Author name code: frisch ADS astronomy entries on 2022-09-14 author:"Frisch, Helene" ------------------------------------------------------------------------ Title: Effects of Angle-Dependent Partial Frequency Redistribution on Polarized Line Profiles Authors: Sampoorna, M.; Nagendra, K. N.; Frisch, H.; Stenflo, J. O. Bibcode: 2019ASPC..519..109S Altcode: Scattering of the solar limb-darkened radiation field on atoms and molecules produces linearly polarized spectrum of the Sun (Second Solar Spectrum). Partial frequency redistribution (PFR) plays a fundamental role in shaping the wings of linearly polarized profiles of strong resonance lines. Here we present the effects of the angle-dependent (AD) PFR on resonance polarization both in the presence and absence of magnetic fields. We consider scattering on a two-level atom with unpolarized lower level, and a one-dimensional isothermal atmosphere. Title: Non-conservative Rayleigh Scattering. A Perturbation Approach Authors: Frisch, H. Bibcode: 2019ASPC..519..101F Altcode: The continuous spectra of stellar and planetary atmospheres may be linearly polarized by Rayleigh or Thomson scattering. The polarization rate follows the Chandrasekhar's law for the Milne or diffuse reflection problems, if scattering processes are the only source of absorption and emission. Deviations from the Chandrasekhar's law due to true absorptions by bound-free or free-free transitions are derived from a perturbation analysis of the polarized radiative transfer equation, using as expansion parameter the ratio ɛ = κc/(κc + σc), with κc and σc the true absorption and scattering absorption coefficients. An expansion in powers of ɛ is proposed for the polarization rate. A comparison with numerical results shows an excellent fit for ɛ up to 10-3 but a systematic underestimation above this value. Title: Nonconservative Rayleigh scattering. A perturbation approach Authors: Frisch, H. Bibcode: 2019A&A...625A.125F Altcode: Context. The continuous spectrum of stellar and planetary atmospheres can be linearly polarized by Rayleigh or Thomson scattering. The polarization rate depends on the ratio κc/(κc + σc), κc and σc being the absorption coefficients due to photo-ionizations and scattering processes, respectively. The scattering process is conservative if κc = 0, and in this case the center-to-limb variation of the polarization rate follows Chandrasekhar's law. Deviations from this law appear if the scattering is nonconservative, that is, if photons have a probability ɛ = κc/(κc + σc) of being destroyed at each scattering.
Aims: Nonconservative Rayleigh scattering is addressed here with a perturbation point of view, using ɛ, assumed to be a constant, as an expansion parameter. The goal is to obtain a perturbation expansion of the polarized radiation field that can be used to measure of the effects of a nonzero ɛ on the polarization rate of the emergent radiation and to check the accuracy of numerical codes.
Methods: The expansion method is an application to Rayleigh scattering of a general perturbation approach developed for scalar monochromatic transport equations. The introduction of a space variable, rescaled by a factor √ɛ, transforms the radiative transfer equation into a new equation from which one can extract simpler equations to describe the field in the interior of the medium and in boundary layers.
Results: The perturbation method is applied to a plane-parallel slab with no incident radiation and an unpolarized primary source of photons. The interior and boundary layer fields are expanded in powers of √ɛ. The expansion of the interior radiation field shows that it is unpolarized at leading order, with an intensity i0(τ∼) satisfying a diffusion equation, and that the polarization appears at order ɛ. The emergent radiation is calculated up to and including order ɛ. The leading term yields the polarization rate of the Chandrasekhar's law. The following one, of order √ɛ, accurately predicts the decrease of the polarization rate for values of ɛ up to 10-3 and shows that it varies roughly as (1 - μ) for any unpolarized primary source. Methods for testing the accuracy of numerical schemes are proposed. The perturbation method is also applied to a slab with an incident radiation field and a polarized primary source of photons. Title: Analytic Results for Rayleigh Scattering and Resonance Polarization Authors: Frisch, H. Bibcode: 2019ASPC..526...19F Altcode: In a plane parallel semi-infinite atmosphere, where the scattering of photons follows the Rayleigh scattering law and all the absorbed photons are reemitted (conservative scattering), the polarized radiation field can be expressed in terms of two functions Hl(μ) and Hr(μ) introduced by Chandrasekhar. It will be shown how the vector radiative transfer problem for the polarized radiation field can be decomposed into two scalar equations leading to exact expressions for Hl(μ) and Hr(μ). For non conservative Rayleigh scattering, the radiation field can be expressed in terms of a matrix H(μ) satisfying a numerically solvable nonlinear integral equation. The method of construction of this nonlinear equation and its generalization to resonance scattering of spectral lines formed with complete frequency redistribution is outlined. Title: Large Area Flat Panel Imaging Detectors for Astronomy and Night Time Sensing Authors: Siegmund, O.; McPhate, J.; Frisch, H.; Elam, J.; Mane, A.; Wagner, R.; Varner, G. Bibcode: 2013amos.confE.105S Altcode: Sealed tube photo-sensing detectors for optical/IR detection have applications in astronomy, nighttime remote reconnaissance, and airborne/space situational awareness. The potential development of large area photon counting, imaging, timing detectors has significance for these applications and a number of other areas (High energy particle detection (RICH), biological single-molecule fluorescence lifetime imaging microscopy, neutron imaging, time of flight mass spectroscopy, diffraction imaging). We will present details of progress towards the development of a 20 cm sealed tube optical detector with nanoengineered microchannel plates for photon counting, imaging and sub-ns event time stamping. In the operational scheme of the photodetector incoming light passes through an entrance window and interacts with a semitransparent photocathode on the inside of the window. The photoelectrons emitted are accelerated across a proximity gap and are detected by an MCP pair. The pair of novel borosilicate substrate MCPs are functionalized by atomic layer deposition (ALD), and amplify the signal and the resulting electron cloud is detected by a conductive strip line anode for determination of the event positions and the time of arrival. The physical package is ~ 25 x 25 cm but only 1.5 cm thick. Development of such a device in a square 20 cm format presents challenges: hermetic sealing to a large entrance window, a 20 cm semitransparent photocathode with good efficiency and uniformity, 20 cm MCPs with reasonable cost and performance, robust construction to preserve high vacuum and withstand an atmosphere pressure differential. We will discuss the schemes developed to address these issues and present the results for the first test devices. The novel microchannel plates employing borosilicate micro-capillary arrays provide many performance characteristics typical of conventional MCPs, but have been made in sizes up to 20 cm, have low intrinsic background (0.08 events cm2 s-1) and have very stable gain behavior over > 7 C cm2 of charge extracted. They are high temperature compatible and have minimal outgassing, which shortens and simplifies the sealed tube production process and should improve overall lifetimes. Bialkali (NaKSb) semitransparent photocathodes with > 20% quantum efficiency have also been made on 20 cm borosilicate windows compatible with the window seals for the large sealed tube device. The photocathodes have good response uniformity and have been stable for > 5 months in testing. Tests with a 20 cm detector with a cross delay line readout have achieved ~50µm FWHM imaging with single photon sub-ns timing and MHz event rates, and tests with a 10 x 10cm detector with cross strip readout has achieved ~20µm FWHM imaging with >4 MHz event rates with ~10% deadtime. We will discuss the details and implications of these novel detector implementations and their potential applications. Title: Performance characteristics of atomic layer functionalized microchannel plates Authors: Siegmund, O. H. W.; Richner, N.; Gunjala, G.; McPhate, J. B.; Tremsin, A. S.; Frisch, H. J.; Elam, J.; Mane, A.; Wagner, R.; Craven, C. A.; Minot, M. J. Bibcode: 2013SPIE.8859E..0YS Altcode: Microchannel plates that have been constructed by atomic layer deposition of resistive and secondary emissive layers, onto borosilicate glass microcapillary arrays provide a novel alternative to conventional microchannel plates for detection of radiation and particles. Conventional microchannel plates can also benefit from atomic layer deposition of highly efficient secondary emissive layers. Our evaluations of these techniques have revealed unique features of atomic layer functionalized microchannel plates, including enhanced stability and lifetime, low background rates, and low levels of adsorbed gas. In addition borosilicate glass microcapillary arrays show enhanced physical and thermal robustness, which makes it possible to successfully fabricate large area devices (20 cm) with good uniformity of operational characteristics. Title: Latest Advancements in Microchannel Plate Detectors Authors: Vallerga, John; Siegmund, O.; McPhate, J. B.; Tremsin, A.; Welsh, B.; Frisch, H.; Wagner, R. G.; Elam, J.; Mane, A.; Varner, G. Bibcode: 2013HEAD...1312315V Altcode: Microchannel plate (MCP) detectors have been used in space-based astronomical detectors for decades (e.g. EUVE, ROSAT, Chandra, COS on Hubble) and their performance continues to improve. We will present the latest technological developments in MCP detectors, including very large format MCPs made from low background borosilicate glass (20cm x 20cm MCPs); functionalized by atomic layer deposition of semiconductor layers and high electron emission coefficient layers which substantially increase the lifetime of the output gain; high speed cross strip readouts (> 4MHz count rate at 10% deadtime); and photocathode development for increased QE in the UV and x-ray bands. We will also present our efforts to raise the technology readiness level of our cross-strip electronics to level 6 by developing low power and low mass ASICs that can be space qualified. This work is supported by NASA Grants NNX12AF46A, NNG11AD54G and DOE grant #DE-AC02-06CH11357. Title: Large Area and High Efficiency Photon Counting Imaging Detectors with High Time and Spatial Resolution for Night Time Sensing and Astronomy Authors: Siegmund, O.; Vallerga, J.; Tremsin, A.; McPhate, J.; Frisch, H.; Elam, J.; Mane, A.; Wagner, R.; Varner, G. Bibcode: 2012amos.confE..92S Altcode: The development of large area photon counting, imaging, timing detectors with high performance has significance for applications in astronomy (such as our sensor on the SAAO SALT 10m telescope), night time remote reconnaissance, airborne/space situational awareness, and high-speed adaptive optics. Sealed tube configurations for optical/IR sensing also have applications in detection of Cherenkov light (RICH), biological single-molecule fluorescence lifetime imaging microscopy and neutron imaging applications. In open faced configurations these devices are important for UV and particle detection in space astrophysics, mass spectroscopy and many time-of flight applications. Currently available devices are limited to sizes of about 5 cm and use either conventional microchannel plates, or dynode multipliers for amplification, coupled coarse pad array readouts. Extension of these schemes to devices as large as 20 cm with high spatial resolution presents significant problems and potentially considerable cost. A collaboration (Large Area Picosecond Photon Detector) of the U. Chicago, Argonne National Laboratory, U.C. Berkeley, U. Hawaii and a number of other institutions has developed novel technologies to realize 20 cm format detectors in open face or sealed tube configurations. One critical component of this development is novel microchannel plates employing borosilicate micro-capillary arrays. The microchannel plates are based on a novel concept where the substrate is constructed from a borosilicate micro-capillary array that is made to function as a microchannel plate by deposition of resistive and secondary emissive layers using atomic layer deposition. The process is relatively inexpensive compared with conventional microchannel plates and allows very large microchannel plates to be produced with pore sizes as small