Giant planet formation
Mtotal
Mrocky core
Mgas envelope
Core accretion
= core nucleation
= core instability
= “bottom-up”
Runaway gas accretion when
Menvelope ~ Mcore
Miguel &
Brunini 08
Critical core
masses
for various
accretion
rates
Rafikov 2006
Core or No Core?
M, R, J2, J4, J6, ...
+ P(ρ)
+ hydrostatic equilibrium (w/rotation)
⇒ ρ(r)
J2
spheric features as they rotated around the planet provide
rotation periods. This method, however, is subject to erro
introduced by winds and weather patterns in the plane
atmosphere. Instead, rotation rates are now found from th
rotation rate of the magnetic field of each planet, genera
as measured by the Voyager spacecraft (radio emissions ar
ing from charged particles in Jupiter’s magnetosphere c
be detected by radio telescopes on Earth). This approach a
sumes that convective motions deep in the electrically co
ducting interior of the planet generate the magnetic fie
and that the field’s rotation consequently follows the rot
tion of the bulk of the interior. Measuring Saturn’s magnet
field rotation rate is particularly difficult because the fie
is nearly symmetric about the rotation axis of the plane
Indeed, in 2006, data from the Cassini spacecraft led to
revision in the previously accepted rotation period by 1%
and the new value, shown in Table 1, may still not refle
the true rotation of the deep interior. monics must be calculated
For the jovian planets, an x (P ) relation is typically guessed,
core
r/R
Relative contribution to
gravitational harmonics
in Saturn
an interior model computed, and the results compared to
Figure 5, which shows the
the observational constraints. With multiple iterations, a
depth from the center of t
variation 2.2
in composition
with pressure that is compatible
of a Saturn model that con
Atmosphere
with the observations is eventually found.
gravitational harmonics J 2 ,
The combination
of
these
three
ingredients,
an
equaprovide
information
about
The observable atmospheres of the jovian planets
provid
FIGURE 2 Illustration of the ways that a planet changes shape tion of state P = P (T , x , ρ), a temperature–pressure recloser to the surface.
further
constraints
on
planetary
interiors.
the of
atm
lation,
T
=
T
(P
),
and
a
composition–pressure
relation,
TheFirst,
construction
com
owing to its own rotation. A nonrotating planet (a) is purely
x = x (P ),spheric
completelytemperature
specifies pressureatas 1only
observational
bara funcpressure,
or T1 , constraints
constraia
pherical. Saturn’s distortion due to its gravitational harmonic J 2 tion of density,
P = P (ρ). Because the jovian planets are
requires several iteration
of thethedeep
interior.
The
tempe
s shown approximately to scale in (b). The J 4 and J 6 distortions believed the
to betemperature
fluid to their centers,
pressure
and
strain interior
modern computers
also distribution
related by the equation
hydrostatic
dozens
of int
ature
of the of
jovian
planets toiscalculate
believed
to follo
of Saturn are shown in (c) and (d), exaggerated by about 10 and density are
equilibrium (with a first-order correction for a rotating
many parameters within
100 times, respectively. (Figure courtesy William Hubbard, Univ.planet): a specified pressure–temperature path
known as an ad
determined boundaries. A
Ariz.)
abat. For an adiabat, knowledge of the
temperature
in the
equations of statean
o
flect the differences betwe
∂P
2 2
pressure
at a single
the temper
= −ρ(r)g(r)
+ rωpoint
ρ(r) uniquely specifies
The size and composition o
∂r
3
ture as a function of pressure at all other
along
th
rom observations of spacecraft or stellar occultations. Disas thepoints
heavy element
enric
varied with different equa
where g is the gravitational acceleration at radius r and ω
J4 x 100
GM
Φ=−
r
J6 x 100
!
