Gravitational Wave Emission from a Bounded Source the Nonlinear Regime

格式：pdf
大小：89.90 KB
文档页数：3

下载文档原格式

Thermal gravitational waves

v12
3kTc . 12
For a core temperature of Tc 10 7 K , the velocity of the particles is of the order of
8 10 5 m / s . For a star of density 200 g / cc 2 10 5 kg / m 3 , the numbeting in these values we get the power of thermal gravitational waves emitted as,
10 9 Watt at a frequency of 1017 Hz . E
2
The flux of thermal gravitational waves from the sun, received at earth is of the order of half a watt. 2. Thermal gravitational waves from compact stars In the case of white dwarfs, the number density is of the order of n1 n2 10 37 m 3 , and the velocity corresponding to the white dwarf temperature of Tc 10 8 K , is of the order of 2 10 6 m / s . The volume of the white dwarf is of the order of 4 1018 m 3 and the frequency corresponding to the temperature Tc 10 8 K is And for a white dwarf,

初三英语宇宙探索发现单选题50题

初三英语宇宙探索发现单选题50题1. The Earth is a ______ that orbits around the Sun.A. starB. planetC. satellite答案：B。

本题考查对宇宙天体的基本概念以及相关英语单词的掌握。

选项A“star”指恒星，像太阳就是恒星，而地球不是恒星。

选项B“planet”是行星，地球是围绕太阳公转的行星，符合题意。

选项C“satellite”是卫星，通常是围绕行星转动的天体，地球不是卫星。

2. Which of the following is a natural satellite of the Earth?A. The MoonB. MarsC. Venus答案：A。

本题重点在于对地球的天然卫星这一概念的考查以及相关天体名称的英语表达。

选项A“The Moon”月球是地球的天然卫星。

选项B“Mars”火星是一颗行星，不是地球的卫星。

选项C“Venus”金星也是一颗行星，不是地球的卫星。

3. Stars are huge celestial bodies that can produce their own ______.A. waterB. lightC. air答案：B。

这题考查恒星的特性以及对应的英语单词。

恒星是巨大的天体，它们能够自己产生光。

选项A“water”水，恒星不会产生水。

选项C“air”空气，恒星也不会产生空气，选项B“light”光，符合恒星的特性。

4. Among the following planets, which one is closest to the Sun?A. MercuryB. JupiterC. Neptune答案：A。

本题考查太阳系行星与太阳的距离相关知识以及行星名称的英语。

在太阳系中，水星是距离太阳最近的行星。

选项B“Jupiter”木星距离太阳较远。

选项C“Neptune”海王星是距离太阳非常远的行星。

引力波观测原文PhysRevLett.116.061102

Observation of Gravitational Waves from a Binary Black Hole MergerB.P.Abbott et al.*(LIGO Scientific Collaboration and Virgo Collaboration)(Received21January2016;published11February2016)On September14,2015at09:50:45UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal.The signal sweeps upwards in frequency from35to250Hz with a peak gravitational-wave strain of1.0×10−21.It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole.The signal was observed with a matched-filter signal-to-noise ratio of24and a false alarm rate estimated to be less than1event per203000years,equivalent to a significance greaterthan5.1σ.The source lies at a luminosity distance of410þ160−180Mpc corresponding to a redshift z¼0.09þ0.03−0.04.In the source frame,the initial black hole masses are36þ5−4M⊙and29þ4−4M⊙,and the final black hole mass is62þ4−4M⊙,with3.0þ0.5−0.5M⊙c2radiated in gravitational waves.All uncertainties define90%credible intervals.These observations demonstrate the existence of binary stellar-mass black hole systems.This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.DOI:10.1103/PhysRevLett.116.061102I.INTRODUCTIONIn1916,the year after the final formulation of the field equations of general relativity,Albert Einstein predicted the existence of gravitational waves.He found that the linearized weak-field equations had wave solutions: transverse waves of spatial strain that travel at the speed of light,generated by time variations of the mass quadrupole moment of the source[1,2].Einstein understood that gravitational-wave amplitudes would be remarkably small;moreover,until the Chapel Hill conference in 1957there was significant debate about the physical reality of gravitational waves[3].Also in1916,Schwarzschild published a solution for the field equations[4]that was later understood to describe a black hole[5,6],and in1963Kerr generalized the solution to rotating black holes[7].Starting in the1970s theoretical work led to the understanding of black hole quasinormal modes[8–10],and in the1990s higher-order post-Newtonian calculations[11]preceded extensive analytical studies of relativistic two-body dynamics[12,13].These advances,together with numerical relativity breakthroughs in the past decade[14–16],have enabled modeling of binary black hole mergers and accurate predictions of their gravitational waveforms.While numerous black hole candidates have now been identified through electromag-netic observations[17–19],black hole mergers have not previously been observed.The discovery of the binary pulsar system PSR B1913þ16 by Hulse and Taylor[20]and subsequent observations of its energy loss by Taylor and Weisberg[21]demonstrated the existence of gravitational waves.This discovery, along with emerging astrophysical understanding[22], led to the recognition that direct observations of the amplitude and phase of gravitational waves would enable studies of additional relativistic systems and provide new tests of general relativity,especially in the dynamic strong-field regime.Experiments to detect gravitational waves began with Weber and his resonant mass detectors in the1960s[23], followed by an international network of cryogenic reso-nant detectors[24].Interferometric detectors were first suggested in the early1960s[25]and the1970s[26].A study of the noise and performance of such detectors[27], and further concepts to improve them[28],led to proposals for long-baseline broadband laser interferome-ters with the potential for significantly increased sensi-tivity[29–32].By the early2000s,a set of initial detectors was completed,including TAMA300in Japan,GEO600 in Germany,the Laser Interferometer Gravitational-Wave Observatory(LIGO)in the United States,and Virgo in binations of these detectors made joint obser-vations from2002through2011,setting upper limits on a variety of gravitational-wave sources while evolving into a global network.In2015,Advanced LIGO became the first of a significantly more sensitive network of advanced detectors to begin observations[33–36].A century after the fundamental predictions of Einstein and Schwarzschild,we report the first direct detection of gravitational waves and the first direct observation of a binary black hole system merging to form a single black hole.Our observations provide unique access to the*Full author list given at the end of the article.Published by the American Physical Society under the terms of the Creative Commons Attribution3.0License.Further distri-bution of this work must maintain attribution to the author(s)and the published article’s title,journal citation,and DOI.properties of space-time in the strong-field,high-velocity regime and confirm predictions of general relativity for the nonlinear dynamics of highly disturbed black holes.II.OBSERVATIONOn September14,2015at09:50:45UTC,the LIGO Hanford,W A,and Livingston,LA,observatories detected the coincident signal GW150914shown in Fig.1.The initial detection was made by low-latency searches for generic gravitational-wave transients[41]and was reported within three minutes of data acquisition[43].Subsequently, matched-filter analyses that use relativistic models of com-pact binary waveforms[44]recovered GW150914as the most significant event from each detector for the observa-tions reported here.Occurring within the10-msintersite FIG.1.The gravitational-wave event GW150914observed by the LIGO Hanford(H1,left column panels)and Livingston(L1,rightcolumn panels)detectors.Times are shown relative to September14,2015at09:50:45UTC.For visualization,all time series are filtered with a35–350Hz bandpass filter to suppress large fluctuations outside the detectors’most sensitive frequency band,and band-reject filters to remove the strong instrumental spectral lines seen in the Fig.3spectra.Top row,left:H1strain.Top row,right:L1strain.GW150914arrived first at L1and6.9þ0.5−0.4ms later at H1;for a visual comparison,the H1data are also shown,shifted in time by this amount and inverted(to account for the detectors’relative orientations).Second row:Gravitational-wave strain projected onto each detector in the35–350Hz band.Solid lines show a numerical relativity waveform for a system with parameters consistent with those recovered from GW150914[37,38]confirmed to99.9%by an independent calculation based on[15].Shaded areas show90%credible regions for two independent waveform reconstructions.One(dark gray)models the signal using binary black hole template waveforms [39].The other(light gray)does not use an astrophysical model,but instead calculates the strain signal as a linear combination of sine-Gaussian wavelets[40,41].These reconstructions have a94%overlap,as shown in[39].Third row:Residuals after subtracting the filtered numerical relativity waveform from the filtered detector time series.Bottom row:A time-frequency representation[42]of the strain data,showing the signal frequency increasing over time.propagation time,the events have a combined signal-to-noise ratio(SNR)of24[45].Only the LIGO detectors were observing at the time of GW150914.The Virgo detector was being upgraded, and GEO600,though not sufficiently sensitive to detect this event,was operating but not in observational mode.With only two detectors the source position is primarily determined by the relative arrival time and localized to an area of approximately600deg2(90% credible region)[39,46].The basic features of GW150914point to it being produced by the coalescence of two black holes—i.e., their orbital inspiral and merger,and subsequent final black hole ringdown.Over0.2s,the signal increases in frequency and amplitude in about8cycles from35to150Hz,where the amplitude reaches a maximum.The most plausible explanation for this evolution is the inspiral of two orbiting masses,m1and m2,due to gravitational-wave emission.At the lower frequencies,such evolution is characterized by the chirp mass[11]M¼ðm1m2Þ3=5121=5¼c3G596π−8=3f−11=3_f3=5;where f and_f are the observed frequency and its time derivative and G and c are the gravitational constant and speed of light.Estimating f and_f from the data in Fig.1, we obtain a chirp mass of M≃30M⊙,implying that the total mass M¼m1þm2is≳70M⊙in the detector frame. This bounds the sum of the Schwarzschild radii of thebinary components to2GM=c2≳210km.To reach an orbital frequency of75Hz(half the gravitational-wave frequency)the objects must have been very close and very compact;equal Newtonian point masses orbiting at this frequency would be only≃350km apart.A pair of neutron stars,while compact,would not have the required mass,while a black hole neutron star binary with the deduced chirp mass would have a very large total mass, and would thus merge at much lower frequency.This leaves black holes as the only known objects compact enough to reach an orbital frequency of75Hz without contact.Furthermore,the decay of the waveform after it peaks is consistent with the damped oscillations of a black hole relaxing to a final stationary Kerr configuration. Below,we present a general-relativistic analysis of GW150914;Fig.2shows the calculated waveform using the resulting source parameters.III.DETECTORSGravitational-wave astronomy exploits multiple,widely separated detectors to distinguish gravitational waves from local instrumental and environmental noise,to provide source sky localization,and to measure wave polarizations. The LIGO sites each operate a single Advanced LIGO detector[33],a modified Michelson interferometer(see Fig.3)that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms.Each arm is formed by two mirrors,acting as test masses,separated by L x¼L y¼L¼4km.A passing gravitational wave effec-tively alters the arm lengths such that the measured difference isΔLðtÞ¼δL x−δL y¼hðtÞL,where h is the gravitational-wave strain amplitude projected onto the detector.This differential length variation alters the phase difference between the two light fields returning to the beam splitter,transmitting an optical signal proportional to the gravitational-wave strain to the output photodetector. To achieve sufficient sensitivity to measure gravitational waves,the detectors include several enhancements to the basic Michelson interferometer.First,each arm contains a resonant optical cavity,formed by its two test mass mirrors, that multiplies the effect of a gravitational wave on the light phase by a factor of300[48].Second,a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometer as a whole[49,50]:20W of laser input is increased to700W incident on the beam splitter,which is further increased to 100kW circulating in each arm cavity.Third,a partially transmissive signal-recycling mirror at the outputoptimizes FIG. 2.Top:Estimated gravitational-wave strain amplitude from GW150914projected onto H1.This shows the full bandwidth of the waveforms,without the filtering used for Fig.1. The inset images show numerical relativity models of the black hole horizons as the black holes coalesce.Bottom:The Keplerian effective black hole separation in units of Schwarzschild radii (R S¼2GM=c2)and the effective relative velocity given by the post-Newtonian parameter v=c¼ðGMπf=c3Þ1=3,where f is the gravitational-wave frequency calculated with numerical relativity and M is the total mass(value from Table I).the gravitational-wave signal extraction by broadening the bandwidth of the arm cavities [51,52].The interferometer is illuminated with a 1064-nm wavelength Nd:Y AG laser,stabilized in amplitude,frequency,and beam geometry [53,54].The gravitational-wave signal is extracted at the output port using a homodyne readout [55].These interferometry techniques are designed to maxi-mize the conversion of strain to optical signal,thereby minimizing the impact of photon shot noise (the principal noise at high frequencies).High strain sensitivity also requires that the test masses have low displacement noise,which is achieved by isolating them from seismic noise (low frequencies)and designing them to have low thermal noise (intermediate frequencies).Each test mass is suspended as the final stage of a quadruple-pendulum system [56],supported by an active seismic isolation platform [57].These systems collectively provide more than 10orders of magnitude of isolation from ground motion for frequen-cies above 10Hz.Thermal noise is minimized by using low-mechanical-loss materials in the test masses and their suspensions:the test masses are 40-kg fused silica substrates with low-loss dielectric optical coatings [58,59],and are suspended with fused silica fibers from the stage above [60].To minimize additional noise sources,all components other than the laser source are mounted on vibration isolation stages in ultrahigh vacuum.To reduce optical phase fluctuations caused by Rayleigh scattering,the pressure in the 1.2-m diameter tubes containing the arm-cavity beams is maintained below 1μPa.Servo controls are used to hold the arm cavities on resonance [61]and maintain proper alignment of the optical components [62].The detector output is calibrated in strain by measuring its response to test mass motion induced by photon pressure from a modulated calibration laser beam [63].The calibration is established to an uncertainty (1σ)of less than 10%in amplitude and 10degrees in phase,and is continuously monitored with calibration laser excitations at selected frequencies.Two alternative methods are used to validate the absolute calibration,one referenced to the main laser wavelength and the other to a radio-frequencyoscillator(a)FIG.3.Simplified diagram of an Advanced LIGO detector (not to scale).A gravitational wave propagating orthogonally to the detector plane and linearly polarized parallel to the 4-km optical cavities will have the effect of lengthening one 4-km arm and shortening the other during one half-cycle of the wave;these length changes are reversed during the other half-cycle.The output photodetector records these differential cavity length variations.While a detector ’s directional response is maximal for this case,it is still significant for most other angles of incidence or polarizations (gravitational waves propagate freely through the Earth).Inset (a):Location and orientation of the LIGO detectors at Hanford,WA (H1)and Livingston,LA (L1).Inset (b):The instrument noise for each detector near the time of the signal detection;this is an amplitude spectral density,expressed in terms of equivalent gravitational-wave strain amplitude.The sensitivity is limited by photon shot noise at frequencies above 150Hz,and by a superposition of other noise sources at lower frequencies [47].Narrow-band features include calibration lines (33–38,330,and 1080Hz),vibrational modes of suspension fibers (500Hz and harmonics),and 60Hz electric power grid harmonics.