fireball 1
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LA-3064
UC-34 , PHYSICS
TID-4500
(30th
Ed.)
LOS
ALAMOS SCIENTIFIC
LABORATORY
OF THE U N I V E R S I T Y OF C L I F O R N I O S L M O S
NEW
M E X I C O
REPORTWRITTEN
Februa ry
1964
REPORT DISTRIBUTED:
June
17, 1964
THEORY OF
THE FIREBALL
bY
Hans
A.
Bethe
This report exp ress es he opinions of the author or
authors and does not necessa rily reflect the opinions
or views of the
Los
Alamos ScientificLaboratory.
Contract
W-7405-ENG.
36 with he
U.
S. Atomic
Energy
Commission
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ABSTRACT
The successive stages of the f i re ba l l due t o a nuclear explosion in
ai r a re defined (Sec. 2). Tnis paper i s ch ie fly concerned
with
Stage C,
from
tne,
minimum
in
t h e
ap-parent
f i r e b a l l
temperature t o t h e point where
th e f i r eb a ll becomes transpa rent .
In
the
f i r s t
part of this s tage
( C I ),
the shock
(which
previously was opaque) becomes tra nspa rent due t o
decreasing pressure.
The
ra dia ti on comes from a region
in
which the
temperature distribution i s given es se nt ia ll y by the Taylor solution; the
radiating layer
i s
given by the condition that the
m e a n
free path i s
about
1/50
of theradius (Sec.
3 ) .
The r ad ia ti ng t q e r a t u r e during t h i s
-1/4
stage ncreases about asp , here p is thepressure.
To supply the energy for the rad iation, a coo ling wave proceeds from
j
the outsid e into the hot interio r (Sec. 5 ) . When this wave reaches the
isothermalsphere, he enperature is close t o its second maxmum There-
af ter , th e character
of
the so lu tio n changes; it
i s
naw dominatedby
the
cooling wave (Stage
C
11). m e temperature would decrease s1mi.y (a s
P I6) i f the problem were one-dimensional, but in f a c t it i s probably
nearly constant f o r the hree-dimensional case (Sec.
6 )
The radiating
surface
shrinks
slowly. The
cooling
wave eat s nto he soth erm al sphere
until t h i s
i s
completely used
up.
The i nne r pas t of the sothermalsphere,
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i.e., th e p a r t which has not yet been reached by the coo ling wave, con-
ti nu es to expand adiaba tically; it therefore cools veryslowly and
remains opaque
.
Af ter th e ent ir e isothermal sphere i s used up, th e f ir e b al l becomes
transparent and the ra di ati on drops rapidly. The ball
w i l l
therefore be
l e f t
a t
a
rather high empe rature (Sec.
7 ) ,
about 5000 .
The cooling wave reache s the isotherma l sphere at a definit e pres-
sure p, M 5(p1/po) 1/3 bars, where p i s the ambient and po the sea level
1
density. The radiati ng tempe rature at this time
i s
about
10,
OOOo
.
The
sl ig h t dependence of physic al prope rties on yield i s exhibited i n approx-
m t e formulae..
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I am
greatly indebted t o Harold Brode of the Rand Corporation for
giving me a patient introduct ion to t h e work on fi r e b a l l dynamics since
World
War
11, fo r sending me t he pertinent Rand publications on the sub-
ject, and
f o r
doing a spe ci al numerical computation f o r
me.
To Burton
Freeman of General Atomic I
am
gratefuloreveralnterest ingiscus-
sions of f i r e b a l l developmentand opacities
of
a i r . I want t o thank
Herman Hoerlin
fo r
much information on the observational material, and
for select in g for me pert inent
Los
A l m s reports. I
am
?specially
gr at ef ul t o Albert Petschek f o r c r i t i c a l reading of th e manuscript
and
for several helpful additions.
My
thanks
are
also due t o
many
other
members
of
Groups
J-10
and
T-12
a t
Los Alamos,
t o J. C, S t u a r t and K.
D.
w a t t a t General
Atomic,
Bennett Kivel
and
J. C.
Keck a t
Avco-Everett,
and t o F. R. Gilmore a t R a n d f o r important in fo ma t on.
.
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1. INTRODUCTION
2. PHASES
IN
DETELOPMENT
3. RADIATIONFRCM BEHIND
THE
SHOCK (Stage
C
I)
a. Role of
H02
b. Temperature Di st rib ut ion behind th e Shock
C. Nean Free Path and Radiating Temperature
d.
Energy
Supply
4.
AESORPTION COiEFFICIENTS
a. Infrared
b.
Visible
c. Ultraviolet
5
.
THE COOLING
WAVE
a.Theory
of
Zel'dovich e t
al.
b. nsideStructure
of
Fireball ,
Blocking
Layer
C. Velocity of
-Coolin@;
Wave
d. Adiabatic Expansion after
Cooling.
Radiating
Temperature
e Beginning of Strong Cooling Wave
f .
Maximum
Ehission
g o Weak Cooling Wave
6.
EFFECT OF
THREE
D ~ E N S I O N S
a. I n i t i a l Conditions and Assumptions
b. Shrinkage
of
Isothemal.
Sphere
c. TheWarm Layer
7.
TRANSPARENT FIREl3ALL (
Stage D)
REFERENCES
7
3
5
9
i2
17
19
24
27
32
32
34
40
44
44
48
51
55
60
64
68
70
72
77
80
84
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1. INTRODUCTION
Tile radiation- rom t h e
f . i r ~ b ~ . - h a s ~ b ~ ~ s ~ ~ ~ ~ a -
by._ x , .,,
many authors.Already i n t h e Summer Study a t Berkeley
i n 1942,
Bethe
and
Telle r found th a t th e energy transmitt,ed by a nuclear explosion
i n t o a i r
i s
m e d i a t e l y converted into x-rays, and studied the qualita-
1
t ive fea tures of the transmission of the se x-rays.
At
Los Alamos,
Marsnakand others showed that thi s ra di ati on propagates
as
a wave,
175th a sharp front.Hirschfelder
and
Magee gave
the f i r s t
comprehensive
treatment of th is ear ly phase of the fireball development,
and also
studied someof the l a te r phases, esp ec ial ly th e ro le of NO2.
2
3
Ma y optic al observa tio ns have been made
in
the numerous tests of
atomic weapons. Some of tine re su l ts a re contained in Effects of Nuclear
Weapons, ' pp. 70-84 (see a l so pp. 316-368)
A
summary of the spectro -
scopic observations up t o 1956 was compiled by D e W i t t .5 Careful scrutiny
of th e extens ive observ ationa l mater ial would undoubtedly give a wealth
of fu rt he r information.
On
the theore tical side, there has been some analytical
a d
good
deal of numerical work. Ana lyt ical work has concentrated on th e ea rl y
phases. One of the most rec en t analyt ica l papers on the ear ly flow of
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6
radiat ion . (Stage
A
of Sec. 2) i s byFreeman.BrodeandGilmore 7 t r e a t
also Stage B, the rad$ation from th e shock fro nt , wi th pa rti cu lar emphasis
on the dependence
on
a l t i tude .
The most complete numerical ca lc ul at io n has been done by Brode 8 on
asea l e v e l megaton explosion.
We
sh al l use h is results extensively,
but f o r convenience we shall tr an sl at e them t o a yield of lmegaton.
Wherever the phenomena are
purely
hydrodynamic,
w e
may simply sc al e th e
linear dimensions and the tim e by the cube root of th e yie ld , and t h i s
is
the
principal.
use
w e
s h a l l make
of
Brode
s
results , e
.g.
Fn
Sec.
3b.
Where radiation i s important, th is sca lin g
w i l l
give only a rough guide.
Brode calculatespressures, emperatures,densities,etc.,asfunctions
of time and radius , f o r scaled (1-megaton) times of about t o 10
seconds. The ca lcula tio ns show clearly he s tages in f i r e b a l l and hock
development, as def ined in Sec. 2, at le as t Stages A
t o
C.
Brode and Meyerott9 have con sidered the physical phenomena involved
in the optical.
"opening"
of the
shock
a f te r t he minimum of radiation,
Stage
C
I
in th e nomenclature of Sec. 2, espe cial lyth e decrease
i n
opacity due t o decreased density and
t o
the dissociat ion of
NO2.
Zel'dovich, KompaneetsandRaizer''have investigated haw the energy
for the radia t ion i s supplied a ft er th e ra d ia ti on minimumand have in tro-
duced the concept of a cooling wave moving in to th e hot f irebdll. The
present report
is
la rg el y concerned with an extension of the ideas of
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Much ef fo r t has been devoted t o
t h e
ca lcula tion of f i r eb a l l s a t
various altitudes. Brode'lmade such ca lcu lati on s
in
1958, in connection
with
th e te st se ri e s of t h a t year. Gilmore12made a prediction
of
the
Bluegill explosion i n
1962.
Since then, manymore ref bed cal cul atio ns
of Bluegi l l have been made.
For ny understanding of f i r e b a l l phenomena
it
i s ess ent ia l to have
a
good
equation of s t a te f o r ai r
and
good ta bl es of absorption coeffi-
cients. For the equation of st at e, we have used Gilmore's t ab le s,
although Hilsenrath'sgive more d e ta il
i n
some respects.
