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By
N. Langer, MPI f. Astrophysik, Karl-Schwarzschild-Str. 1, D-85740 Garching, FRG
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
Recent results of the theory of massive star evolution are discussed. We divide the regime of massive stars in a “low” mass and a high mass part, and show that the evolution, the basic theoretical problems in their modeling, and the display of circuinstellar matter are quite different for stars from both parts.
For stars in the lower considered mass regime, it is shown that our ignorance about, internal mixing processes is the main source of uncertainty in stellar model calculations. Mixing processes related to thermal convection are discussed, and their effect on the observable stellar parameters and presupernova structure are sketched. The role of mixing induced by differential rotation is also briefly described. We argue that the supernova stage is a good possibility to investigate the circumstellar material of these objects and describe its high diagnostic potential for the whole presupernova evolution.
Our understanding of stars in the upper mass range, i.e. the most massive stars, also suffers from uncertainties related to internal mixing. However, we argue that it is the mass loss process which dominates their global evolution. The evidence that those objects do lose the major part of their initial mass before they collapse is discussed, together with the possibility of the display of circumstellar nebulae during their hydrostatic evolution. Finally, the fate of very massive stars is discussed.
By
Tomoharu Suzuki, Department of Astronomy, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan,
Toshikazu Shigeyama, Department of Astronomy, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan,
Ken'ichi Nomoto, Department of Astronomy, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
If a supernova progenitor has undergone significant mass-loss then the expanding supernova ejecta will eventually collide with this circumstellar material (CSM). Shock waves arising from the collision will compress and heat both the ejecta and the CSM. The emission from the shocked material depends strongly on the density distributions of the ejecta and the CSM, thereby providing important information about the nature of the CSM.
SN 1987A
Images from the European Southern Observatory (ESO) (Wampler et al. 1990) and the Hubble Space Telescope (HST) (Jakobsen et al. 1991) revealed the presence of a ring-like structure at ∼ 6 × 1017 cm from SN 1987A. The outermost part of the supernova, ejecta is expanding at ∼ 104 km s−1 (Shigeyama & Nomoto 1990) and so is expected to collide with the ring ∼ 10 years after the explosion.
Hydrodynamical model
The progenitor of SN 1987A went through a red supergiant (RSG) phase, and then contracted to a blue supergiant (BSG) before the explosion (for reviews, see Arnett et al. 1989, Hillebrandt & Höflich 1989, Podsiadlowski 1992, and Nomoto et al. 1993a). This evolutionary scenario implies that the SN 1987A environment was formed as follows: the progenitor blew a stellar wind with a velocity ∼ 10 km s−1 and a mass loss rate ∼ 10−5M⊙ yr−1 during the RSG stage, and with corresponding values of ∼ 550 km s−1 and ∼ 10−6M⊙ yr−1 during the BSG stage (Lundqvist & Fransson 1991).
By
Michel Guélin, IRAM, 300 Rue de la Piscine, F-38406 St. Martin d'Hères, France,
Robert Lucas, IRAM, 300 Rue de la Piscine, F-38406 St. Martin d'Hères, France,
Roberto Neri, IRAM, 300 Rue de la Piscine, F-38406 St. Martin d'Hères, France
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
Recent observations of circumstellar shells at arc second resolutions (i.e., 100 R•, in the case of IRC+10216) reveal clumpy structures, asymmetries and jets. The most recent maps of such objects, observed with the IRAM Plateau de Bure interferometer and Pico Veleta telescope, are presented.
Introduction
The dusty envelopes of late type stars are fascinating objects on their own; they are also interesting for what they teach us about IS chemistry. From their velocity field and density profile, we can study the mass loss during a crucial phase of stellar evolution: since the gas expands at nearly constant velocity in all, but the innermost envelope, the velocity maps yield a 3-D view of the molecule spatial distribution; the distributions of the different molecular species show how photochemical, molecule-molecule and grainsurface reactions proceed with time in a well behaved environment.
The closest massive envelopes lie a few hundred parsecs away and have small angular sizes. The construction of large millimeter-wave interferometers, in particular the IRAM Plateau de Bure interferometer (Guilloteau et al. 1992), has provided a major breakthrough in their investigation.
Molecular emission in IRC+10216
The most remarkable and probably closest massive envelope surrounds the bright IR object IRC+10216 (CW Leo). Its outer radius, observed in the mm lines of 12CO, the most abundant molecule and the best shielded from photodissociation after H2, is R = 3′ (Guélin & Cernicharo, in preparation).
