Mr. Missed Her
, I'm afraid this may be disappointing, but simply using the equation for an ellipse will not help you determine the position as a function of time for an orbit, because the speed is not constant but rather must obey the vis viva equation
. In fact, there is no algebraic expression which will do it. The solutions to the equation of motion for a particle in a gravitational well are transcendental
, meaning they cannot be written as a finite combination of elementary functions. So this is a fairly advanced problem, and a topic of quite some length in celestial mechanics.
To give a hint to the scope of it, I'll briefly overview a standard approach, which is to move to polar coordinates (problems involving radial forces become much easier to analyze in polar coordinates), and solve for the radius as a function of polar angle. The relation between radius and time for a particle in a gravitational field is given by a differential equation:
where the first (negative) term is the gravitational force, the second term is the centrifugal force, l is the angular momentum and [math]
is the reduced mass.
Using the substitution u=1/r and transforming time into polar angle (and skipping all the steps therein), we obtain
which is a differential equation which is much more readily solvable. Transforming the solution back to r(ф),
I won't prove this, but it turns out that this ε is in fact the eccentricity of an ellipse, or a conic section in general. So the orbit is a conic section, and we can determine its exact shape in terms of the masses, gravitational constant, eccentricity, and angular momentum.
Finally, to bring time back into the picture, we relate the angle (or "mean anomaly") with respect to perihelion, back to the other parameters and initial conditions, and essentially are solving r(ф(t)).
So, hopefully that gives some intuition to the problem and how to approach it, though my explanation certainly is not thorough enough to be workable. If this is something you'd like to pursue more deeply or apply to a project, there are some good resources available out there which can help guide you through it.