Blog Post 22, Worksheet 7.1, Problem 3

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 this run-away fusion inside the white dwarf is "ignited," it propagates as a wave traveling 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?

From problem 2, we know that the relationship between the speed of sound (\(c_s\)), pressure (P), and density (\(\rho\)) is given by: \[c_s \sim (\frac{P}{\rho})^{1/2}\] The speed of sound is a speed, or \(\frac{\text{distance}}{\text{time}}\). To solve for time, we thus have: \[\text{time} = \frac{\text{distance}}{c_s} = \frac{R}{(\frac{P}{\rho})^{1/2}}\] From problem 1, we know that the pressure of the white dwarf is given by: \[P = \frac{GM^2}{4πR^4},\] and from the definition of density as \(\frac{\text{mass}}{\text{volume}}\), we know that \[\rho = \frac{\text{mass}}{\text{volume}} = \frac{M}{\frac{4}{3}πR^3}\] Plugging in these expression for P and \(\rho\), we have: \[t = R (\frac{4πR^4}{GM^2})^{1/2} (\frac{M}{\frac{4}{3}πR^3})^{1/2}\] \[t = R(\frac{3R}{GM})^{1/2}\] \[t = (\frac{3R^3}{GM})^{1/2}\] We know that density, \(\rho\), is given by mass/volume, or \(\frac{M}{\frac{4}{3}πR^3}\). If we ignore the constants, we can thus replace \(\frac{R^3}{M}\) in the expression for time with \(\frac{1}{\rho}\): \[t \sim (\frac{1}{G\rho})^{1/2}\] From this, we can see that the relationship between time scale and density is given by: \[t \sim \rho^{-1/2}\]