physics)of)the)planets)...
TRANSCRIPT
Astronomy 6570
Physics of the Planets
Outer Planet Interiors
Giant Planets, Common Features
Mass: 15 – 317 M Radius: 3.9 – 11.2 R Density: 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.: H2 + He (roughly solar)
+ CH4, NH3, H2O, … (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
CondensaJon Levels
Radio Spectrum
T(°K)
P (bar)
Jupiter
λ
• Microwave emission originates from 0.5 – 10 bar levels • NH3 absorption v. strong near 1 cm è minimum in TB.
TB(°K)
• Condensation levels correspond to predicted cloud layers
• Only uppermost clouds observed directly
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 Jupitera Saturna Uranusa Neptunea
Radius, Rplanet (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/MEarth 1b 20,000 600 50 25
Surface MagneJc Field (Gauss)
Dipole Equator, B0 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 Sensec +11.3° -‐9.6° -‐0.0° -‐59° -‐47°
Distance (A.U.) 1d 5.2 9.5 19 30
Solar Wind Density (cm-‐3) 10 0.4 0.1 0.03 0.005
RCF 8 RE 30 RJ 14 RS 18 RU 18 RN
Size of Magnetosphere 11 RE 50-‐100 RJ 16-‐ 22 RS 18 RU 23-‐26 RN
a Magnetic field characteristics from Acuna & Ness (1976), Connerney et al. (1982, 1987, 1991). b MEartth = 7.96 × 1025 Gauss cm3 = 7.906 × 1015 Tesla m3. c Note: Earth has a magnetic field of opposite polarity to those of the giant planets. d 1 A.U. = 1.5 × 108 km.
Planetary Interior Models (general consideraJons)
Assume spherical symmetry (for simplicity only!)
1. Hydrostatic equilib.: dPdr = !"g =
!Gm r( )"r2
2. Mass conservation: dmdr = 4# r 2"
3. Equation of state: P = f ",T; composition( )
4. Heat transfer: k dT
dr = F! conduction
dTdr =
$T$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( ) = 0
2. P R( ) = 0
3.T R( ) =Tsurf (solid surface)Teff (jovian planets)
%&'
('Adjust model parameters (e.g., composition, Mcore, etc.) to fit observables:
M,R,J2 ~ CMR2( ),J4, etc.
Model ) " r( ),P r( ), g r( ),T r( )
Central pressures : dP
dr = ! Gm"
r 2
# roughly Pc !P0
R ! 4$3
GM2
R5
% Pc !GM2
R4
Case I : a uniform-density planet:
m r( ) = 4$3 "r 3 #
g = 4$3 G"r #
dPdr = ! 4$
3( )G"2 r
% P r( ) = Pc !2$3 G"2 r 2
Boundary condition # Pc =2$3 G"2R2 = 3
8$GM2
R4
! (gcm-3)
Case II : P = !!2 (n = 1 polytrope ~ Jupiter)
"Pc =2# 3
9 G!2R2 (see Polytrope notes)
Body ρ(g cm-‐3) R (km) Case I Case II (Mb = 1011 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
Examples
Detailed models: Earth = 3.5 Mb Jupiter = 70 Mb Uranus = 8 Mb
Adiabats
HYDROGEN
H-He
Adiabatic equations of state (EOS) for hydrogen, “ ice”, and “rock”.
Zero-temperature planet models for pure components.
Hydrogen phase diagram.
Giant Planet Models (Stephenson)
Polytropes
Interior structure equations: dmdr = 4! r 2" (1)
dPdr = #g" = # Gm"
r2 (2)
(2) $ m r( ) = # r2
G"dPdr
% dmdr = # 1
Gddr
r2"
dPdr( ) = 4! r 2" ! from (1)
We assume P = " "1+ 1
n " = constant (3)
and write " = "0 &n ' P = P0 &
n+1
with P0 = " "0
1+ 1n
Substitute in the DE:
ddr r 2 d&
dr( ) + 4!G "02
n+1( )P0
()*
+,-
r 2 &n = 0 (4)
' set . = rr0
, with r0 /n+1( )P04!G"0
2()*
+,-
12
dd. .2 d&
d.( ) + .2 &n = 0 ! the LANE-EMDEN equation. (5)
Boundary condiJons:
! = !0 at r = 0 "# 0( ) = 1 (a)
P = 0 at r = R "# $1( ) = 0 (b)
where $1= R
r0
.
Also ! % 0 at all 0 & r < R "$1 = 1st zero of # $( ).Also we require dP
dr'#n d#
d$= (g!) 0 as r ) 0
* 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, r0{ }" P0,! - fit to observed M,R, J2
R = r0 #1
M = m R( ) = $ R2G
1!
dPdr
%&'
()*R
from (2)
= $r02#
12
Gi
n +1( )P0!
0r0
d+d# #
1
i.e., M = $ 4, !0
r03 #
12 d+
d# #1
and !!
0= 3#1
d+d# #
1 so for a given value of n, M,R( )- r0, !0.
Polytropes (cont’d.)
J2 is determined by the moment of inertia, C
MR2. For a spherical planet
A = B = C and A + B +C = 3C = 2 r 2 dm! " C = 2
3 r 2 dm0
R
! = 8#3 r 4 $ r( )dr
0
R
! = 8#
3 $0 r05 %4 &n d%
0
%1!II
! "# $#
Now II = ' (2 dd%
%2 )&*+,
-./ d%!
= ' %14 )&
1+ 2 %3 )&! d%
= ' %14 )&
1+ 2 %
13 &
1' 6 %2 & d%!
= ' %14 d&
d% %1'6 %2 & d%0
%1!
and C
MR2=
'2II
3%14 )&
Summary of ProperJes of Polytropes:
R = !1 r0
" = 3!1
d#d!
$%&
'()1
"0
P0 =4*G"0
2 r02n+1 = + GM2
R4
P.E.= ,- GM2
R
M = , 4* "0 r03 !1
2 d#d!( )
1
Polytropic Models of Jovian Planets:
J2
q= 1
3 k2
!J4
q2
q =
! 2R3
GMConclusions: n ≈ 0.95 … Jupiter 1.3 … Saturn 1.5 … Uranus 1.2 … Neptune
Note: For n = 1,r = !
2!G"#$
%&'
12
, independent of (0 and M .
Podolak et al. (1989) Uranus
Podolak et al. (1989) Uranus
Models fit Current J2 & J4
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 Ru 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 H2 + He.