understanding the properties of binary x-ray pulsars

8/9/2019 Understanding the Properties of Binary X-ray Pulsars

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INDIAN INSTITUTE OF SPACE SCIENCE AND

TECHNOLOGY, THIRUVANANTHAPURAM

UNDERSTANDING THE PROPERTIES OF

BINARY X-RAY PULSARSINTERNSHIP PROJECT REPORT

Submitted by

ABHIMANYU S

IN

ASTRONOMY AND ASTROPHYSICS

INPHYSICAL RESEARCH LABORATORY

January 2010


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Abstract

This report is a summary of the work done by me during my one-month internship programme at

Physical Research Laboratory, Ahmadabad. The topic of the internship project was

Understanding the Properties of Binary X-ray Pulsars. The studies were carried out by

analyzing data from the satellite, Suzaku. This report provides a description of some of the basic

properties of accretion-powered binary X-ray pulsars by reading various articles/books related tothe topic.

This study concentrated on the understanding of some of the fundamental properties (temporal

and spectral properties) of the accretion-powered binary X-ray pulsars. X-ray data in 0.1-10 keV

energy range, obtained from the observation of a binary pulsar GRO J1008-52 with the space-

based observatory SUZAKU. The data was analyzed to understand the properties of the neutron

star in the binary system and the surrounding medium. I carried out the analysis of the pulsar

under the guidance of my supervisor.


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List of Symbols, Abbreviations, and Nomenclature

XRB X-Ray Binary

HMXB - High Mass X Ray Binary

LMXB - Low Mass X Ray binaryXRT X-ray Telescope

XIS - X Ray Imaging Spectrometer

WAM Wide All Sky Monitor

GRO Compton Gamma Ray Observatory

Her Hercules


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Contents

Abstract 3

Nomenclature

Contents 51. Introduction

i. X-Ray Binaries 6

ii. Pulsars 6

iii. Formation and Structure

iv. Periods

v. Magnetic Fields

vi. Emission Processes

vii.Spectra of X-Ray Pulsars 8

2. Observation 10

3. Analysis 11

i. Timing Analysis 11

ii. Phase Averaged Spectral Analysis 12

4. Discussion 17

5. Conclusion 18

Appendices

Suzaku 19

FTOOLS 22

Brehmsstrahlung

References 23


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Introduction

X-ray Binaries

X-ray binaries are a class of binary stars that emit in the x-ray region of the electromagnetic

spectrum. The x-rays are produced when matter from one star (called donor. This will usually be

a normal evolving star) falls to the other (called accretor. This will be a compact object i.e., a

white dwarf, neutron star or a black hole.). The matter accreted from the donor forms a disc like

structure around the accretor called accretion disc. The disc like form is due to the great angular

velocities arising in order to conserve angular momentum. The infalling matter releases its

gravitational potential energy as radiation, which is a very efficient process because several

tenths of its rest mass will be converted into radiation.

On the basis of mass of the donor, x-ray binaries are classified into two groups: High Mass X-

Ray Binaries (HMXB) and Low Mass X-Ray Binaries (LMXB).

In High Mass X-Ray Binaries, the companion star is a massive star usually an O or B star or a

blue supergiant. For such stars, the amount of mass flowing out through stellar wind will be very

high. The compact object accretes the fraction of stellar wind that comes within the gravitationalfield ofthe compact object and the kinetic energy of accreted matter is converted into radiation.

Here, the masses of the accretor and donor may be comparable or the donor may be more

massive than the accretor. HMXB will be bright in both x-ray and visible wavelengths.

In Low Mass X-Ray Binaries the companion star is a low-mass star (a typical low-mass main

sequence star or an evolved star such as a red giant) where mass will be much less than the mass

of the compact object. The companion star will have a weak stellar wind, which will be

insufficient to power the accretion process of the compact star. The companion star, during its

evolution, gradually grows in size and eventually fills the Roche lobe. In such a situation, the

companion star can transfer its mass to the Roche lobe around the compact object, which can be

accreted. LMXB are very bright in x-ray wavelengths but faint in visible region.

Some XRBs where the companion star is an intermediate mass star are sometimes categorized as

Intermediate Mass X-Ray Binaries. The accretion mechanism in such XRBs can be either of the

above-mentioned mechanisms.

