I need to show that $$P_c=\frac {9GM^2}{14\pi R^4}$$ from the equation of hydrostatic equilibrium: $\frac {dP}{dr}=-\frac{GM\rho}{r^2}$

in the exercise before, we show that $\rho_c=\frac {6M}{5 \pi R^3}$
if I use this, and the fact that I know that $\frac{dP}{dr}$ can be exchanged with $-\frac{P_c}{R}$ we get:
$P_c=\frac{6GM^2}{5 \pi R^5}$

This is where I get stuck. I'm not really sure on where to go from this.

  • $\begingroup$ I think you didn't provide sufficient information for answering your question. Is this for a star? Does the sub-script 'c' refer to the central quantities? What equation of state is assumed? How did you derive $P_c=6GM^2/5\pi R^4$ from $dP/dr=-P/R$? $\endgroup$
    – Walter
    Commented Dec 30, 2022 at 17:29
  • $\begingroup$ There also seems to be some confusion in the meaning of the symbol $M$. In the relations for $P_c$ and $\rho_c$, it refers to the total mass, while in the equation for hydrostatic equilibrium it refers to the cumulative mass $M(<r)=4\pi\int_0^r r^2 \rho(r) dr$ interior to $r$. $\endgroup$
    – Walter
    Commented Dec 30, 2022 at 17:33
  • $\begingroup$ This is the entire question: Use the expression for hydrostatic equilibrium to show that $P_\mathrm{c}$ is given by $P_\mathrm{c} = \frac{9 G M^2}{14 \pi R^4}$. Argue, using $L \propto M^{4,8}$, that the mass is 0.76 $M_\odot$ and determine the central pressur for ε indi A. ε indi A is a star. Radius: $0.82R_\odot$, $T_{eff}=4585 K$. This is the only information I have including a bit more, but definitely irrelevant here. @Walter In our book it is stated that we can exchange dP/dr with $P_c/R$ $\endgroup$
    – C H
    Commented Jan 2, 2023 at 17:42
  • $\begingroup$ Well, the question is part of a book and in that book is more information that is also required for the solution. I would assume that this holds for the equation(s) of state and other assumptions made. Exercises and Problems in textbooks are almost never self-contained, but usually require other information/methods/equations/assumptions found in the text. Which book is it? $\endgroup$
    – Walter
    Commented Jan 9, 2023 at 21:12


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