THREE SOLUBLE PROBLEMS IN LINEAR TRANSPORT THEORY* ~ A: r \ I "-",. Ro E. Aamodt Department 6f Physics The University of Michigan Ann Arbor, Michigan August 1962

~- -,, LA. M 0,11.

ACKNOWLEDGMENTS I would like to gratefully acknowledge the continuous instruction and helpful suggestions of Professor Kenneth Case throughout my entire graduate program. I also thank Dr. Francis Harlow for his inspiration, interest and personal concern during the course of this work. I gratefully acknowledge the financial support of the Eastman Kodak Fellowship for the 1960-1961 school year, and the Office of Naval Research and Conductron Corporation during the year, 1961-1962. I humbly dedicate this work to my wife, Donna, without whose love, understanding and silent sacrifice this work would never have been completed. My only hope is that some day these past years will prove worthwhile to her. ii

TABLE OF CONTENTS LIST OF TABLES ABSTRACT CHAPTER I, INTRODUCTION II. THE NEUTRON TRANSPORT EQUATION WITH r-1 CROSS SECTIONS Ao Preliminary Remarks Bo Solutions for the Infinite Medium C. Diffusion Solution for the Shell Source Do Finite Sphere Problem E, Conclusion III, USEFUL IDENITITIES FOR HALF-SPACE PROBLEMS IN LINEAR TRANSPORT THEORY A. Preliminary Remarks Bo Solutions of Equation (III-1) C. Completeness Theorem D. Applications to Half-Space Problems E, Identities for X(z) Fo An Application G. Conclusion IV, DENSITY IN A SIMPLE MODEL OF THE EXOSPHERE A. Discussion Bo Construction of the Solution C, Particle Density D. Conclusion Page iv v I 3 3 6 9 11 17 19 19 19 25 33 37 39 40 41 41 42 45 46 APPENDIX A. B. C. D. REFERENCES FOOTNOTE DERIVATION OF THE I TEGRAL EQUATION (II-2) PROPERTIES OF THE TRANSFORMED KEMEL, K(z) CALCULATION OF THE EXTRAPOLATED RADIUS A USEFUL REPRESENTATION FOR X(z) 47 49 52 55 56 58 iii

LIST OF TABLES Table Page I Cmin9 II PD/Pas) C= 1 11 III ti and t2 28 iv

ABSTRACT In this study three exactly soluble problems in linear transport theory are considered. In the first problem, the one velocity, time independent, isotropic scattering, neutron transport equation with r-1 cross sections is solved exactly by two-sided Laplace transforms. The homogeneous solutions and the Green's functions are exhibited for both the infinite medium and the finite sphere. The diffusion approximation is compared to the exact result. In the second problem, a typical translationally invariant linear transport problem for a half-space is solved by the eigenfunction method. Using some derived identities for the solution of the associated homogeneous Hilbert problem it is shown that the usual cumbersome integrals occurring in half-space problems can be reduced to a simple form. For the third problem the particle density of a simple model of the exosphere is obtained by solving exactly the collisionless Boltzmann equation. The main point of the solution is that it is a discontinuous, multivalued function of the constants of motion. Results, of course, agree with those of other methods based on Newtonian mechanics. v

!CHAPTER I INTRODUCTION The complexity of the integro-differential equations occurring in linear transport phenomena permit few problems to be exactly solubleo However, as has been suggested by Wigner (1), use of invariance properties permit not only simplification of the equation, but may in fact allow complete solutions to be found. The first two problems in this theais are problems which do have the "complete" symmetry necessary for solubility. The first problem that is considered is the one velocity, time independent, isotropic scattering, neutron transport equations with cross sections which vary as r-o. The invariance of this equation under the transformation r + ar, a > 0 suggests (1) simplicity of the equation in spherical coordinates and the use of Mellin transforms, but for convenience the usual transformation is made so that Laplace transforms are applicable0 Solutions to the infinite and finite sphere problems are then shown to be expressible in quaadrature and. are compared. to the usual diffusion approximation. The second problem considered is a typical one dimesnsional translationally invariant transport equation. Instead, of exploiting the full symmetry of the problem we solve the two independent variables equation by the usual eigenfunction technique*, The completeness of these eigenfunetions is then used to solve some typical half-space problems, Several iden.tities derived for the solution of th'e associate homogenteous Tslbert problem aer used to

2 simplify the resulting cumbersome integrals. In the third problem, a simple model of an exospherethe linear collisionless Boltzmann equation for particles in an external gravitational field is solved exactly by using the method of characteristics. The solution s shown to be a discontinuous, multivalued function of the constants of motion.

CHAPLTER -I THE NIEUTRON TRANSPORT EQUA TION WITH r —1 CROSS SECTI1ONS A. PRELIMINARY REMARKS In this chapter the one velocity, time independent, isotropic scattering neutron transport equation with cross sections which vary as r-1 is solved by Laplace transform techniqueso While the r-1 dependence of the cross sections makes this model mainly an exercise in mathematical physics; with little modification the results can be applied to electron transport in velocity space with cross sections which vary as v-2. 'The time independent integral equation for the stationary neutron density, p(r), in the one velocity approximation can be written: p(r) = ' d- 'e __ _.~') + ~ (I-l- ) Where S(r) is the isotropic source density, C(r) is the average number of 00 secondaries emitted after a collsion occurs at r j and, aB') is the probabil.ity that a neutron suffer a collision at r for smal1 ro For -the:modeel being considered C( = C, ~(r) = r ~Zr C and a being dimensionless constants. Using the spherical symmetry of the proglem the homogeneous equation for an infinite medi~i can be reduced to- (see Appendix A),.~(9g) = Co /o dq,,(e) K.( io 9-!) (ll-2a) 3

4 (0G) = rp(r) 1 + aeI2G (II-2b) 1 and K(g) = f d - x a (II-2c) x 1 + x tanh G (a is an arbitrary constant with length dimensions). As two-sided Laplace transforms will be used exclusively in this chapter, the notation and analyticity properties will be standardized and exhibited now. The transform of the kernel, 00 K(z) = e-ZG K(11e) dG (II-3) -00 has the following representation: (see Appendix B) K(z) = i ( ) (ii-4) din for -a < Rez <, = 2(a + 1), and ) (z) (z) is analytic in in the strip -a < Re z < a, and the analytic continuation of K(z) defined by (II-4) is analytic everywhere except for simple poles at z = +(4m + a), m = 0, 1, 2,.... Clearly K(z) is a real, even function of z. The zeros of 1 - Ca K(z) for -a < Re z < a, and its analytic continuation for IRe zi > a are needed for inverting the transforms. These are discussed in Appendix B and can be summarized as follows: (a) if (oC)" > K(0), 1 - Ca K(z) has two real zeros, ~Zo, 0 < zo < a, for -a < Re z < a; (b) if (aC)-l< K(O), 1 - Ca K(z) = 0 has two imaginary roots, ~Zo, for -a < Re z < a;

5 (c) if (aC)1 = K(0), 1 - Ca K(z) = 0 has a double root at z = zo = 0 (d) and for any a, C O0, there are an infinite number of real roots, ~Zn, n = 1,2,53,o. to 1 - Co K(z) = 0, zn > 0, where zn lies in the interval 4(n-l) + a < Zn < 4n + a. As a convenient reference point, we note that the representation of K(z) given by (B-2) simply gives K(2) = 1/a. Therefore, if C > 1, zo < 2, and if C < 1, zo > 2. As K(z) is analytic in the strip -a < Re z < a, the use of Laplace transforms requires that we look for solutions such that Lim |In(0) I < A el9! (II-5) 1Q+oo where p is real and less than a. Then 00 0 f-t( z) - -ze l(8)d and 0 f_(z) = e" (()dG (II-6b) -00 will be analytic for Re z > P, and Re z < -P, respectively (2). Moreover the inversion theorem gives us: c-e+iX for 9 > 0, 7(~) = lim 2-i f+(z) eZ~dz (II-7a) X+oo a - -iX and for G < 0,

