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I would like to calculate the potential energy of a particle which is within a big dark matter halo. The Newton shell theorem says that if I have a particle within a spherical mass distribution (let us say at radius r) the mass with R>r exert no net Force to the particle. But there is a constant potential...

phi(r) = -G*M/R

M is the mass outside r and R is the radius of the dark matter halo. My first problem is that the dark matter halo does not have a sharp edge so what is R? Furthermore the density is not constant -> rho(r) and therefore I assume that the formula above should be modified in my case to

phi(r) = -G *int^infinity_r M(r)/r dr

is this correct???

The potential within r should be

phi(r) = -G*M(r)/r

is this correct?

So I just have to add both components and this should give the net potential?

I am thankful for any advice or comment

best regards

florian