Refining the First Law
Kepler's first law was:
"The orbit of a planet is an ellipse, with the Sun at one focus"
Actually, this is not 100% accurate. Imagine the planet magically getting heavier and heavier, while the Sun is getting lighter and lighter. At some point both of them would be equally heavy: could we then say which is orbiting around which?
To be fully accurate, the first law should have placed the focus of the orbital ellipse at the center of gravity of the planet-Sun system. (The center of gravity will be defined later, but intuitively, if the masses are very unequal, as with a planet and the Sun, it lies close to the center of the heavier object.) Because the Sun is so much heavier than the Mars, the effect on the orbit of Mars (which Kepler studied) was too small to be noted by him. Nevertheless, the Sun also moves in response to motions of its planets, and motions of this type have become an important tool in the search for planets outside the solar system.
An Earth-sized planet orbiting a distant star would be far too dim to be seen with any earthly telescope, especially against the glare of its sun, the star itself. However, as the planet goes around the orbit, its star also moves in a mirror image orbit around the common center of gravity. It is a much smaller orbit and a much slower motion, because the center of gravity is very close to the center of that star (in the Earth-Sun system, it is inside the Sun), but it can still be detected by subtle variations of the starlight.
Recently a few such planets have been found, but most are Jupiter-sized and none seems suitable for life. The search, however, goes on, and you can find out about its current status (and much more) on the Planet Quest web site of NASA's Jet Propulsion Lab (JPL). One recent discovery (15 April 1999) has been the system of upsilon Andromeda which appears to contain at least 3 planets. For another recent discovery associated with a distant planet, click here.
Bodies in the solar system can also move in other conic sections, in parabolas or hyperbolas, whose equations resemble that of an ellipse, but have e equal to 1 or larger. Properly speaking, these orbits do not have a "semi-major axis," because they are not limited in size like ellipses but extend to infinity. Instead, we must write
r = p/(1 + e cos f)
which doe not raise any problems at e=1. Such bodies are not bound to the Sun, but are free to escape it. The denominator in the equation of the trajectory then becomes zero for some values of f, making its r infinite, and as the moving body approaches those values, it moves further and further away, with no limit. Comets in general have an eccentricity e close to 1, suggesting they have come from the very distant fringes of the solar system. The space probe Voyager 2 has e > 1 and is on its way out of the solar system, never to return.
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