Astronomy 6570 Physics of the Planets Outer Planet Interiors
Giant Planets, Common Features Mass: Radius: Density: 15 317 M 3.9 11.2 R 0.69 1.67 gcm - 3 RotaJon period: 9.9 18 hours Obliq.: 3 98 Vis. Surf.: clouds; zonally banded (N?) decreasing contrast: J è S è U Atmos. Comp.: H 2 + He (roughly solar) + CH 4, NH 3, H 2 O, (enhanced by 3 5 in J) Atmos. Struct.: adiabajc below ~ 1 bar warm stratospheres Energy output: ~ 2 * solar input (exc. U) Atmos. Circ n.: Zonal winds of 100 400 ms Mag. Field: 0.14 4.2 Gauss Jlt = 0 59 Satellites: inner, regular sats ( e ~ i ~ 0) outer, irreg. sats Ring system: increasing mass: J è N è U è S assoc. with small satellites ephemeral structures?
Radio Occultation Temperature Profiles
Condensation levels correspond to predicted cloud layers CondensaJon Levels Only uppermost clouds observed directly Radio Spectrum P (bar) Jupiter T( K) Microwave emission originates from 0.5 10 bar levels T B ( K) NH 3 absorption v. strong near 1 cm è minimum in T B. λ
Planetary insolation patterns small obliquity (J) large obliquity (U)
Emitted infrared flux and equivalent brightness temperatures versus latitude for the four outer planets. The radiation is emitted, on average, from the 0.3 to 0.5 bar pressure levels. The equator-to-pole temperature differences are small. The largest temperature gradients occur at the extrema of the zonal velocity profile. (Ingersoll, 1990)
Zonal wind profiles (Voyager data)
Internal circulajon models: Saturn
Magnetic field comparison:
Comparison of planetary magnejc fields Earth Jupiter a Saturn a Uranus a Neptune a Radius, R planet (km) 6,373 71,398 60,330 25,559 24,764 Spin Period (Hours) 24 9.9 10.7 17.2 16.1 MagneJc Moment/M Earth 1 b 20,000 600 50 25 Surface MagneJc Field (Gauss) Dipole Equator, B 0 0.31 4.2 0.22 0.23 0.14 Minimum 0.24 3.2 0.18 0.08 0.1 Maximum 0.68 14.3 0.84 0.96 0.9 Dipole Tilt and Sense c +11.3-9.6-0.0-59 - 47 Distance (A.U.) 1 d 5.2 9.5 19 30 Solar Wind Density (cm - 3 ) 10 0.4 0.1 0.03 0.005 R CF 8 R E 30 R J 14 R S 18 R U 18 R N Size of Magnetosphere 11 R E 50-100 R J 16-22 R S 18 R U 23-26 R N a Magnetic field characteristics from Acuna & Ness (1976), Connerney et al. (1982, 1987, 1991). b M Eartth = 7.96 10 25 Gauss cm 3 = 7.906 10 15 Tesla m 3. c Note: Earth has a magnetic field of opposite polarity to those of the giant planets. d 1 A.U. = 1.5 10 8 km.
Planetary Interior Models (general considerajons) Assume spherical symmetry (for simplicity only!) 1. Hydrostatic equilib.: dp dr 2. Mass conservation: dm dr =!"g =!Gm r = 4# r 2 " r 2 ( )" ( ) 3. Equation of state: P = f ",T; composition 4. Heat transfer: % ' & ' ( k dt dr = F! conduction dt = $T dr $P ( ) s dp dr! convection - in approximate treatments (4) may be dropped and (3) replaced by P = f " 3 first-order D.E.'s ) 3 boundary conditions 1. m 0 2. P R ( ) = 0 ( ) = 0 %' 3.T ( R) = T (solid surface) surf & (' T eff (jovian planets) Adjust model parameters (e.g., composition, M core, etc.) to fit observables: M, R, J 2 ( ~ C ),J MR 2 4, etc. Model ) " ( r ), P ( r ), g ( r ),T ( r ) ( )
Central pressures : dp dr # roughly P c!p 0 R! 4$ 3 =! Gm" r 2 GM 2 R 5 % P c! GM 2 R 4 Case I : a uniform-density planet: m r % P r ( ) = 4$ 3 "r 3 # g = 4$ 3 G"r #! (gcm -3 ) dp =! ( 4$ dr 3 )G" 2 r ( ) = P c! 2$ 3 G"2 r 2 Boundary condition # P c = 2$ 3 G"2 R 2 = 3 8$ GM 2 R 4
Case II : P =!! 2 (n = 1 polytrope ~ Jupiter) " P c = 2# 3 9 G! 2 R 2 (see Polytrope notes) Examples Body ρ(g cm - 3 ) R (km) Case I Case II (Mb = 10 11 Nm - 2 ) Moon 3.34 1738 0.047 0.155 Earth 5.52 6371 1.73 5.68 Uranus 1.32 25,600 1.60 5.25 Jupiter 1.33 71,400 12.6 41.4 Detailed models: Earth = 3.5 Mb Jupiter = 70 Mb Uranus = 8 Mb
Adiabats Adiabatic equations of state (EOS) for hydrogen, ice, and rock. HYDROGEN H-He
Zero-temperature planet models for pure components.