as 10 microns. These provide many performance characteristics typical of conventional microchannel plates, but have been made in sizes up to 20 cm, have low intrinsic background (<0.1 events/sq-cm/sec) and high stability with no observed gain degradation behavior over at least 5 Coul/sq-cm of charge extraction. Initial tests in a 20 cm detector with a cross strip electronic readout have achieved 4k x 4k pixel imaging with single photon sub-ns timing and MHz event rates. In concert with this effort we have made stable, uniform 20 cm bialkali photocathodes with >20% quantum efficiency on borosilicate windows compatible with a large sealed tube device. Other related efforts have also produced small sealed tubes with 30% quantum efficiency GaAs sealed tubes with high resolution imaging and timing that are immediately applicable to current applications, and opaque GaN UV photocathodes directly deposited onto these novel microchannel plates. We will discuss the details and implications of these novel microchannel plates with respect to the realization of novel detectors up to 20 cm format with reasonable cost and performance, robust construction, high flexibility of format and readout, reduction of fabrication effort, dramatically increased lifetime and stability, and their potential applications. Title: Forward-scattering Hanle effect in the solar Ca I 4227 Å line Authors: Frisch, H.; Anusha, L. S.; Bianda, M.; Holzreuter, R.; Nagendra, K. N.; Ramelli, R.; Sampoorna, M.; Smitha, H. N.; Stenflo, J. O. Bibcode: 2012EAS....55...59F Altcode: High sensitivity spectropolarimetric observations of the four Stokes parameters of the solar Ca I 4227 Å line have been performed in October 2010 at IRSOL with the ZIMPOL polarimeter, near the disk center, outside an active region (Bianda et al. 2011). They were analyzed in Anusha et al. 2011 with a combination of detailed radiative transfer modelling of the Hanle effect for the linear polarization and weak field Zeeman approximation for the circular polarization. This approach made possible a unique determination of the magnetic field vector at various positions along the slit of the spectrograph. A summary of the observations and of their analysis is presented here. Title: Search for anomalous production of multiple leptons in association with W and Z bosons at CDF Authors: Aaltonen, T.; Álvarez González, B.; Amerio, S.; Amidei, D.; Anastassov, A.; Annovi, A.; Antos, J.; Apollinari, G.; Appel, J. A.; Arisawa, T.; Artikov, A.; Asaadi, J.; Ashmanskas, W.; Auerbach, B.; Aurisano, A.; Azfar, F.; Badgett, W.; Bae, T.; Barbaro-Galtieri, A.; Barnes, V. E.; Barnett, B. A.; Barria, P.; Bartos, P.; Bauce, M.; Bedeschi, F.; Behari, S.; Bellettini, G.; Bellinger, J.; Benjamin, D.; Beretvas, A.; Bhatti, A.; Bisello, D.; Bizjak, I.; Bland, K. R.; Blumenfeld, B.; Bocci, A.; Bodek, A.; Bortoletto, D.; Boudreau, J.; Boveia, A.; Brigliadori, L.; Bromberg, C.; Brucken, E.; Budagov, J.; Budd, H. S.; Burkett, K.; Busetto, G.; Bussey, P.; Buzatu, A.; Calamba, A.; Calancha, C.; Camarda, S.; Campanelli, M.; Campbell, M.; Canelli, F.; Carls, B.; Carlsmith, D.; Carosi, R.; Carrillo, S.; Carron, S.; Casal, B.; Casarsa, M.; Castro, A.; Catastini, P.; Cauz, D.; Cavaliere, V.; Cavalli-Sforza, M.; Cerri, A.; Cerrito, L.; Chen, Y. C.; Chertok, M.; Chiarelli, G.; Chlachidze, G.; Chlebana, F.; Cho, K.; Chokheli, D.; Chung, W. H.; Chung, Y. S.; Ciocci, M. A.; Clark, A.; Clarke, C.; Compostella, G.; Convery, M. E.; Conway, J.; Corbo, M.; Cordelli, M.; Cox, C. A.; Cox, D. J.; Crescioli, F.; Cuevas, J.; Culbertson, R.; Dagenhart, D.; d'Ascenzo, N.; Datta, M.; de Barbaro, P.; Dell'Orso, M.; Demortier, L.; Deninno, M.; Devoto, F.; d'Errico, M.; Di Canto, A.; Di Ruzza, B.; Dittmann, J. R.; D'Onofrio, M.; Donati, S.; Dong, P.; Dorigo, M.; Dorigo, T.; Ebina, K.; Elagin, A.; Eppig, A.; Erbacher, R.; Errede, S.; Ershaidat, N.; Eusebi, R.; Farrington, S.; Feindt, M.; Fernandez, J. P.; Field, R.; Flanagan, G.; Forrest, R.; Frank, M. J.; Franklin, M.; Freeman, J. C.; Frisch, H.; Funakoshi, Y.; Furic, I.; Gallinaro, M.; Garcia, J. E.; Garfinkel, A. F.; Garosi, P.; Gerberich, H.; Gerchtein, E.; Giagu, S.; Giakoumopoulou, V.; Giannetti, P.; Gibson, K.; Ginsburg, C. M.; Giokaris, N.; Giromini, P.; Giurgiu, G.; Glagolev, V.; Glenzinski, D.; Gold, M.; Goldin, D.; Goldschmidt, N.; Golossanov, A.; Gomez, G.; Gomez-Ceballos, G.; Goncharov, M.; González, O.; Gorelov, I.; Goshaw, A. T.; Goulianos, K.; Grinstein, S.; Grosso-Pilcher, C.; Group, R. C.; Guimaraes da Costa, J.; Hahn, S. R.; Halkiadakis, E.; Hamaguchi, A.; Han, J. Y.; Happacher, F.; Hara, K.; Hare, D.; Hare, M.; Harr, R. F.; Hatakeyama, K.; Hays, C.; Heck, M.; Heinrich, J.; Herndon, M.; Hewamanage, S.; Hocker, A.; Hopkins, W.; Horn, D.; Hou, S.; Hughes, R. E.; Hurwitz, M.; Husemann, U.; Hussain, N.; Hussein, M.; Huston, J.; Introzzi, G.; Iori, M.; Ivanov, A.; James, E.; Jang, D.; Jayatilaka, B.; Jeon, E. J.; Jindariani, S.; Jones, M.; Joo, K. K.; Jun, S. Y.; Junk, T. R.; Kamon, T.; Karchin, P. E.; Kasmi, A.; Kato, Y.; Ketchum, W.; Keung, J.; Khotilovich, V.; Kilminster, B.; Kim, D. H.; Kim, H. S.; Kim, J. E.; Kim, M. J.; Kim, S. B.; Kim, S. H.; Kim, Y. K.; Kim, Y. J.; Kimura, N.; Kirby, M.; Klimenko, S.; Knoepfel, K.; Kondo, K.; Kong, D. J.; Konigsberg, J.; Kotwal, A. V.; Kreps, M.; Kroll, J.; Krop, D.; Kruse, M.; Krutelyov, V.; Kuhr, T.; Kurata, M.; Kwang, S.; Laasanen, A. T.; Lami, S.; Lammel, S.; Lancaster, M.; Lander, R. L.; Lannon, K.; Lath, A.; Latino, G.; LeCompte, T.; Lee, E.; Lee, H. S.; Lee, J. S.; Lee, S. W.; Leo, S.; Leone, S.; Lewis, J. D.; Limosani, A.; Lin, C. -J.; Lindgren, M.; Lipeles, E.; Lister, A.; Litvintsev, D. O.; Liu, C.; Liu, H.; Liu, Q.; Liu, T.; Lockwitz, S.; Loginov, A.; Lucchesi, D.; Lueck, J.; Lujan, P.; Lukens, P.; Lungu, G.; Lys, J.; Lysak, R.; Madrak, R.; Maeshima, K.; Maestro, P.; Malik, S.; Manca, G.; Manousakis-Katsikakis, A.; Margaroli, F.; Marino, C.; Martínez, M.; Mastrandrea, P.; Matera, K.; Mattson, M. E.; Mazzacane, A.; Mazzanti, P.; McFarland, K. S.; McIntyre, P.; McNulty, R.; Mehta, A.; Mehtala, P.; Mesropian, C.; Miao, T.; Mietlicki, D.; Mitra, A.; Miyake, H.; Moed, S.; Moggi, N.; Mondragon, M. N.; Moon, C. S.; Moore, R.; Morello, M. J.; Morlock, J.; Movilla Fernandez, P.; Mukherjee, A.; Muller, Th.; Murat, P.; Mussini, M.; Nachtman, J.; Nagai, Y.; Naganoma, J.; Nakano, I.; Napier, A.; Nett, J.; Neu, C.; Neubauer, M. S.; Nielsen, J.; Nodulman, L.; Noh, S. Y.; Norniella, O.; Oakes, L.; Oh, S. H.; Oh, Y. D.; Oksuzian, I.; Okusawa, T.; Orava, R.; Ortolan, L.; Pagan Griso, S.; Pagliarone, C.; Palencia, E.; Papadimitriou, V.; Paramonov, A. A.; Patrick, J.; Pauletta, G.; Paulini, M.; Paus, C.; Pellett, D. E.; Penzo, A.; Phillips, T. J.; Piacentino, G.; Pianori, E.; Pilot, J.; Pitts, K.; Plager, C.; Pondrom, L.; Poprocki, S.; Potamianos, K.; Prokoshin, F.; Pranko, A.; Ptohos, F.; Punzi, G.; Rahaman, A.; Ramakrishnan, V.; Ranjan, N.; Redondo, I.; Renton, P.; Rescigno, M.; Riddick, T.; Rimondi, F.; Ristori, L.; Robson, A.; Rodrigo, T.; Rodriguez, T.; Rogers, E.; Rolli, S.; Roser, R.; Ruffini, F.; Ruiz, A.; Russ, J.; Rusu, V.; Safonov, A.; Sakumoto, W. K.; Sakurai, Y.; Santi, L.; Sato, K.; Saveliev, V.; Savoy-Navarro, A.; Schlabach, P.; Schmidt, A.; Schmidt, E. E.; Schwarz, T.; Scodellaro, L.; Scribano, A.; Scuri, F.; Seidel, S.; Seiya, Y.; Semenov, A.; Sforza, F.; Shalhout, S. Z.; Shears, T.; Shepard, P. F.; Shimojima, M.; Shochet, M.; Shreyber-Tecker, I.; Simonenko, A.; Sinervo, P.; Sliwa, K.; Smith, J. R.; Snider, F. D.; Soha, A.; Sorin, V.; Song, H.; Squillacioti, P.; Stancari, M.; St. Denis, R.; Stelzer, B.; Stelzer-Chilton, O.; Stentz, D.; Strologas, J.; Strycker, G. L.; Sudo, Y.; Sukhanov, A.; Suslov, I.; Takemasa, K.; Takeuchi, Y.; Tang, J.; Tecchio, M.; Teng, P. K.; Thom, J.; Thome, J.; Thompson, G. A.; Thomson, E.; Toback, D.; Tokar, S.; Tollefson, K.; Tomura, T.; Tonelli, D.; Torre, S.; Torretta, D.; Totaro, P.; Trovato, M.; Ukegawa, F.; Uozumi, S.; Varganov, A.; Vázquez, F.; Velev, G.; Vellidis, C.; Vidal, M.; Vila, I.; Vilar, R.; Vizán, J.; Vogel, M.; Volpi, G.; Wagner, P.; Wagner, R. L.; Wakisaka, T.; Wallny, R.; Wang, S. M.; Warburton, A.; Waters, D.; Wester, W. C., III; Whiteson, D.; Wicklund, A. B.; Wicklund, E.; Wilbur, S.; Wick, F.; Williams, H. H.; Wilson, J. S.; Wilson, P.; Winer, B. L.; Wittich, P.; Wolbers, S.; Wolfe, H.; Wright, T.; Wu, X.; Wu, Z.; Yamamoto, K.; Yamato, D.; Yang, T.; Yang, U. K.; Yang, Y. C.; Yao, W. -M.; Yeh, G. P.; Yi, K.; Yoh, J.; Yorita, K.; Yoshida, T.; Yu, G. B.; Yu, I.; Yu, S. S.; Yun, J. C.; Zanetti, A.; Zeng, Y.; Zucchelli, S. Bibcode: 2012PhRvD..85i2001A Altcode: 2012arXiv1202.1260T This paper presents a search for anomalous production of multiple low-energy leptons in association with a W or Z boson using events collected at the CDF experiment corresponding to 5.1fb-1 of integrated luminosity. This search is sensitive to a wide range of topologies with low-momentum leptons, including those with the leptons near one another. The observed rates of production of additional electrons and muons are compared with the standard model predictions. No indications of phenomena beyond the standard model are found. A 95% confidence level limit is presented on the production cross section for a benchmark model of supersymmetric hidden-valley Higgs production. Particle identification efficiencies are also provided to enable the calculation of limits on additional models. Title: Advances in microchannel plates and photocathodes for ultraviolet photon counting detectors Authors: Siegmund, O. H. W.; Fujiwara, K.; Hemphill, R.; Jelinsky, S. R.; McPhate, J. B.; Tremsin, A. S.; Vallerga, J. V.; Frisch, H. J.; Elam, J.; Mane, A.; Bennis, D. C.; Craven, C. A.; Deterando, M. A.; Escolas, J. R.; Minot, M. J.; Renaud, J. M. Bibcode: 2011SPIE.8145E..0JS Altcode: 2011SPIE.8145E.251S A new method of fabricating microchannel plates has been investigated, employing microcapillary arrays of borosilicate glass that are deposited with resistive and secondary emissive layers using atomic layer deposition. Microchannel plates of this kind have been made in sizes from 33 mm to 200 mm, with pore sizes of 40 μm and 20 μm, pore length to diameter ratios of 60:1, bias angles of 8°, and open areas from 60% to 83%. Tests with single MCPs and MCP pairs have been done and show good imaging quality, gain comparable to conventional MCPs, low background rates (~ 0.085 events sec-1 cm-2), fast pulse response, and good ageing characteristics. The quantum efficiency for bare and alkali halide coated MCPs is similar to conventional MCPs, and we have also been able to deposit opaque GaN(Mg) cathodes directly onto these MCPs. Title: Analysis of the Forward-scattering Hanle Effect in the Ca I 4227 Å Line Authors: Anusha, L. S.; Nagendra, K. N.; Bianda, M.; Stenflo, J. O.; Holzreuter, R.; Sampoorna, M.; Frisch, H.; Ramelli, R.; Smitha, H. N. Bibcode: 2011ApJ...737...95A Altcode: Coherent scattering of limb-darkened radiation is responsible for the generation of the linearly polarized spectrum of the Sun (the Second Solar Spectrum). This Second Solar Spectrum is usually observed near the limb of the Sun, where the polarization amplitudes are largest. At the center of the solar disk the linear polarization is zero for an axially symmetric atmosphere. Any mechanism that breaks the axial symmetry (like the presence of an oriented magnetic field, or resolved inhomogeneities in the atmosphere) can generate a non-zero linear polarization. In the present paper we study the linear polarization near the disk center in a weakly magnetized region, where the axisymmetry is broken. We present polarimetric (I, Q/I, U/I, and V/I) observations of the Ca I 4227 Å line recorded around μ = cos θ = 0.9 (where θ is the heliocentric angle) and a modeling of these observations. The high sensitivity of the instrument (ZIMPOL-3) makes it possible to measure the weak polarimetric signals with great accuracy. The modeling of these high-quality observations requires the solution of the polarized radiative transfer equation in the presence of a magnetic field. For this we use standard one-dimensional model atmospheres. We show that the linear polarization is mainly produced by the Hanle effect (rather than by the transverse Zeeman effect), while the circular polarization is due to the longitudinal Zeeman effect. A unique determination of the full \bm {B} vector may be achieved when both effects are accounted for. The field strengths required for the simultaneous fitting of Q/I, U/I, and V/I are in the range 10-50 G. The shapes and signs of the Q/I and U/I profiles are highly sensitive to the orientation of the magnetic field. Title: Observations of the forward scattering Hanle effect in the Ca I 4227 Å line Authors: Bianda, M.; Ramelli, R.; Anusha, L. S.; Stenflo, J. O.; Nagendra, K. N.; Holzreuter, R.; Sampoorna, M.; Frisch, H.; Smitha, H. N. Bibcode: 