Post on 13-Oct-2020
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