Integral form of the time-dependent radiation transfer equation—III. Moving boundaries

Radiar. Transfer Vol. 38, No. 6, pp. 415-487, Printed in Great Britain. All rights reserved

J. Quant. Spectrosc.

1987 Copyright

0

0022-4073/87 $3.00 + 0.00 1987 Petgamon Journals Ltd

INTEGRAL FORM OF THE TIME-DEPENDENT RADIATION TRANSFER EQUATION-III. MOVING BOUNDARIES ALAIN Commissariat

MUNIER~

a 1’Energie Atomique, Centre d’Etudes de Lime&Valenton, B.P. 27, 94190 Villeneuve-St-Georges, France (Received 14 November 1986)

Abstract-The theory of characteristics is used to give the general time-dependent integral form of the transfer equation in a time-dependent, inhomogeneous medium, submitted to arbitrary boundary and initial conditions. The medium emits and absorbs radiation, but scattering is neglected. The boundary condition is applied to a moving surface. The general solution is given analytically in Cartesian and spherical coordinates for a 1-D configuration.

1.

INTRODUCTION

In this last paper of three, we study the time-dependent integral form of the radiation transfer equation. In the first part,’ we gave a mathematical proof of the formal solution in Cartesian coordinates for the transfer of photons in a non-scattering inhomogeneous slab. In the second part,2 we gave the expression in spherical symmetry. The first two papers were concerned with time-dependent solutions when the equation of transfer is submitted to fixed boundary conditions (BC), the value of the intensity, possibly time-dependent, being given on a motionless surface. Finding the integral form of the transfer equation when the (BC) varies is more complex because this moving surface can sweep through the medium, and it becomes difficult to determine whether a particular photon is on the inner side or the outer side of the boundary at a given time. The general problem of solving the transfer equation is an arbitrary inhomogeneous, time-dependent (although non-scattering) medium is very difficult, but some particular problems can be solved exactly and we give here the solution for a homogeneous medium submitted to a BC moving with a constant velocity. Even this apparently simple situation shows the difficulty of taking into account retardation effects in the emission and propagation of photons. In the first two sections, we determine how a moving plate is seen by a fixed observer. This leads in the following two sections to the determination of the intensity in explicit form, for a homogeneous medium in Cartesian coordinates. Section 5 deals with the homogeneous problem in spherical symmetry, while Sets. 6 and 7 are devoted to the general inhomogeneous case.

2.

THEORETICAL

DEVELOPMENT

2.1. The I-D Cartesian homogeneous problem 2.1.1. The eidographl of a plate (orthogonal displacement). Consider a fixed observer A, at a distance x from the origin 0, looking at an infinite plate perpendicular to the axis OX, and moving with a constant velocity v parallel to OX (Fig. 1). Some of the photons emitted by the plate are received by the observer. Let CL,be the cosine of the direction of observation with respect to OX (i.e. p0 = cos 8 = cos P*AX), and assume that the position of the plate is known at each instant t. At the time t, the position of the plate is z and the observer looking in the direction p,, receives a photon emitted earlier by a point P* on the plate. This retarded time depends clearly on the direction cosine P,,, and is denoted by tz. At that tvisiting Scientist, Los Alamos $From the Greek (eidos: form,

National Laboratory, Los Alamos, shape; graphein: to write). 475

N.M.,

U.S.A.

ALAIN MIJNIER

476

Fig. 1. Eidograph

of a plate (orthogonal

displacement).

retarded time, the position of the plate was z(t,*,) = z zO.For p,, = 1, (i.e. if the observer is looking in the direction OX), the retarded time and position are respectively t: and z: . Since the plate is assumed to move with a constant velocity, we have that z: = z - v(t - t:>

(1)

and this clearly corresponds to the distance travelled by the photon during the time interval defined by the instants of emission and absorption z: --x =c(t

-t:>.

