Stellar Structure - Stony Brook University · Lattimer, PHY 688, Stellar Structure and Evolution...
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Stellar Structure • Hydrostatic Equilibrium: dP dr = − Gmρ r 2 , dm dr =4πρr 2 • Equation of State: P = ρ N o μ k B T • Mean Molecular Weight: μ = 2 4 2X + 3 4 Y + X i>He n i (1 + Z i ) ρN o 3 5 −1 ≃ 4 2+6X + Y Y e = X + Y 2 + X i>He n i Z i ρN o ≃ 1+ X 2 • Radiative Transfer: L = −4πr 2 4ac 3κ R ρ T 3 dT dr • Opacity: κ R ≃ κ o ρT −3.5 ZY e cm 2 g −1 T> 10 4 K, Kramer ′ s κ R ≃ 0.4Y e cm 2 g −1 T> 10 7 ρ 1/3.5 K, electron scattering • Energy Generation: L =4π Z ˙ ǫr 2 dr, ˙ ǫ ≃ ǫ c „ ρ ρ c « λ „ T T c « ν p-p cycle has λ =1,ν ≃ 4 CNO cycle has λ =1,ν ≃ 20 He burning has λ =2,ν ≃ 40 Lattimer, PHY 688, Stellar Structure and Evolution – p.1/15
Transcript of Stellar Structure - Stony Brook University · Lattimer, PHY 688, Stellar Structure and Evolution...
Stellar Structure• Hydrostatic Equilibrium:
dP
dr= −Gmρ
r2,
dm
dr= 4πρr2
• Equation of State:P = ρ
No
µkBT
• Mean Molecular Weight:
µ =
2
42X +3
4Y +
X
i>He
ni(1 + Zi)
ρNo
3
5
−1
≃ 4
2 + 6X + Y
Ye = X +Y
2+
X
i>He
niZi
ρNo≃ 1 + X
2
• Radiative Transfer:L = −4πr2 4ac
3κRρT 3 dT
dr• Opacity:
κR ≃ κoρT−3.5ZYe cm2 g−1 T > 104K, Kramer′s
κR ≃ 0.4Ye cm2 g−1 T > 107ρ1/3.5 K, electron scattering
• Energy Generation:
L = 4π
Z
ǫr2dr, ǫ ≃ ǫc
„
ρ
ρc
«λ „
T
Tc
«ν
p-p cycle has λ = 1, ν ≃ 4CNO cycle has λ = 1, ν ≃ 20He burning has λ = 2, ν ≃ 40
Lattimer, PHY 688, Stellar Structure and Evolution – p.1/15
Standard Solar Model• Assume fraction of pressure due to gas is fixed within the star.
β =Pg
Pg + Pr=
»
1 +aµ
3No
T 3
ρ
–−1
• This yields a polytrope of index n = 3, (P = Kρ1+1/n)
K =
„
No
µβ
«4/3 „
3
a(1 − β)
«1/3
R = 11.2
»
1 − β
µ4β4ρ2c
–1/6
R⊙
M = 18.0
√1 − β
µ2β2M⊙
ρc/ρ = 54.2
Tc = 1.96 · 107 βµMR⊙
RM⊙
K
• For Sun (M = 1M⊙, µ ≃ 0.6):
β ≃ 1−„
µ2
18
«2
≃ 0.9996, ρc =11.23
18g cm−3 ≃ 77 g cm−3, Tc ≃ 1.3·107 K
Lattimer, PHY 688, Stellar Structure and Evolution – p.2/15
Scaling Relations• Dimensional analysis (κ ∝ ρmT−s, ǫ ∝ ρλT ν , ignore composition):
P ∝ M2
R4, T ∝ βM
R, ρ ∝ M
R3
L ∝ RT 4
κρ∝ R4+3mT 4+sM−1−m ∝ Mǫ ∝ M1+λR−3λT ν
• Energy transport only:
L ∝ β4+sR3m−sM3+s−m ∝ β7.5R−.5M5.5 Kramer′s
∝ β4R0M3 electron scattering
• Including energy generation (for Kramer’s):
L ∝ βνR−3λ−νM1+λ+ν
∝ β111/13M70/13 pp, β65/9M232/45 CNO, β121/17M432/85 He
R ∝ β−7/13M3/13 pp, β5/9M31/45 CNO, β65/91M73/91 He
• Stellar lifetimes can be estimated by
τ ≃ τ⊙M
M⊙
L⊙
L, τ⊙ ≃ 1010years
Lattimer, PHY 688, Stellar Structure and Evolution – p.3/15
• Most stars are Main Sequence stars, which burn H into He in their cores.
