( Update 2019: The information that used to be in this post was somewhat outdated. Please see this post instead.)
Back in 2013 I wrote en essay I tried to get published, it is available here:
Newtonian Gravitation with Radially Varying Velocity-Dependent Mass
The whole idea behind the project was to use the Newtonian law of gravity but letting the mass vary implicitely with time and getting to something very similar to the Schwarzschild solution.
As of 2019 I am using this expression:
which is a tiny bit different from the expression I use in my old essay. Using this expression you can, by numerical integration get orbits such as this one:
For more orbits and explanations see this post.
(I have an older essay that I called ”Another road to Schwarzschild”)