(2)

It follows that the retarded time and retarded position are ct:=ct

-

z-x

i-T-q’

and z,*_--.

z-x

1 +v/c

If the direction of observation is not parallel to OX (i.e. for p0 # l), the distance between the point of emission and the observer is [generalizing (1) and (2)] z* = (zFO- x)/j& = [z - v(t - t;J -Xl/&,

(5)

and

z* = c(t And consequently

- t$Q.

(6)

the retarded time and position are ct*flo =ct

z-x --)

PO+

v/c

and z*

_~Jz-x)+x.

PO -

Po+vlc

’

while the retarded distance is z-x l”=-----. cL,+vIc

(9)

Since the retardation depends on the direction of observation, the observer will not see a straight plate, but a curved surface. This surface is still invariant under a rotation around OX, and represents the locus (in space-time) of all the points on the plate which have emitted photons that are received by the observer at the time t. We call this locus the eidograph of the plate with respect to the observer. The equation of the eidograph is very simple in polar coordinates if the origin is

Time-dependent

radiation transfer equation-III

477

the observer. We simply write p = I* and cos 0 = pO, and from Eq. (9) we have P=

z-x coso +v/c’

(10)

or, writing X = p cos 0 and Y = p sin 6, we have the equation in Cartesian coordinates (X2 + Y*)Uz/c2= (z - x - X)“.

(11)

As X + cc this curve admits the two asymptotes y = + [X(1 - v’/c”) - z + x] We may choose the intersection


.

(12)

of the two asymptotes as the new origin, then writing

X=R+n2(z

-x)

(13)

Y=9

(14)

n = (1 - vZ/c2)-“2,

(15)

and substituting into Eq. (1 l), the curve becomes P(1 -n-‘)-X’/Ll”=(l which is a hyperbola. Eq. (11) we have

-/i’)(z

-x)”

(16)

In the limit v/c + 1, this hyperbola degenerates into a parabola and from Y2=(z -x)2-22x(z

-x)

(17)

which is indeed a parabola with p = 1 as an asymptotic direction. In the limit of a static plate (v/c = 0) we recognize that the eidograph is the plate itself, whose equation is given by either X = z - x [from Eq. (1 l)] or 8 = 0 [from Eq. (16)]. Moreover, the position of the plate is a parameter of the eidograph, which is continuously changing. The shape becomes singular when z = x, i.e. when the plate goes past the observer. At that time the eidograph degenerates into its asymptotes. 2.1.2. The eidograph of a plate (oblique displacement). The preceding calculation is slightly more involved if the plate is not orthogonal to the axis OX. In this case, the eidograph does not show any symmetry of revolution around OX. Assume that the plate makes a constant angle a with the direction OX; then Eqs. (3)-(4) are still valid for cl0= 1. However, in a direction p0 # 1 we have, for an observer or coordinates (x, y) (Fig. 2) which is not on the axis OX, I* = c(t - ttO)= (h,*o- x)/pL,,

(18)

where h:O is the projection of the point P* on the axis. We also have z = z;O+ fJ(t - t,*,),

(19)

and tan a = yJ(h,*, - z,*,),

tan 8 = (yP - y)/(hzO - x)

Fig. 2. Eidograph of a plate (oblique placement).

(20)

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ALAIN MUNIER

where yp represents the ordinate substituting into (18) we obtain

P *. Eliminating

of the point z -(x

I* =

y, between

the two Eqs. (20) and

--/tana)

(21)

~~(1 -tane/tancc)+v/c’ Or writing

i = l/tan

tl, the polar equation

with respect to the observer,

z --(x-iv)

P= In Cartesian

of the eidograph

~0~0 -[

(22)

+21/c

sin8

becomes

X = p cos 9, Y = p sin 0, this reads

coordinates,

(A? + Y2)v2/2 =

[z -

(x - [y) + iy -

It is easy to show that this curve admits (for x + co) two asymptotes -[

a=

+ (v/c)

Xl’.