• Other groups are red giants, which have exhausted H fuel and “burn” He into Cand O, supergiants which are burning even heavier elements, and white dwarfs(dead low-mass stars).
• Main Sequence stars can be divided into spectral types O, B, A, F, G, K and Mbased on surface temperature or color
• Note that there are straight lines of constant radius contoursw
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.dav
idda
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Lattimer, PHY 688, Stellar Structure and Evolution – p.4/15
Single StarEvolution
1. H fusion begins
2. Core H exhaustion
3. H shell ignition
4. Hertzsprung Gap traversal
5. Joins Hayashi track
6. He ignition
7.
8.
9.
10. Core He exhaustion
Beyond He burning, neutrinoemission dominates energygeneration.Core burning is followed by shellburning of successive elements.Mass loss plays a significantrole in advanced evolution.
Lattimer, PHY 688, Stellar Structure and Evolution – p.5/15
Single Star Evolution
Lattimer, PHY 688, Stellar Structure and Evolution – p.6/15
Single Star Evolution
Lattimer, PHY 688, Stellar Structure and Evolution – p.7/15
Evolution in Close Binaries• A majority of stars are in binary systems and may undergo mass transfer at some
point in their evolution.
• Mass transfer can occur due to evolutionary expansion (red giant formation) or toorbital shrinkage due to loss of angular momentum.
• Mass transfer occurs when the radius of a star exceeds the Roche lobe, which isthe critical gravitational potential surrounding both stars that pass through theinner Lagrange point L1.
• Binary definitions:M = M1+M2, µ = M1M2/M, a = a1+a2, a1/a2 = M2/M1, e
E = − 1
2
GM1M2
a, J = µ
p
GaM(1 − e2) = µΩa2√
1 − e2
Ω2 = (2π/P )2 = GM/a3 (Kepler’s Law)v1 = Ωa1 sin i (Projected orbital velocity of star 1)f1(M1, M2, i) = (M2 sin i)3/M2 = v3
1/(GΩ) (Mass function)
• Note that mass function depends only on observable quantities.
Lattimer, PHY 688, Stellar Structure and Evolution – p.8/15
Roche LobesΦ = −GM/|~r − ~r1| − GM2/|~r − ~r2| − 1
2(~ω × ~r)2 (Gravitational potential)
L1L2 L3
L4
L5
L1L2 L3
L4
L5
Distances measured in units of a
R2/a ≃ 0.46q/(1 + q) (Kopal 1959), q = M2/M1, b2 = (.5 + .227 log q)a
Lattimer, PHY 688, Stellar Structure and Evolution – p.9/15
Binary Classification1. Detached
• Most binaries this class• Evolve independently
2. Semi-detached• Mass transfer dominates evolution• Inflowing gas can form accretion disc• X-ray binaries, cataclysmic variables
3. Contact• Stellar atmospheres merge into common envelope• Dynamical friction brakes orbital motion
Lattimer, PHY 688, Stellar Structure and Evolution – p.10/15
Mass Transfer and Orbital EvolutionJ = (M1a2
1 + M2a22)Ω = µa2Ω = µ
√GMa
a
a= 2
J
J− 2
M2
M2
− 2M1
M1
+M
M
Conservative mass transfer (M = J = 0)a
a= −2
M2
M2
(1 − q) ⇒ af
ai=
„
M1iM2i
M1fM2f
«2
,Pf
Pi=
„
M1iM2i
M1fM2f
«3
For q < 1, conservative mass transfer to less massive component causes orbit to shrink.In terms of q, one has
da
a=
2
q
q − 1
1 + qdq ⇒ a ∝ q−2(1 + q)4
Angular momentum can be lost due to magnetic braking and to gravitational radiation
JGW = −32
5
G3
c5M1M2M
a4s−1
Lattimer, PHY 688, Stellar Structure and Evolution – p.11/15
Stability of Roche Mass TransferMass transfer will be stable if the radius of the donor star does not exceed the RocheLobe radius after the star thermally and hydrostatically readjusts.
