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NOTE THE LAGRANGIAN FORMALISM IN THE RADIATIVE TRANSFER THEORY R. A. KRIKORIAN”?

and A. G. NIKOGHOSSIAN”,’

de France, Institut d’Astrophysique, 98bis Bd. Arago, 75014 Paris, France, “Byurakan Astrophysical Observatory, 378433, Armenia, and ‘Observatoire de Paris-Meudon, CNRS-URA 326, F-92195, Meudon Cedex, France Collkge

(Received

29 November

1995)

Abstract-The paper is devoted to the application of the variational approach to the radiative transfer problems in the one-dimensional case. To this end the results of the rigorous mathematical theory are used. It is shown that the conservation law due to the form-invariance of Lagrangian with respect to the axes translation transformation entails both the Ambartsumian’s invariance principle and Rybicki’s quadratic relations. Copyright 0 1996 Elsevier Science Ltd

1. INTRODUCTION

The existence of a variational principle for a differential equation or set of differential equations presents a number of theoretical advantages. In particular, it permits to establish a systemic

connection between symmetries and conservation laws. Moreover, in some cases the constructed functionals contain more fundamental information than the corresponding Euler-Lagrange equations. Within the framework of modern functional analysis, the problem of the existence of a variational principle for systems of partial differential equations was solved by Vainberg,’ who showed that this problem is equivalent to determining whether an operator is potential or not. Tonti,’ recognising the importance of Vainberg’s work developed a procedure to determine whether a given operator is self-adjoint. The conditions for the existence of a potential are very stringent. For instance, it is proven (a) that a scalar differential equation whose highest derivative is of odd order, has no potential; (b) for a system of differential equations, the kth equation must have an even derivative with respect to the kth dependent variable. 3.4Furthermore, the fact that the given system is or is not a system of Euler-Lagrange equations may depend on the labeling of the equations. Although the variational approach is widely used in various branches of theoretical physics, it is not the case in the field of radiative transfer theory. To our knowledge, the only exception is a paper of Anderson’ where a variational principle is applied to the integro-differential equation of radiative transfer and, by using a generalisation of Noether’s theorem, due to Tave1,6 conserved quantities are derived. However, some important points of modern functional analysis have been overlooked in this work, as one may infer from the author’s following statement: “The solution of the problem of finding the Lagrangian densities is a question of qualified guessing, as no general method is available for establishing the Lagrangian corresponding to a given equation.” The purpose of our paper is therefore to derive a variational principle for the simplest case of the one-dimensional transfer problem, using Vainberg’s and Tonti’s results.‘.? It will be shown that the conservation laws obtained by applying Noether’s theorem corresponds to the quadratic relations of Rybicki’ for the one-dimensional problem. tTo whom all correspondence

should be addressed. 465

Note

466

2. THE CONSERVATION FOR ONE-DIMENSIONAL

TRANSFER PROBLEM

In this section we shall apply the Lagrangian formalism to the problem of radiative transfer in the one-dimensional medium (two-stream approximation), our purpose being the derivation of appropriate conservation laws. We start with the simplest case of isotropic and monochromatic scattering. Consider the Sobolev’s function p(z) which possesses the following probabilistic meaning: it gives the probability density of the random event consisting in that the photon absorbed at the optical depth 7 will escape (generally, as a result of multiple scattering) the atmosphere. The function p(7) has been widely used in8 and at the present time is well-known in, the radiative transfer theory. Particularly, for the problem at hand it satisfies the following equation N[p] = p,, - (1 - I)p(z) = 0

(1)

where 1 is the single scattering albedo and, for convenience, the operator notation N[p] is introduced. It is known from functional analysis 2.3that N[p] = 0 is the Euler-Lagrange equation of the functional F p(7) dr

F =

s

’ N[op(z)] da

(2)

s0

(where r~is a scalar variable and J dr represents integration over the physical domain) if and only if the Frechet derivative of the operator N[p] is symmetrical, i.e.

s

$(7)X

b(7)l

d7 =

cp(7)K W(7)l d7 s

where A$[cp] = hi {N[p + E(P]- iV[p]}/E= {dlV[p + ccP]/&}<=o.

(4)

It can be easily verified that the operator N[p] given by Eq. (1) satisfies condition (3) and hence the Lagrangian followed from (2) is J%Jl = (1/2)bZ + (1 - A)$l.

(5)

As this Lagrangian does not depend explicitly on 7, we have in accordance with the Noether’s theorem L - pf(lJL/+,) = const

(6)

which, being applied to the Lagrangian (5), yields pf - (1 - A)p2 = const.

As it is often the case, this conservation Eq. (1).

