Download Stellar Masses and the Main Sequence

Stellar Masses and the Main Sequence
Measurements of main-sequence stars demonstrate that there is a
mass-luminosity relationship, i.e., L ∝ Mη. For M > 1 M η ~3.88,
while at lower masses, the relation flattens out. I good rule-of-thumb
is L ∝ Mη, with η ~ 3.5.
Main Sequence Mass-Radius Relation
There is also a mass-radius relation for main-sequence stars.
When parameterized by R ∝ Mξ, ξ ~ 0.57 for M > 1 M, and ξ ~
0.8 for M < 1 M."
Stellar Masses for White Dwarfs
The masses of white dwarf stars
are all less than 1.4 M. Most
are ~ 0.59 M. "
There is also an inverse massradius relation for white dwarfs.
The simple theory says M ∝ Rα,
with α = -1/3."
Numbers to Keep in Mind
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τ #~ 10 Gyr = Main sequence lifetime of the Sun
1 M #~ 2 × 1033 gm = Mass of the Sun"
1 L #~ 4 × 1033 ergs/sec = Luminosity of the Sun
1 R #~ 7 × 1010 cm = Radius of the Sun
Q
~ 6.3 × 1018 ergs/gm = Energy from hydrogen fusion
Δm ~ 27 MeV = mass defect in hydrogen fusion
Δm ~ 0.7% = percent mass defect of hydrogen fusion
X
~ 0.75 = fraction of hydrogen (by mass) in the Sun
Y
~ 0.23 = fraction of helium (by mass) in the Sun
Z
~ 0.02 = fraction of “metals” (by mass) in the Sun
Stellar Timescales
Before starting to discuss how stars work, it is important to have an
order-of-magnitude feel for stellar timescales. They are the shortcut to
everything!
•  The Nuclear Timescale: How long will a star live?
words, how long will it take to use up its nuclear fuel?)"
τ nuc
(In other
Fuel
QM
=
=
Rate of Consumption
L
For hydrogen fusion, Q = 6.3 × 1018 ergs/gm. (If the star is fusing
helium, the coefficient is an order of magnitude smaller.) Notes:
€
•  Main sequence stars will only consume ~ 0.1 of their available
fuel before they must adjust their structure.
•  You can scale to the Sun: τnuc = 1010 years.
Stellar Timescales
•  The Thermal Timescale: How long does it take for a star to adjust
its structure? (How long does it take to release its energy?) This is
also called the Kelvin-Helmholtz timescale"
τ KH
Gravitational Energy GM 2
=
=
Rate of Consumption
RL
This describes how long a star can shine with no nuclear energy
generation. Alternatively, it describes how long it takes energy
€
produced
by gravitational contraction to work its way out. For the
Sun, this number is ~ 108 years.
Stellar Timescales
•  The Dynamical Timescale: How long does it take a star’s interior
to feel changes and its surface? (For instance, if the star accreted
mass, how long would it take the interior to feel the extra weight?)
Alternatively, how long does it take an object to “fall” to the star’s
core?"
τ dyn
1/ 2
3 ⎞1/ 2
⎛
⎛
⎞
Size of the Star
R
R
=
= ⎜
⎟ = ⎜ ⎟
Speed of Pressure Wave ⎝ G M ⎠
⎝ g ⎠
Note: this is equivalent to the free-fall time, and it’s equivalent to
Kepler’s 3rd law. For the Sun, τdyn ~ 30 minutes.
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Stellar Timescales
•  The Mass-Loss Timescale: How long does it take for a star to lose
all its mass via its wind?"
τ dyn
Mass of Star
M
=
= ˙
Mass Loss Rate M
For the Sun today, this is ~ 1013 years.
€
Stellar Structure
The internal structure of most stars can be computed easily(?) by
solving 4 simultaneous differential equations with 4 unknowns.
dM(r)
Mass Conservation :
= 4 π R 2 ρ(r)
dr
dP(r)
GM(r)
Momentum Conservation :
= −gρ = −
ρ(r)
2
dr
R
dL(r)
Energy Conservation :
= 4 π R 2 ρ (r) ε ( ρ,T)
dr
dT(r) dP(r) dT dP T ⎛ d lnT ⎞ dP T
Thermal Structure :
=
=
∇
⎜
⎟ =
dr
dr dP dr P ⎝ d ln P ⎠ dr P
where P = P(ρ,µ,T) and
κ ( ρ,T) = opacity
ε ( ρ,T) = ε nuclear − ε neutrino + ε gravitational = energy generation
d lnT
3κ ( ρ,T)L P
∇=
=
4 or ∇ ad
d ln P 16π acG M T
Digression: Mean Molecular Weight
Mean molecular weight (µ) is defined as mass per unit mole of
material, or, alternatively, the mean mass of a particle in Atomic Mass
Units. Thus, number density is related to the mass density, ρ, by
ρ
ρ N avo
n=
=
µ ma
µ
Often, this is split into two terms, one for ions
x i ρ N avo
n I = ∑ n i = ρ N avo ∑ =
µI
i€
i Ai
⎛ x ⎞ −1
where µI = ⎜∑ i ⎟
⎝ i Ai ⎠
And one for the (massless) electrons
€
€
xi
ρ N avo
n e = ρ N avo ∑ f i Z i =
Ai
µe
i
⎛ Z x f ⎞ −1
where µe = ⎜∑ i i i ⎟
⎝ i Ai ⎠
where xi is the mass fraction of the species, Zi is its atomic number, Ai,
its atomic mass, and fi the fraction of electrons that are free.
