# Harvard Astronomy 201b

## CHAPTER: The Sound Speed

In Book Chapter on February 7, 2013 at 10:00 pm

(updated for 2013)

The speed of sound is the speed at which pressure disturbances travel in a medium. It is defined as

$c_s \equiv \frac{\partial P}{\partial \rho}$,

where $P$ and $\rho$ are pressure and mass density, respectively. For a polytropic gas, i.e. one defined by the equation of state $P \propto \rho^\gamma$, this becomes $c_s=\sqrt{\gamma P/\rho}$. $\gamma$ is the adiabatic index (ratio of specific heats), and $\gamma=5/3$ describes a monatomic gas.

For an isothermal gas where the ideal gas equation of state $P=\rho k_B T / (\mu m_{\rm H})$ holds, $c_s=\sqrt{k_B T/\mu}$. Here, $\mu$ is the mean molecular weight (a factor that accounts for the chemical composition of the gas), and $m_{\rm H}$ is the hydrogen atomic mass. Note that for pure molecular hydrogen $\mu=2$. For molecular gas with ~10% He by mass and trace metals, $\mu \approx 2.7$ is often used.

A gas can be approximated to be isothermal if the sound wave period is much higher than the (radiative) cooling time of the gas, as any increase in temperature due to compression by the wave will be immediately followed by radiative cooling to the original equilibrium temperature well before the next compression occurs. Many astrophysical situations in the ISM are close to being isothermal, thus the isothermal sound speed is often used. For example, in conditions where temperature and density are independent such as H II regions (where the gas temperature is set by the ionizing star’s spectrum), the gas is very close to isothermal.