The
two cal-
13
14
culations agree.Gilmore's equation was approximated an al yt ic al ly by
Brode
8
For the absor ption coeff ici ent we have
used
the tables by Meyerott
e t al.,
15'16
which extend t o
12,000.
These were supplemented by the
18
workof Kivel and Bai1eyl7
and
more recent work by Taylor
and
Kivel on
the
free-free electron t ransi t ions in the f i e ld of neutral atomsand
molecules. A t higher emperatures, here are calculations byGilmore
and Latter,I9 Karzas
and
LatterY2*and curves by Gilmoregl which are
brought to dateperiodically. The most recentcdcu lation on th e
absorption o f a i r a t about 18,000' and above have been done by Stuart
and Pyatt.*2 This temperaturerange i s not of gre at importance f or th e
problem
of
th i s paper, but is important fo r th e expansion of th e is o-
thermal sphere inside
t h e
shock wave before
it
is
reached by the cooling
wave.
8
Brode has used the average of the absorption coefficient over
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frequency, the opaci ty, which
i s
suff ic ient for t rea t ing t h e in ternal
flaw
of radiation. A re a l i s t i c treatment of the flow t o t h e outside
requires the absorption coefficient as a w c t i o n
of
frequency; Brode
merely wanted t o obtai n reaso nable overa l l resul ts for this flow. He
approximated the opac ity by an analytic expression. Also in most
of
t he
ot'ner work cited above an average opacity has been used. An exception
i s
some of the recent workon
the
radiat ion
flow
i n high al ti t ud e explo-
sio ns (B lueg ill ), where th e frequency dependence
mus
be, and
has
been,
ta ke n nt o account.Gilmore21has ca lculated and made av ai la bl e curves
of
effective opacity,
in
which the radia tion mean fr ee pa th was averaged
(using a Rosseland weighting factor) only over those frequencies f o r
which it
i s
l e s s than 1kilometer.
This
l i s t of references on work on the fireball dynamics
and
opac-
i t y
i s
f a r
from complete.
The energy from
a
nuclear explosion
i s
transmitted through the outer
p a rt s of th e weapon, including i t s case, either by radiation (x-rays) or
by shock o r both. Whichever t h e mode of transmission inside the weapon,
once the energy gets in to th e surrounding
air ,
the energy w i l l be trans-
ported by x-rays. This
i s
because
the
a i r
w i l l
be heated t o s u c h a
high
temperature
(a
millio n degre es or more) th at transp ort by x-rays
i s
much
f a s t e r
than
by hydrodynamics.
This
stage
of
energy transp ort (Stage
A )
has
been extensive ly studi ed by many authors (e.g., Hirschfelder and
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Magee
in
Report LA-2000) and w i l l therefore not be further consid,ered . .
here
.
During Stage A, temperatures are very high . The Pianckspectrum
of
t t h e a i r
is
in thex-rayor f a r ult ra v io le t region, and hence is irmnedf-
at el y absorbed by the surrounding a i r . The very hot a i r i s therefore
surrounded by a cooler envelope, and
only t h i s
envelope
i s
v i s i b l e t o
observersat a distance. The obse rvable emperature herefo reh a s l i t t l e
physicalsignificance. It
is
observed that the size
of
the ltrminous
sphere lncreases rapidly, and th e
t o t a l
emission also .Increases,
up
t o
a f i r s t maxmum
When the temperature of the cen tral sphere of a i r has fallen, by
successive emission and re-absorption of x-rays, t o about j0O,00O0,
a
-hydrodynamic shock
forms,
The shock
now moves
f a s t e r than the tempera-
tu re couldpropagate by ra dia tio n tra ns po rt, The shock ther efo re sepa-
rates fromth e very hot, nearly isothenml sphere a t
the
center. This
i s
Stage 13 in th e development. The shock mves by simple
Its front obeys
the
Hugoniot relations, t he density being
Behind the fro nt , the air expands adi aba tica lly, and a t a
of
the
shock
radius, . the density is apt
t o
have fallen.by
o r m r e compared
t o
the shock dens ity, while the temp erature has risen
by a comparable factor, (3.16) Thus th e int er ior
is
a t very
low
density,
and
hence the pressu re m u s t be neasly constant (otherwise there would
be
very large accele rations which soon would equalize the pressure) .
This
greatly simplifies the structure of the shock, and leads t o such simple
. .
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re la t ions as
(3.2)
between shock radius and pressure.
Well inside the shock, t h e isothermalspnere pursues i t s separate
his tory .
It
continues t o engulf more material because radiation flow
continues, though
a t
a educed ra te . H. Brode has kindlycalculated
f o r
me th e temperature h is to ri es of se veral ma teri al poi nts, based on hi s
paper RM-2248. These hi sto rie s cle ar ly ex hib it the expansion
of
the i s o -
thermalsphere
in
materialcoordina tes. Tne expansion can
d s o
be treated
by a semi-analytic method which
I
hope t o discuss in a subsequent paper.
The isothermal sphere remains is ol ate d from the out sid e world un t i l
it
i s reached by the cool ing wave, Sec.
5 .
The rad ia ti on to the ou ts id e now comes from th e shock. E a r l y in
Stage B, the shock has
a
precursor of lower temperature, caused by u l t r a -
vi ol et ra di at io n from th e shock, and the observable emperature
i s s t i l l
below t he shock tempera ture (Stage B I ) However, the observableradius
i s verynear he shock radius.Later,
as
the shock frontcools d m ,
t h e shock rad iate s dir ectl y and
i t s
temperature becomes di r ec t ly observ-
able.
(Stage B
11)
.
The f i r s t
m a x i m u m
in vis ible rad iati on probably
OCCUTS
between Stages
B
I a d
B
11. As the shock cools d m , the radi-
at io n from th e shock front decrea ses, and th e observable temperature
decreases t o a m i n i m u m (Ref. 4, par.
2.lI.3,
p. 75) of about 2000.
When the shock i s suff icie ntly cool,
i t s
front becomes transparent,
and onecan look into it to higher emperatures (Stage
C )
.
The central
isothermal sphere, nmev er, remains opaqueand, f o r some time, invisible.
Because higher temperatures a r e now revealed, the to ta l radia t ion increases
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toward a .second maxmum This stage has been very l i t t l e considered
theoretically, except i n numerical calcula tions, and
forms
the subject
of this report.
We shall
show
tha t f o r
some
time In Stage Cy th e ra di at io n comes
fm h e a i r between isothermal sphere and shock fron t (S tag e
C
I ).
During this
time, the rad iati on can be calcula ted essentia lly from the
temperature di str ib ut io n which
i s
se t up by the adi aba tic expansion of
the ma ter ia l behind th e shock (Secs 3 and 5f ,g) The temperature and
t o t a l
intensity
o f
the radiation increase with time toward
the
larger,
second
maxmum
The
energy fort he ra di at i on
i s
largely sqplied
by
a cooling wave
(Sec. 5 ) which gradually ea ts int o th e hot int er ior .
When
this coollng
wave reaches the isothermal sphere, the radiati on temp eratur e reache s
i t s
maximum
(Sec.
5e); it
then declin es again as the coolin g wave e at s
more deeply in to the sothermalsphere(Stage
C
11). This
process ends
by t h e isothermal sphere being comgletely eaten away (Sec. 6 ).
m e r hi s has happened, th e en tir e fi reb all , isothermal sphere and
cooler envelope, i s t ransparent t o
i t s
own thermal. radiati on (St age
D ) .
The
molecular
bands, which previously appeared i n absorption, now appear
i n emission (Stage D I ). Emission
w i l l
lead to further coollng of the
f i r e b a l l ,
though
more
slowly
than before.
Soon,
when th e temperature
f a l l s below about
6000, the
emission becomes very weak,
and
stibsequent
cooling i s almost en tir el y ad ia ba tic (Stage D 11). A t sea level , the
pressure
may
go down t o
1
a r before
the
temperature fa l l s belaw
6000~;
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in th i s case there i s no Stage D 11.. A t higher elevation, there usually
is.
The
f i r e ba l l w i l l
then
remai n
hot,
a t
about
6000~
o r a somewhat
lower temperature due to ad ia ba ti c
expansion
i n
Stage
D
I1
. The
only
process which can now le ad to fu rt h er cooling
i s the rise
of the f i r e -
ball, which leads t o
further
adiabatic
expansion and,
more
important, t o
turbulent
m i x i n g
at the surfa ce with the ambient a i r (Stage
E).
The
t ime required fo r th i s i s ty p ic al ly 10 seconds o r more, being determined
by buoyancy
A t very high alti tud e,
the
shock wave never plays an important
part
but radiation transport continues until the temperature g ets too
low
for
effective emission.
In
other words, Stage A continues t o t h e end.' O f
course, a shock does
form,
but it i s , so to speak, an afterthought, and
it
plays l i t t l e par t in the dis t r ibut ion of energy, A t medium altitude,
l e t
us say,
10
t o
30
kilometers, the st ag es ar e much the same as at se a
le ve l bu t th e shock wavebecomes tra ns pa re nt ea rli er , i.e., a t a higher
temperature,because
t h e
densi ty
i s
l a t e r ;
t h i s means
t ha t
t h e
minimum
emission comes earlier.
