In this paper we study a three-dimensional photon transport problem in an interstellar cloud, with a localized photon source inside. The problem is solved indirectly, by defining the adjoint of an operator acting on an appropriate space of continuous functions. By means of sun-adjoint semigroups theory of operators in a Banach space of regular Borel measures, we prove existence and uniqueness of

the solution of the problem. A possible approach to identify the localization of the photon source is finally proposed

Lisi, M.

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DOI 10.1393/ncc/i2010-10586-0Colloquia: ICTT2009IL NUOVO CIMENTO Vol. 33 C, N. 1 Gennaio-Febbraio 2010Study of a localized photon source in spaces of measuresM. Lisi(∗)Universita` degli Studi di Siena - Siena, Italy(ricevuto il 29 Ottobre 2009; approvato il 5 Febbraio 2010; pubblicato online il 23 Marzo 2010)Summary. — In this paper we study a three-dimensional photon transport problemin an interstellar cloud, with a localized photon source inside. The problem is solvedindirectly, by defining the adjoint of an operator acting on an appropriate space ofcontinuous functions. By means of sun-adjoint semigroups theory of operators ina Banach space of regular Borel measures, we prove existence and uniqueness ofthe solution of the problem. A possible approach to identify the localization of thephoton source is finally proposed.PACS 28.20.-v – Neutron physics.PACS 98.38.-j – Interstellar medium (ISM) and nebulae in Milky Way.1. – IntroductionThe interstellar space, far from being a vacuum, can be considered a “chemical lab-oratory” responsible of the birth of stars and galaxies. Developments in astrophysicsduring the last decades made this subject in the forefront of general scientific and pop-ular interest.The intergalactic medium is mainly composed of hydrogen (90%), about 10% of theatoms are helium, and a further 0.1% of atoms are carbon, nitrogen and oxygen. Otherelements are even less abundant. Mixed with gas are “dust” grains of silicon and car-bonates [1].Matter is concentrated in big clouds (nebulae or intestellar clouds), whose dimen-sions are of the order of ten lights years, i.e. between 10−1 and 10 parsec (one parsec isabout 3 · 1013 kilometers). Note that the diameter of the solar system is of the order of10−4 parsec. The numerical density of the particles inside an interstellar cloud rangesfrom 103 to 106 particles per cubic centimetre (Earth atmosphere density, at sea level, isapproximately 1019 particles per cubic centimetre, whereas in the intergalactic vacuumone can find 100 particles/cm3). This means that a nebula is extremely rarefied, but notso much as the intergalactic vacuum [2].(∗) E-mail: lisi7@unisi.itc© Societa` Italiana di Fisica 129130 M. LISIMathematical models in this context have been studied only in the last decade, start-ing from the one-dimensional case (see, for instance [3] and the references quoted therein).Also inverse procedures connected to this kind of problems represent a subject of interestbecause of their physical applications ([4] and therein). However, in some cases, the gen-eralization of the model to the three-dimensional problem does not seem to be so easy,because of the difficulties related to the mathematical analysis. In particular, the studyof the case of an emission nebula (like for example the Orion nebula), i.e. an interstel-lar cloud with a star inside (that can be modelled with a pointwise source, representedby a Dirac delta functional), has not been solved. From a mathematical point of viewthe difficulties connected to this kind of model are due to the delta-“function”, whichdoes not belong to a Banach space. The one-dimensional case has been