Simulations of Glitches in Isolated Pulsars
美国X射线脉冲星导航计划及其启示
军 、民、商用 卫 星 系统 仍 然 “ 设 防” 不 ，存
在 固 有 的 脆 弱 性 ， 因 此 某 些 重 要 的 空 间 设 施
可 能成 为敌对 国双方 攻 防对抗 和恐 怖组 织攻
击 的 目标 。
的灾 难 ） 已 经 或 正 在 受 到 各 航 天 大 国 的 关
注 。在 新 一 轮 国 际 太 空 竞 争 中 ，保 护 卫 星 的 安 全 ，确 保 出入资讯
麓
帅 平
（ 国 空 间技 术 研 究 院） 中
口 口 ２ ０ 年 ， 美 国 国 防 高 级 研 究 计 划 局 ０４ （ ＡＲ Ａ） 提 出 了 Ｘ 射 线 脉 冲 星 导 航 计 划 ， Ｄ Ｐ
磁 场 、 超 强 电场 和 超 强 引 力 场 的 天 体 ，其 典 型半 径 约 为 １ｋ ，而 质 量 却 与 太 阳相 当， ０ｍ 核 心 密度达 到 １ ０ ｇ ｃ 。 ×１ ｋ ／ｍ。 脉 冲 星 属 于 高 速 自转 的 中 子 星 ， 其 旋 转 轴 与 磁 极 轴 之 间 有 一 个 夹 角 ， 两 个 磁 极 各 有 一 个 辐 射 波 束 ， 若 星 体 自转 且 磁 极 波 束 扫 过 安 装 在 地 面 或 航 天 器 上 的 探 测 设 备 ， 就 能 获 得 一 个 脉 冲 信 号 ， 犹 如 为 海 上
周 期 范 围 一 般 为 １ ６ ～ １ ０ Ｓ 且 具 有 ． ｍｓ ００ ， 良好 的 周 期 稳 定 性 ， 尤 其 是 毫 秒 级 脉 冲 星 的 自转 周 期 变 化 率 达 到 １ 。 １ ０ ～ ０ 誉 为 自然 界 最 稳 定 的 时 钟 。 ， 被
星爆发 的遗 迹 ，是 一 种具有 超 高 密度 、超 强
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。
风暴的多普勒雷达自动识别
风暴的多普勒雷达自动识别Ξ胡 胜1,2 顾松山1 庄旭东2 罗 慧31南京信息工程大学,南京,2100442广州中心气象台,广州,5100803陕西省气象局,西安,710015摘 要 3种基于雷达的风暴自动识别方法:(1)美国WSR288D Build7.0风暴算法,它利用多个预设阈值来检验回波的强度和连续性,以构造具有三维连续结构的风暴,该方法在风暴合并、分裂以及多个单体相距较近时误差较大。
(2)为美国WSR288D Biuld9.0风暴算法(B9SI),它用7个反射率因子识别阈值替代此前唯一的一个反射率因子阈值,增加了特征核抽取和相近单体处理技术,并保留远距离上的强的2D分量。
该方法在面对成串或成簇多单体时,能够识别出多个单体核,并准确定位。
B9SI没有考虑反射率因子纹理结构和空间梯度的变化,也没有利用径向速度资料,因此无法描述风暴对流的发展状况。
(3)CSI方法,它在降低B9SI反射率因子识别阈值的基础上,利用模糊逻辑技术对B9SI输出结果和雷达基资料做进一步的处理,以计算描述风暴对流发展强弱的对流指数。
CSI首先提取一组最能描述风暴对流性特征的物理量,包括反射率因子纹理结构、反射率因子空间变化率、垂直积分含水,并分配权重;其次,利用每一个物理量的统计结果,结合其物理意义,设计出相应的隶属函数,以计算风暴与该物理量描述的对流性特征相匹配的概率;最后对多个概率值进行加权平均即得对流指数。
此外,计算了2004年8月11日发生在广州的超级单体演变过程中的对流指数,分析表明:对流指数两次加大对应了超级单体的合并增长和辐合增长过程;风暴最强盛时对流指数为0.744;随后对流指数减小,雷达观测到的最大反射率因子对应高度明显降低,地面上开始出现大范围的强降水。
关键词:风暴识别,核抽取,模糊逻辑技术,对流指数。
1 引 言20世纪70年代以来,国内外雷达气象学者在用雷达探测强对流天气领域做了大量的工作,他们深入地分析了风暴发生、发展、成熟和消亡的物理机制,提出了一些风暴的识别方法:Rinehart[1]应用模式来识别风暴单体;Rinehent等[2]利用相关技术分析法对整幅回波图像做处理;刘黎平[3]以面积较大的回波块为研究对象,在等高面中计算云体的中心、面积、强度等,称其为二维矩心识别法;此外还有矩不变量法、边界层探测法、模糊聚类法等。
脉冲星Glitch现象物理本质
• 辐射功率
•, 自转能减慢
Er o t
1 I2 2
m
1 2
BpR3
2
P
• 磁场 • 特征年龄
中子超流涡旋的两种辐射
–– 27年前我们的研究
1) 中微子回旋辐射––For Spin down (Peng, Huang & Huang 1982) 原理:按照粒子物理学中Wenberg – Salam 弱电统一理
И.Φ. MaЛОВ, <PAДИОПУЛЬСАРЫ>, 2019,(p.83)
Log(dP/dt)-15=(1.75 0.56)logP – (0.01 (对 P > 1.25s 脉冲星 (87个) )
对 P > 1s.25 脉冲星
自转减慢只能由中国小组的NSV(中 子超流涡旋)模型描述;
对 0s.1 < P < 1s.25 脉冲星
3P2中子超流涡旋的磁偶极辐射
在较强的外磁场下，中子星内3P2 中子超流体处于各向异性（B相) 状态，在一定温度下磁矩方向沿着外磁场方向排列的3P2 中子 Cooper对数量多于逆磁场方向排列的3P2 中子Cooper对。这就 导致了3P2 中子Cooper对整体的Pauli顺磁磁矩(当温度降低时， 这种Pauli顺磁磁矩增强)。这些处于超流涡旋运动状态的中子绕 着超流涡丝轴线高速旋转。离涡丝轴线愈近的中子其旋转角速度 愈大(距离平方成反比, 最高角速度可达1017/sec以上)。具有磁 矩的中子就会发出磁偶极辐射。 .