[64].Additionally,the detector response to gravitational waves is tested by injecting simulated waveforms with the calibration laser.To monitor environmental disturbances and their influ-ence on the detectors,each observatory site is equipped with an array of sensors:seismometers,accelerometers, microphones,magnetometers,radio receivers,weather sensors,ac-power line monitors,and a cosmic-ray detector [65].Another∼105channels record the interferometer’s operating point and the state of the control systems.Data collection is synchronized to Global Positioning System (GPS)time to better than10μs[66].Timing accuracy is verified with an atomic clock and a secondary GPS receiver at each observatory site.In their most sensitive band,100–300Hz,the current LIGO detectors are3to5times more sensitive to strain than initial LIGO[67];at lower frequencies,the improvement is even greater,with more than ten times better sensitivity below60Hz.Because the detectors respond proportionally to gravitational-wave amplitude,at low redshift the volume of space to which they are sensitive increases as the cube of strain sensitivity.For binary black holes with masses similar to GW150914,the space-time volume surveyed by the observations reported here surpasses previous obser-vations by an order of magnitude[68].IV.DETECTOR VALIDATIONBoth detectors were in steady state operation for several hours around GW150914.All performance measures,in particular their average sensitivity and transient noise behavior,were typical of the full analysis period[69,70]. Exhaustive investigations of instrumental and environ-mental disturbances were performed,giving no evidence to suggest that GW150914could be an instrumental artifact [69].The detectors’susceptibility to environmental disturb-ances was quantified by measuring their response to spe-cially generated magnetic,radio-frequency,acoustic,and vibration excitations.These tests indicated that any external disturbance large enough to have caused the observed signal would have been clearly recorded by the array of environ-mental sensors.None of the environmental sensors recorded any disturbances that evolved in time and frequency like GW150914,and all environmental fluctuations during the second that contained GW150914were too small to account for more than6%of its strain amplitude.Special care was taken to search for long-range correlated disturbances that might produce nearly simultaneous signals at the two sites. No significant disturbances were found.The detector strain data exhibit non-Gaussian noise transients that arise from a variety of instrumental mecha-nisms.Many have distinct signatures,visible in auxiliary data channels that are not sensitive to gravitational waves; such instrumental transients are removed from our analyses [69].Any instrumental transients that remain in the data are accounted for in the estimated detector backgrounds described below.There is no evidence for instrumental transients that are temporally correlated between the two detectors.V.SEARCHESWe present the analysis of16days of coincident observations between the two LIGO detectors from September12to October20,2015.This is a subset of the data from Advanced LIGO’s first observational period that ended on January12,2016.GW150914is confidently detected by two different types of searches.One aims to recover signals from the coalescence of compact objects,using optimal matched filtering with waveforms predicted by general relativity. The other search targets a broad range of generic transient signals,with minimal assumptions about waveforms.These searches use independent methods,and their response to detector noise consists of different,uncorrelated,events. However,strong signals from binary black hole mergers are expected to be detected by both searches.Each search identifies candidate events that are detected at both observatories consistent with the intersite propa-gation time.Events are assigned a detection-statistic value that ranks their likelihood of being a gravitational-wave signal.The significance of a candidate event is determined by the search background—the rate at which detector noise produces events with a detection-statistic value equal to or higher than the candidate event.Estimating this back-ground is challenging for two reasons:the detector noise is nonstationary and non-Gaussian,so its properties must be empirically determined;and it is not possible to shield the detector from gravitational waves to directly measure a signal-free background.The specific procedure used to estimate the background is slightly different for the two searches,but both use a time-shift technique:the time stamps of one detector’s data are artificially shifted by an offset that is large compared to the intersite propagation time,and a new set of events is produced based on this time-shifted data set.For instrumental noise that is uncor-related between detectors this is an effective way to estimate the background.In this process a gravitational-wave signal in one detector may coincide with time-shifted noise transients in the other detector,thereby contributing to the background estimate.This leads to an overestimate of the noise background and therefore to a more conservative assessment of the significance of candidate events.The characteristics of non-Gaussian noise vary between different time-frequency regions.This means that the search backgrounds are not uniform across the space of signals being searched.To maximize sensitivity and provide a better estimate of event significance,the searches sort both their background estimates and their event candidates into differ-ent classes according to their time-frequency morphology. The significance of a candidate event is measured against the background of its class.To account for having searchedmultiple classes,this significance is decreased by a trials factor equal to the number of classes [71].A.Generic transient searchDesigned to operate without a specific waveform model,this search identifies coincident excess power in time-frequency representations of the detector strain data [43,72],for signal frequencies up to 1kHz and durations up to a few seconds.The search reconstructs signal waveforms consistent with a common gravitational-wave signal in both detectors using a multidetector maximum likelihood method.Each event is ranked according to the detection statistic ηc ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2E c =ð1þE n =E c Þp ,where E c is the dimensionless coherent signal energy obtained by cross-correlating the two reconstructed waveforms,and E n is the dimensionless residual noise energy after the reconstructed signal is subtracted from the data.The statistic ηc thus quantifies the SNR of the event and the consistency of the data between the two detectors.Based on their time-frequency morphology,the events are divided into three mutually exclusive search classes,as described in [41]:events with time-frequency morphology of known populations of noise transients (class C1),events with frequency that increases with time (class C3),and all remaining events (class C2).Detected with ηc ¼20.0,GW150914is the strongest event of the entire search.Consistent with its coalescence signal signature,it is found in the search class C3of events with increasing time-frequency evolution.Measured on a background equivalent to over 67400years of data and including a trials factor of 3to account for the search classes,its false alarm rate is lower than 1in 22500years.This corresponds to a probability <2×10−6of observing one or more noise events as strong as GW150914during the analysis time,equivalent to 4.6σ.The left panel of Fig.4shows the C3class results and background.The selection criteria that define the search class C3reduce the background by introducing a constraint on the signal morphology.In order to illustrate the significance of GW150914against a background of events with arbitrary shapes,we also show the results of a search that uses the same set of events as the one described above but without this constraint.Specifically,we use only two search classes:the C1class and the union of C2and C3classes (C 2þC 3).In this two-class search the GW150914event is found in the C 2þC 3class.The left panel of Fig.4shows the C 2þC 3class results and background.In the background of this class there are four events with ηc ≥32.1,yielding a false alarm rate for GW150914of 1in 8400years.This corresponds to a false alarm probability of 5×10−6equivalent to 4.4σ.FIG.4.Search results from the generic transient search (left)and the binary coalescence search (right).These histograms show the number of candidate events (orange markers)and the mean number of background events (black lines)in the search class where GW150914was found as a function of the search detection statistic and with a bin width of 0.2.The scales on the top give the significance of an event in Gaussian standard deviations based on the corresponding noise background.The significance of GW150914is greater than 5.1σand 4.6σfor the binary coalescence and the generic transient searches,respectively.Left:Along with the primary search (C3)we also show the results (blue markers)and background (green curve)for an alternative search that treats events independently of their frequency evolution (C 2þC 3).The classes C2and C3are defined in the text.Right:The tail in the black-line background of the binary coalescence search is due to random coincidences of GW150914in one detector with noise in the other detector.(This type of event is practically absent in the generic transient search background because they do not pass the time-frequency consistency requirements used in that search.)The purple curve is the background excluding those coincidences,which is used to assess the significance of the second strongest event.For robustness and validation,we also use other generic transient search algorithms[41].A different search[73]and a parameter estimation follow-up[74]detected GW150914 with consistent significance and signal parameters.B.Binary coalescence searchThis search targets gravitational-wave emission from binary systems with individual masses from1to99M⊙, total mass less than100M⊙,and dimensionless spins up to 0.99[44].To model systems with total mass larger than 4M⊙,we use the effective-one-body formalism[75],whichcombines results from the post-Newtonian approach [11,76]with results from black hole perturbation theory and numerical relativity.The waveform model[77,78] assumes that the spins of the merging objects are alignedwith the orbital angular momentum,but the resultingtemplates can,nonetheless,effectively recover systemswith misaligned spins in the parameter region ofGW150914[44].Approximately250000template wave-forms are used to cover this parameter space.The search calculates the matched-filter signal-to-noiseratioρðtÞfor each template in each detector and identifiesmaxima ofρðtÞwith respect to the time of arrival of the signal[79–81].For each maximum we calculate a chi-squared statisticχ2r to test whether the data in several differentfrequency bands are consistent with the matching template [82].Values ofχ2r near unity indicate that the signal is consistent with a coalescence.Ifχ2r is greater than unity,ρðtÞis reweighted asˆρ¼ρ=f½1þðχ2rÞ3 =2g1=6[83,84].The final step enforces coincidence between detectors by selectingevent pairs that occur within a15-ms window and come fromthe same template.The15-ms window is determined by the10-ms intersite propagation time plus5ms for uncertainty inarrival time of weak signals.We rank coincident events basedon the quadrature sumˆρc of theˆρfrom both detectors[45]. To produce background data for this search the SNR maxima of one detector are time shifted and a new set of coincident events is computed.Repeating this procedure ∼107times produces a noise background analysis time equivalent to608000years.To account for the search background noise varying acrossthe target signal space,candidate and background events aredivided into three search classes based on template length.The right panel of Fig.4shows the background for thesearch class of GW150914.The GW150914detection-statistic value ofˆρc¼23.6is larger than any background event,so only an upper bound can be placed on its false alarm rate.Across the three search classes this bound is1in 203000years.This translates to a false alarm probability <2×10−7,corresponding to5.1σ.A second,independent matched-filter analysis that uses adifferent method for estimating the significance of itsevents[85,86],also detected GW150914with identicalsignal parameters and consistent significance.When an event is confidently identified as a real gravitational-wave signal,as for GW150914,the back-ground used to determine the significance of other events is reestimated without the contribution of this event.This is the background distribution shown as a purple line in the right panel of Fig.4.Based on this,the second most significant event has a false alarm rate of1per2.3years and corresponding Poissonian false alarm probability of0.02. Waveform analysis of this event indicates that if it is astrophysical in origin it is also a binary black hole merger[44].VI.SOURCE DISCUSSIONThe matched-filter search is optimized for detecting signals,but it provides only approximate estimates of the source parameters.To refine them we use general relativity-based models[77,78,87,88],some of which include spin precession,and for each model perform a coherent Bayesian analysis to derive posterior distributions of the source parameters[89].The initial and final masses, final spin,distance,and redshift of the source are shown in Table I.The spin of the primary black hole is constrained to be<0.7(90%credible interval)indicating it is not maximally spinning,while the spin of the secondary is only weakly constrained.These source parameters are discussed in detail in[39].The parameter uncertainties include statistical errors and systematic errors from averaging the results of different waveform models.Using the fits to numerical simulations of binary black hole mergers in[92,93],we provide estimates of the mass and spin of the final black hole,the total energy radiated in gravitational waves,and the peak gravitational-wave luminosity[39].The estimated total energy radiated in gravitational waves is3.0þ0.5−0.5M⊙c2.The system reached apeak gravitational-wave luminosity of3.6þ0.5−0.4×1056erg=s,equivalent to200þ30−20M⊙c2=s.Several analyses have been performed to determine whether or not GW150914is consistent with a binary TABLE I.Source parameters for GW150914.We report median values with90%credible intervals that include statistical errors,and systematic errors from averaging the results of different waveform models.Masses are given in the source frame;to convert to the detector frame multiply by(1þz) [90].The source redshift assumes standard cosmology[91]. Primary black hole mass36þ5−4M⊙Secondary black hole mass29þ4−4M⊙Final black hole mass62þ4−4M⊙Final black hole spin0.67þ0.05−0.07 Luminosity distance410þ160−180MpcSource redshift z0.09þ0.03−0.04。

Gravitational waves from hot young rapidly rotating neutron stars

II. THE R-MODE INSTABILITY
The r-modes of rotating Newtonian stars are generally deﬁned to be solutions of the perturbed ﬂuid equations having (Eulerian) velocity perturbations of the form
ω=−
(l − 1)(l + 2) Ω. l+1
(2.3)
˜ is related to the imaginary Thus the time derivative of E part of the frequБайду номын сангаасncy 1/τ by ˜ ˜ 2E dE =− . dt τ (2.7)
These modes represent large scale oscillating currents that move (approximately) along the equipotential surfaces of the rotating star. The restoring force for these oscillations is the Coriolis force; hence the frequencies of these modes are low compared to the usual f and p−modes in slowly rotating stars. These expressions for δv and ω are the lowest order terms in an expansion in terms of the angular velocity Ω. The exact expressions contain additional terms of order Ω3 . There may exist other modes of rotating barotropic stellar models with properties similar to these classical r-modes; however, our discussion here is limited to the properties of these classical r-modes. The density perturbation associated with the r-modes can be deduced by evaluating the inner product of v (the unperturbed ﬂuid velocity) with the perturbed Euler equation, and the equation for the perturbed gravitational potential [10]: dρ 2l δρ= αR2 Ω2 ρ dp 2l + 1 r l l+1 R

超大质量双黑洞系统中的吸积盘Formationand

系统喷流方向不变。现有的观测表明，FRII射电星系中的吸积~盘10是8 y标r 准
的而FRI射电星系中的吸积盘则很可能是非标准的
•观测：大约 7%的射电源属于X-形态源 [Leahy & Parma 92] •理论：喷流等离子体遗迹可探测时间大约一千万年，7%的探测概率意味着低亮度的FRII射电星系寿命大约一亿年。这和测得的AGN寿命一致
rL
1
1 m
2/3
rs
m=
if 104
The whole inner region is destroyed by waves
if 104 and H/r<0.1
Opens a gap in the accretion disk The outer boundary of the gap:
evolution merging
Galaxy + galaxy
Normal galaxy
星系与活动星系的大统一模型（？）
gas: starbursts galaxy Evolution Short-lived AGNs
Martini 2004 106 yr tQ a few 108 yr
1.3、Gravitational Wave Astrophysics:
•观测：喷流遗迹与寄住星系盘垂直 •理论：在吸积盘与超大质量双黑洞发生相互作用前，吸积盘因角动量守恒而与星系盘共面，因而产生的喷流与吸积盘和星系盘垂直
•观测：活动的喷流随机取向 •理论：新喷流方向垂直于双黑洞轨道面，而星系并合形成的超大质量双黑洞相对于星系盘取向随机
Random distribution of jet orientations (Liu, Zhang, Wu, 2005, in preparation)

