Problems

practice

1. Geosynchronous Satellite
   There is a special class of satellites that orbit the earth with a period of one day.
   1. Determine the orbital radius at which the period of a satellite's orbit will equal one day. State your answer as …
      1. an approximate fraction of the moon's orbital radius
      2. an approximate multiple of the earth's radius
      3. an "exact" number of kilometers
   2. How will the satellite's motion appear when viewed from the surface of the earth?
   3. What type of satellites use this orbit and why is it important for them to be located in this orbit? (Keep in mind that this is a relatively high orbit. Satellites not occupying this band are normally kept in much lower orbits.)
2. Kirkwood Gaps
   The asteroids are a group of small rocky bodies orbiting the sun in relatively circular orbits. (In comparison, comets are small icy bodies orbiting the sun in highly elliptical orbits.) The number of asteroids currently identified is something on the order of 200,000 but only about half of these are in orbits that are known with enough certainty to receive an official catalog entry in the Minor Planet Center Orbit Database (MPCORB). The vast majority of asteroids lie in the region between the orbits of mars and jupiter known as the main asteroid belt. The graph below shows the distribution of asteroids in the densest part of this region. The values highlighted in red show orbital radii for which there are few or even no corresponding asteroids. The values highlighted in blue show the effective edges of this part of the main belt. These features were discovered by the American astronomer Daniel Kirkwood (1814-1895) in the Nineteenth Century and are now known as Kirkwood gaps.

   [magnify]

   These orbits are empty because they share a simple harmonic relationship with the orbit of jupiter; that is, the ratio of the period of an unoccupied or under-occupied orbit in a Kirkwood gap forms a simple whole number ratio with the period of an orbit of jupiter (something like 2:1 or 3:2 or 5:3). Because of this synchrony the point of closest approach between the two bodies -- the moment when their mutual gravitational attraction is the greatest -- will always take place at the same phase in the asteroid's orbit. Small perturbations applied at just the right moment over and over again reinforce one another until eventually the asteroid enters a new orbit. Repeat this procedure for many simple harmonic ratios and a series of gaps will open up in an asteroid belt that would otherwise be randomly populated.

   Using a statistical or spreadsheet application determine …

   1. the radii of all possible resonant orbits that can be generated using the numbers 1 through 9
   2. the resonance ratios responsible for each of the seven Kirkwood gaps identified above
3. Write something different.
4. Write something different.

conceptual

1. Kepler's first two laws of planetary motion state that "the path of each planet about the sun is an ellipse with the sun at one focus" and "each planet moves so that a line drawn from the sun to the planet sweeps out equal areas in equal periods of time." Explain in one paragraph how these two laws together agree with the more fundamental laws of …
   1. conservation of angular momentum
   2. conservation of energy

numerical

1. In 1984 Marc Davis, Piet Hut, and Richard A. Muller presented the following highly speculative hypothesis in a letter to the British science journal Nature.

   A 26-Myr periodicity has recently been seen in the fossil record of extinction in the geological past. At least two of these extinctions are known to be associated with the impact on the Earth of a comet or asteroid with a diameter of a few kilometres. We propose that the periodic events are triggered by an unseen companion to the Sun, travelling in a moderately eccentric orbit, which at its closest approach (perihelion) passes through the "Oort cloud" of comets which surrounds the Sun. During each passage this unseen solar companion perturbs the orbits of these comets, sending a large number of them (over 1 × 10⁹) into paths which reach the inner Solar System. Several of these hit the Earth, on average, in the following million years….

   This "unseen solar companion" later acquired the name Nemesis, after the Greek goddess of divine retribution (Νέμεσις).

   Determine the average distance from Nemesis to the sun. Give your answer in …

   1. astronomical units
   2. light years
   3. compare it to the distance to the nearest star (4.3 light years)

statistical

1. de-revolutionibus.txt
   Read the following passage from the English translation of De Revolutionibus. Determine the period of each of the five planets known to Copernicus in the Sixteenth Century using his measurements. State your answers in years or days as appropriate. Compare them to the Twenty-first Century values. Compile your results in a table like the one below.

   planet	period (copernicus)	period (contemporary)	per cent deviation
   saturn		29.4580 years
   jupiter		11.8625 years
   mars		686.980 days
   venus		224.701 days
   mercury		87.9708 days
   		average deviation →

2. The asteroids in the following table are reported to be in resonant orbits with a planet. Determine
   1. the ratio of the number of periods of the asteroid to the number of periods of the planet,
   2. the most likely ideal ratio that relates them, and
   3. the per cent deviation between the observed and ideal ratios.

   asteroid			planet		nasteroid : nplanet		deviation
   name		r (AU)	name	r (AU)	observed	ideal	(%)
   1685	toro	1.367	venus	0.723
   1685	toro	1.367	earth	1.000
   1221	amor	1.920	earth	1.000
   3753	cruithne	0.998	earth	1.000
   87	alinda	2.485	jupiter	5.204
   8	flora	2.201	jupiter	5.204
   	pluto	39.482	neptune	30.047

3. Write some sort of resonant orbit problem to go with this nice illustration.

   [magnify]

   	janus	mimas	enceladus	tethys	dione	rhea	titan
   1.120 & 1.190 rs ringlets
   D68, D72, D73 ringlets
   d-c ring interface (guerin division)
   colombo gap & titan ringlet
   maxwell gap & maxwell ringlet
   1.470 rs & 1.495 rs ringlets
   c-b ring interface (bond-dawes gap)
   end of b ring
   cassini division		2:1	3:1	4:1	5:1	9:1
   huygens gap & ringlet
   start of a ring
   encke gap & I, main, O1, O2 ringlets		5:3
   keeler gap
   outer edge of a ring	7:6
   roche division
   R/2004 S 1 & S 2 ringlets
   f ring
   start of g ring
   end of g ring
   start of e ring
   end of e ring