . . . . . . . . . 97

4.10 γ-ray beam surface brightness . . . . . . . . . . . . . . . . . . . . . . 99

4.11 Effects of magnetic alignment . . . . . . . . . . . . . . . . . . . . . . 102

xvi

Chapter 1

Introduction

1.1 Pulsars

Discovered by their radio emission in 1967 (Hewish et al. 1968), identified as rotating

neutron stars the following year (Gold 1968; Pacini 1968), studied at a multitude of

wavelengths in the intervening decades, and yet still shrouded in a great amount of

scientific mystery, pulsars are a fascinating class of objects populating our galaxy. The

reasons that pulsars are so scientifically fascinating are not hard to understand, for

they are home to some of the most exotic conditions in the universe: densities so high

that any further collapse will lead to a black hole, magnetic fields above the quantum

electrodynamical critical field strength, rotation rates so high that corotating particles

travel at relativistic velocities. Within a few years of their discovery in the radio band,

pulsars were observed to emit γ rays as well (Browning, Ramsden, & Wright 1971).

The Energetic Gamma Ray Experiment Telescope (EGRET) on board the Compton

Gamma Ray Observatory (CGRO), launched in 1991, uncovered the beginnings of

the Galactic γ-ray pulsar population with a half dozen firm detections of pulsations

above 100 MeV (Hartman et al. 1999).

The observed emission from these pulsars is powered by the loss of rotational

kinetic energy, as the small initial spin periods increase and the star “spins down;”

1

2 CHAPTER 1. INTRODUCTION

correspondingly, these objects are often referred to as “rotation-powered pulsars.”

Almost two thousand pulsars have been discovered in our galaxy to date, the vast

majority of which were found with radio telescopes. However, the amount of energy

emitted by a pulsar in its radio beam is but a tiny fraction of the total energy output

of the star. For those pulsars which are emitting γ rays as well, the energy output in

γ rays far exceeds that output at radio wavelengths. Indeed, for most γ-ray pulsars,

the energy output in γ rays dominates the entire electromagnetic emission of the star,

which can include optical, ultraviolet, and x-ray emission as well. To best understand

the pulsar engine, then, we must come to understand the γ-ray production mechanism

and site.

1.2 The Magnetosphere

Different theories of γ-ray production have posited different emission geometries and

different observational characteristics of pulsar γ-ray emission. A long-standing de-

bate in the field centered over the location of the γ-ray emitting region, with theories

falling in to one of two main categories: low-altitude, near surface emission or high-

altitude, outer magnetosphere emission. For the former, γ-ray emission was thought

to originate within a couple stellar radii of the neutron star surface (a few dozen

kilometers at most). This close to the star the gravity is so strong that general rela-

tivistic corrections become important and small perturbations in the magnetic field

cause deviation from the otherwise standard dipole structure.

The high-altitude scenario, on the other hand, requires special relativistic correc-

tions, as particles move with magnetic field lines at large fractions of the speed of

light. Indeed, one of the geometric surfaces of interest in outer magnetospheric mod-

els is the so-called “light cylinder.” Centered on the rotation axis, the light cylinder

is the surface on which a particle must be traveling at the speed of light in order to

maintain corotation with the neutron star. For some of the faster pulsars the light

cylinder radius can be as little as a few thousand kilometers.

1.3. THE FERMI GAMMA-RAY SPACE TELESCOPE 3

More than halfway through the first decade of the new millenium, it was still very

difficult to discriminate between these various γ-ray pulsar models. The limited size of

the γ-ray pulsar population, combined with the small amount of data and large errors

for the objects that were known, made it impossible to differentiate observationally

between the predictions of the models. New missions, returning new data, would be

required to continue the hunt for the solution to the pulsar problem.

1.3 The Fermi Gamma-ray Space Telescope

On June 11, 2008 NASA successfully launched the Gamma-ray Large Area Space

Telescope (GLAST) into a low Earth orbit. Later renamed the Fermi Gamma-ray

Space Telescope in honor of Enrico Fermi, the satellite contains two primary instru-

ments: the Gamma Burst Monitor and the Large Area Telescope (LAT). The LAT

is an array of sixteen identical towers, arranged in a 4 × 4 grid. Each tower contains

a converter/tracker and a calorimeter which together measure the properties of the

e+/e− shower created by an incident γ ray. From this shower, the instrument is able

to reconstruct the incident direction and total energy of the γ ray that initiated the

shower. Full information on the Fermi LAT can be found in Atwood et al. (2009).

For γ-ray pulsar observations, the LAT is an incredible observational upgrade

over its closest predecessor, EGRET. The LAT boasts a sensitivity ∼25 times greater

than EGRET, combined with source localization ∼30 times better and the ability to

constantly monitor the full sky. Where EGRET had firm detections of six pulsars

after nearly ten years of operations, the LAT had a catalog of 46 pulsars with only

six months of data (Abdo et al. 2010a). With this new flood of γ-ray pulsar data we

continue the study of the pulsar problem, focusing primarily on the γ-ray production

mechanism and geometry.

4 CHAPTER 1. INTRODUCTION

1.4 Outline

This thesis contains efforts to understand and constrain the pulsar magnetosphere,

especially the geometry of the γ-ray emitting region. In Chapter 2 we develop a pulsar

magnetosphere code that allows us to simulate a variety of γ-ray emission models.

These models are explored over the full range of input parameters, and the results

are compiled graphically as an atlas of γ-ray pulsar light curves. In Chapter 3 we

develop an algorithm to quantitatively compare a model light curve to an observed

light curve. This algorithm is applied to the full model sets described in Chapter 2, in

order to determine the input parameters that create the best model fit to the data. In

Chapter 4 we simulate a full Galactic population of pulsars, onto which we can apply

any of the geometric γ-ray emission models. This allows us to make comparisons not

only for single objects, but for the population as a whole. We study distributions of

observables, trends and correlations, and evaluate for the first time the efficacy of a

model as applied to the sum of the objects. Finally, in Chapter 5, we summarize our

findings and mention prospects for future projects.

Chapter 2

Compiling an Atlas of Model Light

Curves

This chapter is based on “An Atlas for Interpreting γ-Ray Pulsar Light Curves” by K. P. Watters et al. 2009, ApJ,

695, 1289.

We have simulated a population of young spin-powered pulsars and computed the

beaming pattern and light curves for the three main geometrical models: polar cap

emission, two-pole caustic (“slot gap”) emission and outer magnetosphere emission.

The light curve shapes depend sensitively on the magnetic inclination α and viewing

angle ζ . We present the results as maps of observables such as peak multiplicity

and γ-ray peak separation in the (α, ζ) plane. These diagrams can be used to locate

allowed regions for radio-loud and radio-quiet pulsars and to convert observed fluxes

to true all-sky emission.

5

6 CHAPTER 2. COMPILING AN ATLAS OF MODEL LIGHT CURVES

2.1 Introduction

With the successful launch of the Fermi Gamma-ray Space Telescope, formerly

GLAST, many pulsars are expected to be detected in the γ-ray band. This is an

important opportunity since ∼GeV γ-ray emission dominates the observed electro-

magnetic output of young pulsars. Also, some pulsars are expected to be detected in

the γ-ray band but undetectably faint in the radio. The natural interpretation is that

the radio and γ-ray antenna patterns differ. Indeed, the observed pulsars have very

different radio and γ-ray pulse profiles. Since the observed pulse profile is simply a

cut through this antenna pattern for the Earth line-of-sight angle ζE , interpretation

of the radio and γ-ray profile requires a beaming model. When the spin geometry is

known, observed pulses can be used to select the best model. Conversely, for a given

model, the pulse pattern can restrict the spin orientation. Finally, beaming correc-

tions can be used to relate the observed line-of-sight flux to the full-sky emission.

Thus, aided by beaming models, γ-ray pulsar data can give a new window into the

energetics and evolution of neutron stars in the Galaxy.

There are three main geometrical models for the γ-ray pulse beaming, each tied to

locations in the magnetosphere where force-free perfect MHD ( ~E · ~B = 0) conditions

break down and the uncanceled rotation-induced EMF ~E ≈ r~Ω× ~B/c causes particle

acceleration and radiation. These are the so-called magnetospheric gaps. The first

γ-ray (and radio) pulsar models traced emission to acceleration zones on the field

lines at very low altitudes above each magnetic pole (although Daugherty & Harding

1996 argued that such “polar cap” model emission could extend to ∼ 2R∗). Another

major class of models posits “Holloway” gaps above the null charge surface, extending

toward the light cylinder (the “outer gap” model; see Cheng, Ho, & Ruderman 1986;

Romani 1996). More recently, it has been argued that the true null charge location

depends on currents in the gap (Hirotani 2006), allowing the start of the outer gap

to migrate inward towards the stellar surface. Conversely, it has been argued that

acceleration at the rims of the polar caps may extend to very high altitude (Muslimov

& Harding 2004), effectively pushing the polar cap activity outward. Together these

effects suggest that emission can occur over a large fraction of the boundary of the

2.2. SIMULATION METHOD 7

open zone. One model that has been suggested for this geometry makes low-altitude

emission visible from one hemisphere and higher-altitude emission (i.e., above the null

charge surface) visible from the other. This is the “two-pole caustic” (TPC) model

(Dyks & Rudak 2003), which is thus intermediate between the polar cap (PC) and

outer gap (OG) pictures.

2.2 Simulation Method

We start by building a library of pulsar antenna patterns. We follow standard prac-

tice in basing our beaming on the retarded potential (Deutsch 1955) magnetic dipole

fields. This has the appreciable advantage of analytic simplicity, allowing rapid com-

putations. Numerical models of force-free magnetospheres do exist (Spitkovsky 2006),

but such models do not have the resolution required for the sort of beaming compu-

tations pursued here and in any case do not, by definition, include acceleration gaps.

Still, it should be remembered that magnetospheric charges and currents may well

have an observable effect on the field structure and pulse shapes computed here.

We model the last closed field line surface (those field lines tangential to the speed

of light cylinder at r⊥ = rLC = cP/2π), tracing these field lines to the polar cap

surface at the stellar radius r∗. The relevant magnetospheric features in the OG and

TPC models are tied to the light cylinder and locations identified as fractions thereof;

thus, the period of the pulsar is an unnecessary parameter in mapping the structure

of the magnetosphere. Models are computed across all possible magnetic inclinations

α. We follow the field line structure to < 10−4rLC , allowing precision mapping of

the PC emission region even for pulsar periods >1 s. A single magnetosphere model

can serve a range of pulsar periods by truncating the grid at larger fractions of rLC

for shorter period pulsars, thereby maintaining the same physical distances. General

relativistic effects make small changes for r < 3r∗ but are not included in the present

sums.

In all models, the acceleration gap is inferred to arise near the open zone boundary


tied to the last closed field lines. For the PC picture, we simply follow radiation

emitted tangentially to these field lines at altitudes ≤ 1R∗. In the original definition

of the TPC model (Dyks & Rudak 2003) the emission was similarly placed on the last

closed field lines but extended to an altitude r < rLC and a perpendicular distance

r⊥ < 0.75rLC, where rLC is the perpendicular distance to the light cylinder from the

rotation axis. In the outer gap (OG) model the emission is from a full set of open

field lines associated with the closed zone surface, but pair production and radiation

are expected to start above the null-charge surface [~Ω · ~B(RNC) = 0].

In any physical model, however, there should be an open zone vacuum region

separating the surface of last closed field lines from the pair formation front, which

defines the radiating surface of the gap. The OG model of Romani (1996) defined gap

thickness w as the fraction of the angle from the last closed field line to the magnetic

axis that remains a charge-starved vacuum with a large acceleration field. This char-

acteristic gap width w is proportional to the γ-ray efficiency (see below). A physical

TPC model should also have a finite gap thickness; the “slot gap” model of Muslimov

& Harding (2003) provides a possible physical foundation for the TPC picture. These

authors give an estimate for the dependence of gap thickness on spin period P and

surface magnetic field B; again the thickness increases for pulsars putting a larger

fraction of the spindown power into pulsed γ-rays. Thus, for both models, we assume

that the gap thickness can be related to a heuristic γ-ray luminosity

Lγ ≈ ηESD ≈ C × (ESD

1033erg s−1)1/2 × 1033 erg s−1 (2.1)

with C, a slowly varying function of order unity which must come from a detailed

physical model. This law, which assumes a γ-ray efficiency (w =)η ∝ E−1/2SD , is natural

in models that maintain a fixed voltage drop across the acceleration gap. Clearly, as

w −→ 1 this simple geometrical approximation should break down. The original TPC

formulation has w = 0. In a full physical model, the radiating pair formation front

would have a finite thickness and the radiation pattern of the particles would have

a finite width; these effects are not treated here but would serve to smooth out the

beams computed in the basic geometrical model. Sample antenna patterns (sky maps


of the pulsar beam) are shown for the three models in Figure 2.1.

Figure 2.1 Sample antenna intensity patterns for α = 65 with the horizontal axis ofpulsar phase (φ) and vertical axis of viewing angle (ζ). The TPC model for w = 0.05is shown in green, the OG model for w = 0.1 is shown in red and the PC emissionsite is shown in blue. The cyan lines show the locus of the possible high altitude(r = 500 km, here for P = 0.2 s) radio emission, with the radiating front half shownsolid and the back half dashed (see §3.3).

In each case, the antenna pattern is built up by Monte Carlo simulation. Radiation

is emitted uniformly along the gap surface. The photons are aberrated, the time of

flight to a distant observer is computed and the emission is assigned to a bin of pulsar

phase and co-latitude (the sky map antenna pattern; see Romani & Yadigaroglu 1995;

Dyks, Harding, & Rudak 2004). We normalize this antenna pattern by summing

emission at all pulsar phases φ and along all lines-of-sight ζ and scaling by the heuristic

γ-ray luminosity (Equation 2.1).

Although we focus here on the beaming characteristics rather than the population,

to obtain a realistic sample of pulsar spin properties we simulate a Galactic population

of young pulsars following the method of Weltevrede & Johnston (2008a). Pulsars are

assigned a P and P from the parent distribution of Weltevrede & Johnston (2008a).

Since we are interested in energetic pulsars, we retain objects with E > 1034 (i.e.,

a max characteristic age τ = P/2P ≈ 106 y), drawing until we have a sample of

∼175,000 pulsars. This is ∼12× the true Galactic density (given a birthrate of ∼1.5

per century); the over-density serves to decrease statistical fluctuations. Magnetic


inclinations α and Earth viewing angles ζ were drawn isotropically. Given the low

upper limit on age, we ignore any possible magnetic alignment, which, if present,

appears to occur on longer time scales. For each drawn pulsar the appropriate γ-ray

light curve is computed for the drawn line-of-sight ζ using the appropriate antenna

pattern from the library of sky maps.

While we do use a realistic pulsar population, we do not report here on predictions

for the statistical distributions of γ-ray pulse shape, luminosity and Galactic location.

These, and predictions for the relative numbers of radio-loud and radio-quiet pulsars

expected in the LAT sky survey, are deferred to a subsequent paper. We also do not

treat here the millisecond pulsar population, as with small light cylinder radii the

difference between inner magnetosphere and outer magnetosphere emission zones is

less stark.

2.2.1 Radio Emission

Radio beam models have a long history but are, if anything, even more heuristic than

those for the GeV γ-rays. Here we consider two scenarios. One is simple low-altitude

emission, which assumes a circular cone of radio emission along the last closed field

lines with an angular width

2ρ = 10.8P−1/2 degrees (2.2)

centered on the magnetic axis (Rankin 1993; Gould 1994). Here, ρ is the half-opening

angle of the radio cone; the scaling is well supported by observations, which also agree

with the beam being roughly circular, as expected from dipole field line structure at

moderate altitudes. To model this scenario in our baseline computations we assume

radio observability whenever the line of sight passes within ρ of the magnetic axis.

The radio pulse peak is assumed to be at the same phase as this axis.

Recent evidence, however, suggests that young, Vela-like pulsars have a fixed

emission altitude of ∼ 500 − 1000 km or ∼ 30 − 70R∗ (Karastergiou & Johnston


2007). Since these are the dominant denizens of the γ-ray pulsar population, we also

(see §3.3) consider the effect of such wide beams at higher altitude. This wide beam

emission is assumed to be patchy, with some lines of sight crossing the radio zone

but receiving negligible flux (Lyne & Manchester 1988). This is difficult to treat,

except statistically, and is only discussed qualitatively in this work. As we shall see

in §3.3, it is useful to consider the possibility that the leading portion of the radio

zone has higher occupancy than the trailing half. We thus also compute radio pulse

observability for emission only along the leading edge of a 500 km altitude radio

emission zone (Figure 2.1). Aberration and light travel effects are computed for all

radio pulses as well.

2.2.2 Pulse Profile Characterization

We wish to summarize the properties of the γ-ray pulses of the simulated population.

We start by flagging the major peaks for each light curve, generated from cuts at a

given ζ across antenna patterns such as those in Figure 1. Examples of light curves

for these models can be seen in Figure 5 of Dyks, Harding, & Rudak (2004) for the

particular case α = 60. In the Appendix we show two summary figures of the light

curves for a range of α and ζ (and w) for the OG and TPC models.

We have created a peak-finding routine that runs on the γ-ray light curves pro-

duced from the models. The light curves are searched for maxima that rise above the

local minimum by least 20% of the global light curve maximum. Most non-thermal

pulsar peaks are relatively narrow, so we flag “sharp” peaks by requiring that the the

full width at half maximum (FWHM) of the qualifying peak (i.e. measured at 10%

of the full pulse amplitude below the peak maximum) have ∆φ < 0.1. The bright

EGRET pulsars with high S/N light curves (Crab, Vela and Geminga) all show two

peaks using this cut. However, wider maxima may also be of interest, so we addition-

ally flag “broad peaks” with FWHM 0.1 < ∆φ < 0.3. These tend to appear when

either ζ or α is small. Note that some pulsars with both small ζ and α have appre-

ciable γ-ray flux but show only shallow sinusoidal variations with no strong “peaks”


as defined by the above algorithm.

Two additional key values regarding peak locations are also recorded. First, we

record the separation from the first peak to the last peak. Additionally, we measure

the lag of the first peak from the radio pulse. To help the reader understand the

results of this peak flagging, in the Appendix figures we record the number of peaks,

the broad peaks and the pulse width (in %) as determined by this routine for each of

the plotted light curves.

Finally, we compute fΩ, the ratio between the simulated total emission of the

pulsar over all observer angles ζ and the total emission if the observed flux at ζE were

emitted uniformly across the sky.

2.3 Results

We have collected the geometrical pulse properties of the three models into an “atlas”

(Figure 2.2) that compactly summarizes many complex pulse profiles. We plot simu-

lated pulsars in the (α, ζ) plane, color coding the points to show the model properties.

Small dots show objects detected in γ-rays but not in the radio. Radio detectable

objects tend to lie at small β = ζ − α, the magnetic pole impact angle. The dashed

lines show the low-altitude standard radio detection window for P = 0.2 s, the median

for the observable γ-ray pulsar population. Of course, these are not strict boundaries;

pulsars with large P have smaller radio pulse widths and may remain radio invisible

at smaller β while especially fast pulsars may be radio detected well beyond the plot-

ted lines. There are radio-detectable, γ-invisible objects as well. These are generally

found at small α, ζ and are not plotted here.

The top row shows the pulse complexity, the number of major pulse peaks, with

broad peaks appearing in the cyan curve. The second row shows, for each model, the

phase interval ∆ between the most widely separated major peaks. These plots may be

used to predict the γ-ray properties for a given model and known orientation angles.

Conversely, for an observed γ-ray pulse profile one may restrict the allowed viewing

2.3. RESULTS 13

Figure 2.2 An atlas of pulse profile properties for three γ-ray pulsar models. Top row:pulse complexity (peak number); broad peaks lie in zone enclosed by the cyan lines.Middle row: pulse width ∆, the maximum separation of the major peaks. Bottomrow: the “flux conversion factor” fΩ for correction to all-sky flux. For the PC model,the small ranges of ∆ and fΩ require re-normalization of the color scale for thosepanels.


angles for a given model or even rule a model out completely. Of course, objects

with well-constrained viewing geometry and observed profiles are particularly useful

as they allow the most strict discrimination between pulse models. More generally,

the distribution of observed pulse shapes is different between the models. Once one

assumes an underlying pulsar distribution, such population analysis is also a very

powerful discriminant (see Chapter 4).

