Substellar Interiors
PHY 688, Lecture 13
Outline
• Review of previous lecture
– curve of growth: dependence of absorption line
strength on abundance
– metallicity; subdwarfs
• Substellar interiors
– equation of state (EOS)
– thermEodynamics of hydrogen
Feb 23, 2009
PHY 688, Lecture 13
2
Previously in PHY 688…
Feb 23, 2009
PHY 688, Lecture 13
3
Alkali (Na, K) lines
in visible spectra of
late-L and T dwarfs
become saturated!
2005)
Feb(Kirkpatrick
23, 2009
PHY 688, Lecture 13
4
Lorentzian Line Profile at Increasing τ
simulation
for the Hα
line profile
Feb 23, 2009
PHY 688, Lecture 13
5
Lorentzian Line Profile at Increasing τ
simulation
for the Hα
line profile
saturation at
τ>5
Feb 23, 2009
PHY 688, Lecture 13
6
Lorentzian Line Profile at Increasing τ
simulation
for the Hα
line profile
Feb 23, 2009
PHY 688, Lecture 13
7
Lorentzian vs. Gaussian Line
Profiles: Small τ
simulation
for the Hα
line profile
Lorentzian
Gaussian
Feb 23, 2009
PHY 688, Lecture 13
8
Lorentzian vs. Gaussian Line
Profiles: Large τ
simulation
for the Hα
line profile
• core more
sensitive to
Gaussian
parts
• wings more
influenced
by Lorentzian parts
Feb 23, 2009
Lorentzian
Gaussian
PHY 688, Lecture 13
9
Universal Curve of Growth
• each absorbing species follows a different curve of growth
• however, ratio of W to Doppler line width Δλ depends upon the
product of N and a line’s oscillator strength f in the same way for
every spectral line
• useful for determining element abundances
re
a
squ t
roo
1
"W #
log $
%
' &! (
flat
0
1
ar
e
n
li
W "N
!
!
Feb 23, 2009
W" N
W " ln N
v
"# = #
c
# 2kT
=
c m
1
0
1
!
2
3
log (Nf )
PHY 688, Lecture 13
4
!
10
Metallicity
[m/H]* = log N(m)*/ log N(H)*– log N(m)Sun/ log N(H)Sun
• [Fe/H]Sun = 0.0 is the solar Fe abundance
• [Fe/H] = –1.0 is the same as 1/10 solar
[m/Fe]* = log N(m)*/ log N(Fe)*– log N(m)Sun/ log N(Fe)Sun
• [Ca/Fe] = +0.3 means twice the number of Ca atoms per
Fe atom
• Sun’s metal content is Z ≈ 1.6% by mass
Feb 23, 2009
PHY 688, Lecture 13
11
The Early Universe Had No Heavy
Elements
Feb 23, 2009
PHY 688, Lecture 13
(figure credit: N. Wright, UCLA)
12
Metal Enrichment Is Due to Stars
H
He
nuclear
fusion
neutron
capture
Fe
Li
Feb 23, 2009
PHY 688, Lecture 13
13
Stellar Populations
• Young stars (Pop I) are
metal-rich
– [m/H] > –2.0
– Milky Way disk, spiral arms
• Old stars (Pop II) are
metal-poor
– [m/H] < –2.0
– Milky Way halo, bulge
• Ancient stars (Pop III) are
expected to have been
nearly metal-free (at birth)
Feb 23, 2009
PHY 688, Lecture 13
14
Cool Metal-Poor Stars
• M-type metal-poor stars are
most numerous
– M (+L) stars are longest lived
• dM: normal M dwarfs
[m/H] > –1
• sdM: M subdwarfs
〈[m/H]〉 ~ –1.3
• esdM: M extreme subdwarfs
〈[m/H]〉 ~ –2
– i.e., Pop II
• usdM: M ultra subdwarfs
〈[m/H]〉 < –2
– also Pop II
Feb 23, 2009
PHY 688, Lecture 13
15
Cool Metal-Poor Stars
• lower atmospheric opacity of
subdwarfs reveals the deeper, hotter layers of these stars
• a dM and an sdM star of the
same mass have
approximately the same
luminosity, but the sdM star
can be up to 700 K hotter
• subdwarfs : dwarfs ~ 1 : 400
(in Milky Way)
• sd’s identified by their high
velocities relative to Sun
– kinematics characteristic of
galactic halo stars
Feb 23, 2009
PHY 688, Lecture 13
16
Subdwarf SEDs
• signatures of metal deficiencies dM
esdM
L*
• higher gravity in deeper layers?
