Solar System Dynamics

Science is, after all, an attempt to make sense of the world around us.

§1 Structure of the Solar Systems

Kepler’s law
Newton’s law
Titius-Bode “Law”: The mean distance $d$ in AU from the Sun to each of the six known planets was well approximated by the equation:

d = 0.4+0.3(2^i),\ \textrm{where}\ i=-\infty,0,1,2,4,5

Planet	i	a(AU)	TB Law(AU)
Mecury	-inf	0.39	0.4
Venus	0	0.72	0.7
Earth	1	1.00	1.0
Mars	2	1.52	1.6
Ceres	3	2.77	2.8
Jupiter	4	5.20	5.2
Saturn	5	9.54	10.0
Uranus	6	19.18	19.6

Resonance
1. spin-orbit coupling:
  1. Moon (1:1)
  2. Mercury (3:2)
2. orbit-orbit coupling:
  1. Jupiter and Saturn (5:2)
  2. Neptune and Pluto (3:2)
  3. Jupiter System: Laplace relation: $n_{\mathrm{I}} - 3n_{\mathrm{E}} + 2n_{\mathrm{G}} = 0, n = 360/T$ , prevent the triple conjunctions of the threee satellites. When a conjunction takes place between any pair of satellites, the third satellite is always at least 60 degree away
  4. Saturn System: the widest variety of resonant phenomena.
  5. Uranus System
  6. Neptune System
  7. Pluto System
  8. Asteroid Belt
Commensurability

\frac{n_1}{n_2}\approx\frac{i_1}{i_2}

where $n_1$ and $n_2$ are the mean motions of the two objects, using integers $i_1,i_2 \in \{1,2,\dots,i_{\textrm{max}}\}$ with $i_1<i_2$ and $i_{\textbf{max}}=7$ but excluding the case $i_i=i_2=1$ .

However, the observed near-commensurability in the solar system are all of the form

\frac{n_1}{n_2}\approx\frac{p}{p+1}

Recent Developments
1. Why are there pronounced gaps at most of the major jovian resonances in the asteroid belt but a clustering of asteroids at the 3:2 resonance?
2. Where do short-period comets come from?
3. Why do the orbital elements of some groups of asteroids have common values?
4. Why are there numerous resonances among the satellites in the jovian and saturnian systems but none in the Uranian system?
5. Why are the eccentricities and inclinations of some satellite orbits too large to fit in with current understanding of tidal evolution?
6. What produced the Cassini division in Saturn’s rings?
7. How are narrow rings maintained despite the spreading effects of collisions and drag forces?
8. Are planetary rings transient phenomena or can they survive for billions of years?

§2 The Two-Body Problem

Introduction: The interaction of two point masses moving under a mutual gravitational attraction described by Newton’s universal law of gravitation
Equations of Motion

The The gravitational forces and the consequent accelerations experienced by the two masses are
$\mathbf{F_1} = +\mathit{G}\frac{m_1m_2}{r^3}\mathbf{r} = m_1\mathbf{\ddot{r}_1},\ \mathbf{F_2} = -\mathit{G}\frac{m_1m_2}{r^3}\mathbf{r} = m_2\mathbf{\ddot{r}_2}$
where $\mathbf{r}=\mathbf{r_2} - \mathbf{r_1}$ denotes the relative position of the mass $m_2$ with respect to $m_1$ . Thus
$m_1\mathbf{\ddot{r}_1} + m_2\mathbf{\ddot{r}_2} = 0$
which can be integrated directly twice to give
$\begin{aligned} m_1\mathbf{\dot{r}_1} + m_2\mathbf{\dot{r}_2} &= \mathbf{a},\\ m_1\mathbf{r_1} + m_2\mathbf{r_2} &= \mathbf{a}t+\mathbf{b}= \mathbf{a},\\ m_1\mathbf{r_1} + m_2\mathbf{r_2} &= \mathbf{a}t + \mathbf{b} \end{aligned}$
where $\mathbf{a}$ and $\mathbf{b}$ are constant vectors. With substitution of the centre of mass’s position vector $\mathbf{R} = (m_1\mathbf{r_1} + m_2\mathbf{r_2}) / (m_1 + m_2)$ , the eqution above can be written as
$\mathbf{\dot{R}} = \frac{\mathbf{a}}{m_1 + m_2},\ \mathbf{R} = \frac{\mathbf{a}t+ \mathbf{b}}{m_1 + m_2},$
implying that either the centre of mass is stationary (if $\mathbf{a}=0$ ) or it is moving with a constant velocity in a straight line with respect to the origin $\mathit{O}$ .

Now consider the motion of $m_2$ with respect to $m_2$ , the equation of relative motion:
$\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d} t^2} + \mu\frac{\mathbf{r}}{r^3} = 0$
where $\mu = \mathit{G}(m_1+m_2)$ . Taking the vector product of it and we have $\mathbf{r}\times\mathbf{\ddot{r}}=0$ , which can be integrated directly to give the angular momentum integral
$\mathbf{r}\times\mathbf{\dot{r}}=\mathbf{h}$
Now move to polar coordinates, the position, velocity, and acceleration vectors can be written as
$\mathbf{r} = r\mathbf{\hat{r}},\ \mathbf{\dot{r}}=\dot{r}\mathbf{\hat{r}} + r\dot{\theta}\mathbf{\hat{\theta}},\ \mathbf{\ddot{r}}=(\ddot{r}-r\dot{\theta}^2)\mathbf{\hat{r}}+\left[\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d} t}\left(r^2\dot\theta\right)\right]\mathbf{\hat{\theta}}$
hence the angular momentum
$\mathbf{h}=r^2\dot{\theta}\mathbf{\hat{z}}$
The area swept out by the radius vector in time $\delta t$ is
$\delta A \approx\frac12 r(r+\delta r)\sin\delta\theta\approx\frac12 r^2\delta \theta$
by dividing each side by $\delta t$ and taking the limit as $\delta t \rightarrow$ we have
$\frac{\mathrm{d}A}{\mathrm{d} t} = \frac12 r^2 \frac{\mathrm{d}\theta}{\mathrm{d} t} = \frac 12 h$
Since $h$ is a constant this implies that equal areas are swept out in equal times. Note that this does not require an inverse square law of force, but only that the force is directed along the line joining the two masses.
Orbital Position and Velocity