Calculating mean velocity of an orbiting body as it moves towards a point. I'm making a game, in the game planets orbit a central point in circular orbits, they move directly towards their targets and the vector is simply added to their orbital path. Whilst not realistic it avoids many challenges and the resultant efficiency allows me to simulate thousands of planets in real-time.
Though planets aren't affected by gravity they have a tendency to become trapped in the center when their average vectors move towards it. I want to minimize this by placing an arbitrary force outwards on the planets when they get too close to the center but to calculate the appropriate force I must determine the resultant vectors magnitude.
I created a diagram to show what I mean a bit better:


*

*p = planet orbiting

*o = orbit path

*t = target point

*v = orbit velocity

*V = velocity at which planet is moving to target

*d = distance from center to point

*D = orbit radius



I need to calculate D/time though I have no idea how to work it out. Thanks for reading and I really hope you guys can help out.
 A: After some clarifications in the comments, I now understand your question as follows: You have a planet, a centre, and a target. The velocity of the planet has two components, one given by the velocity it would have if it were on a circular orbit about the centre, and the other directed towards the target.
The rate $\dot D$ of the change of the planet's distance $D$ from the centre with time is the sum of the rates of the two changes corresponding to the two velocity components being added. The orbital velocity component is tangential to the orbit and thus doesn't contribute to $\dot D$. Thus $\dot D$ is determined by the velocity component directed at the target. This depends on the angle $\theta$ between the target $t$ and the planet $p$ as viewed from the centre $c$. In cartesian coordinates with the origin at the centre and the target on the positive $x$ axis, the target is at $(d,0)$ and the planet is at $(D\cos\theta,D\sin\theta)$. Thus the velocity component directed at the target is directed along $(d-D\cos\theta,-D\sin\theta)$. The radial component of this vector has magnitude
$$
\left(\pmatrix{d-D\cos\theta\\-D\sin\theta}\cdot\pmatrix{\cos\theta\\\sin\theta}\right)=d\cos\theta-D\;,
$$
so a velocity in this direction with speed $V$ has radial component
$$
\dot D=\frac{d\cos\theta-D}{\sqrt{d^2+D^2-2dD\cos\theta}}V\;,
$$
where the denominator is the norm of the direction vector. In your drawing you have $d\gt D$, and in this case $\dot D$ is positive for some values of $\theta$ and negative for others (as is also clear from the image).