Stage C
proceeds si mi la rly to sea level,but
a t th e second maximum of radiation the pressure
i s
s t i l l much above
ambient, there fore Stage C I1 involves a gr ea te r ra di al expansion of the
isothermal sphere than a t
sea
l e v e l which proceeds simultaneously
with
th e inward motion of he coo ling wave.Moreover, there
i s
much adia-
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W believ e that the theory developed i n t h i s paper w i l l be useful
jn
studying the dependence of phenomenaon al ti tu d e (ambient pre ssu re),
but we have not yet explo ited it
f o r
t h i s purpose.
3. RADIATION FRaM BEHDTD THE
SHOCK
(Stage C I)
a., Role of NO2
The diatomic species in equilib rium air, both neutra ls and ions,
show very l i t t l e absorption
i n
th e vi si bl e at temperatures
Up
t o about
16
Iioooo~. This i s shown clearly
in
theab le s by Meyerott e t We
define the vis ib le fo r th e purposes of t h i s paper, ar bi tr ar il y
and
incorrect ly, as the frequency range
hv
= 1/2 o 2-3/4 ev
i
= 4050 t o 22,300 an
Then,even a t a density as high as lopo p, = density of a i r a t HIT
= 1.29
x
w/cm ), the mean freepath i s never less han 100 meters
at k0OO0, 1000 meters a t
3000,
and s t i l l longer at luwer temperature.
3
Tnese values refer
to
hv
= 2-5/8
ev;
f o r
lawer frequencies, the mean
f r e e
path
is
even longer.
I n
-the sea -level shock wave from a megaton explosion, t h e temper-
a t w e range.
from 3000
t o
4000'
occupies
a
distance of about 10
meters
(
see Sec. 3b).
Thus
this region i s definit ely ranspa rent, even
at
the
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highestpossiblede nsi ty of about l o p For explosions at nigher a l t i -
tude, t h i s conclusion i s even more true.
0.
Tne ta bl es by Meyerott e t
al..
o not include absorption by tria tom ic
(and more complicated)molecules, O f these, NO2
i s
known t o have strong
absorption bands i n th e vi si bl e. Af te r th is paper was completed, I
received new ca lcu lati ons by Gilmoreg3 which include t he ef fe c t of U02.
The effect
i s
very s t r ik ing as
i s
shown
by
Table
I,
which g ive s the
abso rption coef ficient for the typic al frequency hv
=
2-1/8 ev, and
f o r
several
temperatures and dens ities (the abso rption
i s
strong from
PIP.
T
Without
With
Table I. Absorption Coefficients a t hv
=
2-l/8 ev
with and without
NO2
about 1-314 o 2-3/4 ev).
Note: For
each value, th e power of 10 i s indicated by a strperscript.
10 10
10 10 1
Because NO
i s
triatomic, i t s absorption depends strongly on density.
2
As
the shocked gas expands, the NO2 dissociates and the gas becomes trans-
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I
b.
Temperature Distribution behind the Shock
We wish to ca lcula te the temperature distribution behind the shock.
We can do
t h i s
because th e ma te ria l which has gone through
the
shock
expands
very
nearly adiabatically, as
long
as it i s not engulfed by the
internal ,
hot
isothemal sphere. We
are
interested in the pe riod when
the shock temperature goes from about
4000
t o a
few hundred
degrees, i.e.,
un t i l th e strongcooling wave (Sec. 5 ) s tar t s . Fora 1-megaton explosion,
t h i s corresponds roughly t o
t
= 0.05 t o
0.25
second.
A t
a given
time,
the pressure
i s
near ly constant over most of
the
volume ins ide the shock, except f o r t h e Me d i a t e neighborhood of the
shock;
the
shock pressure
i s
roughly twice
th is
constant, inside pressure.
C o n r p a r i n g
two material elements in the inside regio n,
we
may calculate
their relative temperatures i f we
knm.
t.heir . t q e r a t u r e s when t he
she-ck
.
I1
traversed them,andassume ad iaba tic. expansion from there
on.
A
material element which
i s
i n i t i a l l y a t po i n t
r
w i l l
be
shocked
when the shock radius i s r. The shock pressure a t t h i s t ime
is*
Ps( )
= 20(
7Av -
lw 3
where
Y =
yi el d i n megatons
r
=
radius
in
kilometers
P pressure
pE
energy perunit volume
- 1 z - z 3 . 3 )
~
?Notations
f o r
themodp amic quantities
similar
t o those ofGilmore
(Report IN-1543)
.,
19
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and the average of y i s taken over the volume insi de th e shock wave.
Tne
bas is
of
(3.2) i s th a t th e to ta l energy in the shockedvolume, Y,
i s the volume times th e average energy pe r unit volume, t h e la tt e r i s
t h e average pressu re divid ed by y r -
1,
and the average pressure i s
cl os e t o one-half of the shock pressure.
Tne d en si ty at th e shock i s
.. -
\
wnere the sub scri pt
s refers to
shock conditions. Now
an
examination of
Gilmore's tables13 shows t h a t 7 does h o t change very much
along
an adi-
abat.
As
an example, 'we l i s t i n Table I1 certain quanti t ies referring
t o some adiabats which w i l l be pa rti cu lar ly important fo r our theory.
Tnese ar e the adi aba ts for which t h e temperature T i s between
4000
and
12,000 at a density of O . l p o . In th e second l i n e we
list
the temper-
ature Ts fo r th e same entropy
S
at a density ps =
lopo.
This i s close
enough to the shock density ( 3 . 4 ) so w e may consider Ts as th e temper-
ature of the
same
material when the shock wave went through
it.
(Adia-
\
batic expansion,
i.e.,
no radiatio n ransp ort, i s assumed.)The third
line gives y r
- 1
o r the bresent conditions,
p
= 0 . 1 ~ ~nd T, the
fourth l ine
i s the same quantity for the shock conditions. It
i s
evident
t'nat
7
-
1
i s
nearly constant for T
=
4000
and
6000,
not
so
constant
f o r 8000 and
12,000.
On th e average y '
-
1
0.18.
The last
two
l i n e s
i n Table
I1
give the number
of
par t i c le s
(
atoms, ions, electrons) per
o r ig ina l a i r molecule under present If and shock conditions.
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Table
11.
Adiabats
( 'Present density, 0 .1-p shock density,
l opo )
0;
T 4,000 6,000 8, ooo 12,000
T
9,000 10,500
18,000
28,000
y - 1
0
213
0.194
0.144
0 1-53
S
Y; - 1 0.208
0
190 0.190 0.20
z 1-13 1.27 -
1.68 2.06
=S
1-03
.
1.6 2.06
3
Assming
an
adiabat
of
constant
7
=
y ,
the den sity of
a
mass
element i s
P =
PS ( k ) 1 / Y
3.5)
Now
ps i s
a constant, and a t
any
given time,
p i s
t he s&e
for a l l
mass
elements except those very close
t o
the
shock,
hence
4 7
P - Ps
w means proportional- t o )
If
we
now
introduce the abbreviation
3
m = r
(3 .7)
which i s proportional to the mass ins ide the
mass
element considered,
and i f we use
(3.2),
we
f i n d
t h a t a t
any
given time
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The radius R
i s
given
R3
=
I -
P
J ;+ =
m
- 1 const.
7
3 . 9 )
We sha l l set t h e constant equal t o zero whichamounts
to
the ( incorrect)
assmption that (
3.8)
holds d a m t o m = 0. Actually, the isot'nermal
sphere gives the constant a fi ni te , po si ti ve value.
To fin d the tem peratu re dist rib uti on, we note t h a t the enthalpy
H
7 E
Y 1 - l P
We are
us i ng
th e en thalpy rather than the intern al energy because the
in te ri or of th e shock
i s
a t constant pressure, not constant density.
The hermodynamic eqmti on fo r H is
T d S = d H - ~ d p
At givenpressure,
i.e.,
given ime, 3.8) t o ( 3 . 1 0 ) give
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Now the approximate relation between internal energy and temperature i s
given by Gilmore, Fig. 5 ,
viz.,*
4
where T i s the temperature in u ni ts of 10 degrees.
using p =
from
3 . 3 ) and setting y =
1.18,
which i s a reasonable
*
average (see Table 11)w e may
rewrite
(3.13) :
where p i s the pressure in
bars.
Since
H
i s proportional
t o E,
3 . 1 2 )
and (3.14) give
%e thermodynamic relation
@) =
T
g) P
T
V
leads t o a r ela tion between y -
1
and the exponents in the re la t ion
E = &
T
x y
Since we have chosen y = 1.18
and
y = 1.5 th i s re la t ion g ives x = 0.09.
This
i s
i n
sufficient agreement with x
=
0.1 as used in (3.13)
.
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T - R
-10
(3 .16)
8
The numerical calculations
o f
Brode a re i n good agreement
w i t h
t h i s a t
the relevant times,
from
about 0.05 t o 0.5 second.
It
w i l l be noted that (3.16) was obtained without integration of
the hydrodynamic equations; it
follows
simply from the equati on of st at e.
The weakest assumptions are 1) he relation between E and
T, (3.13), and
( 2 ) the neglec t of th e constant
i n ( 3 .9 ) . B u t in
any case, T
w i l l
be a
very hign power of
R.
C. Mean FreePath and Radiating Temperature
The emission of radiation from
a
sphere of va ria ble temperature i s
governed by the absorp tion coeffi cient. For vi si bl e li gh t, th e absorption
coefficient ncreases apidlywith temperature. For any given wave
6
length, tine emission w i l l then come from a layer which i s one optical
mean free path inside the hot mate ria l.*
*Actually the
m a x i m u m
emission comes from deeper insi de the fir eba ll.
