Solar System Dynamics: Exploring Orbital Dynamics and Systems

This assignment delves into the intricacies of solar system dynamics, covering key concepts in celestial mechanics. The solutions address the equation of relative motion in a two-body problem, demonstrating the planar nature of orbital motion through vector product integration. It includes Kepler's laws of planetary motion and applies them to analyze orbital characteristics. The assignment also explores the Lagrange equilibrium points in the circular three-body problem, investigating their stability. Furthermore, it examines orbital angular momentum, tidal forces, and the Kirkwood gaps in the asteroid belt, discussing the theories proposed to explain their formation, including collisional, gravitational, and cosmogonic hypotheses. The solutions provide a comprehensive understanding of orbital dynamics, planetary motion, and the factors influencing celestial bodies within the solar system.

Solar System Dynamics
1. The equation of relative motion in a two-body problem is r + ur/r^3 = 0 where r is
vector orbital radius and u is the constant. Take the vector product of each side of
the equation with r and directly integrate the resulting equation to explain why the
why motion of the two bodies must lie on the same plane.
Soln
cos x ≈ 1 − 1
2 x2 + O(x4) for small x, we obtain
E1 = M + e sin M,
E2 = M + e sin(M + e sin M) ≈ M + e sin M + 1
2e2 sin 2M,
E3 = M + e sin(M + e sin M + 1
2e2 sin 2M)
≈ M + e − 1 8e3 sin M + 1 2e2 sin 2M + 3 8e3 sin 3M
2. Write down Kepler’s first two laws of planetary motion.
Kepler’s third law of motion implies that a planet’s mean
motion, n, is related to its semi-major axis, a, by the equation
n^2 = k/a^3 where k is the constant. Use this equation to
show that a change da in semi-major axis results in a change
dn/n = -(3/2) da/a in mean motion.
Soln
First Law: The planets orbit the Sun in ellipses with the Sun at one focus (the other focus
is empty).
Kepler’s second law: The line joining the Sun and a planet sweeps through equal areas in
an equal amount of time.

Kepler’s Third Law: The square of the period of a planet’s orbit (P) is directly
proportional to the cube of the semi major axis (a) of its elliptical path that is P2∝a3
3. Sketch and label the five Lagrange equilibrium points l1 tol3 in the rotating frame
for the circular three body problem.
Soln
We have seen that the five Lagrange points, to , are the equilibrium points of
mass in the co-rotating frame. Let us now determine whether or not these
equilibrium points are stable to small displacements.
Now, the equations of motion of mass in the co-rotating frame are specified in Note
that the motion in the - plane is complicated by presence of the Coriolis acceleration.
However, the motion parallel to the -axis simply corresponds to motion in the potential.

Hence, the condition for the stability of the Lagrange points to small displacements
parallel to the -axis is simply.
we obtain
(1100)
where ,
, , etc. However, by definition, at a
Lagrange point, so the expansion simplifies to
4. The constant orbital angular momentum per unit mass of an object moving in a
circular orbit in a two-body problem is given by h = na^2 where n is the mean root
and a is semi-major axis. Janus and Epimetheus are two satellites that orbit the
planet in a near circular orbits and have a close approach every four years but their
total orbital angular momentum h2 + hE is constant. Use Kepler’s third law to
show that …
Soln

5. A satellite is in a prograde orbit around a planet and raises a tide on it. If the
satellite is outside the synchronous orbit, sketch the shape of the resulting tidal
bulge and its orientation with respect to the line jointing the centers of the planet
and the satellite. What is the relationship between minimum and maximum tidal
distortion? Give a qualitative explanation for the resulting evolution of the satellite.
Soln
6. What are the Kirkwood Gaps in the asteroid belt? Provide a brief of the four main
types of theory that have proposed to explain them.
Soln
Scattered in orbits around the sun are bits and pieces of rock left over from the dawn of
the solar system. Most of these objects, called planetoids or asteroids — meaning "star-
like" — orbit between Mars and Jupiter in a grouping known as the Main Asteroid Belt.
Within in the asteroid belt are relatively empty regions known as Kirkwood gaps. These
gaps correspond to orbital resonances with Jupiter. The gas giant's gravitational pull
keeps these regions far emptier than the rest of the belt. In other resonances, the asteroids
can be more concentrated.

At the beginning of that period, which coincides with the first long-term numerical
investigations of resonant motion, different hypotheses to explain the origin of the gaps
were still competing with each other: collisional, gravitational, statistical and
cosmological. The statistical hypothesis. This is the claim that the gaps are simply an
illusion. Since the dynamical equations governing the motion of objects in resonance
are similar to those of a pendulum, if an asteroid is in resonance it will spend
most of its time at the extremes of its motion. The collisional hypothesis. This was
originally proposed by Kirkwood. It suggests that the changes in orbital elements at a
resonance cause the perturbed asteroids to collide with nearby objects. The cosmogonic
hypothesis. Cosmogonic theories propose that the gaps are regions where asteroids failed
to form or that they reflect processes that operated in the early stages of the formation of
the solar system but have now ceased. The gravitational hypothesis. According to this
hypothesis the gaps can be understood in the context of the Sun–Jupiter–asteroid three-
body problem.