6 I(~) = Lim 1 -Oa+e +ix f-(z) eZ, E > 0. -a+ -i\ (II-7b) B. SOLUTIONS FOR THE INFINITE MEDIUM In the usual way (2) the homogeneous equation for the infinite medium, Eq. (11-2), defines an analytic continuation of f+(z) and f-(z) in transform space such that f+(z)[l - Cof(z)] = - f(z) [1 - CK(z)] (11-8) for -a < Re z < a, and both sides are analytic in this strip. Hence, in this strip f+(z) = -f_(z) except at the zeros, ~+z, of 1 - CaK(z), and f.(z) is analytic except for possible poles at z = ~Z0. Using the inversion theorem, (II-7), and converting back to r and p(r), we find the general solution of the homogeneous equation is given by: p(r) = Al () zo/21 \a + B(0Zo/2-1, if (co)-' K (O) (II-9a) and p(r) =. A2(a) + B2(i) 2n |, if (Cci in r, if (Ca)- = K(O). (II-9b) For the infinite medium Green's function or shell source solution we introduce into Eq.. (II-1) the source: (r - ro) S(r;ro) = (43)2 r- 2 (4it)2 ro2 ' (II-10) Using the same substitutions as in the homogeneous equation, the integral

7 equation for the shell source solution can be written as: 00 T](o;e) = Co - ' ne '(@oo) K(JG - '|) + K( - Gl) _o (II-l) r = ae-2Go % = 8jvr,. In transform space, this equation again defines an analytic continuation of f+(z), f_(z), such that -Z"o K(z) f+(z)[l - oK(z)] = -f_(z)[l - CoK(z)] + e K(-z (II-12) for -a < Re z < a, and both sides are analytic in this region. Inverting the transform, we obtain the general solution to (II-11): 7+iOO z(9-o ) (9;9o) = A1 eZo + B o9 eZoG + 1 e (z)dz, (II-15) 7 x -ioo 1 - CK(z) For zo < y' < c, (Ca)'" K(0O). To determine A1 and B1 boundary conditions must be applied. If we require rp(r;r ) to finite as r-O and r-oo Eq. (II-13) becomes (for (aC)'- > K(0)): z(e-eo), ^Q(G;00).= 1 i _z( iei (I.-14) -00o 1 - CK(z) For G > 0o (G < GO) the contour integral. can be completed in the left (right) half plane. In terms of the density, Eq. (EI-14) is: p(r;r;) C) = K (z,) i3 > r <znJ (I- r 5a) X0r(Gc)2 I K(zn) ao n=O and

8 <ro Z \ /2 p(r;~ro) = 90 — n -rfor r > r. (II-15b) Xr(Cc) 2 K '(zn) rI 0 n=O In (II-15) we have required (aC)-' > K(O), and used the property ' (z) = -K'(z), K'(z) = d/dz K(z). The boundary condition that we have used is that the total number of collisions per unit radial distance, 4tr2a(r) p(r;ro), be finite at the origin and infinity. If instead of this condition we had required that p(r;ro) be everywhere non-negative and vanish as r -+ o, for all C, then we would have found the same solutions as Eq. (II-15). For (Ca)-l < K(O) the boundary conditions that rp(r;ro) be finite at the origin and infinity are not enough to uniquely determine the solution. In order to specify a unique solution one would need some kind of a "radiation condition." However, it is evident that there are no solutions for (cC)-1 < K() for which the density is everywhere nonnegative. Hence, for C > Cmin = l/o(0) there are no physically acceptable stationary solutions to the transport equation. This is normally referred to as the "criticality" condition. That is, if C > Cmin the sphere is "super-critical" and if C < Cmin the sphere is "sub-critical." In Table I we have calculated Cmin for various values of a. It is interesting to note that unlike the constant cross section case, which has a spectrum consisting of two discrete eigenvalues and a continuum on (Loo) and (-,oo), (3), the r-1 transport equation has a purely discrete unbounded spectrum.

9 TABLE I Cmin a Cmin aC min.25 2.555 2.5 1o050.50 1,571 3.0 1,032.75 1o328 4.0 1.024 1 1.218 5.0 1. 018 1.2 1.171 o~> 1+(3ac2) 1.5 1.118 1.7 1o092 2.0 1.071 C. DIFFUSION SOLUTION FOR THE SHELL SOURCE For most realizable systems (e.g., a reactor) the transport equation is too complex to be solved by analytical techniques; one normally has to resort to a diffusion calculation. It is interesting therefore to see just how the diffusion solution approximates the transport solution for the model that we are considering. The diffusion equation for the r~1 cross sections with a shell source can be derived by the usual methods. The equation is: ~ a a(r - ro) V j + - P(rjro) = Xv, '(II-16a) r PD 4,4vr2 J' 3 V PD (r;r) (II-16b) and aa = a(l-C). In spherical coordinates, and using the substitution x = an ro/r, Eq. (II-16a) becomes: 2d -3r 6(x) +-2 PD() = -(II17) dx2 dx 4-.Cvro

10 where a - 35 oa/Co Solving this equation by Fourier transforms, and requiring that rPD(r;ro) be finite at the origin and infinity, we find: PD (r;ro) -= - (ii, r < r (II-18a) *u C~r ^+0^r0 PD(r;r~) = C3ar ( r-)( C~~~~~r r~~~-fg r >r o (II-18b) It is interesting to everwhere non-negative if note that the solutions a > -1. This condition given by Eq. (II-18) are is just 1 1 1 32< 1 -.1 3a2 (11-19) which is the "sub-criticality" condition for the transport equation when For C < Cmin the asymptotic transport solutions given by Eq. (If-15) are: pas(r;ro) Pas(r:to) grNzo/2 Xr(Co)2 K'(zo) r2 I r z/2 Xr(Co)2 K'(z) K r Ar cc) 1"" I O (II-20a) (IE-20b) In the limit of no absorption, C = 1, zo = 2, a = 0, the asymptotic transport solutions have the same dependence on r as the diffusion solutions. For a>>l Eq, (B-2) gives that K'(2) 1/353, and in this limit (C = 1) the asymptotic solutions become exactly the diffusion solutions. Similarly,

11 if aL<, w'e find K'(2) 2 -, and in this limit we see that the diffusion solution is smaller than the asymptotic solutions by a factor of 3a/2. For C = 1 we have several ratios of p/poas listed in Table II. TABLE II ppas) c = 1 3a c3 K (2) = - = 1 <<1 3a/2 1.70 2.86 3.93 4.96 >> 1... 0 From Table II it is evident that the diffusion approximation is reliable for 0 = 1, and a. 35 If C 1, the:root z,; becomes approximatel.y zo 3aaa 6 1 ' 2 20 \5a3 (K )21) if aa/2a2 CK'(2) <<1. Likewise, for a<<l, the exponent of r in the diffusion solution becomes approximately 1 + 3oaa/2Ce Therefore, if 3o3K'(2) 1, (i.eo. cr 4), and |1 - C|, the diffusion solution gives an excellent asymptotic representation to the neutron densityo Do FINITE SPHERE PROBLEM The equation for the neutron density in a finite sphere of radius "a" surrounded by a vacuum can be derived from (11-1). It is.

12 a 1 rp(r) - C dr1p(rI) -x 1 - x ' (II-22) o Ir-r'I L j r+r for r < a. Using the same substitutions as used to derive Eq. (II-2), Eq. (II-22) becomes: 00 rj(9) C= a dG' r(G') K( | - 9'1), for 9 > 0. (II-25) 0 This equation can be solved by the Wiener-Hopf method (2). Defining an r(G9) for 9 < 0 by the right hand side of Eq. (II-23) and using Laplace transforms as before, we obtain: f+(z) [1 - Ca (z)] = - f_(z) (II-24) for.- < Re z < (X as 1 - Co K(z) has two zeros at z = +z in this strip, we look for a Wiener-Hopf decomposition of the form. X_(z) 2 2 1 - CK(z) X (z) z) (I -z-2) such that X_(z) is analytic and non-zero for Re z < p, z finite, and X+(z) is analytic and non-zero for Re z > -B, z finite. For this purpose, consider: ( [- Ca K(z)].[z2 2] T(z) (II-26) z Z0 We note: (a) T(z) is analytic and non-zero for -cz < Re z < o. (b) T(z) is a real even function of z.