d ln R
d ln M2
≡ α ≥ d ln RL
d ln M2
=d ln a
d ln M2
+1
3= 2
„
2M2 − M
M − M2
«
+1
3= 2q − 5
3
If the outer part of the star has a deep convective envelope, the star responds asR ∝ M−1/3, or α = −1/3, so the stability range is q ≤ 2/3.In the case of stable mass transfer, if the Kelvin-Helmholtz timescale is less than thenuclear timescale, the star will remain in thermal equilibrium during Roche lobe overflow.The Kelvin-Helmholtz timescale is
τKH =GM2
RL∼ 3 · 107
„
M
M⊙
«2 „
R
R⊙
«−1 „
L
L⊙
«−1
yr
On the Main Sequence, LR increases with mass much faster than M2, so donor starsthat are relatively massive generally have stable Roche lobe overflows that occur onthermal timescales. This case applies to supersoft X-ray sources which are accretingwhite dwarfs in binaries.
Lattimer, PHY 688, Stellar Structure and Evolution – p.12/15
Types of Mass Transfer
1. Case A• During core H burning
2. Case B• After core H burning,
but before He ignition• H shell burning
3. Case C• During core He
burning
Lattimer, PHY 688, Stellar Structure and Evolution – p.13/15
Explosive Mass LossTotal energy of binary prior to explosion is
Eo = −GMo1M2
2ao= −GMo
1M2
r+
1
2µoV 2
o
~Vo is instaneous relative velocity of the stars. After the explosion, if the orbit is bound,
E = −GM1M2
2a= −GM1M2
r+
1
2µV 2
~V is new relative velocity of the stars. The kick velocity is ~∆V = ~V − ~Vo, and the massloss is ∆M = Mo
1− M1. Define V 2
c = Gµo/ao.
a
ao=
1 − ∆M/Mo
1 − (2ao/r)∆M/Mo + (V 2o − V 2)/V 2
c
a < ∞ ⇒ ∆M
Mo<
r
2ao
„
1 − V 2o − V 2
V 2c
«
With no kick, ~Vo = ~V , ∆M/Mo < r/(2ao). For a circular orbit, ∆M/Mo < 1/2.For a kick velocity directed at an angle θ with respect to Vo, the stability condition is
∆M
Mo<
r
2ao
»
1 − ∆V (∆V + 2Vo cos θ)
V 2c
–
=1
2
»
1 − ∆V
V 2c
(∆V ± 2Vc)
–
where the final equality is for an aligned or misaligned kick in the initially circular orbitcase. Kicks in the direction of the orbital motion tend to destabilize the system. Note fora misaligned kick in which ∆V ∼ Vo, that ∆M < Mo so no amount of mass lossdestabilizes the system. In the misaligned kick case, it is possible to stabilize the orbitwith some amount of mass loss unless ∆V > (1 +
√2)Vo.
Lattimer, PHY 688, Stellar Structure and Evolution – p.14/15
AccretionA naive model for accretion in a binary invokes spherical symmetry. Mass falling onto anaccreting object liberates gravitational energy and the pressure of the radiation theninhibits accretion. In principle, the maximum accretion rate is reached when radiationpressure balances gravity. Equating the pressure support from hydrostatic equilibriumequations to the outward force of radiation pressure gives
dP
dr= −GMρ
r2= −κρFrad
c= −κρ
c
L
4πr2.
Infalling, hot, H-dominated material will have opacities dominated by electron scattering:κ = σT /mp = κT = 0.2(1 + X) cm2 g−1. The implied limiting accretion luminosity is
LEdd =4πcGM
κT≃ 1.3 · 1038
„
M
M⊙
«
erg s−1 = 3.3 · 104
„
M
M⊙
«
L⊙.
The chemical composition and the spectral energy distribution of the emission will affectthe exact value. In situations with extremely large flows, photons become trapped in thematerial, and transport by neutrinos becomes more efficient. For example, neutralcurrent scattering by free nucleons has a cross section
σn =σo
4
„
Eν
mec2
«2
= 1.7 · 10−44
„
Eν
MeV
«2
cm2.
The neutrino Eddington luminosity is then
LEdd,ν = 4πcGMmp
σn= 5 · 1056 ergs−1
for neutrinos with average energy of 10 MeV.Lattimer, PHY 688, Stellar Structure and Evolution – p.15/15
