Engineering topics

Goldreich, Peter

Publications and source records attributed to Goldreich, Peter.

14 records

NASA NTRS1994

Effect of nonlinear interactions on p-mode frequencies and line widths

We calculate the effect of nonlinear interactions among solar acoustic modes upon the modal frequencies and energy loss rates (or line widths). The frequency shift for a radial p-mode of frequency 3 mHz is found to be about -0.5 microHz. The magnitude of nonlinear frequency shift increases more rapidly with frequency than the inverse mode mass (mode mass is defined as the ratio of energy in the mode to its surface velocity amplitude squared). This frequency shift is primarily due to nonresonant three-mode interactions and is dominated by high l surface gravity waves (f-modes) and p-modes. The line width of a radial p-mode of frequency 3 mHz, due to resonant nonlinear interactions, is about 0.3 microHz. This result is consistent with that of Kumar and Goldreich (1989). We also find, in agreement with these authors, that the most important nonlinear interactions of trapped p-modes involve f-modes and high-frequency p-modes (frequency greater than about 5 mHz) which propagate in the solar photosphere. Thus, using the arguments advanced by Kumar & Goldreich (1989), we conclude that nonlinear couplings cannot saturate the overstable solar p-modes at their small observed amplitudes. Both the nonlinear frequency shifts and line widths, at a fixed frequency, are proportional to the inverse of mode mass which for modes of degree greater than about 100 is approximately l(exp 0.8). Therefore, the frequency of an f-mode of l = 1000, due to nonlinear interactions, is decreased by approximately 0.4%.

Kumar, Pawan↗

NASA NTRS1994

Excitation of neutron star normal modes during binary inspiral

As a compact binary inspirals due to the emission of gravitational waves, its orbital period decreases continuously down to approximately 1 ms, its value at coalescence. During the last part of the inspiral, the two stars are close together, and their tidal interactions become strong. Neutron stars have many normal modes (core g-modes, crustal discontinuity modes, shear modes, etc.) whose periods lie in the range (approximately several ms) swept by the orbital period. Some of these modes are resonantly excited by the tidal force. The amount of energy a mode absorbs is proportional to the square of the overlap integral between its displacement field and the tidal force field. For all modes of interest, this overlap is poor, resulting in relatively weak excitation. For the best case, the absorbed energy is only a small fraction (approximately 10(exp -6)) of the orbital energy, so the orbital phase shift is too weak to be detected by observations of the gravitational wave signal emitted by the inspiraling binary. However, with displacement amplitudes of excited quadrupole modes ranging up to 0.5% of the stellar radius, the possibility of a detectable electromagnetic signature cannot be dismissed. Both the periods of the modes and the energy they absorb depend quite strongly on the internal structure of the star. Their observation could shed light on the correct high-density equation of state.

Reisenegger, Andreas↗

NASA NTRS1994

Excitation of solar p-modes

We investigate the rates at which energy is supplied to individual p-modes as a function of their frequencies nu and angular degrees l. The observationally determined rates are compared with those calculated on the hypothesis that the modes are stochastically excited by turbulent convection. The observationally determined excitation rate is assumed to be equal to the product of the mode's energy E and its (radian) line width Gamma. We obtain E from the mode's mean square surface velocity with the aid of its velocity eigenfuction. We assume that Gamma measures the mode's energy decay rate, even though quasi-elastic scattering may dominate true absorption. At fixed l, E(Gamma) arises as nu(exp 7) at low nu, reaches a peak at nu approximately equal 3.5 mHz, and then declines as nu(exp 4.4) at higher nu . At fixed nu, E(Gamma) exhibits a slow decline with increasing l. To calculate energy input rates, P(sub alpha), we rely on the mixing-length model of turbulent convection. We find entropy fluctuations to be about an order of magnitude more effective than the Reynolds stress in exciting p-modes . The calculated P(sub alpha) mimic the nu(exp 7) dependence of E(Gamma) at low nu and the nu(exp -4.4) dependence at high nu. The break of 11.4 powers in the nu-dependence of E(Gamma) across its peak is attributed to a combination of (1) the reflection of high-frequency acoustic waves just below the photosphere where the scale height drops precipitously and (2) the absence of energy-bearing eddies with short enough correlation times to excite high-frequency modes. Two parameters associated with the eddy correlation time are required to match the location and shape of the break. The appropriate values of these parameters, while not unnatural, are poorly constrained by theory. The calculated P(sub alpha) can also be made to fit the magnitude of E(Gamma) with a reasonable value for the eddy aspect ratio. Our resutls suggest a possible explanation for the decline of mode energy with increasing l at fixed nu. Entropy fluctuations couple to changes in volume associated with the oscillation mode. These decrease with decreasing n at fixed nu, becoming almost zero for the f-mode.

