aa r X i v : . [ a s t r o - ph . H E ] A ug Phase Evolution of the Crab Pulsar between Radio and X-ray
L.L. Yan , , , M.Y. Ge , J.P. Yuan , S.J. Zheng , F.J. Lu , Y. L. Tuo , , H. Tong , S.N.Zhang , Y. Lu , J.L. Han , Y.J. Du Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, ChineseAcademy of Sciences, Beijing 100049, China; [email protected] University of Chinese Academy of Sciences, Beijing 100049, China. Qian Xuesen Laboratory of Space Technology, No. 104, Youyi Road, Haidian District,Beijing 100094, China Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang830011, China National Astronomical Observatory, Chinese Academy of Sciences, Jia 20 Datun Road,Beijing 100012, ChinaReceived ; accepted 2 –
ABSTRACT
We study the X-ray phases of the Crab pulsar utilizing the 11-year obser-vations from the
Rossi X-ray Timing Explorer , 6-year radio observations fromthe Nanshan Telescope, and the ephemeris from Jodrell Bank Observatory. Itis found that the X-ray phases in different energy bands and the radio phasesfrom Nanshan Telescope show similar behaviors, including long-time evolutionand short-time variations. Such strong correlations between the X-ray and radiophases imply that the radio and X-ray timing noises are both generated fromthe pulsar spin that cannot be well described by the the monthly ephemeris fromthe Jodrell Bank observatory. When using the Nanshan phases as references tostudy the X-ray timing noise, it has a significantly smaller variation amplitudeand shows no long-time evolution, with a change rate of ( − . ± . × − periods per day. These results show that the distance of the X-ray and radioemission regions on the Crab pulsar has no detectable secular change, and it isunlikely that timing-noises resulted from any unique physical processes in theradio or X-ray emitting regions. The similar behaviors of the X-ray and radiotiming noises also imply that the variation of the interstellar medium is not theorigin of the Crab pulsar’s timing noises, which is consistent with the resultsobtained from the multi-frequency radio observations of PSR B1540 − Subject headings: stars:neutron — pulsars: individual (PSR B0531+21) — X-rays:stars
1. Introduction
Pulsars are famous for their rotation stability and highly repeatable pulse shapes.However, when examining pulsar’s periodicity with high precision, there appear to betwo main types of irregularities, namely glitch and timing noise. The origin of timingnoise remains controversial in spite of years of studies. Models proposed to explain thetiming noises of pulsars include the random process (Cordes and Helfand 1980), unmodeledplanetary companions (Cordes 1993), the free precession (Stairs et al. 2000), and theinterstellar medium (ISM; e.g. You et al. (2007)). Hobbs et al. (2010) analyzed the timingproperties of 366 pulsars in detail. Their study shows that timing noise is widespread inpulsars and cannot be explained using a simple random walk in the observed rotationalparameters. The timing residuals of PSR B1540 −
06 are consistent at different radiofrequencies, which implies that its timing noise is not caused by ISM (Hobbs et al. 2010).The underlying physical processes that cause timing noise are still unclear.Among all the pulsars, the Crab pulsar is probably the most suitable source to studythe origin of timing noise for its frequent spin irregularities and abundant observationaldata. This pulsar has been comprehensively studied in almost all wavelength bands fromradio to very high energy γ -rays. Its pulse profile shows a double-peak structure in all ofthese wavebands. Generally, the two dominant pulses in the radio band are denoted as mainpulse (MP) and interpulse (IP; Lyne et al. (2013)), and the two X-ray peaks are denotedas P1 and P2 (Kuiper et al. 2001). Detailed studies show that the exact pulse morphologyvaries as a function of photon energy (Abdo et al. 2010; Ge et al. 2012) and the positions ofthe main peak in different energy bands are not exactly aligned, i.e., the optical, X-ray and γ -ray pulses lead to the radio pulses (Kuiper et al. 2003; Rots et al. 2004; Oosterbroek et al.2008; Abdo et al. 2010; Molkov et al. 2010). Recently, secular changes of both the radioand the X-ray profiles were found, though their change rates are different from each other 4 –(Lyne et al. 2013; Ge et al. 2016). These changes were attributed to a progressive changein the magnetic inclination (Lyne et al. 2013; Ge et al. 2016), and such magnetic fieldvariations are also confirmed