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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.

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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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