Theory of the Fireball

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or a radiating temperature of 10,800°. Tnis is a reasonable result, thowh appreciably higher than the teqeratures usually observed. How- ever, as we shall show in Secs. 5e md f, our calculation gives the max- imum temperature reached, md is likely to be somewhat too high due to OUT approxinmtions . e. Bewinning of Strow Coolim Wave 2 Equation (5 -19) gives the inward speed of the cooling wave in @/a sec. Precisely, this is the speed at which the point of teqerzhre T 1 (radiating temperature) moves relative to the material, once fne cooling wave is fully established. But even if there is no cooling wave, i.e., if we have simply adiabatic expansion behind the shock, a point of given temperature T1 will move intrzrd. This "adiabatic motion'' is the minimum velocity vlnich the point T can nave. Therefore, if tne adiabatic speed 1 is greater than (5.19) , it will be tne correct velocity. Of course, there will still be a. cooling wave because this is needed to supply the energy for the radiation; tills 'be&< cooling wave" wil be described in Sec 5g. But its inward motion, more accurately the velocity of its foot (point C in Fig. 2), will not be determined by the requirement of sufficient energy flow, (3.19) , but by the "adiabatic speed" which we shall derive from (3.17) Region IV of Fig. 2 will nar be absent, Tnus, outside the cooling wave, at point Cy region V wil begin Mediately, with the temperature distribution given by (3.15). It is therefore important to determine the time ta (and pressme p ) a 60
at which u of (5 .IS) becomes larger than t'le adiabatic speed of temperature T Before this time ta, we 'have a weak cooling wave . We call 1' this Stage C I; afterwards, the cooling wave is strong, and the radiation is essentially described by the theory of Sec. 5d (Stage C 11) . To determine tine point of separation p between these two stages is a refinea ment of the considerations of Sec. 3d. Tne temperature in the adiiibatically expanding material behind the shock is a function of time and position, given by nydrodynamics and equation of state. The imrard motion of a point of given temperature rel-ative to the material is given by tfnere the subscript r means that the partial derivative must be talcen at given material point r, not at given geometrical radius R. We have from (3.15) 61
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 26 and 27: T-R -10 (3.16) 8 The numerical calc
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 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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