The Physics
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Opus in profectus

# Heliocentrism

## 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 19th 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….

Davis, Hut, Muller, 1984

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

Determine the average distance from Nemesis to the Sun.

1. astronomical units
2. light years
2. Compare your answer to the distance to the nearest star (4.3 light years)

### statistical

1. 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 16th century using his measurements. State your answers in years or days as appropriate. Compare them to the 21st century values. Compile your results in a table like the one below.

For the Earth has 57 revolutions in respect to saturn — we call this the movement of parallax — in 59 of our solar years 1 day 6 minutes of a day 48 seconds approximately: during this time the planet has by its own movement completed two circuits plus 1°6'6".

Jupiter is outrun by the Earth 65 times in 71 solar years minus 5 days 45 minutes 27 seconds: during this time the planet by its own movement has 6 revolutions minus 5°41'2½".

Mars has 37 revolutions of parallax in 79 solar years 2 days 27 minutes 3 seconds: during this time the planet by its own movement completes 42 periods plus 2°24'56".

Venus outruns the movement of the Earth 5 times in 8 solar years minus 2 days 26 minutes 46 seconds. And during this time it has 13 revolutions minus 2°24'40" around the Sun.

Finally, Mercury completes 145 periods of parallax, by which it outruns the movement of the Earth, in 46 solar years plus 34 minutes of a day 23 seconds. And it has 191 revolutions around the Sun in that time plus 34 minutes of a day 23 seconds approximately.

Nicolaus Copernicus, 1543

How good was Copernicus's planetary data?
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 minor celestial bodies 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
3. the per cent deviation between the observed and ideal ratios
Minor objects in orbital resonance with planets
object 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
134340 Pluto 39.482 Neptune 30.047
3. Write some sort of resonant orbit problem to go with this nice illustration.

Magnify

ring feature ↓ moon → Jan. Mim. Enc. Tet. Dio. Rhe. Tit.
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 & 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 S1 & S2 ringlets
F ring
start of G ring
end of G ring
start of E ring
end of E ring