The “interacting wind” model of planetary nebulae is based on the idea that the white dwarf phase of stellar evolution is preceded by a red giant phase. A fast wind from the hot white dwarf overtakes the more slowly moving red giant wind and the region of interaction forms a shell of material which is driven outwards by the fast wind and photoionised by the white dwarf. The emission from this shell produces the observed planetary nebula. Assume that the fast wind switches on instantaneously at the end of the red giant phase at $t = 0$. The mass of swept up material is given by: $$ \frac{dM_s}{dt}=A(V_s-V_{RG}) $$ where $V_s=dR_s/dt$ is the velocity of the shell and $A=\dot{M_{RG}}/V_{RG}$ Assuming that the evolution of the interior of the mass-loss bubble is adiabatic, I have to find out the radius of the swept up shell and the pressure are given by $$ R_s(t) = \left ( 2L_w/3A \right )^{1/3} t $$
Attempted solution
I have started with \begin{eqnarray} \frac{d}{dt} \int_{shell} \rho d^3x = -\int_{S_2} \rho (v_i -u_i) n_i ds + \int_{S_1} \rho (v_i -u_i)n_i ds \end{eqnarray}
Then finally I got the answer of The mass of swept up material. But I have started with Energy driven and momentum
$$ \frac{d}{dt}\left (M \frac{dR_s}{dt} \right )= \frac{3}{2} \frac{L_w}{v_w} \frac{R_s^2}{R_1^2}= 4\pi p_bR_s^2 $$
I think from here I need find out the required equation. Could some one help ?