Pulsars

Pulsars or pulsating stars are astronomical objects, which were found to emit electromagnetic

radiation as pulses in accurate periods. Typically, this emission is in X-Ray or Radio frequencies.

Few pulsars, which emit in the visible region have also been found. Jocelyn Bell Burnell and

Anthony Hewish first observed a pulsar on November 28, 1967. Pulsars were later identified as

rotating neutron stars, powered by its own rotation or accretion of matter from its binary

companion. The words pulsar and neutron star are often used synonymously. However,

neutron stars those do not emit radiation in the direction of earth, are not pulsars. Therefore,

pulsars are a subset of neutron stars.

Formation and Structure

Neutron stars are formed when a star of mass ranging from 1.44 solar mass (Chandrasekhar

limit) to 2-3 solar mass (Tolman-Oppenheimer-Volkoff limit) exhausts its fuel and undergoesstar death. Such a star undergoes a supernova explosion. The outer parts of the star are blown

off and if the core still has a mass 1.44-3 solar masses, it collapses to form a neutron star. The

Fig.2. Structure of a neutron star


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electron degeneracy pressure will be insufficient to stop the gravitational collapse. This process

leads to a point where the electrons and protons combine to form neutrons (an inverse decay

process). If the star is within the Tolman-Oppenheimer-Volkoff limit, it will be supported by

neutron degeneracy pressure against gravity at

this point.

The outermost layer of a neutron star is a solidcrust. Here atoms are crushed to form a solid

lattice of nuclei with a sea of electrons filling the

gaps. The crust of the neutron star may be in a

liquid state for hotter neutron stars. Beneath the

outer crust, electrons and nuclei with high neutron

number exist which will decay under normal

conditions, but are stabilized by extremely high

pressure. Further inward is a region where

neutrons leak out of the nucleus (neutron drip).

Here there is a mixture of free neutrons, nuclei,

and electrons. The core of the neutron star isthought to be a superfluid of neutrons. It may be

possible that mesons and free quarks exist in this region.

Periods

Pulsars are extremely accurate clocks. Their periods are found to be accurate up to 1 in 10^14

parts. Pulsar periods typically range from milliseconds to several seconds.

Pulsar periods are observed to increase very slowly over time. A pulsar can be approximated as a

magnetic dipole with a very large dipole moment. When such an object rotates about an axis

other than its magnetic axis, dipole radiation occurs in the expense of its rotational energy. Thus

the angular velocity of the pulsar drops. A phenomenon where the pulsar period suddenlydecreases has also been observed. This phenomenon is called glitch. Glitches are thought to

occur due to internal disturbances like transition of the superfluid core from one metastable state

to a lower one or starquakes where the crest of the neutron star undergoes a drastic change to

attain a more stable shape.

Magnetic Fields

Surface magnetic fields of pulsars are of the order of 108 - 109 Gauss (LMXB, rotation powered),

1012-1013 G (HMXB) or 1014-1015 G (Soft Gamma Ray Repeaters, Anomalous X-Ray Pulsars).

In such high magnetic fields, atoms near the neutron star surface are constricted in a plane perpendicular to the magnetic field and become cylindrical. Here, the electric forces

perpendicular to the magnetic field are negligible compared to the magnetic force. These atoms

will have huge quadrupole moments leading to very compact packing.

Several models for the origin of pulsar magnetic fields have been proposed. The first and

simplest idea is that of fossil fields. This idea implies that the enormous magnetic fields are

produced by the process of magnetic flux conservation when the progenitor of the neutron star is

compressed during the collapse. Such a process is sufficient to produce a magnetic field of the

order of 1012 G which is about the canonical value of HMXBs.

The lower values for magnetic fields for LMXBs and recycled rotation powered pulsars areunderstood in terms of accretion induced field decay. This is possible by two processes. One

process is an enhanced Ohmic decay, where the heating due to accretion increases the resistivity

Fig. 1. Accretion process in an XRB


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of the pulsar thus reducing the currents. Another process is a thermo magnetic effect. Surface of

a neutron star without accretion will be cooler than its interiors which cause convection currents

that cause the magnetic field. But due to accretion,

the surface becomes hotter than the interiors

causing inverse convection currents which reduce

the magnetic field.

Emission Processes

The emission processes in radio and X-Ray

pulsars are entirely different. The emission in

radio pulsars involves coherent emission of radio

waves powered by the rotation of the pulsar while

the emission in the X-ray pulsars involves mainly

thermal emission due to accretion.

