If the difference ψ − ωt is represented by ρ, Kepler’s equation can be written Successive...

If the difference ψ − ωt is represented by ρ, Kepler’s equation can be written

Successive approximations to ρ can be obtained by expanding sin ρ in a Taylor series in ρ, and then replacing ρ by its expression given by Kepler’s equation. Show that the first approximation by ρ is ρ1, given by

and that the next approximation is found from

an expression that is accurate through terms of order e4.