Abstract
In the last chapter we calculated the lifting force of the Moon by imagining a mass of any small size resting on the Earth’s surface. We looked at the ratio of the Earth’s gravitational force holding that mass fast to the surface to the competing tidal force of the Moon (and Sun) tending pull it away. The result was the proportionate tidal pull of the Moon (and Sun) on the mass. The greater was that ratio, which we called τ, the greater was the tidal pull on that mass. Multiply the pulling effects on the little mass by the great quantity of the Earth’s substance at its surface and the result is seen in the tides and (less evidently) a rise in the Earth’s crust.
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Notes
- 1.
The key to simplifying what results in a cubic equation is approximating. Because Δr << r the terms Δr 2 and Δr 3 can be eliminated, and the resulting fraction is quite close to unity to a reasonable approximation.
- 2.
By substituting GMm/r 3 for ω 2 in this equation, you will recognize it as the same tidal force equation encountered before, but in radial notation and with the coefficient 3 instead of 2: F tidal = 3GMmΔr/r 3. The coefficient is different because we used a different mathematical model (two small spheres) to create it. This “twin sphere” analysis of dynamical forces is found in many texts, but that given by James Van Allen, presented here, is one of the clearest. Van Allen [4].
- 3.
This is so since density is mass per unit volume (ρ = m/V) then m = ρV. Volume is found by the traditional formula, V = 4/3πr 3.
- 4.
See the reference cited in footnote 14, at page 1.
- 5.
The distance data is derived from http://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html. The density information comes from http://ssd.jpl.nasa.gov/?sat_phys_par.
- 6.
See generally, http://en.wikipedia.org/wiki/Rings_of_Saturn#G_Ring. There are rings that extend far beyond the Roche limit for Saturn. Some rings seem to be associated with particles being blasted off the surface of moons by meteorid or micrometeorid impacts, or ejecta from the moons themselves. Saturn’s E ring, for example, appears to be fed by cryovolcanic plumes – ice geysers – from the surface of Enceladus.
- 7.
The Cassini-Huygens mission has added enormously to our knowledge of the Saturn system. See http://saturn.jpl.nasa.gov/index.cfm for a host of pictures and data.
- 8.
See also the JPL website on Planetary Satellite Physical Parameters, http://ssd.jpl.nasa.gov/?sat_phys_par#legend
- 9.
Many moons of each of these planets thus lie beyond the ones shown in the graph. But there are no moons within the inner Roche limit, and most hover close to or are beyond 1.8 radii. Wide density and orbital differences indicate that some moons may have formed at the time their parent planet coalesced, and that others were no doubt captured in passing. A retrograde orbit (moving in reverse direction from the revolution of the planet in its orbit) is good evidence of capture. Jupiter, for example, at the outer edge of the asteroid belt, has captured many rocky, high density moons.
- 10.
Rob Landis [1]. His paper includes a fascinating description of the detailed preparations for the event.
- 11.
- 12.
E.g., D.A. Crawford (Sandia National Laboratories), Comet Shoemaker-Levy 9 Fragment Size and Mass Estimates from Light Flux Observations, http://www.lpi.usra.edu/meetings/lpsc97/pdf/1351.PDF. See also, http://www.nature.com/nature/journal/v370/n6485/abs/370120a0.html.
- 13.
Data was taken from http://ssd.jpl.nasa.gov/?sat_elem.
- 14.
The comets of the Kreutz group all have similar, though not identical orbital elements. Their orbits are highly inclined and eccentric, with semi- major axes of ~100 AU and periods of ~ 1,000 years. The original parent comet of this group is estimated to have had a diameter of perhaps 100 km. The majority of the Kreutz group comets have diameters in the range of a meter to 10 m that are only detectable when they are close to the sun. Gundlach et al. [2].
- 15.
Bortle [3].
- 16.
Ibid., 39.
- 17.
Ibid., 40.
- 18.
This idea was put forth in Gundlach et al. [2]. The authors concluded that the radius of the comet must have been between.2 and 11 km before perihelion. Matthew Knight of the Lowell Observatory and Johns Hopkins Applied Physics Lab estimated that the comet’s core must have been at least 500 m in diameter; otherwise it couldn’t have survived so much solar heating. http://science.nasa.gov/science-news/science-at-nasa/2011/16dec_cometlovejoy/.
References
Landis R (1994) Comet P/Shoemaker-Levy’s collision with Jupiter: covering HST’s planned observations from your planetarium. http://www2.jpl.nasa.gov/sl9/hst1.html. International Planetarium Society Conference, Space Telescope Science Institute
Gundlach B, Blum J, Skorov Yu V, Keller HU (2012) A note on the survival of the sungrazing comet C/2011 W3 (Lovejoy) within the Roche limit (preprint submitted to Icarus) 9 Mar 2012. http://arxiv.org/pdf/1203.1808.pdf
Bortle JE (2012) The remarkable case of comet Lovejoy. Sky Telescope 123(5):36
Van Allen JA (1993) Elementary problems and answers in solar system astronomy, University of Iowa Press, Ames, pp. 213–214
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MacDougal, D.W. (2012). Moons, Rings, and the Ripping Force of Tides. In: Newton's Gravity. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5444-1_18
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