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Lecture 5-4: Solution Methods

This is not part of the course

1

a) Classical approach Basic equations of stellar structure + boundary conditions in §11

Classical (historical) method of solution: Outward integration from center, where Inward integration from surface, where Fitting two resulting sets of curves at intermediate point, requiring all 4 curves to be continuous

Discontinuity in infinite force in infinite radiative flux in infinite density in infinite energy generation

Discontinuities possible, and allowed in other quantities

PTmL(r)

P = Pphot, T = Tphot

L =m = 0

m, L(r), P, T

2

⇒⇒⇒⇒

Example: convective core and radiative envelope

Nuclear reactions: composition of core changes with time Convection: uniform composition throughout core no change in envelope

At rcore: discontinuity in discontinuity in

Similarly: discontinuity in Presence of discontinuities crucial for stellar evolution

This example applies to MS stars with M>1.2M# (see §13d)

P1 = P2T1 = T2

!"#

$#⇒

ρ1µ1=ρ2µ2

∇rad

κ

"

#$

%

&'1

=∇rad

κ

"

#$

%

&'2

∇rad

3

⇒

⇒⇒

ρκ

Define the following dimensionless variables: Write:

Then the equations of stellar structure are

mass continuity

hydrostatic equilibrium

thermal equilibrium

radiative energy transport

adiabatic convective energy transport

b) Schwarzschild variables

p = 4π R4

GM 2 P, t = N0kµ

RGM

T, q = mM, f =

L, x = r

R

κ =κ0ρnT −s, ε = ε0ρ

λT ν

(1) dqdx

= x2 pt

(2) dpdx

= −qptx2

(3) dfdx

= Dx2pλ+1tν−λ−1

(4a) dtdx

= −C f pn+1

x2tn+s+4

(4b) 1tdtdx

=Γ2−1Γ2

1pdpdx 4

Where:

These contain dependence on: - physical constants - two scale factors which are functions of L, R, M and

Boundary conditions:

Most applications: use (convective) zero surface conditions The differential equations (1)-(4) can be integrated by standard numerical techniques (e.g., Runge-Kutta schemes) and require the fitting of core and envelope solutions

C =C(n, s) = 34(4π )n+2ac

N0kG

!

"#

$

%&s+4

κ0µ s+4

LRs−3n

M s−n+3

D = D(λ,ν ) = 1(4π )λ

GN0k!

"#

$

%&

ν

ε0µν M ν+λ+1

LRν+3λ

x = 0 : q = f = 0x =1: p = pphot, t = tphot

5

µ

c) Invariants and the (U,V)-plane Fitting of core- and envelope integrations is conveniently done using the dimensionless (homology invariant) variables:

U =d lnmd ln r

=4π r3ρm

= 3× local densitymean interior density

V = −d lnPd ln r

=Gmr

ρP=32×gravitational energyinternal energy

W =d lnd ln r

=4π r3ρε

= 3× local energy generation ratemean energy production inside r

ne +1=d lnPd lnT

= = effective polytropic index

Limiting values at center: at surface:

Inward and outward integrations must be matched at some intermediate point, but required smoothness of fit depends on whether is continuous or discontinuous (cf §13a)

U = 3, V = 0, W = 3U = 0, V→∞, W = 0

6 µ

continuous Solution must be: - continuous in (U,V,W) space - with continuous slope

discontinuous At fitting point:

Uµ

!

"#

$

%&in

=Uµ

!

"#

$

%&out

Vµ

!

"#

$

%&in

=Vµ

!

"#

$

%&out

Wµε

!

"#

$

%&in

=Wµε

!

"#

$

%&out

7

µ

µ

Fitting often done outside region of energy generation: W=0 Thus, suffices to consider the (U,V)-plane

If core or envelope is polytropic: fit to a single curve

8

U =px3

qt, V =

qtx, W = D x3pλ+1tν−λ−1

f, (ne +1)rad =

1Cqtn+s+4

f pn+1, (ne +1)ad =

Γ2−1Γ2

Relation to Schwarzschild variables

(U,V,W,ne+1) are defined as logarithmic derivatives. Thus, invariant under certain scale transformations (§12,§13e)

These are often applied to Schwarzschild variables, separately in core and envelope fitting convenient in U,V (+W) space

d) Cowling model We will see in §15 that main sequence stars with M>1.2M# generate their energy via the CNO-cycle: steep function of T energy generated in central core

Expect: convective core & radiative envelope (different possible) for r>rc = radius of core

This is the Cowling model 9

⇒

⇒ε ⇒µ

ε r( ) =0

Mechanical structure For solve eqs (1), (2), (4b): polytrope of index n=3/2 solve eqs (1), (2), (4a): numerical solution This is conveniently done using Schwarzschild variables: - Integrate inwards for chosen C - When ne drops to 1.5: core is reached: this defines rc

- Fit, in (U,V) plane to curve of n=3/2 polytrope: possible for only 1 C-value

This shows that no knowledge of is needed! But need to know for envelope integrations Thus, different model obtained for each choice of n and s

Thermal equilibrium Energy generated = energy radiated, eq(3): For each choice of this gives the value of D required by thermal equilibrium Expressions for C and D then give M=M(R, ) or L=L(M, )

1= D x2pλ+1tν−λ−1 dx0

1

∫

0 ≤ r ≤ rcrc ≤ r ≤ R

κ =κ0ρnT −s

10

ε

λ andν

µ µ

Classical solution methods generate one model at the time For construction of series of neighboring models (evolution!) it is more convenient to use ‘relaxation method’ (Henyey) Idea Start with approximate solution (polytrope, or previous model) Use equations of stellar structure to calculate corrections Iterate until convergence

Algorithm (see KW §11.2; HKT § 7.2.4)

Write:

Then eqs (1)-(4):

Divide model in K-1 mass shells, and discretize:

e) Henyey method

y1 = r, y2 = P, y3 = T, y4 =

dyidm

= fi (y1, y2, y3, y4 ) i =1,2,3, 4

0 = Aij =

yij − yi

j+1

mij −mi

j+1 − fi (y1j+1/2,..., y4

j+1/2 )

shell boundaries shell center 11

Values in center of cell can be found in various ways, e.g., Four special equations at center, since Series expansions of §11b give:

r0 =m0 = 0 = 0

r1 − 3m1

4π ρ0"

#$

%

&'

1/3

= A11

P1 −P0 + 3G8π

4π ρ0

3"

#$

%

&'

4/3

[m1]2/3 = A21

[T1]4 −[T 0 ]4 +κ0 (εn

0 −εν0 +εg

0 )2ac

34π"

#$

%

&'2/3

[ρ0 ]4/3[m1]2/3 = A31

lnT1 − lnT 0 +π6"

#$

%

&'1/3

+G∇ad

0

P0[ρ0 ]4/3[m1]2/3 = A3

1

1 − (εn0 −εν

0 +εg0 )m1 = A4

1

Radiative Convective

Only two equations at surface, as fixed by boundary conditions (§11c) yields total of 4K-2 equations

PK , T K

fi (y1j+1/2,..., y4

j+1/2 ) = fi ([y1j + y1

j+1] / 2,...,[y4j + y4

j+1] / 2)

12

Solve by means of a Newton-Raphson technique Assume you have trial solution (e.g. previous model) Since this is approximate, the do not vanish Hence we must calculate corrections Linearize equations (�) by Taylor expansion to first order This gives 4K-2 linear relations between and Write this in matrix form with H the Henyey matrix or or H is of block-diagonal form and can be inverted efficiently so that corrections follow and new solution: Then iterate until convergence; this gives full solution

yi,nj

δ yi,nj

Aij

Aij δ yi,n

j

yi,n+1j = yi,n

j +δ yi,nj

H

δ y1,n1

.

.δ y1

K

δ y4K

!

"

#######

$

%

&&&&&&&

= −

A11

.

.A1K

A4K

!

"

#######

$

%

&&&&&&&

H •U = −A U = −H −1 •A

13

g) Changes in chemical composition

Transmutation of elements in core. Thus, changing chemical composition and model evolves on

yields evolutionary track in the Hertzsprung-Russell Diagram

Henyey method ideally suited for calculation of sequence of models as function of time, as changes are small

Henyey method: Fast and reliable Choose K sufficiently large Take into account physical transitions inside star Be careful with evaluation of average shell values

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τ nuc

Start with composition Xi (i=1, 2, …,I) in mass element m, at time t

Then:

Since , these are I-1 independent differential equations

Eqs (1)-(5) are the equations of stellar evolution

Nuclear transmutation

See §9: with mi = mass of nucleus i rji = reaction rate for j→i rij = idem for i→j

Also:

with �ji = energy generated per unit mass by transmutation j→i Qji = energy generated when unit mass j→i

The Qji are known from nuclear physics, and the can be calculated at each point in star (as function of and T; see §9)

(5) Xi (m, t +Δt) = Xi (m, t)+∂Xi

∂t#

$%

&

'(m,t

Δt (i =1, 2,..., I )

Xi =1i=1

I

∑

∂Xi

∂t=mi

ρrji − rik

k∑

j∑$

%&&

'

())

∂Xi

∂t=mi

ρ

ε jiQji

−εikQikk

∑j∑$

%&&

'

())

[ε = ε jk ]j,k∑

15

εij

ρ

Convection Material in convective zone is mixed: Evolutionary calculations: (no TE)

Þ  ‘instantaneous’ mixing of new material over entire zone

τmix <<< τ KH << τ nucΔt ≈ 0.01τ nuc or Δt ≈ 0.01τ KH

∂Xi

∂t=

1(m2−m1)

∂Xi

∂t#

$%

&

'(nuclear

dm+m1

m2

∫ 1(m2−m1)

∂m2

∂tXi2 − Xi( )− ∂m1

∂tXi1 − Xi( )

*+,

-./

Result: Second term caused by possible moving of zone boundaries See KW §8.2.2 for additional mixing processes 16

Consider the differential equation Forward differencing: (explicit) Back differencing: (implicit)

Explicit scheme: Unstable unless bΔt<1

Implicit scheme: Provides correct limit a/b when Δt large and/or Xn small Generally more accurate

dXdt

= a− bX ⇒ X = ab+ exp −bt( )

Xn+1 − Xn

Δt= a− bXn ⇒ Xn+1 = aΔt + Xn (1− bΔt)

Xn+1 − Xn

Δt= a− bXn+1 ⇒ Xn+1 =

aΔt + Xn

1+ bΔt

Appendix: Explicit and implicit schemes

17