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I am a Physics student and I'd really appreciate if anybody could help me with an exercise my professor gave me a couple of weeks ago.
It goes like this:

Let us consider a spherical symmetrical gas in the interstellar space being in hydrostatic equilibrium, with constant pressure and radius $P=P_0$ , $R=R_0$.
Suppose we perturbate the nebula in a linear way, so that the radius and the pressure become: $P=P_0+\delta P , R=R_0+\delta R$

Use: 1) Newton's 2nd law : $m\cdot \frac{d^2R}{dt^2}=-\frac{GmM}{R^2}+4\pi R^2 P$ , where $m$ is the mass of the outer shell of the gas.

2) Adiabatic equation: $PV^{\gamma}=c\in \mathbb{R}$ , to show that the gas will perform simple harmonic motion which equation is $\frac{d^2\delta R}{dt^2}+[(3\gamma-4)\frac{GM}{R_0^3}]\delta R=0$

The solution my prof. provided has many logical steps which I cannot follow, so I'd really appreciate if anybody can give me an analytic solution
P.S. I have good grades in all other courses.

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  • $\begingroup$ $V\sim R^3$ and Taylor expansion to first order $\endgroup$
    – Bort
    Commented Jan 25, 2016 at 10:17

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Without knowing much about Jean's instability I can probably help you with the derivation of the expression.

You start with the expression in 1) and expand it to first order in $\delta P$ and $\delta R$. Then you use that $R_0$ and $P_0$ are stationary solutions i.e $\ddot{R_0}=0$ thus $GmM/R_0^2= 4 \pi R_0^2 P_0$. You can use this to simplify the linearisation: $ 8 \pi R_0 P_0=2 GmM/R_0^3$. To treat the last term in the linearisation you expand $P=c/V^\gamma$ in $\delta R$ which couples $\delta P$ and $\delta R$. This expression can again be simplified if you use $GmM/R_0^2= 4 \pi R_0^2 P_0$ and $P_0 V_0^\gamma=c$.

The solutions are the usual $\delta R \sim e^{\pm i \omega t}$ or $\delta R \sim e^{\pm \omega t}$ depending on $\gamma$.

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