2011A&A...530L..13B Altcode: 2011arXiv1105.2157B Chromospheric magnetic fields are notoriously difficult to measure. The chromospheric lines are broad, while the fields are producing a minuscule Zeeman-effect polarization. A promising diagnostic alternative is provided by the forward-scattering Hanle effect, which can be recorded in chromospheric lines such as the He i 10 830 Å and the Ca i 4227 Å lines. We present a set of spectropolarimetric observations of the full Stokes vector obtained near the center of the solar disk in the Ca i 4227 Å line with the ZIMPOL polarimeter at the IRSOL observatory. We detect a number of interesting forward-scattering Hanle effect signatures, which we model successfully using polarized radiative transfer. Here we focus on the observational aspects, while a separate companion paper deals with the theoretical modeling. Title: Linear Polarization of the Solar Ca I 4227 Å Line: Modeling with Radiative Transfer and Last Scattering Approximation Authors: Anusha, L. S.; Stenflo, J. O.; Frisch, H.; Bianda, M.; Holzreuter, R.; Nagendra, K. N.; Sampoorna, M.; Ramelli, R. Bibcode: 2011ASPC..437...57A Altcode: To model the Ca I 4227 Å line polarization, radiative transfer effects with partial frequency redistribution (PRD) must be taken into account. The numerical solution of the relevant polarized radiative transfer (RT) equations is computationally very demanding. The “last scattering approximation” (LSA) is a concept allowing faster methods to be devised. It is based on the remark that a single scattering of the radiation field is sufficient for creating most of the polarization. Its key ingredient is the anisotropy of the radiation field. If the anisotropy is extracted from the observed center to limb variation of the intensity profile, only the wings of the Q/I spectrum can be modeled (Sampoorna et al. 2009). We show here that the core region may be modeled as well if one takes into account the depth variation of the anisotropy which is obtained from an unpolarized multilevel RT (Anusha et al. 2010). After a validation of the LSA approach by comparison with a polarized RT calculation, we apply both approaches to model recent observations of the Ca I 4227 Å line polarization taken on the quiet Sun. Apart from a global scaling factor, both approaches give a very good fit to the Q/I spectrum for all the wavelengths. As the LSA is 8 times faster than the RT approach, we can recommend it as an efficient method to analyze other strong resonance lines in the second solar spectrum. Title: Observations of the Solar Ca I 4227 Å Line Authors: Bianda, M.; Ramelli, R.; Stenflo, J. O.; Anusha, L. S.; Nagendra, K. N.; Sampoorna, M.; Holzreuter, R.; Frisch, H. Bibcode: 2011ASPC..437...67B Altcode: Our aim is to understand some interesting polarization features observed in the solar Ca I 4277 Å line. Here we only discuss the observational aspects. Observations have also been made in other chromospheric lines within a few hours of those in the Ca I 4227 Å line, in the same region near the north solar limb, to illustrate the potential of simultaneous observations in different lines. Title: Spectral line polarization with angle-dependent partial frequency redistribution. II. Accelerated lambda iteration and scattering expansion methods for the Rayleigh scattering Authors: Sampoorna, M.; Nagendra, K. N.; Frisch, H. Bibcode: 2011A&A...527A..89S Altcode: Context. The linear polarization of strong resonance lines observed in the solar spectrum is created by the scattering of the photospheric radiation field. This polarization is sensitive to the form of the partial frequency redistribution (PRD) function used in the line radiative transfer equation. Observations have been analyzed until now with angle-averaged PRD functions. With an increase in the polarimetric sensitivity and resolving power of the present-day telescopes, it will become possible to detect finer effects caused by the angle dependence of the PRD functions.
Aims: We devise new efficient numerical methods to solve the polarized line transfer equation with angle-dependent PRD, in plane-parallel cylindrically symmetrical media. We try to bring out the essential differences between the polarized spectra formed under angle-averaged and the more realistic case of angle-dependent PRD functions.
Methods: We use a recently developed Stokes vector decomposition technique to formulate three different iterative methods tailored for angle-dependent PRD functions. Two of them are of the accelerated lambda iteration type, one is based on the core-wing approach, and the other one on the frequency by frequency approach suitably generalized to handle angle-dependent PRD. The third one is based on a series expansion in the mean number of scattering events (Neumann series expansion).
Results: We show that all these methods work well on this difficult problem of polarized line formation with angle-dependent PRD. We present several benchmark solutions with isothermal atmospheres to show the performance of the three numerical methods and to analyze the role of the angle-dependent PRD effects. For weak lines, we find no significant effects when the angle-dependence of the PRD functions is taken into account. For strong lines, we find a significant decrease in the polarization, the largest effect occurring in the near wing maxima. Title: Polymers in Constrained Ceramic Environments Authors: Frisch, H. L.; Huang, L. Bibcode: 2011emst.book.7684F Altcode: There is a great deal of interest in the preparation and characterization of organic-inorganic nanocomposites, particularly in those materials in which an organic phase is ceramic constrained in such a way as to produce novel structures, frequently at the molecular level. Such ceramic-constrained environments include zeolites, clay galleries, and mesoporous silica. In some cases, a polymer is directly introduced into a constraining structure (clay galleries), while in others a constrained monomer is polymerized subsequently, for example in pores, to obtain the final composite material. This review focuses on some specific aspects of the morphology of organic-inorganic nanocomposites and how the confinement affects the properties of the polymer. Title: Spectral line polarization with angle-dependent partial frequency redistribution. I. A Stokes parameters decomposition for Rayleigh scattering Authors: Frisch, H. Bibcode: 2010A&A...522A..41F Altcode: Context. The linear polarization of a strong resonance lines observed near the solar limb is created by a multiple-scattering process. Partial frequency redistribution (PRD) effects must be accounted for to explain the polarization profiles. The redistribution matrix describing the scattering process is a sum of terms, each containing a PRD function multiplied by a Rayleigh type phase matrix. A standard approximation made in calculating the polarization is to average the PRD functions over all the scattering angles, because the numerical work needed to take the angle-dependence of the PRD functions into account is large and not always needed for reasonable evaluations of the polarization.
Aims: This paper describes a Stokes parameters decomposition method, that is applicable in plane-parallel cylindrically symmetrical media, which aims at simplifying the numerical work needed to overcome the angle-average approximation.
Methods: The decomposition method relies on an azimuthal Fourier expansion of the PRD functions associated to a decomposition of the phase matrices in terms of the Landi Degl'Innocenti irreducible spherical tensors for polarimetry T^K_Q(i, Ω) (i Stokes parameter index, Ω ray direction). The terms that depend on the azimuth of the scattering angle are retained in the phase matrices.
Results: It is shown that the Stokes parameters I and Q, which have the same cylindrical symmetry as the medium, can be expressed in terms of four cylindrically symmetrical components I_Q^K (K = Q = 0, K = 2, Q = 0, 1, 2). The components with Q = 1, 2 are created by the angular dependence of the PRD functions. They go to zero at disk center, ensuring that Stokes Q also goes to zero. Each component I_Q^K is a solution to a standard radiative transfer equation. The source term S_Q^K are significantly simpler than the source terms corresponding to I and Q. They satisfy a set of integral equations that can be solved by an accelerated lambda iteration (ALI) method. Title: Generalization of the Last Scattering Approximation for the Second Solar Spectrum Modeling: The Ca I 4227 Å Line as a Case Study Authors: Anusha, L. S.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.; Sampoorna, M.; Frisch, H.; Holzreuter, R.; Ramelli, R. Bibcode: 2010ApJ...718..988A Altcode: To model the second solar spectrum (the linearly polarized spectrum of the Sun that is due to coherent scattering processes), one needs to solve the polarized radiative transfer (RT) equation. For strong resonance lines, partial frequency redistribution (PRD) effects must be accounted for, which make the problem computationally demanding. The "last scattering approximation" (LSA) is a concept that has been introduced to make this highly complex problem more tractable. An earlier application of a simple LSA version could successfully model the wings of the strong Ca I 4227 Å resonance line in Stokes Q/I (fractional linear polarization), but completely failed to reproduce the observed Q/I peak in the line core. Since the magnetic field signatures from the Hanle effect only occur in the line core, we need to generalize the existing LSA approach if it is to be useful for the diagnostics of chromospheric and turbulent magnetic fields. In this paper, we explore three different approximation levels for LSA and compare each of them with the benchmark represented by the solution of the full polarized RT, including PRD effects. The simplest approximation level is LSA-1, which uses the observed center-to-limb variation of the intensity profile to obtain the anisotropy of the radiation field at the surface, without solving any transfer equation. In contrast, the next two approximation levels use the solution of the unpolarized transfer equation to derive the anisotropy of the incident radiation field and use it as an input. In the case of LSA-2, the anisotropy at level τλ = μ, the atmospheric level from which an observed photon is most likely to originate, is used. LSA-3, on the other hand, makes use of the full depth dependence of the radiation anisotropy. The Q/I formula for LSA-3 is obtained by keeping the first term in a series expansion of the Q-source function in powers of the mean number of scattering events. Computationally, LSA-1 is 21 times faster than LSA-2, which is 5 times faster than the more general LSA-3, which itself is 8 times faster than the polarized RT approach. A comparison of the calculated Q/I spectra with the RT benchmark shows excellent agreement for LSA-3, including good modeling of the Q/I core region with its PRD effects. In contrast, both LSA-1 and LSA-2 fail to model the core region. The RT and LSA-3 approaches are then applied to model the recently observed Q/I profile of the Ca I 4227 Å line in quiet regions of the Sun. Apart from a global scale factor both give a very good fit to the Q/I spectra for all the wavelengths, including the core peak and blend line depolarizations. We conclude that LSA-3 is an excellent substitute for the full polarized RT and can be used to interpret the second solar spectrum, including the Hanle effect with PRD. It also allows the techniques developed for unpolarized three-dimensional RT to be applied to the modeling of the second solar spectrum. Title: The Hanle Effect as Diagnostic Tool for Turbulent Magnetic Fields Authors: Anusha, L. S.; Sampoorna, M.; Frisch, H.; Nagendra, K. N. Bibcode: 2010ASSP...19..390A Altcode: 2010mcia.conf..390A The Hanle effect is calculated for a random magnetic field characterized by a finite correlation length and a probability density function of the magnetic field vector. It is shown that linear polarization is essentially independent of the magnetic field correlation length, but strongly depends on the distribution of the field strength. Title: The quiet Sun magnetic field observed with ZIMPOL on THEMIS. I. The probability density function Authors: Bommier, V.; Martínez González, M.; Bianda, M.; Frisch, H.; Asensio Ramos, A.; Gelly, B.; Landi Degl'Innocenti, E. Bibcode: 2009A&A...506.1415B Altcode: Context: The quiet Sun magnetic field probability density function (PDF) remains poorly known. Modeling this field also introduces a magnetic filling factor that is also poorly known. With these two quantities, PDF and filling factor, the statistical description of the quiet Sun magnetic field is complex and needs to be clarified.