by another study on red timing noise (Yi and Zhang 2015).With seven years of observations, it was found that the X-ray pulse in 2–16 keV leads theradio one by 0 . ± . . ± . × − periodper day (Rots et al. 2004), which could also be explained as systematic errors. Given thesecular changes of the radio and X-ray profiles, to study the phase lags between differentenergy bands and their variations is important to uncover the origin of timing noise and theproperties of magnetosphere structure.In this paper, by using the 11-year observations from the Rossi X-ray TimingExplorer ( RXTE ), 6-year radio observations from Nanshan radio telescope at the XinjiangAstronomical Observatory (Wang et al. 2001), and the monthly renewed ephemeris fromthe Jodrell Bank Observatory (Lyne et al. 1993), we investigated in detail the timingbehaviors of this pulsar in the X-ray and radio wavebands. First, the phase comparisonsbetween the Proportional Counter Array (PCA) and the High Energy X-ray TimingExperiment (HEXTE) on board
RXTE are used to estimate the instrumental influences onthe phase determination. Then, the accuracy of the Jodrell Bank ephemeris is checked bythe correlation between the X-ray phases and the radio phases obtained by the Nanshanradio telescope. Furthermore, the X-ray phases are corrected by the new phase indicatorfrom Nanshan radio telescope to study the relationship between the X-ray and radio timingnoises, including the effects of the dispersion measure (DM). 5 –
2. Observations and Data Reduction2.1. Timing Ephemeris from Jodrell Bank
In this study, the time reference for the radio phases from Nanshan and X-ray phasesfrom
RXTE of the Crab pulsar is taken as the times-of-arrival (TOAs) from Jodrell Bankradio ephemeris (JBE; Lyne et al. (1993)). A 13 m radio telescope at Jodrell Bank monitorsthe Crab pulsar daily, offering a radio ephemeris that is used for the analyses of RXTE and Nanshan data. The ephemeris we used is in CGRO format, the format required by the
RXTE data processing, and it contains the following information: R.A. and decl. in J2000coordinates, the first and last dates for valid parameters, the infinite-frequency geocentricUTC TOA of a pulse, rotation frequency and its first two derivatives, the barycentric (TDB)epoch of the spin parameters, and the root-mean-square radio timing residual. Because ofthe uncertainties of the radio receiver system and the calibration, we add a systematic errorof 40 µ s for phases calculated from this ephemeris as suggested by Rots et al. (2004). Allerrors of the phases in this paper are 1 σ , for both statistical and systematic errors. RXTE
Observations and Data Reduction
The X-ray observations used in this paper were obtained by both PCA and HEXTE onboard the
RXTE . The PCA instrument is composed of five Proportional Counter Units witha total photon collection area of 6500 cm . Its effective energy range is 2–60 keV, and thetime resolution is about 1 µ s (in Good Xenon mode; Jahoda et al. (2006)). These propertiesmake PCA an ideal instrument to study the detailed temporal properties of pulsars. Inthis paper, we use the publicly available data in event mode E 250us 128M 0 1s. The time µ s. HEXTE consists of two independent detectorclusters A and B, and each of them contains four NaI(Tl)/CsI(Na) scintillation detectors.This instrument is sensitive in 15–250 keV, with a detection area of 1600 cm and a timeresolution of 7 . µ s (Rothschild et al. 1998). In its default operation mode, the field of viewof HEXTE, each cluster is switched on and off source to provide instantaneous backgroundmeasurements. The HEXTE data used in this paper are in mode E 8us 256 DX0F.The PCA and HEXTE data were analyzed by using the FTOOL from the astronomysoftware HEASOFT (v6.15). The method of data reduction and pulse profile calculationis the same as in Ge et al. (2016), but with the observations selected a little differently.Only 243 observations between MJD 51955 (2001 February 15) and 55927 (2012 January01) are used in this paper, since from each of these observations high statistical PCA andHEXTE pulse profiles can be obtained simultaneously. The pulse profiles were binned into1000 phase bins, in energy bands 2–60 keV for PCA and 15–250 keV for HEXTE. The Nanshan 25 m radio telescope, operated by Xinjiang Astronomical Observatory,started to observe the Crab pulsar frequently in 2000 January. As described in Wang et al.(2001), the two hands of circular polarization at 1540 MHz are fed through a 2 × × .
