HW 8 Written solns

1. Ionization fraction in log T vs. log ρ𝜌 plane, overplotted with strucure for Present, RG, and HB phases of sun

a, b:

sec. 14.2, Saha eqn.:

Let i = 0, so that f ≡ f1≡ n1 /∕nH= ne/ nH, and n0= nH (1-f). Then Saha eqn. can be rewritten:

f 2

1-−f = g+

g0 2mp (

2π𝜋 me kh2 )3/∕2 T

3/∕2

ρ𝜌 ⅇ-−ΔEi/∕kT = 4.0 × 10-−6 T32

ρⅇ-−1.6 105T ≡ b[ρ𝜌,T]

Fom the quadratic formula, we then have:

f = (1/2) (-b + b2 + 4 b )

sec. 14.2, Saha eqn.:

Let i = 0, so that f ≡ f1≡ n1 /∕nH= ne/ nH, and n0= nH (1-f). Then Saha eqn. can be rewritten:

f 2

1-−f = g+

g0 2mp (

2π𝜋 me kh2 )3/∕2 T

3/∕2

ρ𝜌 ⅇ-−ΔEi/∕kT = 4.0 × 10-−6 T32

ρⅇ-−1.6 105T ≡ b[ρ𝜌,T]

Fom the quadratic formula, we then have:

f = (1/2) (-b + b2 + 4 b )

c: coding this in Mathematica gives:

b[ρ_, T_] := 4.0 × 10-−6T3/∕2

ρⅇ-−1.6 105T ; (*⋆ MacDonald notes 07-−IonRecomb eqn. 7.5.5 *⋆)

f[ρ_, T_] := 0.5 -−b[ρ, T] + b[ρ, T]2 + 4 b[ρ, T] ;

fl[lρ_, lT_] := f10lρ, 10lT

p0 = ContourPlotfl[lρ, lT], {lρ, -−10, 4},{lT, 3.6, 6.5}, ContourShading → False, Contours → {.1, .5, .9},ContourStyle → {{Thick, Dotted}, {Medium, DotDashed}, Thick}, ContourLabels → True,FrameLabel → "Log ρ (kg/∕m3)", "Log T (K)", GridLines → Automatic

2 HW8-solns.nb

d, e: Overplot of T vs. ρ𝜌 for present, RG, and HB phases of the sun is:

Show[p0, p123, PlotLabel → {"f = H ion fraction; present sun, RG, HB "}]

Reading dot-dashed intersections give temperature and density for f=1/2 as:present day: log T=4.4, log ρ𝜌= -1.5 kgm3

HB: log T=4.3, log ρ𝜌= -3.9 kgm3

RG: log T=4.1, log ρ𝜌= -4.3 kgm3

HW8-solns.nb 3

2. Stellar evolution with EZ-Web for M=10 Msun

soln

e. M vs. r for it=1 and it=400

0 1 2 3 40

2

4

6

8

10

r (Rsun)

M(M

sun)

{t=, 0., yr}

0 50 100 150 200 2500

2

4

6

8

10

r (Rsun)

M(M

sun)

t=, 2.56559×107, yr

4 HW8-solns.nb

e. M vs. r for it=1 and it=400

0 1 2 3 40

2

4

6

8

10

r (Rsun)

M(M

sun)

{t=, 0., yr}

0 50 100 150 200 2500

2

4

6

8

10

r (Rsun)

M(M

sun)

t=, 2.56559×107, yr

f. Overplots of T vs. M for it=1 (black) and it=400 (blue), on both linear and log scales for T:

0 2 4 6 8 100

1×108

2×108

3×108

4×108

5×108

6×108

7×108

M (Msun)

T(K

)

t=, 2.56559×107, yr

0 2 4 6 8 100

2

4

6

8

M (Msun)

T(K)

t=, 2.56559×107, yr

g. Overplots of log ∇ad (blue) and log ∇rad (purple) vs. r for it=1 and it=400:

0 1 2 3 4

-−0.8

-−0.6

-−0.4

-−0.2

0.0

0.2

r (Rsun)

∇rad

{t=, 0., yr}

Initial ZAMS model has ∇rad > ∇ad only in core, r <1 Rsun

0 50 100 150 200 250

-−1

0

1

2

3

r (Rsun)

∇rad

t=, 2.56559×107, yr

Final stage model has ∇rad > ∇ad in both small core r<1 Rsun, and envelope R > 1.5 Rsun

HW8-solns.nb 5

g. Overplots of log ∇ad (blue) and log ∇rad (purple) vs. r for it=1 and it=400:

0 1 2 3 4

-−0.8

-−0.6

-−0.4

-−0.2

0.0

0.2

r (Rsun)

∇rad

{t=, 0., yr}

Initial ZAMS model has ∇rad > ∇ad only in core, r <1 Rsun

0 50 100 150 200 250

-−1

0

1

2

3

r (Rsun)

