To find the mass and radii of hill sphere of a planet after passing through a region of planetary nebula Consider a planet with initial mass m* kg and radii of hill sphere as r*. 

The  planet it traveling (assume) perpendicular to the nebula at a speed of 13km/s and the nebula is approaching the planet at 15 km/s. 



The nebula has a density of D kg/km^3.

The planet travels through this region for 1.4yrs in which it covers 570.5 million km.  

Is is possible to calculate the mass and volume of hill sphere of the planet at the end of 1.4 yrs? If so how?
The problem I face is that with gain in mass the planets radii of hill sphere increases, hence more mass falls in which in turn again causes the hills spheres radii to increase, this process goes on and on till the end . I'm struggling to imply integration!

(In the hill sphere formula you can consider a and M to be constants)


Thank you!
I have come up with a technique to find a solution to this problem as shown in this paper
A Study to Show : Jupiter : Our Next Star 
My 11th grade research paper 2018