I was wondering if there is an established method to keep track of the orbit of an exoplanet assuming we know a - the semi-major axis of the orbit, e - the eccentricity of the orbit, and i - the inclination of the orbit. Can we track its position once a month?
To describe the position of an orbiting body you need 6 numbers. There are different ways to do this:
You can give the position $(x,y,z)$ and velocity $(\dot x, \dot y, \dot z)$. At a given time $t_0$, and then use Newtons laws to work out the position of the planet at any time in the future.
You can give the orbital parameters:
- Eccentricity (the shape of the ellipse)
- semi-major axis (the *size of the ellipse)
- (inclination, Longitude of ascending node, argument of periapsis) The 3d position of the ellipse. These depend on the particular choice of reference frame. For exoplanets it is common to take the reference plane to be the the plane through the star and perpendicular to the line of sight from Earth. This means that many exoplanets have an inclination of about 90 degrees.
- Mean anomaly at epoch (The position of the planet on the ellipse at time $t_0$)
And then use Keplers laws to work out the position of the planet at any time in the future.
So we can track them theoretically.
However determining these six values from observations is not easy, and in many cases we don't know all six at all well. If we can detect a transit we know that the inclination is close to 90, and we might be able to combind transit observations with radial motion observations to get some limits on the other values. Nevertheless, these six values can't be simply measured from the data.
For most exoplanets they can't be directly imaged (they are too close to the glare of the star, and to dim to be seen) There are a few exoplanets that can be tracked They are usually very large, warm planets that are far from their star.