Hydrogen phase diagram.
Giant Planet Models (Stephenson)
Polytropes Interior structure equations: dm dr = 4! r 2 " (1) dp dr (2) $ m r % Gm" = #g" = # r 2 (2) ( ) = # r 2 G" dm dr = # 1 G dp dr d dr ( r 2 " dp ) dr = 4! r 2 "! from (1) We assume P = " " 1+ 1 n " = constant (3) Substitute in the DE: and write " = " 0 & n ' P = P 0 & n+1 with P 0 = " " 0 1+ 1 n d dr ( ( ) + 4! G " 0 2 * r 2 d& dr ) ( n+1)p 0 ( ' set. = r, with r r 0 0 / n+1 ) * +, - r 2 & n = 0 (4) ( )P 0 4! G" 2 0 ( ) +. 2 & n = 0! the LANE-EMDEN equation. (5) d. 2 d& d. d. +, - 1 2
Boundary condijons: where $ 1 = R r 0.! =! 0 at r = 0 " # 0 P = 0 at r = R " # $ 1 ( ) = 1 (a) ( ) = 0 (b) Also! % 0 at all 0 & r < R " $ 1 = 1 st zero of # ( $ ). Also we require * dp dr '#n d# = (g! ) 0 as r ) 0 d$ d# d$ 0 ( ) = 0 (c) So starting with (a) & (c), the L-E equation is integrated outwards until # = 0 for the first time.
Polytropes (cont d.) Free parameters : { n,! 0, r 0 } " P 0,! - fit to observed M, R, J 2 R = r 0 # 1 M = m( R) = $ R2 G = $ % 1 dp & '! dr r 2 #1 2 0 ( n + 1)P G i 0 d+! r d# # 0 0 1 ( ) * from (2) R i.e., M = $ 4,! 0 r 0 3 #1 2 d+ d# # 1 and!! 0 = 3 # 1 d+ d# # 1 so for a given value of n, ( M,R) - r 0,! 0.
Polytropes (cont d.) J 2 is determined by the moment of inertia, A = B = C and A + B + C = 3C = 2 r 2 dm " C = 2 R r 2 dm 3! = 8# R r 4 $ r 3! ( )dr 0 = 8# $ r 5 3 0 0! % 1 0 % 4 & n d% 0!#" # $ Now II = ' ( 2 d * d% %2 -! &) +,. / d% = ' % 4 &1) + 2! % 3 &) d% 1 II = ' % 4 &1) + 2 % 3 &1 ' 6! % 2 & d% 1 1 % = ' % 4 d& '6! 1 % 2 & d% 1 d% % 0 1 C MR 2. For a spherical planet! and C = MR 2 '2II 3% 4 &) 1
Summary of ProperJes of Polytropes: R =! 1 r 0 " = 3 $ d# '! d! 1 % & ( ) " 0 1 P 0 = 4*G" 0 2 r 0 2 n+1 P.E. =,- GM2 R = + GM2 R 4 ( )1 M =, 4* " 0 r 0 3! 1 2 d# d!
Polytropic Models of Jovian Planets: J 2 q = 1 3 k 2 q =! 2 R 3 GM Conclusions: n 0.95 Jupiter 1.3 Saturn 1.5 Uranus 1.2 Neptune!J 4 q 2 1 "! % 2 Note: For n = 1,r = # $ 2!G & ', independent of (0 and M.
Podolak et al. (1989) Uranus
Podolak et al. (1989) Uranus Models fit Current J 2 & J 4
Hubbard and MacFarlane rock ice gas Density distribution in Uranus, calculated for a model with solar abundances of ice and rock. The temperature distribution corresponds to the present epoch.
Hubbard and MacFarlane rock ice gas Density distribution in Neptune, calculated for a model with solar abundances of ice and rock. The temperature distribution corresponds to the present epoch.
Notes on Giant Planet Interiors: Internal heat sources in J, S & N probably due to conjnued gravitajonal energy release thru contracjon: ~ 30 km/my for Jupiter. Models predict that Jupiter is also cooling at ~ K/My. Lack of internal heat source in Uranus may be a consequence of subtle composijonal gradients inhibijng interior convecjon. ObservaJons can be matched if there is no convecjon interior to ~ 0.6 R u è magnejc field must be generated in the outer icy mantle. D/H rajons in U and N are ~ 10-4, similar to Earth and comets but ~ 10 larger than in J and S è composijon dominated by ices rather than H 2 + He.