Glitch的 观测现象
年轻脉冲星的Glitch现象:
(非常规则缓慢增长的)脉冲周期 (P) 突然变短现象
• 脉冲周期平稳地增长背景上偶然地脉冲周期会突然变短(周期变 化幅度为10-6-10-10) ，随后较之前更迅速地变慢，持续直到恢复过 去的周期增长率。这种现象称为Glitch现象。
袋模型下奇异星的非牛顿引力效应
华中师范大学学报(自然科学版)JOURNAL OF CENTRAL CHINA NORMAL UNIVERSITY(Nat.Sci.)Vol.56No.2Apr.2022第56卷第2期2022年4月DOI；10.19603/ki.1000-1190.2022.02.007文章编号：1000-1190(2022)02-0250-05袋模型下奇异星的非牛顿引力效应皮春梅心（1.湖北第二师范学院物理与机电工程学院，武汉430205；2.湖北第二师范学院天文学研究中心，武汉430205）摘要：该文研究标准袋模型奇异星在考虑非牛顿引力效应下的结构和性质.文章结果显示，对于标准袋模型描述的奇异物质，随着重子数密度的增大,非牛顿引力效应的修正项能量密度越大;非牛顿引力效应的引入使物态变硬,而且较大的非牛顿引力参数4对应较硬的物态方程;非牛顿引p-力效应的引入有效地增大了星体能支撑的最大质量并且当非牛顿引力参数粤》1.93GeV-2时能够解释目前观测到的最大质量脉冲星（PSRJ0470+6620）的数据.关键词：奇异星;物态方程;非牛顿引力中图分类号：P142.5文献标志码：A开放科学（资源服务）标志码（OSID）：物态方程是理论研究致密星结构和性质的重要输入量，它给出了物质内部压强P和密度E之间的关系.结合广义相对论下的流体静力学平衡方程（即TOV方程）和物态方程，可以计算致密星的密度、不同半径处的压强以及质量和半径等物理性质.不同的物态方程会给出不同的致密星内部成分和结构.20世纪60年代Gell-Mann M和Zweig G 建立了强子结构的夸克模型，中子星内部物质组分有了更多的可能性•1984年,Witten E提出了奇异夸克物质设想m:由数量近乎相等的u、d、s夸克组成的夸克物质比"Fe更稳定.根据这个设想，致密星可能是由u、d、s三味夸克物质所组成的奇异星阂.2021年利用Shapiro延迟效应观测发现了毫秒脉冲星J0740+6620具有2.08兰:器M0的大质量匚旳，根据这一观测结果，很多含有奇异粒子的物态方程被排除•一些软物态方程，例如奇异物质的标准MIT袋模型，在经典理论框架下所能支撑的最大质量较小，无法支持观测发现的大质量中子星.但是,在对引力的认识还并不完善的今天，对此还不能完全肯定•在统一引力和其他三种基本相互作用力，即电磁相互作用，强相互作用和弱相互作用的过程中，人们发现描述引力的平方反比关系不再成立.平方反比关系需要根据弦理论预测的其他时空维度的几何效应（或者粒子物理标准模型之外的超对称理论所预言的弱耦合玻色子的交换）做出修正冲].尽管至今尚未确认非牛顿引力的存在，已经有很多地面实验和天文观测对偏离牛顿引力程度的上限给出了限制，相关文献综述见[7].中子星和奇异星的非牛顿引力效应已经得到广泛研究金切，发现在致密天体中这种非牛顿引力可能会具有明显的物理效应，为软物态方程支持大质量致密星（中子星和奇异星）带来了希望.1奇异夸克物质的物态方程夸克物质的状态方程本质上应该由量子色动力学（QCD）来计算，鉴于对低能强相互作用非微扰特性认识的不足，这一计算方法还不能进行•在实际计算中经常采用唯象模型，例如袋模型.此模型忽略夸克间的动力学相互作用，视其为理想气体.各类粒子的巨热力学势分别为”购:4=u9d,（1）Q=—-^2[“3—诚）1/2（“7—号诚）+■I分3一就（2）收稿日期：2021-04-06.基金项目：国家自然科学基金青年项目（11803007）. *通信联系人.E-mail:,cn.第2期皮春梅：袋模型下奇异星的非牛顿引力效应251「越，⑶其中,％和少分别为粒子的质量和化学势.通过热力学关系可以利用巨热力学势计算系统的各热力学量，如各种粒子数密度、压强和能量密度等•第Ki=“,d,s,e)种粒子的数密度是夸克物质通过弱相互作用保持化学平衡•各类粒子化学势M之间满足平衡条件“d=(5)作为一个稳定系统，还应当满足电中性条件要求：91可九一可(加+%)一％=0.(6)重子数密度为n b=-y(n…+n d+n s).(7)不考虑非牛顿引力效应时，能量密度为E q=〉:(fit+円71/)+B,(8)i=u i d,s i e相应地，压强为P q=—工Hi—B,(9)i—u,d,s,e这里B是袋常数.本文忽略u夸克和d夸克的质量,s夸克的流质量取％=93MeV[19],选取具有代表性的袋常数B1/4=140MeV.2非牛顿引力效应根据Fujii理论购，非牛顿引力可以表述为在传统的引力势基础上增加一个汤川型的修正项，即V(r)=_&8加1况2(1+幺貢力)=rV n G)+VVG),(10)其中，Gg=6.6710X10-n N•m2/kg2,a是无量纲的汤川引力强度参数以是短程相互作用的特征长度•利用矢量玻色子交换模型，,=丄=I g2A卩，a士4k G#，其中，土分别代表标量(+)和矢量(一)玻色子，“,g 和分别是玻色子一重子耦合常数，玻色子质量和重子质量.非牛顿引力效应可近似地通过物态方程来描述，而保持爱因斯坦场方程不变.汤川型的修正项对能量密度的贡献为E y=壽j"3角(工1)盍(zi)d工1d丄2,(11)其中,v是归一化常数,『=z|N—云|.上式中重子数密度前的因子3的引入是因为每个夸克的重子数为1/3E14].考虑到m(Hi)=n b(rc2)=«6E21_22],并且取V=4k R3z/3,有E y=reT^dr,(12)通过积分很容易得到，Ey=弊谄口一(1+迹)尹].(13)因为原则上研究对象很大,可以取Rff故Ey=(14)综上,考虑非牛顿引力效应，奇异夸克物质的能量密度为E=E q+E y,(15)其中E q由(8)式给出.相应地，汤川型修正项对压强的贡献为(16)假定玻色子质量与介质密度无关凹，有P y=^~2n b-(17)2V-考虑非牛顿引力效应，奇异夸克物质的压强为P=P Q+P Y,(18)其中，P q由(9)式给出.这里需要指出，非牛顿引力理论是超越了广义相对论的理论.众所周知，平方反比关系是广义相对论在弱场低速情形下的近似.非牛顿引力理论(具体到本文，是在传统的引力势基础上增加一个汤川型的修正项)下，平方反比关系不成立.实际上，超越相对论的其他一些引力理论，如f(R)理论，在弱场低速情形下也不满足平方反比关系⑷.如Shao所述盟】，广义相对论的场方程中有两个部分，其一是时空几何，其二是物质与能量.在对广义相对论做修正时，既可以修正时空几何部分，也可以修正物质与能量部分，两种途径是简并的.