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Stellar Structure and Evolution

Main topics– Basic equations of stellar structure– Physics of stellar matter– Stellar evolution from main sequence to the final stages

including synthesis of the elements

Recommended book– Kippenhahn & Weigert (1989; KW)– Hansen, Kawaler & Trimble (2005; HKT)– Background/supplemental: see separate list

Exam– Material discussed in lectures & problems– Optional: `scriptie’ on topic to be agreed upon

Spring 2007

If simple perfect laws uniquely rule the Universe, should not pure thought be capable of uncovering this perfect set of laws without having to lean on the crutches of tediously assembled observations? True, the laws to be discovered may be perfect, but the human brain is not. Left on its own, it is prone to stray, as many past examples sadly prove. In fact, we have missed few chances to err until new data freshly gleaned from nature set us right again for the next steps. Thus pillars rather than crutches are the observations on which we base our theories; and for the theory of stellar evolution these pillars must be there before we can get far on the right track.

Martin Schwarzschild (1958)

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1. Some facts about stars

Sun – M = 1.9891 1033 gram– R = 6.9598 1010 cm– L = 3.8515 1033 erg/sec– = 1.4086 gm/cm3

Stars– Apparent magnitudes ⇒ colors– Stellar atmosphere models ⇒ L, Teff, R– Certain binaries ⇒ M

p L R T T Keff eff O= ⇒ =4 57802 4π σ ,

10 100 08 1001 800 1500

6 6− ≤ ≤≤ ≤≤ ≤

L LM M

R R

O

O

O

.10 102000 100000

9 6− ≤ ≤≤ ≤ρ ρO

effT K

a) Mass, luminosity, radius, & effective temperature

ρO

Discovery of the Hertzsprung-Russell diagram ≈ a century ago showed that for most stars the absolute magnitude MVand effective temperature Teff (as derived from its spectral type SP) are correlated

MV

Spectral Type

b) The Hertzsprung-Russell Diagram

Most stars on the main sequence, some on the giant branch, and one lone outlier (white dwarf)

3

H-R diagram for the nearby starsLuminosities based on approximate distances: scatter in MVUsing Morgan-Keenan spectral types: discreteness in Teff

MV derived from parallax πobtained by HIPPARCOSfor nearby stars with V<10

V-I color is function of Teff,

Unlike spectral type SP, it is a continuous quantity

H-R diagram for the Solar Neighborhood

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c) Hertzsprung-Russell diagrams for star clusters

Main sequence, turn-off, subgiants, giants, horizontal branch, asymptotic giant branch, and white dwarf sequence

All stars at almost same distance: can use V instead of MV

Use color (B-V or V-I) instead of SP: continuous quantity

Deep H-R diagram of a globular cluster

HB

WDs

Blue stragglers

Extreme HB

MS

Sub-giants

GiantsAGB stars

5

NGC 6397HST/ACS

6

The HyadesIndividual parallaxes corrected for depth of the cluster by using the proper motions

⇒ secular parallaxesThese are a factor 3more accurate than π

Teff from atmospheric models for spectra

⇒ the currently most accurate absolute H-R diagram for any star cluster de Bruijne et al. 2001 A&A 367, 111

Composite H-R diagram for starclusters

Aim is to understand:Position of stars in this diagramEvolution of stars in this diagramDifferences between cluster HRD’sNature of Cepheids

Sandage 1957 ApJ 125, 435

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d) Three Kinds of H-R Diagrams

M M BC L Lbol V O= + = −4 76 2 5. . log

m M d A

d pc AU

V V V− = − +

=′′

=′′

5 51 206265

log

π π

with BC the bolometric correction

Distance modulus; AV extinction

Trigonometric distance d follows from parallax π

Absolute bolometric magnitude Mbol and luminosity L:

Absolute magnitude MV versus spectral type SP

Absolute magnitude MV versus a color, e.g., B-V

Bolometric luminosity versus effective temperature: log L versus log Teff (‘physical’ HR diagram)

For certain binaries (e.g., double-lined eclipsing variables) possible to determine individual masses (e.g. Popper 1980 ARA&AMartin & Mignard 1998 AA 330, 585)

LL

M M M M MM M M M MM M M M MO

O O O

O O O

O O O

=≤ ≤≤ ≤≤ ≤

0 66 0 08 0 50 92 0 5 40300 40 130

2 5

3 55

2

. . .. .