高考英语一轮总复习第一全程攻略 Unit 1 Great scie

手惰市安逸阳光实验学校Unit 1 Great scientists Ⅰ.完形填空(建议用时17′)[2017·河北定州模拟]Carl Kenton is a wealthy businessman. Five years ago, after returning from abroad to his motherland, he __1__ his small company. Speaking of success, Glen often tells us a story about his __2__ expensive “school” fee. He always owes his success to it.At that time, Glen, who already got a Ph.D. Degree, Decided to return to the homeland, starting a company. Before __3__， he bought a Rolex watch with the __4__ made through years of work after school and the scholarships. At the airport he had to accept the routine customs check. The watch on his wrist was also demanded to be __5__ down for inspection. Glen knew that carrying the __6__ goods out had to pay the tax, and he worried about paying __7__ for his watch. So when he was checked, he told a lie that his watch was a __8__ fake(假货). When he was __9__ of his “smarts”， immediately, __10__ the presence of Glen, the officers hit the watch, which __11__ nearly 100,000, into pieces at hearing Glen's words. Glen was 12 . Before he understood why, he was taken to the office to be examined __13__. For many times of entryexit __14__ he knew that only those people in the “blacklist” would “enjoy” this special treatment.The officers looked over everything carefully in the box, and __15__ him no matter what time of entry and exit he must accept the check andif __16__ reusing and carrying fake and shoddy goods, he would be charged according to law! Suddenly, his face turned red, and he had nothing in mind after __17__ the plane for long. After returning to the homeland, he often told the story to his family, and his employees, too. He said that this made a deep __18__ on him, because the additional high “school” fee that he had ever paid made him realize the value of __19__， which he would __20__ as the secret of his success forever.篇章导读：本文是一篇记叙文。

四川师范大学附属中学2023-2024学年高二下学期期中考试英语试题(含答案)

四川师大附中2023-2024学年度（下期）半期考试试题高2022级英语试卷说明：英语考试时间共120分钟，满分150分。

英语试题卷分第Ⅰ卷（选择题）和第Ⅱ卷（非选择题）。

考试做答时，须将答案答在答题卡上，在本试卷、草稿纸上答题无效。

第一部分听力（共两节，满分30分）第一节（共5小题，每小题1.5分，满分7.5分）听下面5段对话。

每段对话后有一个小题，从题中所给的A、B、C三个选项中选出最佳选项，并标在试卷的相应位置。

听完每段对话后，你都有10秒钟的时间来回答有关小题和阅读下一小题。

每段对话仅读一遍。

1. Where does the conversation probably take place?A. At a ticket office.B. In a restaurant.C. On a train.2. How much did the woman pay for the dress?A. 20 pounds.B. 29 pounds.C. 49 pounds.3. What did the woman do?A. She cleaned the table.B. She took out insurance.C. She received letters.4. How does the woman sound?A. Excited.B. Interested.C. Surprised.5. What are the speakers mainly talking about?A. What to eat.B. How to cook.C. Who to invite for dinner.第二节（共15小题；每小题1.5分，满分22.5分）听下面5段对话或独白。

每段对话或独白后有几个小题，从题中所给的A、B、C三个选项中选出最佳选项，并标在试卷的相应位置。

听每段对话或独白前，你将有时间阅读各个小题，每小题5秒钟；听完后，各小题将给出5秒钟的作答时间。

Gravitational Wave

Michelson interferometer
What does Gravitational wave mean to us?
To us 1.It provides us a new way to detect the universe. We are looking forward to a brand new world.(Before we can only use electromagnetic wave.) 2.It proves the validity of general relativity once again.(He has pointed out this almost 100y ago but he can hardly prove it.) To me It is interesting and I do not want to just be one member of goldfish.
This is the end,thank you!
Kody Corn,16.03.10
“Gravitational Wave” in Interstellar
Something about Gravitational Wave
On September 14, 2015 at 5:51 a.m. Eastern Daylight Time (09:51 UTC),the twin Laser Interferometer Gravitational-wave Observatory(LIGO) detectors,located in Livingston,Lousiana, and Hanford, Wahington, USA both measured ripples in the fabric of spacetime--Gravitational waves - arriving at the earth from a cataclysmic event in the distant universe.

Multiple imaging by gravitational waves

Abstract Gravitational waves act like lenses for the light propagating through them. This phenomenon is described using the vector formalism employed for ordinary gravitational lenses, which was proved to be applicable also to a non–stationary spacetime, with the appropriate modiﬁcations. In order to have multiple imaging an approximate condition analogous to that for ordinary gravitational lenses must be satisﬁed. Certain astrophysical sources of gravitational waves satisfy this condition, while the gravitational wave background, on average, does not. Multiple imaging by gravitational waves is, in principle, possible, but the probability of observing such a phenomenon is extremely low.
To appear in Int. J. Mod. Phys. D
1 Introduction
In general relativity (and in all metric theories of gravity) a gravitational ﬁeld deﬂects light rays propagating through it, and gravitational waves are no exception. It is well known that exact solutions of the Einstein equations representing plane gravitational waves exhibit focusing properties on timelike and null geodesics propagating through them [1]. Light propagation through realistic linearized gravitational waves outside the laboratory has also been studied [2]–[12]. Most authors restricted their attention to cosmological gravitational waves; a particular attention was payed to the frequency shift eﬀect induced by cosmological gravitational waves on the microwave background photons [14, 6, 13, 15], expecially after the COBE discovery of anisotropies in the cosmic microwave background. Other works focussed upon the eﬀects of gravitational waves perturbing ordinary gravitational lenses [16]–[18]. In recent papers, the case of gravitational waves generated by astrophysical sources was studied [19]–[22] and, in some cases, optimistic statements were made about the detectability of these waves via the eﬀects induced by their interaction with the light coming from distant objects.

gravitational wave - 英译汉

Gravitational waves were first predicted by Albert Einstein in his famous 1916 paper on general relativity. One of the central, and strangest, tenets of general relativity is that space and time are not separate things but rather are linked together in a single fabric: space-time. Massive objects, like stars, stretch and curve this fabric, sort of like how a bowling ball distorts a rubber sheet. These dips cause objects such as planets, and even light, to take a curved path around those more massive bodies.
In Historic First, Einstein's Gravitational Waves Detected Directly
爱因斯坦提出的引力波史上第一次被直接证明
WASHINGTON — Gravitational waves, thecosmic ripplesthat distortspace-timeitself, have been directly detected for the first time.
此番发现可以被视作为天文学及天体物理学领域上的里程碑。根据LIGO的有关研究，引力波与光波不同，它们不会在穿越空间地过程中被介质扭曲或者改变；因此能传播有关造成其形成的物质及事件的“纯粹”信息。

高三物理研究英语阅读理解30题

高三物理研究英语阅读理解30题1<背景文章>Newton's three laws of motion are fundamental principles in physics. The first law, also known as the law of inertia, states that an object at rest will remain at rest and an object in motion will continue in motion with a constant velocity unless acted upon by an external force. This law was a major breakthrough in understanding the nature of motion.The second law describes the relationship between force, mass, and acceleration. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, it can be expressed as F = ma, where F is the force, m is the mass, and a is the acceleration.The third law, known as the law of action and reaction, states that for every action, there is an equal and opposite reaction. This means that when one object exerts a force on another object, the second object exerts an equal and opposite force on the first object.Newton's laws were developed in the 17th century and had a profound impact on the development of modern science and technology. They have been used to explain the motion of planets, the behavior of machines, and the flight of rockets. In modern technology, Newton's laws are applied invarious fields such as engineering, aerospace, and robotics.1. What is Newton's first law also known as?A. The law of gravityB. The law of inertiaC. The law of accelerationD. The law of action and reaction答案：B。

