Discovery of Delayed Spin-up Behavior Following Two Large Glitches in the Crab Pulsar, and the Statistics of Such Processes

Glitches are typical events of pulsars, observed as sudden jumps in rotational frequency (ν) and spin-down rate ($\dot{\nu }$), usually followed by a recovery stage, in which ν and its derivative $\dot{\nu }$ recover gradually to the extrapolated values of the pre-glitch evolution trend. The behavior of the spin frequency post-glitch could be described by polynomial components and several exponential processes (as described in Equation (1)), such as $\sum {\rm{\Delta }}{\nu }_{\mathrm{di}}\exp (-t/{\tau }_{{\rm{i}}})$, where ν_di and τ_i are amplitude and timescale of the ith component. Most of these exponential processes are positive values of ν_di (called "normal recovery processes"), however some negative ones are observed in the Crab pulsar (Lyne et al. 1992; Wong et al. 2001; Shaw et al. 2018a; Zhan et al. 2018). Here, the exponential process with negative ν_di is called the delayed spin-up process, as defined in Shaw et al. (2018a). The delayed spin-up process may dominate the evolution of ν and $\dot{\nu }$ immediately after the occurrence of a glitch, but is much more difficult to be detected; probably due to that, this process has a much shorter timescale than the recovery process (McCulloch et al. 1990; Dodson et al. 2002; Palfreyman et al. 2018). Vela pulsar is very famous for its large glitches in which some of them have been continuously observed, but besides the ordinary recovery processes, only upper limits of 12.6 s–2 minutes have been obtained for the rising timescale of these glitches, before the recovery starts. No delayed spin-up process has been detected in the Vela pulsar (McCulloch et al. 1990; Dodson et al. 2002; Palfreyman et al. 2018; Ashton et al. 2019). The Crab pulsar is another important object for pulsar glitch study, from which 26 glitches have been detected so far (Espinoza et al. 2011; Wang et al. 2012; Lyne et al. 2015; Shaw et al. 2018a, 2018b). Compared to Vela pulsar, Crab pulsar has two unique features, though its glitch amplitudes are usually smaller than those of Vela pulsar. The first feature is that its Δν and ${\rm{\Delta }}{\dot{\nu }}_{p}$ values are positively and linearly correlated (Lyne et al. 2015; Shaw et al. 2018a). The second feature is that delayed spin-up processes have been observed in its large glitches with time scales of 0.5–3.0 days, such as the glitches of 1989, 1996, and 2017 (Lyne et al. 1992; Wong et al. 2001; Shaw et al. 2018a; Zhan et al. 2018).

Presently, there are mainly two trigger mechanisms for pulsar glitches. One is the star quake model, in which the (outer) crystalline crust of a neutron star (NS) would break as strain in the crust gradually accumulates due to the spin-down of the NS and finally surpasses its maximum sustainable strain. Sudden rearrangement of the stellar moment of inertia caused by the star quake would result in a glitch (Ruderman 1969). The other mechanism invokes neutron superfluidity in a NS, which is expected when the internal temperature of star drops below the critical temperature for neutron pairing. The superfluid neutrons rotate by forming quantized vortices, which can get pinned to nuclei in the outer crust. Once the pinned vortices are released suddenly, glitches are the result of angular momentum transfer between the inner superfluid and the outer crust (Anderson & Itoh 1975; Alpar et al. 1984b). After the glitch, the superfluid vortices would move outwards because of losing angular momentum and subsequently be repinned to the outer crust. The superfluid vortex model has its advantage in understanding pulsar glitches, especially for the post-glitch recovery process (Baym et al. 1969). In addition, it should be noted that sudden crust breaking may also trigger vortex unpinning avalanches (Alpar et al. 1993). The delayed spin-up behaviors observed in some glitches of the Crab pulsar may not be well explained based on a simple star quake or superfluid vortex model, since the timescale for crust breaking and plate motion or unpinned vortices to move radially outward is less than a minute, which is hard to account for in the presence of a 2 day delayed spin-up process (Graber et al. 2018). One possible scenario for the delayed spin-up process might be that it is the initially induced inward motion of some vortex lines pinned to broken crustal plates moving inward toward the rotation axis (Gügercinõglu & Alpar 2019). Other possible scenarios are (1) the excess heating due to a quake in a hot crust induces secular vortex movement (Greenstein 1979; Link & Epstein 1996), or (2) the mutual friction strength in a strongly pinned crustal superfluid region changes due to the propagation of the unpinned vortex front (Haskell et al. 2018).

