GIANT PLANETS & PLANETARY ATMOSPHERES

Transcription

1 GIANT PLANETS & PLANETARY ATMOSPHERES Problem Set 6 due Tuesday 25 October 2018 ASTRONOMY 111 FALL From last lecture INTERIOR TEMPERATURE OF A ROCKY PLANET! "# 'Λ "$ =! $ "$ + -! 1 "$ 3* + $ / "$ # = 'Λ 6* + $ / - $ + 1 where C and D are integration constants. At r = 0: T approaches infinity unless C = 0. At r = R: # = # 2 = / = # / Thus # $ = # 2 + 'Λ 6* + 8 / $ / 25 October 2018 ASTRONOMY 111 FALL

2 From last lecture INTERIOR TEMPERATURE OF A ROCKY PLANET Taking the density to be! = $% &'( ) the heating rate to be that in carbonaceous chondrites today, Λ = erg s 17 g 17 the thermal conductivity 8 9 = < erg s 17 cm 17 K 17, as appropriate for silicate rocks we Bodyget: Orbital radius r [AU] Mass M [g] Radius R [km] Albedo A b T(R) [K] T(0) [K] Earth x Moon x Vesta x T(0) is too high for the Earth and Moon, but it is about right for Vesta. 25 October 2018 ASTRONOMY 111 FALL From last lecture INTERIOR TEMPERATURE OF A ROCKY PLANET Remember, each body is assumed to be uniform in density here. When temperatures exceed 2000 K, they are overestimates:! T increases linearly with increasing T for liquid metals and convection is important for heat transport in liquids as well. Nevertheless, this demonstrates a few important points: If Earth were not already differentiated, it would become so very quickly. If the Moon has any liquid metal in its core, it is just barely liquid. Vesta is solid through and through, and has probably been so for quite some time October 2018 ASTRONOMY 111 FALL T(r) (K) Vesta Moon r/r Earth 2

3 From last lecture THE SMALLEST DIFFERENTIATED BODIES What about at earlier times? At the time of CAl formation Gyr ago, the radioactive heating power and proto-solar luminosity were Λ = *+ erg s *0 g *0 1 = 2.51 = erg s *0 Consider a small uniform sphere with non-porous, carbonaceous-chondrite composition in an orbit like that of 1 Ceres: 6 = 2.7 g cm -3, A b = 0.05, r = 2.77 AU and suppose that is it just barely massive enough that mafic minerals melt in its center: T(0) = 1200 K, so that on average 7 T = 4.30 x 10 5 erg s -1 cm -1 K October 2018 ASTRONOMY 111 FALL From last lecture THE SMALLEST DIFFERENTIATED BODIES Solving (iteratively) for the R which gives T(0) = 1200 K, we get R = 1.34 x 10 5 cm M = 2.70 x g T s = 216 K Thus, it is possible that nonporous bodies as small as a few km in size melted in their centers and became differentiated, if they formed early enough in the Solar System s history. T(r) (K) r/r 25 October 2018 ASTRONOMY 111 FALL

4 GIANT PLANETS & PLANETARY ATMOSPHERES The other giant planets Vitals of Saturn, Uranus, and Neptune Gas giants and ice giants Vertical density and pressure structure of atmospheres All four giant planets on the same scale (Voyager images, JPL/NASA) 25 October 2018 ASTRONOMY 111 FALL SATURN S VITAL STATISTICS Mass x1029 g (95.2 M ) Equatorial radius x109 cm (9.45 R ) Average density g cm-3 Moment of inertia MR2 Bond albedo Orbital semimajor axis x1014 cm (9.582 AU) Orbital eccentricity Obliquity Sidereal revolution period years Sidereal rotation period hours Moons 61+ Rings 7 major ones 25 October 2018 Saturn as seen from Cassini (JPL/NASA) ASTRONOMY 111 FALL

