Substellar Interiors II. Overview of
Nuclear Fusion and Evolution
PHY 688, Lecture 14
Outline
• Review of previous lecture
– equation of state (EOS) in substellar objects
– thermodynamics of hydrogen
• Substellar Interiors
– nuclear fusion
– towards an analytical solution to the EOS and its
evolution
Feb 25, 2009
PHY 688, Lecture 14
2
Previously in PHY 688…
Feb 25, 2009
PHY 688, Lecture 14
3
0.05 Msun Brown Dwarf at 5 Gyr
(X = 100% H)
(1 bar = 106 dyn/cm2 = 0.99 atm)
Feb 25, 2009
PHY 688, Lecture 14
4
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 25, 2009
PHY 688, Lecture 14
5
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 25, 2009
4 /3
)1/ 2 * #
&
M
,1+ %
( /
,+ $ 0.0032MSun ' /.
PHY 688, Lecture 14
cm
6
Sun
R ∝ M0.8
Substellar radius changes by <50%;
always near 1 RJup ≈ 0.12 RSun
Feb 25, 2009
PHY 688, Lecture 14
(Burrows & Liebert 1993)
7
Hydrogen phase diagram
T
Feb 25, 2009
PHY 688, Lecture 14
(Burrows & Liebert 1993)
8
Low-Density Regime (Photosphere)
• 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
• ∆Mphot ~ 10–10 MBD
Feb 25, 2009
PHY 688, Lecture 14
9
Hydrogen phase diagram
7
0.6
ρ
∝
T
T
0
ρ
T∝
.4
7
0.6
ρ
∝
T
Feb 25, 2009
PHY 688, Lecture 14
(Burrows & Liebert 1993)
10
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”
– Coulomb parameter " = Z 2e 2 rskT > 1
– bulk of Jupiter (~85%), Saturn (~50%), brown dwarfs (>99.9%)
!
Feb 25, 2009
PHY 688, Lecture 14
11
Hydrogen phase diagram
T
strongly coupled
Coulomb plasma
Feb 25, 2009
PHY 688, Lecture 14
(Burrows & Liebert 1993)
12
Outline
• Review of previous lecture
– equation of state (EOS) in substellar objects
– thermodynamics of hydrogen
• Substellar Interiors
– nuclear fusion
– towards an analytical solution to the EOS and its
evolution
Feb 25, 2009
PHY 688, Lecture 14
13
substellar contraction stops
– by ignition of thermonuclear fuel
– by onset of degeneracy
•
thermonuclear rates depend
on both T and ρ
– Tignition is a function of ρ
– in lowest mass star, Tc
decreases at final stages
before stabilizing because
of increase in P
•
thermonuclear burning in
brown dwarfs
6
5
Tc / 106 [K]
•
– H for billions of years at
>0.070 MSun (unsustained)
– D (2H), Li for 10–100 Myr
(i.e., also unsustained)
Feb 25, 2009
stars
brown dwarfs
“planets”
R / 109 [cm]
Cooling and
Nuclear Burning
4
3
2
1
PHY 688, Lecture 14
14
(Burrows et al. 2001)
Energy Generation in
Low-Mass Stars and Brown Dwarfs
• From
Lecture 4:
Q = 1.44 MeV
Q = 5.49 MeV
reaction stops here
for <0.25 MSun stars
Feb 25, 2009
PHY 688, Lecture 14
15
Energy Generation in
Low-Mass Stars and Brown Dwarfs
•
unscreened energy generation rates for 1H + 1H (p + p) and 1H + 2H (p + d):
"˙ pp = 2.5 #10 ( $X /T
6
"˙ pd = 1.4 #10
24
2
23
6
($XYd /T
)e
23
6
%33.8 T61 3
)e
ergs gm%1cm%1
%37.2 T61 3
ergs gm%1cm%1
Yd is 2H mass fraction (primordial value is 2 × 10–5)
2 2
but recall that H is in a state of strongly coupled plasma (" = Z e rskT > 1)
•
!
!
Feb 25, 2009
PHY 688, Lecture 14
16
Hydrogen phase diagram
T
strongly coupled
Coulomb plasma
Feb 25, 2009
PHY 688, Lecture 14
(Burrows & Liebert 1993)
17
Energy Generation in
Low-Mass Stars and Brown Dwarfs
•
unscreened energy generation rates for 1H + 1H (p + p) and 1H + 2H (p + d):
"˙ pp = 2.5 #10 ( $X /T
6
"˙ pd = 1.4 #10
24
2
23
6
($XYd /T
)e
23
6
%33.8 T61 3
)e
ergs gm%1cm%1
%37.2 T61 3
ergs gm%1cm%1
Yd is 2H mass fraction (primordial value is 2 × 10–5)
•
2 2
but recall that H is in a state of strongly coupled plasma (" = Z e rskT > 1)
! – proton and deuteron screening decreases Coulomb barrier
– p + p and p + d rates enhanced by factor of
! 1.29 , 1.06#)
S " e H (0), where H(0) " min(0.977#
– at main sequence edge S ~ 2; can be higher at lower masses
!
Feb 25, 2009
"˙ pp # T 6.31$1.28 in core of transition mass object
"˙ pp # T 4 $ in core of the Sun
PHY 688, Lecture 14
18
Analytic Model of Brown Dwarfs
• Brown dwarfs
– are fully convective, hence isentropic
– have an EOS that is polytropic both above and below
PPT
– can thus be solved analytically
• Goal: solve for evolution of substellar L and Teff,
mass burning limits
– and learn something along the way!
Feb 25, 2009
PHY 688, Lecture 14
19
Hydrogen phase diagram
T
Feb 25, 2009
PHY 688, Lecture 14
(Burrows & Liebert 1993)
20
substellar contraction stops
– by ignition of thermonuclear fuel
– by onset of degeneracy
•
thermonuclear rates depend
on both T and ρ
– Tignition is a function of ρ
– in lowest mass star, Tc
decreases at final stages
before stabilizing because
of increase in P
•
thermonuclear burning in
brown dwarfs
– H for billions of years at
>0.070 MSun (unsustained)
– D (2H), Li for 10–100 Myr
(i.e., also unsustained)
Feb 25, 2009
6
5
Tc / 106 [K]
•
stars
brown dwarfs
“planets”
R / 109 [cm]
Cooling and
Nuclear Burning
4
3
2
1
PHY 688, Lecture 14
21
(Burrows et al. 2001)
From Lecture 1: BDs–a Theoretical
Expectation
• Kumar (1963)
– modeling of
<0.1MSun stars
– importance of
electron degeneracy
• minimum mass
below which objects
can not fuse H
Feb 25, 2009
PHY 688, Lecture 14
22