Mean Molecular Weight -- Simplification
In stars, the expressions for mean molecular weight can be greatly
simplified. For example, the fraction of heavy elements in a star is
small (Z ~ 0.02), so
−1
−1
⎛ x ⎞ −1 ⎛ X Y
⎞
⎛
⎞
Z
1
−
X
4
i
µI = ⎜∑ ⎟ = ⎜ + +
⎟ = ⎜ X +
⎟ =
⎝
⎠
⎝
1 4 ~ 14
4 ⎠
1+ 3X
⎝ i Ai ⎠
Also, if we assume that the gas in the interior of a star is (almost)
entirely ionized, fi ~ 1. And, while hydrogen has Zi/Ai = 1, most
€ other elements have Zi/Ai ~ ½. So
−1
−1
⎛ Z x f ⎞ −1 ⎛
⎞
⎛
⎞
1
1
1
−
X
2
i i i
µe = ⎜∑
⎟ =
⎟ = ⎜ X + Y + Z ⎟ = ⎜ X +
⎝
⎠
⎝
2
2
2 ⎠
1+ X
⎝ i Ai ⎠
The combined mean molecular weight is therefore
1 1
1
=
+
€
µ µI µe
Equation of State
To solve the equations of stellar structure, one must also known the
relationship between pressure, density, and temperature and mean
molecular weight. This is called the equation of state. The pressure
comes from 3 sources, i.e., P = Prad + Pion + Pelectron.
•  Radiation pressure: Prad
•  Ion pressure (ideal gas):
1 4
= aT
3
Pion
ρ
=
kT
µi ma
€
•  Electron pressure: For main sequence, electrons will act as an
ideal gas, but in giants, partial degeneracy can occur, and one must
€
solve Fermi-Dirac integrals,
i.e.,
4 π (2me kT )
Pe = 3
3h
me
5/2
∫
∞
0
⎧⎪
⎫⎪
η3 / 2
ne h 3
⎨
3 / 2 ⎬
η −ψ dη where ψ = ln
1+ e
⎪⎩ 2(2π me kT ) ⎪⎭
Opacity
Opacity is the opposite of transmittance. To get the appropriate
(Rosseland) mean, you have to integrate over the blackbody
function. There are several sources of opacity in stars.
•  Electron scattering (Thomson scattering):
n eσ e
κe =
= 0.2(1+ X) cm2 g -1
ρ
•  Free-free absorption (Kramer’s style opacity):
2
Z
κ ff ~ 10 23
ρ T −7 / 2 cm2 g -1
µe µI
€
€
•  Bound-free absorption (Kramer’s style opacity):
κ bf ~ 10 25 (1+ X ) ρ T −7 / 2 cm2 g -1
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•  Bound-bound absorption (complex, but roughly Kramers)
•  H- opacity (H- + h ν  "H + e-): complex, but below ~ 104 K
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κ H − ~ 2.5 × 10 −31 (Z /0.02) ρ1/2 T 9 cm2 g -1
Opacity
Massive supercomputer projects have calculated Rosseland mean
opacities as a function of density and temperature.
€
Opacity Sources
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Energy Transport: Radiation
In theory, there are 3 ways to transport heat:
  Conduction: Only important in white dwarfs and neutron stars
  Convection: mixing
  Radiation: diffusion of light due to absorption and re-emission.
Radiation is nothing more than the random walk of energy, where
the mean free path is lph = 1/κρ. The expression for energy transport
of this type is fairly simple:
⎛
⎞
L
1
dU
1
3 dT
F=
= − c lph ⎜ 4 π T
⎟
2 = − c lph
⎝
4 πR
3
dr
3
dr ⎠
This is usually re-written using thermodynamical relations to
€
dT GMρT
= 2 ∇ rad
dr
RP
where ∇ rad
3κ L P
=
16 π ac G M T 4
Energy Transport: Convection
In general, convection is more efficient, if it exists. If a blob of
material is displaced outward, does it continue to rise (taking its
heat with it) or sink back to where it started? In other words, at
location r + δr, is it denser or less dense than its new environment?
Therefore, for convection to occur
⎛ dρ ⎞
⎜ ⎟ <
⎝ dr ⎠ blob
⎛ dρ ⎞
⎜ ⎟
⎝ dr ⎠ star
After a bit of thermodynamic manipulation, this becomes
∂ lnT
= ∇ ad < ∇ rad
∂ ln P
€
Note: ad is a thermodynamic quantity intrinsic to the gas. For an
ideal, gas (P ∝ ργ, with γ=5/3), ad= 0.4. For pure radiation
pressure (P ∝ ⅓ aT€4) , ad= 0.25.
Nuclear Reaction Rates
Most light-element nuclear reactions
depend on density (squared) and
temperature to some power.
To
compute that power, one has to
integrate reaction rate cross-sections
over the Maxwellian distribution.
The cross sections are energy
dependent, since they depend on
momentum and tunneling.
⎛ 1 ⎞ 1
σ ∝ π λ ∝ ⎜ 2 ⎟ ∝
⎝ p ⎠ E
2
⎧ 2 π Z1 Z 2 e 2 ⎫
⎧ KZ1Z 2 ⎫
⎬ ∝ exp ⎨ − 1/ 2 ⎬
σ ∝ exp ⎨ −
⎩ E
⎭
ν
⎩
⎭
€
14.16 2 2 1/ 3 2
σ v = 1/ 3 ( Z1 Z 2 A) −
T6
3
Nuclear Reaction Rates
Most heavy-element nuclear reactions
involve nuclear resonances, many of
which are poorly known.
This
introduces uncertainty into some
aspects
of
heavy
element
nucleosynthesis.
Proton-Proton Chain
The sun produces energy by fusing 4 hydrogen atoms into one helium
atom, thereby creating 26.73 MeV per helium nucleus. Most of this
happens via the proton-proton chain.
• PP I:
•  1H + 1H 
"2H + e+ + νe
<εν> = 0.263 MeV
3He + γ
•  2H + 1H 
4He + 1H + 1H
•  3He + 3He

