Planetary Dynamics
Dr Sarah Maddison
Centre for Astrophysics & Supercomputing
Swinburne University
OUTLINE:
This lecture will cover the gravitational theory behind
planetary dynamics, including:
• Kepler’s laws and Newton’s laws,
• resonances,
• tides, and
• orbits and orbital elements.
To understand simulations of planetary dynamics, we’ll
also cover:
• the N-body problem.
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Laws of Motion…..
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Kepler’s Laws
Kepler (1609, 1619) presented three empirical laws of
planetary motion from obs made by Tycho Brahe:
(1) Planets move in an ellipse with the Sun
QuickTime™ and a
GIF decompressor
are
needed
to see this picture.
at one focus
(2) The radial vector from the Sun to a
planet sweeps out equal area in equal
time
(3) The orbital period square is proportional to the
semi-major axis cubed (T2  a3)
But empirical laws with no physical understanding of why
planets obey them…
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Newton’s Laws
Newton’s (1687) three laws of motion:
(1) Bodies remain at rest or in uniform motion in a straight
line unless acted on by a force
(2) Force equals the rate of change of momentum
(F = dp/dt = ma)
(3) Every action has an equal and opposite reactions
(F12= -F21)
Plus his universal law of gravitation:
F = Gm1m2 / d2
Probably first derived by Robert Hooke, but Newton used it
to explain Kepler’s laws.
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Newton’s laws revolutionized science and dynamical
astronomy in particular.
E.g. extending Newton’s law of gravitational to N > 2
showed that the mutual planetary interactions
resulted in ellipses not fixed in space
 orbital precession
Planetary orbits rotate in space over ~105 years
QuickTime™ and a
MPEG-4 Video decompressor
are needed to see this picture.
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But it’s an approximation (though a pretty good one!)
Mercury should precess at a rate of 531”/century, but
43”/century greater. Precession of Mercury’s perihelion
explained using Einstein’s theory of General Relativity.
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Resonances…..
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Resonances
Lots of discoveries of minor bodies in the last 50 years:
• ~100 new satellites
• over 10,000 catalogised asteroids
• over 500 reliable comet orbits
• over 1000 KBOs
• dust bands in the asteroid belt
• planetary rings of all giants with
unique characteristics
 All follow Newton’s laws and experience subtle
gravitational effects of resonances
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Resonances result from a simple numerical relationship
between periods:
• rotational + orbital periods  spin-orbit coupling
• orbital periods of N bodies  orbit-orbit coupling
• plus more complex resonances…
Dissipative forces drive evolutionary processes in the
Solar System connected with the origins of some of
these resonances.
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Examples of Solar System resonances:
(1) spin-orbit coupling of the Moon:
Trot = Torb  1:1 or synchronous spin-orbit coupling
same face of the Moon always faces Earth
H
Sun’s
rays
A
G
3rd quarter
New
moon
B
F
Full
E
moon
1st quarter
D
C
Dark side of
the
Moon
A
B
C
Near side of the Moon
(the face that we see!)
D
E
F
G
H
Phases as seen from Earth
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Examples of Solar System resonances:
(1) spin-orbit coupling of the Moon:
Trot = Torb  1:1 or synchronous spin-orbit coupling
same face of the Moon always faces Earth
(2) spin-orbit coupling of Mercury:
3Trot = 2Torb  3:2 spin-orbit coupling
two Mercury years = three
sidereal days on Mercury
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Examples of Solar System resonances:
(1) spin-orbit coupling of the Moon:
Animation decompressor
Trot = Torb  1:1 or aresynchronous
spin-orbit coupling
needed to see this picture.
same face of the Moon always faces Earth
QuickTime™ and a
(2) spin-orbit coupling of Mercury:
3Trot = 2Torb  3:2 spin-orbit coupling
(3) orbit-orbit resonances of planets:
- Jupiter + Saturn in 5:2 near resonance, perturbs both
planet’s orbital elements on ~900 year timescale
- Neptune + Pluto in 3:2 orbit-orbit resonance, maximises
separation at conjunction and avoids close approaches
- other planets involved in long term secular resonances
associated with the precession of their orbits
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Examples of Solar System resonances cont..