The delayed spin-up processes in glitches thus carry rich information on how the glitches progress, and thus offer valuable probes to the inner structure of NSs (Haskell & Antonopoulou 2014; Haskell et al. 2018). Given the small number of known spin-up processes, any new event of this kind will add precious knowledge about glitches and the physics behind. The common feature for the three glitches with delayed spin-up processes happened in 1989, 1996, and 2017 is that their Δν is large, compared to the known glitches of the Crab pulsar. We therefore selected two large glitches in 2004 and 2011, performed detailed analyses about their timing behaviors, and found that they do contain delayed spin-up processes. We name them with G1, G2, G3, G4, and G5, corresponding to the events in 1989, 1996, 2004, 2011, and 2017, respectively. In order to describe different components conveniently, the full spin evolution is divided into four components: the rapid initial spin-up process of the frequency (C1), the delayed spin-up process (C2), exponential decay processes (C3), and the permanent change of the frequency and its derivatives (C4) (Lyne et al. 1992; Wong et al. 2001; Shaw et al. 2018a; Zhan et al. 2018), which dominate the glitch behavior in different stages accordingly. Based on the parameters of these five glitches, we have also carried out a statistical study of the spin-up processes.

This paper is organized as follows. The observations and data reduction are described in Section 2, and the timing analysis results are presented in Section 3. Section 4 includes the physical implications of these results and the main conclusions.

The temporal analyses of these glitches use all the radio, X-ray, and Gamma-ray observations we can access. We use the RXTE, International Gamma-Ray Astrophysics Laboratory (INTEGRAL) and the Nanshan 25 m radio telescope observations, together with the spin frequency and its derivative from radio format of Jodrell Bank¹⁶ (Lyne et al. 1993), to analyze the timing behaviors of G3, and due to the low cadence, only the Fermi Large Area Telescope/Gamma-ray Burst Monitor (Fermi-LAT/GBM) observations are used for the analyses of G4. For G5, the observations from Insight-HXMT, Fermi-LAT/GBM, the Nanshan 25 m radio telescope, and the Kunming 40 m (KM40) radio telescope are used to study the behaviors of G5. We cite the parameters for G1 and G2 from Wong et al. (2001). In order to perform timing analysis, the arrival time is corrected to Solar System Barycentre (SSB) with solar system ephemerides DE405 using the pulsar position of α = 05^h31^m31972 and δ = 22°00'52069 (Lyne et al. 1993). In this section, we first describe the data reduction for observations. Then, the calculation for time of arrival (TOA) and its error are presented. Finally, the description of the timing method for the glitches is given.

2.1. Data Reduction for Radio Observations

2.2. Data Reduction of X-Ray and γ-Ray Observations

In this section, we introduce the data reduction processes of the RXTE, INTEGRAL, Insight-HXMT, and Fermi observations, respectively.

2.2.1. Data Reduction of the RXTE Observations

The RXTE observations used in this paper were obtained by both the Proportional Counter Array (PCA) and the High Energy X-ray Timing Experiment (HEXTE). The detailed introduction of PCA and HEXTE can be found in Rothschild et al. (1998), Jahoda et al. (2006), and Yan et al. (2017). In this paper, the public data (ObsID P80802 and P90802) in event mode E_250us_128M_0_1s in 5–60 keV from PCA and E_8us_256_DX0F in 15–250 keV from HEXTE are used. The Standard RXTE data processing method with HEASOFT (ver 6.25 ) is used to obtain the timing data (i.e., the arrival time of each photon used in the analyses) as follows: (1) generate the Good Time Interval by ftool maketime based on the RXTE filter file; (2) filter the events with the grosstimefilt tools; and (3) convert the arrival time of each photon to the SSB with faxbary. The criteria of the selection and the detailed process can be found in Yan et al. (2017). The TOA for RXTE is integrated from the typical observation.

2.2.2. Data Reduction for INTEGRAL

The INTEGRAL observations of the Crab pulsar are subdivided into the so-called Science Windows (ScWs), each with a typical duration of a few kiloseconds (Winkler et al. 2012). By selecting offset angles to the source of less than 10°, between 2014-03-01 and 2014-04-01, 96 public ScWs are selected for Crab in the data archive at the INTEGRAL Scientific Data Center. The data reduction is performed using the standard Off-line Scientific Analysis, version 10.2. The integration time of TOA for INTEGRAL is about 1 hr.