13 (c) Lim T(z) = 1 in the strip. -+oo Decomposing a C(z) T(z) -= r+( Z) (I1-27) where r_(z), c+(z) have the same properties as X_(z), X+(z) respectively, the unique solution to Eq. (II-27) is given by Cauchy's theorem as: a_(z) = e O+(z) = e -r (z) (II-28a) -r+(z) (II-28b) where for -% < Re z < X X+ico r (z) = 1 z) = 1 I" 3n T(z')d z' - 2r2i2 z' - z X-ioo -X+ico r+(z) 1 f n T(z')dz' 2i z' - z -X-io (II-29a) (II-29b) and and p < X <, a By construction ~n T(z) is single valued in the strip -a < Re z < a, so the branch of In T(z) can be chosen to be the one which vanishes as z -+ ~ ioo Therefore, r_(z) and r+(z) exist and go to zero for Re z going to minus and.plus infinity respectively. The unique solutions for X+(z) and X,(z) are now given by: X(z) = --- z - C (II-30a) and x+(z) = [z + a] o,(z). (II-30b)

14 It is clear that X+ are real functions of z. Also for z real, we have that X_(z) < 0 for z < X, and X+(z) > 0 for z > -\. Inserting (II-30) into (11-24) we obtain: f+(z) [z2 - Zo2] f_(z).....^ ---— ~ = "(II-31) X+(z) X_(z) for -a < Re z < a. The left hand side of (II-31) is analytic for Re z > -B; the right hand side is analytic for Re z < P; they are equal in the strip, and hence define an entire function everywhere, By the extended Liouville theorem, we have: f+(z) (z2 - z22) (I X- ( Pr(z), (IT-32) X+(Z) where Pm(z) is a polynomial of order m. As r(O) is finite for ~ -+ 0 (rp(r) is finite as r + a), then for Re z - + oo f (z) _ q(0)/z + O(zl+e) e > 0. Also X (z) z, hence we find that Pm(z) = constant = i(0). The Re z+oo for f+(z) is therefore given by: n(0) X+(z) f+(z) = -(O) -- (II-33a) z2 - Z2 or 11(0) X_(z) f (z) = -: (II-33b) 1 - CacK(z) Using the inversion theorem, for G > 0, the contour can be deformed into the left half plane where the only singularities of the integrand are the zeros of 1 - CaK(z). We find the density is given by

15 apca) ~ o X______ (, ____2 (ZO) _Nz/2 =O Ca K (zn) Ca X(z0) r for r < a, and (Co)-a1 j (O). The obvious identity: _( —zn) x(z) ~ - - - = (II-35) i= Ca K'(zn) Ca K'(zo) n=O plus the fact that X_(z) < 0, z.real and less than p insure that the neutron density is non-negative for r < a when C < Cmin. Similarly, for C > Cmin the density becomes negative somewhere in the sphere. This is the same "criticality" condition as for the infinite medium, as of course it must be because of the invariance of the equation under the transformation r - Xr, a > 0. As the neutron density is non-negative for C < Cmin, for r > a, we can ask for the distance r~ > a, where the asymptotic solution vanishes. Asymptotically, for r - 0 the solution (II-34) becomes: p(r) [X-(zo) X(-zo)] s1inh -In - (II-36) Ca K'(z )r - ~ 2a (II-36) 'X_(zo) l/zO where r = a (1-37) and (Ca)- > K(O). In the "just critical" limit (C = Cmin), the extrapolation radius, r~, becomes simply: 2X1(O) r~ = a e X(). (II-38) For C = Cmin = 1.218, a = 1, the extrapolated radius is computed approx

16 imately in Appendix C. We find that r~ = (1.64)a We can also find the shell-source solution for the finite sphere. As this problem is so similar to the previous, we will just sketch the method and give the result. Using the same notation and substitutions as before, we find the integral equation: 00 *(QO;Oo) C= Co (';Oo)de' ( ) + K( [G - o +1), (II-39) o for 9 > 0, go > 0 (i.e., shell source inside of the sphere). In transform space we then have: _zG. f+) [ C ( (z) [1 - CK(z) e -fz+ (II-40) for -a < Re z < a. Using the decompositon of 1-Ca K(z) made before, Eq. (II-40) becomes: f+(z) f (z) K(z) e-ZOo --- (Z2- Z2)2 -- --- t - — ' (I'-41) X+(z) _ _(z) + X_(z) Making another decomposition such that K(z) e.ZO - = +(z) + <2.(z) (11-42) XX_(z) where 2+(z) is analytic for Re z > -P and goes to zero as Re z -oo, and n2(z) is analytic for Re z < P, and goes to zero Re z +- (, we find that (o ) X+(z) K(z) eZo ~ (z) X(z) r+(z) = - - CK ) (Ir-43) z2 %z2 [ l- Ca K(z) 1 - CaK(z)

17 The unique solutions for ~+(z) and _(z), for -P < Re z < 5 are given by: -B+ioo ~ z +(z) 1 - f K'(z') ez- 0 dz' (II-44a):+iZI (iI-44a) 2l qpioo <X'_(Z;) [Z- - z] and a+ioo (- ) = 2% -7* (II-44b) Now, using the inversion theorem and requiring that rp(r;ro) be finite at the origin we obtain for C < Cm: p(r;ro) = j r-(zn) (r)Zn/ [ap(a) - )- (-zn)] n=O (II-45a) 00 + 1 X '( ) (r) /, for r < ro < a Xr(Ca)2 X (z) r a n=0 n=O (II-45b) Zrn/2 + r < r < a Xr(Ca)2 Z vy where Zo/2 ap(a) = S (zo) - CaX(z ) ( (-46) E. CONCLUSION In conclusion it should be pointed out that the totally black sphere problem can now be easily done. In this problem, there is a black (a = oo, C = 0) sphere of radius "a" inbedded in a medium with r-1 cross sections. For the homogeneous problem, the solution is simply given by

18 Eq. (11-34) with r/a replaced by a/r ( +- -0). The shell source solution can also be obtained from the previous solution by requiring now that rp(r;ro) be finite at infinity. We also note that for applications to electron transport in velocity space with total cross section which vary as v-2 a class of slowing-down problems can be solved with formally little modification. In particular, problems can be solved in which ca(' + v)dS the probability per sec that an electron with velocity:' suffers a collision and has a final velocity in dv about A, has the invariance property that dv"v"' a(t1'T11) _= -4r'd oa('4), where l" = c', "l = Oc, a > 0, and this "slowing-down" function is isotropic. This invariance requirement on a( + v) simply insures that the transport equation is invariant under the transformation v+ -. Using substitutions similar to those used in the neutron problem, one can derive an integral equation with translational symmetry and solve it by means of the Laplace transform. Evidently, the only significant differences of this clas of electron slowing-down problems from the r"1 neutron problem are the properties of the transform of the kernel, K(z), and of course as a result, a change in the spectrum.

