Douglas Gough, Michael Proctor and Steven Tobias remember the South African-born astrophysicist a... more Douglas Gough, Michael Proctor and Steven Tobias remember the South African-born astrophysicist and mathematician, a former president of the RAS
We consider the effect of the solar cycle on g-mode parametric resonance, and hence on the limiti... more We consider the effect of the solar cycle on g-mode parametric resonance, and hence on the limiting amplitude of the overstable solar g1(ℓ = 1) mode. We find no change in the expected limiting amplitude from that found by Dziembowski (1983), who ignored cycle variations.
My brief from the principal organizers of this colloquium is to draw your attention to some of th... more My brief from the principal organizers of this colloquium is to draw your attention to some of the prejudices upon which the observations that have been discussed here have had or should have had some impact, and to remark on how that might revise our views about physics: it should not be a balanced summary of the meeting, but a prejudiced review.
Rotational splitting Δω(n,l,m) of the eigenfrequencies of a star rotating with angular velocity Ω... more Rotational splitting Δω(n,l,m) of the eigenfrequencies of a star rotating with angular velocity Ω(r, θ) about a unique axis can be represented as a weighted integral of Ω over r and θ, (r, θ, φ) being spherical polar coordinates about the axis of rotation. For high-frequency acoustic modes, Δω/m collapses essentially to a function of ω — ω /(l+1/2) and M = m/(l + 1/2) alone, and the weighting kernel K(r,θ) becomes asymptotically degenerate, each factor being of essentially Abel type. Therefore, formally, the splitting integral can be inverted, once a procedure has been found for extending Δω over the domain of (ω, M) such that the turning points (rt,ωt), given by (c(rt)/rt, sinωt) = (ω, M) where c is sound speed, span the star. Obtaining that representation is the most difficult stage of the inversion. We report on a procedure that treats the inverted two-dimensional Abel integral as a repeated double integral, representing the data successively along a set of parallel lines M =cons...
Helioseismology and asteroseismology put high demands on the accuracy and consistent numerical re... more Helioseismology and asteroseismology put high demands on the accuracy and consistent numerical realization of the equation of state (for a review see Däppen, these proceedings). This is explicitly illustrated by the helioseismic determination of the helium abundance of the solar convection zone in a recent investigation by Kosovichev et al. (1992). In that work it was observed that details of the treatment of the heavy elements matter more than is intuitively expected. Naively, one would expect an uncertainty of less than 10−4 in the key thermodynamic quantity, the adiabatic gradient Γ1. This is because in material of solar composition the heavy-element abundance is less than about 1.5 × 10−3 by number, and under solar conditions the dominant nontrivial contributions to the seismically relevant thermodynamic quantities predicted by modern equations of state agree to a few per cent, even for the much more abundant hydrogen and helium. However, Kosovichev et al. (1992) found that unce...
The component of the frequency splitting of solar five-minute oscillations observed by Duvall, Ha... more The component of the frequency splitting of solar five-minute oscillations observed by Duvall, Harvey and Pomerantz that is even in azimuthal degree measures latitudinal and depth variations in the structure of the sun. We indicate how the data hint that there is a shallow perturbation, possibly associated with a magnetic field, that is concentrated at low latitudes.
Magnetic fields contribute to the splitting of the degeneracy of modes of like order and degree. ... more Magnetic fields contribute to the splitting of the degeneracy of modes of like order and degree. The splitting is estimated for some simple hypothetical toroidal magnetic field configurations in the sun, and the results are compared with previous asymptotic estimates. Splitting by a field confined to a thin layer at the base of the convection zone is found not to agree with recent measurements.
The discrepancy between theoretical eigenfrequencies of standard solar models and the frequencies... more The discrepancy between theoretical eigenfrequencies of standard solar models and the frequencies of solar modes of degree between 2 and 5 measured at Stanford is degree-independent for cyclic frequencies above about 2 mHz. Below that frequency the discrepancy for dotriacontapole modes diverges from that of the modes of lower degree. The differences between eigenfrequencies of a simple solar model containing a cloud of weakly interacting particles in its core and of one without do not reproduce this behaviour.