(4) Galileans satellite’s spin-spin resonances :
- Io + Europa 2:1 resonance
- Europa + Ganymede 2:1 resonance
1 2 3 4 5 6 7 8 9
Io passes Europa every
2nd orbit and Ganymede
every 4th orbit
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Examples of Solar System resonances cont..
- average orbital angular velocity or mean motion
defined as n = 360/T (degrees per day)
- mean motions of the Galileans:
nI = 203.448 o/d, nE = 101.374 o/d, nG = 50.317 o/d
so nI/nE=2.0079 and nE/nG=2.01469 and hence
nI - 3nE + 2nG = 0 (to within obs errors of 10-9 o/d)
This is the Laplace relation, prevents triple conjunctions
- 2:1 Io:Europa resonances results in active volcanism on Io
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Examples of Solar System resonances cont..
(5) Saturn’s satellites have widest variety of resonances :
- Mimas + Tethys 4:2 resonance (nM/nT=2.003139)
- Enceladus + Dione 2:1 resonance (nE/nD=1.997)
- Titan + Hyperion 4:3 resonance (nT/nH=1.3343)
- Dione & Tethys 1:1 resonance with small bodies on their orbits
- Janus + Epimetheus on 1:1 horseshoe orbits (swap orbits every
3.5 years) http://ssdbook.maths.qmw.ac.uk/animations/Coorbital.mov
- 2:1 resonant perturbation of Mimas causes gap
in rings (Cassini division)
- structure of F ring due to Pandora + Prometheus
http://photojournal.jpl.nasa.gov/animation/PIA07712
Cassini division
- spikes in Encke gap due to Pan
Encke gap
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Examples of Solar System resonances cont..
(6) Uranus’s satellites also in resonance:
- Rosalind + Cordelia in close 5:3 resonance
- Cordelia + Ophelia bound to narrow  ring by
24:25 and 14:13 resonances with the inner
and outer ring edge
9 rings of Uranus
- resonances not due to the major satellites, though
high inc of Miranda suggests resonances of the
past, may have produced resurfacing events
Ariel
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Examples of Solar System resonances cont..
(7) Pluto:
- Pluto + Charon in synchronous spin state “totally tidally despun”(both keep same face
towards each other, fixed above same spot)
Ave separation ~17 RPluto
Pluto & Charon
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Examples of Solar System resonances cont..
(7) Pluto:
- Pluto + Charon in synchronous spin state “totally tidally despun”(both keep same face
towards each other, fixed above same spot)
Ave separation ~17 RPluto
Pluto & Charon
(8) Kuiper Belt:
- predicted by Edgeworth (1951) and Kuiper (1951) and observed
in 1992 (Jewitt & Luu)
- three main classes: Classical, Resonant and Scattered
- Third of all KBOs in 3:2 resonance
with Neptune, i.e. Plutinos
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Examples of Solar System resonances cont..
(9) Asteroid Belt:
- Resonant structure found by Kirkwood (1867), noticed gaps at
important Jupiter resonances: 4:1, 3:1, 5:2, 7:3, 2:1
but also concentrations at 3:2 and 1:1
Resonance
2:1
5:3
3:1
7:2
5:2
7:3
a (AU)
3.3
3.7
2.5
2.3
2.83
2.95
Resonances not totally cleared, some asteroids captured by Jupiter
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Tides…..
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Tidal forces
• Small bodies orbit massive object due to gravity, but are also
subject to tidal forces that may tear the satellite apart.
• The satellite feels a
stronger gravitational
force on its near side to
its far side  tidal forces
are differential.
gravity at near surface is
stronger than at far surface
• Oscillations can develop and
deform or disrupt the satellite.
as satellite approaches massive
object, tidal forces get stronger
and satellite is distorted
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The Roche limit
• Maximum orbital radius for which tidal
disruption occurs is the Roche limit.