3. Phase Calculation and the Linear Fitting Method3.1. Phase Calculation for
RXTE
As described in Ge et al. (2012), the two asymmetrical pulses of the Crab pulsar in theX-ray band could be modeled by the formula (1) proposed by Nelson et al. (1970). In thispaper, we only fitted the shape of P1 of the X-ray profile to obtain its peak phase. L ( φ − φ ) = N a ( φ − φ ) + b ( φ − φ ) c ( φ − φ ) + d ( φ − φ ) e − h ∗ ( φ − φ ) + l, (1)where L is the intensity at phase φ , l is the baseline of the light curve, φ is the phase shift, N is the pulse height of the profile, and a , b , c , d and h the shape coefficients. The pulsephase is measured in phase units, of range (0,1). We fitted the shape of P1 with a relativelybroad phase window (–0.055, 0.0355) centered at phase –0.01.The calculation procedures of phases of P1 and the estimation of their statistical errorsare the same as in Ge et al. (2012). The X-ray phases with high precision are obtained afterthe fitting procedure, and we denote the phases of the X-ray pulse P1 as Φ P , Φ H for PCAand HEXTE, respectively. 8 – The calculation procedures of the radio phase of MP for Nanshan (Φ N ) are as follows:(1) Remove the dispersion effect for TOAs using the DM values from JBE (Lyne et al.1993). In this step, we need to reckon the DM values in Nanshan observations using linearinterpolation. The time delay ( t DM ) caused by DM is t DM = D × DM ν , (2)where D is the dispersion constant, D = 4 . × MHz pc − cm s, and ν is the centeredfrequency, i.e. 1540 MHz (Lyne and Graham-Smith 2012). Because the JBE DM valueswere not obtained in the same time as the Nanshan observations, the DM at the time ofNanshan observations were obtained with linear interpolation. (2) Convert the TOAs fromJodrell Bank and Nanshan to the TDB time system. (3) Calculate φ J for Jodrell Bank andΦ N relative to φ J with formula (3) and (4), respectively. φ J = f ( T J − T ) + 12 f ( T J − T ) + 16 f ( T J − T ) , (3)Φ N = mod [ f ( T N − T ) + 12 f ( T N − T ) + 16 f ( T N − T ) − φ J , , (4)where T J and T N are the TOAs in the TDB time system, from Jodrell Bank and Nanshan,respectively, f , f and f are the spin parameters at the reference epoch T , and modis to obtain the residual after removing the integral periods. The final errors of Φ N arefrom the errors of Nanshan TOAs and the systematic uncertainty of 40 µ s as mentionedpreviously. In order to investigate the effect of DM on the timing noises, we also calculatethe Nanshan phases without de-dispersion by skipping the first step above, and denote itas Φ N . Because of the process “mod”, i.e. removing the integral periods, the difference ofΦ N and Φ N is smaller than one period (see Fig. 4). 9 – In order to study the phase variations versus time and the correlations between phasesfrom different data sets, we fit the data points with a linear function. For the variation ofa parameter versus time, if the slope deviates significantly from zero, long-term evolutionsshould exist. For the correlations between two parameters, the slope can also tell usinformation about how these two parameters are correlated, as we will discuss in section 4.In this paper, the fitting method is the robust linear modeling (RLM) from the Rstatistical software package (Feigelson and Babu 2012), which has been used to study boththe phase variations versus time and the correlations between different phases. The
MASS (Modern Applied Statistics with S) library based on R-language has the rlm function forRLM. In this function, the fitting is achieved using an iteratively reweighted least-squarealgorithm. Similarly, the linear fitting for the phase correlations between different datagroups is also achieved by this method, as listed in Table 1 and Table 2.
