29:61 General Astronomy
Fall 2004
Third (and last) Hour Exam ...December 16, 2004
Version with Answers

(1) A satellite orbits a planet at a distance $r$ from the center of the planet. Write down a formula for the orbital period of the satellite.

Answer: The orbital period is given by

$$vP = 2 \pi r$$ (1)

where $P$ is the orbital period, and $v$ is the circular orbit equation. Substituting the circular orbit equation gives us

$$P = \frac{2 \pi r^{3/2}}{\sqrt (GM)}$$ (2)

So you plug in the value for $r$ and (knowing $M$) get the period.

(2) Describe in words and equations how you would find the locations of orbits in 2:1 and 3:1 resonance with the satellite referred to in (1).

Answer: A particle in a 3:1 resonance would have a period 1/3 that of the satellite calculated above. You can then invert the equation above to give a expression for the radius of that orbit,

$$r = \left(\frac{GM (P/3)^2}{4 \pi^2} \right)^{1/3}$$ (3)

For a 2:1 resonance, you would divide the period of the satellite by 2, and solve for $r$ in the same way.

(3) Define what is meant by temperature of condensation. Why is this concept important in our understanding of the formation of the solar system?

Answer: The temperature of condensation of a material is the temperature at which it makes a transition from the vapor phase to the solid or liquid phase. The temperature of condensation depends on (is a function of) the pressure. For temperatures higher than the temperature of condensation, the substance is in the vapor phase. For temperatures less than the temperature of condensation, the material condenses in the solid or liquid phase.

The concept is important in the study of the early solar system because it governed the conditions under which solid material condensed out of the early solar nebula to begin forming the planetesimals, and later the planets. The temperature of condensation for different materials is crucial to understanding the present day chemical compositions of the planets, i.e. the differences between the terrestrial and jovian planets.

(4) Define the term planetesimal.

Answer: Planetesimal is the term used for small, solid objects in the early solar system which were the building blocks of the planets. It usually is reserved for objects in the size range from tens of kilometers to hundreds of kilometers, but is also used for objects from the size of a baseball to the size of the moon. The important point is that they were solid objects, and they later aggregated into the planets we see today.

(5) Comet Machholz, presently visible in the evening sky, 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?

Answer: The eccentricity is defined as the distance between the foci divided by the major axis. Or the distance from the center of the ellipse to the focus ($CF$) divided by the semimajor axis $a$, i.e.

$$\epsilon = \frac{CF}{a}$$ (4)

The distance from the center of the ellipse to the point of perihelion passage is $a$. The distance from the Sun to the point of perihelion passage (equal to the perihelion distance $q$ is

$$q = a - CF = a - \epsilon a = a(1-\epsilon)$$ (5)

We are given $q$ is this problem, $q = 1.2$ au, so we can solve for the semimajor axis,

$$q = a(1-\epsilon)$$ (6)

$$a = \frac{q}{(1-\epsilon)} = \frac{1.2}{(1-0.999502} = \frac{1.2}{4.98 \times 10^{-4}} = 2410 \mbox{ au}$$ (7)

(6) 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 lies inside the Roche limit, or tidal disruption radius. This is the distance of approach to a large body at which a small body is disrupted, or torn apart by the tidal stress from the large object. The equation for this shows that it, for similar densities of the large and small object, it occurs about 2.5 radii from the center of the large object.

(7) 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.

A resonant perturbation is one in which the orbital period of the (smaller) object being perturbed has a mathematical relationship to the orbital period of a large object doing the perturbing, in which an integer ($m$) times the orbital period of the smaller object equals another integer ($n$) times the orbital period of the large object. The perturbation is strongest when both $n$ and $m$ are small integers.

(8) In class I said that comets are visible for only a small fraction of each orbital period. Explain why this is the case. What physical process controls the visibility of a comet, and why doesn't it work all the time?

Answer: Comets become visible when the ices that form their nuclei sublimate and expand into space, forming the coma and tails. Sublimation occurs when the ice heats up from sunlight. The closer a comet comes to the Sun, the warmer it gets. To get the ice to sublimate, a comet must come within a few astronomical units of the Sun. Most comets only spend a tiny fraction of each orbital period close enough to the Sun to sublimate.

(9) Draw a diagram illustrating the structure of a comet. Distinguish between the features we can see in observations from Earth, and those which require a visit from a spacecraft to be observed.

Answer: The features drawn should include the nucleus, the coma, and the two tails, the ion tail and the dust tail. All of these can be seen from Earth except the nucleus. Nuclei of comets have only been photographed by spacecraft that come very close to them.

(10) Briefly describe the kinds of objects which exist beyond the orbit of Neptune. Use names and terms for these objects, and be sure to note the most important ones. Accurate numbers describing the properties of these objects would be, like, awesome.

Answer: We should start with the (a) Kuiper Belt objects, which can be fairly large, probably ice objects up to a couple of thousand kilometers in diameter. The largest ones are Pluto and Quaoar. They are more or less in the plane of the ecliptic, and have orbits from about 30 to 50 astronomical units. (b) Oort Cloud. This is a huge cloud of comets from about 5000 to 50000 astronomical units. It is the source of the long period comets (like Comet Machholz), and shows no preference for the plane of the ecliptic. (c) The recently discovered object Sedna does not fit either category, and is probably representative of a new class of objects. It is on an elliptical orbit between 80 and 900 astronomical units, and is comparable in size to Pluto.

(11) Discuss the evidence for a link between meteorites and asteroids. There are several pieces of evidence, but one is particularly compelling. I want to see it mentioned and briefly discussed.

Answer: The most compelling link is that orbits can be determined for some prominent meteors that have left meteorites. These orbits carried the meteorites out into the asteroid belt, so this is clearly where they came from. Other than that, both asteroids and meteorites can be classified into very similar general categories such as C (carbonaceous), S (stony), M (metallic), etc, so it is clear that in the asteroid belt we are seeing rocks which are essentially the same as those we pick up as meteorites.

(12) How long ago did the solar system form? What role does the radioactive decay of $^{129}$I (an isotope of iodine) to $^{129}$Xe (an isotope of xenon) play in figuring out the history of the solar system?

Answer: The solar system formed 4.55 billion years ago. The Iodine - Xenon radioactive decay is important because it has a very short halflife of 16 Million years. Most meteorites show evidence of having form with the radioactive iodine, which later decayed into Xenon. This shows that the planetesimals formed as solid rocks, including big solid rocks, within a few million years to a few tens of millions of years throughout the solar system.

(13) What is our current understanding of the origin of the Earth's moon (The Moon, with a upper case ``M''). State an observation or property of the present Earth-Moon system which supports this theory of lunar origin. Be sure and give the generally-accepted name for this theory.

Answer: The best theory at the present time is the giant impact theory. According to this view, the Earth had formed with its present mass, and had time to become differentiated, that is that the iron and heavy metals had time to sink and form the Earth's core. At that point, a large object the size of Mars (the second largest planetesimal in our zone) impacted with the Earth and blew off a large part of the mass of the mantle, which subsequently formed the moon. The best evidence in support of this is the fact that the mineralogical content of the Moon looks identical to material from the Earth's mantle that has been highly heated. So it pretty much formed out of the same stuff the Earth is made of, but with the absence of the heavy elements.

Good luck in the remainder of your academic career. I have enjoyed teaching this class and getting to know you. Have a good time during vacation.