Monday, October 02, 2006

The speed of sound in an ideal gas

Reading through today's astronomy preprints on astro-ph I couldn't help but notice one of my pet peeves: the wrong speed of sound!

For an ideal gas, assuming adiabatic compression, a sound wave travels at a speed c_s = sqrt(gamma*pressure/density), where gamma is the ratio os specific heats at constant pressure and constant volume (gamma=5/3 for a monatomic gas).

So for a typical astrophysical plasma with a temperature of T=1,1604,500 Kelvin (kT=1 keV) the sound speed is 511 km/s (the speed of sound in air at STP is about 0.3 km/s). Yet its not uncommon to see the speed of sound for this temperature given as being around 300 km/s. Why the big difference?

Its because many astronomers simply insist on using the isothermal speed of sound, c_s_iso = sqrt(Pressure/density), i.e. ignoring the gamma.

What exactly is the difference?

The isothermal speed of sound is valid in conditions where the period of the distance (i.e. 1/frequency) is long compared to the cooling time of the plasma. You compress the plasma, it heats up adiabatically, but radiation carries the heat away and the plasma cools back the ambient temperature before the next pulse of the sound comes along. Another way of looking at it is that the ratio of specific heats is now ~1.


Basically in any situation where the temperature of the plasma is independent of the density you should use the isothermal sound speed. A good example is an HII region, where the plasma temperature is set by photoionization from the embedded stars.

But if the plasma temperature is not set by an external process and the cooling time is long then then gamma = 5/3 and the isothermal sound speed IS NOT VALID. Yet time after time I see papers always applying the one formula they learnt at grad school (which is typically the isothermal version) irrespective of the actual physical conditions in the plasma. Doesn't anyone actually pay attention to what Spitzer says instead of just using the formulae? Here is Wolfram on the case as well.

OK, I know using the wrong sound speed is not quite as bad a thing as ignoring warnings about increasing terrorist activity and then keeping that information from the 9/11 commission (yes Condi, I mean you), but its still wrong.

[Update] Forgot to add labels to the post.

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