引力波中的相关英语高考考点

引力波中的相关英语高考考点英语可能会在阅读理解中出关于引力波的题目。

Gravitational Lensing of Gravitational Waves from Merging Neutron Star Binaries

a rXiv:as tr o-ph/965140v122May1996Fermilab-Pub-96/091-A,astro-ph/9605???Gravitational lensing of gravitational waves from merging neutron star binaries Yun Wang 1,Albert Stebbins 1,and Edwin L.Turner 21NASA/Fermilab Astrophysics Center,FNAL,Batavia,IL 605102Princeton University Observatory,Peyton Hall,Princeton,NJ 08544(May 22,1996)Abstract We discuss the gravitational lensing of gravitational waves from merging neutron star binaries,in the context of advanced LIGO type gravitational wave detectors.We consider properties of the expected observational data with cut on the signal-to-noise ratio ρ,i.e.,ρ>ρ0.An advanced LIGO should see unlensed inspiral events with a redshift distribution with cut-oﬀat a redshift z max <1for h ≤0.8.Any inspiral events detected at z >z max should be lensed.We compute the expected total number of events whichare present due to gravitational lensing and their redshift distribution for anadvanced LIGO in a ﬂat Universe.If the matter fraction in compact lenses isclose to 10%,an advanced LIGO should see a few strongly lensed events peryear with ρ>5.PACS numbers:98.62.Sb,98.80.Es,04.80.NnTypeset using REVT E XI.INTRODUCTIONAn advanced LIGO may observe gravitational waves produced as distant close neutron star binary pairs spiral into each other.During the last stage of inspiral the binary emits copious gravitational waves,with increasing frequency as the orbital period decreases,until ﬁnally the pair collides and coalesces.LIGO aims to detect the waves emitted during the last15minutes of inspiral when the frequency sweeps up from10Hz to approximately103 Hz[1].In this paper,we discuss the gravitational lensing of gravitational waves from merging neutron star binaries,in the context of advanced LIGO type gravitational wave detectors. Following Ref.[2],we consider properties of the expected observational data with cut on the signal-to-noise ratioρ,i.e.,ρ>ρ0.An advanced LIGO should see unlensed events with a redshift distribution with cut-oﬀat a small redshift z max<1for h≤0.8[2,4].We argue below that there may be a signiﬁcant number of inspiral events detected at z>z max which can be detected because they are magniﬁed due to gravitational lensing.We compute the expected total number of events which are present due to gravitational lensing and their redshift distribution for an advanced LIGO,for plausible choices of cosmological parameters.The aforementioned frequency range over which LIGO can detect neutron star binary inspirals corresponds to a wavelength of gravitational waves from3×104to102km.This wavelength will be much smaller than the characteristic scales of gravitationalﬁelds the gravitational waves are likely to encounter as they pass between the neutron star binaries and the Earth.This means that one may treat the propagation of gravitational waves in the geometrical optics limit[3].In other words the gravitational lensing magniﬁcation will be the same as for optical light and one may use the standard formulae from optical gravitational lens theory.II.OBSER V ATION OF UNLENSED EVENTSNeutron star binary merger rate at redshift z per unit observer time interval per unit volume is˙n m=˙n0(1+z)2η(z),where˙n0is the local neutron star binary merger rate per unit volume,(1+z)2accounts for the shrinking of volumes with redshift(assuming constant comoving volume density of the merger rate)and time dilation,andη(z)=(1+z)βdescribes evolutionary eﬀects.We use the“best guess”local rate density,˙n0≃(9.9+0.6h2)h×10−8Mpc−3yr−1≃10−7h Mpc−3yr−1.[5,6].In the last stage of a neutron star binary inspiral,gravitational radiation energy losses should lead to highly circular binary orbits.In the Newtonian/quadrupole ap-proximation,for a circular orbit,the rate at which the frequency of the gravitational waves sweeps up or“chirps”,is determined solely by the binary’s“intrinsic chirp mass”, M0≡(M1M2)3/5/(M1+M2)1/5,where M1and M2are the two bodies’masses.For a binary inspiral source located at redshift z,the detectors measure M≡M0(1+z),which is referred to as the observed chirp mass.For a given detector,the signal-to-noise ratio is[2]ρ(z)=8Θr01.2M⊙5/6ζ(f max),(1)d L is our luminosity distance to the binary inspiral source.r0andζ(f max)depend only on the detector’s noise power spectrum.The characteristic distance r0gives an overall sense of the depth to which the detector can“see”.For advanced LIGO,r0=355Mpc.ζ(f max) reﬂects the overlap of the signal power with the detector bandwidth(0≤ζ≤1).For source redshift z,ζ≃1for1+z≤10[2.8M⊙/(M1+M2)].ζ≃1is a good approximation in the context of this paper.Θis the angular orientation function,it arises from the dependence ofρon the relative orientation of the source and the detector,0≤Θ≤4.AlthoughΘcan not be measured,its probability distribution has been found numerically in Ref.[7], PΘ(Θ,0≤Θ≤4)≃5Θ(4−Θ)3/256,PΘ(Θ,Θ>4)=0.The luminosity distance d L(z)=(1+z)2d A(z),where d A(z)is the angular diameter distance.In aﬂat Universe with a cosmological constantΩΛ=1−Ω0≥0,[8]d A(z)= cH−10(1+z)−1 z0d w[Ω0(1+w)3+ΩΛ]−1/2.The number rate of binary inspiral events seen by a detector on Earth with signal-to-noise ratio ρ>ρ0per source redshift interval is [2,4]d˙NNL(>ρ0)cH −10 2(1+z )η(z )Ω0(1+z )3+ΩΛC Θ(x ),(2)where C Θ(x )≡∞x dΘP Θ(Θ)is the probability that a given detector detects a binary inspiral at redshift z with signal-to-noise ratio greater than ρ0,it decreases with z and acts as a window function;C Θ(x,0≤x ≤4)=(1+x )(4−x )4/256,C Θ(x,x >4)=0.x is the minimum angular orientation functionx =4cH −10 ,(3)where we have deﬁned parameter A as in Ref.[4],A ≡0.4733 8355Mpc M 0µρ(z ∗),(5)where µis the magniﬁcation.The source can be detected if ρ∗(z ∗)>ρ0,with ρ0denoting the detector threshold.The probability of a source at redshift z being magniﬁed by a factor greater thanµis P(>µ,z)=τL(z)y2(µ),forτL(z)≪1.τL(z)is the optical depth for gravitational lensing, and y2(µ)≃µ−2forµ≫1.For point mass lenses,we usey2(µ)=2 µµ2−1−1 ,µ>351,µ≤35.(6)The above equation leads to underestimation of y2(µ)forµclose to1,which has negligible eﬀect for our purpose.For a source at redshift z∗>z max to be detected,we needµ>µ0, withµ0≡ ρ0Θ 2,(7) where x(z∗)is given by Eq.(3).Note thatµ0depends on the angular orientation function Θ.The largerΘ,the larger the signal-to-noise ratio without lensing[see Eq.(1)],the smaller the magniﬁcation needed to reach the detector thresholdρ0.SinceΘ=x is the minimum angular orientation function needed for a source to be seen without lensing[see Eq.(3)], only events withΘ<x need be considered when we count the number of events which are present due to gravitational lensing.The number rate of binary inspiral events which can be seen due to gravitational lensing by a detector on Earth with signal-to-noise ratioρ>ρ0per source redshift interval isd˙N L(>ρ0)cH−10 2(1+z∗)η(z∗)τL(z∗)Ω0(1+z∗)3+ΩΛx(z∗)0dΘPΘ(Θ)y2(µ0).(8)Note thatΘmin=0,because we take the maximum magniﬁcation to be inﬁnite,which is a reasonable approximation in the context of this paper.We consider two types of lensing:1)“macro-lensing”from the large-scale gravitational ﬁeld of galaxies,and2)“micro-lensing”from the smaller scale gravitationalﬁeld from com-pact objects such as stars.The optical depth from macrolensing is[9]τG L(z)=FcH−103,(9)where F parametrizes the gravitational lensing eﬀectiveness of galaxies[as singular isother-mal spheres].Denoting the matter fraction in compact lenses asΩL,the optical depth of microlensing is[10,11]τp L(z)=3λ(z) z0d w(1+w)3[λ(z)−λ(w)]λ(w)Ω0(1+w)3+ΩΛ,(10)where the aﬃne distance(in units of cH−10)isλ(z)= z0d w(1+w)−2[Ω0(1+w)3+ΩΛ]−1/2.ForΩΛ=0,τp L(z)=35 ,where y=1+z.IV.PREDICTIONS/SPECULATIONTo make more speciﬁc predictions we must choose parameters and to do this we are forced to speculate on the rate of inspiral events and the sensitivity of future gravitational wave detectors.Typically we expect neutron star binaries to have M0=1.2M⊙while the advanced LIGO might have r0=355Mpc,ρ≥ρ0=5then implies A=0.7573.We parametrize evolutionary eﬀects byη(z)=(1+z)β,withβ≥0and a redshift cut-oﬀof z stop=2.5.The typical lifetime of a neutron star binary is about108to109years[5];a signiﬁcant fraction of neutron star binaries formed at z=3would have merged by z=2.5. Since there seem to be a lot of star formation at z>3[12],z stop=2.5is probably reasonable.Weﬁnd that the macrolensing rate is largest when there is a sizable cosmological con-stant,but is still negligible unless there is signiﬁcant evolution.Even with extreme evolution, e.g.ΩΛ=0.8,Ω0=0.2,F=0.05[11]andβ=3macrolensing yields only70%of the num-ber of lensed events of microlensing with more modest parameters:ΩL=0.07andβ=0. Macrolensing of gravitational waves due to galaxies is negligible compared to a plausible microlensing rate.This is partly due to the fact that we have a good idea of the number and properties of galaxies while we are more free to speculate on the number of compact objects and partly due to the fact that point mass lenses are more eﬀective gravitational lenses than galaxies.For the rest of the paper we restrict ourselves to microlensing,which gives a few stronglylensed events per year without much evolution,for a currently acceptable value ofΩL.We have considered two plausible cosmological models:(1)Ω0=1andΩL=0.1;(2)ΩΛ=0.8,Ω0=0.2,ΩL=0.07.We consider expected data with cut on the signal-to-noise ratio,ρ>5.Fig.1shows the expected total number per year of events which are present due to gravitational lensing as function of h,for two cosmological models,withβ=0,1.The solid lines are forΩ0=1 andΩL=0.1,and the dashed lines are forΩΛ=0.8,Ω0=0.2,andΩL=0.07.Fig.2shows the corresponding expected total number per year of events which can be seen without gravitational lensing as function of h,with the same line types as Fig.1.The expected total numbers in both Fig.1and Fig.2increase with increasingβ,as expected.Fig.3shows the redshift distribution of expected events corresponding to Figs.1-2for h=0.8.The dotted lines indicate the distribution of expected events which are present due to gravitational lensing.Note that gravitational lensing leads to tails at high redshift.For each cosmological model,the higher tail corresponds toβ=1.Note that in principle,the evolutionary index can be measured from the region of the redshift distribution dominated by events which can be seen without gravitational lensing.Note also that most of the events which are seen due to gravitational lensing lie beyond the cut-oﬀredshift of the events which can be seen without gravitational lensing.We have used M0=1.2M⊙as the typical intrinsic chirp mass of neutron star binary inspirals.It is expected that M0will fall in the narrow range of1.12−1.26M⊙[2]while an advanced LIGO can measure the observed chirp mass M=(1+z)M0to an accuracy of better than0.1%[7,13].Thus the uncertainty in the redshift of a given event will be very small compared to the large range of z over which the events which are seen due to gravitational lensing are distributed[see Fig.3].Since the redshift distribution of observed events which can be seen without gravitational lensing should terminate at a relatively small redshift z max,an observation of an event with redshift signiﬁcantly greater than z max is a strong evidence for gravitational lensing.One should be able to identify the events which are seen due to gravitational lensing!V.DISCUSSIONWhile most neutron star binary inspiral events detected by an advanced LIGO will probably not be aﬀected by gravitational lensing,there could be a detectable number of events which are signiﬁcantly magniﬁed via gravitational lensing by compact objects.These lensed events will be easily identiﬁable by their high observed chirp masses.For the no-evolution parameters used above one would expect around two events per year which are seen due to gravitational lensing.Even a modest evolution(β=1)of the rate of inspirals can signiﬁcantly increase the rate of events which are seen due to gravitational lensing,and one could imagine even stronger evolution.During the lifetime of a detector,say ten years, one might detect dozens of events which are seen due to gravitational lensing,from which one could estimate the amount of matter in compact lenses,ΩL.The absence of such expected lensed events will place an interesting constraint onΩL.If we can determineΩL in this way,we will have a much better handle on the nature of the dark matter in our Universe.Thus lensing adds utility to the observation of inspiral events,which has already been shown to provide a measure of the Hubble constant,the deceleration parameter,and the cosmological constant[14,15,2,4].Gravitational lensing will also add additional noise to the determination of these cosmological parameters,although this noise is relatively small[15].This is because,as we have seen,most inspiral events are little aﬀected by gravitational lensing.Finally we note that our consideration of lensing for inspiral events is much the same as that which one uses when considering supernovae “standard candles”[16,17].ACKNOWLEDGMENTSY.W.and A.S.are supported by the DOE and NASA under Grant NAG5-2788.E.L.T. gratefully acknowledges support from NSF grant AST94-19400.We thank Josh Frieman for very helpful discussions.REFERENCES[1]K.S.Thorne,in Proceedings of IAU Symposium165:Compact Stars in Binaries,editedby J.van Paradijs,E.van den Heuvel,and E.Kuulkers(Kluwer Academic Publishers).[2]L.S.Finn,Phys.Rev.D53,2878(1996).[3]C.W.Misner,K.S.Thorne,J.A.Wheeler,“Gravitation”(W.H.Freeman and Company,1973).[4]Y.Wang and E.L.Turner,astro-ph/9603034.[5]E.S.Phinney,Astrophys.J.380,L17(1991).[6]R.Narayan,T.Piran,and A.Shemi,Astrophys.J.379,L17(1991).[7]L.S.Finn and D.F.Chernoﬀ,Phys.Rev.D47,2198(1993).[8]S.M.Carroll,W.H.Press,and E.L.Turner,Ann.Rev.Astron.Astrophy.,30,499(1992).[9]E.L.Turner,Astrophys.J.365,L43(1990).[10]E.L.Turner,J.P.Ostriker,and J.R.Gott,Astrophys.J.284,1(1984).[11]Fukugita,M.,and Turner,E.L.,Mon.Not.Roy.Astr.Soc.,253,99(1991).[12]C.C.Steidel,M.Giavalisco,M.Pettini,M.Dickinson,and K.L.Adelberger,Astrophys.J.462,L17(1996).[13]C.Cutler and´E.Flanagan,Phys.Rev.D49,2658(1994).[14]D.F.Chernoﬀand L.S.Finn,Astrophys.J.Lett.411,L5(1993).[15]Markovic,D.,Phys.Rev.D48,4738(1993).[16]Linder,E.V.,Schneider,P.,Wagoner,R.T.(1988),Astrophys.J.,324,786.[17]J.A.Frieman,Comments in Astrophys.,in press(1996).Figure CaptionsFig.1The total number per year of expected events which are present due to gravitational lensing as function of h,withβ=0,1.The solid lines are forΩ0=1andΩL=0.1,and the dashed lines are forΩΛ=0.8,Ω0=0.2,andΩL=0.07.Fig.2The total number per year of expected events which can be seen without gravita-tional lensing as function of h,with the same line types as Fig.1.Fig.3The redshift distribution of expected events corresponding to Fig.1for h=0.8.The dotted lines indicate the distribution of expected events which are seen due to gravitational lensing.For each cosmological model,the higher tail corresponds toβ=1.。

