SPA5241 Planetary Systems - Coursework #4

Useful Information

Object Mass (kg) Radius (km) Semi-major axis Eccentricity 1.98 × 1030 6.96 × 105 ∼ 25 days 3.30 × 1023 2440 1047.51 h 0.3871 AU 0.2056 5.97 × 1024 6,378 23.93 h 1.00 AU 0.0167 7.35 × 1022 1,737 synchronous 384,400 km 0.0549 6.41 × 1023 3,394 24.64 h 1.523 AU 0.0934 1.08 × 1016 11.2 synchronous 9380 km 0.0151 1.90 × 1027 71,398 9.92 h 5.203 AU 0.0484 5.68 × 1026 60,330 10.66 h 9.537 AU 0.0541 Mimas 3.85 × 1019 199 synchronous 185,520 km 0.0202

• The solar mass and radius are denoted by M and R . • G = 6.67 × 10−11 N kg−2 m2 • 1 ly = 9.461 × 1015 metres • 1 Astronomical Unit (AU) = 1.496 × 1011 metres

When computing the Roche radius in the coursework, use the expression that is derived from the Hill radius (i.e., the one containing the factor 31/3).

Exercises

1. and a) Which body causes Earth to gravitationally accelerate more, the Sun or the Moon? Justify your answer with some quantitative or analytic arguments. You may assume circular for both the Sun-Earth and Earth-Moon orbits if needed. b) Which body is responsible for a greater tidal acceleration at the surface of the Earth, the Sun or the Moon? Justify your answer with some calculations. Again, you may assume circular orbits for both the Sun-Earth and Earth-Moon orbits. Is the smaller tidal acceleration negligible (i.e., say ∼ 100 times less), or might it also have a noticeable effect? Justify your answer with some numbers, calculation or mathematical arguments. 2. Another expression for the Roche Radius can be derived from our expression for the Hill radius. Consider a secondary of mass M2 and radius R2 orbiting a primary of mass M1 and R1 with a circular a. This derivation involves setting the radius of a spherical secondary R2 equal to the size of its with respect to a primary. Show that this derivation gives  1/3 3ρ1 rRoche = R1 (1) ρ2 Think about why this differs from the expression derived in class (i.e., what physical pro- cesses are included in this derivation, that are not in the one we did in class).

1 3. Eccentric Orbits, Tides and Geophysics For a secondary on an eccentric (i.e. elliptical orbit), the distance between the secondary and primary goes from a maximum to a minimum and back, once each orbit. a) As we discussed in class, the height of the equilibrium in body 1 due to tidal pertur- bations from body 2 can be estimated from the following expression.

   3 5 M2 R1 H1 = R1 (2) 2 M1 r

where M, R, and r refer to the masses, radii and separation of the bodies. How does an eccentric orbit affect the height of the equilibrium tide raised in the sec- ondary? To answer this compute the height of the equilibrium tide in the Mercury due to the tidal perturbations of the Sun when Mercury is at i. its pericentre ii. its apocentre. b) Discuss and list a few ways that the eccentric orbit might affect the observable and geophysical properties of a real planet or secondary (i.e. a satellite that is not perfectly elastic and/or deformed beyond the elastic limit). 4. Hill sphere and Roche Radius: Phobos at Mars (and planetary satellites more generally) General information regarding Phobos and images of the satellite can be found on Wikipedia [http://en.wikipedia.org/wiki/Phobos (moon)]. a) Compute the Roche radius for bodies orbiting Mars with density similar to that of Phobos (ρ = 1900 kg/m3). Express the answer in metres and Martian radii. b) As we discussed in class, the height of the equilibrium tide in body 1 due to tidal pertur- bations from body 2 can be estimated from the following expression.

   3 5 M2 R1 H1 = R1 (3) 2 M1 r

where M, R, and r refer to the masses, radii and separation of the bodies. Compute the height of the equilibrium tide on Phobos. Express this height H both in metres and body radii (i.e., H/RPhobos). Phobos is shaped like a triaxial ellipsoid with axis lengths A > B > C and A = 13.4-km, B = 11.2-km, and C = 9.2-km (see e.g., wikipedia for the geometric definitions of an ellipsoid http://en.wikipedia.org/wiki/Ellipsoid). How does the estimated tidal height compare with the observed shape of Phobos (A/R)? Does Phobos appear to have internal rock strength sufficiently large to prevent tidal deformation or is the estimated tidal height similar in size to Phobos’ observed dimensions? Note that A ≈ H + R as H is the deviation from the spherical radius R. c) Compute the Hill sphere of Mars’s satellite Phobos with respect to Mars and express this Hill sphere in units of Phobos’s radius. d) How does the eccentricity of the orbit affect the height of the tide raised in Phobos?

2 5. Saturn’s F-ring is a narrow ring orbiting outside the main at a distance of 140,220 km from the centre of the planet (see Figures1 a & b). Curiously the F-ring has satellites interior to it (Prometheus) and exterior to it (Pandora). The Cassini spacecraft has observed large, optically thick clumps in this ring. The gravity of these clumps acting on the rings creates structures in the ring that enable an estimate of their bulk density (ρclump = 250 kg m−3). Using this information, assess the ability of these clumps in the F-ring to survive under, or be disrupted by, tidal perturbations from Saturn. Support your discussion with quantitative arguments. Note, Saturn’s physical parameters are mass M = 5.68 × 1026 kg, radius R = 6.033 × 107 m and density ρ = 687 kg m−3.

(a) Saturn’s ring system (b) structure in the F-ring

Figure 1: (a) Cassini image of Saturn’s rings. The F-ring is the thin narrow ring orbiting outside the main ring system to the right. The moon Pandora can be seen orbiting just outside the F-ring. (b) A closeup image of the F-ring shows a variety of clumps and structure. [images courtesy of NASA]

3