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{\Large ASTR430 Homework \#5 \\
Due Wednesday, November 12, 2003}
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1. Build a Planet!\\ a) Evaluate the moment of inertia $I$ of a uniform
density spherical planet of radius $R$ about its spin axis by
integrating the expression $dI = r^2 dm$ over the volume of the sphere
($dI$ is the differential moment of inertia due to mass element $dm$
at distance $r$ from the spin axis). Write your answer two ways: i) in
terms of $R$ and $\rho$, the mass density and ii) in terms of $R$ and
$M$, the mass of the planet.\\
b) Will the moment of inertia increase or decrease for an oblate
planet (one with an equatorial diameter greater than its polar
diameter)? What about for a differentiated planet with a dense core
and a less dense mantle? Explain your answers.\\
c) Use your answer from a) to get the moment of inertia of a two-layer
planet with a core of radius $R_c$ and density $\rho_c$, and a mantle
of radius $R$ and density $\rho_m$.  Write your answer in terms of
$R$, $R_c$, $\rho_c$, and $\rho_m$. Apply your result to get a
constraint on the interior structure of Mars using the measured
$I_{Mars}=0.365MR^2$.  Write the constraint in terms of $R$, $R_c$,
$\rho_c$, $\rho_m$, and $\rho_{AV}$, where $\rho_{AV} = 3.93$ g/cm$^3$ is
the average density of Mars.\\
d) Write down an expression for the total mass of the planet in terms
of $R$, $R_c$, $\rho_c$, and $\rho_m$. Eliminate mass in favor of 
$\rho_{AV}$ to get a second constraint on the interior structure of Mars.\\
e) Parts c) and d) give two constraints on the three unknowns $R_c$,
$\rho_c$, and $\rho_m$. If we assume a core density for Mars, the
system reduces to two equations in two unknowns.  Eliminate the core
radius from your two equations to get a single equation that relates
the two unknown densities.  Assume an iron core with density
$\rho_c=7.5$g/cm$^3$, and guess different $\rho_m$'s until you find a
solution (This equation cannot be solved analytically). What core
radius $R_c$ does your answer suggest? 


2. Estimate the total amount of energy released when Helium rains out
of Saturn's mantle into its metallic core. For the purpose of this
problem, we will ignore ice and rock components and assume that Saturn
is made of just uncompressed Hydrogen (H) and Helium (He). \\ a) Start
by obtaining the average density of Saturn $\bar\rho$ in terms of
$\rho_H$, using $\rho_{He} = 4\rho_H$ and assuming a solar composition
for Saturn as given in Table 1 on page 25 of the book.\\ b) Next,
write down a general integral for the potential energy of a
spherically symmetric planet in which the planet's density $\rho$ is a
function of radius.  Hint: consider successively adding thin shells of
material to a spherical core by integrating over $r$, the radial
coordinate (see the class handout).  \\ c) Now assuming that the
planet is made of uniformly mixed H and He, calculate the planet's
potential energy.  You've just evaluated the integral for the case of
a planet with uniform density.  This is a test of your answer in part
b); you should get $U = -3GM_S^2/(5R_S)$ as on the class handout. \\ 
d) Now calculate the potential energy of a fully differentiated Saturn
(2 terms). The first and easiest term is the potential energy of a
uniform density Helium core of radius $r_{He} \approx 0.3R_S$ and
density $\rho_{He}$.  Put your answer in terms of $U_{wms}$, the
potential energy of a well-mixed Saturn (your answer to part c). The
second term is the potential energy of the uniform Hydrogen mantle
which overlies the Helium core. Again, put your answer in terms of
$U_{wms}$. Make sure that your answers reduce correctly in the limits
$r_{He} = 0$ and $r_{He}=R_S$.  How much extra energy is available as
heat when Saturn evolves from its well-mixed state to its
differentiated state? Is this a large or small amount of heat?


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