Thermal emission from the neutron stars poles occurs due to heating by bombardment of

particles accelerated by gravity. The basic non-thermal radiative process in X-ray pulsars is the

cyclotron emission at the cyclotron frequency and its harmonics. This occurs by the transition ofelectrons from one Landau level to another. Emission can also occur due to very fast and highly

accelerated movement of charged particles near the magnetic poles through the flux lines which

are curved. This is called brehmsstrahlung. Both thermal and non thermal brehmsstrahlung are

possible in the accretion columns of neutron stars, though thermal brehmsstrahlung is the

dominant process. Thermal brehmsstrahlung is the emission from thermally ionized plasma

where the interactions between ions and electrons causes electron acceleration/deceleration and

hence brehmsstrahlung. Thermal brehmssrtrahlung occurs when the particles populating the

emitting plasma are distributed according to a Maxwell-Boltzman distribution.

The emission in radio pulsars occurs by coherent emission processes like maser emission and

relativistic plasma emission. Maser action occurs when photons are amplified due to negative

absorption in the presence of an inverted population. Relativistic plasma emission process

involves turbulence in plasma induced by a beam of particles passing through it causing

radiation. The turbulence induced can propagate as a wave of appropriate mode from its site of

generation and can be converted into radiation by a process such as scattering, wave-wave

interaction etc.

Spectra of X-Ray Pulsars

Spectra of X-ray pulsars in the X-ray region contains a power law component and other

components that arise due to the reprocessing of the power law component by the surrounding

matter including the accretion disc and the interstellar gas surrounding the pulsar. The following

figure shows the phase-averaged spectrum of Hercules X-1.

Fig.3.


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The dominant component in the X-ray pulsar spectrum is the power law component. The power

law component is a continuous spectrum, which is emitted by the accretion column near the

poles of the pulsar. This is the dominant component in the pulsar modulation and has a large and

near sinusoidal pulse modulation, which suggests its origin from the accretion column near

poles. This power law component is the radiation that is emitted by the accreted matter by

converting its gravitational potential energy to electromagnetic radiation.

The other components of X-Ray radiation found in the observed spectra of pulsars originate from

the reprocessing of the power law component. These additional components may include (i)

modified power law emissions (ii) a reprocessed soft blackbody component (iii) various

components involving elemental spectra, fluorescence etc.

In the spectrum of Hercules X-1, a modified power law component (highly absorbed power law)

is observed. This can be due to forward scattering of the power law component by the matter just

outside the line of sight. The outer edge of the accretion disc is a plausible site for this.

The soft blackbody component has a thermal spectrum with a pulse modulation, which has aphase shifted from that of the power law component. This component is a very common feature

of emission from XBPs. The presence of this component is known as soft excess. The source of

this component can be reprocessing of X-rays at the inner edge of the accretion disc in the case

of luminous pulsars. In the case of less luminous pulsars, this can be explained in terms of

emission by photonized or collisionally heated diffuse gas or thermal emission from the surface

of the neutron star.

Other components in the spectrum are generally emission lines from various elements involving

phenomena like fluorescence, plasma emission etc. These components are specific for each

pulsar.

Observation


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Observations of the x-ray pulsar GRO-J1008 (position: = 56.2960; = -10.2930) were made by

Suzaku (ASTRO-E2) from 30-11-2007 (12:42:34 UT) to 1-12-2007 (13:17:38 UT) with anexposure time of ~110ks. The observations were made by three X-Ray Imaging Spectrometers

(XIS0, XIS1, XIS3) in the energy range 0.4 keV - 10.0 keV. Data in the energy range 1.75 keV -

1.9 keV was excluded from the spectral analysis due to detector errors. The time resolution was

2s. The data was also corrected for the orbital motion of the satellite and the earth.

The observation recorded average count rates of 15.09 counts s-1 (for XIS0), 25.54 counts s-1

(for XIS1) and 28.81 counts s-1 (for XIS3).

Light curve obtained by XIS0 is shown bellow.

Fig 4

Analysis


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Timing Analysis

The folded light curve obtained by XIS0 is shown bellow (fig. 5).

Fig. 5

The pulse period is found to be ~93.73 s.