.

.b gb gb g

e) Mass-luminosity relation

Recent measurements:

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f) Internal structure

HelioseismologyStudy of oscillations of Solar surfaceProvides probe of internal structureExtremely accurate Solar model

AsteroseismologyIdem for other stars, but surface not spatially resolved

Frac

tiona

l err

or in

so

und

spee

d ∝√P

/ρ

g) Nucleosynthesis

Cosmic abundances of most of the elements produced by nucleosynthesis in stars (except H, He, and Li, Be, B)

Mass number of element

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What is the internal structure of stars? What causes the mass-luminosity relation?What sets the range of stellar masses?What generates different classes of stars in HR Diagram?What are the final stages of stellar evolution?How do stars produce the heavy elements?Why do some stars pulsate?What additional processes occur in binary stars?

h) Some questions

To be answered by applying the laws of physics

i) Outline of course

Derivation of four equations of stellar structure - Mass continuity (§ 2)- Hydrostatic equilibrium (§ 2)- Thermal equilibrium (§ 4)- Energy transport by radiation (§6) or convection (§ 8)

Required physics- Thermodynamics (§ 3)- Equation of state including degeneracy, and internal energy (§ 5)- Opacity of stellar material (§ 7)- Nuclear energy generation (§ 9, 10)

Solution methods and simple models (§ 11-14)

Overview of stellar evolution (§ 15-24)

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2. Spherical stars

r RM m R

r m rr

MR

==

=

=

( )

( ) ( )ρπ

ρπ

343

4

3

3

m r r r dr dmdr

r rr

( ) ( ) ( )= ⇒ =z 4 4 12

0

2π ρ π ρ b g

ρ( )( )r

m r

a) Mass-continuity equation

stellar radiustotal mass of star

mean density inside r

mean density of star

density at radius r

mass enclosed inside r

(KW §1)

b) Gravitation

Gravitational acceleration inside a spherical body

g g r Gm rr

ddr

= = =( ) ( )2

Φ

with Φ the gravitational potential, and

Check of accuracy of spherical approximation

Rotation period of the Sun is 27 days ⇒centrifugal acceleration at equator: vc

2/RGravitational acceleration at equator: GM /R2

Ratio is 2 x 10-5 so rotation unimportant, and star can be treated accurately as a sphere

Φ surfaceGMR

= −

(KW § 1)

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dPdr

Gm r rr

= −( ) ( )ρ

2 2a f

1 14

24

2

4

a f

a f

⇔ =

⇔ = −

drdm rdPdm

Gmr

π ρ

π

Equilibrium if and only if for all r ⇒&&r = 0

This is the equation of hydrostatic equilibrium, whichequates the pressure gradient to the gravitational force

Often useful to employ m=m(r)as variable, not r. Then:

c) Hydrostatic equilibriumSmall cylinder of thickness dr, surface 1cm2 ⇒ mass ρ drNewton’s equation of motion:

with P the pressure

&&( )r dP

drGm r

r= − −

12ρ

(KW § 2.1-2.4)

Solutions of (1) and (2) exist for the special case These are so-called barotropic stars, and include the famous polytropes with but also the white dwarfs, both of which we study later (§ 12; § 23)

But generally with T the temperature; this is called the equation of state; it also depends on composition

Example: Ideal gas has

No is Avogadro’s number = 6.02257 1023

k is Boltzmann’s constant = 1.38062 10-16 erg/Kμ is the mean molecular weight (see § 3b)

P K= ργ

P N k Tρ μ= 0

P P= ( )ρ

P P T= ( , )ρ

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d) Time scalesConsider again Newton’s equation: &&

( )r dPdr

Gm rr

= − −1

2ρ

P r g R Rg

g r PR

R RP

Rff

ff

ll

sound

= = = ⇒ =

= ≈ = ⇒ = ≈

0

0

2

2

: && :

: && :exp

exp

ττ

ρ ττ ρ

υ

Two extreme cases:

g GM R P GM R≈ ≈/ /2 2 4τ τexp l ff=

Free-fall time scale

Explosiontime scale

In H.E. both terms contribute equally (but with opposite sign), so we can write . Using (§2f), we find:

This is the hydrodynamical timescale

τ τ τρff l hydro

RGM G

= = = ≈exp

3 12

Stars are in hydrostatic equilibrium

Problem: What is thydro for a neutron star? Can a pulsar be a pulsating neutron star?