Goldreich, Peter↗

NASA NTRS1994

The effects of scattering on solar oscillations

Acoustic modes are scattered by turbulent velocity fluctuations in the solar convection zone. The strongest scattering occurs near the top of the acoustic cavity where the mode changes character from propagating to evanescent. This layer is located at depth z(sub 1) approximately g/omega(exp 2) below the photosphere. The scattering optical depth tau(sub s) is of order M(sub 1)(exp 2), where M(sub 1) is the Mach number of the energy-bearing eddies at z(sub 1). The corresponding contribution to the line width is gamma(exp s) is approximately (omega) M(sub 1)(exp 2)/(pi)(n+1), where n is the mode's radial order. At the top of the acoustic cavity the correlation time of energy-bearing eddies is much longer than omega(exp -1). Also, the pressure scale height H and the eddy correlation length Lambda are comparable to omega/c, where c is the sound speed. Thus scattering couples modes of similar omega and all l and has little effect on the sum of their energies. Observations show that mode energies decline with decreasing n (increasing l) at fixed omega. Consequently, scattering damps p-modes and excites f-modes.

Goldreich, Peter↗

NASA NTRS1993

Interactions among convection, magnetic fields and p-mode oscillations in the sun

Two papers on different aspects of the excitation and damping of solar oscillations were accepted for publication in the Astrophysical Journal. The first paper evaluates the rate at which turbulent convection feeds energy into individual p-modes. It is shown that stochastic excitation by turbulent convection provides a satisfactory fit to the product of the mode energies and linewidths. A somewhat surprising conclusion is that entropy fluctuations are about an order of magnitude more significant than are fluctuations of the Reynolds stress in exciting p-modes. However, entropy fluctuations cannot excite f-modes. This may account for the relatively low energies of the f-modes compared to those of the p-modes. The second paper explores the role of scattering of acoustic modes by turbulent velocity fluctuations. Scattering of a mode is concentrated near the top of its acoustic cavity. Because the turbulence has low Mach number, scattering couples modes having similar frequencies. Its net effects are to increase the linewidths of all modes and to transfer energy from p-modes to f-modes. Scattering is likely to be the dominant source for the linewidths of p-modes. In particular, it may account for the unexpectedly large linewidths measured for low frequency modes. Copies of preprints of the two papers referred to above are attached. The remainder of the report is devoted to a description of unpublished results.

Goldreich, Peter↗

NASA NTRS1992

A new class of g-modes in neutron stars

Because a neutron star is born hot, its internal composition is close to chemical equilibrium. In the fluid core, this implies that the ratio of the number densities of charged particles (protons and electrons) to neutrons is an increasing function of the mass density. This composition gradient stably stratifies the matter giving rise to a Brunt-Vaisala frequency N of about 500/s. Consequently, a neutron star core provides a cavity that supports gravity modes (g-modes). These g-modes are distinct from those previously identified with the thermal stratification of the surface layers and the chemical stratification of the crust. We compute the lowest-order, quadrupolar, g-modes for cold, Newtonian, neutron star models with M/solar M = 0.581 and M/solar M = 1.405, and show that the crustal and core g-modes have similar periods. We also discuss damping mechanisms and estimate damping rates for the core g-modes. Particular attention is paid to damping due to the emission of gravitational radiation.

Reisenegger, Andreas↗

NASA NTRS1992

Magnetic field decay in isolated neutron stars

Three mechanisms that promote the loss of magnetic flux from an isolated neutron star - Ohmic decay, ambipolar diffusion, and Hall drift - are investigated. Equations of motions are solved for charged particles in the presence of a magnetic field and a fixed background of neutrons, while allowing for the creation and destruction of particles by weak interactions. Although these equations apply to normal neutrons and protons, the present interpretations of their solutions are extended to cover cases of neutron superfluidity and proton superconductivity. The equations are manipulated to prove that, in the presence of a magnetic force, the charged particles cannot be simultaneously in magnetostatic equilibrium and chemical equilibrium with the neutrons. The application of the results to real neutron stars is discussed.

