Everything so far has started from known orbital elements. Where do those six numbers come from?
A telescope returns a direction. It says where in the sky the object lies and says nothing about how far away it is.
Anything along that line would look the same. Near and small, or far and large, cannot be told apart from a single observation.
How to get an orbit out of data containing no distance: that question is more than two centuries old.
Six numbers are wanted, whether as orbital elements or as a position and velocity at one time.
A night of observation gives a direction, which is two angles. Three nights give six. Exactly enough.
Having the right count is not the same as having a solution. The six relations are tangled together and do not fall out by substituting one into the next.
Still, three nights suffice. What remains is to untangle them.
The opening is that the three positions must lie in one plane, since the force points at the Sun and the orbit cannot leave its plane.
Lying in one plane, the middle position can be written as a weighted sum of the outer two, and the weights follow from the ratio of the areas swept by the line to the Sun. That is the second law.
Over a short span, the ratio of swept areas is approximately the ratio of the elapsed times. Here, for the first time, something enters that the directions alone could not supply.
Solving that relation together with the observed directions fixes the distance on each of the three nights. Distances give positions, two positions give a velocity, and from those all six elements follow.
On January Piazzi found a new object, later named Ceres.
He followed it for days before it passed behind the Sun and was lost. It would not reappear for nearly a year, and unless someone could say where, it would never be found again. By the methods of the day, days was far too short an arc.
Gauss, then , computed an orbit from those days, using along the way the method of least squares. His prediction differed sharply from everyone else.
On the last night of that year, Zach pointed a telescope where Gauss said, and there it was. That calculation is what made his name.
The procedure has not changed. Every newly found asteroid gets an orbit computed from a few nights of observation.
What has changed is the number and precision of the observations. A first orbit comes from three nights, and it is recomputed as more arrive. The more there are, the narrower the range of orbits that fit.
For objects that come near the Earth, that range is the estimate. Freshly discovered, many orbits still fit, and a nonzero chance of impact can appear. As observations accumulate and the range narrows, it almost always settles at zero.
That closes the second chapter. Elements to describe an orbit, a time to give a position, observations to give the elements. The tools are in place. In the next chapter we set about changing the orbit itself.