solved, going toanalyse the problem in a locally convex space, in particular in a space of distributions,and by using semigroups theory in locally convex spaces [5,6]. In this paper, we proposea method to solve the three-dimensional case, going to study the model in a space ofmeasures and by using sun-adjoint semigroup theory. Applications of semigroup theoryon measure spaces for the study of transport equation are present in the literature: see,for instance, [7, 8] (Chapter XI, sect. 6) [9] and the references quoted therein.We consider a photon transport in a homogeneous cloud, which occupies a closed andconvex region of the space V ⊂ R3 (bounded by the closed regular suface ∂V and withVi the interior of V , so that V = Vi∪∂V ) with a source q (for example, a star), localizedin a point x = x0 inside the nebula. In particular, the pointwise photon source can bemodelled by means of the Dirac delta-“function”. The photon transport equation in theinterstellar space reads as follows [3, 10]:(1)1c∂∂tU = −u · ∇uU − σU +∫S2σsUdu′ + q,where c is the light speed, U = U(x,u, t) is the photon number density, so thatU(x,u, t)dxdu represents the expected number of photons which, at time t, are in the vol-ume element dx centered at x ∈ V and having velocity within the solid angle du aroundthe unit vector u belonging to the unit sphere S2; σ = σ(x,u) and σs = σs(x,u) arethe total and the scattering cross-sections, respectively, represented by two non-negativecontinuous functions on V × S2; q = q(x, t) = q0(t)δ(x− x0) is the photon source local-ized in x0 ∈ Vi, with δ the three-dimensional Dirac delta functional and q0(t) a givencontinuous function of time.In particular, we assume that U satisfies the so called no-reentry boundary conditions:(2) U(x,u, t) = 0, for x ∈ ∂V and ingoing u,i.e. no photons come in the nebula from outside.In order to study the abstract version of problem (1), (2), we define the “standard”Boltzmann operator (we shall call it A). This has been studied usually as an operatoracting on some Lp space. However, since the unknown function U(x,u, t) is a distributionof photons and the source term is modelled with a Dirac delta-“function”, it seems thatthe most natural space in which the Boltzmann operator should act is a space of measureson the set V × S2 ⊂ R6. In particular, we define this operator acting in a Banach spaceof regular Borel measures. We are able to define it indirectly as the adjoint operator ofan operator B, acting on an appropriate space of continuous functions on V × S2. Bymeans of the properties of the preadjoint operator, by using the adjoint of the semigroupSTUDY OF A LOCALIZED PHOTON SOURCE IN SPACES OF MEASURES 131generated by B, we prove existence and uniqueness of the transport equation in a suitablespace of measures. Finally also an inverse procedure can be implemented, in order tolocalized the pointwise photon source inside the nebula.2. – Study of the preadjoint operatorLet E be the following locally compact metric space:(3) E ={(x,u) : x ∈ Vi or x ∈ ∂V and u = 0,u ingoing} ⊂ V × S2.We consider two spaces of continuous functions defined as follows:(4) X = C(V × S2), Y ={f ∈ X : f(x,u) = 0,∀x ∈ ∂V,u outgoing} ⊂ X.In particular, Y is a Banach space with the norm(5) ‖f‖ = sup ∣∣f(x,u)∣∣, ∀(x,u) ∈ E.Remark 2.1. The space Y is the closure of the linear set of all continuous functionswith compact support in the set E. Moreover, by applying the Riesz RepresentationTheorem, we have that the dual space Y ∗ of the space Y is isometrically isomorphic tothe Banach space of all regular Borel measures M(E) on E (which contains the Diracdelta functional). In particular, the correspondence between M(E) and Y ∗ depends onthe fact that for each linear functionals