29:61 General Astronomy
Fall 2005
Third (and last) Hour Exam ...December 16, 2005

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.

Walk with Ursus!!!

(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?

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

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

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

(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.

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

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

(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.

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

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

(11) Some of the 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?

(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.

(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.

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

Formula Sheet

$\bullet$ Exponential decay law:

$$N(t) = N_0 e^{-\alpha t}$$ (1)

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

$$\alpha = \frac{0.693}{T_{1/2}}$$ (2)

$\bullet$ Escape speed from a planet

$$V_{esc} = \left( \frac{2 G M}{R} \right)^{\frac{1}{2}}$$ (3)

where $M$ is the mass of the planet, and $R$ is its radius.

$\bullet$ The Maxwellian distribution function

$$N(v) = \frac{2N_0}{\sqrt 2 \pi} \left( \frac{m}{k_B T} \right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}}$$ (4)

This distribution describes the true distribution for gases in planetary atmospheres, as well as most other astronomical gases.

$\bullet$ 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

$$\vert F\vert = \frac{GMm}{r^2}$$ (5)

where $G$ is the gravitational constant, $G= 6.6720 \times 10^{-11}$ N-m$^2$-kg$^{-2}$.

$\bullet$ 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.

$$v = \sqrt \frac{GM}{r}$$ (6)

$\bullet$ The force acting on a charged particle moving in electric $\vec{E}$ and magnetic $\vec{B}$ fields is given by .

$$\vec{F} = q \vec{E} + q \vec{v} \times \vec{B}$$ (7)

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 Tesla, those of the electric field are Volts/meter.

$\bullet$ The equilibrium temperature of a planet, ignoring the greenhouse effect of its atmosphere (which often is a major correction) is

$$T_{eq} = \left[ \frac{(1-A)S_0}{\sigma}\right]^{\frac{1}{4}}$$ (8)

where $S_0$ is the solar constant for that planet, and $A$ is the albedo.

$\bullet$ Condition for repetition of eclipses:

$$mP_{nod} = nP_{syn}$$ (9)

with $m$,$n$ integers

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

$$\vec{L} = m \vec{r} \times \vec{v}$$ (10)

$\bullet$ The definition of a vector cross product: The magnitude of a cross product is defined as follows. If

$$\vec{C} = \vec{A} \times \vec{B}$$ (11)

then the direction of $\vec{A} \times \vec{B}$ is given by the right hand rule. The magnitude of $\vec{A} \times \vec{B}$ is given by

$$\vert C\vert = \vert A\vert \vert B\vert \sin \theta$$ (12)

where $\theta$ is the angle between the two vectors. $\bullet$ definition of acceleration $\vec{a}$,

$$\vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{d \vec{v}}{dt}$$ (13)

$\bullet$ Roche Limit

$$r_R = 2.44 \left( \frac{\rho_M}{\rho_m} \right)^{1/3}R$$ (14)

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.

$\bullet$ Equation for an ellipse:

$$r(\theta)= a(1- \epsilon \cos\theta)$$ (15)

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.

$\bullet$ Kepler's 3rd Law:

$$a^3 = P^2$$ (16)