| 16 | | This last point is useful for integrating the equation. Since the mass is constant in any shell (by the way, 'shell' here is analogous to a 'concentric sphere'), we can replace [[latex($M_r$)]] in the equation by [[latex($M_r = \frac{4}{3} \pi r_0^3 \rho_0$)]] when integrating this equation for the entire sphere. Integrating this equation under the boundary conditions: the initial velocity is zero, [[latex($\frac{dr}{dt}|_{t=0}=0$)]], and this occurs when [[latex($r=r_0$)]], leads to an equation that describes the position of the radius of the sphere over time (i.e. over the course of collapse), and the velocity at this radius over time. We will look at these equations next. |
| | 19 | This last point is useful for integrating the equation. Since the mass is constant in any shell (by the way, 'shell' here is analogous to a 'concentric sphere'), we can replace [[latex($M_r$)]] in the equation by [[latex($M_r = \frac{4}{3} \pi r_0^3 \rho_0$)]] when integrating this equation for a sphere of initial radius r0. Integrating this equation under the boundary conditions: the initial velocity is zero, [[latex($\frac{dr}{dt}|_{t=0}=0$)]], and this occurs when [[latex($r=r_0$)]], leads to an equation that describes the position of the radius of the sphere over time (i.e. over the course of collapse), and the velocity at this radius over time. We will look at these equations next. |
