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.