I know this question may be a bit more suitable for physics stackexchange, but I'm gonna make a try here to see if you like it.
It is well known that the trajectories of one body attracted by another in space are usually elliptic (periodic planetary motions) or hyperbolic (entry and leave like a lone comet entering solar system and gets slung out again).
Let us play with the thought that we instead have a body which leaks a small percentage of its mass each unit of time. For example if it is a gas planet which does not quite have enough gravity to keep all the mass in.
In such a scenario, what would the resulting equations of motion be and what would the solutions be? I suppose differential calculus and differential equations can be of a big help, but I have not derived anything specific.
Own work
Assuming the mass of the bigger body is so large so it stays fixed, the gravity force on the smaller body will be:
$${\bf F} = G\frac{m_1m_2({\bf v_1-v_0})}{|{\bf v_1-v_0}|^3}$$
Where $\bf v_0$ is a constant vector and $\bf v_1$ is a function of time.
Now using Newtons law of motion ${\bf F} = m{\bf a}$: $${\bf a_1}m_1 = G({\bf v_1-v_0})\frac{m_1m_0}{|{\bf v_1-v_0}|^3} \Leftrightarrow {\bf a_1} = G({\bf v_1-v_0})\frac{m_0}{|{\bf v_1-v_0}|^3}$$
It seems to me that $\bf a_1$ is independent of $m_1$, meaning that even if the object leaks mass, it's acceleration (and therefore motion) will remain unchanged. Am I missing something important here or is it a correct observation?
Edit 2: As mentioned by @GPhys, the formula ${\bf F} = m{\bf a}$ is not valid for when mass changes over time, so my conclusions above are probably not valid. However.. maybe we can use this other formula ${\bf F} = \frac{\partial {\bf p}}{\partial t}$ to get ahead?