(7)

law also may be obtained immediately from the initial

3. CONNECTION WITH THE INVARIANCE PRINCIPLE AND QUADRATIC RELATIONS For the futher discussion, it is expedient to introduce quantities P+(7) and P-(z) which character&e the probability of the photon exit from the boundary 7 = 0 of an atmosphere, if originally it was moving at depth 7 in outward (i.e. to the boundary 7 = 0) and inward directions, respectively. In view of the probabilistic meaning of functions P*(7) one can obviously infer that P(7)

= ww+(7)

+

P-(7)1.

(8)

The functions Z’*(7) are related with one another through the reflection coefficient p as follows P-(7)

= P’(7)p

(9)

Note

467

with p = pm for the semi-infinite atmosphere, and p = p(t,, - z) for finite atmosphere of optical thickness ro. The transfer equations, determining P*(t) are (dP+(r)/dz) - (dP-(r)/dr)

= ‘P’(r)

+ 1/2[P+(z) + P-(z)]

= -P-(t) + (d/z)[Z'+(z) f P-(7)+ P-(r)],

(10)-

@)I.

(10)

It is evident that P+(O) = 1 and P-(O) = pm or ~(7~)dependent on that a semi-infinite or finite atmosphere is treated. Now, in virtue of Eqs. (10) and (8) we have (dp(r)/dt)

= -(1/2)[P+(z)

Finally, with use of Eqs. (11) and (8), the conservation

- P-(z)].

(11)

law (7) may be rewriten in the form

I[P+(z)+ P-(z)]* - 4P+(z)P-(7)= const.

(12)

This equation is the one-dimensional analog of the first fundamental relation obtained by Nikoghossian9 for three-dimensional semi-infinite atmosphere. The fundamental nature of this relation is due to the fact that it generates all the quadratic relations referred to as the Q-integrals of transfer problems.’ Note that in the one-dimensional case Q- and R-integrals coincide. In order to obtain the values of constant in Eq. (12), one may utilise Eq. (9) for finite atmosphere and take 7 = to,bearing in mind that p(O) = 0. As a result, the constant’s value obtained is l[P+(ro)]* so that finally Eq. (12) takes a form 1[P+(r,70)+ P_(?,70)12 - 4P+(z,zo)P-(t,

70) = A[P+(7,70)]2

(13)

where the dependence on optical thickness is indicated explicitly. The quantity q = P+(70,zo)is the transmission coefficient of the atmosphere given by the formula’.” q = (1 - p&)e-kro/(l - pf, edxro)

where k = dm.

(14)

For the semi-infinite atmosphere q = 0 and the conservation law takes a form (15)

A[P+(z)+ P-(z)]* - 4P+(z)P-(r)= 0.

Letting 7 = 0 in Eqs. (13) and (IS), we are led to the Ambartsumian’s equations for finite and semi-finite atmospheres, respectively:

well-known invariance

A[1 + p(zo)12- 4p(70) = &7*(70), A(1 + pxy - 4p,

= 0.

(16) (17)

Thus, it is seen that conservation laws (13) and (15) we obtained are constructed by the similar manner as the corresponding invariance relations, but have more general meaning. Therefore these conservation equations can be regarded as the natural generalisation of the invariance relations to all depths in the atmosphere. Now we shall see that the conservation law is closely connected with the quadratic relations derived by Rybicki’ and Nikoghossian.’ For simplicity, the attention will be confined to the semi-infinite atmosphere, while the generalisation of results to the case of finite medium is straightforward. Consider at first the problem of diffuse reflection from a semi-infinite atmosphere illuminated from outside by radiation of unit intensity. Accounting for the reciprocity principle of optical phenomena, one may infer that the quantities P+(z) and P-(r)can be identified in this problem with the intensities of ingoing and outgoing radiation Z-(7)and Z+(7), respectively. Then Eq. (15) yields Z+(t)Z-(7) = P(7)

or

(1 - A)Z+(z)Z-(7) = AH?(t)

(18)

where J(7)= [Z+(7) + Z-(r)]/2 and H(r) = [Z+(7) - I-(t)]/2. The Eqs. (18) are written in the form of Q- and R-relations obtained by Nikoghossian9 in the three-dimensional case. Now let us turn to the classical transfer problem with distributed sources when the source function S(7) is S(7)= AZ(z)+ (1 - 1)B

(19)

468

Note

where B(7) is the function defining the distribution of radiation sources. We assume that the sources are due to thermal emission and that B(7) is the Planck function. Here we consider the case when B = const. As it was shown in Ref. 11, the solution of this problem is simply related to that for the diffuse reflection. One can write Z*(7) = 1 - [Z*(r)/B]

(20)

where the quantities relevant to the diffuse reflection problem are supplied by asterisk. Now in view of Eqs. (18) the derivation of appropriate quadratic relation for the problem under consideration, is straightforward. Inserting Eq. (20) into Eq. (18), we obtain S’(7) = AZ+(z)Z-(7) + (1 - I)B*.