2.2.3. Data Reduction for Insight-HXMT

Launched on 2017 June 15, Insight-HXMT was originally proposed in the 1990s, based on the Direct Demodulation method (Li & Wu 1993, 1994). As the first X-ray astronomical satellite of China, Insight-HXMT carries three main payloads on board (Zhang et al. 2014, 2020; Cao et al. 2020; Chen et al. 2020; Liu et al. 2020): the High Energy X-ray telescope (HE, 20–250 keV, 5100 cm²), the Medium Energy X-ray telescope (ME, 5–30 keV, 952 cm²), and the Low Energy X-ray telescope (LE, 1–15 keV, 384 cm²). The data reduction for the Crab observations is done with HXMTDAS software v1.0 and the data processes are described in Chen et al. (2018), Huang et al. (2018), and Tuo et al. (2019). One TOA is obtained from the typical observation.

2.2.4. Data Reduction for Fermi-LAT/GBM

The LAT is the main instrument of Fermi, which can detect γ-rays in the energy range from 20 MeV to 300 GeV and has an effective area of ∼8000 cm². It consists of a high-resolution converter tracker, a CsI(Tl) crystal calorimeter, and an anti-coincidence detector, which make the directional measurement, energy measurement for γ-rays, and background discrimination, respectively (Atwood et al. 2009).

In this work, we use the LAT data to perform timing analysis with the Fermi Science Tools (v10r0p5).¹⁷ The events are selected with the angular distance less than 1° of the Crab pulsar and a zenith angle of less than 105° and energy range 0.1–10 GeV (Abdo et al. 2010). After event selection, the arrival time of each event is corrected to SSB with DE405. One TOA is obtained from every two-day exposure.

We also utilize the GBM data around the glitch epoch to refine the timing results. Due to the large field of view (∼2π) of GBM and its relatively high count rate (∼30 cnts s⁻¹) of the Crab pulsar, GBM can also be used to monitor the Crab pulsar continuously like LAT and even has higher cadence as shown in Figure 1. Given the periodicity of the pulse signals, they could be detected when the pulsar is in the field of view of GBM, though the overall background is high due to the large field of view of GBM. As the volume of GBM data is very large, we only select one month of data around G4 and G5, which cover 10 days before and 20 days after each glitch epoch. The events with elevation angle greater than 5° are used to perform timing analysis. Then, one TOA can be accumulated for every 10 minute observation.

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2.3. TOA Calculation for X-Ray and γ-Ray Observations

2.4. Timing Analysis

2.4.1. Part-timing Analysis

We apply the part-timing method to show the spin evolution versus time as described in Ferdman et al. (2015). In order to show spin evolution directly, we divide the data set into several subsets for each glitch. For G3, the time step for ν and $\dot{\nu }$ by TEMPO2 (Hobbs et al. 2006) is about 15 days without data overlapping due to the low cadence of the observations. With high cadence of the observations for G4 and G5, the time steps of the data set for G4–5 are chosen as 1.5 and 0.5 days, respectively. For $\dot{\nu }$, the time steps of the data set for G4–5 are chosen as two times ν, and the overlapping time is set as equal to the time step in order to show more data points in the figures. For each subset, we have taken the center of the time span as the reference epoch for the timing analysis.

2.4.2. Coherent Timing Analysis

A coherent timing analysis of the data set is performed, in order to obtain more precise measurements of the glitch parameters using TEMPO2. The phase evolution of the glitch could be described as Equation (1) considering the glitch parameters (Wong et al. 2001).

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }} & = & {{\rm{\Phi }}}_{0}+\nu {\left(t-{t}_{0})+\displaystyle \frac{1}{2}\dot{\nu }(t-{t}_{0}\right)}^{2}\\ & & +\,\displaystyle \frac{1}{6}\ddot{\nu }{(t-{t}_{0})}^{3}+{{\rm{\Phi }}}_{{\rm{g}}}(t),\end{array}\end{eqnarray} \tag{ 1 }$$

where ν, $\dot{\nu }$, and $\ddot{\nu }$ are the spin parameters at the epoch t₀. Φ_g(t) is the phase description after the glitch as defined in Equation (2).

