Phys333 HW4 Solutions

#1 “Opacity” of people vs. electrons

a.

h = 1.8 m;w = 0.5 m;m = 100 cm;σ = h * w;Framed["σ=" NumberForm[σ, 4]]

σ= 9000. cm2

b.

m = 60 kg;kg = 1000. g;κ = σ / m;Framed["κ=" NumberForm[κ, 2]]

κ=0.15 cm2

g

c.

κe = 0.34 cm2 g; (* Eqn. C2 *)

Frameo["κ/κe=" NumberForm[κ / κe, 2]]

κ/κe= 0.44

#2

#2

Z =.; z =.; ρ =.; m =.; r =.;

a.

n = (ρ / m);"n=" Framed[n]

n=ρ

m

b.

σ = π r2;Framed["σ=" σ]

σ= π r2

2 Hw4-solns.nb

c.

κ = σ / m;"κ=" Framed[κ]

κ=π r2

m

d.

ℓ = 1 / (κ ρ);"ℓ=" Framed[ℓ]

ℓ=m

π r2 ρ

e.

τ = Z / ℓ;"τ=" Framed[τ]

τ=π r2 Z ρ

m

f.

τc = τ / 2; (* center of slab has half the total optical depth *)

L = τc2 ℓ; (* by random walk, total length is τc2 times mfp *)

"L=" Framed[L]

L=π r2 Z2 ρ

4 m

Hw4-solns.nb 3

g.

F =.; σsb =.;Fup = (F / 2); (* flux splits equally up/down *)

Teff = (Fup / σsb)1/4;τs = 0; (* surface optical depth *)

Ts = Teff ((3 / 4) (τs + 2 / 3))1/4 ;(* Eqn. 16.10, applied at surface with τs=0 *)

"Ts=" Framed[FullSimplify[Ts]]

Ts= F

σsb1/4

2

h.

Tc = Teff ((3 / 4) (τc + 2 / 3))1/4;(* Eqn. 16.10, applied at center with optical depth τc *)

"Tc=" Framed[FullSimplify[Tc]]

Tc=1

24 +

3 π r2 Z ρ

m

1/4 F

σsb

1/4

4 Hw4-solns.nb

#3 Hydrogen fusion burning

a. Mass loss from solar luminosity

m =.; s =.; kg =.; J = kg m^2 s2; W = J / s;

c = 3. × 108 m / s;Lsun = 4. × 1026 W;Mdot = Lsun c2;

Framed["Mdot=" NumberForm[Mdot, 2]]

Mdot=4.4 × 109 kg

s

Msun = 2. × 1030 kg;yr = 365.25 * 24 * 3600 s;Framed["Mdot=" NumberForm[Mdot / (Msun / yr) "Msun/yr", 2]]

Mdot= 7. × 10-14 Msun/yr

Hw4-solns.nb 5

b. H-burning rate

ϵ = 0.007;MdotH = Mdot / ϵ;Framed["MdotH=" NumberForm[MdotH, 2]]

MdotH=6.3 × 1011 kg

s

Framed["MdotH=" NumberForm[MdotH / (Msun / yr) "Msun/yr", 2]]

MdotH= 1. × 10-11 Msun/yr

c. Total H burned over main sequence

tms = 1010 yr;MH = MdotH * tms;Framed["MH=" NumberForm[MH, 2]]

MH= 2. × 1029 kg

Framed["MH=" NumberForm[MH / Msun "Msun", 2]]

MH= (0.1 Msun)

This lost H mass was converted to He

d. For initial solar mass fraction X=0.72, what is final X(tams)?

X = 0.72;Xtams = X - MH / Msun;Framed["Xtams=" NumberForm[Xtams, 2]]

Xtams= 0.62

Y = 0.26;Ytams = Y + MH / Msun;Framed["Ytams=" NumberForm[Ytams, 2]]

Ytams= 0.36

6 Hw4-solns.nb

e. Red giant lifetime to burn same H as on main sequence...

L = 5000. Lsun;trg = ϵ MH L c2;

Framed["trg=" NumberForm[trg / (10^6 yr) "Myr", 2]]

trg= (2. Myr)

Note I should have used L(RedGiant) = 500 Lsun (not 5000). In that case, we obtain 10 times longer RG time:

L = 500. Lsun;trg = ϵ MH L c2;

Framed"trg=" NumberFormtrg 106 yr "Myr", 2

trg= (20. Myr)

#4 Planetary nebula emission and expansion

Hw4-solns.nb 7

a. Expansion speed

km =.;yr =.;λr = 656.32 nm;λb = 656.24 nm;Δλ = λr - λb;λo = (λr + λb) / 2;c = 3 × 105 km / s;(* Doppler shift formula eqn. 8.5 *)

vr = (c Δλ / λo ) / 2; (* divide by 2 for expansion speed from center *)

