# Finding the differential function for mass gain over time

1. Mar 31, 2010

### diagopod

This is simply something I've trying to figure out for a long time, not an actual homework question, but felt this was the best place for it.

1. The problem statement, all variables and given/known data

A small gravitational object is surrounded by an expanse of hydrogen in otherwise flat empty space.

At any given instant, due to its gravitation, the object is attracting and absorbing the surrounding hydrogen gas at a particular rate (kilograms/second), presumably a simple function of g and the mass density of the hydrogen in its immediate surroundings.

As a result of the object's continual absorption of the gas that surrounds it, the object is gaining mass over time.

As a result of the object gaining mass over time, its gravitational strength is increasing with time.

Given its current mass and the current mass density of the hydrogen in its surroundings, find the object's approximate mass at any given point in its past, assuming that all of its mass has come from attracting and absorbing the surrounding hydrogen gas.

2. Relevant equations

To calculate the current rate of mass gain I can take the current mass, find g and the free fall velocity at the surface of the object, calculate the volume of surrounding gas "falling" into the object per second, then multiply that times the total mass density of the hydrogen (including both the kinetic energy of the gas and its mass density?).

From there, I'm at a loss.

3. The attempt at a solution

To calculate the mass, g and rate of mass-gain at a given point in the past, I would assume I would start with a graph of a differential function with time on one axis and mass on the other. But not sure what the actual differential function would be that I would be plotting.

I'm also not sure if inevitable counterpart increases in the mass-energy density of the surrounding hydrogen gas over time, as the objects mass and g increases, make the problem extremely complicated or not.

It would seem that this might be related, or even the same as, simple models of star growth out of gas, but not sure of that either. Any guidance would be greatly appreciated.

Last edited: Mar 31, 2010