τ hydroO

ORR

MM

seconds=FHGIKJFHIK1600

3 2 1 2/ /

3 1000sec dayshydro≤ ≤τ

Problem: Solve with r=r0 at t=t0, and show that r=0 is reached for

In Solar units:

Use range of mass and radii for stars

This is similar to the periods of pulsating stars

t G= 3 32π ρ/

&& ( ) /r Gm r r= − 2

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e) Potential energy Eg

Eg = the amount of work needed to bring stellar matter from infinity to the present configuration

Let m(r) be present inside radius r, and add mass dm between r and r + dr. Then:

dE Gmr

dr dm Gmr

dmg

r

= − − = −∞z 2

E Gmr

dm dmg

M M

= − =z z0 0

12

Φ

E r dP r P Pr dr PdV P dmg

MM

MMM

= = − = − = −z zzz4 4 12 3 33

0

30

2

000

π π πρ

⇒

In hydrostatic equilibrium:

We shall see in §4b that Eg ∝ Ei, the internal energy of the star

integrate by parts; twice

f) Order of magnitude estimates

PM

PdmM

m dP GM

mr

dmM M M

= = − =z z z1 140 0

2

40π

P P P R dP G mr

dmc

M M

= − = − =z z( ) ( )040

40π

gM

gdm GM

mr

dmM M

= = =z z1

02

0

TM

TdmM N k

P dmMN k

EM M

g= = = −z z1 130 0 0 0

μρ

μ

E G mr

dmg

M

= = − z0

Mean pressure

Central pressure

Potential energy

Mean grav. acceleration

Mean temperature(for ideal gas)

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I Gmr

dm G m dmM M

σ ν

σ

ν

νν σ νπ ρ,

// /= = FH

IKz z −

0

33

0

343

The integrals on the right-hand side are all of the form

where we have used

Assume the density ρ(r) does not increase outwards,

Then: and

It follows that:

with rc defined by

r m r r3 3 4= ( ) / ( )π ρ

d r drρ( ) / ≤ 0

M rc c=43

3π ρ

ρπ

ρ ρ ρ( ) ( )M MR c= ≤ ≤ =

34

03

33 3

33 3

1 1G MR

I G Mrcσ ν σ ν

σ

ν σ ν

σ

ν+ −≤ ≤

+ −

+ +

a f a f,

Physical interpretation

Consider a mass distribution with total mass M, radius R and arbitrary ρ(r), which does not increase outwards (II)Consider two related configurations with ρ(r) constant (I&III):

P P P

E E E

g g g

T T T

cI

cII

cIII

gI

gII

gIII

I II III

I II III

≤ ≤

≤ ≤

≤ ≤

≤ ≤

Then:

15

320

320

38

38

35

35

34

34

5 5

2

4

2

4

2

4

2

4

2 2

2 2

0 0

GMR

P GMr

GMR

P GMr

GMR

E GMr

GMR

g GMr

N kGM

RT

N kGMr

c

cc

gc

c

c

π π

π π

μ μ

≤ ≤

≤ ≤

≤ ≤

≤ ≤

≤ ≤

GMR

MM

RR

dyne cm

GMR

MM

RR

erg

GMR

MM

RR

cm sec

N kGMR

MM

RR

K

O

O

O

O

O

O

O

O

2

416

2 42

248

2

24

22

0

7

1118 10

3 791 10

2 739 10

2 293 10

= ×FHGIKJFHIK

= ×FHGIKJFHIK

= ×FHGIKJFHIK

= ×FHGIKJFHIK

. /

.

. /

.μ μ

Numbers

Enormous pressures, densities and temperatures!

Specifically, this gives: