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Transfer equation in general curvilinear coordinates

Transfer equation in general curvilinear coordinates. Juris Freimanis Ventspils International Radio Astronomy Centre, Ventspils University College; Institute of Mathematics and Information Technology, Liepaja University e-mail: jurisf@venta.lv. Expanding the Universe

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Transfer equation in general curvilinear coordinates

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  1. Transfer equation in general curvilinear coordinates Juris Freimanis Ventspils International Radio Astronomy Centre, Ventspils University College; Institute of Mathematics and Information Technology, Liepaja University e-mail: jurisf@venta.lv Expanding the Universe Conference devoted to the 200th anniversary of Tartu Observatory Tartu, Estonia, April 27 – 29, 2011

  2. Introduction • Research in the theory of radiative transfer has a long tradition in Tartu Observatory (A.Sapar, T.Viik, R.Rõõm, A.Heinlo, I.Vurm and others): • studies of Ambartzumian / Chandrasekhar and Hopf functions, resolvent functions; • algorithms for the calculation of Voigt and Holtsmark functions; • multilayer atmospheres, Viik’s principle of invariance (Viik 1982); • theoretical studies of Rayleigh and molecular scattering; • scattering within spectral lines with redistribution of radiation over frequencies; • studies of the connection between RTE, classical and quantum electrodynamics (Sapar 1968); • NLTE phenomena in stellar atmospheres; • theory of Compton scattering, • etc.

  3. The differential operator of RTE • Stationary integro-differential radiative transfer equation in a polydisperse medium consists of the following basic terms: • the differential operator, sometimes called “the streaming operator”, • the extinction term, • the integral term describing scattering, • the term describing the primary sources of radiation. • The first of those will be the object of study in this contribution. Just it depends on the chosen coordinate system. • In inhomogeneous and / or anisotropic medium the linear polarization plane of electromagnetic radiation physically rotates around the direction of propagation: • S.M.Rytov. “Dokl. AN SSSR”,1938, vol. 18, pp. 263 – 268 (in Russian); • L.D.Landau, E.M.Lifshitz. Electrodynamics of Continuous Media. Moscow, “Nauka”, 1982, paragraph 85. (In Russian; English translation in 1984.) • L.A.Apresyan, Yu.A.Kravtsov. Radiation Transfer. Statistical and Wave Aspects. Amsterdam, Gordon and Breach Publishers, 1996. In particular, this happens if, due to refraction within inhomogeneous medium, the light ray moves along curve with nonzero torsion.

  4. Polarized radiative transfer equation in curved spacetime (general relativity) – accretion disks around neutron stars and black holes: • C. Fanton, M. Calvani, F. de Felice, A. Čadež. “Publications of the Astronomical Society of Japan”, 1997, vol. 49, pp. 159 – 169. • C.F. Gammie, J.C. McKinney, G. Tóth. “The Astrophysical Journal”, 2003, vol. 589, pp. 444 – 457. • A. Broderick, R. Blandford. “Monthly Notices of the Royal Astronomical Society”, 2003, vol. 342, pp. 1280 – 1290; ibid., 2004, vol. 349, pp. 994 – 1008. • C. Pitrou. “Classical and Quantum Gravity”, 2009, 26, Issue 6, pp. 065006. Broderick and Blandford (2004): due to general relativistic rotation of the polarization basis vectors e||m and e^m along the path of propagation, the linear polarization angle apparently rotates in vacuum: where df /dt is the angular rotation speed of polarization basis vectors.

  5. Some radiative transfer problems in approximately flat spacetime invites the researcher to use curvilinear coordinate systems: J.L.Ortiz et al. Observation of light echoes around very young stars. “Astronomy & Astrophysics”, 2010, vol. 519, A7, 8 pages

  6. Imaging-polarimetricmaps of IRAS 04395+3601in the (a) totalintensity I, (b) polarizedintensity Ip, (c) polarizationstrength P, and (d)polarization position angle. From:T. Ueta et al. HubbleSpaceTelescopeNICMOSimagingpolarimetryofprotoplanetarynebulae.II.Macromorphologyofthedustshellstructureviapolarizedlight. “The Astronomical Journal”, 2007, vol. 133, pp. 1345 – 1360.