To see t h i s we compute the ernigsion normal t o t h e surface, J =
jdr@(R)eD(R)dE? where e =-
i s the emissivity, and the remain-
hv3
e -hv@
C2.e
ing notation
i s
a s i n t h e
text
(below). The integrand has a maximum a t
R"/$* =
+
cuhv/fl*. The op ti ca l depth a t th e maximum is
D* = D(R*) =
n u
@nv'kir*
which i s about 2, rather than 1
in
t h e blue.
B y
steepest
a - 1
descents and with occasio nal use of
1/m
= 0
it turns out t h a t J =
4
B( v,T*),
grea ter by about
a
fac tor 2 than the J used in
e
the t ex t .
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Assume that th e mean f ree path
is
& = G I T
8
-n
where the exponent n
and
the coeff icient 81 depend on the wave length.
Then the optical. depth for a given R
i s
=
I
.e(R')
R
R
R
R
D(R)
= 1
we need
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Since the interesting values of R are of the order of a few hundred
meters, the emitt ing layer w i l l be def ined by t he fa c t th at t he mean
free path
i s
about 5 t o 10 meters.
Equations
(3.18) t o (3.21)
hold f o r emission in the exact ly radial
direct ion.
A t
an angle 8 with he radiu s we get nstead
The apparent emperature of the
emitting
layer i s then, according
t o
where
T ( R , O ) i s
t h e emission emperature f o r forwardemission. The
in te ns ity of emission i s a
known
(Planck) function
of
the wave length
and the emperature .Since th e apparent emperaturedecreases though
slowly)with 0 , according t o (3.23) th er e w i l l be limb darkening.
On
t h e many photographs of atomic explosions, it shouldbe possible t o
observe t h i s Limb darkening and thus check the va lue of
n.
The relation (3.17) needs
t o
hold only in the neighborhood of t he
value
(
3
-21)
of
4
and
i s
therefore quite general as
long
as
4,
decreases
with ncreasing emperature. The exponent nshouldbedetermined at
constantpressure.Forcertain wave leng ths, espe cially in t he u l t r a -
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vio le t , (3.17) i s not valid; these wave len gths are strongly absorbed by
cold or cool a i r (Sec
.
kc)
.
For example, a t p = po and T = 2000, the
mean free path i s less than 1meter f o r all l ight16 of hv > 4.7 ev
(A < 0.26 p ) . Since the emission of l i gh t
of
such short wave length from
such cool a i r i s negl ig ib le , the f i reba l l w i l l not emit such rad iati on at
al.
A detailed discussion of the absorption coefficient i n the vis ib le
w i l l
be given in Sec. 4b. As can
be
seen from th e ta bl es ofMeyerott
e t
al.
and from
our
Table
VI,
f o r a density p = po the mean fr ee pat h
i s of the order of 5 meters a t about 6000~ . This corresponds t o
a
pres-
sure13 of about
25
bars.
For p = O.lpo t he re qu is it e mean f re e pa th of
a few meters i s obtained f o r about 10,OOOo, with p M 7 bars. Thus fo r a
re la ti ve ly modest decrease in pressure, the effe ctiv e temperature of
radiation increases from 6000 t o 10,OOOo, corresponding t o a very sub-
16
J
st an ti al increase
in
radiatio n ntensity . This
i s
the mechanism
of
the
increase in radiation tarard the second
maxmum
A more detailed
d i s -
cussion
w i l l
be given in Sec. 5f.
d. Energy Supply
As l ong
as the radiating temperature i s low, not much energy w i l l
be emitted as radiation, and th i s emission
w i l l
o n l y sl ig ht ly modify the
cooling of the ma teria l due to ad iaba t ic expansion. However,when the
radiating temperature increases, t he radiation cooling w i l l exceed the
adiaba tic cooling t o an increasing extent. It then becomes necessary t o
supply energy from the int er io r to th e radi at ing su rf ac e.
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Most of th e ra di at io n comes from a tnickness ofone o p t ic a l mean
free pa tn near the radiu s a t which ( 3.21) i s sa t i s f i ed .
Let
J be the
radi ation emi tted per unit area per second (which
w i l l
be
of
the order
of the bla ck body radia tion; see belowand Sec. 4 c )
;
nen the
loss of
enthalpy due t o rad iat i on, per gram per second, w i l l be
-
= .ep
J
Adizbaticexpansion,
(%)adi -
A s
long as tne shock
tne
ambient pressure
according t o (3.11),
w i l l
give an entnalpy change
i s
strong, .e. ,
pl,
the pressure
as long as
p
is la rg e compared t o
behaves as
vhere t i s t he time from t h e explosion; herbfore,
,.-
@)adi
- t
1.2 p
Tine rzdia-tion
will be
a r e l a t ive ly smail perturbat ion as long a s
( j. 2h.) -is smaller Clan (3.27) . This
will
st op be% the cas e when
J
P
= 1.2 ,c
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- p - = J
.2 R
50
t ,
(3.29)
Because
of
the steep dependence
of
ternperatwe on
R, (3.16)~
the radi-
a t i ng
surface
R
w i l l
be
close
t o
the shock front
Rs. NOW
the Hugoniot
re la t ionss ta te hat orc lose
to
1
i s
(?)I/*
(3.30)
where p i s the ambient density and
p = 2p, tne
shock' pres sure .Further-
more, f o r the
strong
shock case,
1 S
Rs -
t
0.4
4
The black body ra d ia ti on a t temperature LO
T ' i s
=
5.7
X
~ ~
erg/cm sec
JO
2
3 . 3 3 )
Actually, only tine ra diat ion
up t o
about hvo
=
2.75
ev
can be
emitted t o
large dista nce s because tne absorption
i s
too great for radiat ion
of
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higher requency (above, and Sec.
L C ) .
The
f r a c t i o n of
theblack body
spectrum which can be emitted i s given, in su ff ic ie n t approximation, by
*.
where
U,
0
h V
o w
0
lJ = -
(3.34)
1
1
3 . 3 5 )
'
I n ( 3 . 3 4 ) we have ne gle cte d the fac t tha t the
i nf r ar ed, bel ow
about
1/2 ev, a l s o cannot be emi tted (Sec . 4a),
and
nave approximated
(
eU - 1)
i n
tjne Planck spectrum bye-U; bothcorrectionsaresmall.
To
=
8OWo
has
been chosen as a reasonable average emperature
(
see Sec.
5 ) .
Near
t h i s temperature, va rie s about as T
,
so thatne ctua l adiat ion
to large dis tances i s about
-1.5
Solving
(3.32) f o r p,
with
po =
1-29
(normal ai r
density)gives
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For
p1 =
po,
T = 0.8,
t h i s
i s
14
bars.
A t higher temperature ( T '
> 1)
he ul tra vi ol et can be transported
as
easily as the vis ible a l though i t
w i l l
not escape t o large dis tances
(Sec. kc ). Then
it
i s reasonable t o use the
f u l l
black body radiation
( 3 .33 ) for he emission. Ins ert in g hi s nt o ( 3 .32 ) gives
(,),,
8/3
1
O,
t h i s
i s 40 bars.
( 3 .39 )
Thus for
p
greater than 1 4 t o
40
bars , the radiat ion
i s only
a
f r ac t ion of the adiabat ic coolin g, for lower pressure radiatio n cooling
i s
more important.
A t
the
lower pres sure s hen , energy
m u s t
be supplied
from
the ins i de to maintain the radi at io n. Thi s give s r ise to a cooling
wave moving
inwards
as
w i l l
be discussed in Sec.
5 .
It i s interes t ing that the condit ion
( 3 . 38 ) refers
to the p r e s su re
alone.Neither the oca lden sity nor theequation of sta te en ter s. The
opacity of air ente rs only insofar as
i t
determines th e rad iat in g temper-
a tu r e
T
'
through the condition
(
3
*21)
A more accurate expre ssion for the limit ing pressure w i l l be derived
in
Sec. 5e.
It w i l l
tu rn
out
t o be considerably lower, about
5 bars.
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4.
ABSORPTI ON coEFF1c1ms
a. Infrared
/ .
Themain abso rption in the infra red i s due to free-f ree elect ron
t ran s i t i on s . These are t rea ted incor rec t ly i n th e paper by Meyerott e t
al . , i n whicn it i s assumed that such transitions occuronly i n the
f i e ld of ions .
A t .
tk e important emperatures
of
8000 or les s , the degree
of ionizat ion i s
o r
le ss . Therefore, th eree - f r eerans i t i ons
i n
th e f i e ld of n eut ral atomsand molecules ar e muchmore important than
th os e in th e fi el d of ions, even thougheach indiv idua l atom cont ribute s
fa r le ss th an each ion.
16
The effectiveness
of
neut ra ls in inducing fr ee- fre e t ra nsi t io ns has
18
been measured and int erp ret ed by Taylor and Kivel a t
laboratory. A s compared t o one' ion, theeffectiveness
tant neut ra l spec ies i s
N2
:
2.2
f
0.3
X
N:
0.9
f
0.h x
lo-=
0: O e 2 f
0.3
X lo-=
t h e Avco-Everett
of the most mpor-
Thus nit rog en giv es about t h e same contrib ution wnether molecular o r
atomic, and th econtr ibut ion of oxygen i s verysmall. A s a result, one
atom
of
a i r i s equivalent
t o
about
a
=
0.8
X ion
(of
unit charge).