The final row of the atlas depicts a flux correction factor fΩ, useful in calculation

of γ-ray efficiencies. The pulse profile observed along the line of sight at ζ for a pulsar

with magnetic inclination α is Fγ(α, ζ, φ), which we can compute for a given pulsar

model. This means that the observed phase-averaged flux Fobs for the Earth line-of-

sight ζE is not necessarily representative of the flux averaged over the sky. This has

also been emphasized by Harding, Grenier & Gonthier (2007). Here, we can use our

model to compute the required correction fΩ, so that the true luminosity is

Lγ = 4πfΩFobsD2. (2.3)

where D is the distance to the pulsar and

fΩ = fΩ(α, ζE) =

∫∫

Fγ(α, ζ, φ)sin(ζ)dζdφ

2∫

Fγ(α, ζE, φ)dφ. (2.4)

It has become traditional to assume that the γ-ray beam covers an area of 1 sr

uniformly, so that fΩ = 1/(4π) ≈ 0.08; however, more modern pulsar beaming models

have fΩ ∼ 1 for many viewing angles.

The flux correction factor is crucial for estimating the γ-ray efficiency

η = Lγ/ESD ∝ fΩ. (2.5)

Thus, it appears that many recent papers have underestimated the true γ-ray effi-

ciency (for modern pulsar models) by an order of magnitude.

From our heuristic luminosity law, we expect that the efficiency evolves as η ∝

2.3. RESULTS 15

E−1/2SD . In general, one would expect a larger fraction of the gap to remain charged-

starved for more efficient pulsars, with w ∝ η. In figures 2.3 and 2.4, we see how the

pulse width and sky covering factor evolve with w ≡ η. In particular, for low-ESD,

large-w, high-efficiency pulsars, we see that fΩ decreases. Thus, the correction of

observed phase average flux to true sky flux becomes a function of spindown power.

As w saturates at a large value of order unity, which for this calculation is near

ESD = 1033erg s−1, no radiating surfaces bound the gaps and γ-ray production ceases.

The exact value depends on magnetic inclination α.

For the convenience of other researchers, we have collected approximate fitting

formulae for fΩ, the flux correction factor. Because of the complexity of the beam

patterns, it is always best to use the atlas figures to determine fΩ for a pulsar with

known parameters. However, statistical statements about luminosity and rough pop-

ulation sums can benefit from analytical approximations. For both the TPC and OG

models there is a change in the beaming behavior at large α and ζ . In this region one

sees emission from the hollow cone above the null charge surface that dominates the

pulsation for the classical OG and often provides the late pulses in the TPC picture.

We call this “case I” and thus define an approximate boundary for such objects as

ζ > ζI = (75 + 100w)− (60 + 1/w)(α/90)2(1−w), (2.6)

with ζ in degrees. Any pulsars with ζ < ζI compose case II. For the OG scenario,

< 10% of the modeled pulsars have any emission at smaller ζ , so case II is not a major

contributor to the population. Such objects do, however, exist: principally pulsars

with small w and large inclination α that are observed at small ζ . The emission comes

from diverging high-altitude field lines producing faint γ-ray emission (often with no

strong pulse) over much of the sky. In the TPC model, however, there is substantially

more emission in case II, since the flux at these angles arises from low-altitude field

lines viewed relatively near the polar cap. These light curves tend to show shallow

sinusoidal pulses. They are also missing the caustic peaks (sharp pulses) dominating

the case I objects. Thus, in both the TPC and OG models, Equation 2.6 defines a

boundary past which the pulse emission is fainter and fΩ is larger. Figures 2.2-2.5


Figure 2.3 Beam shape evolution with efficiency for the TPC model. The three rowsshow peak complexity, peak separation and flux conversion factor, as in Figure 2.2.The gap width (and hence γ-ray efficiency) increases left to right.

2.3. RESULTS 17

Figure 2.4 Outer Gap model beam shape evolution as a function of gap width, as inFigure 2.3.


and the light curves in the Appendix make these patterns clear.

Figure 2.5 The pulse width for TPC and OG models at several w. This figure shouldbe used when possible to locate accurately the α and ζ for a given pulsar, for referenceback to Figures 2.3 and 2.4.

For the TPC model, we find in the case I zone

fΩ ≈ 0.8 + 1.2(0.3 − w)cos(2β), (2.7)

where β = ζ −α. This applies for pulsars with w > 0.05 and gives estimates accurate

to 10% for 76% of the pulsars and to 20% for 90% of the pulsars. In the case II zone

the flux correction factors are generally larger, with

fΩ ≈ 0.3 + 1.5(1 − w)[1 + (ζ − ζI)/90]. (2.8)

This region also has more complex behavior, so it is better to use the atlas figures if

possible. The overall accuracy for the TPC model estimates (case I and case II zones

together) is better than 10% for 62% of the modeled objects and within 20% for 82%

of all modeled pulsars.

2.3. RESULTS 19

For the OG we consider the case I zone only, finding

fΩ ≈ 0.17 − 0.69w + (1.15 − 1.05w)(α/90)1.9 (2.9)

for ζ > 60. A ζ-dependent term appears for pulsars viewed closer to the spin axis,

so that

fΩ ≈ 0.17 − 0.69w + (1.15 − 1.05w)(α/90)1.9 (2.10)

−1.35(2/3 − ζ/90) (2.11)

when ζ < 60. These give approximations accurate to better than 10% for 70% of

the simulated case I pulsars with w > 0.05 and are better than 20% for 92% of such

pulsars. The atlas figures give a guide to regions where exceptions occur.

2.3.1 Determining Individual Pulsar Geometries

Many young energetic pulsars produce bright, resolvable X-ray pulsar wind nebulae

(PWNe), the beam dump of relativistic particles and fields flowing out of the magne-

tosphere. Often, high-resolution X-ray images of these PWNe can be interpreted in

terms of cylindrically symmetric structures, with a mildly relativistic (v ≈ 0.3−0.7c)

bulk outflow Doppler-boosting the intensity along the radial flow. The inference is

that the symmetry axis traces the pulsar spin, and fits to such models (Ng & Romani

2008) can provide relatively model-free estimates of ζE. Thus, for such pulsars, the

family of allowed models is given by horizontal slices across the atlas panels.

Since we are concerned with detected γ-ray pulsars, one will inevitably have an

estimate of the pulse width ∆. This is, then, the most robust observable, and provides

a key to determining the rest of the model parameters. Thus, the prescription for

solving a pulsar for a given model is to i) use ESD (Equation 2.1) to estimate w; ii) go

to the atlas pulse width plot for the closest w and locate regions showing a suitable

∆; iii) use an external value for ζE, when available, to locate the relevant point in the

∆ band and confirm that that the pulsar is radio loud or quiet, as appropriate; iv)

use the ζ and α values for this point to read off other pulsar properties (Figures 2-4).


Since the ∆ plot is the “key” to the atlas, and since there are large regions at high α

and ζ with ∆ ∼ 0.4 − 0.5, especially for the TPC model, we provide in Figure 5 an

expansion of the lower right quadrant of the evolving ∆ panels with finer color bands.

This should be used to determine ζ and α when possible, which can be transferred

to Figures 3 and 4.

There is also a well-developed phenomenology connecting the sweep of linear po-

larization within the radio pulses to cuts across a simple magnetic pole. In the classic

rotating-vector model (RVM, Radhakrishnan & Cooke 1969), the emission is assumed

to have a polarization vector locked to the dipole magnetic field (projected on the

plane of the sky) in the emitting region. The shape of the polarization sweep depends

most sensitively on the value sin(α)/sin(β). As β generally takes on small values (or

else the object would not be detected in the radio in the first place), this method

tends to give poor constraints on α and tighter constraints on β. Thus, in principle,

X-ray and radio data together can provide both ζ and α and therefore a prediction

of the γ-ray pulse shape.

In practice, the picture is often not this simple. One possible complication arises if

the altitude of the radio emission is large, where sweep-back and time-delay can alter

the naive rotating-vector model predictions (Everett & Weisberg 2001). Finally, we

should reiterate that our models are purely geometrical (albeit motivated by physics)

and locked to the vacuum dipole field. Field-line perturbations, currents and variation

in emissivity along the gap can all perturb the profiles. Nevertheless, this atlas does

capture the key differences between the models and provides a basis for broad-brush

interpretation of newly discovered pulsars.

2.3.2 The Polar Cap Model

The polar cap picture naturally predicts that only pulsars seen at small β will be seen

in the γ-rays. With the radio emission originating above the surface γ-rays (which

are at r . 2r∗), one would always expect radio emission bracketing the GeV pulse.

“Radio-quiet” objects, such as Geminga, are not possible in this picture unless the

2.3. RESULTS 21

radio cone is very patchy and emission is completely absent along ζE. “Interpulsars,”

with emission visible from both radio poles, are expected at the lower right (α ∼ ζ ∼

90), where the left panel of Figure 2 shows triple- and quadruple-peaked light curves

for this model. Note that there is also a small collection of triply-peaked light curves

at α ∼ ζ ∼ 45 stemming from the notch structure observed in the outline of the

polar cap for inclination angles α ∼ 45 (Dyks, Harding, & Rudak 2004).

The wide double pulses, with ∆ = 0.35−0.45, seen for the bright EGRET pulsars

pose another challenge to the PC model. While a double-pulse light curve is the

dominant mode in the PC zone, a wide separation can only be obtained for very

small α and a small range of ζ . Unless pulsars are born highly aligned, this is

statistically very improbable. Additionally, model-independent measurements of ζ

made through the analysis of PWN tori mentioned above have consistently found

values ζ > 40, well beyond the range for which the polar cap model can produce

large peak separations.

These geometrical arguments, as well as considerations of flux and spectrum,

imply that the bulk of the observed γ-ray pulsar emission seen by EGRET could not

have come from a low-altitude cap. However, if a surface gap does have active pair

production, some γ-rays are nevertheless expected from this region; very sensitive

Fermi LAT observations may be able to uncover such a component.

2.3.3 Observed Pulsars

The atlas lets one read off the expected γ-ray pulse properties for known α and ζ or

estimate from ∆ values for undetermined α and ζ . It is also interesting to compare

the observed properties of specific pulsars with the models’ predictions. Table 2.1 lists

the observed properties of the well known EGRET pulsars and two new detections,

J2021+3651 from AGILE (Halpern et al. 2008) and CTA 1 from Fermi (Abdo et al.

2008). Column 4 lists the externally derived values for the line-of-sight angle ζ , when

available.

Note that for PSR B1055−52 the radio-derived ζ accords well with the value


predicted by the γ-ray pulse width ∆ in the outer gap picture. The TPC picture does

have a solution matching the observed ∆ (albeit only for w much lower than inferred

from ESD), but ζ does not agree with the radio value, and additionally α is too small

to produce the observed radio interpulse. For the two radio-quiet pulsars Geminga

and PSR J0007+7303 in CTA 1 we do not have ζ estimates. For CTA 1, both OG

and TPC models can match the observed ∆. For Geminga, with ∆ = 0.5, the TPC

model has solutions along both the ζ and α axes. For the OG model, Geminga-type

∆ are only natural at small α and large ζ , where one will almost certainly miss the

radio beam.

In addition to the γ-ray pulse width ∆, for radio-detected objects, the lag δ of

the first γ-ray pulse from the radio peak is a second convenient observable. In Figure

2.6 we have plotted in blue a sample of simulated γ-ray pulsars, in the ∆ − δ plane,

assuming that the radio peak is centered along the magnetic axis. If the viewing

angle ζ is too far from the pole (i.e., β large) then no radio pulse will be detected. In

these cases we have plotted the pulsars in green, with δ the phase lag from the closest

approach to the radio pole. For the TPC model, many of these pulsars have a first

γ-ray peak leading the radio pole. Some known γ-ray pulsars with well defined pulse

profiles have been placed on the figure. For the two “radio-quiet” pulsars Geminga

and PSR J0007+7302, no δ is available, so the allowed regions are shown as horizontal

bands. PSR B1509-58 was seen at MeV energies by CGRO, but was not convincingly

seen by EGRET; its broad peak has a small but poorly defined ∆, so it appears as a

diagonal band.

It is clear from Figure 2.6 that many of the observed radio/γ-ray pulsars have

a larger radio lag than expected in either the TPC or OG model if the radio pulse

is centered on the magnetic axis. This suggests that some ingredient is missing

from our basic geometrical model. One possibility is that magnetospheric currents

sweep the γ-ray emitting field lines back to later phase. Alternatively, an extra lag

may be connected with the posited high altitude radio emission of the young pulsars

(Weltevrede & Johnston 2008b). If we invoke patchy emission and make the additional

(new) assumption that the patches are most active on the leading half of the radio

2.3

.R

ESU

LT

S23

Table 2.1 — Geometry and Beaming of Bright EGRET Pulsars

Name log(ESD) w d (kpc)a ζb ∆c δc αTPCd ζTPC

d fΩ(TPC)d αOGd ζOG

d fΩ(OG)d

Crab 38.7 0.001 2.0 ± 0.2 63 0.40 0.111 55-60 63 1.1 70 63 1.0Vela 36.8 0.01 0.287−0.017

+0.019 64 0.426 0.117 62-68 64 1.1-1.15 75 64 1.0B1951+32 36.6 0.02 2.0 ± 0.6 ... 0.48 0.135 55-90 55-90 1.0-1.25 60-90 70-85 0.75-1.1B1706−44 36.5 0.02 2.3 ± 0.7 53 0.2 0.25 45-50 53 1.05 47-49 53 0.7-1.0J2021+3651 36.5 0.02 3.0 ± 1.0 86 0.465 0.165 70 86 1.1 70 86 1.05CTA 1 35.7 0.05 1.4 ± 0.3 ... 0.20 — 55-70 23-45 0.8-1.3 20-50 60-75 0.3-0.9Geminga 34.5 0.18 0.25−0.062

+0.120 ... 0.50 — 30-80,90 90,55-80 0.7-0.9,0.6-0.8 10-25 85 0.1-0.15B1055−52 34.5 0.18 0.72 ± 0.15 67 0.22 0.32 60e 50 0.85 78 67 0.55

aDistance references (in order): traditional for Crab; Dodson et al. (2003); Strom & Stappers (2000); Cordes & Lazio (2002); VanEtten, Romani, & Ng (2008); Abdo et al. (2008); Faherty, Walter, & Anderson (2007); Cordes & Lazio (2002).

bFrom torus fits to X-ray images (Ng & Romani 2008) when available; B1055−52 from radio RVM fit (Lyne & Manchester 1988).

cDerived from light curves in Fierro (1995) except J2021+3651 (Halpern et al. 2008) and CTA 1 (Abdo et al. 2008).

dEstimated from Figures 2.2 and 2.4.

eRequires w . 0.1


Figure 2.6 Pulsar γ-ray peak separation vs. radio lag. Green points are radio-quietγ-ray pulsars, blue points are radio-loud γ-ray pulsars.

zone, then aberration and field-line spreading on this leading half of the cone can

drive the high-altitude radio peak to much earlier phases. The range of allowed lags

is shown in Figure 2.7 for a fixed altitude of 500 km. Note that the phase lead δ

then depends on the pulsar spin period, since this fixed altitude is a variable fraction

of rLC. These are the model tracks in Figure 2.7. Large β will result in a smaller

increase in δ as the radio cone is cut with a small chord, further from the magnetic

axis. Again for the green (radio-quiet pulsar) dots, we plot the δ from the closest

approach to the radio pole. As expected, the wider radio pulse allows fewer green

(radio non-detected) objects.

Finally, we should compare the model pulsar efficiencies η with the observed γ-ray

2.3. RESULTS 25

Figure 2.7 As Figure 2.6, but assuming high-altitude radio emission from the leadinghalf of the radio zone.

fluxes, using our computed fΩ. Of course, this is a purely geometrical comparison,

ignoring real differences in γ-ray spectrum and the radiation processes tapping the

primary electron energy. Nevertheless, the differences in fΩ are substantial (and often

completely ignored in discussions of η). For gaps with near-constant brightness and

pulsars with well determined distances, the fΩ variations can be the biggest unknown,

making our treatment useful.

Table 2.1 contains the estimated fΩ ranges inferred for the TPC and OG models.

In Figure 2.8, we plot the inferred efficiencies of the detected γ-ray pulsars, after

correction with fΩ to simulated all-sky emission for the two models. The distance

uncertainties generally dominate the range in the derived Lγ . The solid line shows

complete conversion of spindown power to γ-rays and the dashed line shows the


heuristic efficiency law assumed above.

In Figure 2.8, two objects near log(E) = 36.5 deserve comment. The first is PSR

J2021+3651 in the “Dragonfly” nebula. This has a very large dispersion measure, and

the corresponding 12.4 kpc distance would imply ≥ 100% efficient γ-ray production

for both TPC and OG pictures. However, Van Etten, Romani, & Ng (2008) argue

that a variety of pulsar and PWN measurements are more consistent with a distance

of ≤ 3 kpc. In fact, the thermal surface emission has a best fit distance of d = 2.1 kpc;

d ∼ 1.6 kpc would bring this pulsar into agreement with the heuristic efficiency law.

The origin of the large dispersion measure (DM) for this pulsar is thus very puzzling.

The next outlier is PSR B1706−44, at similar E, where we have adopted the DM

distance of 2.3 kpc (Cordes & Lazio 2002). Again, smaller distances are implied by

fits to the neutron star thermal emission and the PWN energetics (Romani et al.

2005), with preferred values at d ≤ 1.8 kpc. However, to match the efficiency law

would require distances close to 1 kpc; this is substantially below the statistical lower

bound on the distance estimated from H I absorption (Koribalski et al. 1995). Thus,

the Earth line-of-sight flux of these two Vela-type pulsars seems larger than predicted

from their spindown luminosity and the beaming model. Improved flux measurements

from the LAT and improved distance estimates (e.g., a radio interferometric parallax

for PSR B1706−44) would be particularly valuable in tightening up this argument.

The most interesting region of Figure 2.8 is the low-E range, where the modeled

efficiencies approach unity. For these large w pulsars, fΩ begins to drop significantly

below unity, especially in the OG model, decreasing the effective Lγ . This effect

dominates as pulsars approach the γ-ray death zone, near the end of their lives as high-

energy emitters. The small fΩ values prevent the apparent catastrophe of Lγ > E

for old pulsars. Conversely, while a small fraction of such pulsars, with their γ-beams

closely confined to the spin equator, are visible along the Earth line of sight, those that

are seen have relatively high fluxes and are detectable to relatively large distances.

Geminga is an excellent example.

2.3. RESULTS 27

Figure 2.8 Derived γ-ray luminosity, corrected (TPC – crosses, OG – circles) fromthe observed phase-averaged flux to all-sky, vs. the spindown luminosity. Error barsinclude both the d and fΩ uncertainties (Table 1). The dotted error bar at large Lγ

represents the γ-ray luminosity of J2021+3651 assuming the DM distance of 12.4 kpc.


2.4 Discussion

Our study allows a quick visual summary of model predictions, which may be com-

pared with pulsar discoveries anticipated from the LAT. We have already seen that

the classic polar cap (PC) picture is difficult to reconcile with the data. Both the

TPC and OG pictures, however, seem viable. Certain observations would clearly

discriminate. For example, detection of γ-ray pulsars with α + ζ < 90, but with

nearly constant intensity through the period, would give a clear preference for the

TPC model. In contrast, the prevalence of light curves with simple double pulses

separated by 0.05 < ∆ < 0.3 is a hallmark of high-altitude OG emission. The dis-

tributions in peak widths and in the relative number of radio-loud and radio-quiet

pulsars are also strong discriminants; we address these in Chapter 4.

Also, the predicted pulse profile complexities for the two models differ. OG models

are complex for α ∼ ζ ∼ 90; TPC models have complex profiles at intermediate α.

In general, the radio-faint pulsars are double-peaked in both models, but the TPC

picture has relatively more single- and zero-peaked solutions.

One area of agreement between the TPC and OG models is the preponderance of

large fΩ; this is a principal result of our study: inferred efficiencies should be increased

by roughly an order of magnitude from those commonly assumed. In practice this

means that some young pulsars must have high (> 10%) γ-ray efficiencies. This

argues that a study of the γ-ray emission is an even better probe of magnetosphere

structure and energetics than previously thought. Our fΩ corrections tighten the well-

known trend for efficiency to scale as η ∝ E−1/2, although a few outliers remain. If,

as more pulsars are discovered by the LAT, we find that this trend remains strong, we

may even be able to use observed fluxes, after fΩ correction, for “standard candle”

estimates of pulsar distances. Such arguments suggest relatively low distances for

PSRs J2021+3651 and B1706−44.

Importantly, for models with w(E) increasing with age, fΩ tapers off at small

E. Thus the sky coverage decreases smoothly into a “death zone” where the gap

shuts off. Near shut-off, the small solid angle on the sky ensures that old pulsars

2.5. LIGHT CURVE ATLASES 29

appear relatively bright and are visible to large d. This is an important signature

in the population sums (see Chapter 4). Examination of the distribution of pulse

properties over the simulated pulsar sample is thus the key to testing the models and

to producing the corrected luminosities that will make LAT data powerful in probing

pulsar populations and spindown.

