BC AST 2008 Lecs21-22Post.ppt [Read-Only]physics.bc.edu/Courses/PH510/BC_AST_2008_Lecs21-22Post.pdf · “Bodies that maintain equilibrium between ... Stellar Evolution Solution of

Stellar Astrophysics - BC 2008

Stellar Astrophysics

Chris Engelbrecht

Boston College

Spring 2008


Homework Assignment 4

CO2 Problems 12.3, 12.7, 12.13, 12.15, and 12.18

Due date: Wednesday 19 March, 13:00


Stellar Evolution

The question “What are the stars?” provoked the following

answers (amongst others) when we first met:

The stars are:

“Large balls of hot gas”

“The fabric of the universe”

“Bodies that maintain equilibrium between

gravity

and nuclear reactions”


Stellar Evolution

What have we learned?

HOT

Atmosphere

(thin !)

We are indeed dealing

with hot (spectroscopy!)

gaseous (spectroscopy)

balls (gravity)...

that are opaque (radiation

transfer theory) with a very

thin atmosphere as outer

coating (spectroscopy/rad.

transfer theory)


Stellar Evolution


HOT

Atmosphere

(thin !)

Inside, energy is being transported

either by photons taking a random walk

(ideal gas theory, atomic physics) out-

ward, or, if is high enough, by large-

scale motion of atoms and plasma

(convection theory).

Increases in accompany PIZ’s (i.e.

temperature zones). We can regard the

stellar interior as a sequence of spherical

shells that have individual values of T, P,

, and composition.


Stellar Evolution


Inside, energy is being transported

either by photons taking a random walk

(ideal gas theory, atomic physics) out-

ward, or, if is high enough, by large-

scale motion of atoms and plasma

(convection theory).

Increases in accompany PIZ’s (i.e.

temperature zones). We can regard the

stellar interior as a sequence of spherical

shells that have individual values of T, P,

, and composition.


Stellar Evolution


In the core, quantum tunneling allows

fusion of hydrogen nuclei to occur at

rate sufficiently high that the pressure

gradient can maintain hydrostatic equi-

librium with gravity.

Since the feasible fusion reactions all

have a strong temperature dependence,

the fusion zones have sharp edges.


Stellar Evolution


HOT

Atmosphere

(thin !)

At an optical depth of 2/3, we see the

opaque ball’s surface (photosphere)

as a good approximation to a black body

with temperature Te ;

the layer above the photosphere commu-

nicates information about T, Pe and com-

position to us, via the spectral signature

imposed on the BB flux coming form the

photosphere.


Stellar Evolution

These “thin-shelled balls

of hot gas” are gene-

rated by the collapse

and fragmentation of a

Giant Molecular Cloud

(GMC) or a dense core.



Stellar Evolution


The Jeans approach applied to GMC’s leads to the formation of

stellar clusters and the ZAMS


Stellar Evolution

The following diagram, from a classic paper by Icko Iben, beautifully shows how

the physical parameters in a star “settle down” as the star “reaches” the ZAMS:

Icko Iben (jr), ApJ (1965) 141, 993

log Te

QRC

log R

log ( c / ave)

log L


Stellar Evolution

We have now characterised what stars are, which processes

occur in them, how they form, and what physical characteristics

they are born with. We now turn our attention to their further

evolution:

Icko Iben (jr), ApJ (1965) 141, 993


“The science of stellar evolution describes how the observable

properties of stars may sensibly change as time passes”

- Donald Clayton

Stellar Evolution

Which clues do we have to work with ?


Stellar Evolution

wikimedia

An analogy ....


Stellar Evolution

years 10 6

85 <solar

M

Most stars of a given class more or

less maintain their luminosity

Assumption of hydrogen fusion as

the primary source of luminous

radiation in stellar interiors

years 10 14

07.0solar

M


Stellar Evolution


How can we explain the Post-ZAMS HRD and the observed

content of the galaxies ?

Stellar populations, HII regions, Globular clusters, Open clusters, Dust lanes


Stellar Evolution

http://s

eds.o

rg

ww

w.a

str

onom

ynote

s.c

om

Population I and Population II stars

bluer,

more “metals”

in galactic disk

redder,

“metal-poor”

in galactic halo

and bulge

H II region

IPH

AS

Surv

ey, Jonath

an Irw

in,

Institu

te o

f A

str

onom

y, C

am

bridge

Open cluster (Pop I)

Globular cluster

(Pop II)H

ubble

Space T

el


Stellar Evolution

Dust lanes:

Hubble Heritage


Stellar Evolution

http://seds.org


Stellar Evolution

http://o

utr

each.a

tnf.csiro.a

u

Every open cluster displays a distinct distribution of

stars on the HRD


Stellar Evolution

Madsen e

t al. A

&A

2002


Stellar Evolution

ww

w.a

str

o.c

olu

mbia

.edu

“ZAMS fitting” allows us to situate individual clusters

in a common framework:


Stellar Evolution

outr

each.a

tnf.csiro.a

u

ZAMS fitting produces a “snapshot of cluster ages”


The dependent variables here are P, and Mr . In order to

have a solvable system of differential equations, we need one

more equation relating two of these variables, for instance P

as a function of :

This is called the Barotropic Case.