Gravitational Waves from the First Stars

a r X i v :a s t r o -p h /0603544v 1 20 M a r 2006astro-ph/0603544UMN–TH–2435/06FTPI–MINN–06/07March 2006Gravitational Waves from the First StarsPearl Sandick 1,Keith A.Olive 2,Fr´e d´e ric Daigne 3,and Elisabeth Vangioni 31Department of Physics,School of Physics and Astronomy,University of Minnesota,Minneapolis,MN 55455USA 2William I.Fine Theoretical Physics Institute,School of Physics and Astronomy,University of Minnesota,Minneapolis,MN 55455USA 3Institut d’Astrophysique de Paris,UMR 7095,CNRS,Universit´e Pierre et Marie Curie-Paris VI,98bis bd Arago,F-75014,Paris,France Abstract We consider the stochastic background of gravitational waves produced by an early genera-tion of Population III stars coupled with a normal mode of star formation at lower redshift.The computation is performed in the framework of hierarchical structure formation and is based on cosmic star formation histories constrained to reproduce the observed star for-mation rate at redshift z <∼6,the observed chemical abundances in damped Lyman alpha absorbers and in the intergalactic medium,and to allow for an early reionization of the Uni-verse at z ∼10−20as indicated by the ﬁrst year results released by WMAP.We ﬁnd that the normal mode of star formation produces a gravitational wave background which peaks at 300-500Hz and is within LIGO III sensitivity.The Population III component peaks at lower frequencies (30-100Hz depending on the model),and could be detected by LIGO III as well as the planned BBO and DECIGO interferometers.1IntroductionIn the last few years,there has been signiﬁcant progress in our understanding of the earlycosmic star formation history.Theﬁrst year data obtained by WMAP indicates a large optical depth,implying that the universe became reionized at high redshift in the range11<z<30at95%CL[1].To account for a period of early reionization,it has been arguedthat a generation of very massive stars preceded the oldest observed generation of Population II stars[2].In addition,cosmic star formation rate(SFR)at z 6,has been observed atlevels signiﬁcantly larger than the current rate[3].Taken together,the evidence suggests that the distribution of theﬁrst stars(Population III)are described by a top-heavy initialmass function(IMF),formed in primordial metal-free structures with masses of order107M⊙.As these stars produced heavier elements,the universe achieved a critical metallicity (∼10−4times solar metallicity)[4],at which point the massive mode of star formationyielded to a more normal distribution of stellar masses with a SFR peaked at z≈3.A moredetailed understanding of theﬁrst epoch of star formation will rely on the phenomenological consequences of the models such as element enrichment and supernova rates[5,6,7].One consequence of this new view of star formation is an enhanced rate of core collapse supernovae.The resultant relic neutrino background was investigated in Ref.[8,9].In eachcore collapse supernova explosion,the bulk of the energy released is in the form of neutrinoswhich,because they are weakly interacting,retain information about their origins.Although the neutrino background from a massive mode of star formation(Pop III)at early times isnot likely to be detected due to its redshifted spectrum,the prospects for observation of thespectrum produced by the normal mode of star formation(Pop II)in the near future are good.One possible probe of the massive mode,however,is the stochastic background of gravi-tational waves produced by cosmological core collapse supernovae,which we consider here.Although only a small fraction of the total energy of core collapse is emitted in gravitationalradiation,the improved sensitivities by way of correlation of currently operating ground-based interferometers GEO600[10],LIGO II and III[11],TAMA[12],and VIRGO[13],and of future space-based antennas BBO[14],LISA[15],DECIGO[16],make the positive detection of the accumulated gravitational wave background plausible.The gravitational wave background from core collapse supernovae resulting in black holeshas been calculated in Refs.[17,18],with estimates of the peak of the diﬀerential energy density spectrum ofΩGW h2=10−11−few×10−9reaching its maximum value at frequencies anywhere from a few hundred Hertz to a few thousand Hertz.The calculation has also beenmade speciﬁcally for Population III supernovae resulting in black holes in Ref.[19].They estimate the spectrum to peak atΩGW h2≈10−8at a frequency of O(100)Hz.More recently, the spectra from both a normal mode of star formation,in which all stars collapse to form neutron stars,and a Population III mode,in which all stars collapse to form black holes,was calculated in Ref.[20].Theyﬁnd both peaks to be located at roughlyΩGW h2≈few×10−12 in the most optimistic case,with the peak frequency dependent on the redshift range over which gravitational collapse occurs.As these spectra are highly model dependent,and given that previous estimates of the diﬀerential closure density span four orders of magnitude in therange where detection may soon be possible,it is now important to examine the sensitivity of a detectable signal to the star formation history.Here,we incorporate fully developed chemical evolution models which trace the history of pre-galactic structures as well as the IGM and are based on aΛCDM cosmology with a Press-Schechter model of hierarchical structure formation[21].We adopt the chemical evolution models of Daigne et al.[7]and consider several bimodal star formation histories,each with a normal component of star formation as well as a massive component describing Population III stars.Given an IMF and a respective SFR,we calculate the expected gravitational wave background and compare this result with detector sensitivities.2Calculation of the Gravitational Wave Background Gravitational waves can be characterized by a frequency,f,and an amplitude,h,which is deﬁned by the degree of quadrupole anisotropy and strength of the ing an amplitude determined from the simulation of a15M⊙star from Ref.[22],Buonanno et al.[20]made a generalization to larger stars by using the function’s dependence on the anisotropy and neutrino luminosity during collapse.The shape of the gravitational wave spectrum can then be described by the dimensionless quantityf|˜h(f)|=G Na 3e−f/b,(1)where˜h is the Fourier transform of h,G N is Newton’s constant,Eνis the total energy emitted in neutrinos, q is the average value of the anisotropy parameter,q,deﬁned in Ref.[23],and D is the distance to a typical supernova.Although it is necessary to know the distance in order to quantify the amplitude,we will see that the accumulated energy density does not depend on this parameter.Once the structure of the amplitude is imposed,the constants a and b determine the speciﬁc spectral shape.By roughly reproducing the source spectrum from simulation model s15r in Ref.[22],one obtains a≈200Hz.and b≈300Hz. The sensitivity to these choices will be discussed in section4.A stochastic background of gravitational waves with energy densityρGW and frequency f is best described by the diﬀerential closure density parameter[24],ΩGW(f)=1d log f,(2)withρc=3H20/8πG N the critical density.Given a star formation history,consisting of a star formation rate per comoving volume(SFR),ψ(t),and an initial mass function(IMF),φ(m),and the gravitational wave amplitude,the stochastic background of gravitational waves produced by cosmological core collapse supernovae isΩGW(f)=16π2c3D21+zdtwhere f ′is the frequency at emission,related to the observed frequency,f ,by f ′=f (1+z ),z i is the initial redshift at which supernovae begin to occur,M min and M max are the minimum and maximum masses in each model for which supernovae occur,and τ(m )is the lifetime of a star of mass m [20].UsingEq.1,one obtains ΩGW (f )=16G N dz M max M mindmφ(m )ψ(t −τ(m )) q 2E 2νf ′ 1+f ′15G N d ff 2|˜h(f )|2=16G N a 6e −2f/b .(5)The eﬃciency of gravitational wave production,ǫ,is deﬁned in terms of the remnant mass,M r ,byE GW =ǫM r c 2.(6)Using E ν=3×1053ergs,we determine the eﬃciency to be ǫ=1.5×10−7when a neutron star remnant is produced.When collapse proceeds to form a black hole,we assume,as in Ref.[20],an eﬃciency of gravitational wave production found by Fryer et al.of ǫ=2×10−5[25].One can then determine the quantity q E νfor a star of mass m by assuming only a spectral shape (here,Eq.1)and an eﬃciency of gravitational wave production.In this case it is necessary to specify the mass of the remnant,but not the neutrino luminosity.Note that the eﬃciency of gravitational wave production is much greater when a black hole rather than a neutron star is produced as a remnant.One interesting consequence of choosing the eﬃciency of gravitational wave production to be constant for stars that collapse to black holes is that the anisotropy parameter actually decreases for larger stars,as can be seen in Fig.1.This is because the quantity q E ν∝√M r .The values obtained for q are not unreasonable for larger stars in Models 1and 2b,however it has been noted that one might expect a larger degree of anisotropy due to rotation and/or violent explosions [20].Very little is known about the anisotropy,however there is evidence that stars that collapse to black holes should be more eﬃcient emitters of gravitational waves.Stark and Piran found a maximum eﬃciency of ǫ 7×10−4for an axisymmetric collapse resulting in a black hole [26],and more recently,Fryer et al.obtained the eﬃciency of 2×10−5for a 100M ⊙black hole remnant,which we adopt.Note that since ΩGW ∝ǫ,if the eﬃciency is actually closer to Stark &Piran’s maximum value,the observed energy density in gravitational waves may be as much as 35times larger than that shown in our results below,and even more if the collapse is less symmetric.While it is possible to ﬁx the0.00150.0020.0030.0050.0070.01<q>m (M o ).NS BHBHPISN Figure 1:Anisotropy parameter, q ,as a function of the mass of the progenitor.The energy emitted in neutrinos is taken to be E ν=3×1053ergs for 8M ⊙<m <30M ⊙and E ν=3dz =9.78h −1Gyr ΩΛ+Ωm (1+z )3(7)where ΩΛ=0.73,Ωm =0.27,and h =0.71[1].3Star Formation ModelsThe cosmic star formation histories considered here have been adopted from the detailed model of chemical evolution in Ref.[7].The models are described by a bimodal birthrate function of the formB (m,t,Z )=φ1(m )ψ1(t )+φ2(m )ψ2(Z )(8)where φ1(2)is the IMF of the normal (massive)component of star formation and ψ1(2)is the respective SFR.Z is the metallicity.The normal mode contains stars with mass between 0.1M⊙and100M⊙and has a SFR which peaks at z≈3.The massive component dominates at high redshift.The IMF of both modes is taken to be a power law with a near Salpeter slope so that,φi(m)∝m−(1+x)(9) with x=1.3.Each IMF is normalized independently bym supdm mφi(m)=1,(10)m infdiﬀering only in the speciﬁc mass range of each model.Here we consider two diﬀerent mass ranges for the massive mode,although three mass ranges are presented in Ref.[7],as will be discussed presently.Both the normal and massive components can contribute to the chemical enrichment of galaxy forming structures and the IGM,though the normal mode is not suﬃcient for accounting for the early reionization of the IGM[6].Here,we restrict our attention to the bestﬁt hierarchical model in[7]in which the minimum mass for star formation is107M⊙.The normal mode SFR is given byψ1(t)=ν1M struct exp(−t/τ1),(11) whereτ1=2.8Gyr is a characteristic timescale andν1=0.2Gyr−1governs the eﬃciency of the star formation.In contrast,the massive mode SFR is deﬁned byψ2(t)=ν2M ISM exp(−Z IGM/Z crit),(12) withν2maximized to achieve early reionization without the overproduction of metals or the over-consumption of gas.We adopt Z crit/Z⊙=10−4.We consider three diﬀerent models,labeled Models1,2a,and2b to describe the massive mode.They are distinguished by their respective stellar mass ranges.In Model1,the IMF is deﬁned for stars with masses,40M⊙≤m≤100M⊙.All of these stars die in core collapse supernovae leaving a black hole remnant.Model2a is described by very massive stars which become pair instability supernovae.The IMF is deﬁned for140M⊙≤m≤260M⊙.Finally, the most massive stars are considered in Model2b and fall in the range270M⊙≤m≤500 M⊙,with the SFR as in Model1.These stars entirely collapse into black holes and do not contribute to the chemical enrichment of either the ISM or IGM.The coeﬃcient of star formation,ν2is80,40,and10Gyr−1for Models1,2a,and2b respectively.In each case, star formation begins at very high redshift(z≃30)but peaks at redshifts between10and 15,depending on the model.Note that the absolute value of the SFR depends not only on ν2,but also on the eﬃciency of outﬂow.See[7]for details.In Figure2,we show the SFR,for Models1,2a,and2b(including the normal mode). Also shown by the dotted curve is the SFR for the massive mode alone in Model1.For comparative purposes,we also consider an example of a model(with an IMF as in Model 1)in which the massive mode occurs as a rapid burst at z=16designated as Model1e. This is also shown in Figure2compared with the analogous result for Model1.As one can see,the massive burst SFR is signiﬁcantly larger than the model considered above for theFigure2:The star formation rate for Models1,2a,and2b as labeled by their respective mass ranges,and a rapid burst model.The dashed line in the upper panel shows the SFR of the massive mode of Model1.short duration of the burst,while at lower redshifts the SFR,which is determined byν1is nearly identical.Because of our lack of understanding of gravitational wave production in pair-instability supernovae,we will not consider Model2a any further.The rate of core collapse supernovae can be calculated directly in terms of the IMF and SFRSNR= m sup max(8M⊙,m min(t))dmφ(m)ψ(t−τ(m)),(13)where m min(t)is the minimum mass of a star with lifetime less than t.The diﬀerential energy density parameter for each model is calculated using Eq.4.4ResultsThe diﬀerential energy density in gravitational waves from core collapse supernovae in Model 0is shown in Fig.3.These stars collapse to form either neutron stars or black holes with a mass equal to the mass of the progenitor’s helium core.The spectrum peaks at a frequency of about360Hz,withΩGW h2=3.5×10−10.This spectrum should be similar to the upper curve in Fig.4of Ref.[20],which peaks at roughly300Hz but with a maximum value about two orders of magnitude lower than that in Fig.3.One can see from Fig.4that our supernova rate lies below theirs at high redshift and they are comparable at later times.A large supernova rate at high redshift serves to broaden the gravitational wave spectrum, but in this case it is not the cause of the disparity in peak heights,which is attributable primarily to diﬀerences in the IMFs.Their spectra were calculated under the assumption151050100500100010-1010-1110-1210-13f (Hz)ΩG W h 2Figure 3:Gravitational Wave background from core collapse supernovae in Model 0.that all progenitors collapse to form neutron stars,whereas our Model 0also includes stars that collapse to black holes.As can be seen from Eq.9,the number of stars collapsing to black holes in Model 0is suppressed by an IMF that favors low masses,but stars emit ∼3orders of magnitude more energy in gravitational waves when the collapse yields a black hole than when it yields a neutron star due to the larger eﬃciency and remnant mass.As a result,the gravitational wave background for the normal mode of star formation is enhanced by two orders of magnitude over that in Buonanno et al.[20].The background of gravitational waves from stars that collapse to black holes was cal-culated by Araujo et al.[18]assuming a Springel &Hernquist [28]model of star forma-tion.They ﬁnd that for a Salpeter IMF deﬁned for stars up to 125M ⊙and all stars with 25<m <125collapsing to black holes with M r =10 2.557.51012.515redshift0.250.50.7511.251.51.75S N R x 103 (M p c -3 y r -1)Figure 4:Rates of core collapse supernovae considered in the calculation of the gravitational wave background for Model 0(solid blue),from Buonanno et al.(dotted red)[20],and using the Springel and Hernquist SFR from Araujo et al.(dashed black)[18].still higher frequencies due to signiﬁcant star formation at redshifts between 3and 5.The fact that maximum values of the energy density parameter span two orders of magnitude is attributable primarily to the diﬀering SFRs seen in Fig.2,but also to the diﬀering remnant masses in Model 1and 2b.Stars in Model 1have much smaller initial masses and collapse to form black holes with a mass equal to the progenitor’s helium core,whereas stars in Model 2b,which are larger to begin with,collapse entirely to black holes.This entails larger bulk motions of matter,with ΩGW ∝M r ,as can be seen from Eqs.4-5.The spectrum from supernovae in Model 2b,which includes 300M ⊙progenitors,could be compared with the curves in Figure 8of Ref.[20].Although the frequencies at which the peaks occur are similar,the energy density obtained in our calculation is larger.The diﬀering peak heights are attributed predominantly to the diﬀerence in the assumed supernova rates for the Pop III modes.In Model 2b,the baryon fraction in Pop III stars is f III =7.0×10−2.This is much larger than the maximum value of f III =10−3assumed in [20],yielding ΩGW h 2approaching 10−11.The supernova rate,and therefore the energy density in gravitational waves,scales linearly with f III .In addition,we assume total collapse of massive stars leaving a remnant mass equal to the progenitor mass whereas Buonanno et al.assume that 300M ⊙stars in collapse to form 100M ⊙black holes,and ΩGW h 2is linear in the remnant mass.Finally,Model 2b,is not a delta function of 300M ⊙stars,but rather a power law distribution containing stars up to 500M ⊙.These eﬀects combine to enhance our peak height by a factor of about 300.It is more diﬃcult to compare Models 1and 1e,as the IMF contains stars with masses less than 100M ⊙.Our fraction of baryonic matter in Population III stars is diﬀerent for each model;3.3×10−3for Model 1and 1.2×10−2for Model 1e.The total gravitational wave background for Models 1,1e,and 2b,including the normal151050100500100010-1010-1110-1210-13 f (Hz)ΩG W h 210-92b1e 1Figure 5:Gravitational wave background from Population III core collapse supernovae in Models 1,1e,and 2b.0.1110100100010-1010-1110-1210-13f (Hz)ΩG W h 210-910-142b 1e1Figure 6:Gravitational wave background from core collapse supernovae in Models 1,1e,and 2b for both the normal and massive mode combined.mode,is shown in Figure 6.In the spectrum for Models 1e and 2b,one can clearly see the peak from the normal mode near 360Hz and that from the massive mode at lower frequency.The massive mode contribution to the total background spectrum in Model 1is not large enough to be seen here.Consequently,the spectra for Model 0alone and Model 1(including the normal mode)are identical.However,as discussed in Section 3,Model 0alone could not provide a suﬃcient ﬂux of ionizing photons for reionization at high redshift.As was mentioned in Section 2,the shape of the spectrum of gravitational waves dependson the parameters a and b in Equation 6.While any variation in these parameters can have a dramatic eﬀect on the resultant spectrum when the anisotropy parameter and neutrino luminosity are speciﬁed,the eﬀects are strongly suppressed by our normalization scheme in which we specify the eﬃciency of gravitational wave production,ǫ,for stars that collapse to black holes.The eﬀect of varying the parameters a and b in the spectrum for Model 2b is shown in Figure 7.0.1110100100010-1010-1110-1210-13f (Hz)ΩG W h 210-9Figure 7:Total gravitational wave background from core collapse supernovae in Model 2b including both the normal and massive modes.The three curves correspond to diﬀerent values for the parameters a and b .Our nominal case of a =200Hz and b =300Hz is shown as the solid curve.The dashed (dotted)curve corresponds to a =b =250Hz (a =150Hz and b =350Hz).5DetectionAs the backgrounds calculated here are the collective result of the collapse of all supernovae in the models considered,it is important to know whether the background is continuous.The ratio of the duration of each burst to the time between successive bursts is called the duty cycle,given by DC = z idR SN∆τis the average time duration of a single burst of gravitational wave emission [29].One can approximate the duty cycle by choosing a ﬁxed value for∆τ(1+z )≈1/f ,where fis the observed frequency.Then the duty cycle can also be approximated byDC≈1fz iM maxM minφ(m)ψ(t−τ(m))dzdmdz,(15)where dV/dz=4πr2(z)/H is the comoving volume element and dr=(1+z)dt[20].The duty cycles approximated using these two methods are given in Table5.In column2,the duration of the burst was assumed to be roughly the same for all supernovae,DC(∼DC×f0.180.230.470.34LIGO III and BBO/DECIGO,it is possible that the entire spectrum below a few hundred Hz.may be observed in the near future.-3-2-10123-18-16-14-12-10-8-6Log(f /Hz)L o g (ΩG W h 2)LISALIGO IIIII DECIGOBBO(corr)BBO(uncorr)Figure 8:Sensitivities of proposed and operating gravitational wave detectors BBO,DE-CIGO,LIGO II and III (correlated),and LISA expressed in terms of energy density [30,16].The dashed lines are the expected gravitational wave backgrounds from Models 1e and 2b as in Figure 6.6ConclusionsWe have calculated the gravitational wave background from three diﬀerent star formation histories that reproduce the observed chemical abundances and reionize the universe at high redshift.Each star formation history consists of a Population III mode of star formation coupled to a normal mode that reproduces the observed SFR for z 6.The gravitational wave background was calculated assuming the amplitude given in Eq.1.We obtained three distinct gravitational wave background spectra.In each case,there is a peak due to the normal mode near f =360Hz.as well as a peak due to the massive mode at lower frequency.The location of the massive mode peak depends on the redshift range over which Population III star formation occurred;the earlier star formation reached maximum,the smaller the peak frequency.The gravitational wave background from core collapse supernovae in our models is found to be large enough that it should be detected by the next generation of space-based laser interferometers,BBO and DECIGO,if not sooner by LIGO.The background will constitute a shot noise signal in the frequency range accessible with LIGO,however we expect the signal to be continuous for frequencies that will be probed by BBO and DECIGO.Within the nextfew decades,a detection of the gravitational wave background from cosmological supernovae will provide valuable information about the history of structure formation in the universe.AcknowledgementsWe would like to thank G.Sigl for useful conversations.The work of K.A.O.,F.D.and E.V.was supported by the Project“INSU-CNRS/USA”,and the work of K.A.O.and P.S. was also supported in part by DOE grant DE–FG02–94ER–40823.References[1]A.Kogut et al.,Astrophys.J.Suppl.148(2003)161[arXiv:astro-ph/0302213].[2]R.Cen,Astrophys.J.591(2003)L5[arXiv:astro-ph/0303236];Z.Haiman&G.P.Holder,Astrophys.J.595(2003)1[arXiv:astro-ph/0302403];J.S.B.Wyithe&A.Loeb,Astrophys.J.588(2003)L69[arXiv:astro-ph/0302297];V.Bromm,Publ.Astron.Spc.Pac.116,103(2004)[arXiv:astro-ph/0311292].[3]nzetta,N.Yahata,S.Pascarelle,H.W.Chen,and A.Fernandez-Soto,Astrophys.J.570(2002)492arXiv:astro-ph/0111129; A.M.Hopkins,Astrophys.J.615(2004)209[arXiv:astro-ph/0407170]; A.M.Hopkins and J. F.Beacom, arXiv:astro-ph/0601463.[4]V.Bromm and A.Loeb,Nature425(2003)812[arXiv:astro-ph/0310622];N.Yoshida,V.Bromm and L.Hernquist,Astrophys.J.605(2004)579[arXiv:astro-ph/0310443].[5]J.Tumlinson, A.Venkatesan and J.M.Shull,Astrophys.J.612(2004)602[arXiv:astro-ph/0401376]; B.Ciardi and A.Ferrara,Sp.Sci.Rev.116(2005)625 [arXiv:astro-ph/0409018];R.Schneider,R.Salvaterra, A.Ferrara and B.Ciardi, arXiv:astro-ph/0510685.[6]F.Daigne,K.A.Olive,E.Vangioni-Flam,J.Silk and J.Audouze,Astrophys.J.617(2004)693[arXiv:astro-ph/0405355].[7]F.Daigne,K. A.Olive,J.Silk, F.Stoehr,& E.Vangioni,Ap.J.(in press),[arXiv:astro-ph/0509183].[8]T.Totani and K.Sato,Astropart.Phys.3,367(1995)[arXiv:astro-ph/9504015];T.Totani,K.Sato and Y.Yoshii,Astrophys.J.460,303(1996) [arXiv:astro-ph/9509130];S.Ando,K.Sato and T.Totani,Astropart.Phys.18(2003)307[arXiv:astro-ph/0202450];S.Ando,Astrophys.J.607(2004)20 [arXiv:astro-ph/0401531].S.Ando and K.Sato,New J.Phys.6(2004)170 [arXiv:astro-ph/0410061];M.Kaplinghat,G.Steigman and T.P.Walker,Phys.Rev.D62(2000)043001[arXiv:astro-ph/9912391];L.E.Strigari,M.Kaplinghat,G.Steigman and T.P.Walker,JCAP0403,007(2004)[arXiv:astro-ph/0312346];L.E.Strigari,J.F.Beacom,T.P.Walker and P.Zhang,JCAP0504,017(2005)[arXiv:astro-ph/0502150]; F.Iocco,G.Mangano,G.Miele,G.G.Raﬀelt and P.D.Serpico,Astropart.Phys.23,303(2005)[arXiv:astro-ph/0411545].[9]F.Daigne,K.A.Olive,P.Sandick,&E.Vangioni,Phys.Rev.D72(2005)103007[arXiv:astro-ph/0509404].[10]B.Willke et al.,Class.Quant.Grav.21,S417(2004).[11]E.Gustafson,D.Shoemaker,K.Strain,&R.Weiss,1999,LIGO Document T990080-00-D;D.Sigg,Class.Quant.Grav.21,S409(2004);B.Abbott et al.[LIGO Scientiﬁc Collaboration],Nucl.Instrum.Meth.A517,154(2004)[arXiv:gr-qc/0308043].[12]R.Takahashi[TAMA Collaboration],Class.Quant.Grav.21,S403(2004).[13]F.Acernese et al.[VIRGO Collaboration],Class.Quant.Grav.21,S385(2004).[14]S.Phinney et al.,NASA Mission Concept Study(2004).[15]P.Bender et al.,LISA Pre-Phase A Report,(1998).[16]N.Seto,S.Kawamura,&T.Nakamura,Phys.Rev.Lett.87(2001)221103[arXiv:astro-ph/0108011].[17]V.Ferrari,S.Matarrese,&R.Schneider,Mon.Not.R.Astron.Soc.303(1999)247[18]J.C.N.de Araujo,O.D.Miranda,&O.D.Aguiar,Mon.Not.R.Astron.Soc.348(2004)1373[19]J.C.N.de Araujo,O.D.Miranda,&O.D.Aguiar,Mon.Not.R.Astron.Soc.330(2002)651[arXiv:astro-ph/0202037].[20]A.Buonanno,G.Sigl,G.G.Raﬀelt,H.-T.Janka,&E.M¨u ller,Phys.Rev.D72(2005)084001[arXiv:astro-ph/0412277].[21]W.H.Press and P.Schechter,Astrophys.J.187,425(1974).A.Jenkins et al.,Mon.Not.Roy.Astron.Soc.321(2001)372[arXiv:astro-ph/0005260].[22]E.M¨u ller,M.Rampp,R.Buras,&H.-T.Janka,Astrophys.J.603(2004)221[arXiv:astro-ph/0309833].[23]E.M¨u ller,&H.-T.Janka,Astron.Astrophys.317(1997)140[24]M.Maggiore,Phys.Rept.331,283(2000)[arXiv:gr-qc/9909001].M.Maggiore,[arXiv:gr-qc/0008027].[25]C.L.Fryer,S. E.Woosley and A.Heger,Astrophys.J.550(2001)372[arXiv:astro-ph/0007176].[26]R.F.Stark&T.Piran,Phys.Rev.Lett.55(1985)891[27]A.Heger&S.E.Woosley Astrophys.J.567(2002)532[arXiv:astro-ph/0107037].[28]V.Springel and L.Hernquist,Mon.Not.Roy.Astron.Soc.339,312(2003)[arXiv:astro-ph/0206395].[29]R.Schneider,A.Ferrara,B.Ciardi,V.Ferrari,&S.Matarrese,Mon.Not.R.Astron.Soc.317(2000)385[arXiv:astro-ph/0909419].[30]A.Buonanno,“TASI Lectures on Gravitational Waves from the Early Universe,”[arXiv:gr-qc/0303085].[31]V.Corbin&N.J.Cornish,[arXiv:gr-qc/0512039].[32]A.Cooray,Mod.Phys.Lett.A20(2005)2503[arXiv:astro-ph/0503118].。

