MATHEMATICAL PHYSICS

M. V. MASLENNIKOV

THE MILNE PROBLEM WITH AN ARBITRARY INDICATRIX

(Presented by Academician M. V. Keldysh, 13 VIII 1957)

The Milne problem on the propagation of radiation in a plane-parallel scattering and absorbing medium filling the half-space \(z > 0\) of three-dimensional space leads to the integral equation:

\[ \psi(\tau,\mu)=\hat A\psi(\tau,\mu)+F(\tau,\mu), \tag{1} \]

\[ \hat A f(\tau,\mu)=\int_0^\infty e^{-\rho}Y(\tau-\rho\mu)\,d\rho \int g_0(\vec\Omega\vec\Omega')f(\tau-\rho\mu,\mu')\,d\vec\Omega'. \tag{2} \]

Here \(\tau\) is the optical thickness, measured from the boundary of the medium; \(\vec\Omega\) is the unit vector of the direction of propagation of the radiation; \(\mu=\vec\Omega\mathbf{k}\), \(\mu'=\vec\Omega'\mathbf{k}\), where \(\mathbf{k}\) is the unit vector of the axis \(Oz\).

\(\psi(\tau,\mu)\) determines the phase density of radiation at the point \((\tau,\mu)\); \(2\pi g_0(\nu)\) is the probability density of scattering in one elementary act through the angle \(\arccos\nu\), so that

\[ 1-2\pi\int_{-1}^{1} g_0(\nu)\,d\nu \]

is the corresponding probability of absorption. By \(Y(x)\) is denoted the Heaviside function: \(Y(x)=0\) for \(x\le 0\); \(Y(x)=1\) for \(x>0\). The free term \(F(\tau,\mu)\) is determined by the density of radiation sources and by the boundary condition:

\[ F(\tau,\mu)=\int_0^\infty e^{-\rho}Y(\tau-\rho\mu)\eta(\tau-\rho\mu,\mu)\,d\rho +Y(\mu)B(\mu)e^{-\tau/\mu}. \tag{3} \]

\(\eta(\tau,\mu)\) is the phase density of the sources; \(B(\mu)\) is the angular distribution of radiation incident from outside on the boundary of the medium.

If \(g_0(\nu)\) is a polynomial in \(\nu\), then equation (1) can be reduced to an integral equation with respect to the function

\[ \psi_0(\tau)=\int_{-1}^{1}\psi(\tau,\mu)\,d\mu \]

with a kernel depending on the difference, and studied by the Hopf–Wiener method (for example, \((^{1-4})\)). In the general case such a transition is impossible, and (1) with an arbitrary indicatrix, apparently, has not been studied. Moreover, in the general case there is no expression of \(\psi(\tau,\mu)\) through \(\psi_0(\tau)\). It is therefore expedient to study equation (1) directly.

Here the following assumptions on \(g_0(\nu)\) are adopted:

1) \(g_0(\nu)\ge 0;\quad g_0(\nu)\in L_2(-1,1);\quad 2\pi\int_{-1}^{1}g_0(\nu)\,d\nu=\vartheta\in(0,1]\).

2) There exist \(\alpha>0\) and \([\mu_1,\mu_2]\subset(-1,1)\) such that \(g_0(\nu)\ge\alpha\) for almost all \(\nu\in[\mu_1,\mu_2]\).

If we set

\[ g(\mu,\mu')= \int_{\mu\mu'-\sqrt{1-\mu^{2}}\sqrt{1-\mu'^{2}}}^{\mu\mu'+\sqrt{1-\mu^{2}}\sqrt{1-\mu'^{2}}} g_0(\nu)\left(1-\mu^2-\mu'^2-\nu^2+2\nu\mu\mu'\right)^{1/2}\,d\nu, \tag{4} \]

then \(g(\mu,\mu')\in L_2(\Delta)\), \(\Delta=(-1,1)\times(-1,1)\), and

\[ \hat A f(\tau,\mu)=\int_0^\infty e^{-\rho}Y(\tau-\rho\mu)\,d\rho \int_{-1}^1 g(\mu,\mu')f(\tau-\rho\mu,\mu')\,d\mu' . \]

From (4) one can derive the following important property of \(g(\mu,\mu')\):

1. If the interval \([-1,1]\) is divided into two nonintersecting measurable parts \(E_1\) and \(E_2\), and if almost everywhere on \(E_1\times E_2\) \(g(\mu,\mu')=0\), then either the measure of \(E_1\) or the measure of \(E_2\) is equal to zero.