2.5 Light Curve Atlases

Here we show a selection of light curves for the two-pole caustic (TPC) and outer gap

(OG) models. For both, we arrange the sample light curves on an α, ζ grid so that

the results may be compared with the “atlas” figures (Figures 2.2-2.4). Light curves

are shown for four w values (0.01, 0.1, 0.2, 0.3). The number of peaks (total and

broad only) and the maximum peak separation ∆ (expressed here in %) as flagged

by the automatic peak finder are given near the individual light curves.

These light curves allow the reader to visualize the topological parameters sum-

marized in the color “atlas” figures. For example, in the TPC plot one can see the

broad, sinusoidal profiles that arise at low α and ζ and thus earn the “zero major

peak” classification. One also sees how the TPC model has complex pulses near

α ∼ 40, ζ ∼ 60 and that these sometimes give small separations between the

strongest peaks. For the OG plots, one sees that for α > 60 and small ζ the normal

first peak is often lost; this is a result of the rapid spreading of the high-altitude field

lines contributing to this peak, which prevents formation of the sharp leading caustic.

Also, one sees that the pulse shapes are often complex near ζ = 90 and that the

pulse shapes simplify toward basic double pulses as w becomes large.


Figure 2.9 Collection of sample light curves for the TPC model. Four select w (valuesin bottom right panel) are shown for each panel; the radio pole has closest approachat phase=0. The values for the number of all peaks, the number of broad peaks andthe maximum peak separation (in %) are indicated by each curve. Intensities arenormalized to pulse maximum.


Figure 2.10 OG model light curves, as for Figure 2.9. Four select w (values in bottomright panel) are shown for each panel; the radio pole has closest approach at phase=0.Intensities are normalized to pulse maximum. For small α or ζ , especially at large w,no pulses form.

Chapter 3

Towards a Quantitative Method of

Comparison

This chapter is based on “Constraining Pulsar Magnetosphere Geometry with γ-Ray Light Curves” by R. W. Romani

and K. P. Watters 2010, ApJ, 714, 810.

We demonstrate a method for quantitatively comparing γ-ray pulsar light curves with

magnetosphere beaming models. With the Fermi LAT providing many pulsar discov-

eries and high quality pulsar light curves for the brighter objects, such comparison

allows greatly improved constraints on the emission zone geometry and the magne-

tospheric physics. Here we apply the method to Fermi LAT light curves of a set of

bright pulsars known since EGRET or before. We test three approximate models for

the magnetosphere structure and two popular schemes for the location of the emission

zone, the Two Pole Caustic (TPC) model and the Outer Gap (OG) model. We find

that OG models and relatively physical B fields approximating force-free dipole mag-

netospheres are preferred at high statistical significance. An application to the full

LAT pulsar sample will allow us to follow the emission zone’s evolution with pulsar

spindown.

32

3.1. INTRODUCTION 33

3.1 Introduction

With the successful launch of Fermi, the number and quality of γ-ray light curves

available for comparison with magnetospheric models has substantially improved

(Abdo et al. 2010a). The quality and uniformity of the data make this a good time

to down-select models for pulsed magnetospheric emission, by comparison with the

observed curves. In the present paper we describe a method of attack for this pro-

cess and apply it to the new improved light curves provided by Fermi for several

well-known γ-ray pulsars.

The most popular pulsar models postulate regions in the magnetosphere where

the force-free perfect MHD ( ~E · ~B = 0) conditions break down, and in response to

the uncanceled rotation-induced EMF ~E ≈ −~r × ~Ω × ~B/c, the charges re-arrange so

that non-negligible fields penetrate these lower density ’gap’ zones, causing particle

acceleration and radiation. Traditionally, one approximates the magnetosphere by a

vacuum dipole field.

There are three locations commonly discussed for these magnetospheric gaps. The

earliest pulsar models focused on acceleration at the foot points of the open field line

zone, the so-called “polar caps”. This emission should arise from within about a stellar

radius (Daugherty & Harding 1996), and should therefore suffer strong attenuation

from one photon (γ-B) pair creation. The corresponding hyper-exponential cutoffs are

not seen in the LAT pulsar spectra, and so it has been concluded that the bulk of the

pulsar emission must come from higher altitudes (Abdo et al. 2010a). The first high

altitude model posits “Holloway” gaps above the null charge surface, extending toward

the light cylinder at rLC = c/Ω (the “outer gap” model; Cheng, Ho, & Ruderman

1986; Romani 1996). More recently, it has been argued that acceleration at the rims

of the polar caps may extend to very high altitude (Muslimov & Harding 2004),

effectively pushing the polar cap activity outward. In this ‘slot gap’-type picture

emission can occur over a large fraction of the boundary of the open zone. The

geometrical realization of this model makes low altitude emission visible from one

hemisphere and higher altitude (r > rNC) emission visible from the other. This is the

34 CHAPTER 3. TOWARDS A QUANTITATIVE METHOD OF COMPARISON

“two pole caustic” (TPC) model (Dyks & Rudak 2003).

In Chapter 2, we created a library of light curves for these three emission locations

in a dipole field and classified the phases of the dominant peaks. The results were

presented as an ‘atlas’ of pulse properties which could be used to predict the pulse

multiplicity and phase separation of a given model for a particular viewing geome-

try. While this is useful for an overview of the possible light curves and provides a

convenient reference to read off the allowed angles for a given model, the complexity

of the modeled beams meant that an appropriate phase separation could be found in

several models for any one pulsar.

These computations implicitly assume static co-rotating charges only. Of course,

the presence of radiating pair plasma in the magnetosphere requires some particle

production, and so the vacuum plus co-rotating charge model will be approximate

at best. Therefore an alternative attack is based on numerical solutions for mag-

netospheric charges, currents and fields, assuming that pair production is so robust

that the force-free condition holds everywhere (Contopolous, Kazanas & Fendt 1999;

Spitkovsky 2006; Bai & Spitkovsky 2010b). Such filled magnetospheres lack the ac-

celeration fields required to produce powerful γ-ray beams, so the truth must lie in

between, with some regions experiencing charge starvation and departing from force-

free.

Here we wish to show how the actual light curves from the computations can be

compared with the data. We demonstrate that the present data quality are sufficient

to start comparing predictions between different assumed magnetic field structures

and emission zone locations. In particular, while in many cases several models can

produce plausible matches to the light curves for some angle, when we restrict our

attention to the geometries demanded by external constraints, one can often strongly

exclude certain models.

For our model set, we start by amending certain inadequacies in the vacuum

computations pointed out by Bai & Spitkovsky (2010a). This ends up making modest

differences to the results in Chapter 2. We also introduce a simple model to illustrate

3.2. MODELING METHOD 35

the field structure perturbations resulting from open-zone currents. Comparing with

the Fermi full band light curves, we find that lower altitude TPC-type models have

great difficulty producing acceptable matches for many pulsars for all of these field

geometries (at least for the assumptions made here) and that OG-type models are

statistically strongly preferred. Further, in Fermi-quality data we find that the light

curve matches can often be significantly improved by the magnetospheric current

perturbations. In Chapter 4 we apply such analysis to the full set of Fermi LAT

pulsars, utilizing a population synthesis model to extract additional information on

the pulsar beam structure and its evolution with age as the pulsar spins down.

3.2 Modeling Method

Our basic assumptions are that i) the stable, time-averaged pulse profile is caused

by emission zones locked to a set of ’open’ field lines, ii) these field structures are

dominated by the dipole component of the field anchored in the star, since for young

pulsars this emission occurs at many R∗, iii) that a co-rotation charge density dis-

tributed throughout the bulk of the magnetosphere causes the plasma to follow the

field lines and rotate with the star and iv) in the charge-starved gap zones the un-

canceled field rapidly accelerates e + /e− to highly relativistic energies so that these

γ-ray producing particles also follow the magnetic field lines.

As has been well established, starting with Morini (1983) and discussed explicitly

by Romani & Yadigaroglu (1995), the combination of field line sweep back, co-rotation

of the radiating particles and travel time across the light cylinder tends to pile up

the emission into ’caustics’ in pulsar phase. As noted by Bai & Spitkovsky (2010a)

this reflects the singularity of the Jacobian matrix mapping from the emission surface

to the antenna pattern on the sky. The singularity naturally produces sharp pulses

with emission from a range of magnetospheric altitudes, a property shared by all

competitive magnetospheric emission models. The sharpness of the peaks of the

pulsed emission (Abdo et al. 2009) implies that the ‘width’ of the radiating zone is

modest, at least in a time averaged sense. However, the resolved rise to peaks of the


observed γ-ray pulses suggest either a wide range of emission height, a fundamental

radiation pattern with wings, or a dispersion in the direction of the emission zone,

again in a time-averaged sense.

Once one chooses a prescription for the magnetic field structure, one can follow

radiation from the active emission zone to a (φ, ζ) observer plane, where φ is the pulsar

spin phase, with phase zero when the surface magnetic dipole axis passes closest to

the observer line of sight. The observer line of sight is at viewing angle ζ to the

spin axis. This two dimensional intensity map is the ‘antenna pattern’ for the pulsar

radiation shown in Chapter 2. Once one specifies the inclination α of the magnetic

dipole axis to the rotation axis one has a complete geometrical description. Of course,

the choice of the radiating zone and the (energy-dependent) intensity weight through

that emission zone depend on the radiation physics (Romani 1996). In practice, we

use simplified assumptions that capture the basic behavior, and look for departures

from the resulting predictions as guides to improved modeling of the physics. For

example, it is often assumed that the gaps self-limit to a fixed potential drop (Arons

2006) which means that the γ-ray luminosity will be proportional to the open zone

current (Harding 1981). One natural way to do this is to imagine radiation reaction-

limited particles passing through a charge-starved gap spanning a fraction w of the

open zone of the pulsar magnetosphere. In this paper we take the emission surface to

be relatively thin, lying a fraction w of the way from the last closed field line surface

to the magnetic axis. For w not too close to one, this approximates the constant

voltage limit if one takes the heuristic γ-ray luminosity

Lγ ≈ (1033erg/sESD)1/2. (3.1)

This implies an efficiency η = Lγ/ESD equal to the fractional gap width w. Realistic

models must, of course, have a finite thickness for the emission zone. We find that

this produces relatively small perturbations to the light curve shape if δw . w.

The phase-averaged γ-ray flux for the Earth line-of-sight Fobs is related to the

total γ-ray luminosity by

Lγ = 4πfΩFobsD2 (3.2)


where D is the distance to the pulsar and fΩ is the ‘flux correction factor’; an isotropic

emitter corresponds to fΩ = 1. Notice that we need this quantity to infer the actual

pulsar γ-ray efficiency

η = 4πfΩFobsD2/ESD, (3.3)

for comparison with model predictions. A prescription for fΩ is given in Equation 2.4;

the solid angle corrections are applied correctly in the values in that chapter (albeit

for the ‘Atlas’-style beam computation).

3.2.1 Magnetic Field Structures

To start, one must choose a field line structure. In the earliest light curve modeling

(Morini 1983; Chiang & Romani 1994) static dipole magnetic field lines were assumed.

Emission was assumed beamed along these field lines, which were taken to occupy

the rotating frame, and the resulting aberrations and time delay formed the observed

pulse. While this “static” (Stat) model had the right topology (closed zone plus

flaring open zone), and already predicted (for outer magnetosphere emission) the

generic double γ-ray pulse lagged from the low altitude radio pulsations, it clearly is

not a physically consistent picture.

The next step, introduced by Romani & Yadigaroglu (1995) and used widely

by other modelers, is to use the retarded dipole instead of the static case. This is

commonly referred to as the Deutsch (1955) field approximation. However, at large

radius from the star, this is effectively just a rotating point dipole which is most

simply expressed by Kaburaki (1980) and can be re-written as

~B = −[

~m(tr) + r ~m(tr) + r2 ~m(tr)]

/r3 + r(

r ·[

3~m(tr) + 3r ~m(tr) + r2 ~m(tr)]

/r3)

~m(tr) = m(sinα cosωtr x + sinα sinωtr y + cosα z) (3.4)

with m the dipole magnetic moment evaluated at the retarded time tr = t−r/c. This

is the field structure used in Chapter 2 and so we refer to it as the “atlas” (Atl) field.

It has the virtue of having the intuitively expected ‘sweepback’ as one approaches the


light cylinder (Figure 3.1).

Figure 3.1 Illustrations of the basic dipole field geometry. Left: Static, Right: retardedpoint dipole (as used in the ‘Atl’ and ‘PFF’ computations). Top – the position ofthe open zone boundary (polar cap boundary) in magnetic coordinates θB(φB) formagnetic inclination α = 70. Middle – magnetic field lines in the equatorial planefor α = 90. Bottom – field lines for α = 80 above one polar cap with lighter (blue)showing the lines which cross rNC , the ‘Outer Gap’ lines.

This field is stationary in the non-inertial frame rotating with respect to the lab

frame, where particles are assumed to follow the field lines. In Chapter 2 and earlier

computations by a number of groups, the particle velocity in the lab frame was incor-

rectly computed as a co-rotation induced boost to a relativistic particle following the

rotating field lines. Bai & Spitkovsky (2010a) correctly point out that the particle

velocities in the two frames are instead connected by a simple coordinate transfor-

mation. These authors assumed that the earlier computations (incorrectly) used the

field (4) in the rotating frame. The boost error in Chapter 2 is conceptually different,


but mathematically equivalent to this assumption. In practice, the coordinate trans-

formation allows one to compute the velocity β ′‖ along the magnetic field line for the

particle forced to co-rotate with the star at a highly relativistic lab frame velocity

|β0| −→ 1

~β0 = β ′‖B + ~Ω × ~r/c (3.5)

which gives

β ′‖ = −B · (~Ω × ~r/c) + [B · (~Ω × ~r/c)]2 − (~Ω × ~r/c)2 + 11/2 (3.6)

so that substituting into Equation 3.5 gives the direction of the particle as viewed in

the lab frame. Note that β ′‖ can become very small for particles traveling nearly along

~Ω×~r for r⊥ ∼ rLC , i.e. nearly tangent to the light cylinder in the ‘forward’ direction.

In this case small effects of particle inertia or field line perturbations will ‘break open’

such field lines and the particles should leave the light cylinder and contribute little

to the pulse emission. Although we do not follow these physical effects, such field

lines do not affect our light curves, since we taper the emissivity (see §3.2.3 below).

In any case, to have the plasma static in the rotating frame requires the presence

of the co-rotation charge density that produces the lab frame field

~E = (~Ω × ~r) × ~B/c. (3.7)

This is assumed present in the ‘Atlas’ and similar computations, and is similarly

assumed here. We refer to the revised computation as the ‘pseudo force-free’ (PFF)

case, since it contains co-rotation enforcing charges, but no currents. These PFF

computations, for both TPC and OG-type models, give light curves slightly distorted

from in the ‘Atlas’-type computations and appear to match well to the sample antenna

patterns and light curves of Bai & Spitkovsky (2010a). The principal effect, as noted

by these authors, is that for small w < 0.02 the low altitude emission that dominates

the second pulse of the TPC model has decreased phase folding, making the Jacobian

non-singular. There is a pulse peak but without a true caustic there is no sharp

pulse. This also weakens somewhat the first TPC peak for such high E pulsars. For


pulsars with larger w and for the higher altitude emission of the OG model there are

departures from the ‘Atlas’-style computation, but the effects are more subtle. In the

Appendix we present arrays of example light curves for the TPC and OG models.

These may be compared with, and considered as updates to, the similar light curve

panels in Chapter 2. We also update the figures describing the pulse width and flux

correction factors and their variation with viewing geometry, for comparison with

that paper.

3.2.2 Current-induced B Perturbations

The gap-accelerated radiating charges alone produce some currents in the open zone,

and we expect the gap-closing pair front to produce densities comparable to the co-

rotation value for at least some of the open field lines. This current will perturb the

magnetic field of Equations 3.4.

In the spirit of attempting analytic amendments to the vacuum model, we applied

the approximate perturbation field computed by Muslimov & Harding (2009) for a

pair-starved current flow in the open zone. This is expressed in magnetic coordinates

(rB, θB, φB), where the perturbation amplitude ǫ′ determines B′ = ǫ′ [2m Ω/(r2 c)]

with m the magnitude of the star’s dipole moment

B′rB= B′ s (1 − s2)−1

B′θB= B′[(1 − s2)−1 ∂s

∂θB+ sinθB(1 − s2)3/2−1 ∂s

∂φB]

B′φB= B′[−(1 − s2)−3/2 ∂s

∂θB

+ sinθB(1 − s2)−1 ∂s

∂φB

] (3.8)

and s = cos θrot = cos α cos θB + sin α sin θB cos φB. This perturbation field is defined

with respect to the static vacuum dipole. We take ǫ′ positive for outward-flowing

electrons (i.e. negative current) above the magnetic pole passing closest to the Earth

line-of-sight. This pole is generally inferred to produce the low-altitude radio pulsar

emission. If this pole has the opposite current (i.e. opposite magnetic polarity)

we have ǫ′ < 0. Of course an electrodynamically self-consistent model will have a


particular sign of ǫ′ for, e.g. the positive magnetic pole, but our PFF model includes

no current and so is insensitive to the sign of B; only this perturbation current breaks

the degeneracy.

To approximate the retarded solution, we matched this perturbation to our mag-

netosphere, mapping B(rB, 0, φB) to the magnetic axis of the swept-back dipole.

The perturbation field becomes singular along the rotation axis where s −→ ±1. By

exponentially tapering this singularity (∝ 1− e(|s|−1)/σs , with σs = 0.25) we were able

to integrate the field lines to determine the last closed field line surface (see below).

Unfortunately, for lines near φB = 0, π the residual effects of this singularity still

dominate, making the cap boundary at these azimuths unstable (Figure 3.2).

Accordingly we proceeded with a simpler toy perturbation field computed by

integrating a constant line current passing through the star along the magnetic axis

(the current extends to cylindrical radius r⊥ = rsinθ = 1.2RLC = 1.2c/Ω to minimize

end effects). This perturbation field

~B′(~r) ∝

∫

B(rb,0,φB)

d~I × ~r (3.9)

is treated as static in the rotating frame and is summed with the field of (4). The

effect of this simple perturbation field is dominated by the twisting of field lines along

the magnetic axis, as can be seen by the shift of the ‘notch’ foot-points of the last

closed field line cap boundaries in Figure 3.2. This rotation is also present in the

Muslimov & Harding (2009) field, but is dominated by the singularity effects except

for nearly aligned rotators α ≈ 0. For ease of comparison, we scale the perturbation

amplitude for fields (8) and (9) to the underlying dipole field (4) so that

ǫ = 〈B′(0.5rLC , θ, φ)〉/〈|B(0.5rLC, θ, φ)|〉 (3.10)

where we average over a sphere at r = 0.5rLC. Again the sign of ǫ is determined by

the current; we expect opposite values for the two poles.

None of these fields is fully realistic. In particular the current-induced B field


Figure 3.2 Open zone boundary (polar cap) foot-points: magnetic colattitude θB onthe star surface as a function of magnetic azimuth φB for the α = 70 PFF field withcurrent-induced perturbations. Top: the Muslimov & Harding (2009) pair-starvedfield, mapped to the retarded dipole. Bottom: retarded dipole with perturbationsfrom magnetic axis line currents. Notice that the currents ‘twist’ the field lines forboth models, shifting the ‘notch’ at φ ≈ 0.3. Although the singularity in the Mus-limov & Harding (2009) field (upper panel) has been smoothed, we still see dramaticinstability for lines directed near the rotation axis (φB ≈ 0.4−0.6 and φB ≈ 0.8−0.1).

perturbations used above are not complete consistent field/current systems. Never-

theless, given that the vacuum field geometry has already provided encouraging suc-

cesses in approximating the observed light curves, it is useful to explore how current

systems can perturb the modeled beam patterns and pulse shapes. As an example,

Figure 3.3 shows the ’antenna patterns’ in the hemisphere containing emission from

above the null charge surface (the OG case below). The unperturbed beam pattern

(middle) suffers opposite distortions in the two hemispheres.


eps = +0.2

eps = -0.2

eps = 0

Figure 3.3 Antenna patterns for current-induced perturbations. Pulsar light curvesare obtained from horizontal cuts across these images, after normalization by [sinζ ]−1.Here we show emission from above the null charge surface (OG case) for α = 60,w = 0.1. The middle panel shows the unperturbed PFF structure. The upper(ǫ < 0) and lower (ǫ > 0) panels show the effect of a perturbing line current along themagnetic axis. Note that the front caustic is better closed for ǫ > 0, but for negative ǫthe ’notch’ structure is more pronounced. The dashed lines show the range where theviewing angle ζ passes close to the low altitude emission from the opposite magneticpole, i.e. when one expects a radio pulse. Measurement of light curve perturbationscan thus determine the sign of B for a given magnetic pole.

