Knowing the orbit does not yet say where the body is on the first of March. The calculation from time to position remains.
On a circle it is immediate. The motion is uniform, so the angle goes as the time: a quarter of a year gives degrees.
On an ellipse it does not. The body moves fast at perihelion and slowly at aphelion, so the angle does not advance evenly. Mark it at equal intervals of time and the marks spread out near perihelion and crowd near aphelion.
What goes as the time is not the angle but the area swept, as the second law says. That is the handle to take hold of.
Draw a circle that just encloses the ellipse, of radius equal to the semi-major axis. It is called the auxiliary circle, and the ellipse is that circle squashed vertically.
Take the point where the auxiliary circle sits directly above the planet. The angle to that point, seen from the centre of the ellipse, is the eccentric anomaly .
The angle to the planet itself, seen from the Sun, is the true anomaly . That is the angle that says which way the planet actually lies.
Finally, invent an angle that does go exactly as the time: degrees to a lap, advanced by the fraction of the period elapsed. That is the mean anomaly , and it is the first thing a time gives you.
Rewrite the swept area in terms of the eccentric anomaly. It becomes a sector of the auxiliary circle less a triangle, giving a quantity proportional to .
The swept area also goes as the time, and the angle that goes as the time is . Setting the two equal gives .
This is the equation of Kepler. A time gives , this equation gives , and gives and the distance. That is the route to a position.
But the equation cannot be rearranged to give on its own, because appears both inside and outside the sine. Kepler stopped at the same place four centuries ago, and no closed form has been found since.
Unsolvable is not the same as uncomputable. Start from some value and correct it by the error.
Compute . If it is , the value was right. If not, draw the tangent there and take where it crosses the axis as the next value.
Each round roughly doubles the number of correct digits. Three or four are enough for any precision that is wanted.
For comets with an eccentricity very close to the convergence can slow. Then the starting value is chosen differently, or the equation is rewritten. Four centuries have produced a great many ways of solving this single line quickly.
The toolkit is complete. Hand over the six orbital elements and a time: follows, the equation of Kepler gives , gives and the distance, and the three angles place it in space.
Positions of asteroids and of spacecraft are found along this route. So, at bottom, are the planetary positions printed in an almanac.
The calculation contains no perturbations. It is the two-body answer, with the pull of every other body ignored. That will do over short spans and will not do over long ones.
That closes the second chapter. In the next we set about changing the orbit itself.