Aims: In the present paper, we propose a procedure that combines direct determinations and inversion results to derive the magnetic field vector and filling factor, and their PDFs.
Methods: We used spectro-polarimetric observations taken with the ZIMPOL polarimeter mounted on the THEMIS telescope. The target was a quiet region at disk center. We analyzed the data by means of the UNNOFIT inversion code, with which we inferred the distribution of the mean magnetic field α B, α being the magnetic filling factor. The distribution of α was derived by an independent method, directly from the spectro-polarimetric data. The magnetic field PDF p(B) could then be inferred. By introducing a joint PDF for the filling factor and the magnetic field strength, we have clarified the definition of the PDF of the quiet Sun magnetic field when the latter is assumed not to be volume-filling.
Results: The most frequent local average magnetic field strength is found to be 13 G. We find that the magnetic filling factor is related to the magnetic field strength by the simple law α = B_1/B with B1 = 15 G. This result is compatible with the Hanle weak-field determinations, as well as with the stronger field determinations from the Zeeman effect (kGauss field filling 1-2% of space). From linear fits, we obtain the analytical dependence of the magnetic field PDF. Our analysis has also revealed that the magnetic field in the quiet Sun is isotropically distributed in direction.
Conclusions: We conclude that the quiet Sun is a complex medium where magnetic fields having different field strengths and filling factors coexist. Further observations with a better polarimetric accuracy are, however, needed to confirm the results obtained in the present work.

Based on observations made with the French-Italian telescope THEMIS operated by the CNRS and CNR on the island of Tenerife in the Spanish Observatorio del Teide of the Instituto de Astrofísica de Canarias. Present address: Instituto de Astrofísica de Canarias, vía Láctea s/n, 38205 La Laguna, Tenerife, Spain. Title: The Hanle effect in a random magnetic field. Dependence of the polarization on statistical properties of the magnetic field Authors: Frisch, H.; Anusha, L. S.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2009A&A...501..335F Altcode: Context: The Hanle effect is used to determine weak turbulent magnetic fields in the solar atmosphere, usually assuming that the angular distribution is isotropic, the magnetic field strength constant, and that micro-turbulence holds, i.e. that the magnetic field correlation length is much less than a photon mean free path.
Aims: To examine the sensitivity of turbulent magnetic field measurements to these assumptions, we study the dependence of Hanle effect on the magnetic field correlation length, its angular, and strength distributions.
Methods: We introduce a fairly general random magnetic field model characterized by a correlation length and a magnetic field vector distribution. Micro-turbulence is recovered when the correlation length goes to zero and macro-turbulence when it goes to infinity. Radiative transfer equations are established for the calculation of the mean Stokes parameters and they are solved numerically by a polarized approximate lambda iteration method.
Results: We show that optically thin spectral lines and optically very thick ones are insensitive to the correlation length of the magnetic field, while spectral lines with intermediate optical depths (around 10-100) show some sensitivity to this parameter. The result is interpreted in terms of the mean number of scattering events needed to create the surface polarization. It is shown that the single-scattering approximation holds good for thin and thick lines but may fail for lines with intermediate thickness. The dependence of the polarization on the magnetic field vector probability density function (PDF) is examined in the micro-turbulent limit. A few PDFs with different angular and strength distributions, but equal mean value of the magnetic field, are considered. It is found that the polarization is in general quite sensitive to the shape of the magnetic field strength PDF and somewhat to the angular distribution.
Conclusions: The mean field derived from Hanle effect analysis of polarimetric data strongly depends on the choice of the field strength distribution used in the analysis. It is shown that micro-turbulence is in general a safe approximation. Title: The Hanle Effect. Angle-dependent Frequency Redistribution Function. Decomposition of the Stokes Parameters in Irreducible Components. Authors: Frisch, H. Bibcode: 2009ASPC..405...87F Altcode: We show that the spherical tensor decomposition of the Stokes parameters developed for angle-averaged frequency redistribution functions can be extended to angle-dependent frequency redistribution functions. The irreducible components of the Stokes parameters loose their cylindrical symmetry but one can still write an integral equation for the irreducible components of the source vector. Numerical solution of this equation may be simplified by the introduction of an azimuthal Fourier expansion of the frequency redistribution function. Title: The CDF level 2 calorimetric trigger upgrade Authors: Bhatti, A.; Canepa, A.; Casarsa, M.; Convery, M.; Cortiana, G.; Dell'Orso, M.; Donati, S.; Flanagan, G.; Frisch, H.; Fukun, T.; Krop, D.; Giannetti, P.; Greco, V.; Jones, M.; Liu, T.; Lucchesi, D.; Pagan-Griso, S.; Pantano, D.; Pianori, E.; Piendibene, M.; Pilcher, C.; Ristori, L.; Rogondino, L.; Rusu, V.; Sartori, L.; Schmidt, M.; Veszpremi, V.; Vidal, M.; Zhou, L. Bibcode: 2009NIMPA.598..331B Altcode: CDF II upgraded the calorimeter trigger to cope with the higher detector occupancy due to the increased Tevatron instantaneous luminosity (∼2.8×10cms). While the original system was implemented in custom hardware and provided to the L2 trigger a limited-quality jet clustering performed using a reduced resolution measurement of the transverse energy in the calorimeter trigger towers, the upgraded system provides offline-quality jet reconstruction of the full resolution calorimeter data. This allows to keep better under control the dependence of the trigger rates on the instantaneous luminosity and to improve the efficiency and purity of the trigger selections. The upgraded calorimeter trigger uses the general purpose VME board Pulsar, developed at CDF II and already widely used to upgrade the L2 tracking and L2 decision systems. A battery of Pulsars is used to merge and send the calorimeter data to the L2 CPUs, where software-implemented algorithms perform offline-like clustering. In this paper we review the design and the performance of the upgraded system. Title: Zeeman line formation in solar magnetic fields. Studies with empirical probability distribution functions Authors: Sampoorna, M.; Nagendra, K. N.; Frisch, H.; Stenflo, J. O. Bibcode: 2008A&A...485..275S Altcode: Context: Numerical simulations of magneto-convection and analysis of solar magnetograms provide probability distribution functions (PDFs) for the magnetic field strength.
Aims: In the paper, we explore the effects of these PDFs on Zeeman line formation.
Methods: We calculate the mean Stokes parameters for a Milne-Eddington atmosphere in the limit of optically thin (micro-turbulent) and thick (macro-turbulent) magnetic structures and also the dispersion around the mean profiles in the optically thick limit. Several types of PDFs are considered: (a) Voigt function and stretched exponential type PDFs for fields with fixed direction but fluctuating strength; (b) a cylindrically symmetrical power law for the angular distribution of magnetic fields with given field strength; (c) composite PDFs accounting for randomness in both strength and direction obtained by combining a Voigt function or a stretched exponential with an angular power law. For optically thin structures, explicit expressions are given for the mean values of the Zeeman absorption matrix elements. We also describe how the averaging technique for a normal Zeeman triplet may be generalized to the more common case of anomalous Zeeman splitting patterns.
Results: We show that, for magnetic field rms fluctuations of the order of 6 G, consistent with observational data, Stokes I is essentially independent of the shapes of the PDFs but Stokes Q, U, and V and also the dispersion around the mean values are quite sensitive to the tail behavior of the PDF. We confirm a previous result that Stokes V is less sensitive to the scale of the magnetic structures than Stokes Q and U. The composite PDF proposed for the fluctuations of the magnetic field vector has an angular distribution peaked about the vertical direction for strong fields, and is isotropically distributed for weak fields; it can be used to mimic solar surface random fields. Title: Some aspects of polarized line formation in magneto-turbulent media Authors: Sampoorna, M.; Frisch, H.; Nagendra, K. N. Bibcode: 2008NewA...13..233S Altcode: Observations and numerical simulations of magneto-convection show a highly variable solar magnetic field. Using a statistical approach, we analyze the effects of random magnetic fields on Stokes profiles of spectral lines. We consider the micro and macro-turbulent regimes, which provide bounds for more general random fields with finite scales of variations. The mean Stokes parameters are obtained in the micro-turbulent regime, by first averaging the Zeeman propagation matrix Φ^ over the probability distribution function P( B) of the magnetic field and then solving the concerned radiative transfer equation. In the macro-turbulent regime, the mean solution is obtained by averaging the emergent solution over P( B). It is assumed that B has a Gaussian distribution defined by its mean field B0, angular distribution and dispersion. Fluctuations parallel and perpendicular to B0 are considered. Spectral lines are parameterized by their strength β, which is varied over the range 1-10 4. A detailed comparison of micro and macro-turbulent limit with mean field solution shows that differences are important for β ⩾ 10. When β increases, the saturation behavior of micro-turbulent profiles are significantly different from that of mean field profiles. The Stokes profiles shapes are explained in terms of the non-linear β-dependence of the Unno-Rachkovsky solution using approximate expressions for the mean absorption coefficients. These expressions when inserted in the Unno-Rachkovsky solution can predict Stokes profiles that match with the numerical result to a good approximation. Title: The Hanle effect. Decomposition of the Stokes parameters into irreducible components Authors: Frisch, H. Bibcode: 2007A&A...476..665F Altcode: Context: It has been shown for the weak-field Hanle effect that the Stokes parameters I, Q, and U can be represented by a set of six cylindrically symmetrical functions. The proof relies on azimuthal Fourier expansions of the radiation field and of the Hanle phase matrix. It holds for a plane-parallel atmosphere and scattering processes that can be described by a redistribution matrix where redistribution in frequency is decoupled from angle redistribution and polarization.
Aims: We give a simpler and more general proof of the Stokes parameter decomposition using powerful new tools introduced for polarimetry, in particular the Landi Degl'Innocenti spherical tensors T^K_Q(i,Ω).
Methods: The elements of the Hanle phase matrix are written as a sum of terms that depend separately on the magnetic field vector and the directions Ω and Ω' of the incoming and scattered beams. The dependence on Ω and Ω' is expressed in terms of the spherical tensors T^K_Q(i,Ω) where i refers to the Stokes parameters (i=0,ldots,3). A multipolar expansion in terms of the T^K_Q(i,Ω) is then established for the source term in the transfer equation for the Stokes parameters.
Results: We show that the Stokes parameters have a multipolar expansion that can be written as I_i(ν,Ω)= sumKQT^K_Q(i,Ω)I_Q^K(ν,θ) (K=0,1,2, -K≤ Q≤ +K) where the I_QK are nine cylindrically symmetrical, irreducible tensors, θ being the inclination of Ω with respect to the vertical in the atmosphere. The proof is generalized to frequency-dependent phase matrices. It is applied both to partial frequency redistribution with angle-averaged scalar frequency redistribution functions and to complete frequency redistribution with the Hanle effect in the line core and Rayleigh scattering in the wings. Non-LTE transfer equations for the I_QK and integral equations for the associated source functions S_QK are established. Formal vectors and matrices constructed with I_Q^K, S_Q^K, and T_QK are introduced in order to present the results in a compact matrix notation. In particular, a simple factorized form is proposed for the Hanle phase matrix. Title: Turbulent magnetic field averages for the Zeeman effect . Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2007MmSAI..78..142F Altcode: Stokes parameters measured in the Solar atmosphere are in general time or space averages over a magnetic field probability distribution function. Here we show how to write the Zeeman propagation matrix in a reference frame defined with respect to the direction of a mean magnetic field and how to average over a random magnetic field distribution. We concentrate on the case of a normal Zeeman triplet but indicate how to treat general Zeeman patterns. Numerical results are presented for Gaussian distributions having cylindrical symmetry about a mean field. Different models of probability distribution functions (PDF), are compared. Title: Polarized Spectral Line Formation in Turbulent Magnetic Fields: The Zeeman and Hanle Effects Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2006ASPC..358..126F Altcode: We present a short summary of work carried out on the effects of random magnetic fields with finite correlation length on spectral line polarization. The magnetic field is modeled by a step-wise Markovian random process defined by a probability distribution and a correlation length. Micro- and macro-turbulent limits are recovered when this length goes to zero and infinity, respectively. For the Zeeman effect, explicit expressions have been obtained for the mean emergent Stokes parameters and for their r.m.s. fluctuations. Examples illustrate the dependence of the mean Zeeman propagation matrix on the magnetic field distribution, and the dependence of mean Stokes parameters and their r.m.s. fluctuations on the correlation length of the magnetic field. For the Hanle effect, explicit expressions have also been obtained for the mean Stokes parameters. We outline the approach and give an explicit expression for the mean value of Stokes Q. Title: Stochastic polarized line formation. II. Zeeman line transfer in a random magnetic field Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2006A&A...453.1095F Altcode: Context: .The Zeeman effect produced by a turbulent magnetic field or a random distribution of flux tubes is usually treated in the microturbulent or macroturbulent limits where the Zeeman propagation matrix or the Stokes parameters, respectively, are averaged over the probability distribution function of the magnetic field when computing polarized line profiles.