By
Antonella Nota, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD; Affiliated to ESA, Astrophysics Division, Space Science Department, of ESA,
Laurent Drissen, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD,
Mark Clampin, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD,
Claus Leitherer, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD,
Anna Pasquali, Universita' di Firenze, Arcetri, Italy,
Carmelle Robert, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD,
Francesco Paresce, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD; Affiliated to ESA, Astrophysics Division, Space Science Department, of ESA; Universita' di Firenze, Arcetri, Italy,
Massimo Robberto, Osservatorio di Torino, Pino Torinese, Italy
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
The LMC star S119 is a member of the group of Ofpe/WN9 stars listed by Bohannan and Walborn (1989). The Ofpe/WN9 category, first defined by Walborn (1982), identifies peculiar supergiants whose spectra combine the typical Of characteristic emission lines of He II and N III with equally strong lower ionization emission features, such as those of He I and N II, and are believed to represent a transition phase in the evolution between massive O stars and WR stars.
High Resolution Echelle Observations
We have observed S119 with the high resolution echelle spectrograph EMMI, coupled to the NTT, ESO La Silla, on September 18, 1991. The spectra cover the wavelength range 4100Å- 7800Å, with a spectral resolution of 0.089 Å/pixel at 6563 Å. The selected slit width was 1.5″ × 5″, with a plate scale on the detector of 0.345″/pixel. In the spectrum, previously undetected nebular lines of Hα, Hβ, [NII], [SII] appear strong and spatially extended, an indication that S119 is surrounded by a bright gaseous nebula. We detect clear splitting of all the observed nebular lines. In Figure 1 we show the radial velocity map obtained from the [NII] 6583 Å line profile. During the observation the slit was oriented EW, and the star was not centered in the aperture, so that only the eastern portion of the nebula lies completely within the slit, while the western region is marginally covered (≃ 2″).
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
By
Guillermo Tenorio-Tagle, Instituto de Astrofísica de Canarias, 38200 La Laguna, Tenerife, Spain
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
This is a short summary of several calculations of the evolution of supernova remnants in a constant high density medium n0 ≥ 106−8 cm−3 and an abundance in the range 0.01Z⊙≤ Z ≤ 10Z⊙. The main difference found when comparing them with the standard calculation of a supernova evolving into a constant density medium n0 = 1 cm−3 is that radiative cooling becomes important very early in the life of the remnants. The radiative phase starts well before the ejecta is fully thermalized and while the expansion velocities are still in the range of several thousands of km s−1. Consequently, the remnants miss their Sedov evolutionary phase and, unlike the standard case, in these calculations full thermalization of the ejecta is only completed long after the moment of thin shell formation (see Terlevich et al. 1992, 1994a; hereafter referred to as papers I and II). The cooling event leads to large luminosities (≥ 109 L⊙) in spans of time of only a few years, causing a major rapid depletion of the supernova's stored thermal energy, in only a few weeks or months. Strong radiative cooling leads to an ionizing spectrum (see paper I) and thus to an HII region with multiple components, as it photoionizes the recombining, rapidly-moving swept-up gas and the outer unperturbed matter. The ionizing radiation is also absorbed by the still unshocked and dense expanding ejecta.
Such remnants, hereafter termed “compact SNRs”, are capable of producing the strong ionizing flux that makes them appear as Seyfert I impostors (Filippenko 1989) when occurring in dense regions far away from the nucleus of galaxies.
By
P. J. Huggins, Physics Department, New York University, New York NY 10003, USA,
R. Bachiller, Centro Astronomico de Yebes, E-19080 Guadalajara, Spain,
P. Cox, Observatoire de Marseille, F-13248 Marseille Cedex 4, France,
T. Forveille, Observatoire de Grenoble, B.P. 53X, F-38041 Grenoble Cedex, France
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
Rapid progress has been made over the last few years in the study of the neutral gas in planetary nebulae (PNe), and it is now well established that at least some PNe are surrounded by massive envelopes of neutral gas (see, e.g., the review by Huggins 1993). The neutral envelopes provide a new perspective on the formation and evolution of the ionized nebulae, and allow the study of a range of circumstellar processes with different characteristics than those found in other circumstellar environments. In this paper we summarize the results of a recent survey of millimeter CO emission in PNe to study the molecular component of the neutral gas, and we comment on some of the issues raised by the observations.