∇rad

t=, 2.56559×107, yr

Final stage model has ∇rad > ∇ad in both small core r<1 Rsun, and envelope R > 1.5 Rsun

h. Overplots of log ∇ad (blue) and log ∇rad (purple) vs. M for it=1 and it=400:

0 2 4 6 8 10

-−0.8

-−0.6

-−0.4

-−0.2

0.0

0.2

M (Msun)

∇rad

{t=, 0., yr}

Initial ZAMS model has ∇rad > ∇ad only in core, M <3.5 Msun

0 2 4 6 8 10

-−1

0

1

2

3

M (Msun)

∇rad

t=, 2.56559×107, yr

Final stage model has ∇rad > ∇ad in both innermost core M<0.1 Msun, and envelope M > 3.2 Msun

6 HW8-solns.nb

h. Overplots of log ∇ad (blue) and log ∇rad (purple) vs. M for it=1 and it=400:

0 2 4 6 8 10

-−0.8

-−0.6

-−0.4

-−0.2

0.0

0.2

M (Msun)

∇rad

{t=, 0., yr}

Initial ZAMS model has ∇rad > ∇ad only in core, M <3.5 Msun

0 2 4 6 8 10

-−1

0

1

2

3

M (Msun)

∇rad

t=, 2.56559×107, yr

Final stage model has ∇rad > ∇ad in both innermost core M<0.1 Msun, and envelope M > 3.2 Msun

HW8-solns.nb 7

3. log T vs. log ρ𝜌 domain diagram for ionization, degeneracy, etc.

Calculations

Final results: formulae for a-e, and annotated “master plot”

8 HW8-solns.nb

Final results: formulae for a-e, and annotated “master plot”a. Prad =Pgas : Log T = 7.6 + (1/3) Log ρ𝜌

b. Pressure ionization occurs at: Log ρ𝜌pi= 0.433

c. Ionization fractions set by: Log ρ𝜌i = C + (3/2) LogT 105K) - 105K/T)where C= {0.36, -1.29, -2.50} for f=(0.1, 0.5, 0.9)

d. Equal pressure for ideal gas and non-relativistic degenerate gas requires:Pgas = 2 ne kT = Pnr = ne5/∕3 ℏ2 me

Solving gives critical temperature for degeneracy as a function of density:Tdgnr=ρ mp

2/∕3ℏ2 (2me k)

e. Equal pressure for relativistic and non-relativistic degenerate gas requires:Prel = ne4/∕3 ℏ c = Pnr = ne5/∕3 ℏ2 me

Solving gives critical density for relativisitic degeneracy:ρ𝜌rel = (mec/ℏ )3 mp ; Log[ρ𝜌rel]= 7.46

Putting this all together, here then is an annotated “master plot” of boundaries in the log T vs. log ρ𝜌 plane:

pres

sure

ioni

zed

P rad> Pgas

ionized

neutral

non-−

dege

n

dege

n, no

n-−re

l

dege

n,re

l

P rad< Pgas

partly

ionized

-−10 -−5 0 53

4

5

6

7

8

9

Log ρ𝜌

Log

T

HW8-solns.nb 9

4. EZWeb summary results for M=0.1 to 100 Msun

-−2 0 2 4 6

-−2

0

2

4

6

σ𝜎T4 4π𝜋R2

L

ZAMS

itt

1

-−2 0 2 4 6

-−2

0

2

4

6

σ𝜎T4 4π𝜋R2

L

{it=, 1}

10 HW8-solns.nb

-−4 -−2 0 2 4 6

-−4

-−2

0

2

4

6

σ𝜎T4 4π𝜋R2

L

All data

-−1.0 -−0.5 0.0 0.5 1.0 1.5 2.0

-−2

0

2

4

6

log M/∕Msun

log

L/∕Ls

un

ZAMS

HW8-solns.nb 11

Rsz = dat[[All, itt, 5]];pd = ListPlot[Transpose[{Msz, Rsz}], Joined → False,

Frame → True, FrameLabel → {"log M/∕Msun", "log R/∕Rsun", "ZAMS"}];p0 = Plot[{x, 0.75 x, 0.5 x}, {x, -−2, 2}];Show[pd, p0]

-−1.0 -−0.5 0.0 0.5 1.0 1.5 2.0

-−0.5

0.0

0.5

1.0

log M/∕Msun

log

R/∕R

sun

ZAMS

-−1.0 -−0.5 0.0 0.5 1.0 1.5 2.0

-−0.2

0.0

0.2

0.4

0.6

0.8

1.0

log M/∕Msun

log

T/∕Ts

un

ZAMS

e. Red dotted line, which has slope 3 in log-log plot that is predicted by scaling analysis in sec. 16.2, fits the points quite well.For Mass-Radius plot, the 3 lines are for slopes of 0.5, 0.75 and 1. The steepest, unit-slope line nearly fits the data for lower-mas stars with M<1 Msun.The shallowest, 1/2-slope line near fits the data for higher-mass stars with M > 1 Msun.

12 HW8-solns.nb