Calculating Transit Windows

Assume I have a planet with a known orbital period P, a known periastron ephemeris T_peri in JD, a known eccentricity e, and a known stellar radius R_star. How would I go about estimating when a possible transit would occur (T_transit)? (we're assuming b = 0).

The transit occurs when the true anomaly ν = π/2 − ω (working in radians)

Convert the true anomaly to the eccentric anomaly E using

E = atan2(√(1−e^2) * sin(ν), e + cos(ν))

where atan2(y, x) is the angle coordinate of the point (x, y). (Ensure you check which way round the atan2 function you are using takes the arguments)

Convert to the mean anomaly using the Kepler equation:

M = E − e * sin(E)

The time since periastron is given by

T_transit − T_peri = M/2π * P.

Time of eclipse can be found in a similar way but setting ν = 3π/2 − ω

Thank-you for this. I will have to play around with this and see what I can do. It's much appreciated.