1−
∞ # $2n
"
a
r
n=1
%
J2n P2n (cos θ)
Giant planet formation by
gravitational fragmentation
= gravitational instability
= “top-down”
Requirements: Q ~ 1 and tcool < Ω-1
Could be met at large distance > 70 AU
Uncertainties include
• disk temperature
• mass infall rate from surrounding natal envelope
• final planet masses
More easily fragments into brown dwarfs than planets
Kratter et al. 10
boun
d ele
log M/M☉
ns
ectro
R ( 0.1 RJ )
nd el
u
unbo
ctron
s
Degeneracy pressure
M dwarf L dwarf
75 MJ 65 MJ
T dwarf
30 MJ
The new spectral classes
OBAFGKMLT
Jupiter
1MJ
722
Cooling curves
“hot start”)
Burrows et al.(standard
: Theory of brown dwarfs
Early evolution
uncertain
“cold start”
(core accretion)
“hot start”
Hot Jupiters are inflated
Transit radii > Theoretical radii
Burrows et al. 2007
How much = How long ago
dTc
= −N k
Radiative cooling: L =
dt "
!
kTc
Not completely degenerate: R ∼ RJ 1 +
!F
σTe4 4πR2
Te
Tc
R
Isentrope: se (Te , Pe ∼ g/κe ) = sc (Tc , Pc ∼ GM 2 /R4 )
L ∝ t−24/17
R ↑ Tc ↑ t ↓ L ↑
using
more
accurate
analytic
formulae
from Burrows
& Liebert 93
Cooling tracks
Tc ∝ t−7/17
to increase R by 30% ,
t ∼ 2 × 107 yr
L ∼ 2 × 1026 erg/s
vs. numerical L ∼ 6 × 1026 erg/s
Burrows et al. 07
Log Luminosity (solar units)
3 equations
in 3 unknowns
T e , Tc , R
Log t (Gyr)
“Easy” problem
Compare required L
6 x 1026 erg/s
to
Incident L
L∗
29
2
A
∼
3
×
10
erg/s
πR
p
2
4πa
Even “easier”: When planet is irradiated,
actual required L ~ 4 x 1025 erg/s
jϴ
vφ
Br
Induced Current
Ohmic Power
q
F= v×B
c
εemf = W/q = F "/q
σA
I = εemf /R = εemf
#
Ohmic P = Iεemf
v 2 B 2 σ#A
=
c2
P = I 2R
! ! ! 2
j
dV
P =
σ
copper 6e7 S/m
drinking water 0.0005 to 0.05 S/m
j = σf = σ
!v
c
"
×B+E
Planetary conductivity
Batygin & Stevenson 10, Spiegel et al. 09
j = σf = σ
!v
c
"
×B+E
delta ~ 2.3e8 cm (R ~ 1.05e10)
θ
δ
vm
δ = 0.02 R
1 km/s
r
0
φ
v(r, θ) = vm sin θ φ̂
L∗
2
πR
∼
f
2
4πa
1
2
2
ρv
4πR
h
2
R/v
⇒ v 3 ∝ L∗ /a2
Differential rotation may only be skin deep
If winds extend too deep,
Ohmic power > internal luminosity
δ < 0.03R for Jupiter (maybe)
Also Taylor-Proudman theorem,
plus observed stability of B field,
enforces near solid-body rotation in
convective interior (maybe)
[ P(ρ) v constant on cylinders ]
Liu, Goldreich, & Stevenson 08
see also critique by Glatzmaier 08
j = σf = σ
!v
c
"
×B+E
Assume B(R) = 10 G [cf. Jupiter B(R) = 4.2 G]
Energy flux scaling : B 2 ∝ ρ1/3 q 2/3
σB 2
O(j × B)/c
∝
Elsasser Number =
O(2ρΩ × v)
ρΩ
Internal flux q for Hot Jupiter ∼ 102 q for Jupiter
Elsasser Number ∼ 1 ⇒ B 2 ∝ ρΩ/σ
Br3
To reproduce assumed B,
assume surface dynamo different from Jupiter
Christensen, Holzwarth, & Reiners 2009
Atmospheric Power
j = σf = σ
!v
c
"
×B+E
v
∼σ ×B
c
! ! ! 2
j
dV
P =
σ
σv 2 B 2
2
∼
4πR
δ
2
c
27
∼ 8 × 10 erg/s
Power at Radiative-Convective (RC) Boundary
δ
j
PRC =
! ! !
j2
dV
σ
δRC
j2
∼
2πR × δδRC
σRC
j
σ δRC
∼ 1 × 1025 erg/s
∼P
σRC R
δ
δRC
δ
How much extra power and where?
RC boundary on Jupiter 0.1--1 bar
Where :
convective
interior
Radiativeconvective (RC)
boundary
Specific entropy s = sRC ≈ score
⇒ R(s, M )
Spiegel, Silverio, and Burrows 2009