5 VISITS TO SATURN We have visited Saturn four times. Pioneer 11 (1979) Voyager 1 (1980) Voyager 2 (1981) Cassini ( ) The Voyager family portrait of Saturn and some of its larger moons (JPL/NASA) 25 October 2018 ASTRONOMY 111 FALL SATURN: STRUCTURE & COMPOSITION Like Jupiter, Saturn is a gas giant. It has the lowest average density of all the planets as well as a very low moment of inertia for its mass. Saturn spins almost as fast as Jupiter, and its visible surface is even more distorted by its rotation than Jupiter (polar diameter 10% smaller than equatorial diameter) owing to its lower density and larger rock core. It definitely has a rocky core, ~12 M Saturn s abundance of the elements heavier than H are more abundant than Jupiter s e.g. C/H 10(C/H). Visible constituents: 96.3% H 2, 3.25% He, 0.45% CH 4, 0.013% NH 3, 0.011% HD, % C 2 H 6, % H 2 O. T = 95 K at the cloud tops; 83 K is expected from heating by sunlight. 25 October 2018 ASTRONOMY 111 FALL

6 SATURN: STRUCTURE & COMPOSITION Saturn emits 2.5 times as much power as it receives in sunlight, similar to Jupiter. Is this related to the major abundance difference from Jupiter? There is much less helium in Saturn s upper atmosphere, leading to suggestions of formation and precipitation of liquid helium droplets (helium rain) that gradually raises the density of the interior (thus reducing the potential energy and increasing the heat). Like Jupiter, Saturn has a strong magnetic field, indicating the presence of liquid metallic hydrogen and dynamo action in the surroundings of the rocky core. Despite the muted contrast in many pictures, Saturn s cloud and belt/zone system is much like Jupiter s. 25 October 2018 ASTRONOMY 111 FALL IMAGES OF CLOUDS ON SATURN From Cassini (JPL/NASA) 25 October 2018 ASTRONOMY 111 FALL

7 SATURN S RINGS All the giant planets turn out to have rings. Distinctive feature of Saturn s rings: they are much icier than the others, and the ring particles have very high albedo (so they look much brighter). Saturn s rings as seen by Cassini (JPL/NASA) 25 October 2018 ASTRONOMY 111 FALL URANUS VITAL STATISTICS Mass x1028 g (14.5 M ) Equatorial radius x109 cm (4.01 R ) Average density g cm-3 Moment of inertia MR2 Bond albedo Orbital semimajor axis x1014 cm (19.20 AU) Orbital eccentricity Obliquity Sidereal revolution period years Sidereal rotation period hours (retrograde) Moons 27+ Rings 10 narrow ones 25 October 2018 Uranus, from the Hubble Space Telescope (STScI/NASA) ASTRONOMY 111 FALL

8 NEPTUNE S VITAL STATISTICS Mass x10 29 g (17.1 M ) Equatorial radius x10 9 cm (3.88 R ) Average density g cm -3 Moment of inertia 0.23 MR 2 Bond albedo Orbital semimajor axis x10 14 cm (30.05 AU) Orbital eccentricity Obliquity Sidereal revolution period years Sidereal rotation period hours Moons 14+ Rings 6 narrow ones Neptune, from Voyager 2 (JPL/NASA) 25 October 2018 ASTRONOMY 111 FALL VISITS TO URANUS AND NEPTUNE Only one each, both fly-bys, by Voyager 2. Uranus in 1986 Neptune in October 2018 ASTRONOMY 111 FALL

9 URANUS & NEPTUNE: STRUCTURE AND COMPOSITION Nearly the same size: Uranus is slightly larger, Neptune slightly more massive, so Neptune is significantly denser. Both have substantial cores and a much larger fraction of their mass in the cores than Jupiter and Saturn. Mass [M ] Jupiter Saturn Uranus Neptune Total Core < Atmosphere > They are rich in elements heavier than H and He, compared to Jupiter and Saturn. 25 October 2018 ASTRONOMY 111 FALL URANUS & NEPTUNE: STRUCTURE AND COMPOSITION Their cores are not rocky in the usual sense (silicate and iron): there is a lot of carbon, nitrogen, and oxygen, and a lot of hydrogen in solid and liquid phases. Or, rather, lots of CH 4, NH 3, and H 2O hence the term ice giant, to emphasize this difference from the gas giants. Both have strong magnetic fields that are oriented at large angles from the rotation axis (59 and 47 ) and are off center. The origin of these fields is still a major mystery. And they both have rings and lots of satellites. 25 October 2018 ASTRONOMY 111 FALL