Equation (1) is closely connected with the following characteristic equation:

\[ (1+\lambda\mu)\varphi(\mu)=\hat g\varphi(\mu);\qquad \hat g f(\mu)=\int_{-1}^{1} g(\mu,\mu')f(\mu')\,d\mu';\qquad \lambda\in(-1,1). \tag{5} \]

With the aid of 1), 2), and 1 one can prove that:

1. Equation (5) admits a nontrivial nonnegative solution only for two values of the parameter \(\lambda\): \(\lambda=\pm\lambda_0\), \(\lambda_0\in[0,1)\). To the value \(\lambda=\lambda_0\) (respectively \(\lambda=-\lambda_0\)) there corresponds a unique solution of (5) in \(L_2(-1,1)\), \(\varphi_{\lambda_0}(\mu)\) (respectively \(\varphi_{-\lambda_0}(\mu)\)). The solutions \(\varphi_{\pm\lambda_0}(\mu)\) have the following properties:

a) \(\|\varphi_{\pm\lambda_0}\|=1,\quad \varphi_{-\lambda_0}(\mu)=\varphi_{\lambda_0}(-\mu)\);

b) \(\varphi_{\lambda_0}(\mu)\) is continuous on \([-1,1]\);

\[ \inf_{\mu\in[-1,1]}\varphi_{\lambda_0}(\mu)>0; \]

c) if \(\vartheta=1\), then \(\lambda_0=0\) and \(\varphi_{\lambda_0}(\mu)\equiv \mathrm{const}\); if \(\vartheta<1\), then \(\lambda_0>0\);

d) for \(\vartheta<1\), \((\mu,\varphi_{\lambda_0}^2)<0\) \((x,y)\) denotes the scalar product in \(L_2(-1,1)\).

Let \(k\) be a complex variable. Denote by \(\Gamma(a)\), where \(a\) is a real number, the cut in the \(k\)-plane:

\[ \Gamma(a)=\{k\mid \operatorname{Re} k=k,\ |k|\geq a\}. \]

Put \(g_0^{(1)}=2\pi(\mu,g_0)\).

Then:

1. There exists \(\lambda_1\in(\lambda_0,1]\) such that, whatever the bounded \(\alpha(\mu)\in L_2(-1,1)\), the solution \(\varphi(k,\mu)\in L_2(-1,1)\) of the equation
   \[ (1+k\mu)\varphi(k,\mu)=\hat g\varphi(k,\mu)+\alpha(\mu) \]
   for \(k\in\Gamma(\lambda_1)\), \(k\ne\pm\lambda_0\), exists, is unique, and has the form

\[ \varphi(k,\mu) = -\frac{3}{2k^2}(\alpha,1)\bigl(1-g_0^{(1)}\bigr) + 3\,\frac{(\alpha,1)\mu+(\alpha,\mu)}{2k} + \Phi_{\alpha,g_0}(k,\mu), \quad \text{if } \vartheta=1; \]

\[ \varphi(k,\mu) = \frac{(\alpha,\varphi_{\lambda_0})}{(\mu,\varphi_{\lambda_0}^{2})} \frac{\varphi_{\lambda_0}(\mu)}{k-\lambda_0} + \frac{(\alpha,\varphi_{-\lambda_0})}{(\mu,\varphi_{-\lambda_0}^{2})} \frac{\varphi_{-\lambda_0}(\mu)}{k+\lambda_0} + \Phi_{\alpha,g_0}(k,\mu), \quad \text{if } \vartheta<1, \]