Framed[NumberForm[vr, 2]]

18. km

s

vraubyr = vr * (365. × 24 × 60 × 60) s / yr 150 × 106 km / au;

Framed[NumberForm[vraubyr, 2]]

3.8 au

yr

b. Distance given observed expansion angle is 0.1 arcsec in 10 years

Δα = 0.1 arcsec / 2; (* change in ang. radius is half change in ang. diameter *)

Δr = vraubyr * 10 yr;d = (Δr / Δα) arcsec pc / au ;Framed[NumberForm[d, 2]]

770. pc

c. Physical diameter

diam = 10 arcsec * d au / pc / arcsec;Framed[NumberForm[diam, 2]]

7700. au

8 Hw4-solns.nb

#5 Cooling time for white dwarf with Mwd=1.0 Msun, R=Rsun/100, Tsurf=30,000 K

a. Number of Carbon nuclei

Msun = 2 × 1033 g;mp = 1.67 × 10-24 g;mc = 12 mp ;Nc = Msun / mc;Framed["Nc=" NumberForm[Nc, 2]]

Nc= 1. × 1056

b. Number of electrons

Ne = 6 Nc;(* Each ionized Carbon gives free 6 electrons *)

Framed["Ne=" NumberForm[Ne, 2] ]

Ne= 6. × 1056

c. Total thermal energy (in J)

kbol = 1.38 × 10-23 J / K;Ntot = Nc + Ne; (* Total number of particles *)

Tint = 107 K;Etot = (3 / 2) Ntot kbol Tint;J = kg m2 s2;

Framed["Etot=" NumberForm[Etot / J "J", 2]]

Etot= 1.4 × 1041 J

Hw4-solns.nb 9

d. White dwarf luminosity for R=0.01 Rsun, T=30,000K = 5*Tsun

Tsun = 6000 K;Twd = 30000 K;Rwdbs = 0.01;Lwdbls = Rwdbs2 (Twd / Tsun)4;Framed["Lwd=" NumberForm[Lwdbls "Lsun", 3]]

Lwd= (0.0625 Lsun)

e. White dwarf cooling time

Lsun = 4 × 1026 J / s;Lwd = Lwdbls * Lsun;yr =.;secbyr = 365.25 × 24 × 3600 s / yr;tcool = Etot / Lwd / secbyr;Framed["tcool=" NumberForm[tcool, 2]]

tcool= 1.8 × 108 yr

#6 Carbon white dwarf SN with Mch=1.4 Msun, R=Rearth

10 Hw4-solns.nb

a. Energy release from burning C to Fe

mp = 1.6 × 10-24 g;Msun = 2 × 1033 g;Mch = 1.4 Msun; (* Eqn.19.7 *)

Nnuc = Mch / mp; (* # nucleons = mass/(mass per nucleon) *)

Framed["Nnuc=" NumberForm[Nnuc, 2]]

Nnuc= 1.8 × 1057

J =.ev = 1.6 × 10-19 J;mev = 106 ev;EperNuc = 1 mev; (* figure 18.2 , ΔE/nucleon for C to Fe *)

En = EperNuc * Nnuc; (* total energy is energy/nucleon * # nucleons *)

Framed["En=" NumberForm[En, 2]]

En= 2.8 × 1044 J

b. Compare to gravitational binding energy

km = 105 cm;re = 6400 km;erg = 10-7 J;G = 6.7 × 10-8 erg cm g2;

(* Eqn. 8.3, with Msun → Mch; Rsun → Rearth *)

Eg = (3 / 5) G Mch2 re;

Framed["Eg=" NumberForm[Eg, 2]]

Eg= 4.9 × 1043 J

Framed["En/Eg=" NumberForm[En / Eg, 2]]

En/Eg= 5.7

En > Eg => star will be completely destroyed by explosion

Hw4-solns.nb 11

c. Average Luminosity if 1% of En is radiated over two weeks

week = 7. * 24. * 60 * 60 s;Lsun = 4 × 1026 J / s;Lavg = 0.01 En / (2 week) / Lsun;Framed["Lavg=" NumberForm[Lavg "Lsun", 2]]

Lavg= 5.8 × 109 Lsun

d. Absolute magnitude at peak

Lpeak = 2 Lavg;Mpeak = 4.8 - 2.5 Log[10, Lpeak];Framed["Mpeak=" NumberForm[Mpeak, 3]]

Mpeak= -20.4

e. Maximum distance (in Mpc) at limiting magnitude +20

(* Solve for distance in distance modulus eqn. 3.8 *)

d = 10((20-Mpeak)/5+1) pc 106 pc / Mpc;

Framed["d=" NumberForm[d, 4]]

d= (1180. Mpc)

12 Hw4-solns.nb