The Rayleigh-Jeans Tail

In this post I'll be doing #2d from this week's worksheet on blackbody radiation. I chose the problem because it transforms an ugly equation into a nice relationship using approximations, illustrating the importance of approximations and ballpark mathematics in astronomy. Here's the problem:

Let's consider photon energies that are much smaller than the thermal energy. Use a first-order Taylor expansion on the term e^(hv/kT) to derive a simplified form of Bv(T) in this low-energy regime. Radio astronomers like to talk about the "brightness temperature" of an object, rather than its actual brightness. Why do you suppose they talk about temperatures instead of intensity, like normal people?

Let's start with the expression for Bv(T):

A messy equation, indeed. However, we are concerned with photon energies that are much smaller than thermal energies:

With this assumption, we can Taylor expand the term on the denominator:

Since this is astronomy, let's not worry about higher order terms and focus on the first-order approximation:

Substituting into our original equation:

Thus, using our approximations, we have shown that Bv(T) has a linear relation with temperature!

Now the second part of the question. Why do radio astronomers talk about the "brightness temperature" of an object? Looking at our above expression, the answer is evident. Since the intensity of a blackbody is proportional to temperature, we can immediately know one of them, given the other! In other words, describing a blackbody at low energies with its temperature is equivalent to describing its intensity.

(Note: this still doesn't mean that radio astronomers are normal people...:) )