  7. Near-infrared images of IRAS17150-3224: (a) F160W, (b)F222M, (c) flux-ratio map ofF160W/F606W, and (d)flux-ratio map of F222M/F160W. – From:K.Y.L. Su et al. High-resolution near-infrared imaging and polarimetry of four protoplanetary nebulae. “The Astronomical Journal”, 2003, vol. 126, pp. 848 – 862.

  8. (d): 40% > p > 30% (e): 30% > p > 10% (f): p < 10% Polarimetric results for IRAS17150-3224, displayed in differentranges of the percentagepolarization: (a) 70% > p > 60%, (b)60% > p > 50%, (c) 50% > p > 40%, (d)40% > p > 30%, (e) 30% > p > 10%, (f) p < 10%. – From:K.Y.L. Su et al. High-resolution near-infrared imaging and polarimetry of four protoplanetary nebulae. “The Astronomical Journal”, 2003, vol. 126, pp. 848 – 862.

  9. H2S(1) line emission (red) and nearby 2.15 m continuum intensity (green) in CRL 2688. A logarithmic stretch has been used for both images. – From:R.Sahai et al. The Structure of the Prototype Bipolar Protoplanetary Nebula CRL 2688 (Egg Nebula): Broadband, Polarimetric, and H2 Line Imaging with NICMOS on the Hubble Space Telescope. “The Astrophysical Journal”, 1998, vol. 492, L163 – L167.

  10. (a) NICMOS and (b) ground-based (K-band) polarimetric images of CRL 2688. The tick marks on both axes show offsets in arcseconds from the center. Orange lines indicate polarization and position angle (q ), with a maximum length of p = 85%. Each inverse gray-scale image is the logarithm of the total intensity. Vectors indicating the percentage polarization (length) and position angles are overplotted in all panels and represent the average value in 4 × 4 and 2 × 2 pixel bins in NICMOS and COB, respectively. – From:R.Sahai et al. The Structure of the Prototype Bipolar Protoplanetary Nebula CRL 2688 (Egg Nebula): Broadband, Polarimetric, and H2 Line Imaging with NICMOS on the Hubble Space Telescope. “The Astrophysical Journal”, 1998, vol. 492, L163 – L167.

  11. Below the main results from:J.Freimanis. On vector radiative transfer equation in curvilinear coordinate systems. – “JQSRT”, 2011, in press,will be delivered. The basic assumptions • It is desirable that the coordinate system at least partially reflects the symmetry of radiative transfer problem. Let as assume that: • The 4-dimensional spacetime is a pseudo-Euclidean (i.e. Minkowski) one. Consequently, the 3-dimensional space is Euclidean. It is equipped with the coordinate system {xi}, i = 1, 2, 3, and the corresponding metric tensor gik: The coordinate system is not assumed to be orthogonal. Orthogonal coordinate systems will be reviewed as a special (but very convienent) case. 2) The host medium is homogeneous and isotropic, and it is piecewise homogeneously filled with polydisperse scatterers. Consequently, plane electromagnetic waves propagate along straight lines within each homogeneous part of the space. The real part of the effective refractive index is almost isotropic, and there is no significant birefringence in the medium.

  12. A little reminder Accordingly to G.A.Korn and T.M.Korn (1968) one can define the covariant basis vectors ei, as well as the contravariant basis vectors ei: having the properties Arbitrary vector E(r) can be defined either by its contravariant components Ei(r) or by its covariant components Ei(r):

  13. The direction of propagation of radiation The direction of propagation of radiation can be characterized by the unit vector W: where the derivative is taken along the (straight) line of propagation. It is most convenient to characterize the direction of propagation by the spherical angles (J,j). Usually polar axis J = 0 and zero azimuthal plane j = 0 are tied to the basis vectors of the chosen spatial coordinate system in the given point. Let us designate each of the possible different tensor index values (1, 2, 3) with one of the letters a, b, c, where a¹b ¹c ¹a. Henceforth, let us define that the vector and tensor indices designated by a, b, c are fixed, i.e. the tensor expression of kind Aaibk Baibk involves summation over indices (i, k) but it does not involve summation over indices (a, b).

