This says that each grain should have a net charge of about —e. However, the actual charge is about a factor of ten larger because we have only considered electrons of the average energy. Electrons moving faster than average contribute significantly to the charge buildup. Also, the charge becomes more negative at higher temperatures.
If the photoelectric effect is dominant, the grains will have a positive charge. There must be a balance between the rate at which electrons are being ejected and the rate at which they strike the grain. For positively charged grains, the electrons in the gas are attracted, meaning the tendency for the electrons to strike the grains at a greater rate than positively charged particles is enhanced.
The temperature of a large dust grain is determined by the fact that, on a time average, it must emit radiation at the same rate as it receives radiation. This keeps the temperature constant. The temperature of a dust grain will therefore depend on its environment. If it is very close to a star it will be hot. If it is far from any one star it is cool, receiving energy only from the combined light of many distant stars.
Let's look at the case of a dust grain a distance d from a star whose radius and temperature are R* and T*. We will assume that the albedo is the same at all wavelengths. (If the albedo varies with wavelength, as in more realistic cases, the fraction of incoming radiation absorbed is different at different wavelengths, and the calculation is harder. See Problem 14.11.) The luminosity of the star is d2
The quantity v Ri/d2 is the solid angle subtended by the star as seen from the dust grain. We say that the star acts like a dilute blackbody. It has the spectrum of a blackbody at a temperature T*, but the intensity is down by a factor of (solid angle/4v).
We now look at the rate at which the grain radiates energy. Since it can only absorb (1 — a) of the radiation striking it, it can only emit (1 — a) of the radiation that a perfect blackbody would emit. (A perfect blackbody has an albedo of zero, by definition.) If the grain temperature is Tg, the power radiated is
Equating the power radiated and the power received, and solving for T, gives
Note that the final result does not depend on the size of the grain or the albedo. That is because both enter into the emission and absorption processes. (See Problem 14.10 for a discussion of what happens if the albedo is a function of wavelength.) This result is the same as that derived for a planet in Section 23.2.
Example 14.4 Temperature of a dust grain near a star
What is the temperature of a dust grain a distance 5000 stellar radii from a star whose temperature is 104K?
Using equation (14.16), we have
When dust is sufficiently warm (Tg > 20 K), it is a good emitter in the infrared, and we can
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