arXiv subjects

Jeremy Goodman

Publications and source records attributed to Jeremy Goodman.

83 records · Page 5Linked to original sources

arXiv1998

The Paradox of the Scale-Free Disks

Scale-free disks have no preferred length or time scale. The question has been raised whether such disks have a continuum of unstable linear modes or perhaps no unstable modes at all. We resolve this paradox by analysing the particular case of a gaseous, isentropic disk with a completely flat rotation curve (the Mestel disk) exactly. The heart of the matter is this: what are the correct boundary conditions to impose at the origin or central cusp? We argue that the linear stability problem is ill-posed. From any finite radius, waves reach the origin after finite time but with logarithmically divergent phase. Instabilities exist, but their pattern speeds depend upon an undetermined phase with which waves are reflected from the origin. For any definite choice of this phase, there is an infinite but discrete set of growing modes. Similar ambiguities may afflict general disk models with power-law central cusps. The ratio of growth rate to pattern speed, however, is independent of the central phase. This ratio is derived in closed form for non self-gravitating normal modes. The ratio for self-gravitating normal modes is found numerically by solving recurrence relations in Mellin-transform space.

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arXiv1998

Dynamical Tide in Solar-Type Binaries

Circularization of late-type main-sequence binaries is usually attributed to turbulent convection, while that of early-type binaries is explained by resonant excitation of g modes. We show that the latter mechanism operates in solar-type stars also and is at least as effective as convection, despite inefficient damping of g modes in the radiative core. The maximum period at which this mechanism can circularize a binary composed of solar-type stars in 10 Gyr is as low as 3 days, if the modes are damped by radiative diffusion only and g-mode resonances are fixed; or as high as 6 days, if one allows for evolution of the resonances and for nonlinear damping near inner turning points. Even the larger theoretical period falls short of the observed transition period by a factor two.

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arXiv1997

Tidal Disruption Eddington Envelopes around Massive Black Holes

Optically-thick envelopes may form following the tidal disruption of a star by a massive black hole. Such envelopes would reprocess hard radiation from accretion close to the black hole into the UV and optical bands producing AGN-luminosity flares with duration ~1 year. We show that due to relativistic effects, the envelopes are convective. If convection is efficient, then the structure of the envelopes is similar to that described in previous work; however, the photospheric radius is shown to be very sensitive to the luminosity at the envelope base, suggesting that either the envelope collapses or the envelope expands to a maximum radius at which point a wind may set in. For an envelope without winds, we find a maximum photospheric radius of ~10^16 cm (i.e. minimum effective temperature ~6,000 K). The evolution of the envelopes is described based on simple energy arguments.

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arXiv1997

Two-Component Fokker-Planck Models for the Evolution of Isolated Globular Clusters

Two-component (normal and degenerate stars) models are the simplest realization of clusters with a mass spectrum because high mass stars evolve quickly into degenerates, while low mass stars remain on the main-sequence for the age of the universe. Here we examine the evolution of isolated globular clusters using two-component Fokker-Planck (FP) models that include heating by binaries formed in tidal capture and in three-body encounters. Three-body binary heating dominates and the postcollapse expansion is self-similar, at least in models with total mass M <= 3 x 10^5 M_\odot, initial half-mass radius r_{h,i} >= 5 pc, component mass ratio m_2/m_1 <= 2, and number ratio N_1/N_2 <= 300 when m_2=1.4 M_\odot. We derive scaling laws for ρ_c, v_c, r_c, and r_h as functions of m_1/m_2, N, M, and time t from simple energy-balance arguments, and these agree well with the FP simulations. We have studied the conditions under which gravothermal oscillations (GTOs) occur. If E_{tot} and E_c are the energies of the cluster and of the core, respectively, and t_{rh} and t_c are their relaxation times, then ε\equiv (E_{tot}/t_{rh})/(E_c/t_{rc}) is a good predictor of GTOs: all models with ε>0.01 are stable, and all but one with ε< 0.01 oscillate. We derive a scaling law for εagainst N and m_1/m_2 and compared with our numerical results. Clusters with larger m_2/m_1 or smaller N are stabler.

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arXiv1997

Radio scintillation of gamma-ray-burst afterglows

Stars twinkle to the eye through atmospheric turbulence, but planets, because of their larger angular size, do not. Similarly, scintillation due to the local interstellar medium will modulate the radio flux of gamma-ray-burst afterglows and may permit indirect measurements of their angular sizes. The amplitude of refractive scintillation is of order ten percent at ten gigahertz unless the source size is much larger than the expected size, of order ten microarcseconds. Diffractive scintillation is marginally possible, depending sensitively on the source size, observing frequency, and scattering measure of the interstellar medium.