引力波 gravitational waves

引力波gravitational wavesA new gravitational waves research project is awaiting governmental approval as China steps up its efforts to study the phenomenon, after a team of US scientists announced their historic discovery of gravitational waves on Thursday.11日美国一支科学家队伍宣布发现“引力波”后，中国也加快了研究此现象的步伐，一项关于引力波的新研究项目正等待政府批准。

“引力波”（gravitational waves）也称重力波。

100年前，爱因斯坦首次在广义相对论（General Theory of Relativity）提到了这个词。

石头被投进水中会有波纹产生，而黑洞等质量巨大的天体在剧烈运动时产生的“时空涟漪”（ripples in the space-time），则被称为引力波。

引力波很可能蕴含着宇宙诞生的巨量信息，引力波的发现，将能为大爆炸理论（the big bang theory）和宇宙膨胀理论（the Inflation Theory）等各种预见性理论找到证据。

有人担心，美国激光干涉引力波天文台（LIGO）方面的发现是否意味中国在这一领域的研究“晚了一步”，从而导致中国本土引力波探测工程（China's domestic gravitational waves project）“天琴计划”意义大减。

对此相关人士表示，此次美国发现引力波对“天琴计划”并无太大影响。

LIGO此次发现引力波是孤立事件（an isolated event），仍需其他引力波实验支持（require support from further experiments）。

Gravitational waves from collapsing globular cluster systems

M >> 6.7 · 105
(4) (5)
d < 2.4 · 10−14
where R and m are expressed in solar units. To give a global evaluation of the signal detectable it is necessary, of course, to integrate over all the possible sources, i.e. to cumulate the contribution of all the stars falling into the galactic central black holes in galaxies at any red–shift. 3. Gravitational wave impulses from high density galactic central regions As shown ﬁrstly by Capuzzo–Dolcetta (1993), dynamical friction and tidal disruption are eﬀective mechanisms for globular cluster system (GCS) evolution in galaxies. As a result, a globular cluster loses energy and angular momentum, approaching more and more to the central regions where, eventually, they can merge and grow a ‘super cluster’ and, also, may lose their identity as a cluster due to a strong tidal interaction with a compact central nucleus. Capuzzo–Dolcetta (1993, 1998) computed a series of models of evolution of GCSs in a triaxial external potential, taking into account collective eﬀects (dynamical friction and tidal interaction) to follow the mass loss to the galactic centre and its contribution to the galactic nucleus feeding and growth. We do not deep here the details and characteristics of these models, but just say that we refer to a model where: (i) the nucleus is initially absent; (ii) the GCS system is composed by 1, 000 globular clusters of the same mass,