We have obtained mathematical descriptions of gradients and

conservation laws inside static stars. Two of them look like

this:

2r

MG

dr

dPr= )4(

2r

dr

dMr =

Stellar Evolution

)(PP =


These three equations can be solved to provide the pressure,

density and mass distribution, as functions of radial distance

from the centre of the star.

2r

MG

dr

dPr= )4(

2r

dr

dMr =

Stellar Evolution

)(PP =


If the pressure depends not only on density, but also on tempe-

rature: (as in an ideal gas), we have the

Non-barotropic Case.

Since there are now four dependent variables, we require

another differential equation, describing the temperature

gradient.

These three equations can be solved to provide the pressure,

density and mass distribution, as functions of radial distance

from the centre of the star.

2r

MG

dr

dPr= )4(

2r

dr

dMr =

Stellar Evolution

),( TPP =


The choice of temperature gradient depends on the mode of

energy transport , which in turn depends partly on the mode of

nuclear energy generation. The inclusion of all these factors

greatly complicates the computation of stellar structure (i.e. P, T,

and Mr as functions of r).

However, we have two equations for the temperature gradient:

2r

MG

dr

dPr= )4(

2r

dr

dMr =),( TPP =

Stellar Evolution

2344

3

r

L

Tacdr

dTr=

2

H11|

r

GM

k

m

dr

dTr

ad

μ= ;


We will solve the Barotropic Case for a few special conditions, as

an example of the general approach to fully determining the stellar

structure.

The equation that we will derive is called

the Lane-Emden equation,

and it looks like this:

n

n

nD

d

dD

d

d=2

2

1

Stellar Evolution

CONCEPT 30


Derivation of the Lane-Emden equation:

Start with the equation for hydrostatic equilibrium:

2r

MG

dr

dPr=

)4(2

rdr

dMr =

Re-arrange, then take the

derivative on both sides: dr

dMG

dr

dPr

dr

dr=

2

Stellar Evolution

Then use the second of our

structure equations to simplify (1):

- (1)



Gdr

dPr

dr

d

r4

12

2=Which gives us

Stellar Evolution

KP =

We have eliminated Mr , and are left with P and . Another

equation relating these two physical variables will allow us to

proceed. We have seen that an ideal gas under adiabatic

conditions obeys the rule:

where = 5/3

A system that obeys this relationship in general, is called a

PolytropeCONCEPT 31


Substituting for P , our differential equation becomes:

Gdr

Kdr

dr

d

r4

)(1 2

2=

n

n 1+

Stellar Evolution


i.e.G

dr

dr

dr

d

r

K4

2

2

2=

We now define the polytropic index n as follows:


Substituting for :

Gdr

dr

dr

d

r

K

n

n nn4

1 /)1(2

2=

+

[ ]nnc

rDr )()(

1)(0 rDn

Stellar Evolution


Introducing the dimensionless parameter Dn through:

where

.

,


our equation

=dr

dnr

dr

d

dr

dr

dr

dn

n

/121)/1(2

Gdr

dr

dr

d

r

K

n

n nn4

1 /)1(2

2=

+

Stellar Evolution


is reduced to a simpler form:

Let’s first tackle the derivative itself:

=dr

dDnr

dr

dnn

c

/12


Gdr

dr

dr

d

r

K

n

n nn4

1 /)1(2

2=

+

n

n

n

nn

cD

dr

dDr

dr

d

rG

Kn =+

2

2

/)1(1

4)1(

Stellar Evolution


Substituting in, and collecting constants, our differential equation

is reduced to


Gdr

dr

dr

d

r

K

n

n nn4

1 /)1(2

2=

+

n

n

n

nn

cD

dr

dDr

dr

d

rG

Kn =+

2

2

/)1(1

4)1(

Stellar Evolution


Substituting in, and collecting constants, our differential equation

is reduced to

this is dimensionless


So, this is dimensionless too


n

n

n

nn

cD

dr

dDr

dr

d

rG

Kn =+

2

2

/)1(1

4)1(

Stellar Evolution


Let’s call this factor ( n)2 . Clearly, n is then a parameter of length.

If we now introduce a second dimensionless parameter by defining



2

2

2

1

r

n=


n

n

n

nn

cD

dr

dDr

dr

d

rG

Kn =+

2

2

/)1(1

4)1(

Stellar Evolution


Then

is reduced to

n

n

nD

d

dD

d

d=2

2

1

The Lane-Emden equation.


Solution of the Lane-Emden equation:

This is a second-order ordinary differential equation. In order to

fully solve it, we need two boundary conditions (BC’s).