(23)

of the form Y = aX + b where

1 - v*/c2+ 12 (24)

(d/C*> - i*

and

b=*(Gm-(x-5Y)l (v&/l - d/c’+ c* . 2.1.3. Integral form of the homogeneous section the equation

transfer equation (orthogonal case). Consider

;;+jLg (with K = constant

and B = constant),

condition

-z>

=K(B

submitted

BC: Z[t, x = x,(t), and the initial

(25)

(26)

to the BC

~1=f(t, P)

(27)

(IC) IC: Z(t = 0, x, p) = g(x, /A).

We follow here the method are

and notation

of Part I.’ The two characteristics

p = constant Eliminating

the time variable

in favour

and

The solution

the transfer

of this equation

(26) becomes

/&/ax

= K(B - i).

(31)

B + MY, p)exp(-Kxlp),

equation

(32)

y and p. We can also eliminate

the space

Y, p),

(33)

- i),

(34)

is

ailcat = K(B becomes I= where

(30)

is

I(& x, p) = k

solution

equation

(29)

xv CL)

where & is an arbitrary function of the two characteristics variable in Eq. (26) in favour of y and write

and the corresponding

of the transfer

y = ct -x/p.

equation

I=

in which case, the transformed

(28)

of y, we may write I(& x, II) = ICY,

and in these variables,

in this

$,,(y, cl) is also an arbitrary

B + &(Y, p)exp(-Kct), function

of y and

CL.The relations

(35) (32) and

(35) are two

Time-dependent

radiation transfer equation-III

419

equivalent forms of the general solution of the transfer equation. It follows from Part I that we can introduce the separation parameters to cast the general solution in the more symmetrical form Z(t, x, cl) = B + W--s)+

(L p)exp(-Kc0

+ H(s)4 (Y,p)exp(-Kxlp)

(36)

where H(s) represents the Heaviside distribution of the parameter s(y, p), which in turn depends only on the two characteristics y and p. Since the two functions, 4,, and &,, are arbitary, it is clear that a relation like (36) is also solution of Eq. (26), where $ (y, p) and 4 (y, p) are some arbitrary functions. The separation parameter is constructed in such a way that it is negative at the initial time and positive at the boundary, so that +(y, p) may be matched onto the IC and 4(y, p) onto the BC. We now want to determine the separation parameter s(y, p) so that BC: s(y, /L) > 0

at

x =x,(t)

for

ct 2 0

(37)

IC: s(y,p)
at

ct =

for

x < x,(t = 0).

(38)

0

Assume that the motion of the plate is

I z = 2, + ut.

At the moving boundary

(39)

we have that the characteristic

is

y = ct - x,(t)lK

(40)

but since x,(t) = z, it follows that y = ct(1 -u/c/J)

- ZJU,

(41)

and consequently ct = W + z. -From the condition (37), a suitable candidate boundary) equivalent to the r.h.s. of Eq. (42). At the initial time, the characteristic is

(42)

at the boundary

y = --x/p. Hence the condition x
is s = ct, where ct is (at the

(43)

is equivalent to -py


= z(r = 0) = zg.

(44)

Consequently 0 < .&I+ py.

(45)

Since the direction cosinep is always negative for the photons emitted by the plate and received by the observer, we have that the choice

s-fly+zO P

(46)

-vlc

is correct for both the BC and the IC. This separation parameter may be interpreted in a simple way. Consider the retarded time given by Eq. (7). Since p,, = -p (11, for the observer, p for the emitter), it follows that, using Eq. (9) ct*=ct

+-

z-x

= ct - 1*.

/J -uIc

(47)

This may also be written ct*=PY+z--t

p-v/c

’

(48)

Remembering now that the plate is moving according to Eq. (39) and comparing with (46), it is

480

ALAIN MUNIER

clear that the separation parameter is s=ct*=~y+ZO p -v/c’

(49)

and is expressible solely in terms of the characteristics y and p. Substituting now into Eq. (36), we now have the general solution of the transfer equation for /A< 0 at, x, P) = B + H(-cl*)+

(Y,p)exp(-Kct)

+ H(ct*)$

(y, p)exp(--Kx/p).