The pulse profile shows two dips before the peak. The first dip is shallow and is found at the

phase ~0.15 of the period. The second dip is deep and is found near the phase 0.5 of the period.

The presence of a dip very close to the peak shows the existence of matter or object that strongly

absorbs or scatters the incident radiation when the pole of the pulsar comes close to the line of

sight. Since this effect occurs periodically, it is clear that the absorbing/scattering matter rotateswith the neutron star with the same period. This may be caused by the accreted matter near to the

pole.

The pulse period can be divided into several phases.

a. A sinusoidal increase from 0: Here the radiation from the pulsar is least interrupted.

(0.0-0.2)

b. A small dip: Here the radiation is scattered or absorbed by some matter (0.2-0.35)

c. A large dip: Here the absorption/scattering is very high. (0.35-0.55)

d. Sinusoidal decrease (0.55-0.85)

e. A variation from sinusoidal nature. A steep shoulder like structure is present.

(0.85-1)

Apart from the two dips near the peak, the pulse shape is nearly sinusoidal. But a slight

variation from sinusoidal behavior is observed near 0.9 of the period.


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The ideal light curve that is expected for a pulsar is sinusoidal. Any variation from the sinusoidal

shape is either due to the matter surrounding the neutron star or due to anomalies in the emission

by the neutron star. These properties, however, cannot be studied by analyzing a broad spectrum

(which is 0.4 10.0 keV in this case). Only a study of pulse period which resolves the spectrum

can reveal the exact mechanism involved in producing the observed pulse shapes. It is possible

that various components of the analyzed spectrum are affected differently by the surrounding

matter. This too is out of the scope of this study.

Phase Averaged Spectral Analysis

In this description, I concentrate on the spectral analysis of the observed data obtained during the

entire observation period (i.e., the analysis resolving for different pulse phases are not done).

Due to instrumental errors, the data in the range 1.75 1.90 keV are discarded.

Fig. 6 shows the spectrum obtained using XIS0 in log scale.

Fig. 6

Different models were considered to explain the observed spectrum.

The two models which gave reasonably good fit are described bellow.

1. Power law + Black body + Gaussian


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The basic model consisted of a power law component with a photon index of 0.806693. An

absorption column intensity of nH = 1.03603 X 1022 is introduced in order to explain the

absorption of the radiation by the interstellar matter. Residuals of this model had (1) an excess in

the longer wavelengths and (2) a peak near 6.5 keV. The excess in the longer wavelengths

signifies the presence of a soft blackbody component (soft excess). The ~6.5 keV peak is

possibly an iron emission line. To incorporate these, a blackbody component, with kT = 1.87948

keV and a Gaussian emission with line energy 6.42976 keV were introduced. With thesemodifications, the model showed reasonable agreement with the observed spectrum with a

reduced 2 value of ~1.21 for 2578 degrees of freedom. However, in the lower energy region

(


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Norm 7.805083E-04 0.114389E-03

The model is plotted in fig. 8

Fig. 8

However, the black body temperature that corresponds to kT ~ 1.8 keV is much higher than the

typical values of the temperature of black body emission observed in X-ray pulsar spectra.

6. Power law + Power law + Gaussian

This in this model, the black body component (which gave unrealistic temperature values) was

replaced by another power law component.

The model parameters are given bellow

Parameter Unit Value (Phase averaged)

Power Law

1

nH X 1022 1.28131 +/- 0.578591E-01

Photon Index 1.13945 +/- 0.216020

Norm 7.790261E-02 +/- 0.104662E-01

Power Law

2

nH X 1022 7.35932 +/- 1.61098

Photon Index 1.42088 +/- 0.441893

Norm 9.020678E-02 +/- 0.353927E-01

Iron Line

Line Energy keV 6.41093 +/- 0.691647E-01

Sigma keV 0.898509 +/- 0.802189E-01

Norm 3.094131E-03 +/- 0.677171E-03


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The model is plotted in fig. 9.

Fig. 9

The model along with data and values is plotted in fig. 10 (log scale in both X and Y axes).

Fig. 10


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The reduced 2 value for this model was found to be 1.1814 for 2578 degrees of freedom.


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Discussion

In this section, we discuss about the origins of the various spectral components which were

separated in the last section. Among all the models considered, the model which gave the best fit

is described in the previous section. However, due to the non-acceptable value for the soft black

body temperature in the first model, it is discarded.

