Solar System Perturbations

Remarkable progress in understanding the dynamics of the planets has been possible thanks to their relatively small masses and the overwhelming dominance of the Sun. The figure below shows the relative masses of the Sun, planets and some natural satellites, taking the mass of Earth to be unity.

The Sun is one million times more massive than Earth. Jupiter, the largest planet, is three orders of magnitude smaller than the Sun and three orders larger than Earth. In a logarithmic sense, it is mid-way between Earth and Sun. The Moon is two orders lighter than Earth.

Intelligent Design or Natural Selection?

The problem of two massive bodies, like the Sun and Earth, gravitationally attracting each other was solved completely by Newton. The orbits are elliptical, with the common centre of mass at a focus of the ellipse. With three or more bodies, the problem becomes vastly more complex, and is an ongoing area of research. It is only in very special cases that a solution is possible.

But the configuration of the solar system is such that dramatic simplifications are possible. It is almost as if a beneficent Creator had arranged for a configuration of the solar system that is within humankind’s analytical abilities.

It is by no means trivial to deduce the dynamics of the system, but the problem is not completely beyond our limited mental capacity. Of course, if the planets were larger, with more eccentric orbits, chaos would reign and the evolution of intelligent life would have been unlikely if not impossible.

Two Bodies

Newton’s law of gravitation gives the force between two bodies, of masses m_1 and m_2, separated by a distance r ,

$\displaystyle F = G \frac{m_1 m_2}{r^2}$

where G is the universal gravitational constant. This is the inverse square law of attraction. It is interesting that the force exerted by the Earth on the Sun is precisely equal in magnitude to that of the Sun on the Earth. However, recalling Newton’s law of motion,

F = m a or a = F / m

we see that the acceleration of the Sun is one millionth that of the Earth, since m(EARTH) / m(SUN) is one millionth. Thus, we can treat the Sun as stationary. Earlier, it was Kepler’s deduction from observations that the planetary orbits are ellipses with the Sun at a focus. This is an excellent approximation to reality.

Perturbations

To some degree, each planet is attracted by all the others in addition to the Sun’s attraction. The main effect for the Earth’s trajectory is due to Jupiter. However, since Jupiter is never less that 1AU from Earth (one astronomical unit is the mean Sun-Earth distance), and is one thousand times lighter, its gravitational attraction is minute compared to that of the Sun. Thus we can treat the effect of Jupiter as a small perturbation of Earth’s elliptical orbit. This hugely simplifies the mathematical analysis.

Many other simplifications are possible thanks to the small deviations from ideal cases. Almost all the mass of the solar system is contained in the Sun: the centre of mass of the system is deep within the solar sphere. Most orbits are close to circular, so the eccentricity may be treated as a small parameter. Moreover, all the orbits lie close to a plane, the ecliptic plane.

The Motion of the Moon

Lunar theory has challenged astronomers for centuries. Newton remarked that the motion of the Moon was the one problem that caused him severe headaches. The relative attraction of the Sun and Earth on the Moon is

$\displaystyle \frac{m_{\rm SUN}}{m_{\rm EARTH}}\times\left(\frac{r_{\rm EARTH}}{r_{\rm SUN}}\right)^2 =10^{6}\times\left({\frac{400,000}{150,000,000}}\right)^2 \approx 7$

Thus, the Sun attracts the Moon more strongly than does the Earth. However, the accelerations of the Earth and Moon due to the Sun are close to equal, as they are at comparable distances from the Sun. The difference is a small perturbation, which greatly facilitates the mathematical analysis: we can regard the Earth-Moon system as a single body orbiting the Sun. Higher-order refinements come later!

Finally, we mention the case of an artificial satellite travelling between the Earth and Moon. Its mass is so tiny compared to that of the Earth and Moon that its gravitational effect on them is completely negligible. Once the motions of the Earth and Moon are known, the trajectory of the satellite can be calculated by treating it as a passive test-mass moving within their gravitational potential.

ThatsMaths