CHAPTER III USEFUL IDENTITIES FOR HALF-SPACE PROBLEMS IN LINEAR TRANSPORT THEORY A. PRELIMINARY REMARKS A large class of problems in linear transport phenomena can be formulated as boundary value problems and solved by the eigenfunction technique (7) through (12). Typically, the governing equation possesses translational invariance and can be reduced to the form: x + ) (x)= f() f (') (x.')d', (III-1) \ax 2p where -A < J < I, and f1 () 'fe(G) is a real even function of p. In this chapter we first show that with suitable restrictions on f1(p.)-f2(p.) Eq. (III-1) always generates a complete set of solutions. Using these solutions, some typical half-space problems are considered and it is shown that some identities established for the solution of the associated homogeneous Hilbert problem allow the usual cumbersome integrals occurring in half-space problems to be considerably simplified. B. SOLUTIONS OF EQUATION (III-1) The translational symmetry of Eq. (III-1) suggests looking for solutions of the form: t(x9, ) = e-X/v Vi (0) (III-2) 19

20 For v finite (III-1) becomes: i3 (v - 0,([) = v '-41 f 2() v()di' (III-3) choosing the convenient normalization of /v(4); f2 (t) Ov(t) d = 1 (III-4) (Eq. III-3) can be solved by the standard procedure (8). The usual discrete eigenvalues are then given by the zeros of A(v), where ( ) A(v) = 1 - " f 2() v(t)dt = 1 + V - I2( B 4- ~ V (III-5); We will always assume that fl(O).f2(t) belongs to the class H* (13) on (-5,5), for p finite; and for p infinite that fl(k4).f2(G) belongs to the class H and satisfies: Lim fl () f2() < Cll2, a > 0 (IIIThen A(v) is a real even function of v, sectionally holomorphic with boundary ( -R 3), and asymptotically, A(v) = 1 - V-oo P fl(lL) f2( i)dt + J- ). 1 ~ ~~~~~~~ 2 (III-7) Clearly, if vi is a root of A(v) = 0, so is -vi and ~vi*. Defining vi = -vi, it will be assumed (only for convenience) that the vi can be la

21 beled so that Re vi > 0 i > 0, and if v+l $ vi, then IRe vil < IRe Vi+lJ| For v not on the interval (PAP) A(v) will in general have 2N' zeros, where N' includes the order of the zeros That is: A0-l) (vi) d(j-') A(v) dv(:-l) v=av = 0, j = 1,2, ''mi, ~i = 1,2, e..n (III-8) n' and N' = mi. Also, A~(v) will have 2L zeros for y an [-3,Bp], where i=l A+(v) and A_(v) denote the boundary values of A(v) from above and below respectively. Explicitly, we assume: (j-1) A(+ ()~i) = 0 j = 1,2,' mi, i = 1,2,'', and L = X mi. If we consider the function i=l i=1 (III-9) (III-10) we see that A(v) has no zeros on [-,]~, is analytic, and has 2N' + 2L zeros outside of [-3,]. Choosing arg (+(0) = 0, and using the usual change of argument theorem, we find: N' + L = T A(~,) arg A+(v) arg +) (III-11) (A(L) change along L). If we label the zeros of f (p.) and f (p.) such that:

22 f(l) (vi) = O j = l,2. omin i = l,2,*.n (III-12) f2 (vi) = O j = 1,2, mi i n= + ol, n nl rn2 and 2M = ) mi + ) mi, then clearly L < M, as [f1( ) f2(0) ] = 0 i=1 i=ni + is a necessary condition for ~A) (ji) = 0. Also, as the Im A+(v) = 0, 0 < v < P, M-L times, we have by Eq. (II-1) that N'+L < M-L+1i Throughout this chapter we will assume that N' and L are finite. A sufficient condition for this to be true is that M is finite. With these properties of A(v) we can now construct the solutions of Eq. (III-1), We will separate them into three classes, Class I-Discrete Eigenvalues outside of [-~,3] Corresponding to the 2n' eigenvalues outside of the interval [-,]^ one finds the 2n' solutions: (~) (eX/i Vif ), ~i = 1,2,nvi+ (III-13) i vi -V where A(vi) = 0, for lvii finite. As the ith zero is mi degenerate, the 2n' eigenfunctions given by (I11-13) are supplemented by: (x) = 2() s t g(x,),j = 2,3,( - mi (IIl-14) 1 d1(j-) v=Vi +i = l,2,o.on ' The 2N' solutions given by Eqs.o (III-13) and (III-14) are linearly independent and by Eq. (IIZ-5) clearly satisfy Eq. (III-1). For x = 0, these

23 eigenfunctions will be denoted by (-l) ().?Pvi If | fl(p) f2(1)dt = 1, then A(v) has a double zero at infinity (assuming 2f1l() f2(t)di 4 0). In this case Eq. (III-1) has the 2(N'-l) solutions given by (III-13) and (II-14), and the two linearly independent solutions: (1) (x,>) f f(^) X (2) (xo) = f1(L) [x -]. 00 oo (II-15) Class II-Discrete Eigenvalues on [-3,I] In our previous discussion we observed that in general we have 2. eigenvalues on the interval [-P,P] given by A+(ki) = 0. Corresponding to the zeros of A+(), for k = pi = vi, i = 1,2, on, nl < n we have the n' set of mi linearly independent eigenfunctions: i=1 i)(x,) = [e-X/ ()()] 2, j= 1,2, m (III-16a) ~~dv^1' ~v= i = 1,2,.oon where )(-) = -f.. (III-16b) 1Hi i - - Similarly, corresponding to the zeros of A+(p), for t =i -p vi, i = nl+l, ni 2 -*n', n +1 < n < n, we have the set of 2 mi linearly independent 2 1 2 2 ~eigenfunctions~ i=n +1 eigenfunctions:. 1

24 0i -1) i a(j-1) -~x/-v (0) + (X, t e-( i = 1.~2)*4*mi ~(O) ~ (o alff1(0 '~~ti ~i E: yjti — 1L (III-17a) (III-17b) where We will show in Section C that all of the solutions in Class II are linearly dependent on the solutions of Class III and hence are never explicitly needed. Class III-Continuum Solutions In addition to the solutions of Classes I and II which are associated with the zeros of A(V), there are the continuum of solutions: (x, ) = e(X/ i) (4all "v- < -v < (III-18a) where j~(0) = vf (L) P 1_ + A(v)6(~ - 0) (III-18b) corresponding to the branch cut of A(v) By Eq. (III-4) we find: A+(v) + _(v)?(v) f) = f ( II-19) It should be noted that Eq, (III-19) defines A(v) only up to delta functions in V-ii, i = nl+l,^- n2. Therefore, inplicit in the solutions (III18) are the solutions: i/ (x,) = e-X/ii &( - pi) i = n + 1, n2. (III-20) i 11j

25 These will always be explicitly factored out of the continuum solutions. C. COMPLETENESS THEOREM Theorem: The set of solutions of Class III for 7 < v < 7 1 < 7 2 - < 7Y Y 72 < P plus the solutions in Class I form a complete set of functions on the class of linear continuous functionals, *(i), with compact support in [71,72] and such that: Lim <(-) <C\i - 71 -, C <1 (II1-21) 471 2 2 and for either (or both) y7l y72 infinite Lim k j(0) f2() = 0 (III-22) 1k471 and/or Y2+00 for some k to be specified later. In the following proof we will always assume that none of the zeros of A+(p) coincide with 7 or y2. This condition and the restructions implied by Eqs. (III-21) and (11i-22) can be relaxed within the framework of distribution theory (14-), however, for convenience and simplicity this will not be done here. To prove the theorem we have to show that the singular integral equation (j-i) 2 () = aaij,V (p) + A(v) v(4))dv, (III-23) ij 17

26 has a solution when ([)3 is subject to the restrictions of the theorem. For the present it is required that Yl and 72 be finite. Letting r'(j) = fr(l) - Z ai j i(- )(B), Eq, (III-23) becomes: v,(~) = (f, ) P (v) + X(1) A(n) + bi. - i). (III-24) r2 clearly the bi are given by: ti+E bi = Lim j 8dr(l), all Y71z < 72. (III-25) Assuming that A(.) has the same properties as f(At) the function N(z) 1 2 A() (III-26) 71 has the properties (15): (a) N(z) is analytic except on the interval [7l,72], and N(z) has at most, polar behavior in this interval. (b) Lim N(z) = 0 Z->-oo (c) Lim N(z) < C[z - 7l] z+71 2 z7 2 Multiplying Eq* (III-24) by Bf2(j) and using Eq,. (III-5), the expansion equation becomes the boundary value equation: 7 < /1 -~ ' 2