A Compact Supermassive Binary Black Hole System

a rXiv:as tr o-ph/6442v13Apr26A Compact Supermassive Binary Black Hole System C.Rodriguez 1,2,G.B.Taylor 1,3,R.T.Zavala 4,A.B.Peck 5,L.K.Pollack 6&R.W.Romani 7ABSTRACT We report on the discovery of a supermassive binary black hole system in the radio galaxy 0402+379,with a projected separation between the two black holes of just 7.3pc.This is the closest black hole pair yet found by more than two orders of magnitude.These results are based upon recent multi-frequency observations using the Very Long Baseline Array (VLBA)which reveal two compact,variable,ﬂat-spectrum,active nuclei within the elliptical host galaxy of 0402+379.Multi-epoch observations from the VLBA also provide constraints on the total mass and dynamics of the system.Low spectral resolution spectroscopy using the Hobby-Eberly Telescope indicates two velocity systems with a combined mass of the two black holes of ∼1.5×108M ⊙.The two nuclei appear stationary while the jets emanating from the weaker of the two nuclei appear to move out and terminate in bright hot spots.The discovery of this system has implications for the number of close binary black holes that might be sources of gravitational radiation.Green Bank Telescope observations at 22GHz to search for water masers in this interesting system are also presented.Subject headings:galaxies:active –galaxies:individual (0402+379)–radio continuum:galaxies –radio lines:galaxies1.IntroductionBlack holes are a direct consequence of the physics described in Einstein’s theory of gravity.There is a great deal of indirect astronomical observational evidence for these exotic objects in two mass ranges:stellar mass black holes,with masses of4-15times the mass of our Sun;and supermassive black holes,with masses ranging from105to1010solar masses. There is also some evidence for intermediate-mass black holes,those with masses of a few hundred to a few thousand solar masses(see Filippenko&Ho2003,Gebhardt et al.2005).Since most nearby galaxies harbor supermassive black holes at their centers(Richstone et al.1998),the merging of galaxies,an essential part of the galaxy formation process,is thought to be the prevalent method in which supermassive black hole binaries are formed. Accordingly,such systems should be common in galaxies.An understanding of the evolution and formation of these systems is important for an understanding of the evolution and formation of galaxies in general.The evolution of a binary supermassive black hole involves three stages(Begelman et al. 1980),which are summarized by Merritt&Milosavljevi´c(2005)as follows:(1)As the galaxies merge,the supermassive black holes sink toward the center of the new galaxy via dynamical friction forming a binary;(2)the binary continues to decay mainly due to the interaction of stars on orbits intersecting the binary,which are then ejected at velocities comparable to the binary’s orbital velocity,carrying away energy and angular momentum;(3)ﬁnally, if the binary’s separation decreases to the point where the emission of gravitational waves becomes eﬃcient at carrying away the last remaining angular momentum,the supermassive black holes coalesce rapidly.There is circumstantial evidence that most binary black holes merge in less than a Hubble time(Komossa2003a).Therefore,the most massive systems that are able to coalesce in less than a Hubble time will create the loudest gravitational wave events in the universe(Sesana et al.2004),which might be detectable by a low-frequency gravitational wave experiment such as the Laser Interferometer Space Antenna(LISA).Our ability to resolve both supermassive black holes in any given binary system depends on the separation between them,and on their distance from Earth.It is believed that the longest timescales in the evolution of a supermassive binary black hole system leading up to coalescence is the stage in which the system is closely bound(∼0.1−10pc),meaning that in most of these systems the black hole pair can only be resolved by VLBI observations (see review by Komossa2003a detailing observational evidence for supermassive black holes binaries).Some source properties like X-shaped radio galaxies and double-double radio galaxies,helical radio-jets,double-horned emission line proﬁles,and semi-periodic variations in lightcurves have been taken as indirect evidence for compact binary black holes though other explanations are possible.The BL Lacertae Object OJ287is a candidate for harboringa supermassive binary black hole,inferred from the characteristics of its optical lightcurve, which shows repeated outbursts at11.86y intervals(Sillanp¨a¨a et al.1988).Combining optical as well as radio observations,Valtaoja et al.(2000)presented a new interpretation which suggests that at intervals of11.86y,the secondary black hole crosses the accretion disk of the primary black hole,causing a thermalﬂare visible only in the optical.About a year later,the disturbance propagates down the relativistic jet and results in the growth of new synchroton-emitting shocks visible both in the optical and radio.The observed11.86 y period corresponds to the orbital period of the compact binary black hole.Some wider systems have,however,been found more directly.The ultra luminous galaxy NGC6240, discovered by the Chandra X-ray observatory,was found to have a pair of active supermassive black holes at its center(Komossa et al.2003b),separated by a distance of1.4kpc.Another system that has been known for some time is the double AGN(7kpc separation)constituting the radio source3C75,which was discovered by the VLA to have two pairs of radio jets (Owen et al.1985).In this paper we present further observations of the radio galaxy0402+379,which was discovered by Maness et al.(2004)to contain two central,compact,ﬂat spectrum, variable components(designated C1and C2),a feature which has not been observed in any other compact source.Maness et al.(2004)remarked upon the unusual properties found in this source and proposed several physical explanations.One possible scenario is for one component to be a foreground or background source,instead of being associated with0402+379.However,because of the small separation between C1and C2(7.3pc),and because of a faint bridge of radio emission found connecting components C1and C2,this theory was ruled out.A second explanation was that the nucleus was being gravitationally lensed.However,based on the signiﬁcant diﬀerence in the lightcurves of components C1and C2,and the close proximity of0402+379(z=0.055),this theory was also eliminated.Two other scenarios were proposed by Maness et al.(2004),which could not be conclusively ruled out and remained as possible explanations.Theﬁrst of these suggest that component C2 could be a knot in the southern jet,with C1classiﬁed as the core.To test this hypothesis we performed high frequency,high resolution Very Long Baseline Array(VLBA1)observations of0402+379,designed to resolve any jet component and to look for relative motions.The ﬁnal explanation is that C1and C2are two active nuclei of a supermassive binary black hole system.Throughout this discussion,we assume H0=75km s−1Mpc−1,q0=0.5,and1mas= 1.06pc2.Observations2.1.VLBA Observations from2005VLBA observations were made on2005January24and June13at0.317,4.976,8.410, 15.354,22.222,and43.206GHz.Four IFs with a bandwidth of8MHz were observed in 32channels in both R and L circular polarizations.Four-level quantization was employed at all six frequencies.The net integration time on0402+379was115minutes at0.3GHz, 69minutes at5GHz,69minutes at8GHz,122minutes at15GHz,251minutes at22 GHz,and249minutes at43GHz.Standardﬂagging,amplitude calibration,fringeﬁtting, bandpass calibration(using3C84for bandpass calibration and3C111for gain calibration), and frequency averaging procedures were followed in the Astronomical Image Processing System(AIPS;van Moorsel et al.1996).Opacity corrections were performed for the22and 43GHz data.AIPS reduction scripts described in Ulvestad et al.(2001)were used for a large part of the reduction.All manual editing,imaging,deconvolution,and self-calibration were done using Difmap(Shepherd et al.1995).2.2.Archival ObservationsTo further study this source,we obtained fully-calibrated VLBI data taken in1990(Xu et al.1995),in1996(VCS;Beasley et al.2002),in three epochs(1994,1996,and1999) of the CJ Proper Motion Survey(Britzen et al.2003)and in2003(Maness et al.2004). These data were imaged and modeled in Difmap to aid in analysis of motions,variability, and spectra of0402+379.Further information regarding these observations can be found in Table1.2.3.Green Bank Telescope constraints on H2O MasersObservations were made with the GBT on2005October14.We used the18-22GHz K-band receiver,which uses dual beams separated by3′in azimuth.The GBT beamwidth is∼36′′at22GHz,and pointing uncertainties were∼5′′.Pointing was corrected hourly using0402+379itself,which has suﬃcient continuum emission.The telescope was nodded between two positions on the sky such that one beam was always centered on the position of 0402+379during integration.The spectrometer was conﬁgured with two bandpasses of200 MHz each,overlapped by20MHz at the redshifted H2O frequency of21.075GHz,so that the total coverage was±2700km s−1with respect to the systemic velocity of the galaxy.The spectral resolution was equivalent to0.33km s−1.The zenith system temperature was between40and55K for the duration of the run.Atmospheric opacity at22GHz was estimated from system temperature and weather data,and ranged from0.09to0.12at the zenith.The data were reduced using GBTIDL.We found that the higher frequency IF was subject to a60MHz ripple across the baseband,most likely the result of known imperfections and temperature sensitivities of the IF transmission.To reduce the eﬀects of the bandpass ripple we subtracted polynomials of order4and8from theﬁrst and second IFs respectively. We were left with some small residual ripples,but the period of60MHz is large enough that any maser emission present would have been apparent nonetheless.We Hanning-smoothed the spectra following calibration.The1σrms sensitivity of these observations is∼2mJy per km s−1.No maser emission was detected.2.4.HET SpectroscopyWe obtained a spectrum of the core of0402+379on2004December11with the9.2m Hobby-Eberly telescope(HET;Ramsey et al1998)Marcario Low Resolution Spectrograph (LRS;Hill et al.1998).Two600s exposures were taken,using the G3VPH Grism,a Schott OG515blockingﬁlter and a1.5′′slit placed at the parallactic angle.The resulting spectrum coversλλ=6300−9120˚A at5.6˚A resolution.We applied standard IRAF calibrations and ﬁnd that the spectrum is similar to that obtained by Stickel et al.(1993),with a reddened continuum and Seyfert2emission lines at a redshift of z=0.05523(1)(16,460km s−1).In Figure1we show the Hαregion of the spectrum,where the only strong lines are present.The lines are resolved with Gaussian width12.5±1˚A,after deconvolution of the instrumental resolution.Here the uncertainty is the range in theﬁtted linewidths for the various species;this substantially exceeds the statistical error.The lines appear asymmetric with a red shoulder suggesting two components with a∼7±1˚A separation,i.e.a component velocity separation of∼300km s−1.The residual to the lineﬁt in Figure1also shows signiﬁcant excesses in the line wings,suggesting the presence of a broader Hαcomponent.2.5.The Host GalaxyOptical imaging is at present quite limited,but in the Palomar Sky survey images 0402+379appears as a relatively bright r=17.2elliptical galaxy embedded in a halo of patchy faint emission,extending to an apparent companion25”to the NE.Line emission (O i&N ii)is seen from this region in the HET spectra,suggesting the presence of disturbedphoto-ionized gas.An optical image in Stickel et al.(1993)also shows this faint emission. These authors comment that the elliptical core has a“ﬂat brightness distribution”.The 2MASS(Cutri et al.2003)J−K color of0402+379is1.757,and this is consistent with the value expected for quasars at0402+379’s redshift(Barkhouse&Hall2001)The J−K color could indicate star formation activity from a recent merger,or an obscured AGN consistent with the radio galaxy identiﬁcation for this source.Additional optical imaging would be useful to understand the dynamical state of0402+379’s core and the origin of the excitation of the surrounding nebulosity.A source appears near0402+379in the ROSAT Bright source catalog(Voges et al. 1999).1RXSJ040547.3+380308gives0.21PSPC cnts s−ing the Galactic absorption in this direction N H≈3×1021cm−2,and assuming a typical AGN power law index ofΓ=1.7 this corresponds to an unabsorbed0.1-2keVﬂux of8.5×10−12erg cm−2s−1or a luminosity L0.1−2≈5×1043erg s−1,i.e.∼2×10−26erg cm−2Hz−1.The radioﬂux density of0402+379 at5GHz is1.1Jy(Pauliny-Toth et al.1978;Becker et al.1991)which is a radio luminosity of7×1031ergs sec−1Hz−1.This is in agreement with the X-ray to5GHz radio luminosity correlation of Brinkmann et al.(2000).The ROSAT archive also contains two HRI pointings of this source for a combined exposure of27ks.These show that the X-ray source is largely resolved,extending over a radius of∼15′′and appears to follow the faint diﬀuse emission surrounding the galaxy core. The bulk of the X-rayﬂux lies between the elliptical and its companion,further supporting the interaction hypothesis.The X-ray luminosity for this diﬀuse emission is comparable to the AGN estimate above,e.g.L0.1−2≈3×1043erg s−1for a1keV Raymond-Smith plasma, and likely dominates theﬂux of the ROSAT All-Sky Survey source.Weﬁnd that the X-ray emission of0402+379is unique;of the35known CSO’s(Compact Symmetric Objects)this is the only source detected in the ROSAT All-Sky Survey.Thus, further X-ray observations could abet optical data in probing the nature of this emission and the connection with recent merger and/or nuclear activity.3.Results3.1.Radio ContinuumFigure2shows naturally weighted0.3and5GHz images from the2005VLBA obser-vations.The structure of the source at5GHz reveals the presence of two diametrically opposed jets,as well as two central strong components,one directly between the jets and the other one also between the jets but oﬀset from the center.Following the conventionestablished by Maness et al.(2004),we designated the aligned central component as C2, and the oﬀset central component as C1.As we can see in Figure2,the5GHz image spans ∼40mas(∼40pc),corresponding to a small region in the central part of the0.3GHz image,which shows structure on scales of∼500mas(∼500pc).The orientation of the0.3 GHz image is consistent with that seen by the VLA at5GHz(Maness et al.2004).The VLA image shows extended emission going northwards,whereas the northern jet seen in our 5GHz image is pointing in the northeast direction,suggesting that at some point the jet changes direction and starts moving northwards,probably as a consequence of interactions with the surrounding medium.The VLA1.5GHz image(Maness et al.2004)also shows the extended emission going northwards.The southern jet seen in our5GHz image is pointing in the southwest direction,which is consistent with both the1.5GHz and the5GHz VLA images.Figure3shows naturally weighted8,15,22,and43GHz images from the2005VLBA observations.For both the8and15GHz images,the overall structure of the source is similar to that at5GHz,both jets are present as well as the two central components(C1and C2).It is clear that for higher frequencies the two central components are easily distinguished and remain unresolved,while both jets become fainter and are heavily resolved.At22GHz these eﬀects are readily apparent,and become more prominent at43GHz,where the jets can barely be detected.Before these observations were made,the highest frequency for which data had been taken for this source was15GHz.Elliptical or circular Gaussian components wereﬁtted to the visibility data using Difmap. We obtained an estimate for the sizes of C1and C2based on our15,22,and43GHz model ﬁts.In this case,weﬁt circular Gaussian components for both C1and C2,obtaining an average value of0.173±0.045mas or0.183±0.048pc for C1,and0.117±0.033mas or 0.124±0.035pc for C2.ponent Motions and VariabilityIn order to explore questions pertaining to motion and variability in0402+379,we obtained fully calibrated5GHz VLBI data taken in1990(Xu et al.1995),as well as5GHz VLBA data taken in three epochs(1994,1996,and1999)of the CJ Proper Motion Survey (Britzen et al.2003)and in2003(Maness et al.2004).Combining these data with our2005 observations at5GHz,we were able to probe motion and variability in this source over a time baseline of15y.Motion and variability studies were performed byﬁtting eight elliptical Gaussian com-ponents in Difmap to the2003visibility data.Then,we used this model toﬁt the5GHz data corresponding to the1990,1994,1996,1999,and2005epochs.We let only position and ﬂux density vary;all other parameters were heldﬁxed at the2003values.Results from our ﬁts are listed in Table2,and Figure4shows the components model,where we have labeled each of them and we also added arrows showing the direction of motion of each component, which will be explained below.To study component variability in0402+379,we compared theﬂux density for com-ponents C1and C2,the meanﬂux density of the southern components(S1,S2,S3,and S4),and the meanﬂux density of the northern components(N1and N2)over each of our 6epochs at5GHz.The above regions were chosen primarily on the basis of their isolation relative to other components in the source.Errors for each region were computed on the basis of the rms noise and our estimated absoluteﬂux calibration errors(∼20%for the1990 and1994Mk II VLBI epochs,and∼5%for the1996,1999,2003,and2005VLBA epochs). The resulting fractional variation light curves are shown in Figure5.These light curves were created by dividing each region’sﬂux density at each epoch by the mean regionﬂux density found from averaging all epochs.To aid in readability of our graph,the aligned component (C2),the northern lobe,and the southern lobe are displaced on the y-axis by1,2,and3 units,respectively.From Figure5and Table2weﬁnd that component C1substantially increases inﬂux over the15y baseline,starting from18mJy in1990and increasing in brightness to59 mJy in2005.We alsoﬁnd that component C2is variable,ranging from less than10mJy in1990,to24mJy in1996,and20mJy in2005.Because our1990epoch was observed with Mk II VLBI,this apparent variability could be in part attributed to poor data quality. However,the measuredﬂux for all other components in1990is quite consistent with that in our later epochs,suggesting that the calculated upper limit for component C2’sﬂux in1990 is reliable and that the observed variability in this component is signiﬁcant.For the southern and northern components,weﬁnd that there is no substantial variation in theﬂuxes over the15y baseline.Based on the time variability observed in both C1and C2we can estimate the size of these components.However,in this case the variability time scale found for C1and C2is quite long(roughly5to10y),which gives a weak upper limit on the sizes of the components of a few parsecs,consistent with the estimate made in§3.1.To calculate the relative velocity of the components,we chose component C1as the reference,based on its strength,compactness,and isolation relative to other components in the source.We compared the relative motion for each epoch byﬁtting a line to each component’s relative position,split into x and y components,as a function of time.Resultsof thisﬁtting process are listed in Table3,and plotted in Figure6.The results of this analysis reveal signiﬁcant motion for the northern hot spots N1and N2,yielding a value of0.054±0.008mas/y,or(0.185±0.008)c,and0.033±0.006mas/y, or(0.114±0.019)c respectively.These results show that the northern jet is moving away from the two central components to the northeast.For the southern components S2and S3,signiﬁcant motion was also found,yielding 0.0073±0.0025mas/y,or(0.0251±0.0085)c,and0.016±0.003mas/y,or(0.056±0.010)c respectively.For the other two southern components,S1and S4,even though the values found for the velocities were larger than those found for S2and S3,the values obtained in theﬁtting forχ2were large,a fact that can be veriﬁed by looking at Figures6(d)and 6(g).However,for the projected x position of component S4with time,theχ2obtained was nearly unity,which gives us some conﬁdence in the motion of this component,at least in the x direction.We conclude then that,on average,the southern jet is moving away from the two central components to the southwest,though more slowly than the northern jet.The results obtained for C2show no signiﬁcant motion.The value obtained for the limit on the motion of this component is equal to0.0067±0.0094mas/y,or less than0.088c.In Figure4we draw arrows showing the direction of motion found for each component, as well as their relative magnitude.It is important to note that we placed arrows even for those components for which we do not claim signiﬁcant motion.3.3.Radio Continuum SpectraBy appropriately tapering our22GHz2005data,we obtained an image resolution matched to our8GHz continuum image.These two images were then combined to generate an image of the spectral index distribution across the source(Figure7).In both hotspots of the source,N2and S2,a steep spectrum was found,whereas in both central components,the spectrum isﬂat.A plot ofﬂux density as a function of frequency is included in the image of the spectral index distribution for both C1and C2.The values for theﬂux densities used in order to make these plots were measured from matching resolution images.Details regarding these results are listed in Table4.4.DiscussionFour possible scenarios were proposed by Maness et al.(2004)in order to explain the unusual properties found in0402+379.Two of them were ruled out,but the possibility that C1or C2was an unusual jet component in a dense ISM could not be conclusively eliminated.Our high resolution observations conﬁrm the compactness of component C2,and mea-sure a size of0.124±0.035pc.C2is found to have no signiﬁcant motion,whereas signiﬁcant ﬂux density variability is found.The spectral peak is shown to be at∼10GHz.It is pos-sible that either or both C1and/or C2could be a jet component lit up in a collision with a dense interstellar medium.In this scenario the low observed velocity(<0.088c)is due to the impact with the ISM,and the spectrum is modiﬁed by local acceleration of particles. Diﬃculties with the jet component explanation are(1)it requires a dramatic change in the jet axis on timescales of a few10s of years,while the larger scale emission(see Figure2) indicates that the jet axis has been fairly stable on time scales up to104y;(2)a large gradient in density needs to be invoked to decelerate C1and/or C2but allow the hotspots to advance(Maness et al.2004);(3)if C2is the core responsible for the observed jets and hot spots,and C1is a jet component,then the counterjet is conspicuous in its absence given the orientation of the source close to the plane of the sky indicated by the source symmetry.The absence of a jet associated with C1might atﬁrst seem unusual since only6of 87(7%)sources in the First Caltech-Jodrell Bank VLBI survey(CJ1-the survey in which 0402+379was imaged by Xu et al.1995)show naked cores with no sign of a jet at5GHz. At lowerﬂux levels,however,which are more appropriate since C1by itself would not have made it into the CJ1sample,the fraction of naked cores increases to6of24sources(25%-Taylor et al.2005).Thus the absence of a jet from C1is not by itself evidence against the identiﬁcation of C1as an AGN.The characteristics found in C1and C2are typical of AGN;in a complete survey of32 sources imaged at43GHz,(Lister2001)found no unresolved,isolated jet components.This, together with the morphology of0402+379,leads us to surmise that neither C1nor C2is likely to be a jet component.This leaves the remaining most likely explanation that C1and C2are both active black holes in a compact system.For the remainder of the discussion we assume that this is the case and explore what we can learn about such a system from the present observations.4.1.Constraints on the mass of the black holesWe can use the HET and the VLBA observations to obtain kinematic constraints on the mass of the black hole system.At our observed separation,an orbital velocity of300km/s implies a system mass of1.5×108(v/300km s−1)2(r/7.3pc)M⊙.The suggested line splitting from the optical spectra thus implies a mass of a few times108M⊙.If both radio nuclei are also optically active and show comparable emission line strengths,as expected,then the observed Gaussian line widths put a limit on the line FWHM of∼1300km s−1,implying a limit on the combined black hole mass of∼3×109M⊙.Note that the H i absorption proﬁles of Maness et al.(2004)also found velocity structure of∼1000km s−1extending over a transverse distance of∼20pc.On this scale the mass contribution from gas and stars is not signiﬁcant,so the limit obtained reﬂects the mass of the binary black hole system and indicates a high mass system.The implied mass,∼5×109M⊙,could be dominated by the AGN,if the absorption occurs in the nuclear region.Alternatively the two velocity systems might be probing line-of-sight velocities in the products of a recent merger.An alternative estimate for the central compact object mass can be derived from the blue luminosity of the host bulge(Kormendy&Gebhardt2001).The observed V magnitude of0402+379is∼18.5(Wills et al.1973),after correcting for the Galactic A V∼1.6and a median B-V color index of∼0.9for484ellipticals in the Uppsala General Catalogue (UGC),we obtain a B magnitude of17.8.This corresponds to a central compact object mass of∼7×107M⊙,in reasonable accord with the estimates above,given that this may still be a disturbed system.4.2.Supermassive Binary Black Hole Orbital ParametersUsing our system mass estimate equal to1.5×108M⊙and the projected radial separa-tion between them derived from our2005maps(7.3pc),weﬁnd from Kepler’s Laws that the period of rotation for such a binary supermassive black hole system should be∼1.5×105 y.This period corresponds to a relative projected velocity between components C1and C2 of∼0.001c.The upper limit found for component C1,<0.088c,is consistent with the expected relative velocity between C1and C2,assuming a stable,Keplerian orbit.To actu-ally constrain the masses of the black holes would require observations over a longer time baseline(∼100y).4.3.Gravitational Wave SignalWhat sort of gravitational wave signal might a binary supermassive black hole in 0402+379generate?Assuming the current separation is7.3pc and the total mass is 1.5×108M⊙the natural gravitational wave frequency(Hughes2003)is approximately 2×10−13Hz.This is well below the expected minimum frequency of LISA.Although 0402+379may be a long way from generating a detectable gravitational wave signal it may represent a source of noise important for future observations of cosmologically produced gravitational radiation.Ultra low frequency gravitational radiation generated during inﬂa-tion(Hughes2003)has an upper limit of10−13Hz.Thus,a population of black hole binaries like0402+379may generate substantial noise which could interfere with the detection of the physics of inﬂation.An estimate of this noise contribution requires a population synthesis model(e.g.,Sesana et al.2005)which is beyond the scope of this work.Theﬁnal stage in the evolution of a binary black hole system is the gravitational radia-tion stage,where the semimajor axis decreases to the point at which gravitational radiation becomes the dominant dissipative force.A binary black hole on a circular orbit will merge within the time(Peters1964):t merge(a)=5.8×106 a m1 3m21。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