3.2.3 Gap Locations

For each field structure, we assume the appropriate co-rotation charge density so

that the field pattern is stationary in the rotating frame. We define the open zone

by identifying field lines tangent to the light cylinder (actually at slightly smaller

radius for numerical expediency). These field lines are traced back to the surface

where they define the open-/closed-zone boundary. In Figure 3.1 we show the foot-

point locations, equatorial plane field lines (for α = 90) and the last closed field line

structure for an inclined α = 80 rotator. Our radiating surface is always in the open

zone toward the magnetic axis from this boundary. This plot shows the static (‘Stat’)

and point retarded dipole (‘Atl’ & ‘PFF’) field structures. The active field lines for

the outer gap are lighter (plotted in blue).


With the emitting field lines defined, we must choose the extent of the radiating

surface. For the TPC model, we follow the original definition (Dyks & Rudak 2003;

Dyks, Harding, & Rudak 2004), taking the emission surface to be the full set of last-

closed field lines, but stopping emission once the radial distance is r > rLC or once

the distance from the rotation axis exceeds r⊥ > 0.75rLC. This cut-off is required to

avoid the higher altitude pulse components (as used in the OG picture), otherwise the

modeled light curves generally have three or four pulse components. As in Chapter 2

we make this model more physical by placing the emission surface in the open zone,

separated from the last closed surface by the thickness w appropriate to the pulsar

E (Equation 3.1).

In the outer gap (OG) model radiation is produced on field lines crossing the

null-charge surface (~Ω · ~B(RNC) = 0), with emission starting near this crossing point

and extending toward the light cylinder. Again, we assume a gap thickness w. Field

lines that do not cross the null charge surface before rLC do not radiate.

The outer gap emission should be dominated by a section of the flared cone of field

lines above a given pole. For these simple vacuum-based models, some field lines just

inside the last closed surface extend for a very long distance before exiting the light

cylinder or crossing the null charge surface. For example, some field lines arc over the

star and cross ~Ω · ~B = 0 for the first time far from the star with a path length (in units

of RLC) s ≫ 1. These field lines define a disjoint, ‘high altitude’ second gap surface,

that does not in most cases appear to be active (although the HF pulse components

of the Crab might represent such emission; Moffet & Hankins 1996). Physically, we

expect that such lines arcing near RLC will be opened in a real magnetosphere with

currents and particle inertia. Also, gaps on field lines starting at large rNC will likely

be inactive if the soft emission needed to close the gap arises from other field lines

at much smaller rNC . Accordingly, as in Romani & Yadigaroglu (1995), we apply a

path-length s cutoff in the gap surface emissivity. Here the functional form used (for

s > 1 + 2sNC,min) is

F (s) ∝ e−[(s−2sNC,min−1)/σs ]2, σs = 0.1 (3.11)


with sNC,min the lowest null charge crossing for any active field line and all distances

in units of RLC . A detailed surface emissivity weighting awaits a physically realistic

spectral emission model.

While this work was being prepared for publication, a new discussion of light curve

formation in fully force-free models was presented by Bai & Spitkovsky (2010a). These

authors note the limitations with pure vacuum modeling mentioned above. They

also argue that in fully force-free magnetospheres where the radiation is associated

with the return current sheet, neither TPC nor OG pictures produce realistic light

curves. They propose instead a new ‘Annular Gap’ (AG) geometry that has emission

extending beyond the light cylinder. The foot-points of the AG field lines are the same

as those used here for the TPC geometry (for a given w) but in Bai & Spitkovsky

(2010b) the co-rotating pulsed emission is taken to extend well beyond the the light

cylinder. Thus, this model covers our OG zone plus additional emission at both high

and low altitudes. As we see below, the vacuum dipole field (possibly with current-

induced B perturbations) does a rather good job of matching the observed pulsations

– it will be interesting to see if the force-free numerical fields using the AG or OG

geometries can match the data comparably well.

3.2.4 Computational Grid

We identify the last closed field line surface (those field lines tangential to the speed

of light cylinder) and trace these field lines to the polar cap surface at r∗. We follow

the field line structure to < 10−4rLC , allowing good mapping even for pulsar periods

of ≥ 1s. A single magnetosphere model can serve a range of pulsar periods, since

the OG and TPC model features are computed as distances in fractions of rLC . The

pulsar period only affects R∗/RLC and hence the inner boundary of the emission zone.

Models are computed for all magnetic inclinations α. Photons are emitted along

the particle path in the lab frame (moving tangent to the local field line in the co-

rotating frame) and then phased, including light travel time across the magnetosphere,

to add to the skymap antenna pattern, as would be seen by distant observers. We


assume uniform emissivity along all active field lines (except for the taper at large

pathlength s as noted above). In practice we trace the trajectories for ∼ 104 photons

per field line, which produces adequately smooth light curves for the present compu-

tation. Models are computed for a range of gap widths from w = 0.01 to w = 0.3, to

accommodate a range of pulsar E.

3.2.5 Lightcurve Fitting

Our next task is to take this family of light curves and compare with the observed

pulse profiles. We take our γ-ray data from the full band E > 0.1GeV pulse profiles

published in Abdo et al. (2010a). These use the first six months of Fermi data and give

good statistics for the relatively bright pulsars considered here. In this Fermi pulsar

catalog, a background level is estimated from the surrounding sky. Interestingly, some

pulsars have significant steady (DC) emission. Matching this emission is an important

test of the model – accordingly we fit to the total emission, pulsed and un-pulsed,

after subtracting the background, as estimated in Abdo et al. (2010a). In particular,

we do not fit the un-pulsed flux as a separate degree of freedom – the excess above

the LAT pulsar catalog background is determined by the magnetosphere model.

We take the measured pulsar E and compute w according to Equation 3.1, then

use the closest value from the model grid. For this grid we take all α (in one degree

intervals) and fit the model light curve for each α and ζ . To compare with the data

we bin a given model curve into the phase bins (25 or 50/cycle) defining the light

curve for the pulsar in Abdo et al. (2010a). The total model counts are normalized

to the background-subtracted data counts and we then search for the best-fit model

to the combined (pulsar plus background emission) light curve.

A variety of fitting statistics work well. Simple χ2 fitting generally finds the best

solutions, but is controlled by the more numerous off-pulse and ‘bridge’ emission

phase bins and so occasionally provides poor discrimination against models with

unacceptably weak peaks. Choosing the model with the smallest maximum model-

data difference for any bin Max[Abs(Modi − Di)] does an excellent job of matching


the peaks, but less well on the bridge flux. We find that an exponentially tapered

weighting of the largest model-data differences is robust and produces sensible results.

Thus we minimize

χn =(Modi − Di)

2

Die−i(|Modi−Di|)/n (3.12)

with i(|Modi − Di|) the index of the phase bin differences, sorted large to small. Small

n ≪ 1 just uses the biggest difference bin (and thus focuses on the peaks), large n

goes over to classical χ2 weighting. The method is only weakly sensitive to n and an

e-folding weighting n ≈ 3 worked well.

Our reference phase φ = 0 occurs when the dipole axis, spin axis and Earth line

of sight all in the same plane. It is generally assumed that radio emission arises at

relatively low altitudes (but see Karastergiou & Johnston 2007), and that the radio

pulse occupies a fraction of the open zone field lines at low altitudes. In Abdo et

al. (2010a) phase zero is taken to be the amplitude peak of the main radio pulse.

However, the radio pulse profile often exhibits ‘patchy’ illumination of the emission

zone (Lyne & Manchester 1988). Thus the radio peak may not mark the phase of

the closest approach of the magnetic axis. Additional geometrical information comes

from the linear polarization sweep of the radio pulse, with the maximum sweep rate

associated with the line-of-sight passage past the magnetic axis. This maximum

often occurs well away from the intensity peak. Finally, if the radio emission arises

well above the star surface, relativistic effects can perturb the phases of the closest

approach (ideally pulse intensity maximum) and sweep rate. Blaskiewicz, Cordes, &

Wasserman (1991) argue that the approximate phase shifts are

φInt ≈ −r/rLC φPol ≈ +3r/rLC. (3.13)

Thus polarization information can give additional constraints on the absolute phase,

but altitude (plus mode changing and other sweep perturbations) can lead to substan-

tial additional uncertainty. Accordingly, we have the option to fit light curves with

the model phase free. Generally, we allow φ to shift by as much as ±0.1 of a rotation,

and retain the best fit model. When no radio pulse is known we allow full δφ = ±0.5


freedom in the pulse phase. The phase for the best-fit model is recorded in the (α, ζ)

map, as well as the model light curve and properties of the pulsed emission, for use

in further analysis. Recalling that the radio phase φ = 0 in Abdo et al. (2010a) is set

at the pulse intensity peak, we expect that our fit, which estimates φ0 for the true

magnetic axis, will tend to give small δφ > 0 for emission from finite pulse heights.

However, we expect that δφ will be less than φPol. In practice, patchy pulses and

polarization sweep uncertainties make δφ of any modest amplitude plausible.

In considering the values of the fit statistic χn, we must remind the reader that

these are not complete models. Detailed radiation physics will certainly change the

light curve shape from the uniform weighting assumed here. Also, the lack of self-

consistent magnetospheric currents must, at some level, cause light curve shape errors.

This, coupled with the excellent signal-to-noise of the Fermi light curves guarantees

that these will not be statistically “good” fits. Monte Carlo experiments with the

χn fit statistic show that for Poisson realizations of a 50 bin light curve, the typical

value is 〈χ3〉 = 7.5 while 99% and 99.9% values are 15.5 and 19. Thus fit differences

of ∆χ3 ≈ 8 are statistically significant at the normal 2.5σ level. This gives a guide to

interpreting the relative goodness of fits to the various models.

3.3 Fit Examples

We start with the archetype γ-ray pulsar, Vela. For this pulsar we show the fit

statistic χ surfaces for the three approximate dipole field structures (Stat, Atl, PFF)

discussed above, with darker colors indicating better model fits. The upper row is

for OG models, the lower row for TPC. For OG all field structures show a similar

topology, with best-fit solutions at ζ ≈ 70−80 and α ≈ 60−90, and no solutions at

small α, ζ . The swept-back models (Atl and PFF) have a tail of solutions to smaller

α. The TPC scenario can produce models for nearly all α, ζ . However, except for the

‘Atlas’ geometry which, as also noted by Bai & Spitkovsky (2010a), has unrealistically

sharp caustic pulses for small gap width w, the models produce much poorer fits than

the OG scenario, as they have weak pulses and far too much off-pulse (DC) emission.

3.3. FIT EXAMPLES 49

3.3.1 External Angle Constraints

The model fits in (α, ζ) space are especially useful when we have external constraints

on these angles. Some care is needed, however, in applying angle constraints from

other wavebands. For Vela and a number of other young pulsars with bright X-ray

pulsar wind nebulae (PWNe) we have been able to place relatively model-independent

constraints on viewing angle by fitting for the inclination ζ of the relativistic Doppler-

boosted PWN torus (Ng & Romani 2008). This method does not, however, distinguish

the sign of the spin axis. Therefore the X-ray torus fits allow two possible viewing

angles between the positive spin axis and the Earth line-of-sight ζ = ζ and ζ ′ =

180 − ζ. In fact, in Ng & Romani (2008) the torus position angle was restricted to

be 0 ≤ Ψ < 180, so the angles quoted in this paper may represent either ζ or ζ ′.

In the context of the rotating vector model (Radhakrishnan & Cooke 1969) radio

polarization data can also constrain the viewing angles. In most cases, the small

range of phase illuminated by the radio pulse allows only an estimate of the magnetic

impact parameter

β = ζ − α ≈ sin−1[sinα/(dΨ/dφ)max] (3.14)

where the maximum rate of the polarization position angle sweep Ψ(φ) occurs at

φpol = 0, near the closest approach to the magnetic axis. Here the sign of the sweep

is meaningful, determining whether the line of sight is closer to or farther from the

positive rotation axis than the observed magnetic pole (at inclination α). Thus, when

we have an independent PWN estimate of ζ, we have two possible α values – e.g. for

a negative sweep we get β < 0 and α = ζ−β or α = ζ ′−β. In this paper, we tabulate

the possible values of ζ and ζ ′, but save space by plotting only in the positive rotation

hemisphere. Thus there are two values possible for α from Equation 3.14, but when

α or ζ > 90, we reflect back into the positive hemisphere for plotting purposes.

Occasionally, when the radio pulse is very broad or when the pulse profile presents

an interpulse, the radio polarization can make meaningful estimates of both α and ζ ,


from fits to the full polarization sweep

tan(Ψ + Ψ0) =sinα sin(φ − φ0)

sinζ cosα − cosζ sinα cos(φ − φ0)(3.15)

where the polarization has the absolute position angle Ψ0 at φPol = φ0. Keith et

al. (2009) have recently presented several examples. When the radio illumination

is insufficient for a full solution, we can still break the degeneracy of the fits using

the X-ray ζ constraints. Finally, if the magnetic inclinations orientations are, at

least initially, isotropic we expect the inclination choice giving α closest to 90 to be

preferred on statistical grounds.

In Figure 3.4 the two possible RVM α solutions for Vela (Johnston, et al. 2006)

consistent with the PWN data are shown by the green circles. Note that the OG and

TPC models have minima close to the solution at larger α (bold circle). In general,

with constrained α, ζ values one can make a discrimination between the models. Note

that for the OG model the position of the minimum is closest to the preferred angle

for the relatively physical ‘PFF’ case. The values of the fit statistic are substantially

worse for TPC at all locations near these preferred angles (Table 3.1). Figure 3.5

shows the PFF light curves at the RVM angles, and at the global fit minimum, for

the TPC and OG scenarios.

3.3.2 Testing Field Perturbations

The best-fit models in Figure 3.4 are several σ off of the PWN/RVM angles. This sug-

gests either that there are systematic errors in these angle estimates or that the true

magnetosphere structure is slightly different than the PFF estimate above. The mea-

surement of such offsets for a number of pulsars should enable us to map the required

geometry differences and test physical models for their origin. As an example, here

we illustrate the perturbing effect of magnetospheric currents. Figure 3.6 shows the

χ plane for fits including a magnetic axis current (Equation 3.9). The current shifts

the fit minimum and distorts the light curves. For a best fit in the RVM-determined


50 100 150 200 250 300 350 400 450 500

80604020

80

60

40

20 OG, StatOG, Atl OG, PFF

TPC, Stat TPC, Atl TPC, PFF

Figure 3.4 Goodness of fit χ(α, ζ) surface for Vela with several models. Low χ (dark)represent the best fits. The three columns give results (Left to Right) for Stat, Atland PFF field structures. Top are OG models, bottom are TPC. The green lines showζ allowed by CXO PWN fitting. The ellipses give the combined constraints includingradio polarization data.

region positive currents are required for the OG model. The best models within

∼ 1σ of the PWN+RVM angles also prefer a small positive current. For negative

ǫ the best fit region shifts further from the externally determined angles, although

the match to the pulse shape is quite good, reaching a global minimum (χ = 34) at

(α, ζ) = (82, 71) for ǫ = −0.2. Since the global minimum does not shift precisely

to the PWN/RVM angles, we infer that (unsurprisingly) the true field geometry is

only approximated by this simple model. The TPC model fits are not improved by

including finite ǫ; the best minimum anywhere near the external PWN/RVM angles

has χ ≈ 200.

To summarize, for Vela with the external constraint, PFF is clearly preferred and

the best fit models are relatively close to the known α, ζ . The rotation axis is inferred


Figure 3.5 Pulse profiles for the TPC (left) and OG (right) models for Vela usingthe PFF field structure. The dotted pulse profiles show two periods of the LAT(E > 0.1GeV) Vela data from Abdo et al. (2010a). The first model pulse in eachpanel is for the PWN/RVM-fit α, ζ , with no current perturbation while the secondis for the parameters of the global OG fit minimum. The TPC fit quality varies onlyweakly with angle and does not improve substantially away from the RVM values.

60 80 100 120 140 160 180

OG eps=-0.1 OG eps=+0.2OG eps=+0.1OG eps=0.0OG eps=-0.2

8060

80

60

Figure 3.6 Comparison of the lower right quadrant of the OG fit planes for mod-els with magnetic axis current-induced B-field perturbations. Left to right: ǫ =−0.2,−0.1, 0.0, +0.1, +0.2. The plots show the range 45 ≤ α ≤ 90, 45 ≤ ζ ≤ 90.

to be oriented out of the plane of the sky, we are viewing at ζ ≈ 65 and the magnetic

inclination is large at α ≈ 75. Positive-current field perturbations are preferred.

The γ-ray pulse fits imply that φ0 lies earlier than the radio peak (and earlier than


the maximum sweep rate). This would suggest that Vela has patchy radio emission

dominated by the ‘trailing’ side of the radio zone.

3.3.3 Other EGRET Pulsars

The next most energetic pulsar in our test set is PSR J1952+3253. As for Vela, the

three field structures give quite similar χ-plane maps. In (Figure 3.7) we show the

χ maps for the PFF model with three values of B′. This pulsar has two narrow

peaks separated by ∆ = 0.49 in phase, so we find that the TPC model can produce

reasonable solutions, as indicated by the relatively dark regions in the lower panels.

This object is presently interacting with the shell of its supernova remnant, CTB80,

and so produces a PWN bow shock rather than a torus, precluding a ζ measurement.

However from the max sweep rate of the radio polarization from Weisberg, et al.

(1999), we can infer a constraint on the magnetic impact parameter (two green diag-

onal bands). The sweep maximum appears to lie towards the tail of the broad radio

pulse, consistent with our fit δφ ≈ 0.05; again the radio pulse peak would represent

patchy emission, in this case from the leading edge of the emission region at relatively

low altitudes.

Both models prefer relatively large α and ζ , and each can produce acceptable light

curves within the region allowed by the radio data. If improved radio measurements

or other angle constraints can be obtained, it may be possible to select between these

two models for this object and also measure the altitude of the radio emission zone.

Next and only slightly less energetic is PSR B1706−44 = PSR J1709−4429 (Figure

3.8). This object, in contrast to PSR J1952+3252, has a narrow double pulse with

∆ = 0.25. Here OG models clearly prefer the small α RVM solution. While the best

OG fits have χ ≤ 30, the TPC models in the PWN-determined ζ strip are poor with

χ > 90. Polarization sweep data are limited for this pulsar. The published plots in

Johnston, et al. (2006) give α ≈ 34, which is best matched with negative currents

ǫ ≤ 0. These fits however, imply a surprisingly large offset for the magnetic pole of

δφ = −0.04 = 14, which would put the magnetic axis at the leading edge of the radio


20 30 40 50 60 70 80 90

TPC eps=+0.2

80

OG eps=-0.280604020

60

40

20

OG eps=0.0 OG eps=+0.2

TPC eps=-0.2 TPC eps=0.0

Figure 3.7 Goodness of fit surface for PFF models of PSR J1952+3252, Left to right:ǫ = −0.2, 0.0, +0.2, top: OG, bottom: TPC. The regions indicated by the maximumpolarization sweep rate dΨ/dφ|max are shown as two green wedges.

pulse. Intriguingly, the best fit of all parameters is found with a OG PFF ǫ = 0.05

model at ζ = 53, α = 47 (χ = 19). This very good light curve (Figure 3.9) has the

magnetic axis near the radio peak with δφ = −0.01, but would have a polarization

sweep rate ∼ 3× larger that that seen in present radio data. Again, improved radio

polarization data might help shed light on the preferred inclination angle.

The next pulsar, PSR B1055−52 = PSR J1057−5226, is appreciably older. It

shows a pulse that is nearly a square wave in the full energy band, although the

energy-dependent light curves of Abdo et al. (2010a) indicate that it is likely a tight

double of separation ∆ ≈ 0.2 with strong bridge emission. The pulsar is too old to

show a bright PWN, so no ζ is available from the X-rays. Luckily in the radio it

has both a very broad pulse and interpulse and radio polarization fitting can strongly

constrain both α and ζ . We adopt the values from Weltevrede & Wright (2009) and

note the phase of the maximum polarization sweep is clearly at φ = 0.08 (later) than

plotted in Abdo et al. (2010a). Indeed, we find that the model fits prefer positive


20 40 60 80 100 120 140 160 180

80

80604020

60

40

20

Figure 3.8 Goodness of fit surface for PSR J1709−4429, panels as for Figure 3.7. TheX-ray determined ζ is shown by the horizontal band. For this pulsar the smaller αRVM solution is clearly preferred.