• PP II:
7Be + γ
•  3He + 4He

•  7Be + e-

"
"7Li + νe + γ <εν> = 0.8 MeV
4He + 4He
•  7Li + 1H 
• PP III:
8B + γ
•  7Be + 1H

8Be + e- + ν
•  8B

<εν> = 7.2 MeV
e
4He + 4He%
•  8Be

CNO Bi-Cycle
In higher mass stars, most hydrogen is fused via the CNO by-cycle.
This chain fuses hydrogen to helium using CNO as a catalyst.
•  12C + 1H
 13N + γ
•  13N
 13C + e- + νe
•  13C + 1H
 14N + γ
•  14N + 1H
 15O + γ
•  15O
 15N + e- + νe
•  15N + 1H
 12C + 4He
or (with a relative frequency of ~ 4 × 10-4)
•  15N + 1H
•  16O + 1H
•  17F
•  17O + 1H




<εν> = 0.71 MeV
<εν> = 1.00 MeV
16O
+ γ
17F + γ
17O + e- + ν
e
14N + 4He
<εν> = 0.94 MeV
CNO Bi-Cycle
Note: the net effect of the CNO cycle is 4 1H  4He. In the process,
the abundances of CNO get changed. In particular, since 14N has the
smallest cross section of the set, it tends to get built up.
Nuclear Reaction Rates
Since the proton-proton chain involves ions with the least amount of
charge, it has the smallest temperature dependence of any reaction.
2 −33.80 /T61 / 3
ε pp ∝ ρ X e
ergs s−1 cm−3
Thus, at T6 ~ 5, εpp ∝ T 6, while at T6 ~ 20, εpp ∝ T 3.5. Because CNO
ions have more electrostatic repulsion, the temperature dependence of
23 at T ~ 10 to ε ∝ T 13 at T ~ 50.
its process
is
stronger,
ε
∝
T
pp
6
pp
6
€
The HR Diagram
The Central Temperature and Density for
Main Sequence Stars
Convection on the Main Sequence
Importance of CNO Burning versus Stellar Mass
Stellar Evolution in the HR Diagram
In general, the opacity of
a star is proportional to its
metal abundance (due to
bound-free transitions and
electrons supplied to H-.)
The lower the metal
abundance, the smaller the
opacity, the less energy is
trapped in the star doing
work, the smaller the star,
and therefore the hotter
the star. The metal-poor
main sequence is bluer
than the metal-rich main
sequence.