The second model gave a better fit than the first one and all the model components seem

reasonable.

The nature and possible origin of the components is described bellow.

1. Power Law 1

This component is the dominant component in the pulsar spectrum. The origin of this

emission is the accretion column where the kinetic energy of the matter falling into the

pulsar is converted into radiation.

2. Power Law 2

This component has very similar characteristics as the power law 1 with a higher value

for the hydrogen column intensity (nH ~ 7.36). This component can be a modified

emission of power law 1, which may be due to the circumstellar accreting matter in the

line of sight which has a very high optical depth. It is also possible that this is the forward

scattered version of power law 1 scattered by the interstellar matter just outside the line

of sight.

3. ~6.4 keV Line

This line is an emission from the ionized iron atoms in the accretion disc. The energy~6.4keV confines to the emission due to transition involving the K shell of the iron atom.

Both the power law and the black body components are continuous spectra. The 6.4 keV line is a

local feature.


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Conclusion

The pulse period was found to be ~93.73 s. The pulse shape was sinusoidal with two dips near

the peak and a steep shoulder like structure. The dips may be due to some absorption/scattering

phenomenon.

The phase-averaged spectral analysis showed the presence of three components, (1) power law1(dominant), (2) power law 2 and (3) an iron line at ~6.4 keV.

The power law 1 is emitted by the accretion column near the poles of the neutron star. Power law

2 can be a modified version of power law 1. A possible mechanism for this is forward scattering

by interstellar matter just outside the line of sight. The ~6.4 keV line is an emission from the

ionized iron atoms (K emission).


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Appendix I

Suzaku

Suzaku (formerly ASTRO-E2) is a combined Japanese-US mission for X-ray astronomy. It

initially consisted of three distinct scientific instruments. There is a set of four X-ray Imaging

Spectrometers (XIS), among which three are front illuminated (0.4 - 12 keV) and one is backilluminated (0.2 - 12 keV) with moderate energy resolution. The second instrument is a non-

imaging Hard X-ray Detector (HXD) which is operational in the hard X-ray region 10 - 600 keV.

The non imaging X-ray Spectrometer, which worked in the soft X-ray region is no longer

operational.

An overview of Suzaku parameters are provided in the following table.

Satellite Characteristics

Orbit apogee 568km

Orbital period 96 minutes

Observing

efficiency~ 45%

X-Ray Telescopes (XRT)

Focal length 4.75m

Field of view 17' at 1.5keV

13' at 8keV

Plate scale 0.724 arcmin/mm

Effective area 440cm2 at 1.5keV

250cm2 at 8keV

Angular resolution 2' (HPD)

XIS

Field of view 17.8' X 17.8'

Bandpass 0.2-12 keV

Pixel grid 1024 X 1024

Pixel size 24m X 24mEnergy resolution ~ 130 eV at 6keV

Effective area 340cm2 (FI), 390cm2 (BI) at 1.5keV

(incl XRT-I) 150cm2 (FI), 100cm2 (BI) at 8keV

Time resolution8s (Normal mode), 7.8ms (P-Sum

mode)

HXD

Field of view 4.50 (100keV)

Field of view 34' X 34' (100keV)

Bandpass 10-600keV

- PIN 10-70keV

- GSO 40-600keV

Energy resolution(PIN)

~ 4keV (FWHM)

Energy resolution

(GSO) (FWHM)

Effective area~ 160 cm2 at 20keV, ~ 260 cm2 at

100keV

Time resolution 61s

HXD-Wide-band All-Sky

Monitor (HXD-WAM)

Field of view 2 (non-pointing)

Bandpass 50keV-5MeV

Effective area800cm2 at 100keV / 400cm2 at

1MeV

Time resolution31.25ms for GRB, 1s for All-Sky-Monitor

.


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Suzaku has five X-ray Telescopes. Four of them are used for XISs and one is for XRT, which is

not functional.

All instruments in Suzaku operate simultaneously. The XISs are true imagers with a large field

of view and a moderate spectral resolution. The HXD is a non-imaging device featuring a

compound eye configuration and an extremely low background. HXD is designed to observe

very faint hard X-ray sources. However, it can tolerate brightness up to 10 crab. HXD also

features as the all-sky monitor, which can observe sources like gamma ray, bursts.