The Molecular Gas in PNe
Millimeter CO emission has proved to be an especially useful probe of the neutral gas in PNe, since it can be used to determine the structure and kinematics of the envelopes, and to estimate the mass of the molecular component. In order to systematically study the differences between PNe, particularly evolutionary effects, we have undertaken extensive survey work in the 230 GHz CO (2–1) line using the IRAM 30 m and SEST 15 m telescopes. These provide access to both northern and southern PNe, with angular resolutions of 12″–24″. Our observations considerably extend the earlier survey work by Huggins ≈ Healy (1989), and are up to a factor of ≈ 6 times more sensitive, depending on the angular size of the PNe.
By
P. R. Wood, Mount Stromlo and Siding Spring Observatories, Private Bag, Weston Creek PO, ACT 2611, Australia
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
Observational and theoretical estimates for mass loss rates from AGB stars are discussed. Then models for the evolution of AGB stars including mass loss and the effects of helium shell flashes are presented. Finally, the possibility of mass loss by binary mass transfer is discussed.
Introduction
It is well established that the bulk of mass loss from low and intermediate mass stars occurs during the asymptotic giant, branch (AGB) stage of evolution, leading to the well-defined sequence of mass-losing stars in the IRAS two-colour diagram (van der Veen and Habing 1988) and the formation of planetary nebulae (Abell and Goldreich 1966; Renzini 1981). However, a reliable theoretical understanding of the causes of mass loss is still not available, although progress is being made. An additional complication is that the time history of mass loss during AGB evolution is quite complex since AGB evolution is modulated by helium shell flashes which control the surface luminosity and thereby the mass loss rate. In this paper, mass loss rates from AGB stars are discussed and the effects of helium shell flashes on the mass loss are described and compared with observations. Time dependent winds produced by AGB stars are reviewed. Finally, the evolution of AGB stars that lose mass in binary mass transfer events is briefly described.
AGB mass loss rates
IRAS observations of stars in the solar neighbourhood indicate that those stars with substantial circumstellar shells - those with high mass loss rates - are nearly all AGB stars undergoing large-amplitude pulsation (Habing 1990).
By
J. E. Dyson, Department of Astronomy, The University of Manchester, Manchester M13 9PL, England,
T. W. Hartquist, Max Planck Institute for Extraterrestrial Physics, D-85740 Garching, Germany
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
Mass addition to flowing tenuous plasmas by the ablation of embedded clumps alters the flows and the observational characteristics of both flows and clumps. The boundary layers between the clumps and the flows are the sites of enhanced radiative losses. Flows which in smooth media would be driven by thermal pressure, are instead driven by momentum. There are many possible types of flows and we explore some of them in the context of Wolf-Rayet and planetary nebulae. Flows in which transsonic tenuous plasmas exit from mass-loading cores into smooth haloes are relevant for planetary nebulae. On intermediate scale lengths, the flow-clump interactions produce extended ‘tails’. We give a general discussion of this and describe applications to the cometary tails behind globules in the planetary nebula NGC 7293 and to the tail of the galactic centre red supergiant IRS7. We finally briefly discuss diagnostics of the boundary layers themselves.
Introduction
Practically all diffuse media of astrophysical significance are clumpy media which are responding to energy sources. The most important distinction between flows initiated in clumpy as opposed to homogeneous media, is that in the former, there is mass, momentum and energy interchange at clumptenuous plasma boundaries, i.e. in boundary layers. The consequences are major (Hartquist & Dyson 1993). This interchange reacts back on the dynamical, physical and even chemical state of the global tenuous plasma flow; conversely, the state of the global flow influences the interchange process.
By
George Sonneborn, Laboratory for Astronomy and Solar Physics, Code 681, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, U.S.A.,
Pedro Rodriguez Pascual, IUE Observatory, ESA-VILSPA, Casilla 50727, E-28080 Madrid, Spain,
Willem Wamsteker, IUE Observatory, ESA-VILSPA, Casilla 50727, E-28080 Madrid, Spain,
Claes Fransson, Stockholm Observatory, S-133 36 Saltsjöbaden, Sweden
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
Supernova 1993J in M81 (NGC 3031) was discovered by Spanish amateur astronomers on 28.86 March 1993 (Ripero & Garcia 1993). The first IUE spectra were taken on 30.2 March at VILSPA a few hours after notification of the discovery (Wamsteker et al. 1993) and the supernova was regularly observed by IUE over the next several weeks. This paper summarizes the principal results of the IUE observations (see Fransson & Sonneborn 1994 and Sonneborn et al. 1994 for observational details and more extensive discussion).