The Pearson’s correlation coefficient r (Lee Rodgers and Nicewander 1988) is a suitableparameter to describe the influence of the timing noise on Φ P , Φ H and Φ N quantitatively.Because the Nanshan and RXTE observations were not done simultaneously, and the timeseries of Φ N is serially dependent when checking the autocorrelation function, the Nanshanphases at the time of X-ray observations (Φ ′ N ) are computed by linear interpolation betweenthe neighboring Φ N values. 10 –
4. Results
The X-ray phases Φ P and Φ H are the phases of the X-ray main peaks relative to JBE,from the PCA and HEXTE data respectively. In order to study the relation of the radioand X-ray phases on both long and short time scales, we need to check the accuracy andreliability of these phases first. As shown in Fig. 1, the X-ray phases Φ P and Φ H exhibit simultaneous variations onall time scales. Previous work showed that the X-ray phases from PCA gradually increasewith a change rate of (3 . ± . × − period per day (MJD 50129–52941, Rots et al.(2004)) or (6 . ± . × − period per day (MJD 51955–55142, Ge et al. (2012)). Herewe analyze more observations, in a longer time range MJD 51955–55927, and both PCAand HEXTE showed the same trend with the change rates of (5 . ± . × − and(4 . ± . × − period per day, respectively. Besides the increasing trends, Φ P and Φ H have two kinds of variations on short time scales, which are the slow variations (e.g. inMJD 52600–53000 and 55000–55200) and phase jumps (three points around MJD 53350,corresponding to the JBE in one month).Correlation coefficient is calculated to estimate the degree of correlation between Φ P and Φ H . As shown in Fig. 2 and listed in Tab. 2, Φ H is almost proportional to Φ P , witha slope of 0 . ± .
02 and the Pearson’s coefficient r = 0 .
96, which means that they varysynchronously with the same amplitude. The synchronous variations between Φ P and Φ H imply that they have the same origin. 11 – The Nanshan radio phases Φ N are also obtained by using the same JBE, and they showvariations in different time scales too, as illustrated in Fig. 1e and Fig. 3a. Compared withΦ P in the same time range, Φ N shows similar fluctuations, especially in MJD 55000–55200as in the zoomed in Fig. 3b. For the secular change, Φ N increases linearly with a changerate of (6 . ± . × − period per day in MJD 53500–55688, which is consistent with thechange rate Φ P , (4 . ± . × − period per day in the same time range. These two changerates were obtained in a relatively short period are also consistent with the results obtainedfrom the whole time range for Φ P and Φ H . As shown by the cross marks in Fig. 2, Φ ′ N andΦ P exhibit a strong linear correlation, with the Pearson’s coefficient r = 0 .
78 (Tab. 2) anda slope of 0 . ± .
04. The strong correlation between Φ ′ N and Φ P means that Φ N and Φ P also have a strong correlation.The fitted slope between Φ N and Φ P (0 . ± .
04) is different from 1, the expectedvalues of Φ N and Φ P have the same variation amplitude and an exactly linear correlation.One may think that there is some physics behind this. Nonetheless, we realized that thisresult probably originated from the data handling process. Since the X-ray and Nanshanradio observations are carried out in different times, we obtained the Nanshan phases at thetime of X-ray observations with linear interpolation (Φ ′ N ), so as to study their correlation.This linear interpolation will reduce the amplitudes of the radio timing noises, and thelarger the amplitude is, the bigger the fraction of the variation that will be reduced. As aresult of this linear interpolation process, the slope can be smaller than 1.If the JBE can describe the spin of the Crab pulsar accurately, Φ N should be constantover time, because Φ N is also inferred from the radio data. However, as given above, Φ N hasa secular change with a significance of nearly 6.3 σ , and both its long-time and short-timevariations are similar to the X-ray ones that are also derived from the JBE. It is very likely 12 –that the temporal behaviors of the Crab pulsar cannot be accurately described by thosespin parameters in the JBE, which also causes the apparent variation of the X-ray to radiophase lags. The simultaneous secular changes and fluctuations of the X-ray and Nanshan radiophases imply that they may be caused by the timing noise of the Crab pulsar or theinaccuracies in the JBE. In order to further check the relations between the X-ray andradio phases, here we use the Nanshan phases Φ ′ N as the phase references, and obtain thecorrected X-ray phases Φ ′ P from PCA data, which has a lower variation amplitude (i.e.standard deviation) 0.0013 compared to 0.0020 of Φ P as shown in Fig. 3c. Moreover,Φ ′ P keeps almost constant over the time range MJD 53500–55693 with a change rate of( − . ± . × − period per day. The disappeared secular change of the new X-ray phasessuggests that the JBE is inaccurate.