(50)

We can now match to the IC by setting ct = 0 and using Eq. (28) to obtain

@Ax,PL)= E(x, CL)= B + $(Y, P),

(51)

where the bar indicates that p is negative in I and g. Notice that at ct = 0, y = -x/p Eq. (49) ct* =

and from

-x+z, p -v/c

(52)

-=cO3

so that

g(x, P) = B + $(--X/K

P).

(53)

$(Y, P) =g(-YPL, P) - B.

(54)

Consequently

In the same way, the solution is matched to the BC by setting x =x,(t).

In this case

y = ct - X&)/P = ct - (z. + ut)/l*,

(55)

and it follows that ct*=~y+zo p -v/c

(56)

= ct.

Hence from Eq. (27) T[t, x = xl(t),

PI =_k PL)= B + 4(y, p>ev(-&(t)lpl.

(57)

and by using Eq. (56) to express I in terms of y, (58) Substituting into Eq. (50) we finally have the general time-dependent solution to the moving boundary problem (orthogonal displacement) of the transfer in a homogeneous medium for p < 0 E(-YP,

+H(s){f[k

p)-Blexp(--Kct) (s),p]-B}exp[

-p

+K(:‘y/z)].

(59)

This equation can also be written with the help of the retarded time ct* [Eq. (56)] and the retarded distance to the boundary 1* [Eq. (47)]. We have, using also Eq. (29) f(t,x,p)=

B +H(-cct*)[g(x

-pct,p)-B]exp(-Kct)+

H(ct*)u(t*,p)-B]exp(-KKI*).

Fig. 3. Symmetry in the oblique case.

(60)

Time-dependent

radiation transfer equation-III

481

This relation should be compared with the formal solution given by Mihalas and Mihalas3 [their equation (79.35)]. The remarkable thing is that Eq. (60) can be obtained directly from Mihalas and Mihalas3 by setting K = constant and B = constant and replacing their S, by our I*. This is very interesting because it gives us a clue to the method needed for the time-dependent integral form of the inhomogeneous problem (see Sec. 2.3.2). 2.1.4. Integral form of the homogeneous transfer equation (oblique case). In the oblique case, if we assume that the medium admits a symmetry compatible with the obliquity of the boundary, the radiation transfer equation reads

(61) The solution must be invariant under any displacement parallel to the oblique boundary, so that

where (Fig. 3) y,-_ x2 - XI

tana =I

(63)

i’

or x2 -

iY2 =

XI -

iv,.

(W

It follows that we may choose this quantity as a new variable and write x=x-iy

(65)

I(& x, Y, P> = fk

x

(66)

CL).

In this case, the transfer equation (60) becomes ”

-$+p(l--_tan8)g=K(B-1). Comparing this equation with the orthogonal case of Eq. (26) we see that they are equivalent, provided that x is replaced by X = x - cy and ,Hby ji = ~(1 - c tan 0). The process of integration is then very simple since we can use all the results of the preceding section. The new characteristic is

x

x-iv

Y=ct-_(I_itan8)=ct-p

and from Eqs. (18) and (21), the retarded time is (with ,u~= -p) ct*=

Ml -i tanQ+z, P(l - i tan 0) - v/c

(69)

where x = z, is the position of the plate at the initial time t = 0. We simply have to check that this quantity may be used as a separation parameter. At t = 0, we have

x

-lY

X

’ = - ~(1 - [ tan 0) = - (i

(70)

and substituting into Eq. (68) we obtain ct* =

%-(X-c-Y)

~(1 -i

tan0)--u/c

(71)

this quantity is negative provided 1 - c tan 0 is positive, because z,, > x at y = 0 (the plate is always to the right of the observer) and the photons emitted by the plate and received by the observer have a negative p.

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482

At the boundary x = x,(r) = z. + vt + iv and from Eq. (68) ’ =” Consequently,

-p(l

(zo+

vt>

-i

tan@’

substituting into Eq. (69) we have that ct*=ct

>o.

