Dynamical time scale

The third and the shortest time scale is the dynamical time scale and it is the time it would take a star to collapse if the pressure supporting it against gravity were suddenly removed.

It can be estimated from the time it would take for a particle to fall freely from the stellar surface to the center. That is,

F = mg = \frac{GMm}{R^2}

g = \frac{GM}{R^2} = a = \frac{v}{t} = \frac{s/t}{t} = \frac{R}{t^2}

t^2 = \frac{R^3}{GM}

Thus, the dynamical time scale is given by

t_d \cong \sqrt{\frac{R^3}{GM}}

The dynamic time scale is about 30 minutes for our Sun. For other stars, the dynamical time scale is

t_d \approx \sqrt{\frac{{R/R_o}^3}{M/M_o}} \times 0.5 hours

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