where \(\Phi_{\alpha,g_0}(k,\mu)\) is determined by the functions \(g_0(\mu)\) and \(\alpha(\mu)\) and is an analytic function of \(k\) on the whole \(k\)-plane with the cut \(\Gamma(\lambda_1)\). If \(F\) is a bounded closed set in the \(k\)-plane not containing the points \(\pm\lambda_0\), \(F\cap\Gamma(\lambda_1)=0\), then the solution \(\varphi(k,\mu)\) is bounded on the set \(F\times[-1,1]\) of points \((k,\mu)\).

Denote by \(H\) the set of all points \((\tau,\mu)\), \(\tau\in(-\infty,\infty)\), \(\mu\in[-1,1]\), and by \(H^+\) that part of \(H\) in which \(\tau>0\). Let \(\mathfrak M,\mathfrak M^+\) be the sets of functions defined and measurable on \(H,H^+\), respectively, and let \(L,L^+\) be the parts of \(\mathfrak M,\mathfrak M^+\) consisting of nonnegative functions. Put

\[ \mathfrak M(\delta_1,\delta_2) = \left\{\psi(\tau,\mu)\mid \psi\in\mathfrak M,\ |\psi(\tau,\mu)|\leq \mathrm{const}\cdot Y(-\tau)e^{-\delta_1\tau} + \mathrm{const}\cdot Y(\tau)e^{-\delta_2\tau} \right\}, \]

\[ \mathfrak M^+(\delta) = \left\{\psi(\tau,\mu)\mid \psi\in\mathfrak M^+,\ |\psi(\tau,\mu)|\leq \mathrm{const}\cdot e^{-\delta\tau} \right\}. \]

Then \(\hat A[L]\subset L,\ \hat A[L^+]\subset L^+\) and, if \(\delta>-1,\ \delta_2>-1,\ \delta_1\) is arbitrary, then \(\hat A[\mathfrak M(\delta_1,\delta_2)]\subset \mathfrak M(-1,\delta_2)\) and \(\hat A[\mathfrak M^+(\delta)]\subset \mathfrak M^+(\delta)\). Moreover, \(\hat A\) carries every function from \(\mathfrak M(\delta_1,\delta_2)\) (respectively, \(\mathfrak M^+(\delta)\)) that is continuous jointly in \((\tau,\mu)\) into a function that is again continuous. If \(\chi(\mu)\) is bounded, then, as is easy to compute, for \(\sigma\in(-1,1)\)

\[ \hat A e^{\sigma\tau}\chi(\mu) = e^{\sigma\tau}\frac{\hat g\chi(\mu)}{1+\sigma\mu} - \frac{\hat g\chi(\mu)}{1+\sigma\mu}\,Y(\mu)e^{-\tau/\mu}. \tag{6} \]

Using (6), item 2, and some other properties of \(\hat g\), one can prove the following assertions:

1. If \(\psi(\tau,\mu)\in\mathfrak M^+(\delta)\), \(|\delta|<\lambda_0\), then the series \(\sum_{\nu=0}^{\infty}\hat A^\nu\psi(\tau,\mu)\) converges absolutely and uniformly in every bounded part of \(H^+\) to a function

\[ \hat S\psi(\tau,\mu)=\sum_{\nu=0}^{\infty}\hat A^\nu\psi(\tau,\mu)\in\mathfrak M^+(\delta). \]

1. For every \(\vartheta\in(0,1]\), \(\lim_{n=\infty}\hat A^n 1=0,\ (\tau,\mu)\in H^+\). If \(\vartheta=1\), then the series

\[ \sum_{\nu=0}^{\infty}\hat A^\nu Y(\mu)e^{-\tau/\mu} \]

converges to unity uniformly in every bounded closed part of \(H^+\).