The end of the millennium marks the beginning of the third phase of helioseismology. The first ph... more The end of the millennium marks the beginning of the third phase of helioseismology. The first phase was the establishment of the initial astronomical inferences, such as estimates of the depth of the solar convection zone and the protosolar helium abundance, obtained by comparing the seismic properties of theoretical solar models with the first wave of helioseismic data acquired using instruments that had not been designed for the purpose. The second phase was the determination of the spherically symmetric component of the hydrostatic stratification throughout most of the solar interior, and the angular velocity, using inverse methods to analyse the frequencies of normal modes estimated from data obtained most recently from purpose-built networks of ground-based observatories and from space. We have reached the point beyond which further pursuit of the now-well-tried methods to improve the inferences will be apparently slow. The next era will be characterized by painstaking attenti...
We discuss a generalization of a mixing-length formalism for convection in the presence of a mean... more We discuss a generalization of a mixing-length formalism for convection in the presence of a mean flow, and present the convective fluxes for convective cells with the geometry of rolls and of hexagons.
Solar five-minute oscillations provide a means of testing theoretical models of the sun. By judic... more Solar five-minute oscillations provide a means of testing theoretical models of the sun. By judiciously combining data from low-degree modes, properties of the central and surface regions of the sun can be inferred separately. In principle, it should be possible to draw similar inferences from other stars, once adequate data are available. Recent solar rotational splitting data imply that in the equatorial regions much of the radiative envelope of the sun is rotating more slowly than the photosphere.
A procedure for inverting helioseismic data to determine the hydrogen abundance in the radiative ... more A procedure for inverting helioseismic data to determine the hydrogen abundance in the radiative interior of the sun is briefly described. Using Backus-Gilbert optimal averaging, the variation of sound speed, density and hydrogen abundance in the energy-generating core is estimated from low-degree p-mode frequencies. The result provides some evidence for there having been some redistribution of material during the sun’s main-sequence evolution. The inversion also suggests that the evolutionary age of the sun is perhaps some 10 per cent greater than the generally accepted value, and that the solar neutrino flux, based on standard nuclear and particle physics, is about 75 per cent of the standard-model value.
Dynamical and thermal modulation of the internal structure of the Sun can be manifest at the surf... more Dynamical and thermal modulation of the internal structure of the Sun can be manifest at the surface as changes in irradiance and radius. The relative magnitudes of these changes could provide a diagnostic of at least the location of the primary modulation, if only the mechanism were known. Variations of the frequencies of solar oscillations offer additional potentially valuable diagnostics, but unfortunately at present the mechanisms causing those variations and their relation to the structural and irradiance changes are not yet understood. In this lecture I shall review some of the theoretical conjectures that have been put forward to explain the observations.
Accurate measurement of the frequencies of low-degree acoustic oscillations of sun-like stars is ... more Accurate measurement of the frequencies of low-degree acoustic oscillations of sun-like stars is imminent. We report on our first calculations with proxy data, aimed at assessing the kind of physical information that is likely to be acquired from seismic analysis and the precision with which the frequencies must be measured in order to obtain that information. The results will have an important bearing on future observing strategies, for the duration of observation should be determined primarily by the precision required of the frequency measurements. Our inversions are of eigenfrequencies of modes of an evolved main-sequence star of mass 1.1M⊙. The modes are of degree 0, 1 and 2, with frequencies in the range 1-3 mHz. Thus, by analogy with solar oscillations, they are modes that one should expect to observe in stars similar to the sun. Figure Iadepicts an idealized spectrum of stellar acoustic oscillations as one might expect from intensity variations such as those that could be me...
This is not a review of the literature on solar hydrodynamics, but a discussion of just a few of ... more This is not a review of the literature on solar hydrodynamics, but a discussion of just a few of the Interesting aspects of the subject. It is neither comprehensive, coherent nor well balanced. It concentrates on some of the questions that have been raised recently by new observations and have not yet received much attention, and particularly on topics that were raised at the recent General Assembly of the IAU. Subjects that have been reviewed in the past, such as the propagation of waves in the solar atmosphere and the structure of the solar wind, are avoided. Magneto-hydrodynamics is hardly mentioned at all.
Solar p-mode multiplet oscillation frequencies and expansion coefficients for degeneracy splittin... more Solar p-mode multiplet oscillation frequencies and expansion coefficients for degeneracy splitting have been measured in the summers of 1986 and 1988–1990. Temporal variations not unlike the differences between 1988 and 1986 reported by Libbrecht & Woodard (1990) continue into 1990. These differences are possibly associated with the solar cycle. We report here on our first findings about the variation of the internal angular velocity and asphericity deduced from these frequency differences.
