ASTR 1110, Summer 2000

Homework #2

Motions of the Planets

Due at start of class, June 14 (changed from the syllabus).

Formulas & Numbers

Kepler's 3rd law: (orbital period in yr)² = (average distance in AU)³
Works for bodies orbiting our sun.
Kepler's 3rd law: (orbital period in s)² = 4 pi² (average distance in cm)³ / (G (M₁ + M₂))
Works for all bodies everywhere. M₁ and M₂ are the mass of the central body and the satellite, and G is the gravitational constant.

This homework set shouldn't require any use of a calculator. Homework solutions should be neatly typed, and be sure to fully explain your answers.

1. Fast and Slow

Ida is the first asteroid ever known to have a satellite orbiting it, named Dactyl. The Galileo spacecraft measured that Dactyl's orbital period was about one Earth month. Astronomers might want to know the asteroid's mass, so we could determine what sort of rock it was made of and where it came from.
a) Which of Kepler's laws could be used to determine Ida's mass? How?

Kepler's third law applies here, since it's the one which includes orbital period and mass. We use the second (longer) form of the 3rd law, because Dactyl is orbiting Ida, rather than directly orbiting the Sun. Since we know the orbital period P, and we could measure the distance R between the two bodies, we could plug the numbers into Kepler's 3rd law and get the mass M₁ out. From the pictures, it looks like Dactyl is a lot smaller in mass than Ida, so we can just ignore its mass (M₂) entirely in the equation.

b) Let's assume Ida's mass suddenly doubled -- for instance, if it collided and merged with another asteroid. Would Dactyl suddenly begin to orbit with a longer period, or a shorter one?

P² = 4 pi² A³ / (G (M₁ + M₂))

If M₁ were to go up, P would have to go down to keep the equation balanced. P is the orbital period in seconds: smaller orbital period means that it orbits faster. You can also think of this intuituvely: if Ida's mass were to increase, its gravity could then pull stronger on Dactyl and accelerate it in a faster circle, just like turning a tighter corner in your car if you tug on the wheel a bit harder.

2. Moon Phases

If the moon is full, approximately one time of day does it rise? (e.g., 6 AM, noon, 6 PM, midnight)? For approximately how long will the full moon and the sun be in the sky together? Could this configuration be near a solar eclipse, a lunar eclipse, or neither?

You might want to use a moon ball like we did in the planetarium - just stand in the right place, and the solution will pop out.

We did this exact problem in the planetarium: hold the moon in front of you, with the moon behind you, and you see a full moon. Spin your head around, and you'll see that when the moon just goes out of view, the sun is just coming into view: in other words, one sets when the other rises. If the sun rises at 6 AM and sets at 6 PM, the moon rises at 6 PM and sets at 6 AM. They're essentially never in the sky together. (In reality, in the summer the sun & moon are up for more than 12 hours as we saw in the planetarium, so they might be in the sky together for a couple of hours. Either way, they don't share the sky for 12 or 24 hours - more like 2 or 3 at most, and 0 hours is just fine too.)

If you move your head around in this configuration, you'll block the moon - lunar eclipse! Usually this doesn't happen every month because in the `real' solar system your head would be much smaller, and it's hard to get the Sun, Earth, & Moon to line up perfectly. You could not get a solar eclipse in the configuration - you'd have to wait two weeks for the Moon to move around the Earth until it could block the sun out.

3. Models of the Solar System

Explain several problems with Ptolemy's model of the Solar System, and what led to the heliocentric model. Why do you think that the geocentric model lasted as the preeminent description of the Solar System for 1300 years?

Ptolemy's model had several problems:

The configuration of epicycles was complex: even though circles were geometrically nice, the model required a large number of them.

The model was inconsistent with observations: the complex model no longer fit the positions of the planets accurately, by Copernicus' time.

It was inconsistent with Aristarchus' suggestion that the Sun was at the center of the Solar System.

The orbits of some bodies - for instance, comets - were particularly hard to fit with epicycles.

However, it was held onto as a good model for over a millennium, in part because:

It mostly worked, at least in Ptolemy's time. Although the planetary positions were not accurate, they were pretty good.

No one had done significant observing! There was a dearth of solid, accurate observations of the planets by Western Europeans for most of the ten centures before Copernicus and Brahe. Were they busy doing other things, was there a lack of interest in science, or was Christianity becoming more important and a threat? We can debate all of these.

Survival of the fittest: no one wanted to challenge a model that had been obviously accepted as truth for so long.

The telescope hadn't been invented yet, so there was no direct evidence (like galileo's observations of the Jovian satellites) that the Earth was not the center of the Solar System.

Although Copernicus didn't have Physics at his disposal, he did have Tycho Brahe's extremely accurate naked-eye observations of the planets' positions, and he critically analyzed whether these data could fit the Ptolemaic model or not. He concluded that, unfortunately, they could not, and then proposed - largely unprompted by earlier work - that many of the problems of the geocentric model could be solved with a sun-centered one.

4. Uranian Weather

Uranus is tilted on its side by approximately 90 degrees, so that its `North' pole doesn't point toward Polaris (like Earth's does). Uranus's orbital period is 84 Earth years and its rotational period about one Earth day.
Describe what the days and years would be like on Uranus if you were positioned on a) one of its poles, and b) its equator. How long would the sun be up for in each case, and what would the seasons in each case be like? Would the sun rise and set every day in each case, or not?

You may find it helpful to get a globe (or some fruit), and think this one out. Once you're set up right, it's easy.

Just like if you stood on our North pole day after day and saw the same stars and Sun in the sky, standing on Uranus would give you the same view of the sky day after day as the planet rotates and you stand in place. Slowly, as Uranus moves through its orbit around the Sun, the Sun would appear lower & lower in the sky, until dipping below the horizon and staying there for a half an orbit: 42 years. You'd have a 42 earth-year long summer, followed by a 42 earth-year long winter. (Or, 21 years each of summer, fall, winter, spring.) Moreover, the sun would be up in the sky continually during the summer, and never rise once during the winter.

If you were standing on the equator, the stars would rise & set similar to how they do on Earth. Every day throughout Uranus' orbit, the Sun would rise and set - sometimes high in the sky and sometimes low, depending on the time of year.

See also figure 2.11 in the text.

Dr. Henry Throop, University of Colorado / throop@broccoli.colorado.edu

Last modified 14-Jun-2000