Download - Capturing Oort Cloud Comets
Capturing Oort Cloud Comets
Jeremy A. Miller
Department of Astronomy
University of Maryland
College Park, MD 20742-2421
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