(73)

We may now replace ct* by (69), y by (68) p by fi = p( 1 - c tan 0) and x by X = x - cy into Eq. (50) to obtain for p < 0 i(t,x,~)==++(ct*)~(y,p)exp(-KX/P)+H(-ct*)~(y,P)exp(-Kct). In these new variables, the corresponding

(74)

IC and BC read [from Eqs. (27)-(28)]

BC:Qr, x = iv + 4th y, PI= fP, X = x,(t),PI =f(h IC:Z(t = 0, x,y, p) = i(t = 0, X, fi) =g(x,y,

p) =f(t, F);

(75)

p) =2(X, fi).

(76)

Hence, at t = 0 g(X,fi)=B+$(-x/i&P),

(77)

$(y,G)=$(-Fr,F)-B.

(78)

or In the same way, at the boundary we have that $(y,

fi)

=

[c

fiy+ z. {jl 1 (gqJ’+}exP[ +qJ’:;,p)].

(79)

so that the solution, in the oblique case is

f(t,

x,y,,u) = B + H

+H

1 _f i[$~<~~~e~)J'$],~ tan@}-+ IU II

yp(1 -i tane)+z, {~[-y~(1-~tan8),~(1-~tan8)]-B}exp(-Kct) v/c - ~(1 - c tan 0)

~p(l -i tan@+z, [ tan e) -v/c [ ~(1 -

(x-iv)

~(1 -i

tan e)

-I

zo+Yv/c

+K

~(1-5

tan@-v/c

’

The fact that f(t, x, y, p) is easily expressed in terms of 2 andfinstead offand g [see Eqs. (75)-(76)] implies that the BC and the IC must satisfy the same invariance with respect to a translation parallel to the oblique plane as the transfer equation itself. It would still be possible, although very complicated, to express the intensity Jwith the help offand g by using Eqs. (75)-(76). Notice that, with the provision that p is replaced by fi = ~(1 - 5 tan e), x replaced by X = x - cy, y replaced by its value Eq. (68) and ct* by its definition Eq. (69), then Eqs. (60) and (80) are identical. 2.2. The 1-D homogeneous spherical problem 2.2.1. Spherical symmetry in the homogeneous case. Suppose now that we want to solve the transfer equation in spherical coordinates and symmetry, inside a homogeneous sphere whose radius varies with time. We again assume that the opacity and source terms are constant. More specifically, we treat here the transient regime during the expansion phase of a supernova, the contraction phase being more complicated. The problem can be treated completely if the boundary is motionless (Part II).2 The equation we consider is

= K(B - I),

(81)

where the emission and absorption coefficients are kept constant. We know from Part II [Eq. (41)] that the ‘general time-dependent solution can be written in this case I(& r, CL)= B + H(s)4 (a, y)exp(-Krp)

+ H(-s)$

(a, r)exp(-Kct),

(82)

Time-dependent

where 4 and $ are two arbitrary

radiation transfer equation-III

483

functions of the two characteristics and

Ci=rJm

y=ct-ty

(83)

and where the separation parameter ~(a, y) is a function of the characteristics, and chosen such that it is always positive at the boundary and negative at the initial time t = 0. The conclusion of Part II was that, for a sphere of constant radius R, the separation parameter is for p < 0 s(a, 7) = CC- I = cc *,

(84)

where I is the distance of the point M of coordinate r to the boundary in the direction p. If the radius of the sphere R(t) varies with time, this distance has to be replaced by the distance of the point M to the boundary at the time when the photon was emitted. Hence we have to replace in Eq. (84) the retarded time cc,* by its value in terms of the retarded distance 1* = l*(ct *) = c(t - t*).

(85)

This retarded distance is easily evaluated [see Part II Eq. (12)] and is

Jw.

l(t *) = rp +

(86)

Assuming now that the motion of the boundary is known, hence that R(t) is a known function of t, we can write ct*=ct

-rp

-_R2(t*)-u2

(87)

or c&y-Jz(Fj3.

