Blog #22: Sound-Crossing Timescale - White Dwarf Goes Supernova

3) A white dwarf that exceeds the Chandrasekhar mass will start to fuse carbon in its interior, which releases a great deal of heat, which increases the internal pressure of the white dwarf. However, because the white dwarf is "trying" to support itself using degeneracy pressure, and increasing this pressure doesn't change the star's radius, the increasing temperature leads to more fusion, more energy, and a run-away fusion process is initiated.

Once the run-away fusion inside the white dwarf is "ignited", it propagates as a wave travelling outward at the speed of sound $c_s$. How much time does it take the flame to sweep outward across the radius of the white dwarf? This is also known as the "sound-crossing timescale."

How does this time scale relate to the density of the white dwarf?

In problem 2, the speed of sound, in terms of mass, and radius, is determined to be:

\begin{align*}
c_s = \left(\frac{GM}{5R}\right)^{\frac{1}{2}}
\end{align*}

We also know that speed is defined as a distance over time, which can rearranged to solve for time. The equation for time, therefore, is:

\begin{align*}
t_{sc} = \frac{R}{c_{s}}
\end{align*}

where $c_s$ is the speed of sound, $t_{sc}$ is the "sound-crossing timescale", and $R$ is the radius of the white dwarf. We can substitute the speed of sound $c_s = \left(\frac{GM}{5R}\right)^{\frac{1}{2}}$ and solve for the "sound-crossing timescale" as follows:

\begin{align*}
t_{sc} &= \frac{R}{c_{s}}\\
t_{sc} &= \frac{R}{ \left(\frac{GM}{5R}\right)^{\frac{1}{2}}}\\
t_{sc} &= \left(\frac{5R^3}{GM}\right)^{\frac{1}{2}}
\end{align*}

In order to see how the sound-crossing timescale relates to density, we can use dimensional analysis:

\begin{align*}
t_{sc} &= \left(\frac{5R^3}{GM}\right)^{\frac{1}{2}}\\
t_{sc} &= \left[\frac{m^3}{g}\right]^{\frac{1}{2}}\\
t_{sc} &\propto \sqrt{\frac{1}{\rho}}
\end{align*}

This shows that the sound-crossing timescale is inversely proportional to the square root of the density.