Capturing Oort Cloud Comets

Jeremy A. Miller

Department of Astronomy

University of Maryland

College Park, MD 20742-2421

millerja@wam.umd.edu

Advisor: Douglas P. HamiltonImage from: http://encke.jpl.nasa.gov/comets_short/9P.html

An introduction to comets

• Halley – comets orbit the Sun periodically.• Oort – comets come from a spherical shell

~20,000 AU away.• Kuiper – other comets

come from a flat ring 30-50AU away.

Image from: http://dsmama.obspm.fr/demo/comet.gif

Numerical methodsImpulse Approximation N-body Simulations

Very fast Very accurate

Considers first planet-comet encounter Includes all Solar System effects on comet

Based on simple physics Numerical integrations of DEs

• Everhart 1969 (E69):

• Patched conic method around Jupiter

• Single passes by same comet

• Everhart 1972 (E72):

• Numerical integration with Jupiter and Sun

• Multiple passes by same comet

• Wiegert and Tremaine 1999 (WT):

• N-body simulation with all planets and Sun

• Multiple passes by same comet

Image from: http://science.howstuffworks.com/planet-hunting1.htm

Why simplify the problem?

Image from: http://www.lactamme.polytechnique.fr/Mosaic/images/NCOR.C3.0512.D/display.html

N-body simulations may give more accurate results than analytic methods...

But analytic methods and impulse approximations are easier to understand physically.

Two coordinate systems

The Jupiter-comet system:

*Jupiter bends a comet’s orbit.

*Jupiter can give or take energy.

*Sun’s influence on comet ignored.

The Sun-comet system:

*Comet follows Keplerian orbit around sun.

*Orbital elements used to describe motion.

*Jupiter’s influence on comet ignored.

We approximate the motion of Oort cloud comets by combining these two systems.

The Jupiter-comet system• Comets spending more time in front of Jupiter than behind

are captured.

• The magnitude of the comet’s deflection depends on its speed relative to Jupiter and the distance of closest approach.

– Fast comets barely deflected

– Slow comets deflected quite a bit

Image from: http://analyzer.depaul.edu/paperplate/PPE%20pause/Dynamic%20comet.jpg

The Sun-comet system:Tisserand’s Criterion

*Valid for single planet on a circular orbit

*Valid when comet is far from Jupiter

*Combination of energy and angular momentum

The Sun-comet system:Special case

• io = 90o pericenter encounter

– K=0

– increased inclination = capture

– decreased inclination = escape

– exactly one circular orbit

• Bound orbits have smaller final than initial z angular momentum

• Retrograde comets cannot produce prograde elliptical orbits

General case

Three-body impulse approximation algorithm

Sun-comet coordinate system Jupiter-comet coordinate system (The “target plane”)

The parameters r and uniquely define an interaction

() – Angles that define the comet’s velocity

(r, ) – Position of closest approach on the target plane

Image from: http://nmp.jpl.nasa.gov/ds1/edu/comets.html

The algorithm (sans gory details)

0) Start with parabolic orbit comets velocity v = √2

1) Choose geometry parameters r, and .

2) Convert to velocity vectors.

3) Impulse: rotate velocity vector by in Jupiter-comet coordinate system.

4) Convert final velocity into final parameters.

5) Convert final parameters to heliocentric orbital elements.

Just solve this equation... simple!

Monte Carlo approach:Repeat steps 500 million times

(We used a computer for this part)

We assumed a spherically symmetric distribution of massless comets on parabolic orbits:

, , x=rcos, and y=rsin chosen randomly

rmax chosen to be 200rj.

Image from: http://www.educeth.ch/stromboli/photoastro/index-en.html

Results

*<a> > 0 (51.4% of comets captured)

*<i> > 90o (51.0% of comets retrograde)

-but-

Problem: Tisserand’s constant should be conserved. We find it increases by ~1%.

Lets look at some orbital element distributions...

Image from: http://www2.jpl.nasa.gov/sl9/gif/kitt11.gif

More results for captured cometsFigure 7a: Eccentricity bins

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1

Eccentricity

% in bin

Figure 7b: Semimajor Axis bins

0

0.02

0.04

0.06

0.08

0.1

0.12

0 20 40 60 80 100

Semimajor axis (aj)

% in bin

Figure 7c: Inclination Bins

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0 30 60 90 120 150 180

Inclination (degrees)

% in bin

Figure 7d: Pericenter Bins

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.2 0.4 0.6 0.8 1

Pericenter (aj)

% in bin

Initial q

Final q

Retrograde orbits – Tisserand’s criterion

Low inclination peak – geometry

Comets highly elliptical – weak interactions Most semimajor axes less than 20 aj

Pericenter distribution unchanged

SPCs, HTCs, and LPCs

Figure 9a: Eccentricity Bins

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Eccentricity

% in bin

SPCs

HTCs

LPCs

Figure 9b: Inclination bin

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 30 60 90 120 150 180

Inclination (degrees)

% in bin

SPCs

HTCs

LPCs

SPCs have a larger spread of eccentricities than LPCs

Why? Stronger interactions

SPCs are more strongly peaked at lower inclinations than HTCs

Why? Geometry

Evaluating the impulse approximationImpulse Approximation N-body Simulations Observations

SPC low inclination peak vs. HTC inclination spread

HTCs distinguished from SPCs by their larger inclination (E72)

Only visible HTC, i=28o

Average SPC, i=10o

0.7% of captured comets were SPCs

Many passes by Jupiter required to form SPCs (E72)

----------

51% of captured comets retrograde

Inclination usually increases after single pass by planet (E72)

----------

Larger pericenter less elliptical orbits

---------- q<2AU <e> = 0.59

q>3AU <e> = 0.22

Slight peak in SPCs at e=0.4

---------- Half of SPCs with 0.3<e<0.55

Future work

Our main goal is to find and explain trends in cometary distributions using simple physics.

To further this goal we will:

*Track down error in K

*Compare results to numerical integrations

*Run multiple passes of comets by Jupiter

*Make more graphs for more insight Image from: http://flaming-shadows.tripod.com/gal2.htm

AcknowledgementsI want to thank my advisor, Doug Hamilton, for all of his help on this project. I couldn’t have done it without him (or gotten away with using the royal “we”) .

I also want to thank the astronomy department for the use of their computer labs.

These people helped a bit as well:In 1456 Pope Callixtus III excommunicated Halley’s comet as an agent of the devil and added the following line to the prayer Ave Maria:

"Lord save us from the devil, the Turk, and the comet".

Quote from: http://www.wilsonsalmanac.com/constantinople.html