The fre e-f ree abs orp tio n co eff icie nt in cm'l
i s
then
cLf
=
0.87
x 10
(%)
(e - ) (hv)-3
=
7.0
T
'-I/*(b)
(
e- ) (hv)
3
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where e - i s th e number of el ec tr on s pe r ai r atom, the qu ant ity tab ula ted
i n Gilmore,l3 and hv
i s
the quantum energy i n ev. Table I11 gives some
numbers f o r hv = 1ev, four emperatu res
and
three dens i t ies .
A t
80007
the free-free absorption i s subs t an t i a l , a t
lawer
temperature s negligible .
A t 12,000 the transitions aremostly in th e f i e ld f -7ons i.e.,
Meyerott's numbers need
only
s l ight cor rec t ion , and the absorption i s
l a rge
Table
111.
T
47 OOo
6,000
87 000
12,000
Free-FreeAbsorption Coefficients
c m
)
fo r hv
=
1ev
-1
5.5-&
1.05'5
1.9 7
Note: f o r each value, th e power
of 10
i s
indicated by a superscript.
Another cause of absorption in the infrared i s the vib rat ion al bands
of NO, which have
an
osc i l l a to rt r eng th of
and
hv = l/k ev.
The resu lting abso rption coef ficien t i s about
where (N O ) i s the
percent at p/po
=
cm e While this
1
number of NO molecules
per
a i r atom. Tnis i s
a
few
1
and
T = 4000 t o
8000,
giving p
= 2 X l om3to
i s of the order of magnitude relevant
f o r
emission,
33
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(3.21),
it
i s
small compared to th e fre e-f ree ab sor pt ion at' th i s
low
frequency,except at T
lom3
(mean freepath
less
than 10 meters) o r I L > lom2(4
1 meter).
The
tab le shows that strong absorption covers a partic ula rly
wide
spect ra l
region a t 4000,
shrinking
substant ial ly
a t
6000~. Very strongabsorption,
p > LOm2
m-', occurs i n quite a l w g e spectral . region fo r T
=
4000
but
shrinks to pra ct ic al ly nothing
a t 6000~
nd
t o nothing a t
8000.
A t
12,000, very strong absorption occurs again
but
i s now in the vis ible .
The
strong absorption
i n
t he u l t r av io l e t (hv
> 3.5
w) eans f i rs t
of
al l that
the W i s not emitted t o l a rge distances and can therefore
not be observed; e.g., a t T
=
4000 and
hv
= 3.5 ev, we have
%e should
also
add the photoelectric absorption from exc ited states of
NO,
which should be especially noticeable at
8000.
41
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The fra ct ion of th e Planck s p e c t m beyond u = 10 i s only about Y$, so
t h a t d s s i o n of these frequencies i s negligible.
Table
VII.
Ul tra vi ol et Absorption: Sp ectr al Regions {
hv
in ev)with
Large Absorption as a Function of Temperature for p = O.lpo
T
I L > 10-3 LL > ml
2,000 4.7 - 7.2
5.5
- 7.2
3,000 3.9 .. 7.2 4.7 - 7.2
4,000
3.5
-
7.2
4.6
-
7.2
6,000
4.0
-
7.1
5.8 - 6.0
8,000
2.7
-
6.3
None
12,000
All
2.7 -
3.5
The u lt ra vi o le t can, hawever, be tra nsp orte d qui te easi ly at 8 0 0 0 ~
and even more e a s i l y a t
12,000
i f there i s a temperature gradient
.
Such
a gradient i s always av ai labl e, whether we have adiab atic condit ions
(Secs. 3 5f) or a strong coo ling wave (Sec.5d) Therefore, here w i l l
be a
flow
of ultraviolet radiat ion at the radiat ing temperature, defined
in Sec.
5 ,
which
w i l l
be shown ( S e c s . 5d,
5f ) t o
be about 10,OOOo o r
s l i gh t l y
less . To ca lcu la te th is fl ow we shoulddetermine th e tempera-
tu re gr ad ie nt from consideration s such as Sec. 5d o r
5f,
and then insert
t h i s
into
theradiat ion
fluw
equation.
This
is
similar
t o
(5.3)
except
that
only
the ul traviolet contribution to the flow should be taken into
account.
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When t h i s
i s
done i n a case of constan t
(
frequency-independent
)
absorption coefficient, the ul t ravio le t t ranspor t w i l l be r e l a t e d t o t h e
vis ib le radia t ion t ranspor t as the respec t ive in tens i t ies
i n
the Planck
spectrum. This condit ion seems t o be ne ar ly fu lf i l le d at 12,000. A t
8000, the absorption in the near u l t ravio le t (2.75 t o 4.2 ev) is about
t h r e e t i n e s
that
in the v is ib le ; then the W t ransport
w i l l
be one-third
of t h a t corresponding t o
the
Planck inten sity. Sinc e
the
radiat ing
temperature near the second rad iat ion
maximum is
between
8000 t o 10,OOOo,
t he ac tua l
W
transport w i l l be between one-third and the full Planck
value, relative
t o
the vis ib le radia t ion. According t o
(3 .36) ,
the
W
contains about 43
,of
the Planck i n t e n s i t y a t
8000 ;
hence th e to ta l
radiat ion t ransport a t
t h i s
temperature i s about
of
the
black
body radiation.
A t 12,000
we get
t h e
f u l l
black body value.
For simplicity w e have assumed
the
full
black body
radiat ion in Sec.
5 ,
even though the W i s not emit ted to large dis ta nces . But t h i s problem
could,. and should, be tre at ed more accurately.
Having
discussed the influence of the W on the t o t a l r a d i a t i o n
flow,
we nuw examine w h a t happens t o the W radia t ion after it has gone through
t h e
radiating
layer, i.e., th e la ye r which
emits
t h e v i s i b l e l i g h t t o
large distances. The very near ultrav iolet, 2.75 t o
3.5
ev,
w i U .
be
partially absorbed a t 4000 t o 6000, especial ly i f t he l aye r of matter
a t t h e s e intermediate emperatures becomes th ic k, 0.3 t o 0.5
gm cm
o r SO.
I f
2
43
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The W beyond 3.5
ev
w i l l be strong ly absorbed a t 4000'. Thus the layers
of a i r a t intermediate temperatures get additional heat
which
counteracts
and
may even exceed th e ad ia ba ti c cooling.This
w i l l
tend t o increase
the thickn ess of the medium-temperature laye r. Th is in
turn w i l l (mod-
e ra te ly ) lower th e ra di ati ng temperature, but three-dimensional effects
act the op po sit e way (Sec. 6 ).
a
.
Theory of Z e l dovich
e t
al .
Zelsdovich, Kompaneets,and Raizer
lo
quoted as Z ) have considered
t h e
loss
of ra dia tio n by hot mater ial when th e ab sorptio n coeffic ient
fo r th e ra di ati on in cr ea ses monotonically with temperature. Theyhave
shown t h a t
i n
t h i s case a cooling wave proceeds into the hot material
from the outside . Tnis i s
t o
say, t h e cool emperatureoutsidegradually
ea t s
i t s
way into
the
hot material, while the material
in
the center
remains
unaffected and merely expands adiabatically.
For
simplicity, ZelPdovich
e t al.
consider
a
one-dimensionalcase.
They fu rt he r as,sume th at the sp ec ifi c heat i s constant and express their
theory
i n
terms of the temperature.This
i s
notnecessary; we sh a l l
merely assume that both the enthalpy
H
and the absorp tion coeffic ient
for radia t ion are a rbi t ra ry but monotonically increasing functions
of
the
temperature. Like
2,
we sh a l l assume,
in
t h i s subsection
only,
tha t he
ra di at io n tr an sp or t can be described by an opacity (Rosseland mean) rat he r
than considering each wave len gth separat ely.
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The
fundamental statement
of 2 i s
t ha t the cooling wave keeps
i t s
shape, i.e., that the enthalpy (and other functions of the temperatme)
the cooling wave proceeds tmards smaller x,
i.
.
, nwards.
H
is, of
i s
given by
H = H(x +
Here t i s the time,
Lagrangian velocity
x i s
the Lagrangian coordinate,
and
u
i s
t he
of
th e cooling wave. We have w rit te n
x +
ut
so
t ha t
course,
a
decreasiw function
of x +
ut .
The
Lagrangian coordinate i s
best
measured
in gm/cm
and i s defined by
where X i s the geometrical(Eulerian)coordinate.For given pressure p,
the
density
p i s
a function
of
H
so
that
X x )
can
be
calculated
from
( 5 0 2 )
The Lagrangian vel oci ty
u,
measured i n
gm/cm
sec,
i s
a'constant.
For any given
If,
we
know
the temperature T, hence the opac ity
K
and
the radiat ion flow
where a
i s
the Stefan-Boltzmann constant,
2
4
a
= 5.7 X lom5
erg/cm se c deg
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The equation
of
continuity i s
a H
a J
== a x
5*4)
The
energy in radiati on has been neglected, which
is
jus t i f i ed i n
all practical cases . Using (5.1), (5.4) can be ntegra ted over x t o
give
u H + J = C
5
* 5 )
where
C
is a constant. This i s the fundamental result of 2,
If
the opacity increases monotonically w i t h
H,
then in t h e i n t e r i o r
J w i l l be very nearly zero, and therefore
c =
uH*
5.6)
where
Ho
i s the enthalpy. in th e undisturbed interior, hot region.