δφ, although the value is slightly larger than expected. The peaks on the model

light curves are also stronger than in the data for these parameters; the match can

be significantly improved for models with finite δw (not detailed here). As noted

in Chapter 2, no viable TPC models lie anywhere near the RVM-determined angles

(Figure 3.10). For OG there is a shallow fit minimum centered on the RVM angles.

Superposed on this are χ stripes, caused by the relatively coarse (25 bin) LAT light

curve. With improved LAT data the fits should become more discriminating.

For the TPC model there is a large region with modest χ values at small α

and ζ , with the best solution listed in Table 3.1. However these models are not

only inconsistent with the radio data, but all produce single pulses with one broad,

approximately Gaussian, component and are unlikely to represent the true solution.

If one allows any viewing angle, exceptionally good fits can in fact be found for OG

with χ slightly below the statistical minimum value. Despite an excellent match to

the observed γ-ray pulse, this is also unlikely to represent the true solution. This


Figure 3.9 Light curves for two PFF models for PSR J1709−4429. The model at theRVM position has a rather weak second peak and requires a relatively large δφ. Amodel at the PWN ζ but slightly larger α provides the best fit at δφ ∼ 0.

underlines the fact that sensible solutions must be compatible with external angle

constraints and, when these constraints are applied with care, one may use the multi-

wavelength data to rule out otherwise viable pulse models.

Finally, we discuss Geminga, the archetype of the old γ-selected pulsars, and

the only one known prior to the launch of Fermi. This object does show an X-ray

PWN, but like many old objects it is dominated by bow shock structure (Pavlov,

Sanwal & Zavlin 2006) and so it will be very difficult to extract torus/jet constraints

on ζ . However we do know that despite very sensitive radio searches, no convincing

detection of pulsed radio emission has been found, implying that |β| = |ζ−α| is large.

It has also been suggested that the thermal surface pulsations are most consistent with


50 100 150 200 250

80

80604020

60

40

20

Figure 3.10 Goodness of fit surface for PSR J1057−5226, panels as for Figure 3.7.For this pulsar, interpulse emission allows a good RVM fit for both the inclinationangle α and viewing angle ζ (green circle).

a nearly aligned rotator with small α (Pavlov, Sanwal & Zavlin 2006).

Here, with no information from the radio phase we must allow the φ to be freely

fit. The results are interesting. For all B field structures, and in particular the PFF

structure, the OG model only fits satisfactorily in a small region at small α, large ζ .

This means that one is viewing a nearly aligned rotator from near the spin equator,

i.e. very far from the magnetic axis. This is in good agreement with the expectations

of Romani & Yadigaroglu (1995) and guarantees that the pulsar should be radio faint.

In figure 3.11 we show surfaces computed for w = 0.18 and three trial ǫ values. TPC

shows best solutions over a range of angles and ǫ. However, these have ∆χ > 17

larger than the best OG solution and show only a single peak with bridge emission

(Figure 3.12). Thus while only few OG models are suitable, the light curve shape fits

better and the viewing angles for Geminga are very well constrained. It would be

very interesting to measure ζ and/or α from data at other wavelengths.


In Table 3.1, we collect these model fit results, giving the values for the externally

constrained fits for the unperturbed ǫ = 0 field geometry for both models. In addition

we quote best fits close to the RVM angles and the best global fits, when appropriate.

For Geminga, with no strong radio constraint we only list the global best fits for each

model.

3.3.4 Luminosities

Now that we have estimated the corrections to the all-sky flux fΩ, it is worth checking

how well the pulsars agree with the simple luminosity law Equation 3.1. A major

challenge to this check is the distance imprecision; of the five pulsars considered

here only Vela and Geminga have parallax distance estimates. The other distances

(derived from DM) are subject to appreciable uncertainty.

250 300 350 400 450 500 550 600 650 700

80

80604020

60

40

20

Figure 3.11 Goodness of fit surfaces for PSR J0633+1746 (Geminga), panels as forFigure 3.7. No quantitative external angle constraints are available, although theabsence of radio emission suggests that β = ζ − α > 15. TPC shows a number ofacceptable solutions; OG has few (albeit better) solutions at small α, large ζ .

3.3

.FIT

EX

AM

PLE

S59

Table 3.1 — PFF Model Angles and Efficiencies

Name log(E) w da α/ζ α/ζ′ α/ζ/ǫ χ/δφ fΩ α/ζ/ǫ χ/δφ fΩ F>0.1GeVa ηOG

erg s−1 ←− External −→ ←− TPC −→ ←− OG −→

Vela 36.84 0.012 0.287+0.019−0.017 72/64 122/116 72/64/0 201/+0.02 1.06 72/64/0 120/−0.03 1.03 879.4±5.4 0.013

best in reg 71/64/+0.2 88/−0.02 0.98best global 82/71/−0.2 34/−0.03 0.86

J1952+3252 36.57 0.016 2.0±0.5 b b 71/84/0 19/+0.05 1.10 66/78/0 14/+0.05 0.84 13.4±0.9 0.015best in reg 86/66/-0.2 18/+0.05 1.10 71/71/−0.2 14/+0.05 0.70

J1709−4429 36.53 0.017 1.4−3.6 34/53 108/127 31/49/0 147/+0.10 1.30 36/56/0 30/−0.04 0.89 124±2.6 0.24best global 47/53/+0.2 19/−0.01 0.76

Geminga 34.52 0.175 0.25+0.12−0.06 – – 50/26/0 258/+0.41 0.93 21/89/0 241/+0.87 0.13 338.1±3.5 0.05

J1057−5226 34.48 0.183 0.71±0.2 105/111 75/69/0 414/+0.12 0.91 75/69/0 106/+0.11 0.82 27.2±1.0 0.45best global 41/08/-0.2 37/+0.03 0.38 7/87/-0.2 5/+0.11 0.09

aFrom Abdo et al. (2010a), flux units 10−11erg cm2s−1

b dΨ/dφmax ≈ +3.5/ – Weisberg, et al. (1999).


Geminga is in reasonable agreement with the rule only if it lies at the upper end of

the present distance range. PSR J1057−5226 is nominally somewhat more luminous

than expected, but can be accommodated if the DM distance is a slight overestimate.

PSR J1709−4429, on the other hand, is much brighter than expected unless the true

distance is ∼ 0.7 kpc or the flux correction factor is as small as fΩ ∼ 0.07 (or some

combination of the two factors). Such changes would be quite surprising. It would

be very interesting to obtain a parallax distance estimate for PSR J1709−4429 to

eliminate that source of uncertainty.

Figure 3.12 Light curves for two PFF models for PSR J0633+1746 (Geminga). Theleft model is for the best OG solution, the right model for the best TPC solutionaway from β ≈ 0.

3.4. DISCUSSION 61

Figure 3.13 Pulsar γ-ray luminosity, corrected from the observed phase-averaged fluxto all-sky vs. the spindown luminosity. OG/PFF/ǫ = 0 points.

3.4 Discussion

In Chapter 2 we showed that patterns in the separation of radio and γ-ray pulse

components could be used to find acceptable geometrical parameters in dipole mag-

netosphere models for pulsar emission. Here we show that with high quality LAT light

curves and reference to multi-wavelength information on the inclination and viewing


angles the direct comparison between the model light curve shape is even more con-

straining. Within the restricted context of the present computations – dipole fields,

simple expressions for the gap ranges and γ-ray emissivity on the radiating surface –

we can already make some strong statements.

• Emission starting above the null charge surface (OG model) is strongly statistically

preferred over models which have substantial emission starting from the stellar surface

(eg. the TPC model). For PSR J1952+3252 the fit for the TPC model is not much

worse than that for OG, but for none of the pulsars modeled here is it better, if

external angle constraints are used.

• More realistic dipole field geometries, including sweep back and fields enforcing

co-rotation produce improved matches to the pulsar light curves when one focuses on

solutions near the orientation angles α and ζ implied by lower energy data. Some

additional improvement can be found by using test models for field-induced current

perturbations, but these are evidently not yet sufficiently realistic for detailed fits.

• The correction from observed flux to pulsar luminosity can be substantial. With

these corrections a simple constant Φ, Lγ ∝ E(∝ current) law provides an improved

description of the data. Nevertheless, the large departure for PSR J1709−4429 sug-

gests additional factors are needed to understand pulsar luminosities.

• The best-fit global models are often several degrees off of the externally known ζ

and α. This difference is statistically significant and implies that the LAT data have

the power to reveal perturbations to the simplified field geometry.

The robust success of the pulse profile computations described here for angles near

those available from other data make it likely that the true magnetospheric geometry

is fairly close to our simplified PFF model. It must be re-emphasized that this PFF

model is not a complete physical description – and indeed the whole electrodynamic

picture of ‘Holloway-type’ outer gaps is still of questionable validity. However, other

scenarios need to match the light curves quite closely to be competitive – this likely

forces them into rather similar magnetosphere geometries. Comparison with other

approaches, such as numerical magnetosphere simulations, should be fruitful. With


more pulsar light curves and higher quality data being provided by the Fermi LAT,

we should be able to use the departures from the predictions of these simple beaming

laws to infer geometrical modifications (and work toward their physical origin) and

to follow the evolution of the gap geometry and radiation with pulsar age.

We thank Jon Arons, Sasha Tchekhovskoy and Anatoly Spitkovsky for useful

discussions about the field geometry.

3.5 Light Curve Atlases

We present here figures that summarize the pulse properties of the TPC and OG

models for the PFF dipole geometry assumptions. These may be compared with the

corresponding figures in Chapter 2.


Figure 3.14 Light curves for the Two Pole Caustic (TPC) model. Each panel showscurves for four values of the gap width w. The curves are labeled with the numberof major peaks and the peak separation, in percent.


Figure 3.15 Outer Gap (OG) lightcurves. Labels as for Figure 3.14.


Figure 3.16 Pulse properties in the α−ζ plane for two pulsar emission geometries usingthe PFF field. Contours show the separation ∆ of the principal γ-ray pulse peaks.Bold lines mark ∆ at 0.1 intervals (key bottom middle panel); fine lines mark 0.02intervals. The background shading (key bottom left panel) gives the flux correctionfactor fΩ; large values mean that for the given α and ζ , the pulsar displays less fluxthan the sky average. White regions mean little or no flux from the modeled gaps.The dashed diagonal band in each panel indicates the region where lower altituderadio pulsations are likely detected. Objects seen outside these bands tend to beGeminga-like radio-quiet pulsars.

Chapter 4

Simulating the Galactic Pulsar

Population

This chapter is based on “The Galactic Population of Young γ-Ray Pulsars” by K. P. Watters and R. W. Romani

2010, ApJ, in press.

We have simulated a Galactic population of young pulsars and compared with the

Fermi LAT sample, constraining the birth properties, beaming and evolution of these

spin-powered objects. Using quantitative tests of agreement with the distributions

of observed spin and pulse properties, we find that short birth periods P0 ≈ 50 ms

and γ-ray beams arising in the outer magnetosphere, dominated by a single pole, are

strongly preferred. The modeled relative numbers of radio-detected and radio-quiet

objects agrees well with the data. Although the sample is local, extrapolation to the

full Galaxy implies a γ-ray pulsar birthrate 1/(59 yr). This is shown to be in good

agreement with the estimated Galactic core collapse rate and with the local density

of OB star progenitors. We give predictions for the numbers of expected young pulsar

detections if Fermi LAT observations continue 10 years. In contrast to the potentially

significant contribution of unresolved millisecond pulsars, we find that young pulsars

should contribute little to the Galactic γ-ray background.

67

68 CHAPTER 4. SIMULATING THE GALACTIC PULSAR POPULATION

4.1 Introduction

The Fermi Gamma-ray Space Telescope has proved remarkably successful at discov-

ering spin-powered pulsars emitting GeV γ-rays (Abdo et al. 2010a). The brighter

sources can be studied in great detail and in our quest to understand this emission

we have shown how careful multiwavelength analysis and geometrical modeling of

individual sources can improve our understanding of the γ-ray beaming (Watters et

al. 2009; Romani & Watters 2010). In addition, the Fermi LAT sensitivity has al-

lowed many new pulsar detections; the reported γ-ray pulsar numbers have increased

by more than an order of magnitude since launch in June of 2008. This allows, for

the first time, a serious study of this pulsar population. Such work provides further

guidance to the correct beaming models, and also constrains the birth and evolu-

tion of these energetic pulsars, with important implications for the study of Galactic

supernovae and their products.

Here we focus on the young γ-ray pulsars, whose connection with massive stars

is particularly important. These objects, with ∼ 0.03 − 0.3 s periods, have magneto-

spheres extending to many RNS so that dipole fields should dominate away from the

surface, allowing relatively simple models. Fermi is also discovering a large number

of recycled millisecond pulsars (MSP). We do not treat these objects here, since their

very compact magnetospheres may allow departures from dipole field geometries to

complicate the magnetosphere modeling; additionally their long lives and rare binary

formation channels make population synthesis more challenging. Early attempts to

describe the population of γ-ray MSP (Story et al. 2008; Faucher-Giguere & Loeb

2010) suggest the importance of this source class. However the recent spate of Fermi

detections requires a re-assessment of these models which we defer to future analysis

(see Chapter 5).

We start by describing our simulation method and the process used to build a

model galaxy of pulsars (§4.2). We next describe the different γ-ray models to be

applied to this simulated galaxy and study the distribution of γ-ray pulse profile

morphologies that result (§4.3). These are compared with the observed pulse profiles

4.2. SIMULATING THE GALACTIC YOUNG PULSAR POPULATION 69

from Fermi. We then discuss the modeled evolution of γ-ray emission with age and

decreasing spin-down luminosity, especially near the γ-ray death line (§4.4). Some

possible amendments to the γ-ray model and their observational effects are discussed

(§4.5). We conclude in §4.6 by describing the unresolved young pulsar background and

predicting the properties of pulsars accessible to Fermi with ten years of observations.

4.2 Simulating the Galactic Young Pulsar Popula-

tion

There is a long history of pulsar population synthesis studies using very detailed

modeling of the radio emission properties and the survey selection effects so that

comparison with the large radio samples can be used to constrain the population. The

recent effort of Faucher-Giguere & Kaspi (2006), in particular, has been quite sucessful

at reproducing the properties of the bulk of the radio pulsar population. Of course,

there have been earlier efforts to extrapolate from the radio pulsars and estimate

the numbers of γ-ray detectable pulsars (Romani & Yadigaroglu 1995; Gonthier et

al. 2007). However, with the large increase in pulsar numbers and the improved

understanding of γ-ray beaming supplied by the LAT, it is now possible to use this

sample to produce much more reliable modeling of the young pulsars. This is the

goal of the present paper.

Excluding the recycled pulsars, the LAT sample is very energetic (E & 5 ×

1033 erg s−1) and young (τ . 106 yr). It is, moreover, a more complete sample of the

nearby energetic pulsars than available from the radio alone. For example the ATNF

pulsar catalog contains three pulsars with spin-down luminosity E > 1035 erg s−1 and

d ≤ 2 kpc. The LAT young pulsar sample has at least six such objects and four addi-

tional pulsars with uncertain distances having a large overlap with this region. Thus,

study of the γ-ray pulsars provides an independent, and arguably better, picture of

the birth properties of energetic pulsars.

Such study requires a realistic model of the birth locations, kinematics, and spin


properties of energetic neutron stars. Much of our treatment is quite standard. For

example, we assume the birth velocity distribution of Hobbs et al. (2005) and assign

each modeled pulsar a birth velocity with random isotropic direction and magnitude

drawn from a three-dimensional Maxwellian with σ = 265 km s−1 in each dimension.

The pulsar is then shifted from its birth position to the 3D location expected at

the appropriate simulated age. We safely ignore the Galactic acceleration’s effect on

these trajectories for this young τ . 106 yr sample. We use a conventional birth

magnetic field distribution, log-normal with 〈log(B)〉 = 12.65, σlog(B) = 0.3, with B in

Gauss, (similar to the values used in Faucher-Giguere & Kaspi 2006 and Yadigaroglu

& Romani 1995). We also assume isotropic distributions of magnetic inclination

angle α and Earth viewing angle ζ , and a conventional width for the radio beam ρ.

Indeed, these choices seem quite robust as even modest changes lead to unacceptable

populations. We select the true age uniformly up to 107 yr (to ensure that we cover

the full range over which pulsars may possibly still be active in the γ rays) and

evolve the pulsars at constant v and B to the present, where they are observed at

their appropriate evolved spin-down luminosity E. Because we focus here on only

the most energetic pulsars, we can adopt such simple constant evolution; high spin-

down energy corresponds to low age, and the majority of our simulated γ-ray active

pulsars have τ . 106 yr. However, this means that we do not attempt to model the

old radio pulsar sample, where the effects of gravitational acceleration, field decay

and the approach to the radio death line can affect detection statistics. We do not

simulate here the binary star properties. We also do not attempt to follow the details

of the individual radio surveys whose samples are dominated by these older objects.

However, we do apply a uniform sensitivity threshold (Fmin = 0.1 mJy) and assume

that pulsars are radio-detected if |ζ − α| < ρ and the modeled radio flux exceeds

this sensitivity at the modeled distance from Earth. This is adequate to compare the

radio and γ-ray detectabilities and check the overall population normalization against

the radio surveys, with special attention to the large and uniform Parkes Multibeam

Survey (PMBS) radio sample (Manchester et al. 2001; Lorimer et al. 2006).

Our focus on the young pulsars and our interest in connecting to their massive

star progenitors motivates some amendments to the standard modeling. In the next


section we develop a detailed model of the progenitor distribution to allow us to

connect our pulsar numbers to the OB stars and our local estimates to the Galaxy as

a whole. We also find (in §4.2.2) that the LAT sample requires amendments to the

treatment of the birth spin and the radio luminosity of the pulsars. Finally, we study

in detail the effect of the evolving γ-ray beams on the detection numbers and pulse

properties. These extensions give a new view of the energetic pulsar sample at birth,

with important implications for the neutron star population as a whole.

4.2.1 Galactic Structure

The non-recycled γ-ray pulsars almost exclusively have ages < 106 yr, substantially

less than those of their parents, dominantly ∼ 10M⊙ B stars. Thus it is not unex-

pected that their distribution along the Galactic plane correlates with young star-

forming regions (Yadigaroglu & Romani 1997). To exploit this correlation we need

a detailed model of Galactic birth sites, namely high mass OB stars. On the largest

scale we follow Yadigaroglu & Romani (1997) in using the free electron distribution

(here we use the updated model of Cordes & Lazio 2002) as a proxy for the gas and

ionizing photons associated with massive stars. This model includes the Galactic

spiral arms, the 3.5 kpc ring and an exponential thin disk component with a scale

height of 75 pc, which we take here as the OB star plane distribution. High mass

stars also show a “runaway” component, from disrupted binaries, giving roughly 10%

of the stars peculiar velocities greater than 30 km s−1 (Dray 2006). We account for

this component with a exponential thick disk, as in the Cordes & Lazio model, but

with a scale height of 500 pc.

The LAT pulsar population is apparently quite local (mostly . 2 kpc) and on

these scales we can employ more detailed information on the parent star distribution.

In particular, the Hipparcos catalog gives a nearly complete sample of O and B stars

to ∼ 500 pc with useful parallax information. On somewhat larger scales, catalogs of

OB associations (Mel’nik & Efremov 1995) give the size and locations of massive star

concentrations. We have used these data to generate a 3-D model of the Galactic O


and B0-B2 star density. Locally, we use the direct Hipparcos sample of such stars

with a ≥ 2σ measurement of parallax (smoothed by a 50 pc spherical Gaussian). At

375 pc we make a smooth transition to a sample with uniform disk distributions (thin

+ thick runaway) augmented by the cluster OB star contribution (smoothed over

100 pc). In turn, this population merges smoothly to the large scale (spiral arm plus

disk) distribution with a transition at 1.7 kpc. The overlap between the individual

stars and clusters and between the clusters and the spiral arms allows us to match all

components to the normalization determined by the local, complete OB star sample.

The massive stars (i.e. pulsar birth sites) drawn from this distribution are shown in

Figure 4.1 projected to the nearby Galactic plane. Although the uniform component

is substantial, large OB concentrations and clusters are apparent.

