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{\bf 29:61 General Astronomy\\
     Fall 2005\\
    Third (and last) Hour Exam \ldots December 16, 2005}\\
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Write legibly, preferably in pen.  Start each question on a new page.  It allows me to make comments and generally keeps me in a better mood.  Write legibly. Explain what ideas you are using and what you are trying to do. Each question is worth 5 points; make sure you see them all.  Good luck and no whining.
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{\bf Walk with Ursus!!!}
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\noindent
(1) Comet Machholz, which was  visible in the evening sky exactly one year ago, is in on an orbit with a perihelion distance of 1.2 a.u. and an eccentricity of 0.999502.  How far is it from the Sun at aphelion?     \\

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(2) How long ago did the solar system form? How do we know that? \\

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(3) Explain why the Earth's atmosphere plays a role in determining its temperature.  You should use technical terms and equations.\\

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(4) Briefly describe contributions of the Mars Exploration Rover spacecraft to the question of the climate of Mars in the remote past.  \\

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(5) Describe why the Mars Exploration Landers (``Spirit'' and ``Opportunity'') landed where they did. If you describe the landing site of one of the rovers and give the scientific reasons for its choice, that would be satisfactory.  \\

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(6) Describe three ways in which the Jovian planets differ in a major way from the terrestrial planets.  \\

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(7) Name one attribute or physical property of Jupiter that helps explain why it has such a strong magnetic field.  \\

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(8) If you observe Saturn's ring over a period of many years,  you would conclude that it was very thin compared to its diameter.  Describe what this observation, mentioned in class, is.  \\

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(9) Cassini's Division is a gap in the ring of Saturn.  Why is it there?  \\

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(10) Explain the similarity between the structure of the ring of Saturn and Krkwood's gaps in the asteroid belt.  \\

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(11) Some of the {\em Valley Network} channels on Mars have craters in them, and they flow through moderately cratered landscapes.  What does this observation tell you about the nature of these channels?  \\

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(12) If you go outside tonight at 8PM and look at the stars,  you see a different set of constellations than we saw at the same time at the beginning of the semester, or the constellations we did see are in a different part of the sky.  Describe physically what is causing this, i.e. what is going on?  Diagrams always help in explaining things. \\ 

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(13) Roughly speaking, how big is Mars relative to the Earth?  I am asking how the diameter of Mars compares with that of the Earth.  \\

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(14) What is the significance of the term ``Roche Limit''?\\



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{\bf Formula Sheet}
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\noindent
$\bullet$ Exponential decay law: 
\begin{equation}
N(t) = N_0 e^{-\alpha t}
\end{equation}
where $N_0$ is the number of parent nuclei at $t=0$, $\alpha$ is the decay constant, and $N(t)$ is the number of parent nuclei at time $t$.  The decay constant is related to the half life $T_{1/2}$ by 
\begin{equation}
\alpha = \frac{0.693}{T_{1/2}}
\end{equation}

\noindent
$\bullet$ Escape speed from a planet
\begin{equation}
V_{esc} = \left( \frac{2 G M}{R} \right)^{\frac{1}{2}}
\end{equation}
where $M$ is the mass of the planet, and $R$ is its radius. 


\noindent
$\bullet$ The Maxwellian distribution function 
\begin{equation}
N(v) = \frac{2N_0}{\sqrt 2 \pi} \left( \frac{m}{k_B T} \right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}}
\end{equation}
This distribution describes the true distribution for gases in planetary atmospheres, as well as most other astronomical gases. 

\noindent
$\bullet$ The gravitational force: If two objects possessing masses $M$ and $m$ are a distance $r$ apart, there is an attractive force between them, the magnitude of which is 
\begin{equation}
|F| = \frac{GMm}{r^2}
\end{equation}  
where $G$ is the gravitational constant, $G= 6.6720 \times 10^{-11}$ N-m$^2$-kg$^{-2}$.

\noindent
$\bullet$ The circular orbit equation.  If $M \gg m$, and the orbit of $m$ about $M$ is circular, there is a relation between the radius of the orbit $r$, the orbital speed $v$, and the mass $M$ which is called the circular orbit equation.
\begin{equation}
v = \sqrt \frac{GM}{r}
\end{equation}


\noindent
$\bullet$ The force acting on a charged particle moving in electric $\vec{E}$ and magnetic $\vec{B}$ fields is given by . 
\begin{equation}
\vec{F} = q \vec{E} + q \vec{v} \times \vec{B}
\end{equation}
where $q$ is the charge of the particle.  For a proton, the charge is $1.6022 \times 10^{-19}$ Coulombs, and for an electron, $q = -1.6022 \times 10^{-19}$ Often the symbol $e$ is used for this fundamental charge of an electron or proton.  The units of magnetic field are {\em Tesla}, those of the electric field are {\em Volts/meter}.  



\noindent
$\bullet$ The equilibrium temperature of a planet, ignoring the greenhouse effect of its atmosphere (which often is a major correction) is
\begin{equation}
T_{eq} = \left[ \frac{(1-A)S_0}{\sigma}\right]^{\frac{1}{4}}
\end{equation}
where $S_0$ is the solar constant for that planet, and $A$ is the albedo.  

\noindent
$\bullet$ Condition for repetition of eclipses:
\begin{equation}
mP_{nod} = nP_{syn}
\end{equation}
with $m$,$n$ integers


\noindent
$\bullet$ Angular momentum of an object (with mass m) moving with velocity $\vec{v}$ a distance $\vec{r}$ from the origin of a coordinate system:
\begin{equation}
\vec{L} = m \vec{r} \times \vec{v}
\end{equation}

\noindent
$\bullet$ The definition of a vector cross product: 
The {\em magnitude} of a cross product is defined as follows.  If 
\begin{equation}
\vec{C} = \vec{A} \times \vec{B}
\end{equation}
then the {\em direction} of $\vec{A} \times \vec{B}$ is given by the right hand rule.  
The {\em magnitude} of $\vec{A} \times \vec{B}$ is given by 
\begin{equation}
|C| = |A| |B| \sin \theta
\end{equation}
where $\theta$ is the angle between the two vectors. 
\noindent 
$\bullet$ definition of acceleration $\vec{a}$, 
\begin{equation}
\vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{d \vec{v}}{dt}
\end{equation}

\noindent
$\bullet$ Roche Limit
\begin{equation}
r_R = 2.44 \left( \frac{\rho_M}{\rho_m} \right)^{1/3}R
\end{equation}
with $R$ being the radius of the more massive object, $\rho_M$ being the density of the more massive orbject, and $\rho_m$ being the density of the less massive object. 

\noindent
$\bullet$ Equation for an ellipse: 
\begin{equation}
r(\theta)= a(1- \epsilon \cos\theta)
\end{equation}
where $r$ is the distance from the focus to a point on the ellipse, $\theta$ is the angle between major axis and the point on the ellipse,  and $epsilon$ is the eccentricity of the ellipse.

\noindent
$\bullet$ Kepler's 3rd Law:
\begin{equation}
a^3 = P^2
\end{equation}
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