F ∈ Y ∗, there exists a uniquely determinedmeasure μ ∈ M(E) such thatF (f) =∫Ef(x,u)dμ(x,u), ∀f ∈ Y and ‖f‖ = ‖μ‖,where‖μ‖ = |μ|(E) = supf∈Y,‖f‖=1∫Ef(x,u)dμ(x,u),with |μ| the total variation of μ [11]. Let us define the Boltzmann operator A such that(6) Af = −cu · ∇uf − cσf + c∫S2σsfdu′withD(A) ={f ∈ X : Af ∈ X, f(x,u) = 0,∀x ∈ ∂V,u ingoing}.Note that D(A) “contains” boundary conditions (2).By using A, the abstract version of system (1)-(2) reads(7)⎧⎨⎩dU(t)dt= AU(t) + q(t),U(0) = U0,132 M. LISIwith U0 a given function in X and where U(t) must be considered as a function from[0,+∞) into X [3, 6].To prove existence and uniqueness of the solution of system (7), we use a “preadjointapproach”, i.e. we do not analyse properties of A directly, but we are going to studythe properties of another operator, the preadjoint of the operator A. To this aim, let usdefine the operator B such that(8) Bf = cu · ∇uf − cσf + c∫S2σsfdu′, D(B) = {f ∈ Y : Bf ∈ Y }.Remark 2.2. The operator B is the preadjoint Boltzmann operator, i.e. it is the for-mal preadjoint of the operator A. In fact, its domain “contains” the adjoint boundaryconditions (see (4)). In particular, the operator B can be split into two parts B = S + L, such that(9) Sf = cu · ∇uf − cσf, D(S) = {f ∈ Y : Sf ∈ Y }and(10) Lf =∫S2σsfdu′, D(L) = Y, R(L) = X.We want to study the properties of B (D(S + L) = D(S)), in order to prove that thepreadjoint Boltzmann system(11)⎧⎨⎩df(t)dt= Bf(t),f(0) = f0,has a unique solution in a suitable subset of Y , for a given f0 in such a set.If we extend the domain of the operator S, by defining the operator S˜, such that(12) S˜f = Sf, D(S˜) ={f ∈ Y : S˜f ∈ X},we can extend also the operator B, in such a way that B˜ = S˜ +L. The following lemmaholds.Lemma 2.1. If L is a bounded operator, there exists a linear subset H ⊂ Y , dense in Y ,such that the operator B˜H , resctriction of the operator B˜ on the set H, maps H into Yand it is the generator of a C0−semigroup of operators {T (t); t ≥ 0} in Y .Proof:Since Y is a closed subspace of the Banach space X, S is the generator of a C0-semigroupof operators acting on Y , S˜ is an operator such that its domain is dense in Y , S˜f = Sf ,for any f ∈ D(S) and, finally, (λI − S˜)−1 exists as a bounded operator, for suitable λ(where I is the inclusion Y → X with If = f, for all f ∈ Y ), hypotheses of Lemma 2.2of [12] are satisfied. Hence, the lemma follows. STUDY OF A LOCALIZED PHOTON SOURCE IN SPACES OF MEASURES 133Let us now give a redefinition of B:(13) B = B˜H , D(B) = H ⊂ Y.Theorem 2.1. If L is a bounded operator, then system (11) has a unique solution in Y ,for any f ∈ D(B).Proof:The theorem follows from semigroup theory of operators [13]. 3. – Transport equation in the space of measuresIn sect. 2, we denoted by M(E) the space of regular Borel measures on E (see (3) andRemark 2.2). The Banach space M(E) is dual to the space Y . In this section, in orderto use the properties proved in sect. 2, we shall define the operator A (see (6)) in thespace M(E) as a restriction of the adjoint operator B∗. In fact, since the operator B,defined in (13), is unbounded, closed and densely defined, its adjoint operator B∗ existsand it is closed. In particular, B∗ needs not be densely defined, consequently it is notalways generator of a C0-semigroup. We use sun-adjoint semigroups theory of [14] andits notation.Let M be the subspace in M(E), such that M = D(B∗). Then the operator(14) A : D(A) ⊂ M → M, Aμ = B∗μ,with(15) D(A) ={μ ∈ D(B∗) : B∗μ ⊂ M} ⊂ M,is a restriction of B∗, adjoint operator of the operator B, to the subspace M.Lemma 3.1. If L is bounded operator, the operator A defined in (14), acting in the sub-space M of the space of measures on E, is the generator of a