Figure 4.1 also shows the projected location of the nearby LAT pulsar sample. For

many of these objects the distance estimates are quite poor (range shown by radial

lines). Thus while several pulsars are superimposed on massive star associations, the

poor distance constraints prevent confident assignment. We can, however, form a

‘Figure of Merit’ to test agreement between the distribution of observed pulsar posi-

tions and positions from our detailed model simulation. This is an overlap sum over

the set i of model birthsites with a weight determined from a Gaussian distribution

about the uncertain location of each observed pulsar, j. This distribution combines

the (generally large) uncertainty in the pulsar radial distances with a transverse Gaus-

sian spread of 50 pc to account for the smoothed birthsite distribution and the offset

due to pulsar proper motion. The result is

FoM =∑

j

∑

i

e−

(lj−li)2+(bj−bi)

2

2 tan−1(50 pc/dj)2 × e

−(dj−di)2

2σ2dj . (4.1)

Errors were estimated by boostrap analysis.

Comparisons were made between this smoothed simulation and the LAT pulsars.

The detailed model gives a value of 156 ± 33. This is slightly, but not significantly,

better than the values found from pulsars distributed according to the Cordes &

Lazio (2002) spiral arm model (136± 22) or a simple uniform exponential disk alone


(147 ± 26). We will thus use our detailed model in further analysis, although we

caution that it is not yet demanded by the data. Improved pulsar distance estimates

and/or analysis that uses observed pulsar proper motions to correct back to birth

sites can substantially improve the utility of the detailed Galactic OB distribution.

One relatively robust result from the OB star modeling is a connection with Galac-

tic core collapse rates. Summing up the local, nearly complete, Hipparcos sample for

each luminosity class and using each class’ nuclear evolution lifetime (Reed 2005),

we can calculate the supernova rate contribution from each. We find, including stars

from B2-O5, a local supernova rate of 40 SN kpc−2 Myr−1

(very high mass stars which

might produce black holes contribute ∼ 1% of this rate). In turn our model allows us

to extrapolate this to the Galaxy as a whole, yielding a Galactic O-B2 star birth rate

(i.e. SN rate) of 2.4/100 yr−1. This compares very well with other recent estimates,

eg. 2.30 ± 0.48/100 yr−1 from nearby supernovae (Li et al. 2010), 1.9 ± 1.1/100 yr−1

from 26Al emission (Diehl 2006) and ∼ 1−2/100 yr−1 from a similar analysis of nearby

massive stars (Reed 2005). Thus our model is well normalized and can be used to

connect O-B2 progenitors with their γ-ray pulsar progeny.

4.2.2 Birth Spin Distribution and Pulsar Radio Emission

The new-born pulsar population depends critically on the distribution of spin periods

at birth. This is especially important for understanding the γ-ray sample; the bulk of

the radio population is sufficiently spun down to retain little memory of this adolescent

phase. We describe the birth spin periods with a single parameter truncated normal

distribution:

Prob(P ) ∝ e−

(P−P0)2

P20 . (4.2)

Here the spread equals the mean and we truncate at a minimum P = 10ms.

Faucher-Giguere & Kaspi (2006) find a characteristic birth spin period of P0 =

300ms for the radio population. However, we find that this produces very few short

period, γ-detectable pulsars. Indeed, one misses producing the observed γ-ray pulsar


Figure 4.1 Modeled OB stars (i.e. pulsar birth sites) in the nearby Galactic plane(black dots). Detected γ-ray pulsars are also indicated (radio-selected as green circles,γ-selected as blue squares, for the distinction between these two classes, see theirdefinitions in §4.2.4). Radial lines indicate the uncertainty in the distance estimates(open symbols indicate pseudo- and DM- distance estimates, filled symbols indicateparallax, kinematic, and association-based distances). The Galactic Center lies at (0,0). The overdensity of birth sites at Galactic radii just larger than the Suns’ 8.5 kpcrepresents the Orion spur. Cyg OB2 is prominent toward the right side. Inner armclusters appear toward the bottom of the plotted region.


numbers by a very large factor unless the birthrate is nearly 4 times that allowed

by the observed OB stars and supernovae (i.e. more than 10σ higher than the value

estimated by Li et al. 2010). Accordingly, the birth spin period distribution, for the

γ-ray pulsars at least, must extend to much lower values.

It is, however, possible to reconcile this with the radio results. We find that

the requirement for a large P0 in the radio sample is a consequence of the assumed

radio luminsosity law. We consider here three radio pseudo-luminosity (L = S d2)

laws. The first is a power law distribution that is independent of any other pulsar

parameters (Lorimer et al. 2006). Next, Faucher-Giguere & Kaspi (2006) recommend

a log-normal distribution whose central value scales as a power law with spin-down

luminosity. Finally, we consider the broken power law distribution of Stollman (1987)

for which the power law scaling of the central value stops and is flat above a cutoff

(at E ≈ 1034 erg s−1). We will refer to these functions as LR Flat, LR Power Law,

and LR Broken Power Law, respectively.

To illustrate the effect of the initial spin and radio luminosity laws, we compare

with the young pulsar populations: the LAT pulsar sample (all sky) and the high

energy (E > 1033 erg s−1) PMBS detections (within the PMBS footprint). To guage

pulsar detectability we adopt a standard radio beam width

ρ = 5.8 deg P−1/2 (4.3)

(Rankin 1993), with the assumption of a circular beam of uniform (integrated) radio

flux directed along the magnetic dipole axis; we discuss briefly later the possibility

that young objects have even wider radio beams from high altitude (Karastergiou &

Johnston 2007). We simulate the Galactic pulsar population as summarized above,

assign a pseudo luminosity, check pulsar detectability at the modeled beam orienta-

tion and distance, compute the cumulative radio luminosity functions of the model

detections (Figure 4.2), and compare with the observed samples. In all cases the

‘Flat’ luminosity law does not reproduced the observed luminosities, so we discard

this form. The broken power law and power law distributions are quite similar for the

radio sample (upper panel), but differ for the more energetic radio-detected pulsars


Figure 4.2 Upper panel: Cumulative distribution functions of radio luminosities for apopulation of young radio-detected pulsars. The model CDFs show radio luminositylaws with differing spin-down dependence of the central value: Flat (green), PowerLaw (red) and Broken Power Law (black). These models are compared with theParkes Multibeam sample (in blue); lower panel: same as above, but for the γ-ray-detected set, with the LAT sample in blue. The labels show the Log10 of the KSprobability of agreement for each case.

in the γ-ray sample. We remind the reader that we have not followed the detailed

selection effects of the radio surveys. However, these are unlikely to affect these con-

clusions. As an example, we applied a roll-off of the sensitivity with period, as in

Dewey et al. (1985). Here Smin = S0[9we/(P − we)]1/2 with the effective pulse width

we = [(0.1P )2 + 10 ms]1/2 to approximate the PMBS losses at DM ≈ 200 pc cm−3.

This caused only a small (∼ 6%) decrease in the number of detections for the most


energetic pulsars (Log(E) > 36.5) and negligible loss (<2%) from the sample as a

whole. In particular, we recover the Faucher-Giguere & Kaspi (2006) result that, with

the Power Law model, P0 = 300 ms provides a very good match to the PMBS radio

luminosity function.

However, we see that the Fermi sample (Figure 4.2, lower panel) shows some

discrimination between the models, especially for the high luminosity pulsars which

dominate this set. The numbers of high E pulsars are particularly sensitive to P0.

Accordingly we have explored this sensitivity by comparing the modeled N(E) of

pulsars in the energetic (E > 1033 erg s−1) PMBS and Fermi samples as we vary P0.

Our statistic is a χ2 comparison of the pulsar detections in half-decade bins of E.

We also use the additional measurement of the Galactic core-collapse rate (and error

estimate) from Li et al. (2010), fitting to the total birthrate. The results are shown

in Figure 4.3.

As expected, this shows that with the Power Law luminosity model the radio

sample prefers long initial periods, but P0 = 300ms is completely unacceptable for

the γ-ray sample. This is because of the very small birthrate of energetic pulsars

for this large P0. One can, of course, imagine a γ-ray emissivisity law allowing high

efficiency for detection of these very rare pulsars. The cost is that such pulsars are

then extremely luminous and seen at very large distance. We find that the γ-ray pulsar

sample gives such a sensitive probe of the high E population that changes sufficient

to accomodate P0 as large as 300ms predict a pulsar sample in strong disagreement

with the observed population (see further discussion in §4 for a specific example).

However, if we adopt the Broken Power Law radio luminosity law, we find the

prefered value of P0 for the PMBS sample decreases and that the χ2 at the P0 ≈ 50ms

demanded by the γ-ray pulsars shows only a small increase from the minimum value.

We suspect that a complete treatment of the radio survey selection effects which

should decrease the detectability of short P radio pulsars along the Galactic plane

would further improve the agreement at small P0.

In sum the radio and γ-ray pulsars can come from the same population if


P0 ≈ 50ms and the extreme increase in radio luminosity for the shortest period

pulsars is mitigated with the Broken Power Law. A second quantitative compar-

ison of this agreement is shown in Figure 4.2, where we quote the model-data

Kolmogorov-Smirnov comparison for the two pulsar samples for each luminosity law

(here P0 = 50ms). The Broken Power Law indeed provides the best match to the

detected radio luminosity function, especially for the PMBS sample. For the smaller

Fermi sample the improvement is not large. We have confirmed that both this test

and the N(E) test are insensitive to the chosen γ-ray model (see Section 4.3). We

thus adopt the ‘Broken’ luminosity law as the best fit to the data.

As a consistency check, applying our adopted beaming and flux law to the pop-

ulation, we find that the observed number of PMBS pulsars corresponds to a full

Galactic pulsar birthrate of 1.43/100 yr−1. This is in reasonable agreement with ear-

lier radio population estimates and with the core-collapse estimates in §4.2.1 so we

may turn our attention to the γ-ray-detection criterea.

4.2.3 γ-ray Emission

To complete the population synthesis we require a γ-ray pulsar flux and beaming

model. Such models also depend on the pulsar’s energetics and geometry and a

major goal of the data comparison is to constrain such dependence. All of the γ-ray

models employed in this work utilize the same heuristic efficiency law

η = (1033erg s−1/E)1/2, (4.4)

with γ-ray production ceasing for objects with E < 1033 erg s−1. This law has the-

oretical motivation (Arons 2006) and provides a reasonable match to the observed

pulsar efficiencies (Abdo et al. 2010a).

The portion of the magnetosphere producing γ-rays and hence the fraction of the

sky covered by the beam, however, differs appreciably between models.

In this paper we treat several versions of the two popular vacuum magnetosphere


beaming models. The first is the standard Outer Gap (OG) model, with emission

in each hemisphere produced by field lines above a single pole. In its basic form

the γ-ray production is dominated by the upper boundary of a vacuum gap which is

bounded by the last closed field lines and runs from the null charge surface (Ω ·B = 0)

to the light cylinder RLC = cP/2π. This gap has a spindown-dependent thickness

Figure 4.3 Goodness-of-fit of the model to the observed data as a function of initialspin period. For each model, the pulsar population is assigned an initial spin periodchosen from a normal distribution with centroid and σ P0. Magenta curve (filledcircles) – fit to the γ-ray pulsar sample. Red curve (starred points) – the young radiopulsars, using a Power Law radio luminosity law. Black curve (open circles) – radiopulsar comparison with the Broken Power Law luminosity function.


approximated by

w = η = (1033erg s−1/E)1/2 (4.5)

where w is the fractional distance from the last closed field lines to the magnetic

axis, and the emission peaks ∼ w away from the closed zone. As the pulsar spins

down, w grows; this causes the radiating particles to approach the central field lines,

the null charge crossing to move to high altitudes, and the γ-ray emission to become

more tightly confined to the spin equator. To compute light curves from this simple

version, we compute photon emission from particles traveling in a sheet centered on

w, with a Gaussian cross section. As E decreases, we might expect pair production

to become more difficult and this sheet to thicken. An extreme assumption is that

emission arises from particles spanning the full thickness w inward from the closed

zone. Models computed for this assumption are denoted full gaps (OGFG). In both

versions, this is essentially a ‘hollow’ cone model which tends to produce two caustic

peaks, of varying separation, with a bridge between.

The other popular beaming model has emission extending from the star surface to

high altitude, so that both poles are visible in both hemispheres. By truncating the

emission somewhat before the light cylinder it is possible to produce models lacking

the leading ‘OG’ pulse. This picture most easily produces pulses with ∼ 180 phase

separations and tends to have ‘Off pulse’ flux comparable to the ‘Bridge’ between the

two peaks. In its original form this Two-Pole Caustic (TPC) model (Dyks & Rudak

2003) truncates emission at the lesser of rLC from the star or a distance of 0.75rLC

from the spin axis. This basic model uses the same w relation as the OG model

(Equation 4.5).

Modifications of this geometry have been suggested as better approximations of

the physical realization of such vacuum zones in the Slot Gap model (Muslimov &

Harding 2004; Grenier et al. 2010). For the first version (TPC2), the emission extends

to to 0.95rLC and utilizes a slower, plausibly more physical, w scaling:

w =1

2(1033erg s−1/E)1/6 (4.6)


which has wide w > 0.1 gaps even for very energetic Crab-like pulsars. Finally, we

also define a full gap version of this model (TPC2FG), with γ-ray emission coming

from the full volume between the last closed field lines and the w value calculated

from Equation 4.6.

There are also now a promising set of non-vacuum models, generated by numerical

realizations of the force-free magnetosphere. One version, the ‘Separatrix Layer’ (SL)

model (Bai & Spitkovsky 2010b) generates emission from field lines that start from

w ≈ 0.1−0.15 and merge in the wind zone. In this picture, γ-ray emission arises from

near the light cylinder, extending several 10’s of percent past this distance. While

this model has some attractive features, especially for young pulsars with robust pair

production, it requires numerical realizations. We concentrate here on comparing the

Fermi sample with the analytically-derived vacuum models (OG and TPC) and defer

comparison with the SL model to future work.

4.2.4 Pulsar Samples and Detection Sensitivity

The most uniform γ-ray pulsar sample available at present is the set of 38 young

objects reported in the First Fermi LAT Catalog of γ-Ray Pulsars (Abdo et al. 2010a).

We will refer to this as the “6 Month” sample (because that work was based on 6

months of Fermi LAT data). We also consider a larger sample of 50 young γ-ray

pulsars, including all non-recycled pulsar detections announced as of July 2010. In

particular, this includes the ‘blind search’ detections described in Saz Parkinson et

al. (2010), as well as several individually announced discoveries. We refer to this as

the “Current” sample; typically ∼ 1 year of LAT exposure contributed toward these

detections.

Fermi has two different routes to γ-ray pulsar discovery. When the signal is folded

on an ephemeris, based typically on radio observation, significant pulsations can be

seen to quite low levels. We term these “radio-selected” pulsars. Pulsations can also

be discovered ‘blindly’, by searching directly in the GeV photons, but this method

requires a somewhat higher flux for detection. We use here a threshold increase of


3× for such “γ-selected” detections (Abdo et al. 2010a).

A simple division of pulsars into these two classes is complicated by the discovery

of several γ-ray pulsars with radio luminosities an order of magnitude or more fainter

than those of the typical pulsar population (Camilo et al. 2009; Abdo et al. 2010b).

These sources were detected in deep, targeted integrations of young supernova rem-

nants, unidentified γ-ray sources or detected γ-ray pulsars. Their low flux density

would not be accessible in the typical sensitivity of modern large area sky surveys,

0.1 mJy at 1.4 GHz.

Since we are simulating the properties of sources detectable in large sky surveys,

we wish to assign real detected pulsars to one of these classes based on their observed

fluxes, not the accident of whether they had exceptionally deep radio observation.

Thus “radio-selected” objects must have the survey-accessible flux density above. γ-

selected objects must have the larger GeV flux required for ‘Blind search’ discovery.

This results in a re-classification of several objects. There are three pulsars that had

prior radio detections with 1.4GHz flux densities below 0.1 mJy (PSR J0205+6449,

0.04 mJy; PSR J1124-5916, 0.08 mJy; and PSR J1833-1034, 0.07 mJy). For the

purposes of this work, we do not consider these objects to be radio-selected γ-ray

pulsars. Two of the three objects have high enough γ-ray flux levels that even without

the assistance of a radio ephemeris-guided folding search, they would still have been

detected as γ-selected pulsars (PSR J0205+6449 and PSR J1833-1034). Thus, these

two pulsars are added to the γ-selected subset of the observed population. The third

object, PSR J1124-5916, likely could not have been detected to date in blind γ-ray

searches, and so is removed from the observed population. Finally, PSR J2032+4127

was found in a blind γ-ray search, but radio follow-up observations found a 1.4 GHz

flux of 0.24 mJy. Thus this object “should have” been detected in radio surveys and

is added to the radio-selected subset of the observed population.

With these classification amendments, the “6 Month” sample of 37 young γ-ray

pulsars contains 19 radio-selected and 18 γ-selected pulsars. Similarly the “Current”

sample contains 24 radio-selected and 26 γ-selected objects. These are the samples

against which we will test the γ-ray emission models.

4.3. γ-RAY PULSE MORPHOLOGY 83

In our simulation, any γ-ray pulsar with a radio beam crossing the Earth line-

of-sight with modeled flux density > 0.1 mJy is assumed detectable in a survey and

termed radio-selected. For the γ-ray pulsars we model the phase averaged flux on the

Earth line-of-sight and compare with the estimated detection threshold for each sky

location plotted as a map in Abdo et al. (2010a). We scale the ephemeris detection

threshold with the square root of exposure time from the 6-month estimates in that

paper. To be γ-selected, a modeled pulsar must exceed 3× this ephemeris detection

threshold, and must have a modeled radio flux density < 0.1 mJy at Earth.

4.3 γ-ray Pulse Morphology

In addition to simple detections and luminosities, the observed radio and γ-ray pulsar

profiles contain a wealth of information on the pulsar emitter. We have shown in

Romani & Watters (2010) how detailed analysis of the light curves, radio polarization,

and other multiwavelength data can make strong model comparison statements for

individual sources. Here we wish to treat the statistics of the population as a whole, so

we must rely on relatively crude basic pulse properties that can be measured relatively

uniformly in the discovery data. We thus characterize the pulses by i) the number of

strong, caustic-type peaks in the γ-ray light curve ii) the phase differences between

the radio and γ-ray peaks and iii) the relative strength of γ-ray emission between

(‘bridge’) and outside of (‘off’) the peaks for the common double-peaked profiles. In

the following sections we quantitatively compare each of these Fermi observables with

the model predictions for the pulsar population.

4.3.1 Peak Multiplicity

The first characteristic typically measured for a light curve is the number of peaks

in the pulse profile. To compare with the data, we have taken each pulsar in our

simulated Galaxy, applied the beam shape for each of the five trial γ-ray models

described above, and tagged peaks with an algorithm that identifies peaks and their


Figure 4.4 Peak multiplicity histograms in four different E bins. The black bars arefor the OG model, the red is for the Dyks and Rudak TPC model, and the green isfor the TPC2 model. The distributtion of observed pulsars in each bin is given by themagenta circles. The thick hollow bars use a narrow Gaussian distribution of fieldlines around the w value specified from Equation 4.5 or 4.6, appropriately. The thinfilled bars use the full gap, stretching from the last closed field lines in to the fieldlines specified by the appropriate w value.

phases in the modeled γ-ray light curves. To increase similarity to the actual data we

blocked the model light curves into 50 phase bins, before running the peak tagging

algorithms. The results range from 0 peaks (γ-ray flux on the Earth line-of-sight, but

no strong peak detected) to 4 or more distinct sharp peaks. Since for some models

the peak morphology varies significantly with pulsar age/spin-down luminosity, we

divide the sample into 4 bins of E, logarithmically spaced.

The results are shown as histograms in Figure 4.4, with bars for each of the five

models. For comparison, the measured multiplicities (always 1 or 2) for the “Current”

sample are shown by the circles in each panel. It is immediately apparent that the

original TPC model (central bars) produces many 3 and 4 peak light curves not

seen in the data. The amendments to a slower w evolution (Equation 4.6) and gaps

extending to 0.95RLC (wide right bars) make this disagreement worse. Extending

the emission throughout the full gap (narrow right bars) does blur out some weaker

peaks, mitigating the problem. We will use this blurred version (slow w evolution,


Table 4.1 — Model Probabilities from Peak Number Comparison

log(Poisson Probabilities)

E OG OGFG TPC TPC2 TPC2FG

LOG(E) > 36.5 -4.46 -4.66 -5.58 -8.42 -6.02

36.5 > LOG(E) > 35.5 -3.17 -3.59 -5.82 -11.22 -7.38

35.5 > LOG(E) > 34.5 -2.54 -2.75 -5.52 -8.79 -4.95

34.5 > LOG(E) > 33.5 -3.43 -3.19 -4.97 -7.03 -5.80Σ∆Log(P ) · · · -0.59 -8.29 -21.86 -10.55

fully illuminated gaps) in the remainder of the paper when we refer to ‘TPC’, as it

and the original TPC model give the best match to the observed quantities. The

differences in the outer gap OG predictions between the Gaussian illumination (wide

left bars) and the full gap illumination (narrow left bars) is relatively minor.