10 DISTINCTIVE FEATURES OF URANUS Urauns has an obliquity of 98, causing its rotation axis to be almost parallel to the ecliptic plane instead of perpendicular. Component perpendicular to the ecliptic points in the opposite direction of revolution: retrograde rotation The orbital plane of Uranus moons is similarly tilted: the odd tilt of the planet cannot be explained as the result of one big impact. A very low contrast among the cloud bands leads to a nearly featureless appearance. It took until the Voyager 2 visits for us to be confident in the rotation period. Cloud-top temperature is 59.1 K (should be 60 K from solar heating): no substantial internal source of heat as in Jupiter and Saturn. 25 October 2018 ASTRONOMY 111 FALL VOYAGER 2 IMAGES OF URANUS In true color and contrast (left) and with false color to enhance contrast (right) (JPL/NASA) 25 October 2018 ASTRONOMY 111 FALL

11 DISTINCTIVE FEATURES OF NEPTUNE Neptune has about the same cloud-top temperature as Uranus, 59.3 K. But, it is much further from the Sun and should only be 48 K. Another planet with an internal heat source, like Jupiter and Saturn. Neptune emits about 3.5 times as much power as it receives from the Sun. The upper cloud deck rotates more slowly than the interior, unlike Jupiter, Saturn, and Uranus. The winds are very high (up to 3400 km/hr), and the storms are very violent (e.g. the Great Dark Spot). Related to the internal heat source, as in the case of Jupiter 25 October 2018 ASTRONOMY 111 FALL HYDROSTATIC EQUILIBRIUM If a parcel of air does not move vertically, the forces from gravity and pressure are balanced, a condition called hydrostatic equilibrium. Consider an infinitesimally thin slab with thickness dz, area S, and constant density!. Forces are exerted on it by gravity (its own weight) and by pressure of the air above and below. Pressure = force per unit area. Units: dyne cm -2 (0.1 Pascal) S grsdz PzS ( ) ( + ) Pz dz dzs 25 October 2018 ASTRONOMY 111 FALL

12 HYDROSTATIC EQUILIBRIUM In one dimension (as drawn):! " # = %&# '" +! " + '" #! " + '"!(") '" = '! '" = &% Equation of hydrostatic equilibrium In spherical symmetry, '! ', = &% We will deal mostly with atmospheres one thin, planeparallel layer at a time in Cartesian coordinates. S grsdz PzS ( ) ( + ) Pz dz dzs 25 October 2018 ASTRONOMY 111 FALL EXPONENTIAL ATMOSPHERES & SCALE HEIGHT Suppose that the atmosphere is Plane-parallel: thin compared to the radius of the planet s surface Made of an ideal gas, so! = #$% = &$% ' Has a uniform surface temperature # = molecules per unit volume ( ' = mean mass of molecules in the atmosphere $ = ?@A erg K?@ (Boltzmann constant) Then H! HI = '! $% K 25 October 2018 ASTRONOMY 111 FALL

13 EXPONENTIAL ATMOSPHERES & SCALE HEIGHT Rearrange and integrate from! = 0 to some arbitrary height: %(() ( 1 01 $ -+ = % & +, 23 $ -!, 4 ln +(!) ln + 4 = 01 23! +! = 7 89: ;< (=>? % & 8( = + 4 ;< A 9: P z = ( A D Exponential atmosphere Where E = ;< 9:, the isothermal scale height, is the vertical distance over which the pressure changes by G H. 25 October 2018 ASTRONOMY 111 FALL SCALE HEIGHTS FOR PLANETARY ATMOSPHERES Oddly, the scale heights of the atmospheres for terrestrial and giant planets are not drastically different in size, even though the densities, pressures, temperatures, and masses differ by many orders of magnitude. Typical values are in the tens of kilometers, which is small enough that the plane-parallel approximation is a good one over a few scale heights. Planet Isothermal scale height [km] Venus 15.9 Earth 8.5 Mars 11.1 Jupiter 27 Saturn 59.5 Uranus 27.7 Neptune 20 Pluto October 2018 ASTRONOMY 111 FALL