C0-semigroup {Z(t); t ≥ 0}of positive operators.Proof:Since the operator A is densely defined in M, the theorem follows from sun-adjointsemigroups theory in [14]. Remark 3.1. The semigroup {Z(t); t ≥ 0} generated by A is connected to the semigroup{T (t); t ≥ 0} generated by B as follows:(16) Z(t)μ = T ∗(t)μ, ∀μ ∈ M∗,where {T ∗(t); t ≥ 0} is the adjoint semigroup of Z(t). The subspace M is exactly theset of measures μ ∈ M(S), for which the family is strongly continuous at t = 0. Theorem 3.1. System (7), with A defined in (14), is uniquely solvable, for any initialcondition U0 ∈ D(A).134 M. LISIProof:Since q is continuous in t ≥ 0 and locally Lipschitz continuous in x, uniformly on boundedintervals, there is a Tmax ≤ ∞ such that system (7) admits a unique mild solution on[0, Tmax[, for any U0 ∈ D(A) (see Theorem 1.4, Chapter 6 of [15] on standard argumentsof semigroups theory, and Theorem 4.3.11 of [14] on perturbation theory of sun-adjointsemigroups). Hence, thanks to an indirect approach, which consists on defining A (acting in asuitable Banach space of regular Borel measures) as the adjoint operator of its preadjointoperator B (acting on an appropriate space of continuous functions on V × S2), we areable to prove existence and uniqueness of the solution of the initial system (1).Remark 3.2. Assume that a suitable instrument (for instance, a radio-telescope) is lo-cated at xˆ ∈ R3 − V (where xˆ ∈ R3 − V is “far” from the cloud) and that it mea-sures the photon flux arriving from the source localized at x0 ∈ V . This means thatsuch an instrument detects photons passing through xˆ and having velocity cuˆ, withuˆ = (xˆ−x0)/|xˆ−x0|. It is possible to perform a procedure (inverse problem in trasporttheory [16]), which permits to identify the position x0 of the photon source [17,18]. Alsoa numerical approach similar to that proposed in [4] can be given. ∗ ∗ ∗This work was partially supported by M.U.R.S.T. and G.N.F.M. research funds andby the project PAR 2006 of the University of Siena “Metodi e modelli matematici per leapplicazioni”.REFERENCES[1] Dyson J. E. and Williams D. A., The Physics of the Interstellar Medium (Institute ofPhysics Publishing, Bristol) 1997.[2] Spitzer L., Physical Processes in the Interstellar Medium (John Wiley & and Sons,Toronto) 1978.[3] Belleni Morante A., Lamb W. and McBride A. C., Anal. Appl., 5 (2007) 77.[4] Belleni Morante A., Monaco R., Riganti R. and Salvarani F., Appl. Math.Comput., 154 (2004) 115.[5] Choe Y. H., J. Math. Anal. Appl., 16 (1985) 293.[6] Lisi M. and Totaro S., Math. Meth. Appl. Sci., 29 (2006) 1019.[7] Beals R. and Protopopescu V., J. Math. Anal. Appl., 121 (1987) 370.[8] Greenberg W., van der Mee C. and Protopopescu V., Boundary Value Problems inAbstract Kinetic Theory (Birkha¨user, Basel) 1987.[9] Nelson P. jr. and Victory H. D., Ann. Mat. Pura Appl., 112 (1977) 69.[10] Pomraning G. C., The Equation of Radiation Hydrodynamics (Pergamon Press, Oxford)1973.[11] Rudin W., Functional Analysis (McGraw Hill, New York) 1973.[12] Suhadolc A. and Vidav I., Math. Balk., 3 (1973) 514.[13] Belleni Morante A., Applied Semigroups and Evolution Equations (Oxford UniversityPress, Oxford) 1979.[14] Van Neerven J. M. A. M., The Adjoint of a Semigroup of Linear Operators (SpringerVerlag, Berlin Heidelberg) 1992.[15] Pazy A., Semigroups of Linear Operators and Applications to Partial DifferentialEquations (Springer Verlag, New York) 1983.[16] Zweifel P. F., Transp. Theory Stat. 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Study of a localized photon source in spaces of measures

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Study of a localized photon source in spaces of measures

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