(21)

For the first time, this equation was obtained in Ref. 7. The intriguing nature of this equation, as was pointed out by Rybicki, is its similarity with Eq. (19) which admits a simple physical interpretation, while the physical meaning of Eq. (21) remained obscure until recently. This notion is equally valid also for other quadratic relations. 4. CONCLUSIVE REMARKS In addition to Eqs. (18) and (19), other quadratic relations relevant to the more general laws of distributed sources, can be found on the basis of Eq. (12). This matter for the three-dimensional case is discussed in Ref. 9 at length and this is not the main objective of the present paper. We aimed to develop the Lagrangian approach to the radiative transfer problems, using the rigorous mathematical theory developed by Vainberg,’ Tonti* and others. The results obtained show the profound connection, existing between conservation law, invariance principle and quadratic relations that ensue from the form-invariance of Lagrangian with respect to the axes translation transformation. From this point of view, the conservation law we derived is the analog of the momentum conservation law in mechanics. Note also that the conservation laws of such simple scalar form we obtained, do not always exist as such. For instance, the time-dependent problem leads to the conservation laws of much more complex form (see the Appendix). It should be emphasised that the results of this paper can be generalised to include the three-dimensional case as well as the bilinear relations. These issues will be the matter of further work. Acknowledgemenrs-The authors are grateful to Dr G. B. Rybicki who pointed out Anderson’s paper. This work was completed while one of the authors (A. G. Nikoghossian) was at Laboratory DASOP of Meudon Observatory in the frameworks of the program PICS of CNRS.

REFERENCES 1. M. M. Vainberg, Variational Methodsfor the Study of Non-Linear Operators, Holden Day, San Francisco (1964). 2. E. Tonti, Acad. Roy. Belg. 55, 137; ibid. 55, 262 (1969). 3. R. W. Atherton and G. M. Homsy, Studies in Applied Math. LIV, 31 (1975). 4. F. Bampi and A. Moro, J. Math. Phys. 23, 2312 (1982).

5. D. Anderson, J. Inst. Maths. Applies. 12, 55 (1973). 6. M. Tavel, Transport Theory Statist. Phys. 1, 271 (1971). 7. G. B. Rybicki, Astrophys. J. 213, 165 (1977). 8. V. V. Sobolev, A Treatise on Radiative Transfer, van Nostrand, Princeton (1963). 9. A. G. Nikoghossian, Astrophys. J. (1995). 10. V. A. Ambartsumian, ScientiJic Works, Vol. 1, Izd. Acad. Nauk Arm. SSR, Yerevan (in Russian) (1960). 11. A. G. Nikoghossian and H. A. Haruthyunian, Astrophys. Sp. Sci. 64, 285 (1979). APPENDIX In the case of the one-dimensional time-dependent problem the photon escape probability p(s, u) (u is the dimensionless time), for low densities, satisfies the following partial differential equation8 N(p) = pm,- p,, + (2 - lip. + (I - njp = 0.

(Al)

Although Eq. (Al) is of second order, it does not satisfy the symmetry condition (13). Nevertheless a variational principle may be obtained by introducing the weighting factor e’2- 2)u.The resulting Lagrangian may be written in the form

L[Pl= (1/2)W + (If - (W4)Vl

(AZ)

Note

469

$(T, u) = e’2- ““2p(s,u).

(A3)

where

The Lagrangian (A2) is form-invariant

with respect to the following transformations 5’ = 7 + 6, r’ = r,

u’ = u;

(A4a)

Ii = lA+ c;

(A4b)

7l = r - UC, u, = u- 5C. The corresponding

(DI~){u[$:

(A4c)

conservation laws are (D/Dr)[Sf + $: + (P/4)@ - 2(D/Du)$&

= 0,

(WW[#f

= 0,

+ tit - (~‘/4)+7 - 2(D,‘Dr)$&

+ tit + (~*/4)~3} - (WW{W

+ $3 - (W4)ti’l) - 2(DIDu)(u&$.) - 2(DIDr)(rW.)

(ASa) (AW = 0

(A5c)

where (D/DA+) = (d/W)

+ (a/@)&

+ (a/@&

(i, k = I, 2)

with tir = (d/ax*),

**, =

(ay/aXkaXq.

The Lagrangian (A2) and the corresponding conservation laws (AS) are similar to those found in the theory of the scalar meson tield. In this case transformation in time and transformation of the axes yield respectively the conservation of energy and of momentum.

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