We quantify the data comparison in Table 4.1 and Table 4.2. Rows 1 through

4 of Table 4.1 show the Poisson probability that the model is a fit to the Fermi

data. Each row represents one energy bin, and the best fit model for that bin has

its probability in bold. This emphasizes that the TPC2 model fairs particularly

poorly without the full gap illuminated, and that for each bin the OG models provide

substantially better representations of the peak multiplicity. The modeled presence

of 0 peak sources (which could not, by definition, be discerned in the data) lowers

the probability for all models. Also the absolute normalization is meaningful here,

as these computations are run with the same model-determined pulsar birthrate (see

§4.6). In row 5 we calculate the probability decrement of each model when compared

to the best model (original OG), summed across the four energy bins. Again, in the

rest of the paper we adopt the original OG and TPC2FG version as the representatives

of the two fiducial models.

86C

HA

PT

ER

4.

SIM

ULA

TIN

GT

HE

GA

LA

CT

ICP

ULSA

RP

OP

ULA

TIO

N

Table 4.2 — Model Probabilities from Pulse Morphology Comparisons

∆ - δ : 2-D KS ∆ : 1-D KS Bridge/Off-pulse : FoM

E OG TPC2FG OG TPC2FG OG TPC2FG

LOG(E) > 36.5 -0.96 -3.40 -2.00 -2.05 600 ± 128 462 ± 142

36.5 > LOG(E) > 35.5 -0.92 -3.52 -0.97 -0.72 1928 ± 268 387 ± 143

35.5 > LOG(E) > 34.5 -2.30 -4.52 -1.18 -0.83 552 ± 198 316 ± 159

34.5 > LOG(E) > 33.5 · · · · · · -0.65 -1.52 221 ± 138 143 ± 116Σ∆Log(P ) · · · -7.26 · · · -0.32 · · · -7.37


Figure 4.5 2-D plots in the ∆ − δ phase space, for the same four E bins as in Figure4.4.

4.3.2 Peak Locations

A particularly powerful test of the models arises from comparing the γ-ray peak

separations ∆ (available for all light curves with two or more peaks, taken as the

widest strong peak separation) and the lag of the first γ-ray peak from the magnetic

dipole axis δ. For radio-detected pulsars this axis is typically marked by the main

radio pulse. The “∆ − δ” plot of these quantities is an excellent probe of the model

geometry (Romani & Yadigaroglu 1995; Watters et al. 2009). We produce such plots

here in Figure 4.5, again with four different E bins.

γ-ray pulsars that are not detected in the radio will have an undefined value of

δ; accordingly these objects are plotted in a band to the left, beyond the vertical

dashed line at δ ∼ −0.06. Similarly, pulsars with only a single peak in their γ-ray

pulse profiles have an undefined value of ∆. We assign to these singly-peaked profiles

a value ∆ = 0 and plot in a band along the x axis. Note that in practice limited

signal-to-noise in the γ-ray profiles usually prevents measurements of peak separations

any closer than ∆ ≈ 0.2, so the observed (large dots) single pulse profiles doubtless

include some unresolved tight doubles, such as appear in the models particularly for

small E.


In the plots the data show a clear inverse correlation between δ and ∆. This was

predicted for outer magnetosphere models (Romani & Yadigaroglu 1995) and indeed

appears for the (dark) OG model points. The trend is weak or absent in the (light)

TPC points. In the OG case the correlation is easily understood since both caustic

peaks are generated from the magnetic pole opposite to that producing the radio

emission. As the Earth line-of-sight cuts a smaller chord across this hollow cone, δ

increases and ∆ decreases. In contrast, the TPC model produces the first γ-ray peak

from the same pole as the radio emission. The δ is nearly fixed, showing only small

variation with viewing angle and age.

A less obvious difference is the OG correlation of ∆ and E, again weak or absent

in the TPC model, which as expected shows a typical ∆ = 0.4 − 0.6, with a strong

concentration at ∆ ≈ 0.5, for all spindown luminosities. In the LAT data, if we

consider for each energy bin the fraction of objects with ∆ > 0.4 we find 73% (11/15),

56% (9/16), 38% (5/13), and 17% (1/6), moving through our four energy bins from

high to low.

For a quantitative measure of the goodness of fit between the data and the models,

we employ a two dimensional Kolmorgorov Smirnov test (KS test) in this plane on

the radio-selected samples. We can also run a standard one dimensional KS test on

the γ-selected objects, for which only a ∆ value exists. The probabilities from these

tests are displayed in Table 4.2. With only a single detection, the 2-D KS test is

unavailable for the lowest E bin. The bottom row of Table 4.2 (as for Table 4.1) gives

the summed probability decrement compared to the best (OG) model; although the

OG model itself is not adequate (Prob ∼ 10−4), probabilities for the TPC models are

factors of ≥ 107 lower. Statistically, the OG model is strongly preferred.

Single peaked (x-axis) γ-ray light curves deserve some additional discussion. As

noted above, objects may appear here when the data do not resolve a double. Such

objects in the OG picture are expected to appear at δ ≈ 0.3− 0.4. In the OG model,

a peak may also be single if the first γ-ray peak is missing. Since this caustic forms

at high altitudes, it is rather sensitive to field line distortion from sweep back and to

aberration effects, especially for moderate to large E. In the vacuum models, we find


that this caustic (and peak) are often missing for energetic pulsars when the observer

line-of-sight lies well away from the spin equator. Near the spin equator the caustics

are unaffected, so that large ∆ appear, but smaller ∆ < 0.3 are missing for energetic

pulsars. As a result δ is large (∼ 0.5 − 0.6) since it now measures the distance to

what is normally the second pulse. This effect may be seen in the models and data of

the left panels of Figure 4.5. However, it should be noted that the sensitivity of the

first peak caustic to field line perturbations makes the extent of this effect difficult to

predict. For example Romani & Watters (2010) found that open zone currents can

reduce or eliminate such first peak ‘blowout’, so realistic models containing plasma

may affect the prevalence of large δ.

Finally, it should be noted that if small altitude emission occurs, then the trailing

side of the radio-producing pole may form a caustic (this is the first peak in the TPC

picture). As seen by the heavy concentration of TPC model dots in Figure 4.5, this

results in δ ≈ 0.1 . The Fermi LAT may have in fact uncovered one such object:

PSR B0656+14, which appears near δ = 0.2 in the third panel, has a single pulse,

a peculiar soft spectrum and an apparent luminosity ∼ 30× smaller than seen from

similar E pulsars. Polarization modeling suggests that it has a very small α and ζ , so

that its outer magnetosphere beam should miss the Earth. In this interpretation we

only see the fainter low altitude emission because of this pulsar’s very low distance.

Thus δ provides a useful way of sorting pulsar properties, even when only a single

γ-ray peak is available. Improved S/N, polarization studies and, above all more

physical magnetosphere modeling, should help us calibrate this diagnostic.

4.3.3 Bridge and Off-pulse Emission

The most common γ-ray profiles contain two peaks, and for these profiles we can

introduce a third measure of pulse shape that can be applied to the bulk sample:

the strength of the intra-peak (‘bridge’) and inter-peak (‘off-pulse’) emission. To

estimate these flux levels we divide the double peaked profiles into 4 phase windows.

Two intervals of ∆φ = 0.14 (7 bins in a 50 bin light curve) are centered on the


Figure 4.6 2-D plots in the bridge emission vs. off-pulse emission phase space, forthe same four E bins as in Figure 4.4. Estimated measurements for observed double-peaked pulsars are plotted, assuming a 10% uncertainty in the background levelsreported in the discovery papers. Only objects with high (> 2σ) significance havedots associated with the plotted error bars.

main peaks and the two remaining phase intervals (totaling 0.72 of pulse phase) are

assigned to the ‘bridge’ and ‘off-pulse’. We next measure the mean flux in each of these

phase windows. We then report the bridge fraction as 〈bridge〉/〈(bridge+peaks)〉 and

the off-pulse fraction as 〈off-pulse〉/〈(bridge+peaks)〉. This easily defined measure

generally corresponds well to a visual definition of bridge and off pulse intervals,

although it does not always isolate well the absolute pulse minimum. We plot these

two fluxes for the model light curves, in the usual four panels of spin-down luminosity,

in Figure 4.6.

Clearly these quantities provide good model discrimination. Models radiating only

in the outer magnetosphere (e.g. OG) have little off-pulse flux, especially for older

pulsars, while models radiating at all altitudes have comparable bridge and off-pulse

emission. In particular, the TPC picture produces essentially no light curves with

off-pulse fluxes less than 40% of the pulse phase emission.


Unfortunately, unlike the peak number and separation, such fluxes are not pro-

vided in Abdo et al. (2010a), Saz Parkinson et al. (2010) and the other discovery

papers. One difficulty in making such measurements is the identification of the true

background level. In practice it can be very difficult to distinguish unpulsed “DC”

magnetospheric emission from the contribution of an unresolved background in close

proximity to the pulsar, such as expected from a surrounding pulsar wind nebula. De-

tailed study of source extension and spectrum at pulse minimum can help distinguish

these cases, but this goes beyond the pulsar catalog data.

Nevertheless, the published light curves do plot an estimated background flux

level. Accordingly we entered these light curves, defined the peak and intervening

intervals exactly as above and measure the bridge and off-pulse flux levels. We assume

a systematic 10% uncertainty in the reported background level, and then propagate

this and the Poisson uncertainties in the bridge-, off-, and average-pulse fluxes to

produce the two flux ratios and errors defined above. In a few cases the background

level defined in Abdo et al. (2010a) were above the measured pulse minimum. In those

cases, we reassigned the background flux to this level (and by definition the off-pulse

flux was then 0). In a number of cases, the very large background pedestal prevented

accurate measurements. While we show error bars for all estimates of double pulse

pulsars in Figure 4.6, we mark with large dots only those measurements which had

either a 2σ significance measure of both flux ratios, or a ratio less than 0.1 at 2σ

significance.

Bridge emission varies from ∼ 0.3× to > 1× the average pulse flux at all spindown

luminosities, although a few high E objects seem to have virtually no emission away

from the peaks. In contrast, the majority of the objects have 0.3× or less of the

pulse emission in the off-peak window. Most of the best measured pulsars show very

small values. Values of ∼ 0.1 − 0.3× appear for a number of the most energetic

pulsars, but these measurements are of poor statistical significance. In any case

such objects are most likely to show unpulsed contamination from associated PWNe,

supernova remnants and other diffuse, but unresolved emission. Thus overall, the

general distribution of measured values matches better to the OG predictions, as


shown in Table 4.2.

Nevertheless, several objects are distinctly inconsistent with emission from only

high altitudes. In particular PSR J2021+4026 (panel 3) and PSR J1836+5925 (panel

4), show very strong off pulse emission, which is spectrally consistent with magneto-

spheric emission (hard with a few GeV exponential cut-off). These unusual objects

lie squarely within the TPC model zone. Interestingly both are quite low E, and

both have ∆ very close to 0.5, consistent with a low altitude, two-pole interpretation.

Also, the very energetic PSR J1420-6048 (panel 1) shows strong off pulse emission.

Here, with ∆ = 0.26 a low altitude interpretation is not natural. However, unlike the

other two cases, the very large and poorly determined background (from the bright

PWN emission in the surrounding ‘Kookaburra’ complex) may exceed the catalog

background flux level. However it is clear that, at least in a few cases, an emission

component other than the outer magnetosphere of the OG model contributes sub-

stantially to the pulse profile. An excellent candidate is low altitude emission, such

as posited in the TPC model. An alternative is a current-induced shift of the outer

gap start below rNC (Hirotani 2006). It will be particularly interesting to discover

the physical effect that causes such emission to be significant for a few pulsars, but

negligible for most.

4.4 γ-ray Evolution with E

As illustrated in the four-panel Figures 4.4-4.6, the pulse profile properties evolve as

the pulsar ages and E decreases. In addition there is of course strong evolution in the

pulsar luminosity. Together, these trends strongly affect the pulsar detectability in

the two detections classes (γ-selected and radio-selected) as a function of spindown

flux. We explore these population effects in this section.

In Figure 4.7 we plot histograms of detection numbers versus spin-down luminos-

ity, for both the radio-selected (top panel) and γ-selected (middle panel) subsets. The

blue histogram shows the simulated OG model detections, while the black data points

4.4. γ-RAY EVOLUTION WITH E 93

show the Fermi results, with statistical error bars. In the bottom panel we plot the

“Geminga Fraction,” or fraction of the total number of detected pulsars which are

γ-selected.

The OG model gives a reasonably good fit to the Fermi data, with χ2 Gehrels

fits of 0.587 per degree of freedom for the radio-selected population, 0.251 per degree

of freedom for the γ-selected population, and 0.054 per degree of freedom for the

Geminga Fraction. Ravi et al. (2010) have noted that in the 6 month sample the

most energetic γ-ray-detected pulsars are all radio-detected as well. They argue that

this implies a large radio beam, roughly co-located with the γ-ray emission region

for these most energetic (E > 6 × 1036erg s−1) pulsars. This may be caused by

high altitude emission as posited in Karastergiou & Johnston (2007). Our model,

without high altitude radio beams, predicts 7-8 such very high E pulsars in the 6

month sample, of which 2-3 should be undetectable in the radio (i.e. our line of sight

is outside of the radio beam). The actual LAT sample had 7 such pulsars, all of

which were detected in the radio. We have simulated high-altitude radio emission,

confirming that it prevents the γ-ray only detections at the highest E. However,

we conclude that the present sample is too small to use these numbers to probe the

detailed behavior of such objects with high statistical significance; for the moment the

case for high altitude radio emission still largely relies on the radio pulse properties.

There does exist one moderately significant defect in the histogram for low E radio-

selected pulsars; the model predicts too many such detections. This is reflected as

well in the Geminga Fraction, where the data demand a larger value for the Geminga

Fraction at low energies. Such an increase is not seen in the model, due to the excess

of radio-selected detections. For now we note the discrepancy, but refer to Section

4.5 for discussion of a possible solution.

Spin-down luminosity should, through the γ-ray efficiency, be correlated with the

typical distance of the detected pulsars. Unfortunately, as we have seen, distances

estimates (and hence inferred luminosities) are very poor for many of these objects.

However, Galactic latitude correlates inversely with distance and is a directly ob-

servable property. Accordingly, in Figure 4.8 we show the E − b planes for the


Figure 4.7 Histograms of detection numbers versus E. The model overproduces radio-selected γ-ray pulsars at low E.

radio-selected and γ-selected subsets. The colored points show the locations of Fermi

pulsars and the contours show the density of the OG model detections.

The modeling predicts a dramatic increase in the latitude spread of the lowest

E, (lowest Lγ , nearest) objects. The γ-selected set actually shows this trend quite

well. In contrast, the radio-selected pulsars show a distinct lack of the nearby, lower

luminosity higher latitude objects predicted by the models at log(E) < 34.5. This

is directly connected to the simple lack of low E radio-selected objects noted above.

Also in the radio panel, one notices the very high E Crab pulsar – which is surprisingly

close for such an energetic pulsar and notoriously far from the plane. This distance off


the plane is likely a product of an unusually long-lived low mass runaway progenitor

from the Gem OB1 or Aur OB1 associations.

The model agreement is tested with a two dimensional KS test. For the γ-selected

sample one gets a KS probability of 3.9%, quite acceptable for this population model.

For the radio-selected sample the KS probability is however only 0.5%, which is not

a good agreement. The discrepancy is largely driven by the missing E < 1035 erg s−1

pulsars; if this region is excluded the KS probability rises to 2.6%, which is not

excludable.

Figure 4.8 2-D plot of detections in the E-galactic latitude plane. Note the overpro-duction of model detections at low E in the radio-selected sample.


With the caveat about observational uncertainties noted above we can also com-

pare with the data in the E-d plane (Figure 4.9). Here we expect small d at low

E. In an attempt to illustrate the distance uncertainties, we use different symbols to

plot objects with distance estimates from various sources (drawn from the discussion

in the discovery catalogs). For example, radio detected pulsars have DM distance

estimates. These are shown as open circles. While generally held to be accurate to

∼ 30% for the bulk of the pulsar population, these estimates evidently have much

larger fractional errors for the energetic LAT pulsars near the plane and, in a number

of cases, are clearly substantial overestimates. In some cases we have other distance

estimators ranging in reliability from astrometric parallax (very high) through HI

absorption kinematics to spatial associations (low). These objects are plotted with

filled dots. Note that in the upper panel of Figure 4.9 these distances average lower

than the DM values for all ranges of E.

Finally, for a number of γ-selected pulsars we have no estimate of distance other

than assuming a γ-ray efficiency (Equation 4.4) and isotropic emission and using

the observed γ-ray flux to estimate d. Such estimates are referred to as “pseudo-

distances” (Saz Parkinson et al. 2010), and while they provide helpful consistency

checks, their use is, to some degree, circular if employed to test a model. They are

plotted as crosses.

Examining the 2-D KS test probabilities, we find that the radio-selected sample

returns a probability of 0.7%. As for the E − b set, this is moderately excludable.

Again, a re-test using only objects above 1035 erg s−1 delivers a higher probability of

2.3%. The probability for the γ-selected sample is 19.9%. This large value is certainly

partly due to the use of the pseudo-distances. However, even after removing these

objects, we find a probability of 10.4%, suggesting that the model is a quite good

representation of the data.

Thus our heuristic γ-ray luminosity law (Equation 4.4) and our Broken Power Law

radio-luminosity law provide a good match to the data. Other models provide a much

poorer match. For example, a simple linear γ-ray luminosity law Lγ = 1/3E together

with the straight Power Law radio luminosity law and large birth period 〈P0〉 = 300ms


recommended by Faucher-Giguere & Kaspi (2006) can produce a reasonable number

of γ-detectable pulsars for a Li et al. (2010) birthrate. However the predicted distances

are 2× − 4× those observed and 2-D KS comparisons in the E − b and E − d planes

produce unacceptably low probabilities for the radio-selected γ-ray sample (9× 10−4

and 7 × 10−4, respectively).

Figure 4.9 2-D plot of detections in the E-distance plane. The overproduction ofmodel detections at low E in the radio-selected sample is still evident.


4.5 Possible Amendments to the γ-ray Model

We have shown how to quantitatively compare the observed properties of the LAT

pulsar survey to predictions of population models. While the agreement is generally

quite good, in particular for the models dominated by outer magnetosphere radiation,

some discrepancies are already seen. Our ability to study such discrepancies will

certainly improve as the size of the LAT sample and the quality of the individual light

curve measurements increase. Thus we wish to discuss sensitivities of the observed

data sample to details of the beaming model and amendments to the modeling that

may further improve the fidelity of the synthesized population.

As an example, we should consider how beaming and luminosity evolve at low

E as w and the pulsar efficiency approach a maximum and the gap saturates and

shuts off. In particular, for the OG model, since emission starts above the null charge

surface, large w drives this start toward the light cylinder and produces a decreasing

γ-ray beam solid angle.

If one follows the simple efficiency law (Equation 4.4) into such a regime then

phase average pulse intensity for the few observers seeing all of ηE in this very small

angle would become very large just before the beam shuts off. In principle, this allows

a small number of E < 1034 erg s−1 pulsars to be seen at very large distances. The

more physical alternative, adopted here, is to link ηE to the ‘surface brightness’, the

thickness-integrated emissivity per unit area of the gap volume. For all E producing

modest w this means Lγ ∝ w or ∝ w3, as usual. However as w approaches unity and

the active gap area contracts laterally, the total sky-integrated luminosity smoothly

goes to 0 at gap saturation, although the surface brightness continues to grow. To

illustrate the sensitivity of the data-population comparison to such effects we illustrate

(Figure 4.10) the different prediction of these two cases. The left panels reproduce

the low E end of Figure 4.9, while the right panels show the increased distance reach

– from . 1 kpc to & 3 kpc – if all the power is forced into the decreasing beam. As it

happens the effect is largely concentrated in the γ-selected pulsars, since to produce

emission from such large w, α must be moderate to small (i.e. α . 50), but the

4.5. POSSIBLE AMENDMENTS TO THE γ-RAY MODEL 99

viewing angle ζ must be very near the equator (i.e. ζ & 85). For such objects the

radio beam misses Earth. This and even more subtle evolution effects will become

increasingly subject to test as the LAT sample grows.

Figure 4.10 Figure showing the effects of forcing the model to pump ηE power intothe γ-ray emission region, regardless of how small that region becomes. Note that wehave zoomed in here on the region of interest, E < 1035ergs−1.