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{{\rm{g}}}(t) & = & {\rm{\Delta }}{\nu }_{{\rm{p}}}{\rm{\Delta }}t+\displaystyle \frac{1}{2}{\rm{\Delta }}{\dot{\nu }}_{{\rm{p}}}{\rm{\Delta }}{t}^{2}\\ & & +\,\displaystyle \sum _{i=0,1,2}{\rm{\Delta }}{\nu }_{\mathrm{di}}{\tau }_{{\rm{i}}}(1-\exp (-{\rm{\Delta }}t{/\tau }_{{\rm{i}}})),\end{array}\end{eqnarray} \tag{ 2 }$$

where ${\rm{\Delta }}t=t-{t}_{{\rm{g}}}$ is the time after glitch, Δν_p and ${\rm{\Delta }}{\dot{\nu }}_{{\rm{p}}}$ are the permanent changes of ν and $\dot{\nu }$ for C4, τ₀, τ₁ and τ₂ are the time scales for C1, C2 and C3, Δν_d0, Δν_d1, and Δν_d2 are the amplitudes of the three components. i = 1 refers to the delayed spin-up process, and i = 2 refers to the conventionally observed exponential recovering process. In the following timing analysis, the effect of C1 is neglected as its timescale is too short, which will be analyzed in Section 3.2.

In order to obtain the net evolution of a glitch, we subtract the pre-glitch spin-down trend and then fit the frequency residuals δν with Equation (3) (Lyne et al. 1992; Wong et al. 2001; Xie & Zhang 2013), which consists of a linear function and two exponential functions,

$$\begin{eqnarray}&&\delta \nu ={\rm{\Delta }}{\nu }_{{\rm{p}}}+{\rm{\Delta }}{\dot{\nu }}_{{\rm{p}}}{\rm{\Delta }}t+\sum _{i=1,2}{\rm{\Delta }}{\nu }_{\mathrm{di}}\exp (-{\rm{\Delta }}t{/\tau }_{{\rm{i}}}),\end{eqnarray} \tag{ 3 }$$

where the parameters have the same definition as in Equation (2).

The coherent timing analysis for different instruments is performed simultaneously using parameter "JUMP" to describe the time lags between different energy bands because the peak position of the Crab pulsar evolves with energy as reported in Kuiper et al. (2003), Molkov et al. (2010), and Ge et al. (2012). Setting the position of the radio peak as phase 0, the values of JUMP are −0.340 ms (Insight-HXMT/RXTE/GBM), −0.275 ms (INTEGRAL), −0.250 ms (LAT), compared to radio band, respectively.

The residual $\delta \dot{\nu }$ can be described by

$$\begin{eqnarray}&&\delta \dot{\nu }={\rm{\Delta }}{\dot{\nu }}_{{\rm{p}}}+\sum _{i=1,2}{\rm{\Delta }}{\dot{\nu }}_{\mathrm{di}}\exp (-{\rm{\Delta }}t{/\tau }_{{\rm{i}}}),\end{eqnarray} \tag{ 4 }$$

where ${\rm{\Delta }}{\dot{\nu }}_{\mathrm{di}}=-{\rm{\Delta }}{\nu }_{\mathrm{di}}/{\tau }_{{\rm{i}}}$. The total frequency and frequency derivative changes at the time of the glitch are ${\rm{\Delta }}{\nu }_{{\rm{g}}}\,={\rm{\Delta }}{\nu }_{{\rm{p}}}+{\rm{\Delta }}{\nu }_{{\rm{d}}1}+{\rm{\Delta }}{\nu }_{{\rm{d}}2}$ and ${\rm{\Delta }}{\dot{\nu }}_{{\rm{g}}}={\rm{\Delta }}{\dot{\nu }}_{{\rm{p}}}+{\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}1}+{\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}2}$, respectively; and the degree of recovery can be described by parameters: $\hat{Q}={\rm{\Delta }}{\nu }_{{\rm{d}}2}/({\rm{\Delta }}{\nu }_{{\rm{g}}}+| {\rm{\Delta }}{\nu }_{{\rm{d}}1}| )$, as suggested by Wong et al. (2001).

3.1. The Timing Results for G3–5

We first analyze G3, the second largest glitch, by using the coherent timing method. The timing residuals are shown in Figure 2(a) and the timing parameters are listed in Table 1. The parameters of C3 are consistent with the result from Wang et al. (2012). After subtraction of pre-glitch evolution, the residuals δν and $\delta \dot{\nu }$ are plotted in Figures 3(a) and (b). Due to the observational coverage, no spin-up process has been detected for δν and marginally for $\delta \dot{\nu }$. However, the observational data can not be acceptably fitted without C2 with reduced χ² 1.3 (dof = 54), which means that the delayed spin-up process is needed. Fitting the data with both C2 and C3 gives the timescale τ₁ of G3 as 1.7 ± 0.8 days and ${\rm{\Delta }}{\nu }_{{\rm{d}}1}=-0.35\pm 0.05$ μHz for the delayed spin-up process. We note here that the delayed spin-up process of G3 could be quantified in more detail by using data such as the daily radio monitoring observation at Jodrell Bank observatory.