  14. Option A: W ec J S [ecea] / |[ecea]| = Se abc(gaaeb– gabea) / (gaag ccG) 1/2 ea j In this case the contravariant components of W are as follows:

  15. Option B: W ec J S [ecea] / |[ecea]| = Seabc (g aaeb– g abea) (G/(gaagcc)) 1/2 ea j Now the contravariant components of W are as follows:

  16. Orthogonal coordinate systems: In this case both options A and B are equivalent, and they lead to one and the same expressions

  17. Dyadics as tensors Dyadics used as the basic mathematical objects in M.I.Mishchenko, L.D.Travis and A.A.Lacis (2006) are essentially tensors (Morse & Feshbach 1953; Arfken 1970), obeying all the corresponding transformation laws in case of spatial coordinate transformation. They can be objects of absolute (covariant) differentiation etc. Two-point dyadics of type E(r) Ä E*(r’) are split tensors with contravariant components Ei(r)E*k(r’), covariant components Ei(r)E*k(r’), and mixed components Ei(r)E*k(r’) and Ei(r)E*k(r’) accordingly to Goląb (1974). Two-point dyadics implying parallel displacement of vectors and tensors such as the identity dyadic I(r,r’) and the differential dyadic (ÑrÄÑr ) × I(r,r’) in case r ¹r’ are uniquely defined if and only if the three-dimensional space is Euclidean, with zero Riemann curvature tensor.

  18. Propagation of the ladder specific coherency tensor in a polydisperse medium Rewriting dyadics defined in Mishchenko, Travis and Lacis (2006) as tensors, we have: the ladder coherency tensor and the ladder specific coherency tensor S(L)ik(r,W) as its directional component having the properties Due to transversality of S(L)ik(r,W) it is convenient to project it onto two orthogonal polarization basis unit vectors (e||, e^):

  19. The projections rik(r,W) of the ladder specific coherency tensor onto polarization basis vectors are the elements of the 2x2 specific coherency matrix: The process of radiative transfer is a true physical interaction between the scatterers, the primary sources, and the electromagnetic field. Consequently, radiative transfer equation contains the absolute derivative of the ladder specific coherency tensor: with Christoffel symbols of the second kind (Goląb (1974)) and the propagation tensor kmj(W):

  20. and with the understanding that or The last equation gives the derivatives of directional angles along the line of propagation:

  21. Option A:

  22. Option B:

  23. Orthogonal coordinate systems:

  24. Projecting the transfer equation for the ladder specific coherency tensor onto polarization basic vectors (e||, e^) and making up the Stokes vector from the usual linear combinations of the elements of the specific coherency matrix, one obtains radiative transfer equation for the Stokes vector in arbitrary curvilinear coordinate system: where the last term in the left hand side contains the speed of rotation dy/ds of polarization basis vectors around the direction of propagation W:

  25. and matrix U1 is the polarization reference plane rotation matrix: The rotation speed of polarization reference plane dy/ds is given by the following expressions: and for orthogonal coordinate systems

  26. Examples 1. Elliptic conical coordinate system Its standard definition (Morse and Feshbach 1953; Korn & Korn 1968) uses three coordinates (u, v, w) defining mutually orthogonal families of spheres, elliptic cones around z axis, and elliptic cones around x axis. These coordinates are 8-fold degenerated, and they have several other disadvantages. Alternative parameterization of essentially the same coordinate system: where the arithmetic values of all square roots are taken. Limitations on the parameter b and coordinates (r, Q, F):

  27. The “conical” viewpoint: polar angle J is measured from the normal to the conical surfaces. The differential operator of radiative transfer equation: where

  28. 2. Oblate spheroidal coordinate system with limitations The differential operator of radiative transfer equation:

  29. Conclusions • General expressions for the differential operator of vector radiative transfer equation in arbitrary curvilinear coordinate system have been obtained. • 2) It has been shown that the plane of linear polarization seemingly rotates in the vacuum if the polarization reference frame is tied to the basis vectors of general curvilinear coordinate system. • 3) Feasibility of the general method has been demonstrated by derivation of RTE in elliptic conical and oblate spheroidal coordinate systems. This study was financed from the basic budget of Ventspils International Radio Astronomy Centre, as well as from co-sponsorship by Ventspils City Council and from Latvian Council of Science grant No. 11.1856. Some financing was provided by Institute of Mathematical Sciences and Information Technologies of Liepaja University as well. The participation of the author at this conference is financed from ERDF’s grant “SATTEH” being implemented in Ventspils University College. The author expresses his gratitude to all these entities.

  30. Thank you for your attention!

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