Radial Kepler equation

Description

In orbital mechanics, Kepler’s equation relates various geometric properties of the orbit of a body subject to a central force.

It was first derived by Johannes Kepler in 1609 in Chapter 60 of his Astronomia nova, and in book V of his Epitome of Copernican Astronomy (1621) Kepler proposed an iterative solution to the equation.The equation has played an important role in the history of both physics and mathematics, particularly classical celestial mechanics.

There are several forms of Kepler’s equation. Each form is associated with a specific type of orbit. The standard Kepler equation is used for elliptic orbits (0 ≤ e < 1). The hyperbolic Kepler equation is used for hyperbolic trajectories (e ≫ 1). The radial Kepler equation is used for linear (radial) trajectories (e = 1). Barker’s equation is used for parabolic trajectories (e = 1).

When e = 0, the orbit is circular. Increasing e causes the circle to become elliptical. When e = 1, there are three possibilities:

a parabolic trajectory, a trajectory going in or out along an infinite ray emanating from the centre of attraction, or a trajectory that goes back and forth along a line segment from the centre of attraction to a point at some distance away.

A slight increase in e above 1 results in a hyperbolic orbit with a turning angle of just under 180 degrees. Further increases reduce the turning angle, and as e goes to infinity, the orbit becomes a straight line of infinite length.

The Radial Kepler equation is shown here, where t is proportional to time and x is proportional to the distance from the centre of attraction along the ray.

Related formulas

Variables

tproportional to time (dimensionless)
xproportional to distance from the centre of attraction along the ray (dimensionless)