Hence:

1. If in (3) \(B(\mu)\) is bounded, \(\eta(\tau,\mu)\in\mathfrak M^+(\delta)\), \(\delta>-\lambda_0\), then (1) has an everywhere finite solution \(\psi(\tau,\mu)\), representable by the Neumann series: \(\psi(\tau,\mu)=\hat S F(\tau,\mu)\). If \(|\delta|<\lambda_0\), then \(\hat S F(\tau,\mu)\in\mathfrak M^+(\delta)\).

2. For \(\vartheta\in(0,1]\),

\[ \hat S\varphi_{\lambda_0}(-\mu)Y(\mu)e^{-\tau/\mu} = \varphi_{\lambda_0}(-\mu)e^{-\lambda_0\tau}, \qquad (\tau,\mu)\in H^+. \]

With the aid of item 6 it is not difficult to find a nontrivial solution of the homogeneous equation (1):

1. If \(\vartheta\in(0,1)\) (respectively, \(\vartheta=1\)), then the function

\[ \psi_{g_0}(\tau,\mu) = \varphi_{\lambda_0}(\mu)e^{\lambda_0\tau} - \hat S\varphi_{\lambda_0}(\mu)Y(\mu)e^{-\tau/\mu} \]

\[ \left( \text{respectively, }\ \bar\psi_{g_0}(\tau,\mu) = \tau+\frac{1-\mu}{1-g_0^{(1)}} - \hat S\,\frac{1-\mu}{1-g_0^{(1)}}\,Y(\mu)e^{-\tau/\mu} \right) \]

is a continuous, nonnegative, nontrivial solution on \(H^+\) of the homogeneous equation (1); \(\psi_{g_0}\in\mathfrak M^+(-\lambda_0)\); \(\bar\psi_{g_0}\in\mathfrak M^+(-\varepsilon)\), \(\varepsilon>0\).

1. An everywhere finite function \(\psi(\tau,\mu)\) is not a solution of the homogeneous equation (1) if there exist \(\tau_1\ge0,\ A_0>0,\ \lambda>\lambda_0\) such that \(\psi(\tau,\mu)\ge A_0e^{\lambda\tau}\) for \(\tau>\tau_1\). The homogeneous equation (1) has no nontrivial solutions in the classes \(\mathfrak M^+(\delta)\), \(\delta>-\lambda_0\) for \(\vartheta<1\), and \(\delta\ge0\) for \(\vartheta=1\).

Let in (3) \(\eta(\tau,\mu)=0\), and let \(B(\mu)\) be bounded. Then (1) has a unique bounded solution \(\psi(\tau,\mu)=\hat S B(\mu)Y(\mu)e^{-\tau/\mu}\). From (1) it is not difficult to derive that there exists \(\lim_{\tau=0-0}\psi(\tau,\mu)=\psi(0,\mu)\). Put, for \(\tau<0\), \(\psi(\tau,\mu)=0,\ F(\tau,\mu)=-\hat A\psi(\tau,\mu)\). Then \(\psi(\tau,\mu)\in\mathfrak M(-\infty,\lambda_0)\), \(F(\tau,\mu)\in\mathfrak M(-1,1)\), and \(\psi(\tau,\mu)\) satisfies (1) for all \((\tau,\mu)\in H\). Let

\[ \tilde\psi(k,\mu)=\int_{-\infty}^{\infty}e^{k\tau}\psi(\tau,\mu)\,d\tau. \]

From (1) we find that

\[ (1-k\mu)\tilde\psi(k,\mu) = \hat g\,\tilde\psi(k,\mu)+\mu\psi(0,\mu), \qquad \operatorname{Re} k\in(-1,\lambda_0). \tag{7} \]

\(\tilde\psi(k,\mu)\) is analytic in \(k\) for \(\operatorname{Re} k<\lambda_0\). With the aid of item 3, \(\tilde\psi(k,\mu)\) can be

analytically continued to the region of the \(k\)-plane mentioned in item 3.

