Substellar Evolution
PHY 688, Lecture 15
Outline
• Review of previous lecture
– fusion in brown dwarf interiors
– towards an analytical solution to the EOS
• Substellar Evolution
– analytical solution for L(t) and Teff(t)
Feb 27, 2009
PHY 688, Lecture 15
2
Previously in PHY 688…
Feb 27, 2009
PHY 688, Lecture 15
3
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 27, 2009
stars
brown dwarfs
“planets”
R / 109 [cm]
Cooling and
Nuclear Burning
4
3
2
1
PHY 688, Lecture 15
(Burrows et al. 2001)4
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 27, 2009
PHY 688, Lecture 15
5
Energy Generation is Enhanced 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 27, 2009
"˙ pp # T 6.31$1.28 in core of transition mass object
"˙ pp # T 4 $ in core of the Sun
PHY 688, Lecture 15
6
Analytic Model of Brown Dwarfs
• Brown dwarfs
– are fully convective, hence isentropic
– have an EOS that is polytropic both above and below
the plasma phase transition
– can be solved analytically
• Goal: solve for evolution of substellar L and Teff,
mass burning limits
– and learn something along the way!
Feb 27, 2009
PHY 688, Lecture 15
7
Hydrogen phase diagram
Adiabats are tracks of η~const
T
Feb 27, 2009
PHY 688, Lecture 15
(Burrows & Liebert 1993)
8
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 27, 2009
6
5
Tc / 106 [K]
•
stars
brown dwarfs
“planets”
R / 109 [cm]
Cooling and
Nuclear Burning
4
3
2
1
PHY 688, Lecture 15
(Burrows et al. 2001)9
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 27, 2009
PHY 688, Lecture 15
10
Outline
• Review of previous lecture
– fusion in brown dwarf interiors
– towards an analytical solution to the EOS
• Substellar Evolution
– analytical solution for L(t) and Teff(t)
Feb 27, 2009
PHY 688, Lecture 15
11
Hydrogen phase diagram
Evolution is towards:
• lower entropy S
• higher degeneracy η
T
H+
Feb 27, 2009
PHY 688, Lecture 15
(Burrows & Liebert 1993)
12
0.05 Msun Brown Dwarf at 5 Gyr
(X = 100% H)
(1 bar = 106 dyn/cm2 = 0.99 atm)
Feb 27, 2009
PHY 688, Lecture 15
13
Evolution of Effective Temperature
stars
brown dwarfs
“planets”
2MASS 0535–0546 A/B
M
13
10
M
Ju
p
M
Ju
p
5M
Jup
Gl 229B
1M
Jup
Jupiter
(Burrows et al. 2001)
Feb 27, 2009
PHY 688, Lecture 15
14
Evolution of Luminosity
Feb 27, 2009
PHY 688, Lecture 15
(Burrows et al. 2001)
15