Resonance behaviour and partial averaging in a three-body system with gravitational radiation damping

In a previous investigation, a model of three-body motion was developed which included the effects of gravitational radiation reaction. The aim was to describe the motion of a relativistic binary pulsar that is perturbed by a third mass and look for resonances between the binary and third-mass orbits. Numerical integration of an equation of relative motion that approximates the binary gives evidence of such resonances. These ( m : n ) resonances are defined for the present purposes by the resonance condition,   m ω= 2 n Ω  , where m and n are relatively prime integers and ω and Ω are the angular frequencies of the binary orbit and third-mass orbit (around the centre of mass of the binary), respectively. The resonance condition consequently fixes a value for the semimajor axis a of the binary orbit for the duration of the resonance because of the Kepler relationship  ω= a ^−3/2  . This paper outlines a method of averaging developed by Chicone, Mashhoon and Retzloff, which renders a non-linear system that undergoes resonance capture into a mathematically amenable form. This method is applied to the present system and one arrives at an analytical solution that describes the average motion during resonance. Furthermore, prominent features of the full non-linear system, such as the frequency of oscillation and antidamping, accord with their analytically derived formulae.     

彗木相撞中的惯性引力波

苏梅克-列维9号彗星(SL9)与木星相撞后，在木星上观测到的以常速度(～450m／s)向外扩展的圆环意味着这是碰撞在木星大气中引起的线性波动．我们选取：非旋转、无粘性、密度分层、不可压缩的木星大气模型，而且木星大气以水平速度U=b az运动；给出初始扰动压力P(r；0)作为碰撞的初始条件，用流体力学方程组求解了彗木相撞中的惯性引力波．结果表明：当木星大气以速度U=U0(～170m／s)运动时，彗星碎片的大部分能量都用来产生内波，同时还得到彗星碎片的撞击深度H与水平相速Vp的关系式．当木星大气以速度为U=b az运动时，木星大气的扰动能量不再是在动能和势能间均分。     

Exploiting magnification bias in ultradeep submillimetre-wave surveys using ALMA

The surface density of populations of galaxies with steep/shallow source counts is increased/decreased by gravitational lensing magnification. These effects are usually called 'magnification bias' and 'depletion', respectively. However, if sources are demagnified by lensing, then the situation is reversed, and the detectable surface density of galaxies with a shallow source count, as expected at the faintest flux densities, is increased. In general, demagnified sources are difficult to detect and study: exquisite subarcsec angular resolution and surface brightness sensitivity are required, and emission from the lensing object must not dominate the image. These unusual conditions are expected to be satisfied for observations made of the dense swarm of demagnified images that could form very close to the line of sight through the centre of a rich cluster of galaxies using the forthcoming submillimetre-wave Atacama Large Millimeter Array (ALMA) interferometer. The demagnified images of most of the background galaxies lying within about 1 arcmin of a rich cluster of galaxies could be detected in a single 18-arcsec-diameter ALMA field centred on the cluster core, providing an effective increase in the ALMA field of view. This technique could allow a representative sample of faint,  10–100 μJy  submillimetre galaxies to be detected several times more rapidly than in a blank field.     

On the streaming motions of haloes and galaxies

Little is known about the statistics of gravitationally lensed quasars at large (7–30 arcsec) image separations, which probe masses on the scale of galaxy clusters. We have carried out a survey for gravitationally lensed objects, among sources in the FIRST 20-cm radio survey that have unresolved optical counterparts in the digitizations of the Palomar Observatory Sky Survey. From the statistics of ongoing surveys that search for quasars among FIRST sources, we estimate that there are about 9100 quasars in this source sample, making this one of the largest lensing surveys to date. Using broad-band imaging, we have isolated all objects with double radio components separated by 5–30 arcsec that have unresolved optical counterparts with similar BVI colours. Our criteria for similar colours conservatively allow for observational error and for colour variations due to time delays between lensed images. Spectroscopy of these candidates shows that none of the pairs are lensed quasars. This sets an upper limit (95 per cent confidence) on the lensing fraction in this survey of 3.3×10⁻⁴, assuming 9100 quasars. Although the source redshift distribution is poorly known, a rough calculation of the expected lensing frequency and the detection efficiencies and biases suggests that simple theoretical expectations are of the same order of magnitude as our observational upper limit. Our procedure is novel in that our exhaustive search for lensed objects does not require prior identification of the quasars in the sample as such. Characterization of the FIRST-selected quasar population will enable use of our result to constrain quantitatively the mass properties of clusters.     

Ellipsoidal Harmonic expansions of the gravitational potential: Theory and application

Small bodies of the solar system are now the targets of space exploration. Many of these bodies have elongated, non-spherical shapes, and the usual spherical harmonic expansions of their gravity fields are not well suited for the modelling of spacecraft orbits around these bodies. An elegant remedy is to use ellipsoidal harmonic expansions instead of the usual spherical ones. In this paper, we present their mathematical theory as well as a real application: the simulation of a landing on the surface of a kilometer-sized comet. We show that with an ellipsoidal harmonic expansion up to degree 5, the error on the landing position is at the meter level, while the corresponding error for the spherical harmonic expansion can reach tens of meters.This revised version was published online in October 2005 with corrections to the Cover Date.     

Depletion of background galaxies owing to the cluster lens CL0024+1654: U- and R-band observations

We have obtained U - and R -band observations of the depletion of background galaxies resulting from the gravitational lensing of the galaxy cluster CL0024+1654 ( z =0.39). The radial depletion curves show a significant depletion in both bands within a radius of 40–70 arcsec from the cluster centre. This is the first time that depletion is detected in the U band. This gives independent evidence for a break in the slope of the U -band luminosity function at faint magnitudes. The radially averaged R -band depletion curve is broader and deeper than in the U band. The differences can be attributed to the wavelength dependence of the slope of the luminosity function and to the different redshift distribution of the objects probed in the two bands. We estimate the Einstein radius, r _E, of a singular isothermal sphere lens model using maximum-likelihood analysis. Adopting a slope of the number counts of α =0.2 and using the background density found beyond r =150 arcsec, we find r _E=17±3 and 25±3 arcsec in the U and R bands, respectively. When combined with the redshift of the single background galaxy at z =1.675 seen as four giant arcs around 30 arcsec from the cluster centre, these values indicate a median redshift in the range 〈 z _S〉≈0.7 to 1.1 for the U _AB≥24 mag and R _AB≥24 mag populations.     

Lensing magnification effects on the cosmic shear statistics

Gravitational lensing causes a correlation between a population of foreground large-scale structures and the observed number density of the background distant galaxies as a consequence of the flux magnification and the lensing area distortion. This correlation has not been taken into account in calculations of the theoretical predictions of the cosmic shear statistics but may cause a systematic error in a cosmic shear measurement. We examine its impact on the cosmic shear statistics using the semi-analytic approach. We find that the lensing magnification has no practical influence on the cosmic shear variance. Exploring the possible shapes of the redshift distribution of source galaxies, we find that the relative amplitude of the effect on the convergence skewness is 3 per cent at most.