Goldreich, Peter↗

NASA NTRS1991

Thermal and mechanical damping of solar p-modes

Nonadiabatic effects associated with the transfer of energy and with turbulent stresses add small imaginary parts, omega-i(1) and omega-i(2), to solar p-mode eigenfrequencies. Numerical calculations have shown that these quite different processes make comparable contributions to omega-i at frequencies well below the acoustic cutoff at omega-ac. Analytic expressions are derived which reveal the connection between omega-i(1) and omega-i(2). The estimates yield omega-i proportional to omega exp 8 for omega much less than omega-ac in good agreement with the numerical calculations. However, the observed line width is proportional to omega exp 4.2 at low frequencies. It is suspected that there is an unmodeled component of perturbed convective energy transport or of turbulent viscosity that makes an important contribution to omega-i at omega much less than omega-ac.

Goldreich, Peter↗

NASA NTRS1991

Implications of solar p-mode frequency shifts

An expression is derived that relates solar p-mode frequency shifts to changes in the entropy and magnetic field of the sun. The frequency variations result from changes in path length and propagation speed. Path length changes dominate for entropy perturbations, and propagation speed changes dominate for most types of magnetic field peturbations. The p-mode frequencies increased along with solar activity between 1986 and 1989; these frequency shifts exhibited a rapid rise with increasing frequency followed by a precipitous drop. The positive component of the shifts can be accounted for by variations of the mean square magnetic field strength in the vicinity of the photosphere. The magnetic stress perturbation decays above the top of the convection zone on a length scale comparable to the pressure scale height and grows gradually with depth below. The presence of a resonance in the chromospheric cavity means that the transition layer maintains enough coherence to partially reflect acoustic waves even near cycle maximum.

Goldreich, Peter↗

NASA NTRS1990

Wave generation by turbulent convection

Wave generation by turbulent convection in a plane parallel, stratified atmosphere lying in a gravitational field is studied. The turbulent spectrum is related to the convective energy flux via the Kolmogorov scaling and the mixing length hypothesis. Efficiencies for the conversion of the convective energy flux into both trapped and propagating waves are estimated.

Goldreich, Peter↗

NASA NTRS1989

The formation of sharp edges in planetary rings by nearby satellites

Equations are derived which govern the shapes of the perturbed streamlines near the 'sharp edge' boundaries between regions of high and low planetary ring optical depth; these are maintained by the shepherd satellites, which transfer angular momentum to and from ring particles. The results obtained by these equations' solution with a simple numerical model, whose parameters resemble those of the Encke division, are found to faithfully reproduce the sharp edges bounding the division; they imply that the ring thickness in the unperturbed regions far from the edges is of the order of 10 m, and that the angle-averaged surface density varies on a much shorter radial length scale than that over which the satellite torque is applied. This feature's relationship to the local reversal of angular momentum viscous transport, in the most strongly perturbed regions, is demonstrated.

Borderies, Nicole↗

NASA NTRS1989

Nonlinear interactions among solar acoustic modes

The rates at which nonlinear interactions transfer energy among the normal modes of a plane-parallel, stratified atmosphere are evaluated. It is shown that every p-mode in the 5-minute band is involved in many near-resonant triplets, and, as a consequence, the energy transfer rates are independent of the mode line widths. It is also found that nonlinear mode coupling cannot limit the growth of overstable p-modes, which favors the hypothesis that the sun's p-modes are stochastically excited by turbulent convection.

Kumar, Pawan↗

NASA NTRS1987

Shepherding of the Uranian rings. I - Kinematics. II - Dynamics

The possible kinematical relationships between the Uranian rings and all ten newly discovered satellites of Uranus are examined. Observational evidence is presented for interactions between 1986U7 and 1986U8 and the epsilon, delta, and gamma rings. The likelihood that these two satellites are shepherding the epsilon ring and are responsible for some of its internal structure is assessed, and the results are used to propose a reduction in the ring radius scale. It is also proposed that 1986U7 and 1986U8 are the inner and outer shepherds for the epsilon ring, that 1986U7 is the outer shepherd for the delta ring, and that 1986U8 is an outer shepherd for the gamma ring. The results of theoretical investigations of planetary ring dynamics are then applied to these associations, and their dynamical significance is evaluated. The resonant torques which the satellites exert on each ring are calculated and compared to the viscous torque which transports angular momentum outward through the ring and to the atmospheric-drag torque which causes the ring's orbit to decay.

Porco, Carolyn C.↗

NASA NTRS1986

Nonlinear density waves in planetary rings

The steady-state structure of planetary rings in the presence of density waves at the Lindblad resonances of a satellite is indicated. The study is based on the dispersion relation and damping rate for nonlinear density waves, derived by Shu et al. (1985) and by Borderies, Goldreich, and Tremaine (1985). It is shown that strong density waves lead to an enhancement of the background surface density in the wave zone.

Borderies, Nicole↗