Aims: .To overcome these simplifying assumptions we consider the Zeeman effect from a random magnetic field which has a finite correlation length that can be varied from zero to infinity and thus made comparable to the photon mean free-path.
Methods: .The vector magnetic field is modeled by a Kubo-Anderson process, a piecewise constant Markov process characterized by a correlation length and a probability distribution function for the random values of the magnetic field. The micro and macro turbulent limits are recovered when the correlation goes to zero or infinity.
Results: .An integral equation is constructed for the mean propagation operator and explicit expressions are obtained for the mean values and second-order moments of the Stokes parameters at the surface of a Milne-Eddington type atmosphere. The expression given by Landi Degl'Innocenti (1994) for the mean Stokes parameters is recovered. Mean values and rms fluctuations around the mean values are calculated numerically for a random magnetic field with isotropic Gaussian fluctuations. The effects of a finite correlation length are discussed in detail. Various extensions of the Milne-Eddington and magnetic field model are considered and the corresponding integral equations for the mean propagation operator are given.
Conclusions: .The rms fluctuations of the Stokes parameters are shown to be very sensitive to the correlation length of the magnetic field. It is suggested to use them as a diagnostic tool to determine the scale of unresolved features in the solar atmosphere. Title: The Hanle effect in a random medium Authors: Frisch, H. Bibcode: 2006A&A...446..403F Altcode: This paper considers the Hanle effect produced by a turbulent magnetic field. To overcome the simplified microturbulent treatment whereby the Hanle phase matrix is locally averaged over some magnetic field distribution, we consider a turbulent magnetic field with a finite correlation length. We assume that the magnetic field along each individual photon path can be represented by a Kubo-Anderson process (KAP) and study the stationary solution as time goes to infinity. A KAP is a discontinuous Markov process. The random magnetic field is characterized by a correlation length and a distribution function of the magnetic field vector; both can be chosen arbitrarily. The microturbulent limit is recovered when the correlation length goes to zero. A non-stochastic integral equation of the Wiener-Hopf type is obtained for a mean conditional source vector. This integral equation yields explicit expressions for the mean Stokes parameters, provided one makes physically realistic approximations, namely neglect the effect of the magnetic field on Stokes I, keep only the contributions from I and Q in the source terms for Stokes Q and Stokes U and solve the integral equation for Q with a two-scattering approximation. The final expressions involve mean values and correlation functions of some of the elements of the Hanle phase matrix and show the dependence on the correlation length of the random magnetic field. The combined effects of a turbulent velocity field and a turbulent magnetic field with finite correlation lengths is also studied. The velocity field is represented by a KAP with the same correlation length as the magnetic field. Some of the velocity field effects are treated with an effective medium approximation as in Frisch & Frisch (1976, MNRAS, 175, 157). Explicit expressions are obtained for the mean Stokes parameters. They can account for correlations between velocity field and magnetic field fluctuations. Title: Stochastic polarized line formation. I. Zeeman propagation matrix in a random magnetic field Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N. Bibcode: 2005A&A...442...11F Altcode: This paper considers the effect of a random magnetic field on Zeeman line transfer, assuming that the scales of fluctuations of the random field are much smaller than photon mean free paths associated to the line formation (micro-turbulent limit). The mean absorption and anomalous dispersion coefficients are calculated for random fields with a given mean value, isotropic or anisotropic Gaussian distributions azimuthally invariant about the direction of the mean field. Following Domke & Pavlov (1979, Ap&SS, 66, 47), the averaging process is carried out in a reference frame defined by the direction of the mean field. The main steps are described in detail. They involve the writing of the Zeeman matrix in the polarization matrix representation of the radiation field and a rotation of the line of sight reference frame. Three types of fluctuations are considered : fluctuations along the direction of the mean field, fluctuations perpendicular to the mean field, and isotropic fluctuations. In each case, the averaging method is described in detail and fairly explicit expressions for the mean coefficients are established, most of which were given in Dolginov & Pavlov (1972, Soviet Ast., 16, 450) or Domke & Pavlov (1979, Ap&SS, 66, 47). They include the effect of a microturbulent velocity field with zero mean and a Gaussian distribution. A detailed numerical investigation of the mean coefficients illustrates the two effects of magnetic field fluctuations: broadening of the σ-components by fluctuations of the magnetic field intensity, leaving the π-components unchanged, and averaging over the angular dependence of the π and σ components. For longitudinal fluctuations only the first effect is at play. For isotropic and perpendicular fluctuations, angular averaging can modify the frequency profiles of the mean coefficients quite drastically with the appearance of an unpolarized central component in the diagonal absorption coefficient, even when the mean field is in direction of the line of sight. A detailed comparison of the effects of the three types of fluctuation coefficients is performed. In general the magnetic field fluctuations induce a broadening of the absorption and anomalous dispersion coefficients together with a decrease of their values. Two different regimes can be distinguished depending on whether the broadening is larger or smaller than the Zeeman shift by the mean magnetic field. For isotropic fluctuations, the mean coefficients can be expressed in terms of generalized Voigt and Faraday-Voigt functions H(n) and F(n) introduced by Dolginov & Pavlov (1972, Soviet Ast., 16, 450). These functions are related to the derivatives of the Voigt and Faraday-Voigt functions. A recursion relation is given in an Appendix for their calculation. A detailed analysis is carried out of the dependence of the mean coefficients on the intensity and direction of the mean magnetic field, on its root mean square fluctuations and on the Landé factor and damping parameter of the line. Title: An operator perturbation method for polarized line transfer. VI. Generalized PALI method for Hanle effect with partial frequency redistribution and collisions Authors: Fluri, D. M.; Nagendra, K. N.; Frisch, H. Bibcode: 2003A&A...400..303F Altcode: A generalized iteration method is presented to solve the polarized line transfer equation for a two-level-atom in an arbitrarily oriented, weak magnetic field. The polarized redistribution matrix employed accounts self-consistently for collisions as well as the presence of a weak magnetic field responsible for the Hanle effect. The proposed numerical method of solution is based on a Polarized Approximate Lambda Iteration (PALI) method. A Fourier decomposition of the radiation field and of the phase matrix with respect to the azimuthal angle reduces the complexity of the problem. A generalized core-wing technique is proposed, which permits an efficient implementation of the frequency domain structure inherent in the polarized redistribution matrix. The numerical method is tested for its accuracy and efficiency by comparing with the existing methods. Title: Numerical Methods for Solving the Polarized Line Transfer Equations with Partial Frequency Redistribution Authors: Nagendra, K. N.; Frisch, H.; Fluri, D. M. Bibcode: 2003ASPC..307..227N Altcode: No abstract at ADS Title: Hanle effect with angle-dependent partial redistribution Authors: Nagendra, K. N.; Frisch, H.; Faurobert, M. Bibcode: 2002A&A...395..305N Altcode: The polarized line transfer equation for the Hanle effect is solved in the framework of an exact partial frequency redistribution (PRD) theory developed by Bommier (1997a,b). In that theory the effect of collisions on the Hanle effect is considered self-consistently. We follow that approach in the line transfer computations presented here. The theory formulated by Bommier clearly recognizes two levels of approximations for exact PRD, in order to facilitate the solution of the line transfer equation. The second level employs angle-dependent redistribution functions, and numerically represents a more difficult problem compared to the third level, which involves only the use of angle-averaged frequency redistribution functions. We present a method which can solve the problem in both the levels of approximation. The method is based on a perturbative approach to line polarization. Although computationally expensive, it offers the only practical means of solving the angle-dependent Hanle PRD problem. We discuss the numerical aspects of assembling the so called ``frequency domain dependent redistribution matrices'', and also an efficient way of computing the scattering integral. Some examples are presented to illustrate the interesting aspects of the Hanle-PRD problem with angle-dependent frequency redistribution. A comparison of the emergent profiles computed under angle-averaged and angle-dependent redistribution is carried out, and the effect of collisions is investigated. We show that it is necessary to incorporate an angle-dependent redistribution mechanism especially in the computation of the Stokes U parameter. We demonstrate that the use of simple frequency domains is good enough in practical applications of the Hanle PRD theory. Title: Investigation of weak solar magnetic fields. New observational results for the SrI 460.7 nm linear polarization and radiative transfer modeling Authors: Faurobert, M.; Arnaud, J.; Vigneau, J.; Frisch, H. Bibcode: 2001A&A...378..627F Altcode: Scattering polarization measurements were obtained with THEMIS in July 2000, close to the solar south Pole and to the east Equator and in a period of maximum solar activity. Using the THEMIS multi-lines spectro-polarimetric mode (MTR), we observed simultaneously four spectral domains containing the 460.7 nm Sr i line, several molecular lines around 515.9 nm and the Na i D1 and Na i D2 lines. This allows us to scan different altitudes in the solar atmosphere at the same time and provides us with a large set of constraints to study the behaviour of the magnetic field. This paper is devoted to the Sr i line which exhibits quite a strong linear polarization peak outside active regions. A detailed radiative transfer modeling is performed in order to interpret the observed center-to-limb variations of the line intensity and polarization. It was shown previously (Faurobert-Scholl \cite{Faurobert-Scholl1}) that this line, which is sensitive to the Hanle effect, can be used as a diagnostic tool for the presence of weak turbulent magnetic fields in the solar photosphere outside active regions. The line polarization rates that we measured in July 2000 are 25% lower than what has been reported previously, for observations near the minimum, or in the increasing phase, of the activity cycle (Stenflo et al. \cite{Stenflo1}). They are in agreement with other observations performed with a different observational set-up in August 2000 (Bommier & Molodij \cite{Bommier4}). We show that they are consistent with the presence of a weak turbulent magnetic field with an average strength between 20 G and 30 G in the upper solar photosphere. This is about twice the value which was derived from previous observations. This result raises the possiblity of a long-term variation of the turbulent photospheric magnetic field with the activity cycle. Title: Scattering Polarization and Hanle Effect: On the Importance of Angle-Dependent Frequency Redistribution Authors: Faurobert, M.; Frisch, H.; Nagendra, K. N. Bibcode: 2001ASPC..248..145F Altcode: 2001mfah.conf..145F No abstract at ADS Title: The Hanle Effect with Angle Dependent Redistribution Functions Authors: Frisch, H.; Faurobert, M.; Nagendra, K. N. Bibcode: 2001ASPC..236..197F Altcode: 2001aspt.conf..197F No abstract at ADS Title: An Operator Perturbation Method of Polarized Line Transfer V. Diagnosis of Solar Weak Magnetic Fields Authors: Nagendra, K. N.; Frisch, H.; Faurobeet-Scholl, M.; Paletou, F. Bibcode: 2000JApA...21..255N Altcode: No abstract at ADS Title: Fast Numerical Methods for Polarized Line Radiative Transfer in the Presence of Hanle Effect Authors: Faurobert, M.; Frisch, H.; Nagendra, K. N. Bibcode: 1999ASPC..184...28F Altcode: The Hanle effect provides a diagnostic tool for weak magnetic fields which do not give rise to a measurable Zeeman effect, such as turbulent fields or magnetic canopies. The lines which are sensitive to the Hanle effect are formed under non-LTE conditions, by scattering of photons. Inversion methods for such diagnostics require to solve the non-LTE polarized transfer equation for a large number of magnetic configurations. Fast numerical methods are thus highly required. We present an Approximate Lambda Iteration method to treat the Hanle effect for lines formed with complete frequency redistribution. Referred to as PALI-H, this method is an extension of ALI methods first developed for non polarized line transfer. The starting point is to recast the polarized transfer equation into a vectorial integral equation for a 6-component source function. We show that the convergence of the method is independent of the strength and direction of the magnetic field. The method is very fast and allows to handle any type of depth-dependent magnetic field. Title: An operator perturbation method for polarized line transfer IV: Applications to the Hanle effect with partial frequency redistribution Authors: Nagendra, K. N.; Paletou, F.; Frisch, H.; Faurobert-Scholl, M. Bibcode: 1999ASSL..243..127N Altcode: 1999sopo.conf..127N No abstract at ADS Title: Resonance polarization and Hanle effect: The integral equation formulation and some applications Authors: Frisch, H. Bibcode: 1999ASSL..243...97F Altcode: 1999sopo.conf...97F No abstract at ADS Title: The Hanle effect. The density matrix and scattering approaches to the protect sqrt epsilon -law Authors: Frisch, H. Bibcode: 1998A&A...338..683F Altcode: