Kawaler - Lecture 1 p Learning Physics from the Stars: It...

Kawaler - Lecture 1 p

Learning Physics from the Stars: It's All In the Coefficients

Steve Kawaler, Iowa State University

• Lecture 1: Overview / discussion of the basic equations of stellar structure

• Lecture 2: It's all in the coefficients - input physics and weaknesses that we need to fix

• Lecture 3: Driving and Damping - nonadiabiatic pulsation theory and practice

• Lecture 4: Seismology to the rescue - feedback between pulsation studies and basic input physics

1


stellar evolution in a nutshell• a “reminder” of basic stellar structure and evolution

2

Kawaler - Lecture 1 p Dependent / Independent variables• Independent variable - a measure of position

• distance from center - r

• mass fraction within - Mr

3

Kawaler - Lecture 1 p Dependent / Independent variables• Independent variable - a measure of position

• distance from center - r

• mass fraction within - Mr

• Things that specify conditions within a (hydrostatic) star:

• velocity

• density: ! or n = NA !/µ

• pressure: P

• temperature: T

• chemical composition (fraction by mass): Xi

• ion / charge balance: Yi , ne

• internal energy (per unit mass): U

• entropy (per unit mass): S

• heat flow parameters/ x-sections (/mass): "rad, "cond,

• energy flow: Lr , Fconv

• energy generation/loss (per unit mass): #nuc , #$

4


the ‘core four’... plus• primary mechanical quantities

• r (or Mr)

• P

• primary thermal quantities

• T• Lr

• necessary extra information

• composition (element mass fraction Xi)

• necessary derived quantities

• Equation of state: !, µ, U, S, !ad ,Yi ,etc.

• Atomic physics: yi , "rad, "cond

• Nuclear physics: #nuc , #$• confusing physics: Fconv

• NOTE - for pulsation, need partial derivatives of these (see Sarbani and/or Joergen)

5


Mass conservation• ASSUME - spherical symmetry EVERYWHERE

• P, T, !, etc. all functions of r only

• define Mr as mass contained within radius r

• mass of a (thin!) spherical shell:

• or, simply rearranging,

• this also provides a ‘coordinate transformation’

dMr = Mr+dr − Mr = 4πr2ρ(r)dr

dMr

dr= 4πr2ρ(r)

dF

dr= 4πr2ρ(r)

dF

dMr

6


Mechanical Equilibrium

• gravitational force downwards:

• pressure (force) imbalance upwards

• ... equation of motion (vertical)

• at equilibrium, the above = 0, so we have HSE:

ρ(r)g(r) = ρ(r)GMr

r2

dP

dMr= −GMr

4πr4or

dP

dr= −GMr

r2ρ(r)

ρ(r)d2r

dt2= −ρ(r)

GMr

r2− dP

dr

P (r) − P (r + dr) = −�

dP

dr

�dr

7


mechanical structure - special case

• HSC and Continuity contain only P and !• consider a functional relation between

P and ! only, i.e.

• P(r) = K ! "(r)• with such an Equation of State (EOS)

•

•

dP

dr=

GMr

r2

1K1/γ

P 1/γ(r)

dMr

dr= 4πr2 1

K1/γP 1/γ(r)

8


Energy conservation• If no energy lost or produced in a zone:! Lr = Lr+dr - or - Lr+dr - Lr = 0

• more generally Lr+dr − Lr =

�−dQ

dt+ �

�dMr

(nuclear) energy production / loss rate per gram:

#(!,T, Xi)• nuclear energy production

• neutrino losses

9

heat gain / loss rate per gram:

internal energy change PdV work

•

•"the only explicit time-dependence in equations of stellar structure

dQ

dt=

∂E

∂t− P

∂

∂t

�1ρ

�

dQ

dt= T

∂S

∂t


Thermal Equilibrium• if energy is being lost, then energy must flow...

• and for photon diffusion to transport energy, there must be a gradient in the photon energy density aT4/3 Fr = energy flux (flow per unit area) = energy density gradient x speed x mean free path

• mean free path = [cross section (per gram) x density]-1

• $ = (%!)-1

• combining:

• multiply by surface area, and rearrange:

Fr = −43

ac

κρT 3 dT

dr

dT

dr= − 3κρ

16πacr2

Lr

T 3−or− dT

dMr= − 3κ

4(4πr2)2ac

Lr

T 3

10

Fr = − d

dr

�aT 4

3

�× c× λ


Thermal Equilibrium

•

• more generally:

• define , then

• when radiation carries flux via diffusion

• or

dT

dr= − 3κρ

16πacr2

Lr

T 3−or− dT

dMr= − 3κ

4(4πr2)2ac

Lr

T 3

dT

dr=

dT

dP

dP

dr=

T

P

�d lnT

d lnP

�dP

dr

∇ ≡ d ln T

d ln P

∇rad ≡3κ

16πac

Lr

T 4

P

GMr

dP

dr= −GMr

r2ρ(r)

dT

dr= −∇GMr

r2

ρT

P

dT

dr= −∇rad

GMr

r2

ρT

P

dT

dMr= −∇rad

GMr

4πr4

T

P

11

Kawaler - Lecture 1 p summarizing

• Continuity

• HSE

• Energy conservation

• Energy transport

• Equation of state

• !(P, T, Xi)• ! (P, T, !)