Douglas Gough, Michael Proctor and Steven Tobias remember the South African-born astrophysicist a... more Douglas Gough, Michael Proctor and Steven Tobias remember the South African-born astrophysicist and mathematician, a former president of the RAS
We consider the effect of the solar cycle on g-mode parametric resonance, and hence on the limiti... more We consider the effect of the solar cycle on g-mode parametric resonance, and hence on the limiting amplitude of the overstable solar g1(ℓ = 1) mode. We find no change in the expected limiting amplitude from that found by Dziembowski (1983), who ignored cycle variations.
My brief from the principal organizers of this colloquium is to draw your attention to some of th... more My brief from the principal organizers of this colloquium is to draw your attention to some of the prejudices upon which the observations that have been discussed here have had or should have had some impact, and to remark on how that might revise our views about physics: it should not be a balanced summary of the meeting, but a prejudiced review.
Rotational splitting Δω(n,l,m) of the eigenfrequencies of a star rotating with angular velocity Ω... more Rotational splitting Δω(n,l,m) of the eigenfrequencies of a star rotating with angular velocity Ω(r, θ) about a unique axis can be represented as a weighted integral of Ω over r and θ, (r, θ, φ) being spherical polar coordinates about the axis of rotation. For high-frequency acoustic modes, Δω/m collapses essentially to a function of ω — ω /(l+1/2) and M = m/(l + 1/2) alone, and the weighting kernel K(r,θ) becomes asymptotically degenerate, each factor being of essentially Abel type. Therefore, formally, the splitting integral can be inverted, once a procedure has been found for extending Δω over the domain of (ω, M) such that the turning points (rt,ωt), given by (c(rt)/rt, sinωt) = (ω, M) where c is sound speed, span the star. Obtaining that representation is the most difficult stage of the inversion. We report on a procedure that treats the inverted two-dimensional Abel integral as a repeated double integral, representing the data successively along a set of parallel lines M =cons...
Helioseismology and asteroseismology put high demands on the accuracy and consistent numerical re... more Helioseismology and asteroseismology put high demands on the accuracy and consistent numerical realization of the equation of state (for a review see Däppen, these proceedings). This is explicitly illustrated by the helioseismic determination of the helium abundance of the solar convection zone in a recent investigation by Kosovichev et al. (1992). In that work it was observed that details of the treatment of the heavy elements matter more than is intuitively expected. Naively, one would expect an uncertainty of less than 10−4 in the key thermodynamic quantity, the adiabatic gradient Γ1. This is because in material of solar composition the heavy-element abundance is less than about 1.5 × 10−3 by number, and under solar conditions the dominant nontrivial contributions to the seismically relevant thermodynamic quantities predicted by modern equations of state agree to a few per cent, even for the much more abundant hydrogen and helium. However, Kosovichev et al. (1992) found that unce...
The component of the frequency splitting of solar five-minute oscillations observed by Duvall, Ha... more The component of the frequency splitting of solar five-minute oscillations observed by Duvall, Harvey and Pomerantz that is even in azimuthal degree measures latitudinal and depth variations in the structure of the sun. We indicate how the data hint that there is a shallow perturbation, possibly associated with a magnetic field, that is concentrated at low latitudes.
Magnetic fields contribute to the splitting of the degeneracy of modes of like order and degree. ... more Magnetic fields contribute to the splitting of the degeneracy of modes of like order and degree. The splitting is estimated for some simple hypothetical toroidal magnetic field configurations in the sun, and the results are compared with previous asymptotic estimates. Splitting by a field confined to a thin layer at the base of the convection zone is found not to agree with recent measurements.
The discrepancy between theoretical eigenfrequencies of standard solar models and the frequencies... more The discrepancy between theoretical eigenfrequencies of standard solar models and the frequencies of solar modes of degree between 2 and 5 measured at Stanford is degree-independent for cyclic frequencies above about 2 mHz. Below that frequency the discrepancy for dotriacontapole modes diverges from that of the modes of lower degree. The differences between eigenfrequencies of a simple solar model containing a cloud of weakly interacting particles in its core and of one without do not reproduce this behaviour.