27 Ctf (ji) p '(i) N,(Ct) A,([i) N_ (kt) A_ (i) (III-27) Introducing and N(z) =.)Ii - z)J N(z) A1(z) = A(z) 11'([j - z)mA ' J (III-28) (III-29) (where bj is a zero of A+([i) of order mj, and the product in (III-28) and (III-29) are over all j such that 71 < Pj < Y2) into Eq. (II1-27), the equation can be put in the conventional form: 7(~i) p '(1i) - N+([) X.:(I) - _(t) X_(), X_ ( ) f2( ) 7 () = ---,-1 and X(z) Al W (III-3o) has the following properties: (a) X(z) is sectionally holomorphic with boundary (7y ^2). (b) X(z) is non-zero in the cut plane. X,+() A +(l ) O X+([) (C) = = = G() xe(-) A.(k) Z_(~) (d) X(z) vanishes more slowly than |z-71 and Iz-y21 as z + yl, Y respectively. (e) Lim X(z) = Class (1) const. Z-coo Class (2) zkl, k1 positive integer. Class (3) zk2, k2 positive integer.

28 The problem of finding X(z) subject to these conditions is the "classical" homogeneous Hilbert problem (13). The solution is: X(z) = (z - )tl (z - <y)t2 eP(z) 21 2jd' l 71 LLZ (III-31) (III-52) and t l t2 are integers chosen such that l>t2 + > 0 2 qj - (I-L-33) ( > > ) = ag 1 > t ~ > 0, G() = arg +(k) For the entire-space and half-space problems one can easily obtain t and t, from Eqo (III-11). They are listed belowo 7' -P 0 -P 72 0 ti 1 - (N'+L) 0 -(N' +L) TABLE III t, and t2 t2 -(' +L) -(N'+L) 0 t +t = -k2 -2(N' +L) - (N'+L) - (N'+L) If N'+L > 0, then X(z) for the half-space and entire-space problems belong to Class (3). Solving Eq. (II1-30) for this class, we find: 1 r-k 2(z) 1 dk 7 ( ID? (1i) N(z) = + _ X(z) lr(l j-z) 2rci X(z) ( j- z) m- (III-34) i i 71,

29 Here Pj(z) is an arbitrary polynomial of degree I, and r = mjo This is J a solution to (III-30) only if the k2-r additional conditions /2 X 7y()) }'([)dkt = 0, = 0,1, ok - r (III-35)?1 are satisfied. For the entire-space and half-space problems as r = 2L, k2 = 2(N'+L); r = L, k2 = N'+L, respectively, we find that Prl.k (Z) - 0 in both cases To satisfy (III-35) we include k2-r discrete eigenfunctions of Class I in the sum contained in V(i)o As k2-r < 2N' there are always enough eigenfunctions to do this- Even so, it still must be shown that Eq, (III-35) does not impose any restrictions on \( ot) other than those stated in the theorem, Using the representation of X(z) derived in Appendix D it will be shown that (III-35) is consistent with the theorem, Before proceeding it should be remarked that for X(z) belonging to either Class (1) or (2), an N(z) can be found which satisfies (I-30) without restrictions such as (.I1T-35) In these cases no discrete eigenfunctions are needed and A(v) is easily found to satisfy the assumed conditionso Therefore, in the following, only X's belonging to Class (3) will be considered Eq. (III-35) is explicitly T l y(7) GL/ ()di = X ai r2 y r( ) l (I)d7J i,3j 7 (III-56) ~ = 0,1, l~k2-r-1, and the sum is over k2-r aij t's In order to express

30 (III-36) in its simpliest form, we first consider the integral (j-i) Ii lf A y(n) ( 01) ( ) 4, 71 ( III-37) which is: (j-i) I~ i d(j-l) - = -- _ ) v E72 Iy ( ) f(n) doi -~ J -V 71 v=v. a ( II-38) Using the representation of X(z) given by (D-3), (III-38) becomes: (j- ) dU 1) Y, [lid ' (,,i-,,)r[j 14( - X-Gl) ] (III-)9) or (j-l) 1,i - (j-') - v 4.(1 -v)mJx() J J V=Vi-v (III-40) If V=-o is a zero of A(V) we may want or have to include (1)(p) and (2) (0 ) in the sum over the discrete solutionso In this case we have to calculate the integrals: (0) IQ-, 19 Qr( ) +(1)( ) 7y(11) ( )(I) = ~,2,(k) (2)([)a., 72 Yi (IIi-4la) and (1) IQ, Y2 1 - - r '^7(4i) fl(i)di. 7 1 (Ill-4lb) By the same method as above we have:

31 [yt(k) fl(L)dl = 1 z ( z -JZ)J X(z)dz, (I:I-42) 1 where C encircles the (7r 7y) cut in the negative directiono We have therefore r+r = (-) t = k - r - 1 (I-45) k r -I~im zzr = C = -Lim z[zk2 X(z) ( z)- - (-r] Z+oo if t = k2 - ro Using the representation of X(z) given by (III-31) we find: C = (-)r[yltl/e2t2+ E j + 1 / dt In G(t)] (III-44) -1) It is now evident that the integral I is never zero for all i and all j. Therefore the conditions (I-36) can be satisfied without further restrictions on *(k), It will be shown in Section D that in fact the aij's can be uniquely determinedo We also note that the linear dependence of the solutions of Class II on Class III is now evident. Letting Y (Gi) = (J1' ) it is easy to show that the integral Qi2

32 Hence, we can choose aij = 0, and the linear dependence follows. Furthermore it is evident from N(z) given by Eq. (III-34) that vA(v) = N+( v)-I_(v) has a principle value part and a delta function contribution at v = i. Putting this A(v) into the expansion Eq. (III-24) and factoring the delta functions out of the integral we obtain the usual expansion expression (7) which from the beginning includes all of the discrete eigenvalues, and not the continuum solutions where there is a discrete eigenvalue in the continuum. (The principle value contribution at p = Hi.) For the entire space problem we can construct X(z) by inspection. The solution is given, up to constant, by: X(z) = n. - (III-46) n mi tf (vi2-z2) l=1 It has been specified that y1 and y2 be finite in the above procedure. It is clear that if P = oo, the proof is still true with suitable restrictions on3f(p). For 7y = -oo, Y2 = +O, the solution for X(z) is given by (III-46). If only one end point is finite, say yl, the end point condition for X(z) is applied only to the finite end yl. X(z) is then given by: X(z) (z a y) t er( (III-47) and 00 r(z) = n ( )d (III-48) i ~-z where In G(p) is that branch which vanishes at infinity. It is clear

33 that N(z) will exist with the correct properties if in (III-2) we choose k > 2L + 2 + k1 (k1 given by Class (2)) or k > 2, if X(z) belongs to Class (1) or (2). It is now evident that A(v), given by vA(v) = N+(v) - N_(v) does satisfy the assumed conditions. Therefore, the completeness theorem has been proven. D. APPLICATIONS TO HALF-SPACE PROBLEMS In this section the theorem of Section C will be used to obtain solutions to Eq. (III-l) for 0 < x < oo. For simplicity it is assumed that Vi 4 ~~a N' = n' > 0, L = 0, and f2(G) 4 0 for 0 < 1 < 3, except possibly at t = o if = oo, We first consider the albedo problem, where we want to find a solution to (III-1) for 0 < x < o subject to the boundary condition: *(0o,) = 6( - [o), 0 < < p, o < go < p (IIz-49) and Lim t(x,J1) = 0 o (III-50) X** 00 x~oo Using the theorem of Section C we can write: 8(x2I11a.) = Zai (xi) + A(v) (x)dv d i i o (I1:-51) The sum is over N' discrete eigenfunctions and (III-50) implies that the sum is over i = 1,2. oN'o Condition (III-49) gives: [ > 0