A sqrt epsilon -law was demonstrated by Landi Degl'Innocenti & Bommier (1994) for resonance polarization in a magnetic atmosphere where the primary source of photons is of thermal origin (isotropic and unpolarized). In this paper we propose a generalized form of this law by dropping the hypothesis on the primary source of photons. We restrict ourselves to the case of weak magnetic fields (Hanle effect). For spectral lines formed with complete redistribution, it has been shown by Landi Degl'Innocenti et al (1990), using the density matrix theory in its irreducible tensorial operator version, that the Hanle effect can be reduced to an integral equation of the convolution type for a six-component source vector. As shown by Faurobert-Scholl (1991), a similar equation can be obtained by performing an azimuthal Fourier decomposition of the transfer equation for the Stokes parameters. In the first part of the paper we recall the main steps of the two methods and establish the correspondence between the convolution equations that they provide. In the second part we use these equations to obtain a generalized sqrt epsilon -law. For the equation coming from the density matrix formalism, we essentially follow the original proof of Landi Degl'Innocenti & Bommier (1994). For the equation coming from the Fourier decomposition, because of a lack of symmetry in operator describing the action of the magnetic field, we use as intermediate step the Hopf-Bronstein-Rybicki relation established by Ivanov (1995) for transport operators which are not self-adjoint. Title: An operator perturbation method for polarized line transfer. III. Applications to the Hanle effect in 1D media Authors: Nagendra, K. N.; Frisch, H.; Faurobert-Scholl, M. Bibcode: 1998A&A...332..610N Altcode: In this paper we present an Approximate Lambda Iteration method to treat the Hanle effect (resonance scattering in the presence of a weak magnetic field) for lines formed with complete frequency redistribution. The Hanle effect is maximum in the line core and goes to zero in the line wings. Referred to as PALI-H, this method is an extension to non-axisymmetric radiative transfer problems of the PALI method presented in Faurobert-Scholl et al. (1997), hereafter referred to as Paper I. It makes use of a Fourier decomposition of the radiation field with respect to the azimuthal angle which is somewhat more general than the decomposition introduced in Faurobert-Scholl (1991, hereafter referred to as FS91). The starting point of the method is a vector integral equation for a six-component source vector representing the non-axisymmetric polarized radiation field. As in Paper I, the Approximate Lambda operator is a block diagonal matrix. The convergence rate of the PALI-H method is independent of the polarization rate and hence of the strength and direction of the magnetic field. Also this method is more reliable than the perturbation method used in FS91. The PALI-H method can handle any type of depth-dependent magnetic field. Here it is used to examine the dependence of the six-component source vector on the co-latitude, azimuthal angle and strength of the magnetic field. The dependence of the surface polarization on the direction of the line-of-sight and on the magnetic field is illustrated with polarization diagrams showing Q/I versus U/I at line center. The analysis of the results show that the full six-dimension problem can be approximated by a two-component modified resonance polarization problem, producing errors of at most 20% on the surface polarization at line center. Title: An operator perturbation method for polarized line transfer. I. Non-magnetic regime in 1D media. Authors: Faurobert-Scholl, M.; Frisch, H.; Nagendra, K. N. Bibcode: 1997A&A...322..896F Altcode: In this paper we generalize an Approximate Lambda Iteration (ALI) technique developed for scalar transfer problems to a vectorial transfer problem for polarized radiation. Scalar ALI techniques are based on a suitable decomposition of the Lambda operator governing the integral form of the transfer equation. Lambda operators for scalar transfer equations are diagonally dominant, offering thus the possibility to use iterative methods of the Jacobi type where the iteration process is based on the diagonal of the Lambda operator (Olson et al., 1986, JQSRT 35, 431). Here we consider resonance polarization, created by the scattering of an anisotropic radiation field, for spectral lines formed with complete frequency redistribution in a 1D axisymmetric medium. The problem can be formulated as an integral equation for a 2-component vector (Rees, 1978PASJ...30..455R) or, as shown by Ivanov (1995A&A...303..621I), as an integral equation for a (2x2) matrix source function which involves the same generalized Lambda operator as the vector integral equation. We find that this equation holds also in the presence of a weak turbulent magnetic field. The generalized Lambda operator is a (2x2) matrix operator. The element {11} describes the propagation of the intensity and is identical to the Lambda operator of non-polarized problems. The element {22} describes the propagation of the polarization. The off-diagonal terms weakly couple the intensity and the polarization. We propose a block Jacobi iterative method and show that its convergence properties are controlled by the propagator for the intensity. We also show that convergence can be accelerated by an Ng acceleration method applied to each element of the source matrix. We extend to polarized transfer a convergence criterion introduced by Auer et al. (1994A&A...292..599A) based on the grid truncation error of the converged solution. Title: Transfert de rayonnement hors-ETL et applications en physique solaire. Authors: Faurobert-Scholl, M.; Paletou, F.; Frisch, H. Bibcode: 1996JAF....53...24F Altcode: No abstract at ADS Title: Partial Frequency Redistribution of Polarized Radiation Authors: Frisch, H. Bibcode: 1996SoPh..164...49F Altcode: Resonance polarization, which is created by the scattering of an anisotropic radiation field in regions of zero or weak magnetic fields, is strongly dependent on the frequency redistribution taking place during the scatterings. Here we discuss the frequency redistribution matrix relevant to resonance lines, concentrating on linear polarization. First we analyze in detail the redistribution matrix in a zero magnetic field given by the theory of Omont, Smith and Cooper (1972), revisited by Domke and Hubeny (1988). We explain that the linear polarization maxima which may appear in the wings of the Stokes Q profiles of strong resonances lines such as the Ca I 4227 Å line are coherent frequency redistribution effects. Various approximate forms of the frequency redistribution matrix are also examined. For resonance polarization in a weak magnetic field, we suggest a new expression for the redistribution matrix which can be used at all line frequencies, and is consistent with the condition that the Hanle effect acts only in the line core. Title: CDF Results on the W Mass and the Search for the Top Authors: Frisch, H. J. Bibcode: 1995psc..conf...81F Altcode: No abstract at ADS Title: Monte-Carlo simulation of a radiative transfer problem in a random medium: application to a binary mixture. Authors: Audic, S.; Frisch, H. Bibcode: 1993JQSRT..50..127A Altcode: This paper considers monochromatic radiative transfer in a diffusive three dimensional random binary mixture. The absorption coefficient, along any line-of-sight is a homogeneous Markoy process, which is described by a three-dimensional Kubo-Anderson process. The transfer equation is solved numerically by Monte-Carlo simulations on a massively parallel computer (a Connection Machine) by attaching one or several photons to each processor. The implementation of the simulations on the machine is discussed in detail, in particular the association between photons and processors and the storage of the data concerning the photons and the realizations of the statistics. Title: Quelques problèmes de transfert en physique solaire. Authors: Frisch, H.; Faurobert-Scholl, M. Bibcode: 1991sed..conf..151F Altcode: (1) Introduction. (2) Formation ETL des raies: (2.1) Intensité du rayonnement et équation de transfert. (2.2) Approximation de diffusion. (2.3) Equation de transfert ETL pour les raies. (3) Formation non-ETL des raies: (3.1) Fonction source d'un atome à deux niveaux. (3.2) Fonction de redistribution. (3.3) Equation de transfert non-ETL. (3.4) Analyse asymptotique. (3.5) Transfert non-ETL dans une couche plane. (3.6) Probabilité d'échappement. (3.7) Méthodes numériques en transfert non-ETL. (3.8) Construction de modèles d'atmosphères. (4) Transfert de rayonnement polarisé. (4.1) Les paramètres de Stokes. (4.2) Mécanismes de polarisation des raies spectrales. (4.3) Equation de transfert d'un rayonnement polarisé. (4.4) Effet Hanle. (4.5) Effet Zeeman. Title: An exact analytical solution of a radiative transfer problem in a binary mixture. Authors: Frisch, H.; Pomraning, G. C.; Zweifel, P. F. Bibcode: 1990JQSRT..43..271F Altcode: The authors give an exact analytical solution of a statistical radiative transfer problem. This problem consists of time-independent, non-scattering transport in a binary statistical mixture. The mixing statistics are taken as homogeneous. The chord-length distribution of each constituent is assumed to be representable as a Laplace transform. Markov statistics are a special (degenerate) case of the class of statistics the authors are able to treat. Two different techniques, a singular eigenfunction analysis and an integral equation method, are applied to the problem. Title: Asymptotic analysis of resonance polarization and escape probability approximations Authors: Faurobert-Scholl, M.; Frisch, H. Bibcode: 1989A&A...219..338F Altcode: Polarized transfer asymptotic and first order escape probability methods developed for the nonpolarized case are generalized to include linear polarization produced by the scattering of anisotropic radiation in the absence of magnetic fields. The analyses are based on a coupled integral equation for two-angle-dependent source functions. Some general properties, such as the order of magnitude of the surface polarization and approximations for the source functions, are deduced. The escape probability approximations are compared with exact solutions and found to be more accurate than first order perturbation techniques for lines of moderate optical thickness in solar isothermal prominences. Title: A Model for the Penetration of Lyman Alpha in the Solar Chromosphere Authors: Faurobert, M.; Frisch, H.; Skumanich, A. Bibcode: 1988ApJ...328..856F Altcode: A multilevel calculation of the hydrogen spectrum by Skumanich and Lites, followed by a two-level representation of each transition, shows that all the solar Lyα photons are created in an optically thick layer in the high chromosphere above τ0 ≈ 102. The authors give a simplified model that demonstrates the penetration of Lyα into the lower chromosphere below the creation region, i.e., for τ0 > 102. A second-order escape probability approximation yields a simple analytical expression which accurately reproduces the behavior of the Lyα source function for 102 ⪉ τ0 ⪉ 5×106 provided the destruction probability, ɛ is set close to 10-6. Title: A Cauchy integral equation method for analytic solutions of half-space convolution equations. Authors: Frisch, H. Bibcode: 1988JQSRT..39..149F Altcode: This paper is devoted to a method of Cauchy integral equation for the solution of half-space convolution equations. It was introduced by Frisch and Frisch to solve Wiener-Hopf integral equations with algebraically decreasing kernels, arising in non-coherent transfer with complete frequency redistribution. The author shows here that coherent transfer may also be treated by the Cauchy integral method. Various examples chosen among standard problems are used to illustrate the method. Title: Radiation in moving gaseous media : eighteenth Advanced Course of the Swiss Society of Astrophysics and Astronomy Authors: Kudritzki, R. P.; Yorke, H. W.; Frisch, Helene Bibcode: 1988rmgm.book.....K Altcode: 1988STIA...8950499K The propagation of EM radiation in gaseous media is characterized in chapters based on lectures presented at Leysin, Switzerland, on March 6-12, 1988. The emphases are on the fundamental astrophysics and the experimental and observational methods employed. Topics addressed include hot-star atmosphere models and their application to observations, radiation in diffuse matter (dust properties, gas microphysics, static nebula models, radiation hydrodynamics, and the formation and evolution of H II regions), and radiative transfer with frequency redistribution (asymptotic methods, scaling laws, and approximate solutions). Title: Radiative Transfer with Frequency Redistribution - Asymptotic methods scaling laws and approximate solutions Authors: Frisch, Hélène Bibcode: 1988rmgm.conf..339F Altcode: 1988SAAS...18..339F General Introduction Asymptotic methods. Some general ideas Outline of the lectures Diffusive and Non-Diffusive Large Scale Behaviour Introduction Asymptotic analysis of the integral equation for the source function Specific intensity in the interior Transfer with Partial Frequency Redistribution Introduction The ingredients of a partial redistribution problem The redistribution function The Asymptotic of Partial Redistribution Introduction RI and RIII partial redistribution RIV partial redistribution Coherent scattering and RII partial redistribution Hints for the calculation of subordinate and resonance lines Boundary Layers Introduction Complete redistribution Diffusive type processes Transport of continuous radiation in stars Scaling Laws for Spectral Line Formation. A Summary Introduction Complete redistribution RII redistribution and coherent scattering Time-dependent transfer Escape Probabilties for Static and Moving Media Introduction Escape probability approximation procedures in a static medium Line formation in moving media Numerical Methods Introduction Iterative methods with exact operators Operator perturbation methods Preconditioning; necessity and technique Analytic Solutions of Half-Space Transport Equations Introduction A Cauchy integral equation method of solution Utilizations of exact analytical solutions Monochromatic scattering The H-function in the Wiener-Hopf type methods References Title: The Lyman α line in the solar chromosphere: penetrative model andapproximations. Authors: Faurobert, M.; Frisch, H.; Skumanich, A. Bibcode: 1987PAICz..66..205F Altcode: 1987eram....1..205F The authors present a model for the