5. Origin of the Phase Variation and Timing Noise
There are several factors that can result in the variabilities of the observed X-rayphases: the instability of the time system, the change of the instrument response, the timingnoise of the pulsar, the inaccuracies of the ephemeris, and the intrinsic variation of theX-ray emitting region relative to the radio ones. The effects of these factors are discussedin the following. 13 –
The Mission Operations Center of
RXTE performs clock calibrations several timesa day, using the User Spacecraft Clock Calibration System method, and the timingaccuracy was improved from 4.4 to 2.5 µ s on 1997 April 29 (Jahoda et al. 2006). Besides,the instrumental delay correction for the PCA is 16–20 µ s and for the HEXTE it is 0–1 µ s. The barycenter corrections by FTOOL has an accuracy of better than 1 µ s and hasalso subtracted 16 µ s to account for the instrumental delay in the PCA. Therefore, themaximum timing uncertainties is √ . + 1 + 4 = 6 . µ s, which has a much smaller impacton the timing measurement for X-ray photons than the 40 µ s systematic error from JBE(Rots et al. 2004).If the time systems of RXTE and the Nanshan telescope were inaccurate, wrongtiming recorders would be assigned and thus would cause the abnormal phases. For PCAand HEXTE, the consistent variations might have been caused by the irregularity of thetime system because they use the same time information from the satellite . However,considering that Φ P and Φ N have very similar variations and they are based on twoindependent time systems, the inaccuracy of the time systems cannot account for theobserved phase fluctuations.The aging of detectors would also have impacts on the timing recorders. As the X-rayphase lag of the Crab pulsar evolves with energy (Molkov et al. 2010; Ge et al. 2012), thephase lag will change if the detection efficiency curve varies due to the instrument agingor other factors (Garcia et al. 2014). However, the change of the response functions of theX-ray instrument cannot explain the correlation between Φ P and Φ N , because the detectors http://heasarc.gsfc.nasa.gov/docs/xte/abc/time.html http://heasarc.nasa.gov/docs/xte/time news.html 14 –are totally different. The inaccuracy in the ephemeris have direct impacts on the phase calculations for theX-ray and Nanshan phases. The effect of ISM, pulsar proper motion, glitches, as well astiming noises could all generate inaccurate parameters.
Because of the existence of ISM, the arrival time of radio pulses is dependent onfrequency. Both the mismeasurement of DM and the scattering of ISM have an impact onradio observations and phase results.During MJD 55050–55350, the variation amplitudes of DM are larger than in theother time periods, which is apparently consistent with the questionable points in this timerange. The phase change caused by DM could be obtained using Φ N -Φ N , and its impacton the Nanshan phase could be evaluated by comparing it with Φ N . However, as shownin Fig. 4, the phase change caused by DM is smaller than the phase fluctuation in bothX-ray and Nanshan phases in MJD 55050–55350, as the DM effect has been removed in theJBE (Lyne et al. 1993). So, the large fluctuation in the X-ray and Nanshan phases in MJD55050–55350 could not be explained by the DM effects.As for the scattering of ISM, its influence on TOA could be evaluate by the followingformula (Lyne and Graham-Smith 2012). t scatt = ( DM1000 ) . ( 400 ν MHz ) (5)For the Crab pulsar, DM = 56 .