The first photographic detection of the supernova was on 28.30 March at magnitude 13.6 (Merlin & Neely 1993). Modelling of the supernova V light curve indicates that the explosion occurred on 27.8 March and that shock breakout should have occurred at ∼ 28.0 March (Shigeyama et al. 1994). Careful analysis of pre-outburst plates and images has identified the progenitor and shown that its colors are consistent with a late-type supergiant (cf. IAU Circular No. 5739) and the supernova's Type II classification.
In Fig. 1 we show the first UV spectra of SN 1993J from 30.2 March to 3.5 April, where the very rapid cooling of the exploding photosphere is readily apparent. On 30.2 and 31.2 March the temperature was ∼22,500K and ∼14,500K, respectively. Here we have assumed a Galactic extinction law and EB–V = 0.18, as determined from a fit of the 2200 Å dust feature on 30.2 March.
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
By
M. Perinotto, Dipartimento di Astronomia e Scienza dello Spazio, University of Firenze, Largo E. Fermi, 50125 Firenze, Italy,
B. Balick, Astronomy Department, University of Washington, FM-20, Seattle, WA 98195, USA; Dipartimento di Astronomia e Scienza dello Spazio, University of Firenze, Largo E. Fermi, 50125 Firenze, Italy,
Y. Terzian, Astronomy Department and NAIC, Cornell University, Ithaca, NY 14853, USA,
A. Hajian, Astronomy Department and NAIC, Cornell University, Ithaca, NY 14853, USA,
A. Maccioni, Dipartimento di Astronomia e Scienza dello Spazio, University of Firenze, Largo E. Fermi, 50125 Firenze, Italy
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
In several elliptical PNe, a number of structures have been observed which are called : rims, shells, caps, ansae, knots, etc. in addition to the haloes. Some are macrostructures (rims, shells, haloes) constituting in a sense the bulk of the nebula itself. They can either be homogeneous or contain themselves smaller structures. Others are microstructures, and some of them qualify as FLIERs (see below).
Balick et al. (1993; hereafter paper I) explored the spectra of microstructures in three elliptical PNe : NGC 3242, 7662, and IC 2149. They found that NGC 3242 and 7662 contain pairs of low ionization knots moving at supersonic velocity relative to the ambient gas. They have been named FLIERs (fast low ionization emitting regions). Various interpretations were discussed, but a convincing explanation was not found.
Here we present a preliminary study of microstructures in three more elliptical PNe : NGC 6543, 6826 and 7009.
Observations
The observations were performed using the Palomar 5-m telescope and double spectrograph at dispersions of 2.1 Å/pixel in the blue (3400-5150 Å) and 3.1 Å/pixel in the red (5150-7600 Å). The effective resolution corresponds to about 300 km s−1. Along the slit each pixel is 0″.58 for the red and 0″.78 for the blue. Typical seeing was about 1.5 arcsec.
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
Edited by
R. E. S. Clegg, Royal Greenwich Observatory, Cambridge,I. R. Stevens, Imperial College of Science, Technology and Medicine, London,W. P. S. Meikle, University of Birmingham
In this review, we consider a quantum Coulomb fluid made of charged point particles (typically electrons and nuclei). We describe various formalisms which start from the first principles of statistical mechanics. These methods allow systematic calculations of the equilibrium quantities in some particular limits. The effective-potential method is evocated first, as well as its application to the derivation of low-density expansions. We also sketch the basic outlines of the standard many-body perturbation theory. This approach is well suited for calculating expansions at high density (for Fermions) or at high temperature. Eventually, we present the Feynman-Kac path integral representation which leads to the introduction of an auxiliary classical system made of extended objects, i.e., filaments (also called “polymers”). The familiar Abe-Meeron diagrammatic series are then generalized in the framework of this representation. The truncations of the corresponding virial-like expansions provide equations of state which are asymptotically exact in the low-density limit at fixed temperature. The usefulness of such equations for describing the inner regions of the sun is briefly illustrated.
Abstract
Dans cette revue, nous considérons un fluide coulombien quantique constitué de charges ponctuelles (typiquement des électrons et des noyaux). Nous décrivons différents formalismes s'appuyant sur les premiers principes de la mécanique statistique. Ces méthodes permettent de calculer les quantités d'équilibre de manière systématique dans des limites particulières. La méthode des potentiels effectifs est d'abord évoquée, ainsi que son application aux développements à basse densité.