X-ray Imaging Spectrometers

Suzaku has four X-ray Imaging Spectrometers (XISs). These employ X-ray sensitive siliconcharge-coupled devices (CCDs) operated in a photon-counting mode. In general, X-ray CCDs

operate by converting an incident X-ray photon into a charge cloud, with the magnitude of

charge proportional to the energy of the absorbed X-ray. This charge is then shifted out onto the

gate of an output transistor via an application of time-varying electrical potential. This results in

a voltage level (often referred to as ``pulse height'') proportional to the energy of the X-ray

photon.

The fourSuzaku XISs are named XIS0, 1, 2 and 3, each located in the focal plane of an X-ray

Telescope. Those telescopes are known respectively as XRT-I0, XRT-I1, XRT-I2, and XRT-I3.

Out of these, three sensors (XIS0, XIS1, and XIS3) are currently usable.

Each CCD camera has a single CCD chip with an array of 1024 X1024 picture elements

(``pixels''), and covers an18 X 18region on the sky. Each pixel is 24m square, and the size of

the CCD is 25mm X 25mm. One of the sensors, XIS1, uses a back-side illuminated CCD, while

the others use a front-side illuminated CCD.

A CCD has a gate structure on one of the surfaces to transfer the charge packets to the readout

gate. The surface of the chip with the gate structure is called the ``front side''. A front-side

illuminated CCD (FI CCD) detects X-ray photons that pass through its gate structures, i.e., from

the front side. Because of the additional photo-electric absorption at the gate structure, the low-

energy quantum detection efficiency (QDE) of the FI CCD is rather limited. Conversely, a back-

side illuminated CCD (BI CCD) receives photons from ``back,'' or the side without the gate

structures. For this purpose, the undepleted layer of the CCD is completely removed in the BI

CCD, and a thin layer to enhance the electron collection efficiency is added in the back surface.

A BI CCD retains a high QDE even in a sub-keV energy band because of the absence of the gate

structure on the photon-detection side. However, a BI CCD tends to have a slightly thinner

depletion layer, and the QDE is therefore slightly lower in the high energy band. The decision to

use only one BI CCD and three FI CCDs was made because of the slight additional risk involved

in the new technology BI CCDs and the need to balance the overall efficiency for low- and high-energy photons.


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Appendix 2

FTOOLS

The data analysis for this project was done using FTOOLS forSuzaku.

FTOOLS is a collection of utility programs used to create, examine, or modify the contents ofFITS data files. There are also user friendly GUI tools which allow interactive browsing of FITS

files and provide a more intuitive interface for running the FTOOLS. The FTOOLS package

forms the core of the HEASARC software system for reducing and analyzing data in the FITS

format.

The XSELECT software, which provides a command user interface for accessing the FTOOLS,

was used for accessing FTOOLS. FTOOLS and XSELECT are UNIX-based programs.

Therefore the analysis can only be done with a UNIX operating system like Fedora.


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References

i. Origin of the Soft Excess in X-Ray Pulsars, Hickox, R.C., Narayan, R., Kallman,

T.R., 2004

ii. Pulse Phase Resolved Spectroscopy of Hercules X-1 with ASCA, Endo, T., Nagase

F., Mihara T., 1999

iii. Accretion Powered X-Ray Pulsars, Nagase, F., 1987

iv. Spectral Formation in X-Ray Pulsar Accretion Columns, Becker, P.A., Wolff, W.T.,

2005

v. The Suzaku Technical Description (version Oct 2009)

http://www.astro.isas.ac.jp/suzaku/doc/suzaku_td.pdf

vi. www.wikipedia.org

a. http://en.wikipedia.org/wiki/Accretion_(astrophysics)b. http://en.wikipedia.org/wiki/Accretion_disc

c. http://en.wikipedia.org/wiki/Pulsar

d. http://en.wikipedia.org/wiki/Neutron_star

e. http://en.wikipedia.org/wiki/X-ray_binary

vii. What is the Physics of Pulsar Radio Emission? , Hankins, T.H., Rankin, J.M., Eilek,

J.A., 2009 http://www8.nationalacademies.org/astro2010/DetailFileDisplay.aspx?id=269

viii. http://heasarc.gsfc.nasa.gov/lheasoft/ftools

ix. Rotation and Accretion Powered Pulsars, Ghosh, P. 21

understanding the properties of binary x-ray pulsars

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