Kepler's ODE: Solar Coordinates

Let (rho, nu) and beta be as developed above. Then as we have seen,

[1] rho*cos(nu) = e + cos(beta)
[2] rho*sin(nu) = b*sin(beta)
[3] rho ~= 1 + e*cos(beta)
[4] nu-dot = K/rho2 (K a constant)
These equations mix solar coordinates (rho, nu) and the Kepler angle, beta. If we may express rho as a function of nu only, for points on the ellipse, then [4] may be transformed into a single first order autonomous ODE for nu, which will determine the motion of the planet along the ellipse,
[7] nu-dot = K* (e*cos(nu) - 1)2 / (e2 - 1)2 = K* (e*cos(nu) - 1)2 / b2

Derivation of [7]

But [3] gives rho as a function of beta alone. So let us use [1] to replace beta by nu in [3].

Multiply [1] by e and replace the term e*cos(beta) on the right by (rho - 1) according to [3], obtaining,

[5] e*rho*cos(nu) = e2 + (rho - 1)
Now solving for rho in [5], we have,
[6] rho = (e2 - 1)/(e*cos(nu) - 1) = b2/(1 - e*cos(nu))
as b2 = 1 - e2. This is the parametric equation for the ellipse in solar coordinates. Substituting this value for rho in [4], we have,
[7] nu-dot = K* (e*cos(nu) - 1)2 / (e2 - 1)2 = K* (e*cos(nu) - 1)2 / b2
This is Kepler's ODE in solar coordinates. Solving [7] for nu, we may use [6] to find the corresonding value of rho.
Revised 31 dec 2001 by ralph abraham