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arXiv1997

Fast Tides in Slow Stars: the Efficiency of Eddy Viscosity

The turbulent viscosity of convection is believed to circularize the orbits of close binary stars. When the tidal period is shorter than the turnover time of the largest eddies, turbulent viscosity is believed to be suppressed. The degree of suppression is disputed. We re-examine these beliefs via direct perturbative calculations, linearizing the fluid equations on a turbulent background, and by numerical integration of a chaotic dynamical system subject to periodic forcing. We find that (i) dissipation of rapid tides is severely suppressed; and (ii) turbulent convection does not circularize binaries while the stars are on the main sequence. The latter conclusion seems secure (though weaker) even without the former.

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arXiv1995

Nonlinear Damping of Oscillations in Tidal-Capture Binaries

We calculate the damping of quadrupole f and low order g modes (primary modes) by nonlinear coupling to other modes of the star. This damping is orders of magnitude more rapid than direct radiative damping when the primary amplitude is large, as in tidal capture. Primary modes destabilize high degree g-modes of half their frequency (daughter modes) by 3-mode coupling in radiative zones. In sunlike stars, the growth time $\equivη^{-1}\approx 4 E_{0,42}^{-1/2}$ days, where $E_{0,42}$ is the initial energy of the primary mode in units of $10^{42}~$erg, and of order $10^{10}E_{0,42}^{5/4}$ daughters are unstable. The growth rate is approximately equal to the angular frequency of the primary mode times its dimensionless radial amplitude, $δR/R_*\approx 0.002E_{0,42}^{1/2}$. Although the daughter modes are limited by their own nonlinearities, collectively they absorb most of the primary mode's energy after a time $\sim 10η^{-1}$ provided $E_{0}> 10^{40}~\mbox{erg}$. In fact nonlinear mode interaction may be the dominant damping process if $E_0\gtrsim 10^{37}~\mbox{erg}$. Our results have application to tidally captured main sequence globular cluster stars of mass $\ge 0.5 M_{\sun}$; the tidal energy is dissipated in the radiative core of the star in about a month, which is less than the initial orbital period.

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arXiv1995

Global Aspects of Elliptical Instability in Tidally Distorted Accretion Disks

Tidally distorted accretion disks in binary star systems are subject to a local hydrodynamic instability which excites $m=1$ internal waves. This instability is three dimensional and approximately incompressible. We study the global aspects of this local instability using equations derived under the shearing sheet approximation, where the effects of the azimuthal variation along distorted orbital trajectories are included in source terms which oscillate with local orbital phase. Linear analyses show that the excitation of the instability is essentially local, i.e. insensitive to radial boundary conditions. The region of rapid growth feeds waves into the region of slow or negligible growth, allowing the instability to become global. The global growth rate depends the maximum local growth rate, the size of the rapid growth region, and the local group velocity. We present an empirical expression for the global growth rate. We note that the local nature of the instability allows the excitation of waves with $m\ne 1$ when the local growth rate is large.

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arXiv1995

Geocentrism reexamined

The universe is nearly isotropic on very large scales. It is much more difficult to show that the universe is radially homogeneous (independent of distance), or equivalently, that it is isotropic about distant points. This taken as an axiom, since if it were not true, then we would occupy a preferred position. This paper considers several empirical arguments for radial homogeneity based on the cosmic microwave background (CMB). The tightest limits on inhomogeneity on the scale of the horizon are of order ten percent but will improve soon. These limits involve the Sunyaev-Zel'dovich effect in clusters of galaxies, excitation of low-energy atomic transitions, and the accurately thermal spectrum of the CMB. Weaker limits from primordial nucleosynthesis are discussed briefly.

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arXiv1995

MEASURING SPIRAL ARM TORQUES: RESULTS FOR M100

Spiral arms, if they are massive, exert gravitational torques that transport angular momentum radially within galactic disks. These torques depend not on the pattern speed or permanence of the arms but only on the nonaxisymmetric mass distribution. Hence the torques can be measured from photometry. We demonstrate this using $gri$ CCD data for M100 (NGC 4321). Since we find consistency among the three bands, we believe that dust and young stars in the arms do not seriously bias our results. If the present epoch is representative, the timescale for redistribution of angular momentum in M100 is $5-10$ Gyr, the main uncertainty being the mass-to-light ratio of the disk.

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arXiv1994

Influence of the Stellar Mass Function on the Evaporation Rate of Tidally Limited Postcollapse Globular Clusters

We study the rate of escape of stars (``evaporation'') from tidally-limited postcollapse globular clusters having a power-law distribution of stellar masses. We use a multi-mass Fokker-Planck code and assume a steady tidal field. Stellar-dynamical processes cause the inner parts of the cluster to expand, which in turn causes stars to overflow the tidal boundary. Mass loss by stellar evolution is assumed to be unimportant in these later evolutionary stages. The fraction of the cluster mass lost per half-mass relaxation time ($\trh$) is roughly constant, in agreement with simple homologous models with equal-mass stars. If $\trh$ is computed in the conventional way from the mean stellar mass, however, a broad stellar mass function can double the loss of mass per $\trh$. We discuss implications of our results for the evolution of globular-cluster systems in our own and other galaxies. In particular, the number of Galactic clusters destroyed by evaporation alone may be as large or larger than the present cluster population.

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