Part One: Stellar Evolution

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Part One Stellar Evolution

• II. The Stellar Structure Equations
• A. Hydrostatic Equilibrium
• Definition Hydrostatic equilibrium is the
  condition of force balance throughout a star.
• Simplest case Gravity force vs. Gas Pressure
  force.
• Other forces may effect equilibrium
  centrifugal, magnetic, exterior pressure, other
  pressures

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Part One Stellar Evolution

• Spherically symmetric, uniform density star
• No exterior pressure
• Variables depend only on r.
• Consider a mass element dm with V Adr pictured
  below

F P,top
A

• NII on mass element
• dm(d2r/dt2) FG FP,T FP,B. (II.A.1)

dm, r
dr
A
M(r) mass interior to radius r.
F P,bottom
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Part One Stellar Evolution

• Spherically symmetric, uniform density star
• Mass of volume element
• dm d(rV) d(rAr) rAdr. (A.2)

• So Eqn A.1 becomes
• rAdr(d2r/dt2) -G(rAdr)M(r)/r2d(FP). (A.3)

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Part One Stellar Evolution

• Spherically symmetric, uniform density star
• f) The force due to pressure is PA, so
• d(FP) d(PA) AdP A(Ptop - Pbottom).
• Combining A.1- 3 gives

• rAdr(d2r/dt2) G(rAdr)M(r)/r2AdP, or
• d2r/dt2 GM(r)/r2 (1/r)dP/dr. (A.4)
• Eqn A.4 is the Equation of Motion for dm.

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Part One Stellar Evolution

• Spherically symmetric, uniform density star
• g) For HE, acceleration is zero, and so
• dP/dr -GrM(r)/r2. (A.5)

HE (A.5) is a statement about the pressure
gradient inside a star with a given mass
distribution.
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Part One Stellar Evolution

• Example Find P(r) for a uniform density star
  with M(r) Cr2.
• dP/dr -GrM(r)/r2
• ?dP -?Gr(Cr2)/r2 dr. Note integral from
  r 0 to r.
• P(r) - P(0) -GrC?dr -GrCr.
• Thus, P(r ) Pcentral -GrCr.

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Part One Stellar Evolution

• Example Find P(r) for a uniform density star
  with M(r) Cr2.
• P(r) Pcentral -GrCr.
• Checks
• Units! What are the units for C?
• Cr2 g, so C g/cm2.
• What are the units of GrCr?
• GrCr (dyne-cm2/g2)(g/cm3)(g/cm2)(cm)
• dyne/cm2 units of pressure.

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Part One Stellar Evolution

• Example Find P(r) for a uniform density star
  with M(r) Cr2.
• P(r) Pcentral -GrCr.
• Solve for the central pressure Integrate from
  0 to R.
• P(R) - P(0) -GrC?dr -GrCR.
• Note for an isolated star, P(R) 0
• P(0) Pcentral GrCR.

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Part One Stellar Evolution

• II. The Stellar Structure Equations
• B. Mass Continuity
• Definition Expression for enclosed mass as a
  function of position in the star.
• Spherical star
• dM(r) dMr 4pr(r)r2dr. (II.B.1)

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Part One Stellar Evolution

• II. The Stellar Structure Equations
• B. Mass Continuity
• 2. Example Find Mr for a star with r r0r2.
• Mr 4p?r(r)r2dr 4pr0?r2r2dr (4p/5) r0r5.
• Check units r0 g/cm5.
• Mr (g/cm5)(cm5) g.

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Part One Stellar Evolution

• II. The Stellar Structure Equations
• B. Mass Continuity
• 2. Example Find Mr for a star with r r0r2.
• Mr 4p?r(r)r2dr 4pr0?r2r2dr (4p/5) r0r5.
• Express Mr in terms of total mass M, R, and r
• M (4p/5) r0R5 Mr M(r/R)5.

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Part One Stellar Evolution

• II. The Stellar Structure Equations
• Pressure Equation of State (EOS)
• 1. Definition EOS is a relation between gas
  pressure and other state variables, usually r, m,
  or T.
• 2. Barotropes and Polytropes
• P P(r) Barotropic EOS (C.1)
• P Krg, Polytropic EOS (C.2)
• Where K Polytropic Constant.
• g 1 1/n,
• n Polytropic Index.

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Part One Stellar Evolution

• 3. Singular Isothermal Sphere n 8.
• a) P Kr.
• K T.
• b) For large radii, r r-2.
• Ideal Gas Equation of State (IGEOS)
• a) Definition
• PV Nkt (C.3)
• P rkT/ m (C.4)
• where k is the Boltzmann constant, m is the
  average mass of a gas particle, N total
  number of particles

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Part One Stellar Evolution

• Ideal Gas Equation of State (IGEOS)
• b) Mean molecular weight, m
• m (total atomic mass)/(total number of free
  particles) (Nm/mH). (C.5)
• Alternate IGEOS
• P (rkT/mmH) (C.6)

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Part One Stellar Evolution

• Ideal Gas Equation of State (IGEOS)
• c) Examples
• Completely ionized, hydrogen gas
• m 0.5 gt 1/2 electrons, half nuclei
• P (2rkT/mH)
• Completely Neutral monatomic H gas
• m 1 gt Every particle has mass mH
• P (rkT/mH)
• Completely Neutral diatomic H2 gas
• m 2 gt Every particle has mass 2mH
• P (0.5rkT/mH)

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Part One Stellar Evolution

• Radiation Pressure
• Prad 1/3 aT4. (C.6)
• In practice, a stars internal pressure is due to
  the ideal gas law and photons, i.e.,
• Ptot (1 - b)rkT/ m b(1/3aT4), (C.7)
• Where 0 lt b lt 1.