在研究非牛顿引力对奇异星质量一半径关系(图3)的影响时采用公式(15)和(18)所给出状态方程，这实际上是修正了广义相对论的物质与能量部分，而广义相对论的时空几何部分保持不变.于是，仍然可以采用原来的广义相对论所推导出的TOV方程.252华中师范大学学报(自然科学版)第56卷3数值计算结果与讨论图1为考虑非牛顿引力后奇异星的物态方程,其中非牛顿引力参数4分别取0,2,5,11 GeV-2.圏1表明,非牛顿引力效应的引入使物态变硬，而 且非牛顿引力参数越大，对应的物态方程越硬.5(4(京201 OC U J • A o s y d500I 000 1 500 2 000£/(MeV ' fnf ，)注；曲线旁边的数值代表非牛顿引力参数少的取值，单位 是 GeV-2.图1考虑非牛顿引力后MIT 物态方程的密度-压强关系Fig. 1 Relation between pressure and energy density in MIT model of quark matter with the nonrNewtonian gravity图2给出了不同参数下汤川型非牛顿引力效 应的修正项对能量密度的贡献随重子数密度的变 化.随着重子数密度的增大，修正项能量密度越大.其实从方程(14)中就可以看出修正项能量密度随着重子数密度的平方单调增加的.质量半径关系是星体最重要的性质之一，图3 给出了引入和没有引入非牛顿引力效应的情况下 奇异星的质量半径关系.从图中可以发现，随着非牛顿引力参数粤的增大，相应可支撑的最大奇异星质量也增大.当4 = 0 GeV-2,即没有引入非牛 顿引力效应时，可支持的奇异星最大质量约为1. 9M® ,而当^ = 11 GeV'2时支持的最大质量大约为2. 56M®.这表明越大的非牛顿引力参数对应的物态方程越硬,支持的奇异星最大质量越大.对于奇异物质的标准袋模型状态方程，加入非牛顿引 力效应并且非牛顿引力参数4^1-93 GeV 一2能够P-解释目前观测到的最大质量脉冲星(PSR J0470 +6620)的数据.08649-00064 22 11111(c w -a s h w 2 4 6 « 10 12 14nji'o注:no = 0.17 fm-3是标准核饱和密度.图2汤川型非牛顿引力效应的修正项对能量密度的贡献随重子数密度的变化Fig. 2 The extra density due to the nonrNewtoniancomponent as the function of —注:红色实线对应于£ = 1. 93 GeV"2,此时理论给出 的奇异星最大质量是2. 08M®.绿色实线给出了目前观 测中发现的最大质量脉冲星(PSR J0470 + 6620)的数据，它的质量是2・08M®・图3引入和没有引入非牛顿引力效应的情况下奇异星的质量一半径关系Fig. 3 The mass-radius relation of strange stars withseveral typical sets of model parameters图4给岀了观测到的脉冲星最大质量(PSRJ0470 + 6620,2. 08M @ )对非牛顿引力参数空间的限制.图中标号为“1”至“9”的黑色曲线对应于其他 不同实验对非牛顿引力参数空间的限制曲线“1”和“2”分别对应于质子一中子散射实验在标量第2期皮春梅：袋模型下奇异星的非牛顿引力效应253玻色子和矢量玻色子情形下的限制⑵］;“3”和“4”分别对应于原子核荷半径和束缚能的限制［旳；“5”和“6”的限制分别来自He原子光谱和208pb散射实验［旳；“7”的限制来自对卡西米尔力的测量沏1；“8”的限制来自中子一氤气散射实验血打“9”的限制来自于金和硅组分的转动源与待测质量间引力的测量㉔.红色实线对应于粤=1.93GeV"2,此时奇异星最大质量是2.08M®.作为参考,红色虚线对应于4=11GeV",此时奇异星最大质量是2.56M©.在图中红色实线上方的区域（对应于粤4 >1.93GeV'2）能够允许的奇异星最大质量大于2.08M®.这个区域符合一些其他实验（如“5”）给出的限制，但是却不能符合另一些实验（如“6”“8”“9”）所给出的限制.bg4图4观测到的脉冲星最大质量(2.08M®)对非牛顿引力参数空间的限制Fig.4Upper bounds on the strength parameter|a|respectively,the bosonrnucleon coupling constant g asa function of the range of the Yukawa force fi andmass if hypothetical bosons,set by differrent experiments 4总结本文主要研究了考虑非牛顿引力效应下标准带模型奇异星的结构和性质，包括奇异物质的密度一压强关系、非牛顿引力效应的修正项对能量密度的贡献随重子数密度的变化以及星体的质量一半径关系.结果表明，非牛顿引力效应的引入使物态变硬，而且较大的非牛顿引力参数对应较硬的物态方程;随着重子数密度的增大，修正项能量密度越大;星体能支撑的最大质量在引入非牛顿引力效应的情况下有效地增大了.而且，对于奇异物质的标准袋模型状态方程，加入非牛顿引力效应并且非牛顿引力参数粤$1.93GeV'2能够解释目前观测到卩的最大质量脉冲星(PSR J0470+6620)的数据.参考文献：[1]WITTEN E・Cosmic separation of phases[J]・PhysicalReview D,1984,30(2):272-285・[2]ALCOCK C,OLINTO A V.Exotic phases of hadronicmatter and their astrophysical application]J].Annual Review of Nuclear and Particle Science»1988^38(8)j161-184・[3]CROMARTIE H・Relativistic Shapiro delay measurementsof an extremely massive millisecond pulsar[J].Nature Astronomy,2020,4：72-76.[4]FONSECA E.Refined mass and geometric measurements ofthe high-mass PSR J0740+6620口/OL]・The Astrophysical Journal Letters,2021r915(1)[2021-09-10X https；//doi, org/10,3847/2041-8213/ac0368,[5]ADELBERGER E G,GUNDLACH J H?HECKEL B R,etal・Torsion balance experiments j a low-energy frontier of particle physics]Jl Progress in Particle and Nuclear Physics’2009,62(1):102-134.