In Barbara Ryden's Intro to Cosmology, she states that the gravitational potential energy of a cluster of galaxies, that is given by $$ W = -\frac{G}{2}\sum\limits_{i,j,\ i\neq j}\frac{m_im_j}{|\vec{x}_j-\vec{x}_i|} $$ can be written in terms of the total mass of the cluster, and the half-mass radius, that is the radius of a sphere, located at the center of mass of the cluster, that contains a mass $\frac{M}{2}$, $$ W = -\alpha\frac{GM^2}{r_h} $$ where $\alpha$ is some numerical factor. I have tried to derive this expression with no success, and was wondering if any reference to it exists.


First consider two identically shaped galaxies, but one of them having the mass of each part scaled by some factor. If the masses are $M_1$ and $M_2$, then the scale factor is $M_2/M_1$.

Gravitational potential energy of the second galaxy is then: \begin{equation} W_2=-\frac{G}{2}\sum_{i,j,i\ne j} \frac{\frac{M_2}{M_1}m_i\frac{M_2}{M_1}m_j}{|\vec x_i - \vec x_j|} =\left(\frac{M_2}{M_1}\right)^2 W_1 =M_2^2\frac{W_1}{M_1^2} \end{equation}

If, in addition to the different mass, one galaxy is also uniformly stretched, one having radius $r_{h,1}$ and the other having radius $r_{h,2}$, it can be similarly shown (using $|\vec x_i - \vec x_j| \rightarrow |\frac{h_{r,2}}{h_{r,1}}\vec x_i - \frac{h_{r,2}}{h_{r,1}}\vec x_j|$) that the gravitational potential energy of the second galaxy is now: \begin{equation} W_2=\frac{M_2^2}{r_{h,2}}\frac{W_1 r_{h,1}}{M_1^2} \end{equation}

One can therefore define: \begin{equation} \frac{W_1 r_{h,1}}{M_1^2}=\frac{W_2 r_{h,2}}{M_2^2}\equiv-G\alpha \end{equation} When defined like that, $\alpha$ is a positive dimensionless number that depends only on the shape. This immediately leads to the desired equation: \begin{equation} W=-\alpha\frac{GM^2}{r_h} \end{equation}


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