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Part One Stellar Evolution

• D. Energy Transport
• 1. Radiative Transport
• Opacity is important
• Electron scattering
• Photoionization
• b) Derivation
• Consider a thin shell of a BB emitter
• At r, F(r) sT4.
• At r dr, F(rdr) s(T dT)4
• s(T4 4T3dT).

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Part One Stellar Evolution

• Note The Flux absorbed in the shell is
• dF F(r dr) - F(r) 4sT3(r)dT. (D.1)
• Absorption of Flux is due to opacity k
• dF -k(r)r(r)F(r)dr. (D.2)
• Recall that luminosity is defined as
• L(r) 4pr2F(r ). (D.3)

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Part One Stellar Evolution

• Combining D.1-D.3
• L(r) -16psr2T3(r)/(k(r)r(r))(dT/dr).
• The final form of the radiative transport
  equation
• dT/dr -3k(r)r(r)/64psr2T3(r) L(r).
• (II.D.4)

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Part One Stellar Evolution

• Opacity
• k k(P,T,m)
• Kramers Opacity
• k rT-3.5 (ignoring composition effects).
  (D.5)

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Part One Stellar Evolution

• 2. Convective Transport
• dT/dr (1 - 1/G)T(r)/P(r)dP/dr,
• (II.D.6)
• Where G ratio of specific heats
  cP/cV
• 5/3 for fully ionized ideal gas.

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Part One Stellar Evolution

• 3. Conductive Transport
• F -KdT/dr, (II.D.7)
• Where K is the thermal conductivity.

Conductivity is typically unimportant in stellar
interiors.
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Part One Stellar Evolution

• E. Energy Conservation e(r)
• e 0 except in core.
• Thermal Equilibrium
• dL/dr 4pr2r(r)e(r). (II.E.1)
• Reaction rates are extremely temperature
  sensitive

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Part One Stellar Evolution

• 3. Thermonuclear Reactions in Stars
• a) Deuterium Burning
• PMS stars
• T 106 K
• No ignition for M 0.03 MSUN
• Li depletion is a signature of YSO
• D isotopes of Li, Be, B gt 3He and 4He.

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Part One Stellar Evolution

• b) Hydrogen Burning
• Main Sequence Stars
• PP Chains
• T 15 - (25) x 106 K
• Low Mass Stars
• Rate T4
• Basically 4H gt4He yield 26.7 MeV
• CNO Cycles
• T 30 x 106 K
• High Mass Stars
• Rate T20

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Part One Stellar Evolution

• F. Stellar Structure Models
• 1. Homology Relations Estimates from the SSEs.
• 2. The Mass-Luminosity relation
• a) Approximate the luminosity using Eqn
• Write all derivatives as (Xsurface - Xcenter)
• Ignore constants use average values r(r ) gt
  ltrgt
• dT/dr -3k(r)r(r)/64psr2T3(r) L(r) gt
• (Ts - Tc)/(R - 0) kr/R2Tc3 L
• Tc/R kr/R2Tc3 L.

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Part One Stellar Evolution

• F. Stellar Structure Models
• And so the total luminosity depends on R, Tc, k,
  r.
• L RTC4/kr.
• b) Replace r with M/R3
• L R4TC4/kM.
• c) Use IGEOS and HE to replace Tc
• P rkT/m gt Tc P/r PR3/M.

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Part One Stellar Evolution

• F. Stellar Structure Models
• dP/dr -GM(r)r(r)/r2 gt
• (Ps - Pc)(R - 0) -Mr/R2 gt
• Pc R(M/R2)(M/R3) so
• Pc M2/R4, and (II.F.1)
• Tc P/r (M2/R4)(R3/M) M/R. (II.F.2)

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Part One Stellar Evolution

• Now replace TC
• L M3/k. (II.F.3)

Note Luminosity is determined by mass and
opacity gt composition. gt L varies with
metallicity.
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Part One Stellar Evolution

• Stellar Property Estimates
• a) Estimate of the Suns Central Pressure
• Use II.A.5
• Pc/R GMltrgt/R2
• Pc GMSUNltrSUNgt/RSUN 109 atm.
• Actual value Pc 1011 atm.

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Part One Stellar Evolution

• b) Estimate of the Suns Central Temperature, Tc
• TC PCmmH/ltrSUNgtk 12 x 106 K. Actual value
  15 x 106 K
• c) Estimate of the Suns Luminosity
• L -64psRSUN2Tc3/3kltrSUNgt (-TC/RSUN)
• (9.5 x 1029/k) Js-1, where k 10-3 - 107, so
• L 1022 - 1032 Js-1. Actual value 4 x 1026
  Js-1.

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Part One Stellar Evolution

• 3. Finite differencing
• dP/dr -GMr/r2
• gt DP/Dr -GMr/r2,
• Where differencing (D) is made over a grid.

a) Accuracy of solution will depend on
the number of grid zones.
b) Order of solution will depend on
the Differencing scheme.
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Part One Stellar Evolution

• 4. The Vogt-Russell Theorem
• HE, TE, Nonmagnetic Star
• Mass
• Composition
• Stellar Equations
• Gas Effects
• Boundary Conditions
• Unique Stellar Model

Examples polytropic models, Main
Sequence Stars (Pop I or II).