• Volume 77, Issue 1

July 2011,   pages  1-232

• Preface

• Hydrodynamic stability and stellar oscillations

Chandrasekhar’s monograph on Hydrodynamic and hydromagnetic stability, published in 1961, is a standard reference on linear stability theory. It gives a detailed account of stability of ﬂuid ﬂow in a variety of circumstances, including convection, stability of Couette ﬂow, Rayleigh–Taylor instability, Kelvin–Helmholtz instability as well as the Jean’s instability for star formation. In most cases he has extended these studies to include effects of rotation and magnetic ﬁeld. In a later paper he has given a variational formulation for equations of non-radial stellar oscillations. This forms the basis for helioseismic inversion techniques as well as extension to include the effect of rotation, magnetic ﬁeld and other large-scale ﬂows using a perturbation treatment.

• Disks around young stellar objects

By 1939, when Chandrasekhar’s classic monograph on the theory of Stellar Structure was published, although the need for recent star formation was fully acknowledged, no one had yet recognized an object that could be called a star in the process of being born. Young stellar objects (YSOs), as pre-main-sequence stars, were discovered in the 1940s and 1950s. Infrared excess emission and intrinsic polarization observed in these objects in the 1960s and 1970s indicated that they are surrounded by ﬂattened disks. The YSO disks were seen in direct imaging only in the 1980s. Since then, high-resolution optical imaging with HST, near-infrared adaptive optics on large groundbased telescopes, mm and radiowave interferometry have been used to image disks around a large number of YSOs revealing disk structure with ever-increasing detail and variety. The disks around YSOs are believed to be the sites of planet formation and a few such associations have now been conﬁrmed. The observed properties of the disk structure and their evolution, that have very important consequences for the theory of star and planet formation, are discussed.

• Beyond the Chandrasekhar limit: Structure and formation of compact stars

The concept of limiting mass, introduced by Chandrasekhar in case of white dwarfs, plays an important role in the formation and stability of compact objects such as neutron stars and black holes. Like white dwarfs, neutron stars have their own mass limit, and a compact conﬁguration would progress from one family to the next, more dense one once a mass limit is crossed. The mass limit of neutron stars depends on the nature of nuclear forces at very high density, which has so far not been determined conclusively. This article reviews how observational determinations of the properties of neutron stars are starting to impose signiﬁcant constraints on the state of matter at high density

• Extracting science from surveys of our Galaxy

Our knowledge of the Galaxy is being revolutionized by a series of photometric, spectroscopic and astrometric surveys. Already an enormous body of data is available from completed surveys, and data of ever-increasing quality and richness will accrue at least until the end of this decade. To extract science from these surveys, we need a class of models that can give probability density functions in the space of the observables of a survey – we should not attempt to ‘invert’ the data from the space of observables into the physical space of the Galaxy. Currently just one class of model has the required capability, the so-called ‘torus models’. A pilot application of torus models to understand the structure of the Galaxy’s thin and thick discs has already produced two signiﬁcant results: a major revision of our best estimate of the Sun’s velocity with respect to the local standard of rest, and a successful prediction of the way in which the vertical velocity dispersion in the disc varies with distance from the Galactic plane.

• Hydrodynamic and hydromagnetic stability of black holes with radiative transfer

Subrahmanyan Chandrasekhar (Chandra) was just eight years old when the ﬁrst astrophysical jet was discovered in M87. Since then, jets have been uncovered with a wide variety of sources including accretion disks orbiting stellar and massive black holes, neutron stars, isolated pulsars, 𝛾-ray bursts, protostars and planetary nebulae. This talk will be primarily concerned with collimated hydromagnetic outﬂows associated with spinning, massive black holes in active galactic nuclei. Jets exhibit physical processes central to three of the major research themes in Chandrasekhar’s research career – radiative transfer, magnetohydrodynamics and black holes. Relativistic jets can be thought of as exhausts’ from both the hole and its orbiting accretion disk, carrying away the energy liberated by the rotating spacetime and the accreting gas that is not radiated. However, no aspect of jet formation, propagation and radiation can be regarded as understood in detail. The combination of new 𝛾-ray, radio and optical observations together with impressive advances in numerical simulation make this a good time to settle some long-standing debates.

• Chandrasekhar–Kendall functions in astrophysical dynamos

Some of the contributions of Chandrasekhar to the ﬁeld of magnetohydrodynamics are highlighted. Particular emphasis is placed on the Chandrasekhar–Kendall functions that allow a decomposition of a vector ﬁeld into right- and left-handed contributions. Magnetic energy spectra of both contributions are shown for a new set of helically forced simulations at resolutions higher than what has been available so far. For a forcing function with positive helicity, these simulations show a forward cascade of the right-handed contributions to the magnetic ﬁeld and nonlocal inverse transfer for the left-handed contributions. The speed of inverse transfer is shown to decrease with increasing value of the magnetic Reynolds number.