Equation (5.5) becomes
J = u(H0 -
H)
(5.7)
which
can
be integrated t o give x(T), since H(T)
md
K( T , p ) are
known
fun ctio ns. We have
put i n
evidence the f a c t
that K
depends on pressure
in addition
t o T.
Overmost of the range of
T,
K(T)
i s
the most
r ap id ly
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varying (increasing) function
o f T
and the variat ion of H
-
H is
less
important; therefore,
T x)
becomes s te ep er as
T
increases;
but
when
H
g e t s
veryclose t o
II
th e most rap idlyvaryingfunction-
in (5.8)
i s
Ho
- H, and
H
approaches
Ho
exponentially as ea f o r
s m a l l X.
The
qual-
i ta ti v e behavior of T(x)
i s sham
in
Fig.
1, in accord with
Z.
To obtain
t h i s shape
it
i s essent ia l
that
K( T ) increase much fa st er th an
T .
0
0
3
*O
Fig.
1.
Temperature di s tr ib u ti on in ' Cooling wave.
On the outs ide, we fi n a l l y come t o
a
point xlwhere
on l y
one
optical
mean f r e e path
i s
outside xloFrom th i s po in t we get black body emission,
4
J(xl)
J1 =
aTl
5 . 9 )
Using
t h i s
in 5.7)
we
f ind
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J,
U
A
Ho
- H(T1)
To
determine
u
w e therefore
1. Find the temperature T
1
have t o proceed as follows:
a t which the opacit y
K(T1)
is such th at
there
i s one optical mean freepathoutside x i.e.,
1'
For t h i s purpose we
m u s t haw
of course,
the
temperature distribution
T(x)
f o r T
0. An in teres t ing inter -
mediate state
i s
when the cooling wave has just reached the blocking
la ye r. Tnen, using Tb
J1 -
Jb,
"b = 1% -
H1
= 1.8 and 5.14),
Clear ly th i s m a k e s sense only i f the subtracted term in the numerator i s
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smaller than th e f i r s t term, i .e , when the pressure i s suff ic ient ly
high.
A s
the pressure decreases
below
about
10
bars, he blocking ayer
opens
up
and ceases t o block t'ne
flow
of radiation.
In th e very beginning, when 'ne cooling wave ju s t s ta r t s , t'ne top
temperature
of
the cooling wave,
i s
close to the rad iat in g tempera-
ture
T hen
(5.18)
becomes
TO'
1'
1 =
-
(EX
1
If
we use
(5.12)
fo r J, assume
d
log T/d
log R and R t o
be constant,
wd
use
(3.14),
tinen
For further
discussion,see Sec. 5f.
A s
T increases,
u
decreases from (5.23) via
(5.21) to
(5.19).
Af'ter
th e cooling wave has penetrated t o t h e isothermal sphere, u
i s
apt t o
increase aga in because T i n t h e denominator of
(5.19)
w i l l decrease due
t o ad iabati c expansion
of
the sothermalsphere. Thus th e ve lo cit y u i s
apt t o be a m i n i m w n vhen the cooling wave has just reached the isotherma l
sphere
C
The variation of u with time
i s
not very grea t. Likewise, t he shape
of the cool ing wave changes onlyslowlywith ime. Tne shape i s obtained
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by integrating
5 . 8 ) ; it
depends on timebecause K
i s
a (rather slowly
variable) function
of
thepressure, and
H
i s a (slow) function of time.
This
jus t i f ies
approximately th e bas ic assumption
( 3 .l) of 2:
While the
0
cooling wave does not preserve i t s shape exactly, it does so approximately.
The picture i s th en th at at any given
time
t there a re up t o f ive
regions behind th e shock. I n Fig. 2 th e temperature
i s
plotted schemat-
icallyagainst heLerangecoordinate r. Starting from thecenter,
there i s
f i r s t t h e isothermal sphere I
n
Fig.
2 ).
Fig.
2.
Schematic temperaturedistribution. I, isothermal
sphere;
111,
cooling wave; VI, undisturbed air;
E,
shock front
11,
IY,
and
V
are expanding adiabat
-
ical ly.
This may be followed by a region I1 i n which the temperature di str ib u-
t i o n
i s
ess en tia lly th at est ab lis he d by adiab atic expansion behind th e
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shock, Eq. (3.15)
.
Nextcomes 'ne cooling wave I11 in whicn t e emper-
ature f a l l s more steeply,according t o (3.8)
.
( A t
la t e t imes,region I1
i s wiped out and I11 follows immediately upon
I
.
Region IV includes
th e mat er ia l which has gone through tn e cooling wave,and now cools adi-
abatically; nence the temperature falls
slowly
with r (Sec
.
jd)
D
i s
the ma teri al poi nt from whicn t h e cooling wave sta rt ed or ig in all y; region
V, outside hat point, i s also expanding adi aba tica lly, but
f r o m
snock
conditions; hus it is the continuation of region 11. Finally region VI
i s t h e a i r not yet shocked. A s time goes on, th e cooling wave I11 moves
inward, wiping
out
region
I1 and
then eating nto region I. Region IV
accordingly g r o w s toward the inside, but i t s outer end D stays fixed.
Region
V
expands into V I by shock.
We
note once more that u i s the ve loc i ty in Lagrange coordinates,
and in
gm/cm
sec. The problem i s xnade omewhatmore complicated by the
th re e dimensions and th e ad ia ba ti c expansion, cf . Sec.
6,
but tne
pr in -
cipal features remain the same
d. Adiabatic Expansion a f t e r
Cooling.
Radia ting Temperature
When a given mater ia l element has gone through t he cooling wave, it
i s l e f t at he rad iati ng temperature
T1.
Thereaf'ter, it
w i l l
expand
adiabatically. Equation (3.25) shows that
f o r
adiabatic expansion
or,
using (3.10)
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a l o g I1
- y -
1
l og p
t y ' a t
A s long as the
snoek
is strong, (3.26)
holds
md therefore
GiI-more'sl3
Tables 1.1 and
13
shoy that
f o r T = 5000 to
8000 and
p =
1 o 1 0 bars, y var ies f romabout,
1.13
t o 1.20, so
a log-1
=
- 0.lh t o -
0.20
A t l a t e tjm-es, p no longerdecreases as
fast
a s t , o
H
nlso decreases
more s l o ~ , ~ l - yut (
5.25)
remains valid.
-1.2
We use now t h e
approximate
equation
of
s t a t e 3 o l k ) (together with
B =
y
'E) and find
= P
w L t h
= 0.136
t o 0.167
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using 5 .25 )
m d y
= 1.13 t o 1.20. A given material element vllich went
through the cooling wave
a t
temperature
T '
and pressure
p T r i l l
now ( a t
pressure
p)
have the temperature
I
m m
T t = oe
m
If
w e
assume Tmt o be independent
of time,
th en at a given tfme in Cne
adiabatic region
(T v ,
of
Fig. 2
as
defined in Sec. l;b. Since weow use materialcoordinates
i n gm/cm
,
we should use mean free paths n he same unit. Table V I and (4. 10) m d
(lk.12)
give tine required nformation. We m i t e (4.12)
in
t he form
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2
vitn A =
0.1G
q/ cm and n = 6.3.
The optical depth
is,
according t o (5 .32) and (7 .33)
x1
4
n
1 u t Y
* P -
1
0
(5. 34)
Set t ing
D
= 1,
y
= 1. 15, A = 0.10, and
n
= 6.5 gives
Tnis
i s
m
expl ici t express ion f o r
tne
radicting temperature
i n terms
of
tne ve loc i ty
of the
cooling wave.Tae l a t t e r depends i n
turn on
the
radiating temperature,
i ncreasi ng
with T p 4 so t n a t T
'
occurs altogether
in t n e
10.2
power
and t'nus
can be determined very accu rate ly.
1'
1
Equation 3.s)
an
be fu rt he r reduced by usi ng tne rel at ion between
pressure
and time
wIxLch
i s , f or
a st rong shock, approximately
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where
t i s i n
seconds, p in bars , a d Y i n megatons, a d
pl i s
the den-
s i t y of tine mbient,undisturbed aLr. Then ( > 3 6 ) becomes
Tne
right hand side of
5 . 3 8 )
gi ve s th e co&plete dependenceon
Y
and
p1
since,deriving
(5.36),
we have only used
tne
opacity
'law
3 . 3 3 ) and
th e ad isb ati c cooling of a i r,
5 . 3 2 ) ,
both of which zre independent
of
the explosive yield
Y
and of
t ne
ambient density of t h e a i r .
iY0r.r insek-t u from
5 .
X,9); hen w e obtain
(5 .39)
Tnis
equation gives he radiating emperature i n
t e a s
of
t ne cen t r a l
temperature
T
and
of
thequant i t ies
on
ther igh t
hand
side. The ra di -
a t
n2
temperature
i s
proportional t o a low power
of
th e cent ral temper-
e t u r e
(
about
tne
1/6 '
pover); tnus as th e ins ide coo1s, tne radi atio n
dec reases see Sec. 5e
f o r
d e t a i l s ) .
