PC1672 Advanced dynamics

3.8 The solar system

To a very good approximation, objects in the solar system move in a fixed inverse-square-law force provided by the gravitational field of the Sun.

The planets follow elliptical orbits with small eccentricities. For example, Mars has $\epsilon\simeq 0.1$ . An ellipse with a small eccentricity is very nearly circular in shape, but the centre of force is displaced from the centre by $\epsilon a$ . Deviations from Kepler's laws are very small, 0.1-0.2%. The reduced-mass correction is most significant for Jupiter since it has the largest mass and so the Sun's recoil is most noticeable. Planets which are close to Jupiter, in particular Mars, have their orbits perturbed by the larger planet. Even when all of the other planets are taken into account, there is still a minor discrepancy in Mercury's orbit.

At the other extreme, long-period comets have highly elliptical orbits with eccentricities of between about 0.9 and 1.

Astronomers usually quote distances within the solar system in astronomical units (AU). The AU is defined as the semimajor axis of the Earth's orbit: $1\ \hbox{\rm AU}=150\times 10^8\ \hbox{\rm km}$ . The semimajor axes of the planets orbits range from about 0.4 AU for Mercury to nearly 40 AU for Pluto. Long-period comets tend to have semimajor axes that are hundreds of AU long.

Similarly the year provides a convenient unit of time. In these units the constant in Kepler's third law is simply

$\begin{displaymath}{2\pi\over\sqrt{GM_{\rm Sun}}}=1\ \hbox{\rm yr AU}^{-3/2}\end{displaymath}$

Orbital periods range from about a quarter of year for Mercury to 250 years for Pluto, and they can be thousands of years for comets.

Mercury: The direction of the axis of Mercury's orbit is not fixed in space (as it should be for a pure inverse-square-law force) but precesses by 575 arcseconds in a century. This about 40 arcseconds more than can be explained by taking into account the effects of the other planets. Suggestions of another planet even closer to the Sun, a dust cloud or significant deviations from a spherical shape for the Sun were ruled out by observations. General relativity leads to corrections to Newton's inverse-square law in strong gravitational fields. Einstein was able to show that these were just what was needed to explain the behaviour of Mercury.

Textbook references

K&K 9.6
F&C 6.6
B&O 5.3
M 8.7, 8.8 table 8.1
M&T 8.7

Mike Birse
17th May 2000