Kinetic and Related Models (KRM)

Identification of photon sources, stochastically embedded in an interstellar cloud

Pages: 425 - 432, Volume 2, Issue 3, September 2009      doi:10.3934/krm.2009.2.425

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S. Aiello - Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo Fermi 2, 50125 Firenze, Italy (email)
Luigi Barletti - Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy (email)
Aldo Belleni-Morante - Dipartimento di Ingegneria Civile, Università di Firenze, Via S. Marta 3, 50139 Firenze, Italy (email)

Abstract: Photon transport is considered in an interstellar cloud containing one or several photon sources (stars), defined by $q_i\delta( x- x_{\i})\,i=1,2,\ldots,$ where the locations $x_i$'s are given in a stochastic way. First, the case is examined of a single source of intensity $q_1$ and located at $x_1$ with a probability density $p_1 = \p(x_1)$, such that $\p(x_1)\geq 0$ and $\int_V \p(x_1)\dx_1 = 1$, where $V \subset \R^3$ is the region occupied by the cloud. Then, a Boltzmann-like equation for the average photon distribution function < n >$(x,u;x_1)$ is derived and it is shown that $\p(x_1)$ can be evaluated starting from a far-field measurement of < n >. Finally, the case of two or more photon sources is discussed: the corresponding results are reasonably simple if $\p(x_1,x_2) = \p_1(x_1)\p_2(x_2)$, i.e. if the locations of the two photon source are "independent".

Keywords:  radiation transport, interstellar clouds, inverse problems.
Mathematics Subject Classification:  85A25, 35R30.

Received: November 2008;      Revised: November 2008;      Available Online: July 2009.