34 6 ( Jt- [10) N' ai i=1 B (0) (k) + 0 A(v) v (Gt)dv. (III-52) The ai s can be determined by the N' condition of Eq. (III-35). The equa tions are explicitly: NV V (a) - ai i,' = 0, N'-l. i=1 (III-53) It is easy to show that (et (o) Det I I 1 N'Y N -= ( -) T- v1iX(vi) i=l I';7-, (vI >j =1 - v j), (III-54) which is clearly non-zero. Using Cramer's rule the ai's are found to be: Y (-io) "i~vx~i N'J J4i 10 - j vi - vj i = i1, -N'. (III-55) It should be mentioned that if vi is mi degenerate (N' 4 n') Eq. (111-55) becomes AiIY (p10) KN = 1 biui~ (III-56) here bj = a1j, j = l,2,**.ml = a2j-ml j = ml+l,-. m2+ml, etc., and det uij = (_)N [viX(vi)]mi i=l nt 7T> (v I>j=i - Vj) mmj (III-57)

55 which is never zero and therefore the aij's can be uniquely determined in this case also. In applications we normally want to know the discrete eigenfunction coefficients, because they usually give the asymptotic form of t(x,t), and the emerging "angular" distribution (O,,p), pt < 0. The emerging "angular " distribution for the albedo problem is: O(P o~) = X ai ()() + A(v) v()dv, B < 0 (III-55) i=1 For p < 0, X(|) = and Eq. (III-55) becomes V-p. N' (0,B;pO) = X a^i / (~)) + 2tifl(n) N(p). (III-54) i=l Expressing N(pJ) for p. < 0 by using the representation of X(z) given by Eq. (D-3) we obtain: N(=) ----- 1 [X(+) - X(vi)] 2rtiX(p) o-^ Vij-p i=I ( -55) The emerging "angular" distribution now reduces to: p. < 0 7(=o) 1 p o-p ( III-o6) ^(0,n~no) -- o (III-56) X(O) 1-k i vi-[ y Vi-Vj The Milne problem can be readily solved by using the formulas derived for the albedo problem. In the Milne problem we want to find the solution to Eq. (III-1) subject to the boundary conditions: (vN, 4 c)

36 Lim r(x, ) = i) (x,) Re vi > 0 (III-57) X-0oo and (0,1) = o, p > 0 (III-58) USsing the completeness theorem we can expand *(x,l.) subject to (III-57) as: N' 5 (xp) = 0 (xo ) + ai, ( ) + A(v).(x,)dv (III-59) i=1 O and then (III-58) becomes: i _ 0, -~_v(0) (0) O- aji j( N'-' -^ (n) = j a-)v(p ) + [ A(v) XV(p.)dv. (III-60) j=l 0 In order to calculate the ails and the emerging "angular" distribution we only need integrate the albedo solutions times -^) ([lo) over P-o. This can again be done using (D-3) and we find: Vi X( —i) f i +1 ak = V X (-v k = 12 1. ON', (111-61) Vk x(vk) j4k vj-vk and the emerging "angular" distribution is: 4p < 0 (O ) X(-v ) + vif() + ViV62).=1 ":v ( 7p.. (k4 Vk62) Before computing some of the moments of the emerging distribution for the Milne problem we first derive some useful identities for X(z).

37 E. IDENTITIES FOR X(z) In Section D we showed that the representation (D-3) for X(z) allows the usual combersome integrals involved in a half-space problem to be reduced to a form which involved only X(z) and other, in principle, known functions. In this section we will derive two more identities for X(z), both of which may serve as a means to numerically determined X(z), Qr as will be shown by way of an example in Section F, permit some quantities of interest to be trivally determined. Defining C = f1i(G) (fp2()dL, and for convenience assuming C 4 1, we want to prove: A(z) X(z)X(-z) A) ( = n, L 0) (III-63) (N-C) ~(-v2_z2) (1-C) ( V1^ '2) i=1 To prove this consider the function R(z) = Az) (..I.-64) N' [l-C]X(z)X(-z) (v 2_-z2) i=It is clear that R(z) is analytic in the plane cut from -B to B. Across the cut R+([i)/R_( t) = 1, and also Lim R(z) = 1. Therefore, R(z) 1, Z-roo and the identity (III-63) is proved. From (III-63) we see immediately: X2(o) = 1 1 (III-65) [1-C] (1 iV2 i==i

38 The phase of X(O) is uniquely determined by the fact that Lim X(z) Z-+- oo = (-)N' Iz N (z real) and that X(z) is a non-zero, real function of z. Therefore N'' 2-1/2 X(O) = (-) [(1-C) ' vi2 ] (iiI-66) i=L1 1,' A particularly simple result of this is that [1-C] Vi2 must be real i=1 and non-negative. Of course, if C = 1, an identity similar to (III-63) can be derived. The identity (III-63) may be useful for calculations X(z) for large z. Expanding both sides of Eq. (III-63) in powers of z-1, and equating coefficients, one finds a set of coupled albegraic equations for the coefficients of z-n in X(z). In particular cases a truncated solution may rapidly converge and the coefficients of low inverse powers of z be easily obtainable. To obtain the second useful identity for X(z), we use the representation of X(z) given by (D-3) and the identity (III-63). We find: 1 0 di f1() f2() X(z) = L-] -- (III-67) [1-C] iI1N' ([,z) x(G) 7(it-^) i=Li From (III-67) we first notice that if we know X(I) for -I < < 0, we can readily determine X(z) everywhere. Such being the case, by letting z =,ip -P < i<D in (III-67) we obtain a non-linear integral for X(p.), -p < p < O for which an iterative solution may rapidly converge. More

39 important is the property that (iII-67) allows us to express integrals over X(>)~1 (which always occur in the emerging "angular" distributions) in terms of X(z), and it is this property that we will exploit in the next section. F. AN APPLICATION To illustrate the usefulness of the identities derived in the preceding section we will calculate some moments of the emerging "angular" distribution for the Milne problem, While moments for the general emerging "angular" distribution of Eq. (III-62) can be at least simplified by the identities of Section E we will only consider the usual case; N' =1. The emerging "angular" distribution now reduces to: 2v12X(-v ) f (1),(o,j) = 2V -,-r-l(III-68) X(W) (v,2..2) The emerging "density" is then given by: o o p_(O) = d (tf2) x(0p) - 2v12X(-v1) f () 2(_) d (III-69) _ X(~) (v 2 However, the integral in (III-69) is simply X(O)[1 - C] (from II-67), and so we find: p_(O) = -2X(-v1) [v2(l - C) ]1/2 (III-70) The emerging "current" is.

0 0 j_(o): d f2(f ) *(o,01) = 2v12(-v1) f1 ). (III-71).- _p x() (v.12- From (11-67) we immediately see that the integral in (I-71) is; Lim [1 - C] z X(z) = (1 - ) (II-72) Therefore we have: j_(O) = 2v X(-v1)(1 - C) o (III-73) We note that the average emerging 1tfbrthe half-space problem is given by: =' = j = - ['v2(l1 -[ C)] C (III-74) p_(O) Unfortunately, un for n > 1 is not so easily calculable, however the identities of Section E do help to simplify the integrals for these higher moments, For a more illustrative example of the usefulness of the identities we refer to Reference (16) G. CONCLUSION While we have considered only the Equation (IIl-l) it is clear that for any equation or system of equations which have a spectrumn similar to Eq. (III-1) and for which a completeness theorem of the form of Section C is valid there are similar identities for the solution of the associated homogeneous Hilbert problem for the half-space, The judicious use of these identities will always simplify the formulas of interest.