• The origin of the solar magnetic cycle

After summarizing the relevant observational data, we discuss how a study of ﬂux tube dynamics in the solar convection zone helps us to understand the formation of sunspots. Then we introduce the ﬂux transport dynamo model and assess its success in modelling both the solar cycle and its departures from strictly periodic behaviour.

• Chandrasekhar’s book An Introduction to the Study of Stellar Structure

For me, and for many astrophysicists of my generation, Chandrasekhar’s book An Introduction to the Study of Stellar Structure was very important. I could not have done my PhD (1962–1965) without it. Much more recently (1998) I realized that I could not have written my lecture course on thermodynamics and statistical mechanics without much of it, particularly the ﬁrst chapter. I shall present anecdotal evidence that the inﬂuence of his discussion on the second law of thermodynamics has been important not just for astrophysics but for a much wider range of physics.

Chandrasekhar’s discussion of polytropes was masterly. Even today polytropes play an important role as an aid for understanding stellar structure. I believe that to the list of analytic solutions of the polytrope only one more has to be added: a curious $n = 5$ model of Srivastava (1962).

Stellar structure is nowadays a very computationally intensive subject. I shall illustrate this with a couple of topics from my experience with Djehuty, a supercomputer code for modelling stars in 3D. Nevertheless it remains true, I believe, that analytical mathematical entities like polytropes are fundamental as aids for understanding what the computers churn out.

How close are we to seeing a book with the title The Last Word on the Study of Stellar Structure’? Not very, although much has been learned in 70 years. I shall discuss a few of the aspects of stellar evolution that are problematic today.

I shall discuss a couple of aspects where I believe analysis of piecewise polytropic’ structures sheds light on the question Why do stars become red giants?’

• Chance and Chandra

A few examples are given of Chandra’s work on statistical and stochastic problems that relate to open questions in astrophysics, in particular his theory of dynamical relaxation in systems with inverse-square interparticle forces. The roles of chaos and integrability in this theory require clariﬁcation, especially for systems having a dominant central mass. After this prelude, a hypothetical form of bosonic dark matter with a simple but nontrivial statistical mechanics is discussed. This makes for a number of eminently falsiﬁable predictions, including some exotic consequences for dynamical friction.

• Gravitational waves from binary black holes

It is almost a century since Einstein predicted the existence of gravitational waves as one of the consequences of his general theory of relativity. A brief historical overview including Chandrasekhar’s contribution to the subject is ﬁrst presented. The current status of the experimental search for gravitational waves and the attendant theoretical insights into the two-body problem in general relativity arising from computations of gravitational waves from binary black holes are then broadly reviewed.

• Stellar dynamics and black holes

Chandrasekhar’s most important contribution to stellar dynamics was the concept of dynamical friction. I brieﬂy review that work, then discuss some implications of Chandrasekhar’s theory of gravitational encounters for motion in galactic nuclei.

• Statistical mechanics of gravitating systems ... and some curious history of Chandra’s rare misses!

Chandra’s academic life had several phases each culminating in a monograph describing that subject. I shall deal with aspects of his work in the two earliest phases. I shall describe the overall structure of statistical mechanics of gravitating systems, the relevance of isothermal sphere in the mean-ﬁeld approximation and issues related to collisional relaxation and dynamical friction in self-gravitating system of particles. There are several curious features in the history of these topics which I comment upon.

• Multiple scattering polarization – Application of Chandrasekhar’s formalisms to the atmosphere of brown dwarfs and extrasolar planets

Chandrasekhar’s formalisms for the transfer of polarized radiation are used to explain the observed dust scattering polarization of brown dwarfs in the optical band. Model polarization profiles for hot and young directly imaged extrasolar planets are presented with specific prediction of the degree of polarization in the infrared. The model invokes Chandrasekhar’s formalism for the rotation-induced oblateness of the objects that gives rise to the necessary asymmetry for yielding net non-zero disk integrated linear polarization. The observed optical polarization constrains the surface gravity and could be a tool to estimate the mass of extrasolar planets.

