So far the Sun has been treated as standing still. That is not right.
As the previous article showed, if the Sun pulls the planet then the planet pulls the Sun just as hard. Anything that is pulled moves. The Sun moves.
The swing goes inversely as the mass. The Sun is about times heavier than Jupiter, so it swings about a thousandth as far.
That is not zero. The wobble Jupiter imposes on the Sun carries it a little beyond its own surface.
The two bodies go round a common barycentre. Where it sits is fixed by the ratio of the masses alone, and it lies nearer the heavier one.
The Moon is an st of the Earth, so the barycentre is km from the centre of the Earth: inside it. The Earth does wobble under the pull of the Moon, but the wobble stays within itself.
For Pluto and Charon the ratio is nearer an eighth, and the barycentre lies out in the open between them. Saying that each circles the other comes closer to the truth.
With no outside force, the barycentre itself does not move. What moves is the pair going round it.
Stop following the two positions separately and follow only the separation between them.
The problem becomes a single body going round a centre that does not move. The mass of that body is neither of the two but , the reduced mass.
Because of that substitution, every answer obtained so far survives intact. The ellipse, the law of areas, the shape that came out of the inverse square: nothing has to be given up.
When one body is far heavier, is very nearly the mass of the lighter one. For the Sun and the Earth the difference is three parts in a million, so holding the Sun still causes no trouble worth mentioning.
This view turns straight into an instrument. The planet need not be visible: if the wobble of the star is visible, a planet is there.
As the star approaches and recedes, the wavelength of its light shifts slightly. Read the shift and you have the velocity along the line of sight. That velocity repeats as a wave, and its period is the orbital period of the planet.
The height of the wave tells how heavy the planet is, since a heavier one swings the star further.
In such a wave was found for Pegasi: a planet about the size of Jupiter, going round in days. It was the first planet found around another ordinary star.
The two-body problem can be solved in closed form. Give the initial positions and velocities and where the bodies will be is a matter of writing a formula.
It comes out because enough quantities are conserved. The energy does not change, the angular momentum does not change, and there is a further quantity that keeps pointing at the perihelion without turning.
That third one is why orbits close and why the ellipse has a fixed orientation. When the previous article said that only an inverse square closes, this is the reason: only for an inverse square is that quantity conserved.
This is the quietest part of celestial mechanics. Add a third body and the quiet ends at once, which is the subject of the fourth chapter.