usdM
models:
(Jao et al. 2008)
Feb 23, 2009
PHY 688, Lecture 13
17
sdL7
L7
sdM’s
+
sdL’s
• enhanced
hydrides,
CIA H2
(Burgasser 2008)
Feb 23, 2009
PHY 688, Lecture 13
18
Outline
• Review of previous lecture
– curve of growth: dependence of absorption line
strength on abundance
– metallicity; subdwarfs
• Substellar interiors
– equation of state (EOS)
– thermEodynamics of hydrogen
Feb 23, 2009
PHY 688, Lecture 13
19
0.05 Msun Brown Dwarf at 5 Gyr
(X = 100% H)
(1 bar = 106 dyn/cm2 = 0.99 atm)
Feb 23, 2009
PHY 688, Lecture 13
20
From Lecture 4: Why Are Low-Mass Stars
and Brown Dwarfs Fully Convective?
•
radiation
– photons absorbed by cooler outer layers
– efficient in:
• >1 MSun star envelopes
• cores of 0.3–1 MSun stars
• all stellar photospheres
•
convection
– adiabatic exponent γ = CP/CV
– important when radiation inefficient:
• interiors of brown dwarfs and <0.3 MSun stars
• cores of >1 MSun stars
• envelopes of ~1 MSun stars
•
" dT %
3)*Lr
$ ' =(
2
3
# dr & rad
64 +r ,T
" 1 % T dP
" dT %
$ ' = $1( '
# dr & ad # - & P dr
in low-mass stars and brown dwarfs:
– large opacity κ ⇒ (dT/dr)rad steepens
– more d.o.f. per gas constituent
n = d.o.f / 2; γ = 1 + 1/n ⇒ (dT/dr)ad becomes shallower
!
Feb 23, 2009
PHY 688, Lecture 13
21
Convective Interiors Mean That:
• entropy (S) is constant throughout
– dS = dQ / T = 0 in convective interior (disregarding
radiative atmosphere)
• equation of state is adiabatic
– ∆S = 0 holds for a reversible adiabatic process
• P = Kργ (γ = 1+1/n, n = 1.5)
• brown dwarfs are polytropes of index n = 1.5
Feb 23, 2009
PHY 688, Lecture 13
22
!
From Lecture 4: A Special Solution
to the EOS – Stars as Polytropes
• P ≡ P(ρ) = Kργ, γ = 1+1/n
K - constant, n - polytropic index
• Lane-Emden equations:
1 d $ 2 d" ' n
&"
) + # (" ) = 0
2
" d" % d# (
1/ 2
– dimensionless forms of equation of
hydrostatic equilibrium
– solutions:
.
K (1-n )/ n 1
" * r /a, a * 0( n + 1)
,c
32
/
4 +G
# - dimensionless potential
P = P0" n +1, # = # 0" n , T = T0"
Boundary conditions :
• important polytropes:
– n = 3: normal stars
– n = 1.5: brown dwarfs, planets, white
dwarfs (all are degenerate objects)
– n = 1: neutron stars
– n = ∞: isothermal proto-stellar clouds
Feb 23, 2009
# (" = "1 ) = 0 stellar surface
# (" = 0) = 1 stellar core
$ d# '
& ) = 0 stellar core
% d" (" = 0
PHY 688, Lecture 13
!
