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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,

    $\displaystyle 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).

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Derek C. Richardson 2005-11-23