[6]LI B A,KRASTEV P G,WEN D H,et al.Towardsunderstanding astrophysical effects of nuclear symmetry energy[J].European Physical Journal A,2019,55(7)s 117-191・[7]MURATA J,TANAKA S.Review of short-range gravityexperiments in the LHC eraEJ/OL]・Classical and Quantum Gravity,2015,32(3)[2021-09-10].https；///10.1088/0264-9381/32/3/033001・[8]KRIVORUCHENKO M I,SIMKOVIC F,FAESSLER A.Constraints for weakly interacting light Bosons from existence of massive neutron stars[J/OL].Physical Review D,2009,79[2021-09-10https：//dot org/10.1103/ PhysRevD.79.125023.[9]WEN D H,LI B A,CHEN L W.Supersoft symmetry energyencountering non-Newtonian gravity in neutron stars[J/ OL]・Physical Review Letters,2009f103(21)[2021-09-10],https；//doL org/10.1103/PHYSREVLETT.103,211102・[10]SULAKSONO A,KASMUDIN M・Effects of in-mediummodification of weakly interacting light Boson mass inneutron stars]J]・Modem Physics Letters A,2011.926(5〉：367-375.[11]ZHANG D R,YIN P L,WANG W,et al.Effects of aweakly interacting light U Boson on the nuclear equation ofstate and properties of neutron stars in relativistic modelsEJ/0L1Physical Review C,2011,83(3)[2021-09-10],https；//doi,org/10,1103/PhysRevC,83,035801・[12]YAN J,WEN D H.R-mode instability of neutron star withnon-Newtonian gravityCJ],Communications in TheoreticalPhysics,20139$9(1儿47・52・254华中师范大学学报（自然科学版）第56卷[13]LIN W,LI B A,CHEN L W,et al.Breaking the EOS-gravitydegeneracy with masses and pulsating frequencies of neutronstarsUJ/OL].Journal o£Physics G,2014,41(7)[2021-09-10H.https；///10.1088/0954-3899/41/7/075203.[14]LU Z Y,PENG G X,ZHOU K.Effects of non-Newtoniangravity on the properties of strange stars[J].Research in.Astronomy and Astrophysics,2017,17(2):11-16.口5]YU Z,XU Y,ZHANG G Q,et al.Effects o£a weakly interacting light U Boson on protoneutron stars includingthe hyperon-hyperon,interactions[J].Communications in.Theoretical Physics,2018»69(4)；417-424.：16]YANG S H,PI C M,ZHENG X P,et al.Non-Newtonian gravity in strange quark stars and constraints from theobservations of PSR J0740+6620and GW170817[J丄TheAstrophysical Journal,2020,902(1)：32-3&「17]FARHI E,JAFFE R L.Strange matter E J].Physical Review D,1984,30(11)：2379-2390.[18]WEBER F.Strange quark matter and compact stars[J].Progress in Particle and Nuclear Physics»2005,54(1)；193-28&「19]ZYLA P A,BARNETT R M,BERINGER J,et al.,Review of particle physics[J/OL].Progress in Theoretical andExperimental Physics,2020(8)[2021-09-10].https://doi.org/10.1093/ptep/ptaal04.[20]FUJII Y.Dilaton and possible non-Newtonian gravity[J].Nature Physical Science,1971,234(44)：5-7.[21]ALCOCK C,FARHI E,OLINTO A.Strange starsEJl TheAstrophysical Journal,1986,310；261-272.[22]MADSEN J.Physics and astrophysics of strange quarkmatter[J].Lecture Notes in Physics,Berlin SpringerVerlag,1999,516:162-203.