638)

This implicit equation for t * can be solved in some simple cases, in particular, if the radius varies linearly with time, i.e. if R(t) = R, + vt

(89)

Substituting R (t *) into Eq. (88), squaring and taking the roots of the second order equation in cc*, we obtain the two solutions ct*

=

y

+

zWc 5

,/(Y + @,/c)~- (1 - ~zIc2)(~2 - Ri + a”) 1 - V’IC’

(90)

The minus sign is chosen so that Eq. (90) reduces to (88) with R(t*) = & in the limit v/c + 0 (the plus sign corresponds to an advanced time). It is straightforward to check that at the boundary for r = R(t) = R(t*), this quantity reduces to cc* = cc, implying that I* = 0 as it should. In the same way, at the initial time for cc = 0, it can be shown that ct * = -I*. Hence ct * is positive at the boundary and negative at cc = 0, and can therefore be used as a separation parameter.2 Substituting now the value s = cc * in Eq. (82) we obtain the time dependent solution to the transfer equation in a homogeneous medium for p < 0. We just have now to determine the two arbitrary functions r$ and $ in order to satisfy the BC and IC. Assuming that BC: F[t, r = R(t), ~1 =f(t,

CL),

IC: I(t = 0, r, p) = jj(r, j.i),

(91) (92)

where the bar refers to p < 0. We have from Eq. (82) at cc = 0 g(r, P) = B + 6 (r Jm,

-rp).

(93)

-rl,/‘&?>-B

(94)

Consequently

17;(w)=&/~, and at the boundary

(since in this case t = t *),

?(t*, p) = B + 6 [R(t*)dm,

ct* -,uR(t*)]exp[-KpR(t*)].

(95)

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484

Hence 4 (fx,y) = (Jl{[y -J-]/c,

-Jl

= {f[t *, (ct * - y)lB(t*)l-

- LxZ/R2(t*)) - B)exp[-KJ@?j-S]

B)exp[+K(rp

-

l*>l,

(96) (97)

where t* is given by the smaller of the two roots of Eq. (88) and I* is given by Eq. (86). With these notations, the solution of the equation of transfer for ,u < 0 reads T(t, r, /A)= B + H(ct*){f[t*,

(ct* - y)/R(t*)] - B}exp(--KI*)

+ H(-ct*)[g(,/m,

--y/J=)

- B]exp(-Kct).

(98)

It is clear from Eq. (88) that the retarded time t* may be expressed in terms of the characteristics only, which behave as constants for the transfer equation. Consequently f is a solution of the transfer equation. We now turn to the solution of the transfer equation for p > 0. We know from Part II that the separation parameter in this case is* s = y.

(99)

This separation parameter can be interpreted as the time it takes to a photon with positive p to move along its characteristic from H to M (Fig. 4). If this distance is called 1; and the corresponding retarded time tt, we have Eh*= c(t - tz) = rp;

(100)

hence y = ct,*.

(101)

s=y

(102)

Consequently =ct,*,

may still be interpreted as the retardation from A4 to the boundary. [we remind the reader that the BC for ,U> 0 is precisely given at the point H (see Part II).]* The BC is the value of the intensity at the point H along the same direction. Due to the symmetry of the problem this value is Ih(t, a) = T(t, r = a, p = 0) = B + H(ct,*)(f + H(-ctz)[g(dm,

[t,*, (ct,* - ct)/R(t,*)]--B}exp[-KJv1 -et/,/m)-BB]exp(-Kct)

(103)

and is obtained by replacing in Eq. (98) y by ct and ct* by ct,*, where this last quantity is the solution of ctz = ct - ,/RZ(t,*) -a*.

(104)

We then replace s by its value according to Eq. (102) in the general solution of the transfer equation (82) and evaluate the two arbitrary functions $(a, y) and +(a, y) to match the BC I,,(t, a)

Fig. 4. The spherical problem

Time-dependent at r = a, p = 0 and

radiation transfer equation--III

485

the IC for positive Z.LBy definition IC: Z+(t = 0, r, p) = g +(r, p)

(105)

and at ct = 0, y = -t-p is negative for p positive so that solution (82) reduces to

g’tr, PL)= B + $(rJl -P’, -w).

