Blog #22, Worksheet #7.1, Problem #2, Sound Speed of White Dwarf

2. Use the previous result to derive the sound speed, \(c_s\), within the white dwarf, where \(c_s\) can be related to pressure, \(P\), and mass density, \(\rho\), using dimensional analysis (make the units work).

We begin with our expression from question 1 for Pressure \[P = \frac{GM^2}{4 \pi R^4}\]We then look at the units for this expression \[P = \frac{[F]}{m^2} = \frac{kg \cdot m}{m^2 \cdot s^2} = \frac{kg}{m \cdot s^2}\]We want to solve for \(c_s\) which has units \(\frac{m}{s}\), so we must find some exponent for both density and pressure that gives us the \(\frac{m}{s}\) relationship we need. Density, \(\rho\) has the units \[\rho = \frac{kg}{m^3}\]So we end up with the relationship \[(\frac{kg}{m \cdot s^2})^{\alpha} \cdot (\frac{kg}{m^3})^{\beta} = \frac{m}{s}\]We then need to figure out what exponents for \(\alpha\) and \(\beta\) lead us the the correct units for velocity. After a quick inspection we can clearly see the the density should be in the denominator of our equation and after canceling, we need to take the square root of both equations to get the proper units for velocity. Thus \[\alpha = \frac{1}{2} \, and \, \beta = -\frac{1}{2}\]Thus our final answer gives us \[c_s \propto (\frac{P}{\rho})^{\frac{1}{2}}\]