[23]SHAO L J.Degeneracy in studying the supranuclearequation of state and modified gravity with neutron starsEJ/OLD.AIP Conference Proceedings,2019,2127(1)[2021-09-10].https；///10.1063/1.5117806.[24]KAMYSHKOV Y,TITHOF J,VYSOTSKY M.Bounds onnew light particles from high-energy and very smallmomentum transfer np elastic scattering data[J/OL].Physical Review D,2008,78(11)[2021-09-10J.https：///10.1103/PhysRevD.78,114029.[25]XU J,LI B A,CHEN L W»et al.Nuclear constraints onnon-Newtonian gravity at femtometer scale[J/OL].Journalof Physics G,2013,40(3)[2021-09-10D.https：///10.1088/0954-3899/40/3/035107.[26]POKOTILOVSKI Y N.Constraints on new interactionsfrom neutron scattering experiments]〕].Physics of AtomicNuclei,2006,69(6)：924-391.[27]KLIMCHITSKAYA G L,KUUSK P,MOSTEPANENKOV M.Constraints on non-Newtonian gravity and axionlikeparticles from measuring the Casimir force in nanometerseparation rangeEJ/OL],Physical Review D,2020,101(5)[2021-09-101https：///10.1103/PhysRevD.101.056013.[28]KAMIYA Y,ITAGAKI K.TANI M,et al.Constraints onnew gravitylike forces in the nanometer range E J/OL J.Physical Review Letters,2015,114(16)[_2021-09-10D.https：///10.1103/PhysRevLett.114.161101. [29]CHEN Y J,TH A M W K,KRAUSE D E,et al.Strongerlimits on hypothetical Yukawa interactions in the30-8000nm range[J/OLH.Physical Review Letters,2016,116(22)[2021-09-10].https；///10.1103/PhysRevLett.116.221102.Non-Newtonian gravity in MIT strange quark starsPI Chunmei1,2(1.School of Physics and Mechanical&Electrical Engineering,Hubei University of Education,Wuhan430205,China；2.Research Center for Astronomy,Hubei University of Education,Wuhan430205,China)Abstract:The effects of non-Newtonian gravity on the properties of strange quark starsis investigated with MIT bag model.It is shown that the non-Newtonian contributedenergy density increases with increasing baryon density.It is also found that,for thestandard MIT bag model of strange quark matter,the inclusion of non-Newtoniangravity leads to stiffer EOSs with bigger parameters告and higher maximum masses ofcompact stars,when non-Newtonian gravity parameters is bigger than93GeV-2.Key words:strange quark stars；equation of state；non-Newtonian gravity。
一类动力学方程及流体力学方程解的Gevrey类正则性
Boltzmann 方程 . . . . . . . . . . . . . . . . . . . . . . . . 碰撞算子 Q(f, f ) 的基本性质 . . . . . . . . . . . . . . . . . Fokker-Planck 方程、Landau 方程以及 Boltzmann 方程线性 化模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Navier-Stokes 方程 . . . . . . . . . . . . . . . . . . . . . . . Gevrey 函数空间 . . . . . . . . . . . . . . . . . . . . . . . .
研究现状及本文主要结果 . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.2 1.2.3 1.2.4 存在性及唯一性 . . . . . . . . . . . . . . . . . . . . . . . . . 动力学方程的正则性理论: 空间齐次情形 . . . . . . . . . . . 动力学方程的正则性理论: 空间非齐次情形 . . . . . . . . . . Navier-Stokes 方程的正则性理论 . . . . . . . . . . . . . . .
