sec. 18 a. owocki/phys333/QQ-secs18-24.pdfآ  b. In[$]:= c=.; v = z c Out[$]= 1. c c. In[$]:= ckms...

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Transcript of sec. 18 a. owocki/phys333/QQ-secs18-24.pdfآ  b. In[$]:= c=.; v = z c Out[$]= 1. c c. In[$]:= ckms...

  • sec. 18

    a.

    Ein= Eout

    σ T⊙4 4πR⊙2 πR2

    4πd2 = σ T4 4π R2

    T= T⊙2 2R⊙ d

    for Earth, d=1 au

    Te= T⊙ 2

    2R⊙ au

    In general,

    T(d) = Te au d

    b.

    pcbau = (180. / π) 60 * 60; Tpcsun = 280 K / Sqrt[pcbau]

    Out[$]= 0.616518 K

  • c.

    In[$]:= Tpchotstar = (60 000 / 6000) Tpcsun

    Out[$]= 6.16518 K

    a.

    In[$]:= mp = 1.67 × 10-24 g; n = 100 cm-3; ρ = 2 mp n

    Out[$]= 3.34 × 10-22 g

    cm3

    b.

    In[$]:= a =.; σ = π a2; m = (4 / 3) π a3 ρd; κ = Xd σ / m

    Out[$]= 3 Xd

    4 a ρd

    In[$]:= μm = 10-4 cm; κd = κ /. Xd → 2 × 10-3, ρd -> 1 g  cm3, a -> 0.1 μm 

    Out[$]= 150. cm2

    g

    2 QuickQuestions-secs18-24.nb

  • In[$]:= aucm = 1.49 × 1013 cm;

    ℓ = 1

    κd ρ pc / (pcbau * aucm)

    Out[$]= 6.49458 pc

    c.

    In[$]:= d = 30 pc; τuv = d / ℓ

    Out[$]= 4.61924

    d.

    In[$]:= Auv = 1.08 τuv

    Out[$]= 4.98878

    In[$]:= λuv = 0.1 μm; nm = μm / 1000; λv = 500 nm; τv = τuv (λuv / λv)

    Out[$]= 0.923848

    In[$]:= λir = 2 μm; τir = τuv (λuv / λir)

    Out[$]= 0.230962

    sec. 19

    In[$]:= G =.; R =.; g = G M  R2;

    tg = t /. SolveR ⩵ g t2  2, t[[2]]

    Out[$]= 2 R3/2

    G M

    QuickQuestions-secs18-24.nb 3

  • = π

    2

    R3

    GM ; same except for factor 2 ≈ 1.41 vs.

    π

    2 ≃ 1.57

    tg (R)

    tff =

    π / 2

    2 = 1.11

    In[$]:= π  2 2. 

    Out[$]= 1.11072

    sec. 20

    In[$]:= s =.; aukm = aucm km  105 cm;

    pckm = pcbau * aukm ; kpckm = 1000 pckm; Rgalkm = 8 kpckm; Vo = 220 km / s; Ps = 2 π Rgalkm / Vo; yrs = 365.25 * 24 * 60 * 60 s; PMyr = Ps Myr  106 yrs

    Out[$]= 222.513 Myr

    In[$]:= E21 = 1.2 μm eV / λ /. λ → 21 cm

    Out[$]= 5.71429 × 10-6 eV

    4 QuickQuestions-secs18-24.nb

  • a.

    In[$]:= au =.; d =.; s = au ( α / arcsec) (d / pc) /. {α -> 1 arcsec , d -> 8000 pc}

    Out[$]= 8000 au

    b.

    In[$]:= au =.; s =.; d =.; αc = arcsec (s / au) / (d / pc) /. {s -> 1 pcbau au , d -> 8000 pc}

    Out[$]= 25.7831 arcsec

    In[$]:= pcbau

    Out[$]= 206265.

    sec. 22

    c2

    2 ≡ GM Rs

    ; Rs= 2 GMc2

    ϵ ≡ Eg

    mc2 = GMm/Racc

    mc2 = GM c2 Racc

    = Rs2Racc ; for Racc= 5Rs , ϵ=0.1

    QuickQuestions-secs18-24.nb 5

  • a.

    In[$]:= λLyα = 91.5 nm; λobs = 183 nm; λem = λLyα ; z = λobs - λem

    λem

    Out[$]= 1.

    b.

    In[$]:= c =.; v = z c

    Out[$]= 1. c

    c.

    In[$]:= ckms = 3 × 105 km / s; v = z ckms; Ho = 67. (km / s) / Mpc; d = v / Ho

    Out[$]= 4477.61 Mpc

    sec. 24

    a.

    For empty universe, R(t) = 1 + Hot , so age where R-tage)=0 gives tage = 1 /Ho=

    10 Gyr ho

    where ho≡ Ho

    100 kmsMpc

    For Ho=67 (km/s)/Mpc, ho=67/100 = 2/3, so tage =

    10 Gyr ho

    = 10 Gyr 2/3

    = 15 Gyr

    6 QuickQuestions-secs18-24.nb

  • For empty universe, R(t) = 1 + Hot , so age where R-tage)=0 gives tage = 1 /Ho=

    10 Gyr ho

    where ho≡ Ho

    100 kmsMpc

    For Ho=67 (km/s)/Mpc, ho=67/100 = 2/3, so tage =

    10 Gyr ho

    = 10 Gyr 2/3

    = 15 Gyr

    b.

    For critical universe R(t) = ( 1 +( 3/2) Hot )2/3 so age where R-tage)=0 gives tage = (2 / 3) /Ho=

    10 Gyr ho

    = (2/3) 10 Gyr 2/3

    = 10 Gyr

    Appendices

    QuickQuestions-secs18-24.nb 7