• Neglecting internal satellite forces, disruption
occurs when differential tidal force exceeds
the satellite’s self-gravitation:
where Ms and Mm are the masses of the satellite and central body;
r is their separation; and Rs is the radius of the satellite.
• Substituting average densities the equation becomes:
where Rm is the radius of the central body.
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The Hill radius
• For an N-body system a satellite can feel tidal forces from
several massive bodies, e.g. the Moon feels a tidal force from
the Earth and from the more distant (but more massive) Sun.
Forces on the near side of the
Moon from the Sun and Earth
Forces on the far side of the
Moon from the Sun and Earth
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• The Hill radius is the radius of a sphere around a planet within
which the planetary tidal forces on a small body are larger
than the tidal forces of the Sun.
• For one test particle and two massive bodies (e.g. the Sun
and a planet), the Hill radius, RH, is:
2 is the reduced mass of the second body
given by 2 = M2/(M2+M1)
As a rough guide, the Hill radius is:
- 0.35 AU for Jupiter,
- 0.44 AU for Saturn,
- 0.47 AU for Uranus, and
- 0.78 AU for Neptune.
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Orbits…..
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b
The Geometry of Ellipses
a
ae
Equation of the ellipse:
r2
r1
In Cartesian coordinates:
Let:
Thus :
Eccentricity of the ellipse defined by:
Simple algebra shows that the following relations hold:
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(x,y)
y
r
Specifying a point on the ellipse
Cartesian coordinates with the
origin at the centre of the
ellipse, we have:
(0,0)
c

f
F1
2a
x
From the equation of the ellipse, and by substituting the
equations that define x, y, b and e, it is possible to show that:
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Orbital elements
Orbits are uniquely specified in space by six orbital elements.
The size and shape of an orbit determined by the
semi-major axis, a, and eccentricity, e
• semi-major axis a
• eccentricity e = c/a
c
a
i
The inclination, i,
describes tilt of orbital
plane with respect to
reference plane
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The argument of pericentre*, w, and longitude of the ascending
node, W, determine the orientation of the orbit and where the
line of nodes crosses the reference plane.
descending
node
w
c
a
P
W
0o in
i
ascending
Pisces
node
* Pericentre = periastron, perihelion, periapse
depending on system in question - point of closest
approach to the focus
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The true anomaly, f, tells where orbiting body is at a particular
instant in time and is measured from pericentre to orbiting body.
node
w
c
a
P
W
true anomaly
0o in
i
Pisces
node
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• a, the semi-major axis of the ellipse;
• e, the eccentricity of the ellipse;
• i, the inclination of the orbital plane;
• w, the argument of pericentre;
• W, the longitude of the ascending node; and
• (say) time T when planet is at perihelion
node
w
c
a
P
W
0o in
i
Pisces
node
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Cartesian vs Keplerian orbital elements
• The Cartesian orbital elements are:
(x,y,z)
– position (x, y, z), and
– velocity (vx, vy, vz).
(0,0)
(vx,vy,vz)
• Cartesian & Keplerian are equally precise ways of describing
an orbit.
• Relatively simple equations exist for transforming between the
two coordinate systems.
Cartesian
x
y
z
vx
vy
vz
Keplerian
a
e
i
W
w
f
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Orbital Energy…
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Energy and Orbit Types
• The shape of an orbit depends if body is bound or unbound,
which depends on system total energy of the system.
• Total energy is the sum of the kinetic energy, KE, and the
gravitational potential energy, U:
where:
and
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Thus total system energy is:
• If E < 0, orbiting body m2 does not have sufficient velocity to
escape from the gravitational field of m1  the orbit is bound.