• Symmetry in stochasticity: Random walk models of large-scale structure

This paper describes the insights gained from the excursion set approach, in which various questions about the phenomenology of large-scale structure formation can be mapped to problems associated with the ﬁrst crossing distribution of appropriately deﬁned barriers by random walks. Much of this is summarized in R K Sheth, AIP Conf. Proc. 1132, 158 (2009). So only a summary is given here, and instead a few new excursion set related ideas and results which are not published elsewhere are presented. One is a generalization of the formation time distribution to the case in which formation corresponds to the time when half the mass was ﬁrst assembled in pieces, each of which was at least $1/n$ times the ﬁnal mass, and where $n \geq 2$; another is an analysis of the ﬁrst crossing distribution of the Ornstein–Uhlenbeck process. The ﬁrst derives from the mirror-image symmetry argument for random walks which Chandrasekhar described so elegantly in 1943; the second corrects a misuse of this argument. Finally, some discussion of the correlated steps and correlated walks assumptions associated with the excursion set approach, and the relation between these and peaks theory are also included. These are problems in which Chandra’s mirror-image symmetry is broken.

• Magnetohydrodynamics turbulence: An astronomical perspective

Early work on magnetohydrodynamic (MHD) turbulence in the 1960s due, independently, to Iroshnikov and Kraichnan (IK) considered isotropic inertial-range spectra. Whereas laboratory experiments were not in a position to measure the spectral index, they showed that the turbulence was strongly anisotropic. Theoretical horizons correspondingly expanded in the 1980s, to accommodate both the isotropy of the IK theory and the anisotropy suggested by the experiments. Since the discovery of pulsars in 1967, many years of work on interstellar scintillation suggested that small-scale interstellar turbulence must have a hydromagnetic origin; but the IK spectrum was too ﬂat and the ideas on anisotropic spectra too qualitative to explain the observations. In response, new theories of balanced MHD turbulence were proposed in the 1990s, which argued that the IK theory was incorrect, and made quantitative predictions of anisotropic inertial-range spectra; these theories have since found applications in many areas of astrophysics. Spacecraft measurements of solar-wind turbulence show that there is more power in Alfvén waves that travel away from the Sun than towards it. Theories of imbalanced MHD turbulence have now been proposed to address interplanetary turbulence. This very active area of research continues to be driven by astronomy.

• The origin of the strongest magnetic ﬁelds in dwarfs

White dwarfs have frozen in magnetic ﬁelds ranging from below the measurable limit of about $3 \times 10^3$ to $10^9$ G. White dwarfs with surface magnetic ﬁelds in excess of 1 MG are found as isolated single stars and relatively more often in magnetic cataclysmic variables. Some 1253 white dwarfs with a detached low-mass main-sequence companion have been identiﬁed in the Sloan Digital Sky Survey (SDSS) but none of these shows sufﬁcient evidence for Zeeman splitting of hydrogen lines for a magnetic ﬁeld in excess of 1 MG. If such high magnetic ﬁelds in white dwarfs result from the isolated evolution of a single star then there should be the same fraction of high ﬁeld white dwarfs among this SDSS binary sample as among single stars. Thus, we deduce that the origin of such high magnetic ﬁelds must be intimately tied to the formation of cataclysmic variables (CVs). The formation of a CV must involve orbital shrinkage from giant star to main-sequence star dimensions. It is believed that this shrinkage occurs as the low-mass companion and the white dwarf spiral together inside a common envelope. CVs emerge as very close but detached binary stars that are then brought together by magnetic braking or gravitational radiation. We propose that the smaller the orbital separation at the end of the common envelope phase, the stronger the magnetic ﬁeld. The magnetic cataclysmic variables (MCVs) originate from those common envelope systems that almost merge. Those common envelope systems that do merge are the progenitors of the single high ﬁeld white dwarfs. Thus all highly magnetic white dwarfs, be they single stars or the components of MCVs, have a binary origin. This accounts for the relative dearth of single white dwarfs with ﬁelds of $10^4$ – $10^6$ G. Such intermediate-ﬁeld white dwarfs are found preferentially in cataclysmic variables. The bias towards higher masses for highly magnetic white dwarfs is expected if a fraction of these form when two degenerate cores merge in a common envelope. From the space density of single highly magnetic white dwarfs we estimate that about three times as many common envelope events lead to a merged core as to a cataclysmic variable.

• The legacy of S Chandrasekhar (1910–1995)

Subrahmanyan Chandrasekhar, known simply as Chandra in the scientiﬁc world, is one of the foremost scientists of the 20th century. In celebrating his birth centenary, I present a biographical portrait of an extraordinary, but a highly private individual unknown to the world at large. Drawing upon his own A Scientiﬁc Autobiography,” I reﬂect upon his legacy as a scientist and a great human being.

• List of Participants

• # Pramana – Journal of Physics

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• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019