第二章 预备知识 2.1 2.2 2.3 基本记号
Fourier 变换 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 基本函数空间及常用不等式 . . . . . . . . . . . . . . . . . . . . . . 2.3.1 2.3.2 Lp 空间及其性质 . . . . . . . . . . . . . . . . . . . . . . . . Sobolev 空间及其性质 . . . . . . . . . . . . . . . . . . . . .
群星脉冲星代码
群星脉冲星代码脉冲星（pulsar）是一种自然界中的奇特天体，在1983年获得了诺贝尔物理学奖。
它们是由质量较大的恒星在耗尽核心燃料后，通过引力坍缩而形成的。
脉冲星通常具有非常强烈的磁场，并且以极高的自转速度旋转，所以它们被称为“自转的磁星”。
下面是一个简单的群星脉冲星的代码示例：```pythonimport numpy as np#定义脉冲星类class Pulsar:def __init__(self, mass, radius, period):self.mass = mass # 质量（单位：太阳质量）self.radius = radius # 半径（单位：太阳半径）self.period = period # 自转周期（单位：秒）def calculate_pulse_period(self):#计算脉冲周期（单位：秒）G=6.67某10某某(-11)#万有引力常数c=3某10某某8#光速pulse_period = 2 某 np.pi 某 np.sqrt((self.radius某某3) / (G 某 self.mass)) / creturn pulse_perioddef display_info(self):#打印脉冲星的信息print("Mass: {} M☉".format(self.mass))print("Radius: {} R☉".format(self.radius))print("Period: {} seconds".format(self.period))print("Pulse Period: {}seconds".format(self.calculate_pulse_period())#创建一个脉冲星对象并打印信息pulsar = Pulsar(1.4, 10, 0.01)pulsar.display_info。
基于实测星载原子钟数据的脉冲星守时方法
omi
cc
l
o
ck,t
hepe
r
gned
f
o
rmanc
eo
ft
hepu
l
s
a
rt
imi
ngs
t
emi
ss
imu
l
a
t
edandana
l
z
ed
. Thes
imu
l
a
t
i
onr
e
su
l
t
sshowt
ha
t
ys
y
/30d,t
i
ft
hea
c
cu
r
a
c
ft
hepu
l
s
ea
r
r
i
va
lt
imeo
fapu
l
s
a
于存在较大频率漂 移 的 数 据,需 要 先 去 除 频 率 漂 移
钟)剔除粗差后的频率数据和钟差数据。
1
.2 原子钟钟差模型
多项式模型是 目 前 最 常 用 的 钟 差 模 型 之 一,对
于一组钟差序列 xi(
i =1,2,…,
N ),
xi 可 用 多 项
式模型表示为 [16]
n
xi =a0 +a1ti +a2ti2 + … +anti
设备老化等因素的 影 响,其 钟 差 和 频 率 数 据 中 可 能
存在粗差或中断 [1-2]。因此,在进行星载原子钟性能
评估前,需要对原 始 钟 差 数 据 进 行 预 处 理。 对 于 数
据中断的情 况,可 采 用 线 性 插 值 进 行 拟 合 [2]。对 于
超级计算机
¾ 超级计算机...............................................................................................................................1 超级计算环境 2007 年 3 季度运行情况简报 .........................................................................1
深腾6800
● 共197名用户，3季度增加用户6名。 ● 有134名用户利用LSF提交作业，共完成.51000多个作业，用户作业平均规模为5.9个CPU，累计 使用机时112万CPU小时（按Walltime计算）。 ● 2007年3季度，深腾6800的磁盘阵列系统与QsNet网络系统先后发生故障，导致深腾6800的平均 整体使用率有所下降，为83.5%（按Walltime计算），平均CPU利用率69.1%（按CPUtime计算）。CPUtim e与Walltime之比平均为82.7%。 ● 2007年3季度，作业平均等待时间为23.3小时。 ● 已完成作业按规模分布情况：串行作业数量占62.6%，4处理器节点内并行作业数量占21.1%。 而根据作业使用的CPU小时计算，占用机时最多的并行作业规模分别为16处理器、33-63处理器、32处理 器、64处理器，其比例分别为21.8%，18.1%，17.9%和12.4%，串行作业仅使用总机时的2.1%，表明深腾6 800的计算机时还是主要用于较大规模的并行作业计算。
1. Introduction ...................................................................................................................2
基于X射线脉冲星的深空探测器自主导航方案
生 垦窒 ！ 型 兰 壁 查 旦
！ 旦 生
任务 ，规划 出最有 利的 运行轨 道 ；并利用 轨道 动力学 的知 识获 得相应 的轨 道信息 ，建 立探测 器 的轨 迹预报 器 ，预报探 测器 的位置 、速 度导航 量 ；并 以此 轨道 动力学 方程 作 为系统 的状 态 方程 。然后 根
３ 自主导 航 算 法模 型
３ １ 状 态 方 程 ． 以 三 维 惯 性 空 间 中 的 位 置 和 速 度 建 立 ６维 的 系 统 状 态 量 ｘ，
Ｘ一 ［ Ｙ
状 态 方 程 的 一 般 形 式 为
： ］
（） １
Ｘ（ ）一 ／（ （ ） （ ） ）＋ ｒ ｔ ￡ ，“ ，ｆ Ｘ ｌ ） （
Ｘ ＮＡＶ应 用 于深空 探测 器的导 航 系统 方 案 ，将 其 作 为 一种 全 新 的 深空 探 测 导航 解 算 手段 ，具 有 重
要 的理 论 意义和工 程应 用价值 【 。 ５ ｊ
２ 基 于 Ｘ射 线 脉 冲星 的 自主导 航 方 案
ｘ射 线脉 冲星是 一种 能不 断地发 出稳定 且 有 周期 性 脉 冲 信号 的特 殊 天 体 ，通 过测量 单 个 ｘ 射
对航 天 器的轨 道进 行 自主估 计确 定 。论 文 阐述 了该 方案 的导航 原理 ，在借 鉴现 有航 天 器导
航 系统 的 基 础 上 ，提 出 了基 于 Ｘ 射 线 脉 冲 星 导 航 的 方 案 ， 介 绍 了 方 案 的 硬 件 组 成 、 系统
结构 ，并针 对方 案 中导 航敏 感 器 多 冗余 的 特征 ，给 出 了基 于 多传 感 器组 合 导航 技 术 的 结 构 ，最后 用仿 真 资料对 方案做 了性能概 算和 精 度 估计 。
- 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)。
arXiv:astro-ph/0105461v1 25 May 2001Mon.Not.R.Astron.Soc.000,1–10(2001)Printed1February2008(MNLATEXstylefilev1.4)SimulationsofGlitchesinIsolatedPulsarsMichelleB.LarsonandBennettLinkDepartmentofPhysics,MontanaStateUniversity,Bozeman,MT.59717