• If E > 0, orbiting body m2 has sufficient velocity to escape
 the orbit is unbound
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Different types of orbits:
Ellipses and circles 0  e < 1
Bound
Total energy is negative
Parabola
e=1
Unbound
Total energy is zero
Hyperbola
e>1
Unbound
Total energy is positive
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N-body Problem…
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N-body Problem
• Analytic solution exists for the 2-body problem.
• But no solution for the 3-body problem and stable orbits
difficult to obtain.
– Can simplify to a restricted 3-body problem
(two bodies in circular orbit about COM and third
body with m3 << m1,m2)
• Numerical simulations needed to studying systems of 3
or more objects  N-body problem.
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Setting up a Numerical Experiment
The basic steps involved in using a computer to find a
numerical “solution” to an N-body problem are:
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The Mathematical Model
Two main parts of codes for solving N-body problems:
• the force calculation and
• the time evolution.
Both can be described by a mathematical model - a set of
mathematical equations which tell of the future state of the
system, given a set of initial conditions.
The relevant equations for a dynamical N-body code are just:
• Newton’s law of gravitation for the forces; and
• the equation of motion for the time evolution.
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Gravitational Forces
Newton’s universal law of gravitation between two bodies is:
m1
r
F1
F2
m2
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What about an N=5 system?
The force exerted on body 1 by the other 4 bodies
would be given by:
m4
m3
F13
m2
F14
F12
m1
F15
m5
the sum of the individual forces acting on it:
F1 = F12 + F13 + F14 + F15.
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Also need to calculate the force on particle 2 due to the other 4
particles:
and the force on particle 3 due to
all the other particles:
and the force on particle 4 due
to all the other particles:
m4
m3
m2
m1
m5
and the force on particle 5 due to all the other particles:
A computer would be helpful :-)
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The force equation becomes:
For each N particle i we need to sum over all the other N-1
particles.
The mathematical model for gravitational force is quite easy to
discretise for N particles. (Note that this is an Nx(N-1) or O(N2)
calculation).
However...
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(1) Force is actually a vector quantity, so it has a magnitude and
a direction.
(2) As the particles get closer together, the forces get larger. As
particle i approaches j the denominator rij of the force equation
approaches zero so the force become infinite.
 Need to soften the gravity
Equation becomes:
The softening parameter  must be carefully chosen
- if too large it affects the physics (like an outward force)
- if too small the forces become large (and time must slow down)
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The Equations of Motion
The time evolution of the system is governed by the equations
of motion:
We can easily discretise by writing the differential as a finite
difference:
where
i = initial f = final
The  symbol represents a small but finite change.
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Newton’s second law relates force to
acceleration via the equation:
Substituting F by Fgrav from Newton’s
law of gravitation gives:
Need to solve for the position and velocity of the system at the
next timestep. Hence:
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Taking a small timestep t between the
old and new states of the system, the
final velocity and position are given by:
Once we’ve solved for the gravitational force, F, at the initial
state of the system, we can work out the position and velocity
for each body in the system at the next timestep.
In practice there are many more sophisticated ways to
discretise the equations of motion that produce more accurate
time stepping, but the essential principles have been
described here.
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Accuracy and Stability
• Two more things that we need to be careful about:
our choice of t and N.
• Timestep t controls the stability. If t = constant, we
get large errors when two particles get close. Need a
numerical scheme with a variable timestepping which
automatically decreases t if particles are too close and
increases t as particles move apart.
• The particle number N gives the resolution. Ideally we
want N to be as large as possible, but this means more
calculations. Supercomputers can help us here.
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The N-body Algorithm
We’re now armed with our mathematical model for the
gravitational force and equations of motion; we have a discrete
algorithm for the mathematical model, and we’re ready to write
our computer code to run our computer experiments.
Our computer algorithm will look like:
Set initial conditions
Choose N and t, set initial
particle mi, ri, vi, Fi
Solve equations of motion
ai = vi/ ti
vi = ri/ ti
Calculate forces
Fi = j Gmimj/rij2
Update time counter
tnew = told + t
Output data
rnew, vnew, Fnew, tnew
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