29:61 General Astronomy
Fall 2005
Third (and last) Hour Exam ...December 16, 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. Each question is worth 5 points; make sure you see them all. Good luck and no whining.

Walk with Ursus!!!

(1) Why does Saturn have a ring at that location from the planet, rather than another moon? Use technical terms introduced in lecture. Use of equations is also highly encouraged.
Answer:The ring of Saturn is inside the Roche Limit for that planet. That means that any moon would be torn apart by tidal stresses. The Roche Limit is given by $2.44 \left( \frac{\rho_M}{\rho_m}\right)^{1/3} R$, or about 2.5 planetary radii.

(2) Define the conditions for resonant perturbation. Make clear in your answer what is doing the perturbing, and what is being perturbed. By ``conditions'', I mean a mathematical relation.
Answer: A resonant perturbation occurs when a celestial object with period $P_n$ is acted upon by an external force (perturbed) which is also periodic with period $P_p$. Resonance occurs when the condition $nP_n = mP_p$ is satisfied. The smaller the integers $n$ and $m$, the stronger the perturbation.

(3) What determines whether we can see a hot object glow? Use equations to describe the physical principle you are invoking. Numbers would be great.
Answer: An object at finite temperature $T$ (blackbody) emits electromagnetic radiation over a range of wavelengths with a maximum at wavelength $\lambda_{max} = \frac{hc}{3k_BT}$. To see the object glow, $\lambda_{max}$ must be close to, or in the range of wavelengths to which the human eye is sensitive, i.e. 400 - 700 nm.

(4) Describe why objects in the outer solar system are colder than objects in the inner solar system. Use equations.
Answer:The equilibrium temperature of an object is given by

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

where $S$ is the solar constant. The solar constant is inversely proportional to the square of the distance from the Sun. The further a planet is from the Sun, the smaller is $S$ and the lower is $T_{eq}$

(5) Why does the gas carbon dioxide play an important role in determining the surface temperature of the Earth and other planets?
Answer:Carbon dioxide plays a role via the Greenhouse Effect. Carbon dioxide absorbs radiation in the infrared, a part of the spectrum where a planet is radiating to cool itself. Since the cooling mechanism is impeded by the Greenhouse Effect, the planetary temperature increases. Carbon dioxide and other Greenhouse gases are responsible for the surfaces of planets being warmer than they would be otherwise.

(6) Describe a piece of observational evidence that indicates that the climate on Mars might once have been much more hospitable for life than it presently is.
Answer: One of the best examples is the Valley Networks and Outflow Channels, which are dried-up water channels on Mars. This seems to indicate that when these features formed, the atmosphere of Mars would support liquid water. This would only be possible if the pressure and temperature were higher than they are now.

(7) Comment on the following assertion. The atmosphere of Mars is composed almost entirely of carbon dioxide, which is very different from the atmosphere of Earth. Therefore it makes no sense to think that life like that on Earth could have arisen there.
Answer:This assertion is not correct, because for the first 2 - 2.5 Gyr on Earth, its atmosphere was also mainly CO$_2$. Life on Earth arose during this period.

(8) In the homework problems, we discussed how you could calculate the time necessary to make an interplanetary voyage from the Earth to, say, Saturn. Describe how this calculation works. Use equations and scientific terms. No specific numbers or calculations are necessary.
Answer:You consider an elliptical orbit (having a diagram drawn up helps focus the mind) which has the distance of Earth at perihelion, and the distance to the planet at aphelion. An object on such an orbit could travel from the Earth to that planet. The major axis of the ellipse is the sum of the Earth-Sun and planet-Sun distances. The semi-major axis is half of the major axis. You use Kepler's 3rd Law to calculate the period of such an orbit. The travel time from the Earth to the planet is half of the period.

(9) Describe the difference in the chemical composition of Jupiter versus the Earth.
Answer:The Earth is primarily made of oxygen and heavier elements such as silicon and iron, elements that rocks are made of. Jupiter is primarily made of hydrogen and helium.

(10) Most of the mass of Jupiter is in the form of a highly unusual substance. What is it? By ``substance'', I mean not only the element, but the phase (solid,liquid,gas) and other physical properties.
Answer:Most of the mass of Jupiter is in the form of liquid metallic hydrogen.

(11) A fascinating discovery of the past ten years is that the Earth has apparently gone through a number of ``snowball phases'', in which the polar ice caps grew and grey until the whole planet was encased in ice. Using ideas developed in class, explain why this runaway glaciation would be a ``positive feedback'', in climate terms. That is, once the polar ice caps began to grow, they would tend to grow at a faster rate until they covered the whole planet. You can use words in the answer, but they must utilize concepts introduced in class.
Answer:Ice caps have a high albedo, so most of the energy incident on them is reflected back into space without being absorbed. When the ice caps expand, the mean albedo of the Earth goes up, so the equilibrium temperature goes down, i.e. the Earth cools. When the Earth cools, it favors the expansion of the ice caps so the albedo increases further, etc.

(12) Describe what is meant by the term ``tidal disruption'', and how it can be understood on the basis of elementary physics principles.
Answer:Tidal disruption is when the tidal stress on an object is so great that it becomes stronger than the force of gravity holding the object together. When this happens, the object is torn apart. The basic physical principle behind this is the inverse square law for gravity, $F=\frac{GMm}{r^2}$, where $m$ is the mass of a part of the object being disrupted. The inverse square dependence means the force on the front side is larger than the back side, leading to a stress which can pull the object apart.

(13) Describe Kirkwood's Gaps. First describe what they are observationally, and then why they occur.
Answer:Kirkwood's Gaps are gaps in the distribution of semimajor axes of asteroid orbits. These asteroids are in the Asteroid Belt between the orbits of Mars and Jupiter. The gaps occur for orbits which are in resonance with Jupiter. The asteroid orbits which are resonantly perturbed by Jupiter are vacated, leading to Kirkwood's Gaps.

(14) Roughly how far from the Sun are Venus, Jupiter, and Saturn? Since I don't expect you to memorize distances like these, tell me in your response how you reasoned out the values you choose.
Answer:Venus is about 0.7 astronomical units from the Sun. You can figure this out because it is a terrestrial planet and is closer to the Sun than the Earth. Jupiter and Saturn are at 5.2 and 9.5 a.u. You could figure out they must be at several au from the Sun because they are Jovian planets, and Saturn is further out than Jupiter.