Stella Revolution 1

7/29/2019 Stella Revolution 1

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Stellar Structure and Evolution

z 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

z Recommended book Kippenhahn & Weigert (1989; KW)

Hansen, Kawaler & Trimble (2005; HKT)

Background/supplemental: see separate list

z 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 havemissed 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

z Sun

M

= 1.9891 1033 gram

R

= 6.9598 1010 cm

L

= 3.8515 1033 erg/sec

= 1.4086 gm/cm3

z Stars Apparent magnitudes colors

Stellar atmosphere models L, Teff, R

Certain binaries M

p L R T T K eff eff O= =4 57802 4

,

10 10

0 08 100

1 800 1500

6 6

L L

M M

R R

O

O

O

.

10 10

2000 100000

9 6

O

effT K

a) Mass, luminosity, radius, & effective temperature

O

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)


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H-R diagram for the nearby stars

Luminosities based on approximate distances: scatter in MVUsing Morgan-Keenan spectral types: discreteness in Teff

MV derived from parallax

obtained by HIPPARCOS

for nearby stars with V


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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 MVUse 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


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NGC 6397

HST/ACS


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The Hyades

Individual parallaxescorrected for depth of

the cluster by using

the proper motions

secular parallaxes

These are a factor 3

more accurate than

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:

z Position of stars in

this diagram

z Evolution of stars in

this diagram

z Differences between

cluster HRDs

z Nature 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 5

1 206265

log

with BC the bolometric correction

Distance modulus; AV extinction

Trigonometric distance d

follows from parallax

Absolute bolometric magnitude Mbol and luminosity L:

z Absolute magnitude MV versus spectral type SP

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

z 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&A

Martin & Mignard 1998 AA 330, 585)

L

L

M M M M M

M M M M M

M M M M MO

O O O

O O O

O O O

=

0 66 0 08 0 5

0 92 0 5 40

300 40 130

2 5

3 55

2

. . .

. .

.

.

b gb gb g

e) Mass-luminosity relation

Recent measurements:


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f) Internal structure

Helioseismology

z Study of oscillations

of Solar surface

z Provides probe of

internal structure

z Extremely accurate

Solar model

Asteroseismologyz Idem for other stars, but surface not spatially resolved

Fractionalerrorin

sound

speed

P/

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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z What is the internal structure of stars?

z What causes the mass-luminosity relation?

z What sets the range of stellar masses?

z What generates different classes of stars in HR Diagram?

z What are the final stages of stellar evolution?

z How do stars produce the heavy elements?

z Why do some stars pulsate?

z What additional processes occur in binary stars?

h) Some questions

To be answered by applying the laws of physics

i) Outline of course

z Derivation of four equations of stellar structure

- Mass continuity ( 2)

- Hydrostatic equilibrium ( 2)

- Thermal equilibrium ( 4)

- Energy transport by radiation (6) or convection ( 8)

z Required physics- Thermodynamics ( 3)

- Equation of state including degeneracy, and internal energy ( 5)

- Opacity of stellar material ( 7)

- Nuclear energy generation ( 9, 10)

z Solution methods and simple models ( 11-14)

z Overview of stellar evolution ( 15-24)


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2. Spherical stars

r R

M m R

rm r

r

M

R

=

=

=

=

( )

( )( )

3

4

3

4

3

3

m r r r dr dm

drr r

r

( ) ( ) ( )= =z4 4 120

2 b g

( )

( )

r

m r

a) Mass-continuity equation

stellar radius

total 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 rGm r

r

d

dr= = =( )

( )2

with the gravitational potential, and

Check of accuracy of spherical approximation

z Rotation period of the Sun is 27 days

centrifugal acceleration at equator: vc2/R

z Gravitational acceleration at equator: GM/R2

z Ratio is 2 x 10-5 so rotation unimportant, and star

can be treated accurately as a sphere

surface

GM

R=

(KW 1)


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dP

dr

Gm r r

r=

( ) ( )2

2a f

11

4

24

2

4

a f

a f

=

=

dr

dm r

dP

dm

Gm

r

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

This is the equation of hydrostatic equilibrium, which

equates the pressure gradient to the gravitational forceOften useful to employ m=m(r)

as variable, not r. Then:

c) Hydrostatic equilibrium

Small cylinder of thickness dr, surface 1cm2 mass dr

Newtons equation of motion:

with P the pressure

&&( )

rdP

dr

Gm r

r=

12

(KW 2.1-2.4)

Solutions of (1) and (2) exist for the special case

These are so-called barotropicstars, and include the

famous polytropeswith 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 gashas

No is Avogadros number = 6.02257 1023

k is Boltzmanns constant = 1.38062 10-16 erg/K

is the mean molecular weight (see 3b)

P K=

P N kT

=

0

P P= ( )

P P T= ( , )


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d) Time scales

Consider again Newtons equation:&&

( )

r

dP

dr

Gm r

r=

12

P r gR R

g

g rP

R

RR

P

R

ff

ff

l

l

sound

= = = =

= = =

0

0

2

2

: && :

: && :exp

exp

Two extreme cases:

g GM R P GM R / /2 2 4 expl ff=

Free-fall

time scale

Explosion

time scale

In H.E. both terms contribute equally (but with opposite sign),

so we can write . Using

(2f), we find:

This is the hydrodynamicaltimescale

ff l hydro

R

GM G= = = exp

3 1

2

Stars are in hydrostatic equilibrium

Problem: What is thydro for a neutron star?

Can a pulsar be a pulsating neutron star?

hydroO

OR

R

M

Mseconds=

F

HGI

KJFH

IK1600

3 2 1 2/ /

3 1000sec dayshydro

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 32 /

&& ( ) /r Gm r r = 2


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I Gmr

dm G m dm

M M

,

/

/ /= = FH IKz z

0

3

3

0

343

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

where we have used

Assume the density (r) does not increase outwards,

Then: and

It follows that:

with rc defined by

r m r r 3 3 4= ( ) / ( )

d r dr ( ) / 0

M rc c=4

3

3

( ) ( )MM

Rc= =

3

40

3

3

3 3

3

3 3

1 1G M

R

IG M

rc

+

+

+ +

a f a f,

Physical interpretation

Consider a mass distribution with total mass M, radius R

and arbitrary (r), which does not increase outwards (II)

Consider two related configurations with (r) constant (I&III):

P P P

E E E

g g g

T T T

c

I

c

II

c

III

g

I

g

II

g

III

I II III

I II III

Then:


15/15

320

320

3

8

3

8

3

5

3

5

3

4

3

4

5 5

2

4

2

4

2

4

2

4

2 2

2 2

0 0

GMR

P GMr

GM

RP

GM

r

GM

RE

GM

r

GM

Rg

GM

r

N k

GM

R T N k

GM

r

c

c

c

g

c

c

c

GM

R

M

M

R

Rdyne cm

GM

R

M

M

R

Rerg

GM

R

M

M

R

Rcm sec

N k

GM

R

M

M

R

R K

O

O

O

O

O

O

O

O

2

4

16

2 4

2

248

2

2

4

2

2

0

7

1118 10

3 791 10

2 739 10

2 293 10

= FHG

IKJ

FH

IK

= FHG

IKJ

FH

IK

= FHG

IKJFH

IK

= F

HG

I

KJ

F

H

I

K

. /

.

. /

.

Numbers

Enormous pressures, densities and temperatures!

Specifically, this gives:

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Stella Revolution 1