(106)

Consequently -rldm)-B.

$(@,y)=g+(#?, At the boundary

(107)

r = a and p = 0 so that s = y = ct is positive, Zh(f, a) = B + 4(a, ct),

(108)

$(a, Y) = Z&/c, a) - B.

(109)

and it follows that This value Zj,(y/c, a) is obtained from Eq. (103) by replacing in it ct by y and ct,* by ct*, since in this case the equation for ct,* [Eq. (106)] goes over to the equation for ct * [Eq. (8X)]. We are now able to write down the solution of the time-dependent homogeneous problem with moving boundaries for p > 0 in the form Z’(t, r, p) = B + H(y)H(ct*){f

(ct* - y)/R(t*)]

-[t*,

+ H(y)H(-ct*)[g-(,/m,

- B}exp(-KZ*)

--y/,/m)-

+H(-y)[g+(,/m,

B]exp(-Kct)

-y/J->-

B]exp(-Kct).

(110)

Notice that Eq. (98) for I- and Eq. (110) for I+ can be obtained directly from the general solutions with a fixed boundary [Part II, Eqs. (64) and (77)] by replacing the fixed radius R by its time-dependent value R(t *) at the retarded time t *. 2.3. The I-D inhomogeneous case 2.3.1. Spherical symmetry. We now turn to the general problem of solving the transfer equation (81) when the opacity and the source terms depend on t, r and p. The analytical solution was given for a medium with constant radius [Part II, Eqs. (120) and (123)]. This general solution is still valid when the boundary varies with time, provided that the fixed radius R is replaced everywhere by the moving radius R(t *) at the retarded time t * obtained from Eq. (88). The solution is Z-(t, r, p) = H[y -,/WI

(

f-

i

- J(l

k [y - ,,/-I,

- a*)/R*(t*)

I

I

-sr

BK(r ‘, -)exp

W)

+H(,/--y)

r’ .[sI

g-(,/m,

x exp

‘BK(t’,

-)exp

c [s

K(t”, -) 0

and

+

- y/dm)exp

’ K(t’, +) dt’

-c [

H(y)H[,/p-

y&(,/m,

s

s0 -y/J’=)

V/C

K(r’, +)dr’-c

K(t’, -) dt’

0

OS R.T

38,&-F

K(t’, -)dt’ s0

I’

1

dt”

1I

1

1

dr’

*

[

s0

1>

--c

i

Z’(t, r, p) = H( -y)g+(Jm,

R(r*)

K(r”, -) dr”

-

-y/,/m)exp

+c

K(r”, -) dr’

+

[S

dt’

1 (111)

ALAIN MUNIER

486

-,/-If

+H(y)Hb

E[y -,/‘~I-,/~

-

rK(r’,+)dr’+ 1’ +H(-y)c

’

K(t”, +)dt”

s0 + H(y)H(,/w

- y)c [”

Jo

BK(t’,

II K(r’, -) dr’ sR

1

1

dt’

-)

s 1’

K(t”, -)dt” Y/C

K(r’, +)dr’+c

+ H(Y)

s

’BK(r’, c(

r’ K(r”, +)dr”

+) x exp

- H(y) H [y - J-1

a BK(r’, s WI

dr’

-)

(112)

s r’

[S -

dt’

1

r

x exp

1

K(r”, +) dr” OL

LI

K(r”, -)

dr”

1

dr’.