Accepted2001xxxxxxReceived2001xxxxxx;inoriginalform2001xxxxxxABSTRACTManyradiopulsarsexhibitglitcheswhereinthestar’sspinrateincreasesfractionallyby∼10−10−10−6.Glitchesareascribedtovariablecouplingbetweentheneutronstarcrustanditssuperfluidinterior.Withtheaimofdistinguishingamongdifferenttheo-reticalexplanationsfortheglitchphenomenon,westudytheresponseofaneutronstartotwotypesofperturbationstothevortexarraythatexistsinthesuperfluidinterior:1)thermalmotionofvorticespinnedtoinnercrustnuclei,initiatedbysuddenheatingofthecrust,(e.g.,astarquake),and2)mechanicalmotionofvortices,(e.g.,fromcrustcrackingbysuperfluidstresses).Bothmechanismsproduceacceptablefitstoglitchob-servationsinfourpulsars,withtheexceptionofthe1989glitchintheCrabpulsar,whichisbestfitbythethermalexcitationmodel.Thetwomodelsmakedifferentpredictionsforthegenerationofinternalheatandsubsequentenhancementofsur-faceemission.Themechanicalglitchmodelpredictsanegligibletemperatureincrease.Forapureandhighly-conductivecrust,thethermalglitchmodelpredictsasurfacetemperatureincreaseofasmuchas∼2%,occurringseveralweeksaftertheglitch.Ifthethermalconductivityofthecrustisloweredbyahighconcentrationofimpurities,however,thesurfacetemperatureincreasesby∼10%aboutadecadeafterathermalglitch.AthermalglitchinanimpurecrustisconsistentwiththesurfaceemissionlimitsfollowingtheJanuary2000glitchintheVelapulsar.Futuresurfaceemissionmeasurementscoordinatedwithradioobservationswillconstrainglitchmechanismsandtheconductivityofthecrust.
Keywords:stars:interiors—stars:neutron—stars:evolution—stars:rotation—superfluid—densematter
1INTRODUCTIONManypulsarsexhibitglitches,suddenjumpsinspinrate,superimposedonthegradualspindownduetoelectromagnetictorque(see,e.g.,Lyne,Shemar&Smith2000).Glitchesinvolvefractionaljumpsinspinrateof∆ν/ν≃10−10to10−6,withrecoverytothepre-glitchspin-downrateoccurringoverdaystomonthsinmostcases.Somepulsarsshownoobviousrecovery,andcontinuetospindownfasterthanhadtheglitchnotoccurred.The1989glitchoftheCrabpulsar(∆ν/ν≃7×10−8)waspartiallytime-resolved(Lyne,Smith&Pritchard1992).Thisglitchshowedaquickriseonatimescaleofhourswithadditionalspin-uptakingplaceoverapproximatelyoneday.Incontrast,theVela“Christmas”glitch(∆ν/ν≃2×10−6)observedinDecemberof1988(McCullochetal.1990)showedmuchdifferentbehaviour.Inthiscasetheglitchwasnottime-resolved,andoccurredinundertwominutes.TheJanuary2000glitchintheVelapulsar(∆ν/ν≃3×10−6)wassimilartotheChristmasglitch(Dodson,McCulloch&Costa2000).Anumberofpulsars(theCrabinparticular)exhibitpermanentincreasesinspin-downrateafteraglitchoccurs,typically∆˙Ω/˙Ω≃10−4.IntheCrab,theseoffsetsproducemuchlargercumulativetimingresidualsthantheglitchesthemselves.Inadditiontoglitches,nearlyallpulsarsexhibitlowlevelfluctuationsintheirspinrate,timingnoise,believedtobeofadifferentoriginthanglitches(seee.g.,D’Alessandroetal.1995).Glitchesarethoughttorepresentvariablecouplingbetweenthestellarcrustandthesuperfluidinterior.Twoquestionsconcerningtheglitchphenomenaare:1)whereinthestarthecouplingoccurs,and,2)howthecouplingistriggered.Therotationoftheneutronsuperfluidinteriorisgovernedbythedynamicsofvortexlines;aspinjumpofthecrustwouldresultfromsuddenmotionofvorticesawayfromtherotationaxis.Intheinnercrust,vorticescouldpintothelattice(Anderson&