We have already alluded to another deficiency in our modeling of the low E

population; the substantial over-prediction of radio-selected objects. In fact, there is


an evolutionary process that can be amended to our model to produce exactly this

effect: magnetic alignment. Recent work has suggested that a secular decrease in

α may occur on timescales as short as 1 Myr (Young et al. 2010). Given the high

sensitivity of the γ-ray beaming to α for some models, even partial alignment can

have a large effect on our young pulsar population.

We describe here qualitatively the main effects, deferring detailed population pre-

dictions to a study including the alignment kinematics. As α decreases with age,

there are two main effects on the pulse observables in the OG model: a decrease in

the average ∆ and a decrease in the number of total detections, with an especially

strong elimination of radio-selected objects. These should show up principally at the

largest ages (i.e. lowest E values) of the γ-ray pulsar population.

The first OG effect, a decrease in typical ∆ values, results from the fact that the

highest ∆ values are always produced by large α models. Eliminating such models

would have the effect of shifting the clump of ∆ ∼ 0.4, δ ∼ 0.15 models in Figure

4.5 panel 4 to ∆ ∼ 0.2, δ ∼ 0.25. Of course we need substantially more low E pulsar

detections before any such evolution can be tested.

The second OG alignment effect is a reduction in γ-ray detections at low E. In

outer gap geometries, the approach of the null charge crossing to the magnetic axis

and light cylinder occurs at higher E for aligned pulsars. Thus gaps saturate and

shut off at younger ages (i.e. higher E values) the more aligned a pulsar is. This

has pulse shape effects, but most importantly causes more rapid depletion of the

smaller α pulsars. In fact, the process preferentially eliminates radio-selected γ-ray

pulsars. We illustrate this in Figure 4.11. The main field in that figure shows the

α−ζ plane, populated with detected γ-ray pulsars. γ-selected pulsars are shown with

open black circles, while radio-selected pulsars are shown with filled red circles. The

radio-selected objects must have small |β| = |ζ − α|.

As alignment proceeds, we expect that the largest α values will migrate to lower

α. Note that as angles from 80−60 are depleted (hatched zone), many more radio-

selected objects than γ-selected objects are lost. Additionally, many radio-selected

4.6. DISCUSSION 101

objects actually become γ-selected objects, while the converse is much more rarely

the case. This is illustrated in the figure inset, showing the effects of an increasingly

large shift in alpha values. The number of radio-selected detections drops much faster

than the γ-selected detections (which in fact increases at some points, for the reason

mentioned above) and the Geminga fraction grows toward unity.

Since the principal alignment effects amend discrepancies seen in the data com-

parison, a more complete study seems warranted. In particular, the high sensitivity of

outer magnetosphere models to α can make the effects of even subtle alignment more

easily detected than in the radio data, which is largely controlled by β. Note that

lower altitude models, such as TPC, are much less affected by alignment. In practice

we might expect some spread in the alignment rates with a few objects persisting at

large α to relatively late times so a detailed kinematic study may be required.

4.6 Discussion

We have simulated a model Milky Way Galaxy filled with young pulsars, and by

comparing the distributions of various observables we have been able to constrain

properties of the true population and evaluate the relative fidelity of several γ-ray

pulsar models. To start, we found that a radio luminosity distribution scaling with E

can provide a good fit to the observations, but only if the scaling relation is broken and

plateaus at the very highest luminosities. We also find that the γ-selected sample has

so many pulsars with large spin-down luminosity that a short birth period distribution

(〈P0〉 ≈ 50 ms) is required. These parameters also provide a good description of the

young radio pulsar population.

In comparing with several γ-ray emission models, all based on the vacuum dipole

field structure, we find that Outer Gap (OG) models perform the best in almost

all circumstances. Thus, for the population as a whole there is a very statistically

significant (formally ≥ 107×) preference for the OG model – i.e a model dominated

by radiation above the null charge surface. However,, there are a handful of individual


Figure 4.11 Figure showing the effects of magnetic alignment on the detection frac-tions of older (E ≤ 2.5 × 1034 erg s−1) γ-ray pulsars.

objects that do not fit easily into the predicted OG population. Such objects may well

have significant lower altitude emission, as posited by Slot Gap (TPC-type) models.

Discerning the physical parameters which select when such emission is present will

be particularly interesting. It should also be remembered that while these vacuum

calculations give a clear preference for OG geometries over TPC geometries, they are

themselves not perfect fits and do not represent complete physical models. This is

4.6. DISCUSSION 103

evident since even the OG model does not produce χ2/DoF=1. This is not surprising

since any real magnetosphere must include some currents and plasma affects will

certainly perturb the vacuum conclusions. Thus it will be of interest to extend this

population model/data comparison to the other extremum, the fully force free plasma-

dominated SL models (Bai & Spitkovsky 2010b).

We conclude this discussion by making future predictions of the pulsar birthrate,

detectable numbers and background contribution for the preferred OG model. The

6 Month sample from Fermi contains 37 γ-ray pulsars, with a Geminga Fraction of

49%. At the 6 Month sensitivity level, the OG model produces 2194 γ-ray detections

(normalized to one pulsar birth per year) with a Geminga Fraction of 46%. Thus

the observed pulsar count directly gives the pulsar birthrate: 2194/37 ≈ 59 years per

pulsar birth, or 1.69±0.24(stat) pulsars century−1. This is in excellent agreement with

the core collapse rate (Diehl 2006) and OB star birthrate (Reed 2005) measurements

quoted in Section 4.2.1. It is also only 1.2σ lower than the SN rate computed by Li

et al. (2010), which has the smallest formal error bars.

It is interesting to compare this birthrate estimate to the independent O-B2

star birthrate estimate made in this work, based on the local massive star density

(2.4 OB century−1). The first thing to note is that our γ-ray pulsar birthrate is a

large fraction of these progenitor rates. For example, at 71% of the O-B2 star rate,

a large fraction of these stars must produce γ-ray pulsars. Indeed we check if culling

the lowest mass class (B2, ∼ 9M⊙) is acceptable. With this cut the local massive star

birthrate drops from 40 OB kpc2 Myr−1 to 21.5 OB kpc2 Myr−1, which corresponds to

a Galactic rate of 1.3 OB century−1. This is only 75% of the rate needed to produce

our inferred γ-ray pulsar population. This is also 2.1σ lower than the Li et al. (2010)

Galactic supernova rate. Thus we conclude that B2 stars should produce SNe and

contribute to the young pulsar population.

Perhaps even more interesting is the direct comparison between our pulsar

birthrate and the core collapse rate. The observed γ-ray pulsars contribute > 75%

of the (relatively high) inferred SN rate in this study. Thus we conclude that the

γ-ray pulsar sample must be an excellent census of the local core collapse products.


These rates imply that no more than 25% of the core collapses can produce slow P0

injected pulsars, RRATS, Central Compact Sources in Supernova Remnants, magne-

tars and other exotica. Similar conclusions have already been reached by Keane &

Kramer (2008) using the radio surveys. Thus our new population estimate, derived

from the Fermi sample and sensitive to γ-ray rather than radio beaming effects, sup-

ports the picture that energetic radio pulsars dominate the neutron star birthrate,

and that more exotic objects cannot contribute a dominant fraction of the neutron

star population unless they represent a later phase in the evolution of energetic, γ-ray

producing pulsars.

In outer magnetosphere models there are, of course, some pulsars whose bright

γ-ray beam misses the Earth while the radio beam does not. These objects can be

detected in the radio but will be absent in the γ rays, despite the fact that they are γ-

ray active. This especially occurs for geometries with small α and small ζ (see Figure

4.11 where radio-only pulsars continue to the upper left); for a few objects, known

angles place them in this region, e.g PSR J1930+1852 (Ng & Romani 2008; Abdo

et al. 2010a). Of course, fainter low altitude emission aligned with the radio could

still be visible; PSR J0659+1414 could be such a case. Our population estimates

suggest that there should be a modest number of objects in this radio loud, γ-ray

missed category. At the sensitivity of the 1 year sample we would expect ∼ 10 radio-

detected, γ-ray undetected pulsars with E > 1033.5 and E1/2/d2 as large as that of the

faintest detected γ-ray pulsars; again, these are objects not seen in the γ-rays simply

due to beaming effects. Most of these have E < 1035 erg s−1. As the γ-ray sensitivity

grows, the fraction of such objects decreases. However there are also pulsars whose

radio and γ-ray beams both miss the Earth. At the 1-year sensitivity nearly half

of the pulsars with E1/2/d2 above that of the faintest detections are such ’Isolated

Neutron Stars’ (INS). Such radio-only pulsars and INS are implicitly included in the

birthrate estimates above.

Given the successes of the OG model at matching the observed population, it

is natural to ask what this scenario predicts as we acquire additional observations.

For the birthrate above, we can lower the γ-ray threshold and make predictions of

4.6. DISCUSSION 105

the number of detections as a function of Fermi observation time. Our “Current”

Fermi sample is based on approximately one full year of data; extrapolating from

the 6-month normalization, we find that the model predicts 23 γ-selected pulsars and

26 radio-selected pulsars. The actual observed sample contains 24 γ-selected pulsars

and 26 radio-selected pulsars, an excellent match to the observations. Similarly after

five years exposure the model predicts 49 and 44 detections and after ten years 65

and 55, γ-selected and radio-selected detections, respectively. With the improved 10-

year γ-ray sensitivity, the number of radio-detected, γ-ray undetected pulsars grows

at a similar rate; we expect ∼ 27 such objects with E1/2/d2 large enough that we

would expect detection. The number of INS whose γ-ray beam would be detectable,

if directed at Earth, grows to ∼ 150 objects, still about 1/2 of the total population.

Most are again low E pulsars whose relatively narrow radio and γ-ray beams only

sweep a small fraction of the sky.

Note that we can hope that the actual γ-ray pulsar numbers will be larger by a

modest fraction as analysis and search techniques improve and as deep radio observa-

tions lower the effective survey threshold. Note also that these numbers do not include

the very substantial population of recycled pulsars (MSP) now being detected.

The trend towards higher Geminga Fraction above in these future predictions is

due to two effects. First, as Fermi probes larger distance scales our radio pulsar

sample will not be as complete. In practice, as just noted, deeper radio searches

will mitigate this trend. Of course, we still expect a substantial number of very

low flux density sources will still remain non-detected, since the present very deep

radio searches on many LAT sources indicates that these are a substantial part of the

Geminga population. The second effect increasing the Geminga Fraction is a change

in the E distribution of the detected pulsars. The five and ten year model samples

have a higher proportion of mid to low luminosity objects (E . 1035 erg s−1), and

thus proportionately fewer energetic pulsars. With longer periods at lower E come

narrower radio beams and fewer detections. If alignment, as discussed above, is

included in the models this contribution to the Geminga fraction will be even larger.

The final number to report is the γ-ray background supplied by unresolved pulsars.


To compute this we simply sum up the emission from sources not individually detected

(at a given LAT exposure time). To estimate the spectral content of this contribution,

we assign each modeled pulsar a power-law spectral index Γ and exponential cut-off

energy Ec. The assigned value is inferred from the Γ(E) and Ec(BLC) trends visible

in Figures 7 and 8 of Abdo et al. (2010a) and approximated here by

Ec = [−0.45 + 0.71 log(BLC)] GeV

Γ = −4.10 + 0.156 log(E). (4.7)

These model spectra can be used to sum up the effective young pulsar background

spectrum. In practice we compute 0.1-1GeV and 1-10GeV sky maps of this emission

to look for any interesting spatial distribution. Some evidence of cluster structure is

seen in these maps, but in any event the contribution to the total Galactic background

is small.

At the 6 month sensitivity limit we find that pulsars supply 4.4×10−9 erg cm−2 s−1

(1.8%) of the total background flux in the 0.1-1 GeV band and 3.6×10−9 erg cm−2 s−1

(2.8%) of the flux in the 1-10 GeV band (integrated over the full sky and com-

pared against the Fermi Collaboration’s publicly available background skymap,

gll iem v02). Of course, as time progresses and more pulsars are individually de-

tected, the total unresolved flux from background pulsars goes down. After 10 years

of observation, approximately half of the total flux unresolved at 6 months will have

been detected as individual sources. We would then find that pulsars contribute only

1.0% and 1.5% of the 0.1-1 Gev and 1-10 GeV background bands, respectively. We

remind the reader that this does not include the contribution of recycled pulsars.

As noted, some studies (Faucher-Giguere & Loeb 2010) estimate that this may be

quite substantial at intermediate latitude. A treatment with improved luminosity,

beaming and evolutionary effects, as presented here for the young pulsar population,

seems very desirable.

The work in this chapter and the previous chapters was supported in part by

4.6. DISCUSSION 107

NASA grants NAS5-00147 and NNX10AD11G. K.P.W. was supported by NASA un-

der contract NAS5-00147. This work made extensive use of the ATNF pulsar catalog

at http://www.atnf.csiro.au/research/pulsar/psrcat/ (Manchester 2005).

Chapter 5

Conclusions

5.1 Summary

We have studied in detail the γ-ray production of young pulsars. We have considered,

by simulation, several different emission zone geometries, various perturbations to

the magnetosphere, and the details of photon emission itself. With all this, we have

been able to scrutinize in detail the form of the modeled γ-ray emission. We have

determined when a pulsar has active γ-ray emission, from which observational angles

is the γ-ray emission detectable, the shape and structure of the γ-ray pulse, the

efficiency of converting rotational kinetic energy into γ rays, and more. Comparison

of all these model outputs against the observations has been shown to be a powerful

tool in understanding the likely workings of the pulsar machine (i.e., when model and

observation agree), as well as in discriminating against models (i.e., when model and

observation fail to agree). In order to facilitate the comparison of these γ-ray models

to the data, we have constructed atlases that visually display common model outputs

in a two dimensional parameter space. This parameter space spans the possible

geometric orientations of a pulsar, varying both the inherent magnetic inclination

and the observer’s viewing angle.

In Chapter 2 we determined with high likelihood that pulsar γ-ray emission is

108

5.1. SUMMARY 109

outer magnetospheric in origin and that the polar cap model is not a viable emission

geometry (as mentioned in Chapter 3, this was also reinforced by the spectral studies

in Abdo et al. 2009). The remainder of the work focused on the study of variations

of two of the main outer magnetospheric emission geometries – the outer gap model

and the two-pole caustic model.

The atlases of Chapter 2 allowed one to do a crude look-up of the possible ranges

of parameter space whose outputs conformed to observation for a given object, but

in Chapter 3 we strove to improve on this method. To that end, we developed an

algorithm capable of quantitatively determining the goodness-of-fit of a model light

curve to an observed light curve. This process now used the full information sample

provided by an observed light curve, as opposed to reductive representative values.

We applied this algorithm to both the OG model and the TPC model, for several

of the brighter LAT pulsars. These individual objects showed, almost exclusively,

that the OG model produced better light curves, especially when the model input pa-

rameters were constrained by multiwavelength observations. These multiwavelength

constraints proved crucial in many cases, as given the freedom to search the full input

parameter space, both models are often capable of reproducing light curves that look

fairly similar to any observed light curve.

Finally, in Chapter 4, we applied the OG and TPC models to an entire simulated

galaxy of pulsars, allowing for comparisons against the pulsar population at large. By

using the Fermi LAT population of several dozen young pulsars, we were able to see

how well the models faired when matching not only the properties of single objects,

but distributions of properties across a large sample. We took care to try to reproduce

as faithfully as possible our galaxy’s underlying young pulsar population. In doing so,

we discovered that these young γ-ray pulsars must be born with average rotational

rates several times higher than had often been assumed previously. Additionally, our

study suggested that given a radio luminosity law that scales with pulsar spin-down

energy, one must include a break in the luminosity scaling that is followed by a plateau

at the highest energies. Again, we found that the OG model was repeatedly the best

available fit to the observed data.

110 CHAPTER 5. CONCLUSIONS

5.2 Future Directions

It is important to note here that even the best-fit model may not necessarily be a

“good” model. Our best fits are often several sigma off from observed values. Pertur-

bations and additions to the models bring us closer to agreement with observation,

suggesting that we have yet to include all of the relevant details. Indeed, there are

some physical details that we know remain unmodeled, because of the computational

difficulty of proper modeling or even because the exact underlying physics is unknown.

A few such augmentations or perturbations which it might be possible to incorporate

in future iterations of outer magnetospheric models are discussed below. As the LAT

continues to gather photons it will continue to uncover new pulsars, and perhaps even

more importantly for shedding light on these model corrections, it will continue to

improve the quality of the light curves for those pulsars already detected. As errors

shrink, the exact nature of a model’s deviation from observation may become appar-

ent, and we may be provided the crucial clue to furthering our understanding of the

pulsar machine.

5.2.1 Magnetospheric Currents

Two different toy calculations of magnetospheric currents were utilized in §3.2.2,

where they were shown to have potential in our efforts to realize a physical model of

the magnetosphere. Currents almost certainly have non-trivial effects on the structure

of the magnetosphere, but the lack of their inclusion in the models is unfortunately

due to both of the difficulties mentioned above: uncertainty regarding the underlying

physics and computational difficulty. The exact form taken by the currents threading

the magnetosphere is uncertain; estimates have been made for the vacuum approxi-

mation (Muslimov & Harding 2009) as well as for the fully force-free magnetosphere

(Bai & Spitkovsky 2010b). A true model, however, must likely lie somewhere in be-

tween these two extremes – incorporating both plasma-filled force-free regions and

charged-starved vacuum gaps. The analytic solution style used throughout this work

does not lend itself well to force-free magnetospheres, and so it is likely that the best

5.2. FUTURE DIRECTIONS 111

approach to simulating realistic currents will lie with numerical simulations. These

simulations will need to find a way to integrate regions of low charge density into an

otherwise force-free magnetosphere, but if they are able to do so the results may well

prove valuable.

5.2.2 Gap Widths

As alluded to in §4.2.3, even when the inner surface of the vacuum gap is determined,

w, there are still different ways to treat the emissivity between the last closed field

lines and this surface. The assumptions made about this emissivity will be reflected

in the modeled γ-ray light curves. We experimented in Chapter 4 with two different

prescriptions for illuminating this region between the last closed field lines and the

surface defined by the w parameter. The first was to only illuminate a narrow region

near the inner surface, at w, whereas the second was to illuminate the full gap,

uniformly from the last closed field lines inward towards the magnetic axis.

At the most basic level, a larger emission volume will tend to produce wider peaks

in the pulsar light curve. This correlation, along with the high time resolution of the

LAT data, can be used to place constraints on the thickness of the magnetospheric

emission region of LAT puslars. Precision measurements of the width of the γ-ray

pulse peaks are already being made (Abdo et al. 2009), and these can be compared

against simulations with varying emission region thicknesses to try to nail down the

true geometry of the γ-ray active zone.

5.2.3 Magnetic Alignment

The possible effects of magnetic alignment were discussed briefly in §4.5. As was done

with magnetospheric currents, a toy calculation was performed, but further study is

warranted. A true kinematic study (featuring gradual secular alignment, over an

appropriately varied timescale, for a large population of objects) could shed light on

the form and magnitude of any alterations to the γ-ray pulsar population.

112 CHAPTER 5. CONCLUSIONS

5.3 Millisecond Pulsars

Finally, there are the recycled millisecond pulsars to consider. These objects compose

an obviously related, and yet distinct class of sources from the young pulsars studied

here. With rotational rates that average almost two orders of magnitude faster than

young pulsars, there are a different set of assumptions necessary and quite possibly

different physics at play when working with the compact magnetospheres of these

MSP. We say compact, because the light cylinder radius is directly related to pulsar

period – at 2 ms it is already less than 100 km. General relativistic or multipolar cor-

rections that might have safely been ignored in the magnetospheres of young pulsars

must be taken into consideration for these objects.

A population based study like the one we perform here in Chapter 4 is rapidly

becoming viable thanks to the LAT. Eight MSP were detected in the first six months of

LAT data (Abdo et al. 2010a) and early conference results show that that number has

continued to grow since. A careful study of the Galactic MSP population would also

prove very useful to the study of the galactic γ-ray background. Long since removed

from their supernova birth sites in the Galactic plane, the MSP population occupies

a much wider spread in Galactic latitude than does the young pulsar population.

Additionally, the MSP spin-down luminosity distribution is shifted more towards the

lower end of the young pulsar distribution, making these objects more difficult to

detect and thus more likely to be contributing to the γ-ray background as unresolved

point sources.

Of course these objects are also very interesting in their own right; the first eight

LAT MSP show a wide variety of γ-ray pulse morphologies, with only one object

displaying the classic wide double pulse plus bridge light curve that is characteristic

of most of the young pulsars. Careful study of these MSP may very well improve our

understanding of the young pulsars as well, as we uncover both the differences and

the similarities between the two classes of objects and the physics of the machine that

powers them.

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