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Table 1.  Parameters of the Crab Pulsar for G1–G5

Parameters	G1a	G2a	G3	G4	G5
Epoch(MJD)	⋯	⋯	53067	55867	58038
ν(Hz)	⋯	⋯	29.796943484(8)	29.706916048(2)	29.6375626144(3)
$\dot{\nu }$(10−10 Hz s−1)	⋯	⋯	−3.73308(5)	−3.70743(3)	−3.686433(3)
$\ddot{\nu }$(10−20 Hz s−2)	⋯	⋯	1.3(2)	0.96(17)	0.81(3)
Glitch epoch (MJD)	47767.4	50259.93	53067.0780b	55875.67(1)	58064.548(2)
Δνp (μHz)	2.38(2)	0.31(3)	0.93(2)	0.49(3)	5.7(9)
${\rm{\Delta }}{\dot{\nu }}_{p}$ (pHz s−1)	−0.155(2)	−0.083(6)	−0.19(1)	−0.09(1)	−0.483(8)
${\rm{\Delta }}{\ddot{\nu }}_{p}$ (10−20 Hz s−2)	⋯	0.09(6)	⋯	⋯	⋯
Δνd1 (μHz)	−0.7	−0.31	−0.35(5)	−0.43(5)	−1.23(1)
τ1 ( day)	0.8	0.5	1.7(8)	1.6(4)	2.56(4)
Δνd2 (μHz)	2.28	0.66	5.67(4)	1.26(3)	9.91(9)
τ2 ( day)	18	10.3	24(1)	10.6(3)	45.9(3)
Residuals (μs)	⋯	⋯	214	311	113
χ2/dof (dof)	⋯	⋯	0.99(52)	1.17(873)	1.35(1269)

Notes.

^aThe parameters are obtained form Wong et al. (2001). ^bGlitch epoch is adopted from http://www.atnf.csiro.au/people/pulsar/psrcat/glitchTbl.html. The confidence interval is 68.3%.

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G4 is also analyzed using the coherent timing method. The timing residuals are shown in Figure 2(b) and the timing parameters are listed in Table 1. As shown in Figures 3(c) and (d) for G4, after subtraction of the pre-glitch evolution, the spin frequency residual δν increases first and then decreases with time, which is just the feature of the delayed spin-up process. From the coherent timing analysis, we can obtain that τ₁ = 1.6 ± 0.4 days with Δν_d1 = −0.43 ± 0.05 μHz. With the fitted parameters, $\delta \dot{\nu }$ can also be described by Equation (4) with the same parameters as shown in Figure 3(d).

G5 is re-analyzed using the coherent timing method as well. The timing residuals are show in Figure 2(c) and the timing parameters are listed in Table 1. As shown in Figures 3(e) and (f), the evolution of frequency residual δν is consistent with the results reported in Shaw et al. 2018a and Zhan et al. 2018. From the fitting result, the timescale τ₁ for the delayed spin-up process is 2.56 ± 0.04 days and Δν_d1 = −1.23 ± 0.01 μHz, which are also consistent with the result of Shaw et al. (2018a) and Zhan et al. (2018). The rest of parameters are listed in Table 1.

Our analysis shows that G3 and G4 also have delayed spin-up process. Including G1, G2, and G5, there are five glitches with delayed spin-up process. From Table 1, the mean timescale τ₁ of C2 is ∼1.4 days while the mean amplitude Δν_d1 of C2 is around −0.6 μHz (Lyne et al. 1992; Wong et al. 2001; Shaw et al. 2018a; Zhan et al. 2018)

3.2. The Rising Time Constraint of C1

The rising timescale of C1 is very important to study the pinning process between the inner superfluid and outer crust (Haskell et al. 2018). We make use of the observations from Fermi-GBM to constrain the rising time of C1 for G5. Unfortunately, the Crab pulsar were occulted by earth at the right time for G5 in Fermi-GBM observation. In order to constrain the rising timescale of C1 of G5, Equation (5) is used to describe its frequency evolution (Haskell et al. 2018).

