what 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

\begin{equation}

p(\vec{x})\mathrm{d}\vec{x} = \frac{1}{\sqrt{(2\pi)^n\mathrm{det}(C)}}\exp\left(-\frac{1}{2}\vec{x}^tC^{-1}\vec{x}\right)\mathrm{d}\vec{x}

\end{equation}

have to be positive definite (2 reasons)?

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

with that, CQW wishes all readers a happy holiday season and a successful year full of new discoveries. we'll be on vacation until 09.Jan.2013.

positive definiteness of a matrix $C$ implies positive definiteness of the inverse $C^{-1}$ (easy to see in the diagonal system), so the quadratic form $\vec{x}^t C^{-1}\vec{x}$ is always positive, meaning that the pdf is of Gaussian shape and has a finite integral, i.e. normalisation. likewise, positive definiteness of $C$ implies that the determinant is positive and it is possible to take the square root. positive definiteness of a covariance matrix for a multivariate process is guaranteed by the Cauchy-Schwarz inequality.

ReplyDeletefun bonus question: with that, the number 2 would be its own opposite!

ReplyDeleteit is in principle possible to explain the large virial velocities of galaxies inside a cluster by postulating a different gravitational law. the virial theorem implies that

ReplyDelete\begin{equation}

2T = -nV

\end{equation}

with the kinetic energy $T$ and the potential energy $V$, for a system with a potential $\Phi\propto 1/r^n$. in order to account for Zwicky's observation that the potential energy $V$ is too small to account for the large kinetic energies $T$ for the Newton-case $n=1$, it would suffice to use steeper power laws, $n\gg1$: the mass to light-ratio is about a factor 10 larger in clusters than in galaxies, so the choice of $n$ needs to be accordingly high.