<p>I need to show that <span class="math-container">$$P_c=\frac {9GM^2}{14\pi R^4}$$</span>
from the equation of hydrostatic equilibrium: <span class="math-container">$\frac {dP}{dr}=-\frac{GM\rho}{r^2}$</span></p>
<p>in the exercise before, we show that <span class="math-container">$\rho_c=\frac {6M}{5 \pi R^3}$</span>
<br>
if I use this, and the fact that I know that <span class="math-container">$\frac{dP}{dr}$</span> can be exchanged with <span class="math-container">$-\frac{P_c}{R}$</span> we get:
<br>
<span class="math-container">$P_c=\frac{6GM^2}{5 \pi R^5}$</span></p>
<p>This is where I get stuck. I'm not really sure on where to go from this.</p>

CC BY-SA 4.0
2022-12-30T17:29:50.403Z
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 $dP/dr=-P/R$?
Walter
CC BY-SA 4.0
2022-12-30T17:33:39.657Z
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 $r$.
Walter
CC BY-SA 4.0
2023-01-02T17:42:36.493Z
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 $P_c/R$
C H
CC BY-SA 4.0
2023-01-09T21:12:50.657Z
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?
Walter
<star><hydrostatic-equilibrium>
How to show that $P_c=(9GM^2)/(14\pi R^4)$ from equation of hydrostatic equilibrium
C H
# How to show that $P_c=(9GM^2)/(14\pi R^4)$ from equation of hydrostatic equilibrium

(Asked by: username_0 on 2022-12-28)

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.

**Tags:** star, hydrostatic-equilibrium
> **Comments:**
> **username_1**: 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 $dP/dr=-P/R$?  
> **username_1**: 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 $r$.  
> **username_0**: 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 $P_c/R$  
> **username_1**: 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?  

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## 0 answers