by b0(∞). This asymptotic conﬁguration actually corresponds to the Schwarzschild solution with mass M∞ = m0A30 exp(3b0(∞)/2). Indeed, such a feature feature is in agreement with previous analytical studies on RT spacetimes[5]. (ii) For small values of the initial mass of the distribution, Minit, more rapidly
1 −1
Pj(x)Pk(x) dx
=
2 δkj 2k+1
.
The
evolution
equations
for
the
modal
coeﬃcients
bk (u)
are
derived
in
following
way: we substitute (3) into the equation for λ(u, x), both into (2), and ﬁnally projecting the resulting
equation into each basis function Pk(x). Then,
b˙ n (u)
=
(2n + 1) 12m0A20
e−Q(u,x) [(1 − x2) λ′]′, Pn(x) .
(4)
with n = 0, 1, 2.., N . Therefore, the dynamics of RT spacetimes is reduced to this system of (N +1) equations.
2
in a rather distinct way in order to take into account the nonlinearities of the ﬁeld equations. For this proposal, we shall consider the Galerkin projection method[4] in which we adopted the following decomposition
the obvious Killing vector ∂/∂ϕ. As a consequence of Einstein equations, we have the following evolution
equation
−6m0
K˙ K
+
(λθ sin θ)θ 2K2 sin θ
=
0.
(2)
In the above the subscript θ denotes derivatives with respect to θ. Eq. (2) is denoted the RT equation,
ds2 = λ(u, θ) + B(u) + 2r K˙ du2 + 2dudr − r2K2(u, θ)(dθ2 + sin2 θdϕ2),
(1)
r
K
where r is an aﬃne parameter deﬁned along the shear-free null geodesics determined by the vector ﬁeld
on a given null surface u = u0 (except in the case m0 = 0). A particular and important solution of the ﬁeld equations is the Schwarzschild metric obtained when K(u, θ) = K0 = const, where λ(u, θ) = K0−2 and the corresponding Schwarzschild mass is MSchw = m0K03. This expression will be of particular importance in our characterization of the mass function of the conﬁguration. Concerning this point, we note that in general the function M (u, θ) = B(u)K3(u, θ) is invariant under the coordinate transformation that reduces B(u) to a constant −2m0; also this transformation induces that the type-D O(1/r3) curvature scalar associated with the mass aspect of the spacetime has a (u, θ) dependence given exactly by BK3.
bk(u) are the modal coeﬃcients, and we have adopted the Legendre polynomials Pk(x) as the basis functions
of the projective abstract space. The internal product deﬁned in this abstract space is Pj(x), Pk(x) ≡
N
K2(u, x) = A20 eQ(u,x) = A20 exp
bk(u)Pk(x) ,
(3)
k=0
where we have introduced a new variable x = cos θ such that −1 ≤ x ≤ 1; N is the order of the truncation,
lowing basic features of the nonlinear dynamics of RT spacetimes emerged. (i) All modal coeﬃcients
tend to zero asymptotically; the exception is b0(u) that approaches to a constant value which we denote
The initial conditions bk(u0) are determined by the initial data K(u0, x) ≡ k(x) since bj(u0) =
2 ln k(x),Pj Pj ,Pj
,
j = 0, .., N . We integrate numerically the set of equations (4) for distinct families k(x) and the fol-
∂/∂r,
dot
means
derivative
with
respect
to
u;
λ(u, θ)
=
1 K2
−
Kθθ K3
+K3
cot θ
is
the
Gaussian
curvature
of the surfaces (u = const, r = const.). The geometry is nonstationary and axially symmetric, admitting
I. Damia˜o Soares‡ Centro Brasileiro de Pesquisas F´ısicas R. Dr. Xavier Sigaud, 150. CEP 22290-180
Rio de Janeiro, RJ, Brazil (Dated: February 5, 2008)
Foster and Newman[3] exhibited the linearized solution of the ﬁeld equations. We generalize their approach
∗ Regular Associated of The Abdus Salam International Centre for Theoretical Physics - ICTP, Trieste, Italy †Electronic address: oliveira@dft.if.uerj.br ‡Electronic address: ivano@cbpf.br
We study the dynamics of a bounded gravitational collapsing conﬁguration emitting gravitational waves, where the exterior spacetime is described by Robinson-Trautman geometries. The full nonlinear regime is examined by using the Galerkin method that allows us to reduce the equations governing the dynamics to a ﬁnite-dimensional dynamical system, after a proper truncation procedure. Amongst the obtained results of the nonlinear evolution, one of the most impressive is the fact that the distribution of the mass fraction extracted by gravitational wave emission satisﬁes the distribution law of nonextensive statistics and this result is independent of the initial conﬁgurations considered.

财务报表英文翻译大全

页数:24
(完整版)财务报表英文翻译大全(2)

页数:24
财务报表英文翻译大全

页数:22
财务报表英文翻译大全

页数:22
企业资产负债表_利润表中英文对照大全

页数:7
财务报表英文翻译 Financial Statement

页数:10
财务报表英文翻译

页数:8
中英财务报表对照翻译 financial report English-Chinese terms

页数:7
专业整理资产负债表损益表现金流量表中英文对照翻译

页数:4
财务报表英文翻译大全含资产负债表现金流量表利润表等等

页数:19

Gravitational Wave Emission from a Bounded Source the Nonlinear Regime

合集下载

Thermal gravitational waves

初三英语宇宙探索发现单选题50题

引力波观测原文PhysRevLett.116.061102

Gravitational waves from hot young rapidly rotating neutron stars

超大质量双黑洞系统中的吸积盘Formationand

高考英语一轮总复习第一全程攻略 Unit 1 Great scie

四川师范大学附属中学2023-2024学年高二下学期期中考试英语试题(含答案)

Gravitational Wave

Multiple imaging by gravitational waves

gravitational wave - 英译汉

高三物理研究英语阅读理解30题

引力波中的相关英语高考考点

Gravitational Lensing of Gravitational Waves from Merging Neutron Star Binaries

Gravitational Waves from the First Stars

引力波 gravitational waves

Gravitational waves from collapsing globular cluster systems

A Compact Supermassive Binary Black Hole System

文档推荐

最新文档

Gravitational Wave Emission from a Bounded Source the Nonlinear Regime

合集下载

Thermal gravitational waves

初三英语宇宙探索发现单选题50题

引力波观测原文PhysRevLett.116.061102

Gravitational waves from hot young rapidly rotating neutron stars

超大质量双黑洞系统中的吸积盘Formationand

高考英语一轮总复习 第一 全程攻略 Unit 1 Great scie

四川师范大学附属中学2023-2024学年高二下学期期中考试英语试题(含答案)

Gravitational Wave

Multiple imaging by gravitational waves

gravitational wave - 英译汉

高三物理研究英语阅读理解30题

引力波中的相关英语高考考点

Gravitational Lensing of Gravitational Waves from Merging Neutron Star Binaries

Gravitational Waves from the First Stars

引力波 gravitational waves

Gravitational waves from collapsing globular cluster systems

A Compact Supermassive Binary Black Hole System

文档推荐

最新文档

高考英语一轮总复习第一全程攻略 Unit 1 Great scie