Theory of the Fireball

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T-R -10 (3.16) 8 The numerical calculations of Brode are in good agreement with this at the relevant times, from about 0.05 to 0.5 second. It will be noted that (3.16) was obtained without integration of the hydrodynamic equations; it follows simply from the equation of state. The weakest assumptions are (1) the relation between E and T, (3.13), and (2) the neglect of the constant in (3.9) . But in any case, T will be a very hign power of R. C. Mean Free Path and Radiating Temperature The emission of radiation from a sphere of variable temperature is governed by the absorption coefficient. For visible light, the absorption coefficient increases rapidly with temperature. For any given wave length, tine emission w ill then come from a layer which is one optical mean free path inside the hot material.* *Actually the maximum emission comes from deeper inside the fireball. To see this we compute the ernigsion normal to the surface, J = 2hv3 e -hv@ jdr@(R)e -D(R)dE? where e = - is the emissivity, and the remain- C2.e ing notation is as in the text (below). The integrand has a maximum at R"/$* = a + cuhv/fl*. The optical depth at the maximum is D* = D(R*) = ncu + @nv'kir* which is about 2, rather than 1 in the blue. By steepest a-1 descents and with occasional use of 1/m = 0 it turns out that J = e 4- B( v,T*), greater by about a factor 2 than the J used in the text. 24 16
Assume that the mean free path is &=GIT 8 -n where the exponent n and the coefficient 8 1 depend on the wave length. Then the optical. depth for a given R is we need = I dR' .e(R') R D(R) = 1 R R 25
Page 1 and 2: , - Ip.L ,. : CIC-14 REPORT COLLECT
Page 3: LA-3064 UC-34, PHYSICS TID-4500 (30
Page 6 and 7: i.e., the part which has not yet be
Page 9: 1. INTRODUCTION 2. PHASES IN DETELO
Page 12 and 13: 6 radiation .(Stage A of Sec. 2) is
Page 14 and 15: frequency, the opacity, which is su
Page 16 and 17: elations as (3.2) between shock rad
Page 18 and 19: in this case there is no Stage D 11
Page 20 and 21: highest possible density of about l
Page 22 and 23: \ and the average of y' is taken ov
Page 24 and 25: The radius R is given R3 = I"- P P
Page 28 and 29: Since the interesting values of R a
Page 30 and 31: Most of the radiation comes from a
Page 32 and 33: higher frequency (above, and Sec. L
Page 34 and 35: 4. ABSORPTION coEFF1c1ms a. Infrare
Page 36 and 37: (3.21), it is small compared to the
Page 38 and 39: hv (ev) P/Po = 1 ff N2 0- Table N.
Page 40 and 41: w Q P/Po 1 0.1 0.01 T 4,000 6,000 8
Page 42 and 43: Table VI. Average Mean Free Paths (
Page 44 and 45: The fraction of the Planck spectm b
Page 46 and 47: The W beyond 3.5 ev will be strongl
Page 48 and 49: The equation of continuity is aH aJ
Page 50 and 51: U" J, A Ho - H(T1) To determine u w
Page 52 and 53: temperature is aromd Tb, and in whi
Page 54 and 55: internal energy in that sFhere; in
Page 56 and 57: y integrating (5.8); it depends on
Page 58 and 59: a log I1 - y' - 1 a log p at - y' a
Page 60 and 61: 2 vitn A = 0.1G q/cm and n = 6.3. T
Page 62 and 63: or a radiating temperature of 10,80
Page 64 and 65: assuming the strong shock relation
Page 66 and 67: pressure is larger than pa, (5.50)
Page 68 and 69: inconsistency in our apyroximations
Page 70 and 71: G. We& Cool-ing Wave In Stage C I1
Page 72 and 73: 6. EFFECT OF THREE DDENSIOUS a. Ini
Page 74 and 75: 6. Sinrinkage of Isotnemal Sphere W
Page 76 and 77:
Pa f(x) = - P and obtain the simple
Page 78 and 79:
From y = 1.18, y=l-A Y= -1 2.2 - 0*
Page 80 and 81:
where Re is the outer edge of the l
Page 82 and 83:
esult of the three-dimensional cons
Page 84 and 85:
4 dF: = - 4KaT at where a is the St
Page 86 and 87:
1. 2. 3. 4. 5. 6. 7. 8. 9. 10 . REF
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Theory of the Fireball

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