23
Solutions for Substellar Polytropes
" c # M 2n /(3$n )
Pc # M 2(1+n )/(3$n )
•
•
•
n = 1.5: R ∝ M–1/3
n = 1.0: R ∝ M0 = const
R # M (1$n )/(3$n )
(Burrows & Liebert 1993)
– important for M < 4 MJup objects, in which there are Coulomb corrections to P(ρ)
degenerate !
EOS
analytic fit for brown dwarfs and planets (Zapolsky & Salpeter 1969)
1/ 3
# MSun &
R = 2.2 "10 %
(
$ M '
9
Feb 23, 2009
4 /3
)1/ 2 * #
&
M
,1+ %
( /
,+ $ 0.0032MSun ' /.
PHY 688, Lecture 13
cm
24
Sun
R ∝ M0.8
Substellar radius changes by <50%;
always near 1 RJup ≈ 0.12 RSun
Feb 23, 2009
PHY 688, Lecture 13
(Burrows & Liebert 1993)
25
EOS and Thermodynamics of
Hydrogen
• 10 orders of
magnitude
change in
pressure
• 3 orders of
magnitude
change in
temperature
• expect different
phases of
hydrogen
Feb 23, 2009
(X = 100% H)
PHY 688, Lecture 13
26
Hydrogen phase diagram
T
Feb 23, 2009
PHY 688, Lecture 13
(Burrows & Liebert 1993)
27
Low-Density Regime
• temperature is sufficient to excite rotational levels of H2
into equipartition, although not the vibrational levels
– d.o.f. of H2 molecules:
• 3 (spatial motion) + 2 (rotation around short axes) = 5
– polytropic index n = d.o.f./2 = 2.5 ⇒ γ = 1+1/n =1.4
– P = Kργ ; P = ρkT/µ ⇒ T ∝ ργ – 1 = ρ0.4
• compare with T ∝ ρ0.67 for n = 1.5 polytrope, as in:
– high-density interiors of brown dwarfs
– Jupiter, which does not have sufficient temperature in
atmosphere to excite H2 rotations into equipartition
Feb 23, 2009
PHY 688, Lecture 13
28
Hydrogen phase diagram
7
0.6
ρ
∝
T
T
0
ρ
T∝
.4
7
0.6
ρ
∝
T
Feb 23, 2009
PHY 688, Lecture 13
(Burrows & Liebert 1993)
29
High-Density Regime
• expect phase change at low T, sufficiently high ρ, P
– plasma phase transition (PPT)
– pressure ionization and metallization of H or H+He mixture
– occurs at ρ ~ 1 gm/cm3, P = 1–3 Mbar
• interior is “strongly coupled Coulomb plasma”
– bulk of Jupiter (~85%), Saturn (~50%), brown dwarfs (>99.9%)
Feb 23, 2009
PHY 688, Lecture 13
30
Onset of Coulomb Dominance
•
•
•
•
•
•
Γ – (dimensionless) ratio of Coulomb
1/ 3
2 2
$
'
energy per ion to kT
Z e
#
5/3
"
=
=
0.227
Z
T6
& )
Ze – ionic charge (Z=1 for H)
rskT
µe (
%
T6 = T / 106 K
µe – number of baryons per electron
1
1
1+ X
ne = ρNAµe – electron density
=X+ Y=
relevant length scales
µ
2
2
rs – radius of sphere that contains one
nucleus on average
re – electron spacing parameter
•
Γ << 1 : weekly coupled regime
(Debye-Hückel limit)
– Sun, blood plasma
•
Γ ~ 1 : transition to strong coupling
e
$ 3Z '1 3 $ 3µe Z '1 3
rs = &
) =&
)
4
*
n
4
*
N
#
%
%
e(
A (
$ 3 '1 3 $ 3µe '1 3
rs
re = &
) =&
) = 13
Z
% 4 *n e (
% 4 *N A # (
– brown dwarf matter, planetary
interiors : 1 < Γ < 50
Feb 23, 2009
PHY 688, Lecture 13
!
31
Hydrogen phase diagram
T
strongly coupled
Coulomb plasma
Feb 23, 2009
PHY 688, Lecture 13
(Burrows & Liebert 1993)
32