Astronomy Physics of the Planets. Outer Planet Interiors

Transcription

1 Astronomy 6570 Physics of the Planets Outer Planet Interiors

2 Giant Planets, Common Features Mass: Radius: Density: M R gcm - 3 RotaJon period: 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 ms Mag. Field: 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?

3 Radio Occultation Temperature Profiles

4 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 bar levels T B ( K) NH 3 absorption v. strong near 1 cm è minimum in T B. λ

5 Planetary insolation patterns small obliquity (J) large obliquity (U)

6 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)

7 Zonal wind profiles (Voyager data)

8 Internal circulajon models: Saturn

9 Magnetic field comparison:

10 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) MagneJc Moment/M Earth 1 b 20, Surface MagneJc Field (Gauss) Dipole Equator, B Minimum Maximum Dipole Tilt and Sense c Distance (A.U.) 1 d Solar Wind Density (cm - 3 ) R CF 8 R E 30 R J 14 R S 18 R U 18 R N Size of Magnetosphere 11 R E R J R S 18 R U R N a Magnetic field characteristics from Acuna & Ness (1976), Connerney et al. (1982, 1987, 1991). b M Eartth = Gauss cm 3 = Tesla m 3. c Note: Earth has a magnetic field of opposite polarity to those of the giant planets. d 1 A.U. = km.

11 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 ) ( )

12 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

13 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 = Nm - 2 ) Moon Earth Uranus , Jupiter , Detailed models: Earth = 3.5 Mb Jupiter = 70 Mb Uranus = 8 Mb

14 Adiabats Adiabatic equations of state (EOS) for hydrogen, ice, and rock. HYDROGEN H-He

15 Zero-temperature planet models for pure components.

16 Hydrogen phase diagram.

17 Giant Planet Models (Stephenson)

18 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 = " " 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

19 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.

20 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# # ( ) * 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.

21 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 ! % 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

22 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!

23 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.

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29 Podolak et al. (1989) Uranus

30 Podolak et al. (1989) Uranus Models fit Current J 2 & J 4

31 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.

32 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.

33 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.