We are given velocity, direction of velocity, distance between the two bodies and the mass of massive body. The effect of the mass of the orbiting body can be ignored.
I am looking for parametric equations with inputs of velocity, angle, distance, mass and time which return the 2D coordinates of the orbiting body, in the form:
\begin{align} x& = f_x(v,a,d,M,t) \\ y& = f_y(v,a,d,M,t) \end{align} I have had my own attempt.
let $A$ = semi-major axis, $$ A = \left(\frac{v^2}{GM} + \frac2d\right)^{-1} $$
let $B$ = semi-minor axis, $$ B = \cos(a) v r\sqrt{\frac{\vert A\vert}{GM}} $$
where $G$ = gravitational constant
\begin{align} x &= A \sin(t)\cos(a) - B\cos(t)\sin(a)\\ y &= A\sin(t)\sin(a) + B\cos(t)\cos(a) \end{align} It doesn't work. For example, it is not circular when $v = \sqrt{GM/d}$
Do such equations exist?