78 pc cm − , and when ν MHz = 1540 MHz (Nanshan radio 15 –band), t scatt ≃ . µ s. In the X-ray band t scatt would be much smaller. It is thus clear thatthe influence of ISM scatting on the Nanshan and X-ray phases could be disregarded. With accurate measurement by the Hubble Space Telescope, the proper motion ofthe Crab pulsar has been obtained as µ α = -11.8 mas yr − for R.A. and µ δ = 4.4 mas yr − for decl. (Kaplan et al. 2008). However, the JBE uses a constant position for the Crabpulsar, and the influence of proper motion on the relative phases should be considered.If the pulsar position is changing, the timing residuals should show oscillations withgradually increasing amplitude (Helfand et al. 1977), which could be roughly described by∆Φ pm = h · sin α · ∆ θ/c · f , where h is the distance between the Earth and the Sun, c isthe speed of light, and α is the decl. of the Crab pulsar. Taking into account the propermotion, the maximum value of the amplitude is 0.0035 periods for 10 years. However, theimpact of proper motion is counteracted when the time is longer than one month, becausethe X-ray and Nanshan phases are the relative phases to JBE that were updated monthly.We check the power spectrum of these relative phases to see whether variation power existson the time scale of about a month, which could be the impact of pulsar proper motion onthe relative phases, and eventually no significant signals in the power spectrum of Φ P andΦ H have been found. Thus, the proper motion is not the main reason for the long-termvariation of the X-ray and Nanshan phases. Similarly, the inaccuracy of solar systemephemeris could not account for the long-term variation of the X-ray phases. 16 – Inaccurate spin parameters could lead to phase deviation. There are some outliersin X-ray and Nanshan phases, especially the X-ray phases around MJD 53350, which areobtained by using one ephemeris. These results remind us to check the reliability of the JBEparameters, from the aspects of glitches, rotation frequencies, and radio reference TOAs.Because the parameters of glitches have not been included in the JBE directly(Lyne et al. 1993), we need to check whether the significant residuals are caused by theglitches. However, we find that the obviously abnormal phases are not coincident with theglitch epoches. The effects of glitches after their occurrence month are greatly reduced. InMJD 53341-53372, there is no glitch, so the outliers in this period did not resulted fromglitches.Furthermore, we check the parameters of JBE in MJD 53341–53372 in case the JBEparameters are inaccurate. First, we compare the rotation frequencies inferred from JBEand those we searched from the three
RXTE observations in this period. The maximaldifference between them is (0 . ± . × − Hz for PCA and (2 . ± . × − Hz forHEXTE, where the uncertainties are only from our frequency searching process, whichshow that the frequencies we calculated from these three observations are consistent withthe JBE predictions. Second, we calculate the TOAs of these three X-ray observationsusing the frequencies we obtained above and the software TEMPO2 (Edwards et al. 2006;Hobbs et al. 2006), and then compare them with the TOAs inferred from the JBE timesolution and the FTOOL command FASEBIN. As shown in Fig. 1, TOAs obtained withthe above two methods are consistent. Therefore, both the JBE frequencies and the TOAcalculation process are reliable, and the remaining possibility is that the reference TOA ofJBE in MJD 53341–53372 is inaccurate, which causes the abnormal lags between the X-rayand radio phases. We note that the inaccuracy of JBE TOAs has also been found by the 17 –Jodrell Bank Observatory, as on the web page of JBE it is pointed out, “DO NOT trust thegeocentric pulse arrival times yet!” . It is important for pulsar physics to find out whether the fluctuations and long-termevolutions of the X-ray phases are intrinsic, i.e., due to the relative geometric variationsbetween the X-ray and radio emitting regions. We find that there are two observationalfacts contradicting with this hypothesis. As shown in Fig 2, the Nanshan radio phases arehighly correlated with the X-ray phases derived from the PCA observations, which meansthat the two phases wander simultaneously. Furthermore, by using the Nanshan phases asa reference, the X-ray phases then have a smaller fluctuation amplitude and the long-termevolution disappears (Fig. 3c), which also implies that the X-ray and radio emitting regionsdo not have significant relative changes. Thus, the X-ray and radio phase fluctuations areboth dominated by the pulsar spin.
The almost constant phase-lag between the X-ray and radio bands also suppliesinformation about the origins of timing noises. Variations of the DM and thus the variationsof ISM between the earth and pulsars have been detected, which can lead to the radiotiming noises of those pulsars (You et al. 2007, and references there in). However, becausethe ISM has no effect on the X-ray TOAs, the constant value of the X-ray phase Φ ′ P means −
06 obtained from the multi-frequency radio observations (Hobbs et al. 2010).Thus ISM variation cannot account for all the timing noises of the Crab pulsar.
6. Summary
Utilizing the 11-year X-ray observations from the RXTE, 6-year radio observationsfrom Nanshan Telescope, and the ephemeris from the Jodrell Bank Observatory, we studythe evolution of the X-ray and radio phases of the Crab Pulsar. The X-ray phases fromPCA and HEXTE exhibit synchronous variations on all time scales, and X-ray and Nanshanphases also have a strong correlation with a Pearson’s coefficient r = 0 .
