SCCC 115
Homework 8 - Due 06 Nov. 2005
Prof. Christina Lacey
R&D: 11-20, 12-16, 15-8, 15-10, 15-12     P: 12-3, 12-5, 13-3

R&D: 11-20:
Water is relatively uncommon among the terrestrial planets. Is it common among the moons of Jupiter.

Yes, there is much water on many of the moons of Jupiter and even in Jupiter itself. The cooler temperatures in the outer part of the solar system aided the formation of the solid water and made it possible for the water to remain.

R&D: 12-16:
Why does Titan have a dense atmosphere when other large moons in the solar system do not?

The most important reason is Titan's greater distance from the Sun. Titan's temperature is 94 K, which comes from heating of Titan by solar radiation. At 94 K, the escape speed is low and methane and ammonia condense or freeze, which helped Titan retain these molecules as did absorption of methane and ammonia by water ice. As Titan's internal radioactivity warmed the moon, the ice released the trapped gases into the atmosphere. Sunlight split the ammonia into hydrogen and nitrogen. The hydrogen was able to escape, but nitrogen is heavier and remained. Methane is hard to break apart, so it too remained in the atmosphere. The argon present in Titan's atmosphere probably arose from the out-gassing from Titan's interior.

R&D: 15-8:

Why do giant planets ``migrate?''

Giant planets migrate inward due to friction between the planets' rotation and the solar nebula material. The friction causes the giants planets' rotation to slow down, and then conservation of angular momentum causes the giant planet to move closer to the center of mass (gravity) of the system,which is inward.

R&D: 15-10:
What influence did Earth's location in the solar nebula have on its final composition?

The Earth's location relative to the Sun is critical to understanding its composition. At 1 AU, temperatures were high enough to initially destroy dust grains (silicate and carbonaceous). When cooling occurred, rocky silicate grains formed. This provided the seed material and makes up most of the Earth. Also, Earth's location in the inner solar system subjected it to a significant bombardment of comet-like material that gave Earth its water and helped modify the atmosphere.

R&D: 15-12:
What happened to the outer planets as the solar system was cleared of icy planetesimals?

Uranus and Neptune, in particular, probably moved outward as the icy planetesimals were cleared from the solar system due to gravitational interactions. Neptune captured Pluto and other plutinos and entered into a 3:2 resonance with Pluto and the plutinos. Jupiter is believed to have move inward and Saturn moved outward, but not by much. All of this migration occurred after all the outer giant planets moved inward due to friction with the solar nebula.

P: 12-3:
What would the mass of Saturn be if it were composed entirely of hydrogen at a density of 0.08 kg/m$^3$, which is the density of hydrogen at sea level on Earth? Assume Saturn is spherical. Compare your answer with Saturn's actual mass and the mass of the Earth.

$$R_{Saturn} = 6.0 \times 10^4~ km$$

$$V = \frac{4}{3} \pi R^3_{Saturn}$$

$$\rho = density=\frac{M}{V}$$

$$M_{Saturn} = \rho V = \rho \frac{4}{3} \pi R^3_{Saturn} = 0.08 kg/m^3~ \frac{4}{3}\pi (6.0 \times 10^4~ km ~\frac{1000~m}{1~km})^3$$

$$= 0.08 kg/m^3~ \frac{4}{3}\pi (2.16 \times 10^{23}~m^3 = 7.2 \times 10^{22}~kg$$ (1)

Saturn's actual mass is $5.68 \times 10^{26}$ kg and the Earth's mass is $6.0 \times 10^{24}$ kg. If Saturn were made of hydrogen then it would be much less massive than the real Saturn and much less massive than the mass of the Earth, even though it is much larger in size than the Earth. In fact, a Saturn made out of hydrogen would have roughly the same mass as the Moon.

P: 12-5:
If Saturn's surface temperature is 97 K and the planet radiates 3 times more energy than it receives from the Sun, use Stefan's Law to calculate what the surface temperature would be in the absence of any internal heat source.

Stefan's Law (pg. 74):

$$F=\sigma T^4$$ (2)

So the energy per area per time F, is 3 times more than it should be if Saturn did not have an internal heat source:

$$F_{observed}=\sigma T^4= 5.67 \times 10^{-8} \frac{W}{m^2~K^4} (97~K)^4 = 5.0 \frac{W}{m^2}$$ (3)

$$F_{sunlight}=\frac{1}{3}~ F_{observed}= \sigma T_{sunlight}^4$$ (4)

So:

$$T_{sunlight}^4 = \frac{F_{observed}}{3\sigma}$$

$$T_{sunlight} = \left ( \frac{F_{observed}}{3\sigma} \right )^{1/4} = \left ( \fr... ...m^2~K^4}} \right )^{1/4} = \left ( 3.1 \times 10^7~ K^4 \right )^{1/4} = 74.6~K$$ (5)

P: 13-3:
What is the angular diameter of the Sun, as seen from Uranus? What is the angular diameter of Titania, as seen from Uranus? Will you see solar eclipses from Uranus?

Values:

$$D = Diameter ~Sun_\odot = 2 R_\odot = 1.4 \times 10^6 ~km$$

$$d = distance~ from~ Sun~ to~ Uranus= 2,871 \times 10^6 ~km =2.9 \times 10^9 ~km$$

$$D = Diameter ~ Titania = 1.6 \times 10^3~ km \mbox{Table 3.1, pg. 337}$$

$$d = distance~ from~ Uranus~ to~ Titania = 4.4 \times 10^5~km$$ (6)

Angular diameter of the Sun, as seen from Uranus:

$$\alpha = angular~diameter = \frac{D}{d} ~57.3^\circ = \frac{1.4 \times 10^... ...s 10^9 ~km }~57.3^\circ = 2.9 \times 10^{-2\; \circ} \frac{60'}{1^\circ}= 1.7 '$$

Angular diameter of Titania, as seen from Uranus:

$$\alpha = angular~diameter = \frac{D}{d} ~57.3^\circ = \frac{1.6 \times 10^3~ km}{4.4 \times 10^5~km }~57.3^\circ = 0.2^\circ \frac{60'}{1^\circ}= 12.5 '$$

You will see solar eclipses from Uranus caused by Titania, but Titania will completely block the Sun as Titania has a much larger angular size.