• Energy generation

• #nuc(!, T, Xi)• #&(!, T, Xi)• S (P, T, !)

• Energy transport

• !rad ! %rad (!, T, Xi)

• !cond ! %cond (!, T, Xi)

• !convective ! ????

dMr

dr= 4πr2ρ(r)

dP

dMr= −GMr

4πr4

dT

dMr= −∇GMr

4πr4

T

P

12

dLr

dMr= −T

∂S

∂t+ �


Equation of State: µ(!,T,Xi) ; P(!,T, µ)

• Assume LTE, isotropy (photon mean free path << R)

• ideal neutral, unionized gas - general form

• P =' ni k T where ni = NA !Xi/Ai • define µ-1 ( ' Xi/Ai

• then P = !/µ NAkT ; E=3/2 P/!• Mix of ions and electrons

• P = Pe + Pi = Pe + !/µi NAkT

• Pe = !/µe NAkT

• yes, there are complications...

• and, there’s Prad too: Prad = aT3/3

always PI even when electrons are degenerate

• if electrons are non-degenerate• µe depends on ionization state

13

Kawaler - Lecture 1 p Energy Generation• An “n)v” problem for a given reaction

• v (E) = nucleus relative velocity (E " mv2 " kT)

• )(E) = reaction cross-section, function of energy

• functional form can be specified, but laboratory measurements essential for use in stellar models

• usually tabulated as ‹)v›, an average over the energy distribution corresponding to temperature T.

• non-resonant reactions

• resonant reactions:

• �ij(ρ, T, X) = Qij ρN2A

Xi

Ai

Xj

Aj�σv�ij

�σv�ij = K1 gΓiΓj

ΓT −3/2e−K2/T

�σv�ij =K0S(0)ZiZj

T −2/3e−K3T −1/3

14

Kawaler - Lecture 1 p Energy Transport - radiative• As posed, it is contained in !:

• photon diffusion (‘radiative’ heat transport): != !rad

• %r - the radiative opacity:

• flux:

• integrate:

• where

• so

dT

dMr= −∇GMr

4πr4

T

P

Fν = −4π

31

ρκν

∂Bν

∂T

dT

dr

∇rad ≡3κr

16πac

Lr

T 4

P

GMr

Frad =� ∞

0Fνdν = −4π

31

ρκ̄r

dT

dr

� ∞

0

∂Bν

∂Tdν

=acT3/*1κ̄r

≡� ∞0

1κν

∂Bν∂T dν

� ∞0

∂Bν∂T dν

Fr = −43

ac

κrρT 3 dT

dr

15

Kawaler - Lecture 1 p Energy Transport - conductive• As posed, it is contained in !:

• heat conduction via free electrons: != !cond

• %c - the conductive opacity:

• flux:

• opacity:

• skipping some steps...

• so

dT

dMr= −∇GMr

4πr4

T

P

∇cond ≡3κc

16πac

Lr

T 4

P

GMr

Fc = −43

ac

κcρT 3 dT

dr

κc ≡4acT 3

3ρDe; De ≈

cvveλ

3

κc ∝ µ2e

µIZ2

c

�T

ρ

�2

F = Fr + Fc = −4acT 3

3ρ

dT

dr

�1κr

+1κc

�

16

Kawaler - Lecture 1 p Energy Transport - convective• As posed, it is contained in !:

• heat transport via bulk turbulentmotion: != !conv

• instability criterion !rad > !adiabatic

• blob moves adiabatically upward through material with !>!rad

• blob hotter than surroundings after move (!adiabatic)

• higher T means lower density than surroundings

• accelerates upwards (carrying excess heat)

• eventually disperses, delivering heat upward, reducing !

• if perfectly efficient: redistribution until != !ad <!rad

• otherwise !rad > ! >!ad

• departure from !ad requires some (schematic) model

dT

dMr= −∇GMr

4πr4

T

P

17

P, T, !

"T < "Tad !ad

!>!ad

" ! > " !ad

P=PsurroundT>Tsurround!<!surround

Kawaler - Lecture 1 p Energy Transport - convective• if perfectly efficient: redistribution until != !ad <!rad

• !ad = !ad(P, T, µ)

• assume complete compositional mixing

• otherwise !rad > ! >!ad

• departure from !ad requires some (schematic) model

• mixing length theory:

• blob scale coupled to distance moved before dispersal

• scale taken as proportional to pressure scale height

• single parameter (set) tuned to fit observations

• mixing length theory variants:

• insert your favorite here

• hydrodynamic simulations

• targeted simulations for specific model age, mass, etc.

• ‘calibration’ of mixing length parameter to mass, age, etc.

18


other issues (non-coefficient)• time evolution of abundance

• composition changes via nuclear burning

• direct impact through dS/dt term

• composition changes via chemical diffusion

• computed separately ‘between’ time steps

• diffusion coefficients via atomic physics

• composition changes via turbulence

• instantaneous mixing via convection

• partial mixing via semiconvection, other processes

• rotational mixing

• mass loss / accretion

• rotation

• magnetic fields

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