The end of the millennium marks the beginning of the third phase of helioseismology. The first ph... more The end of the millennium marks the beginning of the third phase of helioseismology. The first phase was the establishment of the initial astronomical inferences, such as estimates of the depth of the solar convection zone and the protosolar helium abundance, obtained by comparing the seismic properties of theoretical solar models with the first wave of helioseismic data acquired using instruments that had not been designed for the purpose. The second phase was the determination of the spherically symmetric component of the hydrostatic stratification throughout most of the solar interior, and the angular velocity, using inverse methods to analyse the frequencies of normal modes estimated from data obtained most recently from purpose-built networks of ground-based observatories and from space. We have reached the point beyond which further pursuit of the now-well-tried methods to improve the inferences will be apparently slow. The next era will be characterized by painstaking attenti...
We discuss a generalization of a mixing-length formalism for convection in the presence of a mean... more We discuss a generalization of a mixing-length formalism for convection in the presence of a mean flow, and present the convective fluxes for convective cells with the geometry of rolls and of hexagons.
Solar five-minute oscillations provide a means of testing theoretical models of the sun. By judic... more Solar five-minute oscillations provide a means of testing theoretical models of the sun. By judiciously combining data from low-degree modes, properties of the central and surface regions of the sun can be inferred separately. In principle, it should be possible to draw similar inferences from other stars, once adequate data are available. Recent solar rotational splitting data imply that in the equatorial regions much of the radiative envelope of the sun is rotating more slowly than the photosphere.
A procedure for inverting helioseismic data to determine the hydrogen abundance in the radiative ... more A procedure for inverting helioseismic data to determine the hydrogen abundance in the radiative interior of the sun is briefly described. Using Backus-Gilbert optimal averaging, the variation of sound speed, density and hydrogen abundance in the energy-generating core is estimated from low-degree p-mode frequencies. The result provides some evidence for there having been some redistribution of material during the sun’s main-sequence evolution. The inversion also suggests that the evolutionary age of the sun is perhaps some 10 per cent greater than the generally accepted value, and that the solar neutrino flux, based on standard nuclear and particle physics, is about 75 per cent of the standard-model value.
Dynamical and thermal modulation of the internal structure of the Sun can be manifest at the surf... more Dynamical and thermal modulation of the internal structure of the Sun can be manifest at the surface as changes in irradiance and radius. The relative magnitudes of these changes could provide a diagnostic of at least the location of the primary modulation, if only the mechanism were known. Variations of the frequencies of solar oscillations offer additional potentially valuable diagnostics, but unfortunately at present the mechanisms causing those variations and their relation to the structural and irradiance changes are not yet understood. In this lecture I shall review some of the theoretical conjectures that have been put forward to explain the observations.
Accurate measurement of the frequencies of low-degree acoustic oscillations of sun-like stars is ... more Accurate measurement of the frequencies of low-degree acoustic oscillations of sun-like stars is imminent. We report on our first calculations with proxy data, aimed at assessing the kind of physical information that is likely to be acquired from seismic analysis and the precision with which the frequencies must be measured in order to obtain that information. The results will have an important bearing on future observing strategies, for the duration of observation should be determined primarily by the precision required of the frequency measurements. Our inversions are of eigenfrequencies of modes of an evolved main-sequence star of mass 1.1M⊙. The modes are of degree 0, 1 and 2, with frequencies in the range 1-3 mHz. Thus, by analogy with solar oscillations, they are modes that one should expect to observe in stars similar to the sun. Figure Iadepicts an idealized spectrum of stellar acoustic oscillations as one might expect from intensity variations such as those that could be me...
This is not a review of the literature on solar hydrodynamics, but a discussion of just a few of ... more This is not a review of the literature on solar hydrodynamics, but a discussion of just a few of the Interesting aspects of the subject. It is neither comprehensive, coherent nor well balanced. It concentrates on some of the questions that have been raised recently by new observations and have not yet received much attention, and particularly on topics that were raised at the recent General Assembly of the IAU. Subjects that have been reviewed in the past, such as the propagation of waves in the solar atmosphere and the structure of the solar wind, are avoided. Magneto-hydrodynamics is hardly mentioned at all.
Solar p-mode multiplet oscillation frequencies and expansion coefficients for degeneracy splittin... more Solar p-mode multiplet oscillation frequencies and expansion coefficients for degeneracy splitting have been measured in the summers of 1986 and 1988–1990. Temporal variations not unlike the differences between 1988 and 1986 reported by Libbrecht & Woodard (1990) continue into 1990. These differences are possibly associated with the solar cycle. We report here on our first findings about the variation of the internal angular velocity and asphericity deduced from these frequency differences.
Uploads
Papers by D. Gough