physics)of)the)planets)...

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.