78. We find thatthe simultaneous secular changes and fluctuations of the X-ray phases Φ P , Φ H and theNanshan phases Φ N are quite possibly caused by the unreliable reference TOAs in the JBEparameters. Using the Nanshan phases as timing reference, the corrected X-ray phases Φ ′ P show lower variation amplitude and remain almost constant over time with a change rate of( − . ± . × − period per day.Based on the results above, we conclude that the distance of the X-ray and radioemission regions on the Crab pulsar does not show detectable secular changes, and thetiming noises are not the result of any unique physical processes in the radio or X-rayradiation regions. In addition, the variation of the ISM is not the origin of Crab pulsar’stiming noises, which is consistent with the results obtained from the multi-frequency radioobservations of PSR B1540 −
06 (Hobbs et al. 2010). 19 –
Acknowledgments
We appreciate Dr. Michael Smith, Lorenzo Natalucci, Craig Markwardt, YuanyuePan, Liming Song, Jinlu Qu, Li Chen, and Jian Li for their useful suggestions. Thiswork is supported by the National Key Research and Development Program of China(2016YFA0400802), National Science Foundation of China (11233001, 11503027, and11303069), the Strategic Priority Research Program on Space Science, and the ChineseAcademy of Sciences, grant No. XDA04010300 and XDB23000000. This work is alsopartially supported by the Trainee Program of Qian Xuesen Laboratory of Space Technology.We thank the High Energy Astrophysics Science Archive Research Center (HEASARC)at NASA/Goddard Space Flight Center for maintaining its online archive service thatprovided the data used in this research. 20 –
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RXTE −0.015 −0.010 −0.005 0.000 0.005Φ P (period)−0.015−0.010−0.0050.0000.0050.010 Φ H , Φ ′ N ( p e r i o d ) Φ ′N vs. Φ P Φ H vs. Φ P Fig. 2.— Correlations between the X-ray and interpolated Nanshan phases. The dotsrepresent the correlation of X-ray phases from PCA and HEXTE. The cross marks representthe correlation of X-ray phases from PCA and interpolated Nanshan phases in MJD 53500–55693. The solid line is the fitting result of dots, while the dash line is the fitting result ofcross marks. The typical errors are plotted to clearly show the correlations. 24 – Φ P , Φ N ( p e r i o d ) (a) 54800 55000 55200 55400 55600-0.01-0.000.01 (b)53500 54000 54500 55000 55500Epoch(MJD)-0.015-0.010-0.005 Φ ′ P ( p e r i o d ) (c) Fig. 3.— X-ray, Nanshan phases and corrected X-ray phases for PCA in MJD 53500–55693.Panel (a) shows the variations of X-ray phases from PCA (black dot points) and radio phasesfrom Nanshan (red square points). The oblique solid lines are the fitting results. The inset(b) shows the X-ray and Nanshan phases in MJD 54700–55700 as marked with the gray beltin panel (a). The X-ray phases are shifted upward with 0.007 to compare them more clearly.The Nanshan phases are averaged in a time window of 1.5 days to show them tersely. Panel(c) shows the corrected X-ray phases from PCA. 25 – P ha s e ( pe r i od ) Φ N0 Φ N0 − Φ N Fig. 4.— The Nanshan phases without de-dispersion and the phase differences betweenthem and the Nanshan phase. Black points: the Nanshan phases without de-dispersion.Red points: the phase difference between Nanshan phases with and without de-dispersion.Φ N is computed by using the steps (1)-(3) in Section 3.2, and in order to investigate theeffect of DM, Φ N is computed by skipping the step (1). 26 –Table 1: The change rates of X-Ray and radio phases Instrument MJD Energy Band Change Rate Intercept a (10 − period/day) (10 − period)PCA 51955–55927 2–60keV 5 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . b − . ± . − . ± . c − . ± . . ± . d . ± . e . ± . Notes. a These intercepts correspond to the values at MJD 54000. b The parameters for X-ray phases from PCA corrected by data of the Nanshan Telescope. c The parameters for the phase lags between HEXTE and PCA. d The result from Rots et al. (2004). e The result from Ge et al. (2012).
Table 2: The correlation coefficients of the X-Ray and radio phases
Instrument versus Instrument MJD Slope r PCA versus HEXTE 51955-55927 0 . ± .
02 0 . . ± .
04 0 ..