BC 1: Let when , i.e. when the density

drops to zero (in practice, when it drops to a sufficiently low

value)

Stellar Evolution

1= 0)( =

nD

BC 2 requires some talking first:


Let represent an infinitesimally small radial distance from the

stellar centre. Then, the mass contained inside a sphere with this

radius is:

2r

MG

dr

dPr=

3

3

4=

rM

Stellar Evolution


If hydrostatic equilibrium holds in this small volume, we can write

0 as 03

4 2= G

dr

dP

, and



0=d

ndD

0=

0dr

dP

Consequently, it is convenient to choose our second boundary

condition as

Stellar Evolution

when

Since we are dealing with polytropes, if

then as well.0dr

d

.

As a final measure, we normalise Dn so that c is the central density:

1)0( =n

D .


Stellar Evolution

The Lane-Emden equation applied to polytropic spheres provides

a useful practice ground for the solution of the full set of stellar

structure equations for physically realistic stellar models.

Let’s play a bit...


Stellar Evolution

The Lane-Emden equation applied to polytropic spheres provides

a useful practice ground for the solution of the full set of stellar

structure equations for physically realistic stellar models.

Let’s play a bit... Let n = 0:

Cd

dD

dd

dDd

Dd

dD

d

d

+=

=

==

3

3

102

202

0

002

2

,

,11

where our BC’s

make C = 0


Stellar Evolution

So:

CD

d

dD

+=

=

20

0

6

1)(

,3

1

and the application of our BC’s and the normalisation we chose, deliver


Stellar Evolution

61)(

2

0 =D

Two other choices of n that deliver simple solutions,

are n = 1 and n = 5:

[ ] 2/125

1

3/1)(

;sin

)(

+=

=

D

D


Stellar Evolution

61)(

2

0 =D

These three solutions are illustrated in the diagram:

[ ] 2/125

1

3/1)(

;sin

)(

+=

=

D

D


Stellar Evolution

The two values of n that are relevant to stellar structure are

n = 1.5 and n = 3.

The first of these corresponds to an adiabatic ideal gas.

The second corresponds to the Eddington standard model and

describes a star in radiative equilibrium:

say ,Pm

kTP

H

gμ

==

fraction of total pressure contributed by the gas


Stellar Evolution

say ,Pm

kTP

H

gμ

==

PaTPr

)1(3

1 4==The contribution from radiation pressure is

3/4

4

,)1(3

1 μ

KP

Pk

mPa

H

=

=

Eliminating T from the equation for gas pressure, we find

(i.e. n = 3)


As mentioned earlier, this is the full set of equations describing

stellar structure in the spherically symmetric approximation:

2r

MG

dr

dPr= )4(

2r

dr

dMr =

),( TPP =

Stellar Evolution

2344

3

r

L

Tacdr

dTr=

2

H11|

r

GM

k

m

dr

dTr

ad

μ= or

)4(2

rdr

dLr =; ;

and

There are five dependent variables appearing in these four

equations. The required fifth equation is the equation of state

(EOS), usually expressed as


Stellar Evolution

As a star fuses hydrogen in its core, the value of the mean

molecular weight and, consequently, the values of opacity and

reaction rate, are forced to change on a continuous basis.

Therefore, the solution to the equations of stellar structure must

also change on a continuous basis.

Stellar evolution is mapped out theoretically by solving these

equations in discrete time steps, feeding the changes occurring in

each step into the equations for the following step.

The first (or at least best-known) detailed solution of this kind was

done by Icko Iben (jr) in a series of seminal papers in the 1960’s.


Stellar Evolution

I. I

be

n (

jr),

ApJ (

19

65

) 1

42

, 1

14

7


Stellar Evolution

I. I

be

n (

jr),

ApJ (

19

66

) 1

43

, 5

16


Stellar Evolution

I. I

be

n (

jr),

An

n.R

ev.

Astr

on

. A

str

op

hys.

(19

67

) 5

71


Stellar Evolution

Marigo

et

al. A

&A

(2001)

(low

-meta

l sta

rs)


Stellar Evolution

Marc

oni et

al. A

&A

(2001)

isochro

ne


Stellar Evolution

ww

w.a

str

o.c

olu

mbia

.edu

“ZAMS fitting” allows us to situate individual clusters

in a common framework:


Stellar Evolution


Ludwig

Boltzmann

Bart Bok Guillermo

Haro

AIP Niels Bohr Library

Wikipedia

Arthur Stanley

Eddington

Credit: Charles LadaUC Santa Cruz

George

Herbig

ww.umpa.ens-lyon.frwww.dpg-physik.de

Meghnad

Saha

www.jiast.cz

James

Jeans

Chushiro

Hayashi

C H

ayashi

BC AST 2008 Lecs21-22Post.ppt [Read-Only]physics.bc.edu/Courses/PH510/BC_AST_2008_Lecs21-22Post.pdf · “Bodies that maintain equilibrium between ... Stellar Evolution Solution of

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