On the other hand, from (7) it follows that:

1. Uniformly with respect to all \(\mu \in [-1,1]\) and \(k\) satisfying the condition \(\left|\dfrac{\operatorname{Im} k}{\operatorname{Re} k}\right| \geqslant \mathrm{const}>0\),

\[ \lim_{|k|=\infty} \widetilde{\psi}(k,\mu)=0. \]

Hence the applicability of Jordan’s lemma to the integral follows:

\[ \psi(\tau,\mu)=\frac{1}{2\pi i}\int_{k_1-i\infty}^{k_1+i\infty} \widetilde{\psi}(k,\mu)e^{-k\tau}\,dk,\qquad k_1\in(-1,\lambda_0). \]

Deforming and shifting the contour of integration in the proper way, with the aid of the formulas of item 3 we find:

1. If \(\vartheta\in(0,1)\) (respectively, \(\vartheta=1\)), then in the problem under consideration

\[ \psi(\tau,\mu)= \frac{(\mu\varphi_{-\lambda_0}(\mu),\psi(0,\mu))} {(\mu,\varphi_{-\lambda_0}^{\,2})} e^{-\lambda_0\tau}\varphi_{-\lambda_0}(\mu) +o(e^{-\lambda_2\tau}),\qquad \lambda_2\in(\lambda_0,\lambda_1) \]

\[ \text{(respectively, }\psi(\tau,\mu)= \frac{3}{2}(\mu^2,\psi(0,\mu))+o(e^{-\lambda_2\tau}),\quad \lambda_2\in(0,\lambda_1)\text{).} \]

In addition, from (7) it is not difficult to obtain theorems on the vanishing of the total flux, on the constancy of the \(K\)-integral, and their generalizations to the nonconservative case (for example, \((\mu\varphi_{\lambda_0}(\mu),\psi(\tau,\mu))=0\) for \(\tau>0\)).

The arguments carried out are also applicable to the homogeneous equation (1). Here the following results are obtained:

1. If \(\psi(\tau,\mu)\geqslant 0\) is a nontrivial solution of the homogeneous equation (1) and \(\psi(\tau,\mu)\in\mathfrak{M}^{+}(\sigma)\), \(\sigma>-1\), then for \(\vartheta\in(0,1)\) (respectively, \(\vartheta=1\)) \(\psi(\tau,\mu)\), up to normalization, coincides with \(\psi_{g_0}(\tau,\mu)\) (respectively, with \(\overline{\psi}_{g_0}(\tau,\mu)\)).

Assertions 9 and 12 form a uniqueness theorem for the solution of the homogeneous equation (1).

\[ \psi_{g_0}(\tau,\mu)= \varphi_{\lambda_0}(\mu)e^{\lambda_0\tau} - \frac{(\mu\varphi_{-\lambda_0}(\mu),\psi_{g_0}(0,\mu))} {(\mu\varphi_{\lambda_0}(\mu),\psi_{g_0}(0,\mu))} e^{-\lambda_0\tau}\varphi_{-\lambda_0}(\mu) +o(e^{-\lambda_2\tau}), \]

\[ \lambda_2\in(\lambda_0,\lambda_1); \]

\[ \overline{\psi}_{g_0}(\tau,\mu)= \tau-\frac{\mu}{1-g_0^{(1)}}+ \frac{3}{2}(\mu^2,\overline{\psi}_{g_0}(0,\mu)) +o(e^{-\lambda_2\tau}),\qquad \lambda_2\in(0,\lambda_1). \]

From this it is easy to obtain certain other relations (for example, formulas for the spatial distribution of radiation, extrapolated lengths, etc.).

I express my gratitude to E. S. Kuznetsov and T. A. Germogenova for discussing the work.

Department of Applied Mathematics
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
8 VIII 1957

REFERENCES

1. E. Hopf, Mathematical Problems of Radiative Equilibrium, Cambridge Tracts, No. 31, 1934.
2. V. Kourganoff, Basic Methods in Transfer Problems, Oxford, 1952.
3. J. Kuščer, J. Math. and Phys., 34, 256 (1956).
4. P. Bouvier, Arch. d. Sci., 8, No. 1, 87 (1955).