# virial theorem in clusters

19 Dec 2012What would be the virial theorem for a different gravitational law, i.e. \(\Phi\propto 1/r^n\) instead of the Newtonian potential? Could you explain Zwicky’s conjecture for the presence of dark matter in a galaxy cluster just by postulating a different value for \(n\) in the potential while assuming that only light-producing matter is gravitating, and what would that slope need to be?

bonus question: why does the covariance matrix \(C\) of a multivariate Gaussian probability density \(p(\vec{x}) = \frac{1}{\sqrt{(2\pi)^n\mathrm{det}(C)}}\exp\left(-\frac{1}{2}\vec{x}^tC^{-1}\vec{x}\right)\) have to be positive definite (2 reasons)?

fun bonus question: aren’t the factorials somehow the opposites of prime numbers?