It
alsodecreases slowly with time
C
due t o t h e pressure factor on tile righ t nand side ,
T
1 - P
For given
p and T the radiation emperature
i s
higher
f o r
lower yield,
T
- Y
and for n igher a l t i tude ,
Ti
-
p,
-0.032
C'
1
9
-1./21
.
For sea level , for
Y = 1,
and for
p
=
5
bar s (cf . Sec. >e
f o r
t h i s
choice)Brode'scalculationsgive T' M 3.6; then
5 . 3 9 )
yields Ti
=
1.08,
C
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o r
a radiating emperature
of
10,800.
Tnis
i s a
reasonableresult,
tho wh appre cia bly hig he r ha n he eqera tur es us ua lly observed. How-
ever,
as we
shall show in Secs.5e
md f ,
our calculation gives he
max-
imum
temperature reached,
md i s
l i k e l y to be somewhat too hig h due t o
OUT
approxinmtions
.
e. Bewinning
of
S t r o w Coolim Wave
2
Equation
5-19)
ives the inward speed
of
t he cooling wave in
@/a
sec. Precisely,
t h i s
i s
th e speed a t which the poin t
of
t e q e r z h r e
T
(radiating temperature ) moves rel ativ e to the ma ter ial , once fne cooling
1
wave
i s
ful ly est ab lish ed . But even
i f
there
i s
no cooling wave, i.e.,
i f we have simply ad iaba tic expansion behind th e shock,
a
point of given
temperature
T1
w i l l move intrzrd. Th is ad iaba tic motion''
i s
t he minimum
velocity vlnich thepoint T can nave. Therefore , i f tneadiabatic speed
i s
greater than (5.19)
, it
w i l l be tne correct velocity. O f course, there
1
w i l l
s t i l l
be
a.
cooling wave because t h i s i s needed
t o
supply the energy
for
the radiat ion;
tills
'be&< cooling wave
will
be described i n Sec 5 g .
But
i t s
inward motion, more acc ura tel y the ve loc ity of i t s foot (point
C
i n Fig. 2 ) , w i l l not be determined by the requirement of sufficient energy
flow,
(3.19), bu t
by the ad iab atic speed which we sha ll de riv e from
( 3 . 1 7 ) Region
I V of
Fig.
2
w i l l nar be absent, Tnus, outside he
cooling wave, at po in t
C y
region
V
will
begin Med iately , with the
tem pera ture distri butio n given by (3.15).
It i s the ref ore important t o determine th e time ta (and pressme p
)
a
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a t which
u
of ( 5
.IS)
becomes larger than t'le ad iab at ic speed of temper-
ature T Before th i s time
ta,
we 'have
a
weak cooling wave
.
We c a l l
this Stage C
I;
afterwards, the
cooling
wave
i s
strong, and
the
radiation
i s ess ent ial ly described by the theory of Sec. 5d (Stage C 11). To
determine tine point of separation p between these two stages i s
a
refine-
ment of the considerat ions of Sec.
3d.
1'
a
Tne temperature in th e ad ii ib at ic al ly expanding mate rial behind t he
shock
i s
a
function of time and posi tion , given by nydrodynamics
and
equation of s ta te . The imrard motion of a point of given emperature
rel-ative t o the material i s given by
tfnere the subscript
r
means that
the
pa r t i a l de r iva t ive
must
be talcen at
given material
point
r,
not a t given geometrical radius
R .
W
have
from
(3.15)
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assuming
t he s t rong
shock
r e l a t ion p
-
-1 2
. The ratio (5.40) i s tnen
I-Iere w e
use the rela tion (3.10)
P -
yIt - 1 eE
and f ind
(5 945
where
H
has been labeled
3
ecause
it
ref ers to th e ra dia t in g tempera-
tu re . % re we
may
i n s e r t (3.30)
and (3.31-),
viz.,
We may then compare t he r e su l t w i th 5 IS),
the
veloc i ty of t h e cooling
vave (set t ing J = 0 )
0
u =
Ho
-
5.
The comparison gives
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Insert ing
(3.33) f o r JIJ and
changing the un it
of p
fromdynes/cm 2 t o
bars = LO dpes/cm
we
get
2
Setting nar
p1
= po and C ~ O O S ~ ~ ,s
in
Sec. 5d, Tc =
3.6
and T1 =
1.08
yields
p1 = 5.0 bars
Thus the cr i t ica l pressure i s 5 bars, which was tne reason f o r the choice
of t h i s , number a t
t'ne
end of Sec. 5d. Tnere i t vas
shown
t h a t 5 bars and
T '
=
3.6 l eads t o Ti = 1.08 so t h a t our numbers are cons ist en t.
C
We
had to rely on Brode ' s so lu t ion to f i n d Tc for a given
p;
t h i s
could
only
be avoided by obtaining an ana lytic solutio n
fo r the
i s o t n e m l
sphere which w i l l be discussed i n a subsequent paper. Apart from this
our treatment i s analytical, making us of the equation
of
st at e and the
absorption characteristics of a i r .
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pressure i s larger than pa, 5 .50)
.
Then the radiatin g surfa ce i s i n
the adiab atic regio n desc ribed n Sec. 3c.
The
condition for i t s posi-
tion is, (3.21), t =
R/5O. From Brode's curves, th e pos it io n of
a
point
of
temperature T
'
near
1 s
given approximately by
R = 0.78 Y T
P
1/3 r-0.10 -1/4
This expression comes pu rely
from
the numerical calcu lation , except th c t
th e co rr ec t dependence on yi el d
i s
inserted. Y
i s
i n megatons, p in ba rs ,
R in kilometers. Equation (5.51) holds
from p =
5 t o 100 bars within
about
5%.
The absorptioncoefficient
i s
given in (4.10), which yields
using t'ne equation
of
s t a t e (4.11). Equating th i s t o 1/50 of
(5.51)
gives
Tr408 = 4.5 y
-1/3
P 1.20
Thus
the radiating temperature increases
as
the pre ssu re decreases.
b
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Since p
- ,
- 12
9
en t ire ly due t o
tihe
opening up
of
the shock,
i.e.,
the decrease in
absorption with decreasing d ensity (pressure) .
For
given pressu re,
the
radiating temperature
(5.54)
is s l i gh t l y
higher
f o r
smaller yield, an ef fe ct which
has
been observed. A f ac tor
of
1000 i n
yield corresponds t o
a
f ac tor
1.6
i n
temperature, hence
a
f ac tor
7 i n
radiation per
unit
area.
The
tot al rad iat ed ' power(assuming
blac k body) a t given p
i s
proport ional t o
The
re la t ively law power
of
t he y i e ld i s remarkable i n t h i s formula,
vnichdescribes Stage
C I.
Since th e to ta l energy
radiated
shouldbe
approximately proportional t o
Y,
the dura t ion
of
Stage C
I
i s then pro-
por t iona l t o Yo*6o . The observed time t o th e second radiat ion maximum
i s
abotrt proportional t o
Y . O u r
theoretical dependence
of
T
on yield
.7 1
i s there fore sornewnat to o strone;. The time dependence of
(3.56)
i s quite
strong, about
as t
2
A s ve nave snam in Sec.
5e,
Stage
C
I
ends when
the
pressure reaches
5 bars.
For
th i s value
of p,
and
f o r Y
= 1, ( 7 S ) gives
T
=
0.92.
Tnis
i s
s l i zh t ly l e s s
t han
t he
T ' = 1.08
deduced
i n
Sec. 5d f o r
t he
same
1
p
m d
Y f r o m
the cooling
??me.
The discrepancy must
be
due
t o
a small
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inconsistency in our apyroximations .
In our one-diniensional neoqy, tne end
of
Stage C
I
narks -the
m a x -
Snrrm.
of
rad iati on, bot h n temperature and t o t a l emission. I n Stage C I
because th e mat er ia l becomesmore transparent. I n Stage C I1 the reverse
i s
the case, (5.39), because t he ma te ri al which has gone through the
cooling wavebecomes th ic ke r wi th time,(5,.32).Tnis ma ter ial provides
opaci ty for the v is ib le l ight from the f i reba l l ; s ince it becomesmore
opaque, the rad ia t io n must nowcome from a layer
of
smaller absorption
coefficient , (5.35), and therefore of lower temperature.
A s (5.36)
s h a ~ s ,
the increase of thickness t
2n
the denominator) i s more important than
th e continued decrease of de ns ity (f ac to r p1/3
).
Tnese results w i l l be
modified in t h e three-dimensional heory, Sec. 6.
k
The maxi mum temperature has been calc ula ted as T = 0.92 or 1.08,
from
o m
two
calculations;
it
i s
clearly close
t o
1,
i.e.,
~ O , O O O ~ .
This number i s not to o mucn out of l i n e with observation considering
t n a t we have calc ula ted a
maximum.
In fac t , tne t rans i t ion
from
Stage
C I
t o
C
I1 cannot be sudden as
we
have assumed; the cooling wavemust
beg in gradua lly, and tn er ef or e th e temperature peak which we have calcu-
la ted w l l actua l ly
be
cut off (Fig.
3 ) .
The observable
maximum
may
easi ly be 1000 lower than our calcula tion.
66
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f
CALCULATED
LOG
t
Fig.
3.
The calculat ed ncrease n empe rature i n Stage C
I,
and decrease in Stage
C I1
( s t r a igh t l ine s ) ,
and
the
estimated actual behavio r.
