29:61 General Astronomy
Fall 2005
Second Hour Exam ...November 18, 2005
VERSION WITH ANSWERS

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. There are 10 questions. Be sure and note that they are not the same amount of credit. Good luck and no whining.

Walk with Ursus!!!

(1)...(5pts) One of the homework problems asked you to pick some years which would be favorable for human spacecraft missions to deep space. Other years were less favorable. Explain the basis for making such choices.

Answer: Choices were based on the phase of the 11 year solar cycle. At times of solar maximum, there are more solar flares and coronal mass ejections, which enhance radiation levels in space.

(2) ...(5pts) If the molecular speed $V_{rms}$ of a gas is less than the escape speed from a planet, why isn't it always the case that the planet can retain an atmosphere composed of that gas?

Answer: $V_{rms}$ is a mean of a distribution of speeds of molecules, not the speed at which all molecules move. Many of the molecules in the distribution will be traveling faster than $V_{rms}$ and could be moving faster than $V_{esc}$, and thus be lost.

(3) ...(10pts) In class, I discussed a theoretical basis for explaining why some solar system objects have atmospheres, while others do not. Explain why Mars and Titan represent a problem for this explanation.

Answer: The basis for determining if a planet can retain an atmosphere is the retentivity, $R$, which is the ratio of the escape speed $V_{esc}$ to $V_{rms}$. If $R \gg 6$, a planet can retain an atmosphere for a long time. If $R \ll 6$, the atmosphere will escape.

Mars and Titan have nearly equal $R$ factors, so should be equally good at retaining or losing an atmosphere. But Titan has a dense atmosphere while Mars has a tenuous one. Something else is going on.

(4) ...(5pts) Briefly describe (in about one paragraph) what we mean by the Van Allen Belts.

Answer: The Van Allen Belts are donut-shaped regions of trapped, high intensity particle radiation near the Earth. The innner one is composed of energetic protons, and the outer one is energetic electrons. They are trapped by the Earth's magnetic field. The Lorentz force $\vec{F} = q \vec{v} \times \vec{B}$ keeps particles from crossing the magnetic field and getting out.

(5) ...(10pts) Explain how we can determine the age of formation of a rock. Use technical terms and concepts.

Answer: The age of formation of a rock is determined by radioisotope dating, in which a radioisotope A decays into daughter isotope B. When a rock forms, it has a number of A, $N_A=N_0$. Over time, the number of A declines according to an exponential law, $N_A=N_0 \exp (-\alpha t)$, where $\alpha$ is related to the half life.

Every A that decays becomes a B; by measuring the ratio of B/A you can determine the time since the rock formed.

An important correction muyst be made for the number of B present when the rock formed. This is done by also measuring the number of another isotope of B, B1.

(6) ...(10pts) Radioisotope A decays to isotope B with a half life of 3.0 Gyr. A rock forms with 1000 atoms of A and none of B. 7.0 Gyr later, now many atoms of A and B will there be in the rock?

Answer: Put the ideas in the preceding problem to work. We have

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

In this problem, we have $N_0=1000$, $\alpha=\frac{0.693}{T_{1/2}}=0.693/3.0=0.231$ inverse Gyr.

After 7 GYr we have

$$N_A(t) = 1000 e^{-0.231(7.0)}=198.$$ (2)

If there are 198 atoms of A, there must be 1000-198=802 atoms of B.

(7) ...(5pts) What are the primary similarities between the Earth and Venus? Discuss the most striking difference or differences.

Answer: The primary similarities are in mass, where the planets are nearly the same ($\sim$ 80 %), and in diameter, which are even more similar. The most important differences are in properties of the atmosphere, such as the temperature, (750K for Venus vs. 290K for the Earth), atmospheric pressure (Venus has 90 times the surface pressure of Earth), and composition of the atmosphere. The atmosphere of Venus is nearly entirely carbon dioxide.

(8) ...(10pts) The solar constant is proportional to the inverse square of the distance from the Sun. In other words, the solar constant $S$ at a planet $r$ astronomical units from the Sun would be

$$S=\frac{S_0}{r^2}$$ (3)

where $S_0$ is the solar constant at the Earth. How does the equilibrium temperature of an object depend on distance from the Sun. Use equations!!!! Hint: In answering this, think of keeping the albedo of an object the same.

Answer: Use equations to get an algebraic answer. The equation to start with is that of the equilibrium temperature,

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

If $S=S_0/r^2$, upon substituting in, we have

$$T_{eq} = \left[ \frac{(1-A)S_0}{\sigma}\right]\frac{1}{\sqrt r}$$ (5)

so the equilibrium temperature is inversely proportional to the square root of the distance.

(9) ...(5pts) A so-called Near Earth Asteroid has an orbital period of 1.0 year and is in the plane of the ecliptic. During the course of its orbit, it moves from outside the orbit of the Earth to inside. Draw a sketch of its orbit, and indicate the one orbital element to which you can attach a number with the information above.

Answer: The diagram should show the Earth as an almost-circular orbit with a semimajor axis of 1 au. The orbit of the asteroid should be elliptical, with the Sun at one focus. Perihelion should be inside the orbit of the Earth, and Aphelion outside. The semimajor axis of this orbit should also be 1au, as given by Kepler's Laws. This should be clearly indicated on the drawing.

(10) ...(5pts) Mars is currently at opposition. Draw a diagram of the solar system from above. Correctly indicate the positions of the Sun, Earth, and Mars. Extra 2 Points!: Indicate the position of Venus as well, with a cartoon word-bubble giving the reason why you put it there.

Answer: For opposition, the Sun, Earth, and Mars should be on a straight line, and located in that order, starting from the Sun. These days, Venus should be located to the left of the Sun, as seen from Earth. The reason you should know this is that it is the bright object visible in the evening sky.