We are considering a point mass $ M $ through a large cloud of dust with mass density $ \rho $. As the point mass moves, it leaves a higher density trail in its wake. This over dense region exerts gravitational force on the point mass and slows it down. In this page we will estimate that force.

Suppose the point mass moves at a velocity $ v $. After time $ \Delta t $ the mass would have moved a distance $ \Delta r = v \Delta t $. The mass of the excess dust concentrated where the point mass used to be is all the dust whose free fall time is below $ \Delta t $. The radius of the sphere from which dust could have come is

$ \Delta t \approx r_{ff} / \sqrt{\frac{G M}{r_{ff}}} \Rightarrow r_{ff} = \left( G M \Delta t^2 \right)^{1/3} $

The excess mass of the collected dust is

$ \Delta m \approx r_{ff}^3 \rho = \left(G M \Delta t^2 \right) \rho $

And the force that it exerts on the point mass is

$ F_{DN} = -\frac{G M \Delta m}{\Delta r^2} = -\frac{G^2 M^2 \rho}{v^2} $