The
radius of
the radiat ing surface increases w i t h time i n Stage
C
I.
Inser t ing
( 5
.?4)
i n t o
(5.51)
gives
R - Y
P
0.34
-0.225 t0.27
5057)
In Stage
C 11,
Yne surface moves rath er rap idl y inward re l a t i v e t o .t h e
material, due
t o
thecooling wave. Inaddition,
f o r
sea evelexplosions
at leas t , the pressure
i s
no longer much above ambient,
so
t h a t t h e out-
ward
motion
o f
the material
slows
d m . Thus the geometr icradius
of
the
rad iat ing su rface no longe r inc rease s much, and
soon
begins
t o
decrease.
Tnerefore, the tot a l rad iat io n
w i l l
reach
i t s
maximum
a t t h e
same
time
as, o r very soon af ter , the maxi mum of the temperature.
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G . We&
Cool-ing Wave
In
Stage
C I1
tne coo lin g wave dominates th e ra d ia ti o n (Sec. 5d)
.
I n
Stage C
I t h e
cooling
mme als o ex is ts , because t he energy
of
t h e
radiztj-on
m u s t
be provided. Hovever, t h e speed a t wnich th e trave proceeds
inward i s nov governed by tn e ad ia bat ic expansion, (3 .LO
a
I n order t o
obtain he correct f lux of radiat ion
J a t
the radiat ing surface, we
must therefore use (5.18) i n reve rse: Tine tem perature
T
a t he nne r
edge
of
Cne
cool i ng
77ave (point
B
in Fig. 2) will reg ula te i t se l f in such
a
vay
t h a t
(5*18)
i s sa t i s f ied , wi th ugiven by 5.lJr5). Using 5.461,
w e
thus
get the condit ion
1
0
or solving for
To and
in se rt in g numbers:
13
y - 0.9
P
3/2
T i 4
5
059)
PlTeglecting
Jo ,
we f in d th at Ho increases rapidly as p decreases.
Wemay in s e r t (5.31 -); then the r ight nand sid e varie s as p
T h u s
the cooli ng wave s t a r t s very weak and the n rapid ly incre ases i n
strengtn. Its head (point B inFig. ' 2 )
i s
a t
f i r s t
c lose to
i t s foot
(point C).
A s
time goes on, it movesmore deeply in to t'ne ho tmaterial,
ea t ing up region
I1
o f Fig. 2. The energy which i s made available
f o r
5/2 - t3*
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radiat ion i s essent ial ly the difference H
: Tne
material drops
suddenly
i n
temperature from
T
t o
T
as hecool ing wave weeps over
0 -
0
1
i t ,
and the energy difference
i s
s e t f r e e
f o r
radiation.
The term J depends
on H
on p, and on th e temperatu regradient
i n th e region nside th e
cooling
wave (region 11). If th i s reg ion
is
adia bat ic, he temperaturegradient can be ca lcu la ted from (3.15). Since
0 0
J 1 i s also related t o the temperature gradient in the adiabatic region
I V, t he r a t io J
/J
tends to uni t y as Ho -.
H1.
Thus the
l e f t
hand side
of
(7 .39)
will have a certain
m i n i m w n
value.Tnis seems to in di ca te
tha t t he re i s no cooling wave a t all until the pres sure has f al le n below
0 1
a ce rt ai n cr it ic al value. We have not in vestigate d thi 's
point
in deta i l .
It i s poss ible that it
i s
simply
r e l a t e d t o tne break-away of the lumi-
nous front from the shock wave,
i.e.,
the beginning
of
Stage
C
.
Equation (5 .99) de sc ribe s the weak cooling wave
in
Stage
C I
no
matter what t he di str ibu tio n of tempe,rature i n region
11.
m e s t a g e
comes t o an endwhen H reaches the maximum possible value,
Hc .
There-
af te r, th e wave described
by 5 .39 )
becomes inadequate
t o
supply the
0
ra di at io n energy: Since th eentha lpycm no longer ncreas e, he speed
of t he cdoling wave has to increase. Tni s speed i s
tnen
given
by ( 5.l9),
and Stage
C
I1 has begun.
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6.
EFFECT OF THREE DDENSIOUS
a. I n i t i a l Conditions and Assumptions
The f ac t t na t t he f i r eb a l l i s spherical causes some deviations f r om
the one-dimensional tiieory of Sec. 5 . In th is se ct io n we shal l
di scuss
Stage C 11, tine penetration of the coolin g wave in to th e isothermal
sphere, i n t h r e e dimensions. I n it i a l l y , we assume tn at he cooling wme
has ju st reached the sotnermal sphere; we c a ll t h e corresponding ti n e
t
=
ta,
and the pressure
i s
t he c r i t i c d p r e s s u r e
p
a
=
5
bars derived
i n
Sec. 5do Subsequently, t h e mass of tne sothermal sphere decreases due
t o tine coo ling wave.
We assume that the ma ter ial , tfnich i s a t temperature T > Tm = +OCOo
a t t he i n i t i a l time t
w i l l
s tay in t h i s temperature ange t'hroughout
Stage C 11. Tnis i s reasonable because th is ma ter ial w i l l be heated by
t h e
ul tra vi ole t ra dia tio n coming
from
the insi de, because
of
tne strong
absorption of a i r
of
medium temperature
(3000
t o
6 0 0 0 ~ )
o r
W
(Sec. 4c)
The W heating i s expected t o compensate approximately th e cooling due
t o a diaba tic expansionof th i s mater ia l ; tn i s i s confirmed by
rough
estimates
of
the he ati ng and cooling.
We
do not knowhow tine temperature
i s di st ri bu te d in th e '%arm layer between the rad iati ng temperature T1
and Yne temperature Tm
=
4000'; tnis could
o n l y
be determined by a detailed
calcu lation of the
UV
radiat ion
flow
i n
t h i s region. We assume th at th e
d i s t r ibu t ion i s smooth, as
it
i s i n Stage C 1, and tha t the refo re the
thickness of the 'harm layer in gm/cm i s d i r e c t l y proportional t o the
a'
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required mean free path of v isi bl e lig nt at th e ra di at in g temperature
(a ls o in gm/cm
)
which in tu rn determines
T~
i t se l f .
2
8
We take he i n i t i a l conditions fromBrode, using h is e r n e s
a t
a
time when the ns ide pre ssu re i s
4 .1
bars , th i s being c los es t to p = 5
bars
of all
the curves he naspublished.Thiscorresponds
t o
a scaled
(1-megaton) time
tc =
Oo32> sec. A t t h i s time, mportantp'hysical
guan-
tit es are as given i n Table
V I 1 1
(dimensions scaled t o 1megaton) o The
l a s t column
o f
the table
i s
not given by Brode but
w i l l
be explained i n
Sec.
6c. In
the table , we have defined a quantity proportion al t o th e
a
mass,
0
J
where
R is
measured
in
hundreds
of
meters. It
i s
in teres t ing that the
mass
of the w a r m layer i s much (4.6 times) larger than that
of
the isothermal
sphere. I t s
volwne
i s about
2.4
tfmes la rg er in Brode
's
calculations.
Table V I I I . Conditions When Cooling Wave Reaches Iso thermal
Sphere, According t o Brode
1
Isothermal Warm Layer
Sphere Brode Sec. 6c
Outer radius, meters
38.5 517 438
Mean temperature
T '
3.0
0.70
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6 . Sinrinkage of I s o t n e m a l Sphere
We denote t h e 'tmass''
of
the isothermal sphere,
as
defined by
(6. 1)~
by 5 Then t h i s
mass
w i l l decrease, due t o t h e progress
of tne
cod ing
wave inward, according t o
d l
d t =
- PO Rf
Note the po n the denominator and the absence of the fac tor
& f l y
both
due t o the de f in i t io n
(6.1)
The
density
of
t h e isothermal sphere
i s
nearly uniform
and vi11
bedenoted by
p
I n i t i a l va lues a t time tar
w i l l bedenotedby
a
subscript a. The isothermal sphere expands adia-
bat ically, hence
i s
and th er ef or e at any time t
The speed of t h e
cooling
wave i s given by 5.LO), thus
5
u =
-
Hl
(6.4)
where J has been
set
equal t o zero because there
i s
no appreciable flow
0
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of radiati on nside he sphere.
H i s
th e enthalpy inside ne sphere,
0
which decreases adiabatically
H
i s th e enthalpy at he radiat ing surface .
It w i l l
be shown i n See. 6c
th at th e temperature T
i s
nearly constant with
tlme,
so t n a t in good
approximation and J in (6.5) ar e constant.
We
shall a lso
make
the
poorer approximation that
3
Ho. Then
(6.5)
and
(6.6)
give
1
1
1
We define
x = E -
t
a
73
J
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Pa
f ( x )
= -
P
and obta in the
simple
di f fe re nt ia l equation
ay
- - Af(x)
- l / 3 Y
,
dx-
vni ch yie lds , tog eth er wit n the boundary conditions,
v
(6.10)
(6.11)
1
The constant A can bedetermined from Sec. ?e. Tne condition here
i s
dr
u = - p E
a
(6.12)
where p is th e de ns it y ou ts id e t'ne cooling wzve. According t o ( 3
&)
Tnen
u s i n g
(5.43)
R
a
E
a
(6.14)
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which i s a pure number, and sma