In these equations, we have defined c

K(t’, +)dt’=c

K(t’,,/or*+(ct’-y)*,

-+,,/l -c?/[cz*+(ct’-y)*]}dt’

s

(113)

and

s

K(r’, +) dr’ =

K[(y & ,/m)/c, s

r’, +“-I

r’ dr’ Jpzg’

with similar replacements for j BK(r’, +) dr’ and J BK(t’, +) dt’. One can prove that this is indeed the solution since the BC and IC are matched in the same way as for the homogeneous case of Eqs. (98) and (1 lo), and that the source term (all that is left when f -,g -,g + are zero) is now evaluated by performing integrals over a radius which is now R(t *). However, due to the definition of the retarded time [Eq. (88)], it is clear that this retarded time depends only on the characteristics y and c1 (and of course the motion of the boundary). Consequently R(t*) is a function of the characteristics and behaves as a constant for the 1.h.s. of the transfer equation. For Eqs. (111) and (112) to be a solution of the transfer equation, the source term must be evaluated by integrating over a constant domain. It follows that Eqs. (111) and (112) are indeed the general time-dependent integral forms of the 1-D spherical transfer equation in a timedependent inhomogeneous nonisotropic medium, submitted to arbitrary IC and BC, with the BC given on a moving surface! 2.3.2. The Cartesian case. The same procedure applies of course, to the Cartesian inhomogeneous case if we want to obtain the integral form to the transfer equation (26) for time-dependent inhomogeneous emission and absorption terms B(t, x) and K(t, x). We simply have to use the general solution (for a fixed boundary) given in Part I and replace the fixed value xlimby x,(t*) evaluated at the retarded time [Eq. (56)]. We have for /Anegative (implying that the boundary to the right is moving)

Time-dependent

radiation

transfer

equation-III

) N(Y

487

+ x’lp)lc, x’, PI dx’

K[(y + x”/,u))/c, x”, p] dx”

K[t’, PW’ - ~1,pl dt’

s I

+c

BK[t’,p(ct’-y),plexp

f’

~[t”,

,u(ct” - y), p] dt”

0

.

(115)

This equation represents the general time-dependent solution of the transfer equation, in 1-D Cartesian coordinates, in a time-dependent inhomogeneous nonisotropic medium, for arbitrary BC and IC. The BC is applied to a moving surface and, in our case always to the right of the observer. A similar solution may also be obtained from p > 0 when the medium is submitted on the left to a moving BC. 3. CONCLUSION We derived the general solution of the 1-D transfer equation in a time-dependent, inhomogeneous, non-scattering medium, submitted to arbitrary BC and IC, for a boundary possibly moving with time. An analytical solution in Cartesian and spherical coordinates was obtained by the repeated use of the theory of characteristics. We defined a separation parameter depending on the time and space characteristics, and behaving like a constant for the 1.h.s. of the equation. This separation parameter was seen to be simply the retarded time to the boundary. In the simple case of a sphere expanding with a constant velocity, it was possible to give the separation parameter in explicit form. In most cases however, one has to solve the implicit Eq. (88). Although the solution we gave here represents some progress in the understanding of transient phenomena in radiation transfer, the model still suffers from two major drawbacks: the absence of scattering and the fact that the emission and absorption are to be given explicitly. In fact, since solving a transfer problem also means solving the energy equation in order to determine the state of the matter involved in the emission, absorption and scattering of the photons, we must substitute the formal expression for the intensity into the r.h.s. of the energy equation. In doing so, integrals over the direction cosine are to be performed, but due to the various Heaviside functions appearing in the integral form of the intensity, the domain of integration for ~1is not longer [ - 1, + l] but is split up into time-dependent sub-domains. As a consequence, in a time-dependent configuration, the energy equation becomes an integro-differential equation lacking many of the numerous properties that Milne’s integral equation has, and to which it reduces in the asymptotic limit. Notice that the spherical solution can be used also for a 1-D cylindrical geometry if we are interested only in the radial propagation of photons. Acknowledgements-The authors thanks his colleagues of the Centre d’Etudes de Limeil and Los Alamos National Laboratory for numerous discussions, and Los Alamos National Laboratory for hospitality and support during a visit when part of this work was done.

REFERENCES 1. A. Munier, 2. A. Munier, 3. D. Mihalas

JQSRT 38, 447 (1987). JQSRT 38, 457 (1987). and B. W. Mihalas, Foundations of Radiation Hydrodynamics. Oxford

University

Press, New York (1984).