CHA.PTER IV DENSITY IN A SIMPLE MODEL OF THE EXOSPHERE A. DISCUSSION We consider the following simple model of the planetary exosphere: Exterior to a sphere of radius r. we have a gas so rarified that collisions may be neglected. The only force acting on the particles then is the gravitational force, due to the total mass M within ro, Within the sphere collisions are so frequent that particles emerging from r - r, have a MaxwellBoltzmann velocity distribution. The problem Is to determ1ine the particle density in the region r r. subject to the condition that there are no particles present which have not come from within the sphere. This problem has been treated by Opik and Singer (.) and, Brandt and Cha tbserlain. (1S) 8 le first authors find particlea dens1t'es by straight forward ki netic theory calculation of the nuamtbe rs of.- parti loes whieah ra h a given point in space. Th'e second, author uta rt from the collisonleee Boltzmann (i..e., LioWuille) eqjuation.* SinJ:ce the latter is n, eL-aly a st:teeiieit wof ewton s of mtion the two ap pro ches. bou hold agree. How ever, there see to be some confusion on n this points In Order to clarif the sit-uaetion we oonetruAlt (a sim1iFple explTelct siolu titon of the Boltzmann equatioLn slb ject tor tho g ivqn?bounary eontitions. om- tie t hio paticle dnstty iV TF rivleil_-y obtAitefd by qa.dt-Ort.. t4l

B. CONSTRUCTION OF THE SOLUTION The distribution function, *(r,v), in the exosphere is to satisfy the collisionless Boltzmann equation: 4 -(v oV + FV ) t(rv) = O, (IV-1) r m 9 where G Mm r F = -. (IV-2) r3 (Here m is the mass of the gas molecules, and r is the radius vector from the center of the sphere.) At r = ro we have the boundary condition that the emerging distribution has the Maxwell form, ioe. *(ro,) = N(m/2r)3/2 exp-[ mv2/2] for v ~ ro > 0. (IV-3) (3 = l/kT) Further we have the condition that all particles exterior to the sphere shall have come from within it. Since the problem has spherical symmetry we know that 4(r,v( ) = V(r,v,[), (IV-4) where r = 11l, v = 1vl, and = (r.v)/rv (IV-5) In terms of these coordinates Eq. (IV-1) becomes rta a ( + (v G M (1 ) a T (r, o fV (. - V) (1 - 2) (rvk) = 0 rr dr rr22 (Iv-6)

43 The method of characteristics (19) shows that the only content of Eq. (IV-6) is that 4 is to be an arbitrary function of E and L2. Thus i = l(E,L2) where E = - mr --, (IV-7) 2 r and t2 = m2v2r2(1 - ) (IV-8) ioe, i * depends only on the constants of motion-which are the energy and the angular momentum. We still have to fit the boundary conditions. To do this we note there is no reason for i to be a continuous or singlevalued function of its arguments. Consider, therefore, the function *(r,v,1) = N(mp/2r)/2 exp-_[G~m/ro + E] (IV-9) (E + GMm/ro - L2/2mr2) [1- e (E) e (-)] Here 0 x < 0 e(x) = 1 x > O (IV-10) 0 x<O. The significance of the 6 functions is readily seen. Consider first a particle at ro with energy E. We have L2/2m 2 = (E + GMm/ro)(l - t2), (IV-11) Hence -012 2 E Ga/o L' / 2 0 E rm o) - (IV-12)

Thus, for pi > 0 the function of Eq. (IV-9) does reduce to that of Eq. (IV-3). Further, the factor 8(E+GMm/ro-L2/2mr02) guarantees that we have no particles whose orbits do not intersect the sphere of radius ro. The remaining factor 1 - e(E) e(-[) arises from the requirement that there be no particles incident from infinity. For E < 0 the particle orbits do not reach to infinity and the factor is 1. However, for those orbits which reach to infinity the factor e(-4) guarantees them to be outgoing., The function of Eq. (IV-9) thus satisfies all the boundary conditions. It only remains to see whether it satisfies the Boltzmann equation. Except for the dependence on 4 this is trivial (since it is a function of the constants of motion). Thus on inserting the function of Eq. (IV-9) into Eq. (IV-6) we need only worry about terms arising from differentiation of e(-[e). Using the result (-p) = - 5() (IV-13) (where 6 denotes the Dirac delta function) we obtain (v.Vr + F * v), = N(m /2X)3/2 expp [ + E] 6(E) e(E + GMnx/r0 - 2/2mro2) (IV-14) (v GM (1 p2) b(p. \$ vr2,

We note the identity 6(t) e(E) e(E + GMm/ro - L/2mo2) = 56() e(E) e F[E(l +-) + L -1 r. (IV-15) But for r > ro the argument of the second step function is negative (for E positive). Hence the function of Eq, (IV-15) is identically zero and the Boltzmann equation is satisfied. We conclude that Eq. (IV-9) does indeed yield a function satisfying all requirements. C. PARTICLE DENSITY The calculation of the total position density in the exosphere is now straightforward. i pr00 p(r) = d (rv) = 2jt d t v2dv *(r,v,). (IV-16) 0 The result is: p(r) = 2N/ (i)1/2 exp d(l - I 00 - (1 - X2)/2 +X ah_/22 e-2 + y2dy e-y2 0 dy y2 e-2 (1 x 1/2 y2 (l+x), 2 1/2 fa \ 12 vlx dy y2 e2 _ y )1/2] dy y2 e- - - (IV-17) where

46 a = aeGM/r, x = ro/r This agrees (up to constants) with the 'lballistic density" calculated by Opik and Singer (17) in the 1961 reference (p. 226, formula 31). The expression can be considerably simplified and written in terms of Error functions, x D(x) = (-)/2 / e dy2 dy (IV-18) 0 as p(r) = exp o(1 - l/x) 2 fl + o(aD1/2) - (1 -x2)12 exp(-c/l + x) + ( + l +,# ( '~)1//2e'~ ex)1/2j]} (v+ x -19) + Ce- [(1 - x) /2 _ 1 o (IV-19) D. CONCLUSION It has been shown, as might be expected, that there is no difficulty in writing down the solution of the collisionless Boltzmann equation for our simple model of the exosphere. This method, while completely equivalent to any other solution of the problem based on Newtonian mechanics, is probably the quickest and possibly most elegant approach.

APPENDIX A DERIVATION OF THE INTEGRAL EQUATION (11-2) To derive the integral equation given by (II-2) we start from the Equation (II-1) with the appropriate cross sections for the modelo First consider the integral in the exponent, f a (r _- (rJ )s> ds = a f ds s This integral can be done and is: oIr-' I r ) a g ( -- r) ds = n rLrr-r (A-2) Substituting this back into (II-1), for the shell source, we have: p(r) ~= / r'2dr? d )' 2 ^ J r'rr - r.(r-r') rr-r' r-* ' -p(r) + 41rc]' 4r Letting X = - t, X2 = r2+r2 -2r r1, and then X = (r+rc)y, Eq0 (A-3) becomes: p(r) J d p(r') -_+ r 4 p Yr (A-4) 2 J r ri 4Wv r2 Ty (+y) o ~ ~" IO -' ' |r+r' [ 47

Now introducing r = a e-20 etc., we find: l(eG) 00 C= C II(e') J -00 ta 1+ 1 tanh8itrv tanhje-[ol 1c,1 dl -yL y Ln IoE)-o'I k17 L1 (A-5) dy $-y,1" Ly from which Eq. (11-2) immediately follows.