By
G. Fontaine, Département de Physique, Université de Montréal, C.P. 6128, succursale A, Montréal, Québec, H3C 3J7, Canada.,
P. Brassard, Départment de Physique, Université de Montréal, C.P. 6128, succursale A, Montréal, Québec, H3C 3J7, Canada.
We present the results of numerical experiments aimed at demonstrating how the g-mode period spectra of pulsating DA white dwarfs depend on the various components of the input physics. We take advantage of recent developments on many fronts of physics (equation of state, opacity, convection) to compare the theoretical pulsation periods of models with different pieces of the constitutive physics, but with otherwise fixed values of their stellar parameters. This exercise is necessary to assess the reliability of the pulsation analyses of white dwarfs which have started to come out.
Nous présentons les résultats de simulations numériques pour déterminer comment les périodes de pulsation (type g) des étoiles naines blanches DA dépendent des différentes composantes de la physique constitutive. A cet effet, nous avons utilisé des résultats récents au niveau de la physique de base (équation d'état, opacité, convection) pour comparer les périodes de pulsation de modèles stellaires ayant des paramètres fixes, mais qui différent au niveau de leur physique constitutive. Notre démarche est essentielle afin de pouvoir quantifier les premiers résultats d'analyses d'étoiles pulsantes qui commencent à être publiés.
Introduction
It is now well established that white dwarf stars become intrinsically variable during certain phases of their evolution. For the majority of them, the so-called DA white dwarfs (with atmospheres dominated by hydrogen), luminosity variations are observed when the stars have effective temperatures in the rather narrow interval 13,000 K ≳ Teff ≳ 11,000 K (Wesemael et al. 1991).
The numerous complexities underlying large tables of thermodynamic quantities act as a deterrent to a careful evaluation of their reliability. As a consequence, equations of state are often used as black boxes. To clarify this situation, some of the more critical issues of equation of state physics are discussed from the point of view of the user. They are illustrated by a comparison of four equations of state for hydrogen. The flaws and disagreements thus brought into light are explained and evaluated with simple physical arguments.
Les tables d'équations d'état utilisées en astrophysique découlent de modèles d'une complexité telle qu'il est souvent difficile d'en évaluer la fiabilité. Il en résulte une situation où les équations d'état sont souvent utilisées sans une analyse critique de leur contenu physique ni de leur précision. Dans le but de remédier à cette situation, une discussion des principaux éléments physiques des équations d'état est présentée dans l'optique de l'utilisateur. Quatre équations d'état de l'hydrogène développées pour être appliquées à des problèmes d'astrophysique stellaire sont comparées de façon critique. Cette comparaison illustre l'importance de certains éléments clés des équations d'état et la nature des problèmes qui subsistent. Les déefauts et les différences observés entre ces quatre équations d'état sont élucidés en termes de physique de base.
Introduction
The richness of stellar phenomena exposed by modern observational techniques calls for a quantitative understanding of more subtle, “second order” effects in stellar structure.
By
A. G. Muslimov, Department of Physics and Astronomy and C. E. Kenneth Mees Observatory, University of Rochester, Rochester NY 14627-0011 USA,
H. M. Van Horn, Department of Physics and Astronomy and C. E. Kenneth Mees Observatory, University of Rochester, Rochester NY 14627-0011 USA
We consider a simple model for the evolution of a poloidal magnetic field initally trapped in a region containing normal npe matter within the outer liquid core of a neutron star. We have performed numerical computations for neutron stars with masses of 1.4, 1.6, and 1.7 M⊙ that undergo very rapid cooling due to the direct Urea process. Because the timescale for the magnetic field decay is directly proportional to T2, such a cooling history produces a rapid decline in the magnetic-field strength B, even for B as low as ∼ 1012 G. In particular, we show that an initially quasi-homogeneous magnetic field of strength B = 1012 G declines during the first ∼ 1 Myr.
Introduction
The calculations of Baym, Pethick, and Pines (1969a) have shown that the electrical conductivity of matter in the core of a neutron star is too large to permit ohmic decay of the magnetic field within the age of the Universe. Recently, Haensel, Urpin, and Yakovlev (1990; hereafter HUY) have pointed out that the magnetic-field strength |B| ∼ 1012 G typical of pulsars is sufficiently strong that the anisotropy of the transport coefficients cannot be neglected and that the “resistivity” for current flow perpendicular to B is many orders of magnitude larger than that for current flow parallel to B. Using a simple “toy” model, they found that internal fields B ≥ 1013 G can decline to ∼ 1012 G in times ∼ 107 years, but that fields ≤ 1012 G remain practically unchanged on this timescale.