planetary dynamics dr sarah maddison centre for astrophysics & supercomputing swinburne...
TRANSCRIPT
Planetary Dynamics
Dr Sarah MaddisonCentre for Astrophysics & SupercomputingSwinburne University
AstroFest 2007
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…..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 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
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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…..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
A
BC
D
E
FG
H
A B C D E F G HPhases as seen from Earth
Sun’s
rays New moon
Full moon
3rd quarter
1st quarter
Sun’s
rays
Dark side of the Moon
Near side of the Moon (the face that we see!)
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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:
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
(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
Io passes Europa every 2nd orbit and Ganymede every 4th orbit
1 2 3 4 5 6 7 8 9
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- 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
Examples of Solar System resonances cont..
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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
- spikes in Encke gap due to Pan
Cassini division
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
- resonances not due to the major satellites, though high inc of Miranda suggests resonances of the past, may have produced resurfacing events
Ariel
9 rings of Uranus
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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) Pluto & Charon
Ave separation ~17 RPluto
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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)
(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
Pluto & Charon
Ave separation ~17 RPluto
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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
Resonances not totally cleared, some asteroids captured by Jupiter
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Tides…..Tides…..
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• Small bodies orbit massive object due to gravity, but are also
subject to tidal forces that may tear the satellite apart.
Tidal forces
• The satellite feels a stronger gravitational force on its near side to its far side tidal forces are differential.
• Oscillations can develop and deform or disrupt the satellite.
gravity at near surface is stronger than at far surface
as satellite approaches massive object, tidal forces get stronger
and satellite is distorted
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• Neglecting internal satellite forces, disruption occurs when differential tidal force exceeds the satellite’s self-gravitation:
The Roche limit
• Maximum orbital radius for which tidal disruption occurs is the Roche limit.
• Substituting average densities the equation becomes:
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.
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.
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.
• 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)
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Orbits…..Orbits…..
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The Geometry of Ellipses
r2 r1
a aeb
Equation of the ellipse:
In Cartesian coordinates:
Let: Thus :
Eccentricity of the ellipse defined by:
Simple algebra shows that the following relations hold:
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Specifying a point on the ellipse
Cartesian coordinates with the origin at the centre of the ellipse, we have:
F1
2a(0,0)
y
x f
r
c
(x,y)
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.
ecliptic planeorbit p
lane
i
ac
• semi-major axis a
• eccentricity e = c/a
The inclination, i, describes tilt of orbital plane with respect to reference plane
The size and shape of an orbit determined by the semi-major axis, a, and eccentricity, e
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ecliptic planeorbit p
laneP
i
ac
0o in Piscesascending
node
descending node
The argument of pericentre*, , and longitude of the ascending node, , determine the orientation of the orbit and where the line of nodes crosses the reference plane.
* Pericentre = periastron, perihelion, periapse depending on system in question - point of closest approach to the focus
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ecliptic planeorbit p
laneP
i
ac
0o in Piscesnode
node
The true anomaly, f, tells where orbiting body is at a particular instant in time and is measured from pericentre to orbiting body.
true anomaly
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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;
• , the argument of pericentre;
• , the longitude of the ascending node; and
• (say) time T when planet is at perihelion
ecliptic planeorbit p
laneP
i
ac
0o in Piscesnode
node
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• The Cartesian orbital elements are:
Cartesian vs Keplerian orbital elements
– position (x, y, z), and
– velocity (vx, vy, vz).
• Cartesian & Keplerian are equally precise ways of describing an orbit.
• Relatively simple equations exist for transforming between the two coordinate systems.
(x,y,z)
(vx,vy,vz)
(0,0)
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Orbital Energy…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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• 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
Thus total system energy is:
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Different types of orbits:
Ellipses and circles 0 e < 1BoundTotal energy is negative
Ellipses and circles 0 e < 1BoundTotal energy is negative
Parabola e = 1UnboundTotal energy is zero
Parabola e = 1UnboundTotal energy is zero
Hyperbola e > 1UnboundTotal energy is positive
Hyperbola e > 1UnboundTotal energy is positive
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N-body Problem…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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Physical phenomena choose the physical system that you wish to investigate
e.g. the motion of N planets around a star, where N ≥ 3
The basic steps involved in using a computer to find a numerical “solution” to an N-body problem are:
Setting up a Numerical Experiment
the physical system is approximated by a
mathematical model, which uses some simplifying
assumptions to describe the workings of the physical systemMathematical model
the mathematical model must be converted from a continuous or differential equation into an algebraic approximation which
computers can solve. Both time and space must be
discretised, which can produce numerical errors
Discretise the model
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Numerical algorithm Choice of discretisation is often related to the algorithm chosen to solve the discrete system. Need to be able to solve the discrete problem rapidly otherwise having a computer is no help at all!
The next steps are:
Computer program
writing the computer code is where most of the hard work lies. The code needs to be well engineered to capitalise on the available computing power, and it should be be easy to use and modify.
Computer experiment
finally you get to run your computer experiments, but you need to know what you’re testing for, what you’re trying to find, and how to do analysis on the data that your experiment produces. Pretty pictures are of course vital at this stage!
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The Mathematical Model
Two main parts of codes for solving N-body problems:
The relevant equations for a dynamical N-body code are just:
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.
• Newton’s law of gravitation for the forces; and
• the equation of motion for the time evolution.
• the force calculation and
• the time evolution.
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Newton’s universal law of gravitation between two bodies is:
Gravitational Forces
rm1 m2
F2F1F1
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What about an N=5 system?
m1
m2
m3m4
m5
F12
F13
F15
F14
The force exerted on body 1 by the other 4 bodies would be given by:
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:
m1
m3m4
m5
m2
and the force on particle 3 due to all the other particles:
and the force on particle 4 due to all the other particles:
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.
Need to soften the gravity
Equation becomes:
(2) As the particles get closer together, the forces get larger. As particle i approaches j the denominator r ij of the force equation
approaches zero so the force become infinite.
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:
wherei = 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:
Need to solve for the position and velocity of the system at the next timestep. Hence:
Substituting F by Fgrav from Newton’s
law of gravitation gives:
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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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• 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.
Accuracy and Stability
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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:
The N-body Algorithm
Set initial conditions
Solve equations of motion
Calculate forces
Update time counter
Output data
Choose N and t, set initial particle mi, ri, vi, Fi
ai = vi/ ti
vi = ri/ ti
Fi = j Gmimj/rij2
tnew = told + t
rnew, vnew, Fnew, tnew