How to calculate an elliptical orbit angle based on trajectory time?

Question

Given state variables are:
time, t
orbital period, T
elliptical eccentricity, e
gravitational constant, u (mu)

need to find angle to central mass, Theta as a function of time.

Answer 1

First, use the above to find the semi-major axis(a) of the orbit:

T = 2*PI*(1/(mu^(1/2)))*a^(3/2)

Once you have "a", you can find the "mean motion" (n).

n = (mu/a^3)^(1/2)

After you've got Mean Motion (n), you use Kepler's equation to find the Eccentric Anomaly (E).

n*(t-T) = E - e*sin(E)

Now compute the True Anomaly (v) from:

cos(E) = (e + cos(v))/(1+e*cos(v))
(Note that v and E are always in the same "half-plane". I.E. when E is between zero and PI, so is v.)

This should give you a full set of the classic orbital elements.

Use the above variables to find the current radius vector (Rbar) and velocity vector (Vbar) of the body with respect to (WRT) the central mass as a function of time.

After you've done that, the "specific angular momentum" of the body WRT the central mass is given by the Hbar vector which is simply the cross product of Rbar and Vbar:

Hbar = Rbar X Vbar

Use the magnitude of the above vectors H, R and V in the following equation to get the "flight path angle" (gamma) WRT the central body.

h = r*v*sin(gamma)

Answer 2

There is no simple closed form solution to your problem. A procedure for the solution follows (all angles are in radians):

Theta is the angle from perihelion (closest approach to the Sun) to the current position

M is the Mean Anomaly which is the Theta angle that the planet would achieve if it were in a circular orbit of the same period as the planet of interest.

e is the eccentricity.

First, solve the Equation of Kepler:

E = M + e*sin(E)

You solve for E by iteration - if you need more on how to do that, say so. Once you know E:

Theta = 2 atan(sqrt((1+e)/(1-e))*tan(E/2))

Of course, the arctangent function has the usual problems with picking the right quadrant. For typically small eccentricities, M and E normally lie in the same quadrant.

Answer 3

I do it this way.

Assume an orbit in the X-Y plane with the Sun at 0,0.

Start the planet at any x,y with any velocity vx,vy.

Compute distance to the Sun as r=SQRT(x*x+y*y).

Compute acceleration of gravity as ax=-x/r/r/r and ay=-y/r/r/r.

Compute new velocity vx=vx+ax and vy=vy+ay.

Compute new position x=x+vx, y=y+vy.

Compute a new r and do it all again. Repeat forever.

This will give you some kind of orbit that depends on the starting position and velocity, but doesn't really make it easy to figure out what particular starting conditions you need for a particular period and eccentricity. You can try different values and by trial and error get pretty much any period and eccentricity you want though. And playing with it is fun, especially of you plot the changing position in real time as the program runs. Don't even think about doing this by hand, it takes thousands of steps to complete one orbit. After you get an orbit you like, the angles and distances at any time are easy to compute.