c2001RAS2M.B.LarsonandB.LinkItoh1975;Alpar1977;Alpar,Langer&Sauls1984;Epstein&Baym1988),allowingthesuperfluidtostoreangularmomentumasthecrustspinsdownunderelectromagnetictorque.Asavelocitydifferencebetweenthesolidandthesuperfluiddevelops,vorticescreepthroughthecrustataratethatishighlysensitivetotemperature(Alpar1977;Alpar,Cheng&Pines1989;Link&Epstein1991;Link,Epstein&Baym1993;Chau&Cheng1993a;Chau&Cheng1993b).BasedonanideabyGreenstein(1979a,b),Link&Epstein(1996)haveproposedathermalglitchmechanisminwhichatemperatureperturbationcausesalargeincreaseinthevortexcreeprate;inconsequence,thesuperfluidquicklylosesangularmomentumanddeliversaspin-uptorquetothecrust.Acandidatemechanismforprovidingtherequiredheatisastarquakearisingfromrelaxationofcrustalstrainasthestarspinsdown.Starquakescoulddepositasmuchas∼1042ergsofheatinthecrust(Baym&Pines1971;Chengetal.1992).Ruderman(1991)hasproposedadifferentmodelinwhichvorticesstronglypinnedtotheinnercrustlatticestressthecrusttothepointoffracture,resultinginoutwardmotionofvorticeswithplatesofmattertowhichtheyarepinned.Inthecore,pinningmayoccurbetweenthevorticesandfluxtubesassociatedwiththesuperconductingprotonfluid(Chau,Cheng&Ding1992),allowingthecoresuperfluid,oraportionofit,tostoreangularmomentum.Ruderman,Zhu&Chen(1998)haveproposedacore-drivenglitchmechanisminwhichtheexpandingvortexarrayofthecoreforcesthemagneticfluxintothehighly-conductivecrust,stressingittofracture.Inthismodel,crustcrackingallowsthecorevortexarraytosuddenlyexpandoutward,spinningdownaportionofthecoresuperfluidandspinningupthecrust.Carter,Langlois&Sedrakian(2000)havesuggestedthatcentrifugalbuoyancyforcesaretheoriginofpressuregradientssufficienttocrackthecrust,allowingoutwardvortexmotion.Otherproposedglitchmechanismsincludecatastrophicunpinningofvorticesinthecrust(Chengetal.1988;Alpar&Pines1993;Mochizuki&Izuyama1995),andvortexmotionatthecrust-coreboundaryduetoprotonfluxtubeannihilationthere(Sedrakian&Cordes1999).Inanyofthesecrustorcore-drivenglitchmodels,dissipationthataccompaniesoutwardvortexmotiongeneratesheatthatmightproducedetectableemissionastheheatarrivesatthestellarsurface.Quantitativecalculationsarenecessarytodistinguishamongdifferentmodelsfortheglitchphenomenon.ThethermalglitchmodelofLink&Epstein(1996)producedgoodqualitativefitstoglitchobservationsintheCrabandVelapulsars.Thispaperisanextensionofthatworkwithmorerealisticphysicalinputsanddetailedmodelingofthetimingdata.Weincludenonlinearthermaldiffusionwhichhastheeffectofslowingtheglitchspin-up.Wealsoincludetheeffectsofsuperfluidheatingduetodifferentialrotationbetweenthesuperfluidandthecrust(Greenstein1975;Harding,Guyer&Greenstein1978;Alparetal.1987;Shibazaki&Lamb1989;VanRiper1991;VanRiper,Epstein&Miller1991;Umedaetal.1993;VanRiper,Link&Epstein1995;Larson&Link1999)andstudythepropagationofheattothestellarsurface.Weconsidertwotypesofrearrangementofthesuperfluidvortices:1)thermalexcitationofvorticesovertheirpinningbarriers(athermalglitch)and2)mechanicalmotionofvortices(amechanicalglitch).Thefirstcasemodelstheresponsetosuddenheatingofthecrust,e.g.,fromastarquake.Thesecondcasemodelscatastrophicunpinningevents,vortexmotionasaresultofcrustcrackingduetosuperfluidstresses,orcore-drivenglitchesinvolvingvortexmotionnearthecrust-coreboundary.Inadditiontosimulationsoftherotationaldynamics,wepredictthecharacteristicsoftheemergingthermalwavewhichcould,insomecases,bevisiblefromthesurfaceoftheneutronstarweekstoyearsafteraglitchoccurs.Thispaperisorganizedasfollows.InSection2weprovideanoverviewofthephysicalsettinganddiscussthetreatmentofthecoupledrotationalandthermaldynamicsweuseinoursimulationsofpulsarglitches.InSection3wediscussthedetailsofournumericalmodels.InSection4wepresentoursimulationsofthespin-upprocessandtheemergenceofthethermalwaveatthestellarsurface.WecompareoursimulationswithspinobservationsoffourpulsarsandsurfaceemissiondatafollowingarecentglitchintheVelapulsar.InSection5weconcludewithdiscussion.