ASTR430 HW#6 (due 12/12/05)

Your work should be handed to me (or placed under my door) by 5 pm Monday, December 12. I will have office hours as usual that day, so you could hand it in then (and discuss your oral presentation from the previous week if you like).

1. The Maxwell-Boltzmann distribution.

   From the kinetic theory of gases, the fraction of particles with speeds between $v$ and $v + dv$ in an ideal gas at thermal equilibrium is given by the Maxwell-Boltzmann distribution function,

   $$p(v)\,dv = \left( \frac{2}{\pi} \right)^{1/2} \left( \frac{m}{kT} \right)^{3/2} v^2 e^{-mv^2/2kT}\,dv,$$
   where $m$ is the molecular mass and $T$ is the temperature ($k$ is the Boltzmann constant).
   1. Verify that this distribution is normalized (i.e., show that $\int_0^\infty p(v)\,dv = 1$). What are the units of $p(v)$?

   2. Derive expressions for the most probable speed $v_{\mathrm{max}}$ (i.e., the speed corresponding to the peak in the distribution), the mean speed $v_{\mathrm{avg}} = \int_0^\infty v\,p(v)\,dv$, and the root-mean-square speed $v_{\mathrm{rms}} = [\int_0^\infty v^2\,p(v)\,dv]^{1/2}$. Show that $v_{\mathrm{max}} \le v_{\mathrm{avg}} \le v_{\mathrm{rms}}$. Evaluate $p(v)$ at each of these values.

   3. Plot the distribution functions for hydrogen and oxygen ($^{16}$O) atoms at 900 K, for $v$ between 0 and 25 km/s. For what $v > v_{\mathrm{max}}$ is $p(v)\,dv \lesssim 10^{-12}$ in each case? Express your answers in km/s, and also in terms of $v_{\mathrm{max}}$, $v_{\mathrm{avg}}$, and $v_{\mathrm{rms}}$ for both species.

2. Atmospheres.
   1. Do problem #9 from Ch. 11 (p. 347).

   2. Also do problem #10.

   3. Infrared observations of 90377 Sedna show a peak at 88 $\mu$m. What is the temperature of Sedna? At that temperature, familiar gases like N$_2$ and CO$_2$ are frozen ices. Helium (He) and neon (Ne) would still be gaseous. Assuming Sedna has a radius of 800 km and a bulk density of 3000 kg/m$^3$, can this body retain an atmosphere of He? What about Ne? Comment.

3. Habitable zones. Do problem #9 from Ch. 12 (p. 369).