APPENDIX B PROPERTIES OF THE TRANSFORMED KERNEL, K(z) The transform of the kernel, K(z), is explicitly: 00 1 K(z) e-z ' f }. (B-1) -z9oo t ItanhiG' ~ Letting e-t l-y, and '9 = G/2, one integral in (B-l) can be done, and l+y we find: 00 IVK~z) 2 COrt (z) = ^2 dt.e-t sinh ztL2 (B2) Z J sinht 0 This function can be written in terms of known functions (4)^ for -a < Re z < a: Z(z). IL Z) ( Z (5], (B-3) a= 2(a+1), andY(z) = d/dz in r(z). As 4 (z) is analytic except for simple poles (4) at z = 0,-1,2,2., K(z) is analytic in the strip -a< Re z < a. Defining the analytic continuation of K(z) for |Re zi >a by the right hand side of (B-3), we can say that K(z) is analytic except for simple poles at z = +(4m+a), m = O,l2,2.**. Using the formula (4): (z) - (-z) - Ctn rz - (B-4) we obtain the asymptotic form of K(z):

50 K(z) t - r ctn z + 0 z. (B-5) Iz>>X 4 z2 The zeros of 1 - Co K(z) are also of interest. The simpliest representation of K(z) for determining these zeros is given by (4) 00 K(z) =2 X V -- 2 B2 6) n=0 (n+ C) - (Z) Using (B-6) it is easy to show that 1 - Coa S(z) = 0 has roots only on the real and imaginary axis. For z real and in the strip -a < z < a, we know that K(z) has poles at z = ~a. Also a(z)/6z > 0 for z > 0, and K(z) is and even function of z. Hence K(z) has a relative mininum at z = 0. Therefore, if (Co)-1 > K(0), there are two real roots, ~zo, in -a < Re z < a. If (Ca)-1 = K(O), there is a double root at z = ~+z = O. For z = iy, y real, we similarly find: K(iy)/ly < 0 for y > 0. Also K(iy) is continuous and finite for all real yo It is therefore evident that there will be two imaginary roots, ~zo, to 1 - Co K(z) = 0, if and only if (Co)- < K(O). For IRe zl > a, the analytic continuation of 1 - Co K(z), defined by (B-3) and B-6) has an infinite number of real zeros. As K(z) has poles at z = +(4m+a), m = 0,1,2,,, and aK(z)/6z > 0, for z > 0, z real, there must be one and only one zero to 1 - Ca K(z) in each interval 4(m-l) + a < z < 4m+a, and similarly for z negative and real. The root in this interval is labeled Zm, and zm+1 > zm > Zm-l. It is also clear from Eqo (B-5) that

51 Lim (zn - Zn_ ) = 4. (B-7) n+-oo

APPENDIX C CALCULATION OF THE EXTRAPOLATED RADIUS An exact calculation of the extrapolated radius r~ (Eq. II-37), is clearly a computer task. However, in this appendix we exhibit two approximation methods which converge very rapidly for the r-1 cross-section and make this calculation amenable to a slide rule. For simplicity we have taken the case a = 1, C = Cm = 1.218 (zo = 0) The extrapolated radius 2X- () /x (o) is then given by Eq. (11-38), i.e., r~/a = e By examining Eqs. (II-29) and (11-30), it is easy to prove that X.(-z) = - X (c-l) Equation (II-25) then becomes: X_(Z) X1(-)) (C-2) X. (z) X. (- z) = - Cm(z) (C-) z2 Expanding 1 - CmK(z)/z2 in a Taylor series about the origin, we find: 1 - CrnK(z) 2 + Ur 1 CmK(z) = 2 + 4 4z2 + ez4 4 aZ6 + a1oZ8 +..., (C-3) z2 where the a's can be computed by using the representation (B-6) of 'K(z) and the tables of Reference (5), They are: a2 =.4112 x 10-1 04 = o2416 x 10O2 6e =.1490 x 10"3 52

53 oa =.9286 x 10-5 cio =.5799 x 10-6 Now, X_(z) is expanded in a Taylor series around the origin and coefficients of like powers of z in Eq. (C-2) are equated. The exact value of X_(O) =- 2 is found. If X(n)(O)/nt is neglected for n > 3, we find: X'(O) =.253. If X(n)(O)/n, for n > 5 are neglected, a quartic equation must be solved and in this approximation we find: XL(0) = 2.54 x 10-1. Therefore, this approximation method converges rapidly and we find that the extrapolated radius is: = 1.64. (C-4) Another method of approximating X_(z) is the use of continued fractions. This method is in general more useful than the previous one as the Taylor series expansion may be slowly convergent for z = zo + 0. The function 1 Ccr K(z) has a continued fraction representation which converges in the entire plane (6) (except at the poles) and the NIth approximation is given by: N Z anZ2n 1 - CmK(z) ~n=1 (C-5) N bnz2n n=O As K(z) O0(1/z,) then we must have aN = bN. Using this condition and Z-t c calculating an and bn by an expansion around the origin, or equivalently

54 by the method of Ref. (6), we may approximate 1 - CmK(z) to any order that we desire. Having made this approximation the Wiener-Hopf decomposition given by (II-25) can be trivally done and X_(z) uniquely determined. Explicitly, we have in first approximation: 1 - CK(z) (C-6) A-z2 Using this approximation, we find ro/a = 1.50. Going to the next approximation, where only a quadratic equation has to be solved, i e.,, 1 - Cm K(z) fAz, (C-7) Bz2+Az4-1 gives ro/a = 1.64. So, this method has rapidly converged to the previous result and is, computational wise, easierO It should be pointed out that in the constant cross-section case, the convergence of the later approximation scheme is not as rapid as in the r-1 caseo The extrapolated end in this case is exactly Xo =.7104, (3) In the same approximation as (C-6), we find Xo 5 o577o In the same approximation as (C-7); we find Xo =.694; and in the next approximation, where a cubic equation must be solved, we obtain Xo = ~706.

APPENDIX D A USEFUL REPRESENTATION FOR X(z) If X(z) belongs to Class (3), i.eo, Lim X(z) = z-k2 k2 > 0, and Z-o o zinteger, we have by CauchyVs theorem: X(z) = 1 X(z))dz ('(D-l) X(z) = - f, (D-1) 2rti c z - z where c encircles the (717Y2) cut in the negative direction. Equation (D-l) becomes: 72 = 72 Iit A>AI (P-2) X(z)= 1 d [X+()-X()] =1.*, i '] (D-2) 2iri 71.1Z 2i Y (l(k-z)Al-(i) or X(z) = 7 ~ r flG-L) (I-5) 71 41-Z ( )mj If X(z) is not in the Class (3), a similar representation of X(z) can. be derived 'by making suitable subtractions. 55

REFERENCES 1. E. P. Wigner, Phys. Rev. 94, 17 (54). 2. E. C. Titchmarsh, Theory of Fourier Integrals, 2nd Edition, Oxford Press (48). 3. K. M. Case, F. deHoffmann, and G. Placzek, "Introduction to the Theory of Neutron Diffusion," I, LASL (53). 4. Erdelyr, etc., Higher Transcendental Functions, I, p 16, McGrawHill (53). 5. H. T. Davis, Tables of Higher Mathematical Functions, Vol. II, The Principla Press Inc. 6. H. S. Wall, Analytic Theory of Continued Fractions, D. van Nostrand Co. 7. K. M. Case, Ann. Phys. 9, 1 (60). 8. Ibid., 7, 349 (59). 9. I. Cercignani, Ann. Phys. (to be published), 10. F. Shure, Univ. of Mich. Thesis (unpublished). 11. H. S. Wang, Nat, Bureau of Standards Report (I). 12. R, S. Zelazny, Case Institute of Techo Report No. Ao6o 13. N. I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groninger, 19535 14. I. M. Gelfand, and G. E. Schilow, Verallgemeinerte Funktionen (Distributionen), I, DVW, Berlin, 1960o 15. H. J. Brennmermann, L. Durand, Jour. of Math. Phys. 2, 240 (61). 16. R. E. Aamodt, and K. M. Case, Ann, of Phys. (to be published). 17. E. J. Opik, and S. F. Singer, Phys. Fluids, 2, 653 (59); 39 486 (60); 4, 221 (61). 56

REFERENCES (Concluded) 18. Jo Co Brandt, and Jo W. Chamberlain, Phys. Fluids 3, 485 (60). 19. See e.g., A, G. Webster Partial Differential Equations of Mathematical Physics, G. Eo Steehert and Co., New York (33). 57

FOOTNOTE *Supported in part by the Office of Naval Research, Department of the Navy. 58

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