formation of Lyman α in the chromosphere below the creation region. The hydrogen atom is treated as a two-level atom and the chromosphere as a semi-infinite medium free of primary sources and illuminated by an isotropic and frequency independent radiation field at r0 ≅ 102. Title: First results from a 1.1-m-diameter superconducting monopole detector Authors: Incandela, J.; Frisch, H.; Somalwar, S.; Kuchnir, M.; Gustafson, H. R. Bibcode: 1986PhRvD..34.2637I Altcode: We present the design and performance of a superconducting induction magnetic-monopole detector with 1.1-m-diameter gradiometer loops. The detector demonstrates that gradiometers can be overlapped with no mutual inductance to yield high redundancy without increasing shield volume. One of two detector units was sensitive to the passage of monopoles through two overlapped gradiometers with 98% coincidence efficiency for 161 hours. The unit has a coincident sensitive area averaged over 4π sr of 4400 cm2. No candidate events were observed, setting an upper limit on the flux of cosmic-ray magnetic monopoles of f<~7.1×10-11 cm-2 sr-1 sec-1 (90% C.L.). The detector was operated in ambient magnetic fields of ~5-125 mG. Title: The status of searches for magnetic monopoles Authors: Frisch, H. J. Bibcode: 1986NYASA.461..652F Altcode: 1986awph.conf..652F The fundamental principles, apparatus, procedures, and results of ongoing terrestrial searches for the superheavy magnetic monopoles (MMs) predicted by GUTs are reviewed. Approaches examined include methods based on the MM magnetic charge (Faraday induction experiments and the Parker bound from Galactic magnetic fields), MM ionization properties (MM velocity estimates; detection in scintillation and gas detectors), and the ability of MMs to catalyze nucleon decay (proton-decay experiments and cosmic-ray-shower hadronic cross section measurements). Diagrams, drawings, graphs, and tables of numerical data are provided, and plans for future MM detectors are indicated. Title: Line transfer with complete frequency redistribution in an absorbing medium - Scaling laws and approximation Authors: Faurobert, M.; Frisch, H. Bibcode: 1985A&A...149..372F Altcode: This paper discusses the influence of a continuous absorption produced by dust grains or photoionization on subordinate or weak resonance lines. The lines are being formed with complete Doppler or Voigt frequency redistribution in a one dimensional slab of finite thickness. An asymptotic analysis in the limit of small β (ratio of continuum to line opacity coefficient) shows that the large scale behaviour of the radiation field is described by a singular integral equation identical to that obtained by Frisch and Frisch (1977) for collisional destruction, but with an exponentially decreasing kernel in the case of the Voigt profile. Asymptotic scaling laws for the mean path length, the mean number of scatterings and the fraction of created photons which escapes the medium are given in the limits of weak and strong absorption. A first order escape probability method is used to evaluate net radiative brackets and line cooling rates, taking into account a possible emission by the continuum in the frequency domain of the line. Finally, a definition of the mean number of scatterings properly incorporating this emission term is proposed. Title: Asymptotic properties of complete and partial frequency redistribution. Authors: Frisch, H. Bibcode: 1985ASIC..152...87F Altcode: 1985pssl.proc...87F Radiative transfer problems with frequency redistribution may be investigated by asymptotic methods when the mean number of scatterings undergone by photons is very large. These methods provide scaling laws for characteristic parameters of the line radiation field. These methods also provide asymptotic transfer equations which describe the large scale behaviour of the radiation field away from boundaries. Complete redistribution and the four standard types of partial redistribution is discussed. Implications for numerical calculations are briefly considered. Title: Pannel discussion on partial redistribution. Authors: Freire Ferrero, R.; Frisch, H.; Linsky, J.; Oxenius, J.; Simonneau, E. Bibcode: 1985ASIC..152..143F Altcode: 1985pssl.proc..143F No abstract at ADS Title: Flux limit on cosmic-ray magnetic monopoles from a large area induction detector Authors: Incandela, J.; Campbell, M.; Frisch, H.; Somalwar, S.; Kuchnir, M.; Gustafson, H. R. Bibcode: 1984PhRvL..53.2067I Altcode: The design and performance of a superconducting induction detector with two 60-cm-diameter superconducting loops is presented. During eight months of data taking, no candidate events were observed, and an upper limit on the flux of cosmic-ray magnetic monopoles of about 6.7 x 10 to the -12th/sq cm sr sec (90-percent confidence level) is set. The detector demonstrates the possibility of operating large induction detectors in ambient magnetic fields greater than 1 mG. Title: Boundary layer conditions for the transport of radiation in stars Authors: Frisch, H.; Faurobert, M. Bibcode: 1984A&A...140...57F Altcode: An expansion technique used by Larsen et al. (1983) for describing coherent scattering is applied to characterizing an outer thermal boundary condition for radiation transport in stellar interiors. Modifications are introduced to cover conditions of non-gray radiative equilibrium, which is analyzed asymptotically. An expansion parameter is defined in terms of coupled transfer and energy equations and carried out to first order to obtain a ratio of the photon free path to the stellar characteristic dimension. The ratio provides a scale for the space variable with which a singular perturbation problem is solved for interior and boundary conditions, the latter being in the stellar atmosphere and the sum of the interior solution and the boundary layer correction. A Robin boundary condition results which is only 5 percent higher in temperature than the interior in the case of a gray opacity coefficient. Title: Mean Escape Probabilities and Mean Numbers of Scatterings for Resonance Lines Authors: Frisch, H. Bibcode: 1984mrt..book...25F Altcode: No abstract at ADS Title: First results from the Chicago-Fermilab-Michigan cosmic ray magnetic monopole detector. Authors: Incandela, J. R.; Campbell, M.; Frisch, H.; Somalwar, S.; Kuchnir, M.; Gustafson, H. R. Bibcode: 1984mono.conf..461I Altcode: No abstract at ADS Title: Scaling laws for the scattering of resonance line photons. Authors: Frisch, H. Bibcode: 1984PhST....7..110F Altcode: 1984PhyS....7..110F Resonance lines of ions with a large Z are broadened essentially by natural damping, and therefore photons with frequencies larger than a few Doppler widths are scattered almost coherently. This "coherent" scattering of resonance-line photons can be described by a space and frequency diffusion process when the mean number of scatterings undergone by the photons is sufficiently large. Title: Mean escape probabilities and mean numbers of scatterings for resonance lines. Authors: Frisch, H. Bibcode: 1984mrt..book...65F Altcode: 1984mrt..conf...65F A mean escape-probability approximation for resonance lines, which encompasses both the effectively thin and the effectively thick limits, is described. Global conservation of photons and the large scale diffusive behavior of the radiation field are reasonably well preserved in this approximation. This approximation is tested on a two-level atom by comparison with accurate numerical results. Title: Dust-driven winds. I - A two-fluid model and its numerical solution Authors: Berruyer, N.; Frisch, H. Bibcode: 1983A&A...126..269B Altcode: A model of dust-driven wind relevant to red giant stars is investigated, in which the usual hypothesis of 'momentum coupling' (amounting to neglect of grain particle mass) is relaxed. When the momentum coupling approximation is abandoned, the sonic point is shifted outwards, and the gas and grain expansion velocities are reduced. In the supersonic part of the wind the diminution is of the order of 30 percent, but it may reach several orders of magnitude below the sonic point. An asymptotic analysis of the flow at the base of the wind shows the existence of a boundary layer where the grains suffer a very strong acceleration, the gas velocity staying almost constant. The coupling between the grains and the gas is maximum just outside the boundary layer, and decreases gradually outwards. Beyond a thousand inner envelope radii the grains and the gas are fully uncoupled. The coupling is still very strong when the velocities have saturated the almost constant values. Title: Escape probabilities, mean number of scatterings and net radiative bracket for resonance lines Authors: Frisch, H. Bibcode: 1982A&A...114..119F Altcode: An examination of the multiple scattering of resonance-line photons in nonconservative media, where the photons have a small probability of destruction at each scattering, yields an approximation for the mean number of scatterings on the basis of scaling arguments which takes into account both the destruction of photons and escape through boundaries. The approximation is used to show that the fraction of the emitted energy escaping from the medium, known as the mean escape probability, is nearly independent of the value of the destruction probability. Analytical solutions which are valid deep within the medium, and accurate numerical solutions of the transfer problem, are used to check the validity of the stated results and examine the errors made when scaling laws for the mean escape probability are transformed into depth variation scaling laws. Title: A method of Cauchy integral equation for noncoherent tranfer in half-space. Authors: Frisch, H.; Frisch, U. Bibcode: 1982JQSRT..28..361F Altcode: No abstract at ADS Title: Non-LTE transfer with complete redistribution. Scaling laws for a slab. Authors: Frisch, H. Bibcode: 1982JQSRT..28..377F Altcode: No abstract at ADS Title: Hydrogen line spectrum in quasars. I - Approximation procedures for line transfer versus an exact treatment Authors: Collin-Souffrin, S.; Delache, Ph.; Frisch, H.; Dumont, S. Bibcode: 1981A&A...104..264C Altcode: Approximation procedures frequently used in handling self-absorption effects in the hydrogen emission lines of quasars are discussed and compared with an exact numerical treatment of line transfer. The model here is a finite slab with prescribed density and temperature. It is noted that if a medium has finite thickness and is such that subordinate lines are not thermalized, then a semi-infinite 'local escape probability,' which implicitly assumes that photons emerge from only one surface of the slab, will overestimate the line fluxes. In this way, the intensity ratio H-alpha/H-beta can be greatly underestimated. A modified form of the 'local escape probability' is proposed which allows for the possibility of photons escaping from both sides of the medium. With this probability, it becomes possible to obtain much more accurate line ratios and fluxes. It is also found that natural broadening effects (representable by a Voigt profile) may be important in subordinate lines. Title: Diffusion approximations for the scattering of resonance-line photons: interior and boundary layer solutions. Authors: Frisch, H.; Bardos, C. Bibcode: 1981JQSRT..26..119F Altcode: No abstract at ADS Title: Broadening of non-LTE lines by a turbulent velocity field with a finite correlation length Authors: Froeschle, Ch.; Frisch, H. Bibcode: 1980A&A....91..202F Altcode: Profiles of non-LTE lines broadened by a turbulent velocity field with a finite correlation length are computed by the 'effective source function' technique. This method can reformulate stochastic transfer as a standard non-LTE problem with an effective escape probability determined by averaging all realizations of velocity fields. Effective source functions and mean emergent profiles were calculated for a two-level atom with two choices of a thermal source, and choices between the Doppler and Voigt line profiles. It is concluded from correlations between the macroturbulent limit, the thermalization length, and variations of emergent profiles in isothermal and chromospheric atmospheres, that a micro-macroturbulent model cannot reproduce a turbulent velocity field with a finite correlation length. Title: Scaling laws for resonance line photons in an absorbing medium Authors: Frisch, H. Bibcode: 1980A&A....87..357F Altcode: Resonance line scattering in the presence of a source of continuous absorption is studied for very small values of the ratio of the continuous to line opacity, beta. Scaling laws for the thermalization length, the thermalization frequency, the mean number of scatterings and the mean path length are extracted from an asymptotic analysis of the equation of transfer in the limit as beta approaches 0. An interpretation is given for asymptotic scaling laws inferred from numerical data by Hummer and Kunasz (1978) and Bonilha et al. (1979). Title: Non-LTE transfer. V - The asymptotics of partial redistribution Authors: Frisch, H. Bibcode: 1980A&A....83..166F Altcode: The large-scale behavior of noncoherent radiative transfer with partial frequency redistribution is examined. Scaling laws for four basic scattering processes are extracted from an asymptotic analysis of the integral equation for the source function in the limit, epsilon approaches zero (where epsilon is the probability of photon destruction) to describe the dependence of characteristic lengths and frequencies on epsilon. It is shown that partial frequency redistribution simplifies drastically for small epsilon and angular redistribution is irrelevant in all cases. For all scattering processes the source function, after the rescaling of optical depths and frequencies, depends on fewer variables than the original source functions. The main results are discussed with reference to implications for numerical methods currently used in the astrophysical literature. Title: Renormalization-group approach to noncoherent radiative transfer Authors: Bell, Thomas L.; Frisch, U.; Frisch, H. Bibcode: 1978PhRvA..17.1049B Altcode: In noncoherent transfer, a photon undergoes multiple scattering with frequency changes, the frequency profile φ (0<=φ<=Λ) after each scattering is prescribed, the mean free path varies like φ-1, and there is a small probability of destruction ɛ. In contrast with the coherent (monochromatic) case, diffusive behavior is not obtained as ɛ-->0. The scaling law of the intensity of radiation can be obtained with a technique suggested by the renormalization-group approach to critical phenomena. The equation of transfer for "core" photons (say, Λ>=φ>12Λ) is solved in terms of "wing" photons (Λ2>=φ>0). The solution is substituted wherever core photons appear in the equation for the wing photons. A closed equation for wing photons is thus obtained. After rescaling of the variables (wave number and φ) it resembles the original equation (φ again runs from 0 to Λ) but has a nonlocal scattering operator. The procedure is iterated and the successive equations are found to approach a fixed form with suitable choice of the rescaling factors. This choice fixes the scaling law. It depends only on the asymptotic behavior of the profile at large frequencies. Known results for the dependence of the "thermalization length" on ɛ are recovered. Title: Rapport d'activité de l'A.D.I.O.N. par le Secrétaire Général et rapport financier. Report 1975 and 1976. Authors: Delache, P.; Schatzman, E.; Frisch, H. Bibcode: 1978BONic..13...33D Altcode: No abstract at ADS Title: Non-LTE transfer - III. Asymptotic expansion for small epsilon . Authors: Frisch, U.; Frisch, H. Bibcode: 1977MNRAS.181..273F Altcode: Radiative transfer with complete frequency redistribution in a one-dimensional full- or half-space with no incoming radiation is analyzed in the limit of small epsilon. It is shown that an asymptotic expansion can be performed in the case of complete frequency redistribution. The coherent-scattering problem is also discussed for purposes of comparison. An asymptotic expansion for specific intensity is derived from an asymptotic expansion for the source function. Title: Non-LTE transfer - IV. A rapidly convergent iterative method for the Wiener-Hopf integral equations. Authors: Frisch, H.; Froeschle, C. Bibcode: 1977MNRAS.181..281F Altcode: It is shown that a simple approximation previously derived for the source function in the isothermal case of non-LTE line transfer with complete frequency redistribution can be extended into a rapidly convergent iterative scheme for solving Wiener-Hopf integral equations. The convergence of the method and its numerization are discussed, and the iterative scheme is used for the singular integral equation of the interior asymptotic expansion as well as the homogeneous Wiener-Hopf equation for the boundary layer. Analogs for these equations based on the approximation for the source function are presented. Consideration is given to the extent to which the leading terms of the asymptotic expansions arising when epsilon tends to zero may be employed in practice to determine the source function for finite epsilon. Title: Applications of Fourier analysis to broadening of stellar line profiles. V: Effects of finite sized eddies on solar lines. Authors: Smith, M. A.; Frisch, H. Bibcode: 1976SoPh...47..461S Altcode: Turbulence dominated by eddies of a finite size produces effects on a line profile which are similar to both macro- and micro-turbulence but which are at the same time neither. It is suggested that one of these effects in the Fourier transform domain, namely the narrowing of the first natural sidelobe relative to the width of the main lobe, can be used as an indicator of the dominance of finite-sized eddies (`mesoturbulence'). An examination of some existing solar data shows that finitesized eddy models fit the observed sidelobe widths better than do models computed with any reasonable combination of micro- and macro-turbulent broadenings (Paper III). Title: Non-LTE transfer-11. Two-level atoms with stochastic velocity field Authors: Frisch, H.; Frisch, U. Bibcode: 1976MNRAS.175..157F Altcode: Following previous work on LTE stochastic transfer (Auvergne et aL; sFrisch), transfer with incoherent scattering is considered for two-level satoms in the presence of turbulent velocity fields with finite seddy-size. Assuming that the velocity along each individual photon path smay be represented by a Markov Process in time, we obtain a snon-stochastic equation of transfer for the `mean conditional sintensity', from which the mean (observable) intensity may be recovered sby integration over the velocity distribution. An integral equation of sthe Wiener-Hopf type is obtained for the `mean conditional source sfunction' from which it is shown by techniques introduced in Paper I s(Frisch & Frisch) that the rms surface source function assumes the susual value EB for uniform thermal source B. An `effective kernel and ssource function' approximation is introduced by which a standard stransfer problem is recovered and worked out explicitly in a special scase. It is shown that finite eddy-size effects can change the effective ssource function and the emergent profile by a factor z or more. Title: LTE and non-LTE formation with turbulent velocity fields Authors: Frisch, H.; Frisch, U. Bibcode: 1976pmas.conf..113F Altcode: No abstract at ADS Title: On the inverse problem in statistical mechanics Authors: Frisch, H. L.; Fesciyan, S. Bibcode: 1976GReGr...7...83F Altcode: We review briefly the status of some inverse problems in classical equilibrium statistical mechanics. Title: Non-LTE transfer. V C revisited Authors: Frisch, U.; Frisch, H. Bibcode: 1975MNRAS.173..167F Altcode: Assuming a semi-infinite atmosphere, a two-level atom and complete frequency redistribution, we give, in the isothermal case, a new derivation of the surface value of the source function based only on the integral equation for the source function. This derivation has a straightforward extension to the case where a uniform velocity gradient is included, to the time-dependent case (thermal source B switched on at time t = o) and to the determination of the mean number of scatterings undergone by an escaping photon (with Ambartsumian's definition). Our derivation suggests also a simple approximate differential equation for the complete source function = /2 B/ r where P = +(i- )K2(r) and K2(r) is the direct escape probability for outwards emitted photons. This approximation is tested both for isothermal and exponential atmosphere. As long as the scale of B is large compared to the photon mean free-path at the centre of the line, our approximation reproduces all the qualitative features and is usually slightly in excess of the exact solution. The approximation is easily extended to non-uniform E and multilevel atoms. Title: Finite eddy-size effects on centre-to-limb variations; an alternative to anisotropic microturbulence. Authors: Frisch, H. Bibcode: 1975A&A....40..267F Altcode: An attempt is presented to explain the center-to-limb increase in width of solar lines on the basis of the theory of line formation in a turbulent medium where finite eddy-size is taken into account (Auvergne et al., 1973). Hydrodynamic velocities along the line of sight are represented by a step-wise constant stochastic process (Kubo-Anderson Process), with Gaussian isotropic velocity distribution and an arbitrary variation of the eddy-size with the continuum optical depth. For the simplest case of constant geometrical eddy-size, the profiles are found to be wider at the limb than at the center; this is because the optical size of the eddies decreases towards the limb, bringing us closer to the microturbulent limit. An application to the forbidden Mg I 4571 A line shows that this effect accounts for much of the observed widening, without recourse to ad-hoc assumptions such as depth variation of the microturbulent velocity or anisotropic microturbulence. Title: Spectral Line Formation in a Tur- bulent Atmosphere Authors: Auvergne, M.; Frisch, H.; Frisch, U.; Froeschle, Ch.; Pouquet, A. Bibcode: 1973A&A....29...93A Altcode: . In order to bridge the gap between the usual microturbulent and macroturbulent approximations, we study the formation of spectral lines in a turbulent medium in which the eddies have a finite scale. The variation of the velocity along the line of sight is represented by a Kubo-Anderson process for which the probability distribution and the correlation length, or its inverse the eddy density n, can be adjusted arbitrarily. Exact analytical expressions are obtained for the mean line profile, its statistical dispersion, and also for the mean curve of growth. The macro- and microturbulent approximations are recovered respectively for n =0 and n = . It is found that in the far wings the mean profile becomes independent of the eddy density. It is shown that the position of the plateau of the curve of growth depends both on the eddy density and on the r.m.s. turbulent velocity a unique determination of the latter is therefore not possible. Key words: spectral line formation - turbulence - stochastic processes - radiative transfer Title: Coupling between Thermal Conduction and Radiative Transfer in a Moving Atmosphere Authors: Frisch, H. Bibcode: 1973A&A....27..223F Altcode: Summary. A singular perturbation expansion method commonly used in boundary layer analysis is applied to the study of the coupling between thermal conduction and radiative transfer in a moving atmosphere. The expansion parameter is the conduction coefficient assumed to be small in suitable dimensionless units. The model consists of a plane slab of finite thickness heated by thermal conduction and loosing energy through emission of radiation. Boundary temperatures, incoming intensities and matter flux are given. In order to bring out the essential ideas the asymptotic expansion is performed on a linearized version of the equations. This restriction is however not essential. Conditions under which the equations may be linearized are discussed. It is shown that thermal conduction is essentially limited to a boundary layer near the hot end. A separate equation is obtained for the exterior region which is then used to find the conditions under which convection dominates over radiative transfer. Matched interior and exterior asymptotic expansions are give in special cases. Key words: radiative transfer - thermal conduction - chromosphere - singular perturbation expansion Title: The Solar Chromosphere and Its Transition to the Corona Authors: Frisch, H. Bibcode: 1972SSRv...13..455F Altcode: Our present knowledge on the average physical properties of the chromosphere and of the transition region between chromosphere and corona is reviewed. It is recalled that shock wave dissipation is responsible for the high temperatures observed in the chromosphere and corona and that, due to the non-linear character of the dissipation mechanism, no satisfactory explanation of the structure of the outer solar layers has yet been given. In this paper, the main emphasis is on the observations and their interpretation. Evidence for the non-spherically symmetric structure of the atmosphere is given; the validity of interpreting the observations with the help of a fictitious spherically symmetric atmosphere is discussed. The chromosphere and the transition region are studied separately: for each region, the energy balance is considered and recent homogeneous models derived from ultra-violet, infrared and radio observations are discussed. It is stressed that although in the chromosphere, a study of the radiative losses may lead to the determination, as function of height, of the amount of mechanical energy dissipated as function of height, a more detailed analysis of the velocity field is necessary to find the periods and the wavelengths of the waves responsible for the heating. The methods used for wave detection and some results are presented. Observational and theoretical evidence is given for the non-validity of the assumption of hydrostatic equilibrium which is commonly used in modeling the transition region. We conclude that a better understanding of the heating mechanism will come through a higher spatial resolution (less than 0.2″) and more accurate absolute measurements, rather than from sophisticated hydrodynamical calculations. Title: Ultraviolet effects on the chemical composition and optical properties of interstellar grains. Authors: Greenberg, J. M.; Yencha, A. J.; Corbett, J. W.; Frisch, H. L. Bibcode: 1972saim.conf..425G Altcode: No abstract at ADS Title: Non-linear Coupling between Thermal Conduction and Radiative Transfer Authors: Frisch, H. Bibcode: 1971A&A....13..359F Altcode: The problem of coupled thermal conduction and radiative transfer is solved numerically for a plane slab, of finite thickness, heated by thermal conduction, the temperatures at the boundaries and the radiation entering the slab being given. The absorption coefficient is taken of the form T , where T is the temperature and n a positive number. An iterative method is used to solve this non-linear differential problem. Th the optically thick case a simple physical interpretation of the solution can be given in terms of a region of pure radiative equilibrium and transition regions where absorption (but not emission) of radiative energy is negligible. Key words: radiative transfer - thermal conduction - solar chromosphere. Title: Growth, Distribution, and Chemical Composition of Interstellar Dust. Authors: Greenberg, J. M.; Yencha, A. J.; Corbett, J.; Frisch, H. L. Bibcode: 1971BAAS....3Q.250G Altcode: No abstract at ADS Title: Coupling between Radiative Transfer and Thermal Conduction in the Chromosphere Authors: Frisch, H. Bibcode: 1970A&A.....9..269F Altcode: The influence of the transfer of radiative energy on the temperature structure of a plane slab of finite thickness heated by thermal conduction is studied. The temperatures at the boundaries and the radiation entering the slab are given. In some cases, constant absorption coefficient per cm and coefficient of thermal conductivity proportional to the third power of the temperature, the problem can be solved analytically. When the absorption coefficient goes to zero or infinity the temperature distribution in the slab becomes asymptotically identical to that of a slab heated by thermal conduction only. In intermediate cases the temperature inside the slab may become greatly inferior to that of the boundaries with very rapid variations in the vicinity of those boundaries. A generalization of the results to the case of a non-constant absorption coefficient is outlined. Title: Couplage du transfert radiatif et de la conduction thermique dans la chromosphère solaire. Authors: Frisch, H. Bibcode: 1970CRASB.270..918F Altcode: No abstract at ADS Title: Contribution à l'étude de la température du surface du soleil et de la remontée chomosphérique de température Title: Contribution à l'étude de la température du surface du soleil et de la remontée chomosphérique de température Title: Contribution to the study of solar surface temperature and chomospheric temperature rise; Authors: Frisch, Hélène Bibcode: 1966PhDT........52F Altcode: No abstract at ADS