$$\begin{eqnarray}&&\delta \nu ={\rm{\Delta }}{\nu }_{0}(1-\exp (-{\rm{\Delta }}t/{\tau }_{0})),\end{eqnarray} \tag{ 5 }$$

where Δν₀ is the amplitude of the frequency jump, τ₀ is the rising timescale, and ${\rm{\Delta }}t=t-{t}_{{\rm{g}}}$ is the time after glitch. As shown in Figures 4(a) and (b), δν could be fitted with Equation (5). As shown in Figure 4(b), the rising timescale τ₀ is less than 0.0202 day (0.48 hr), which is much less than the upper limit of 6 hr given by Shaw et al. (2018a) but still longer than the theoretical value of 0.1 hr suggested by Graber et al. (2018) and Haskell et al. (2018). For G3, the rising timescale could not be constrained because no high cadence observations could be obtained in high energy bands and radio bands from Nanshan 25 m radio telescope around the glitch epoch. For G4, the errors of δν is close to the frequency step with short integrated time 10 minutes.

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3.3. The Correlations Between the Parameters of C2 and Other Components

We first compare the relationship between G1–5 and the other glitches of the Crab pulsar, to see how these five glitches differ from the other ones. The most conventional comparison is to study the jump amplitudes of their frequencies and frequency derivatives. As shown in Figure 5, the Pearson coefficient between $| {\rm{\Delta }}{\dot{\nu }}_{{\rm{g}}}|$ and Δν_g is 0.81. Hence, $| {\rm{\Delta }}{\dot{\nu }}_{{\rm{g}}}|$ and Δν_g show strong linear correlation for all the glitches of the Crab pulsar, including those with delayed spin-up processes. We also compare the correlation between $| {\rm{\Delta }}{\dot{\nu }}_{{\rm{p}}}|$ and Δν_g, which is similar to Figure 5 in Lyne et al. (2015). The value of $| {\rm{\Delta }}{\dot{\nu }}_{p}|$ is obtained from Wong et al. (2001), Wang et al. (2012), and this work because the calculation process in Lyne et al. (2015) is different from the rest ones. As shown in Figure 5, $| {\rm{\Delta }}{\dot{\nu }}_{{\rm{p}}}|$ has strong linear correlation with Δν_g as the Pearson coefficient is 0.98.

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As shown in Figure 6, the Δν_g and $| {\rm{\Delta }}{\dot{\nu }}_{{\rm{g}}}|$ values for the five glitches with delayed spin-up process locate in the higher wing of the overall distribution of all the glitches and are not well separated from those with the rest glitches. This unified positive correlation suggests that the physical mechanism of the five glitches with delayed spin-up processes is probably the same as that of all the other glitches, and it is worth checking the archival data whether the glitches occurred in 1975, 2000, 2001, and 2006 also have delayed spin-up processes, as they have amplitudes comparable to those of G2.

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To understand more characteristics for G1–5, we examine the Pearson and Spearman correlations between their parameters as listed in Tables 2, 3 and plotted in Figures 7, 8. The relationships between τ₁, $| {\rm{\Delta }}{\nu }_{{\rm{d}}1}|$, τ₂, Δν_d2, Δν_p, $| {\rm{\Delta }}{\dot{\nu }}_{{\rm{p}}}|$, and Δν_g have positive correlations as shown in Figures 7 and 8, some of which are consistent with the result of Wang & Zheng (2019). These positive correlations mean that C2 has a larger amplitude and longer timescale when a glitch has a larger spin frequency jump. If C2 also exists for the smaller glitches, from the positive correlation between τ₁ and τ₂, τ₁ should be less than 0.5 days if τ₂ < 10 days, which indicates that C2 might not be easily observed due to the low cadence of most previous observations.

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Table 2.  Pearson Correlations Between the Parameters for G1–G5

${\rm{R}}$	τ1	$| {\rm{\Delta }}{\nu }_{{\rm{d}}1}|$	${\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}1}$	τ2	Δνd2	$| {\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}2}|$	Δνp	${\rm{\Delta }}{\dot{\nu }}_{p}$	Δνg	$\hat{Q}$
τ1	1	0.67	−0.55	0.83	0.87	0.80	0.84	−0.82	0.83	−0.45
$| {\rm{\Delta }}{\nu }_{{\rm{d}}1}|$	⋯	1	0.24	0.87	0.78	0.44	0.88	−0.90	0.90	−0.42
${\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}1}$	⋯	⋯	1	−0.10	−0.25	−0.50	−0.14	0.10	−0.10	0.22
τ2	⋯	⋯	⋯	1	0.98	0.76	0.95	−0.99	0.99	−0.43
$| {\rm{\Delta }}{\nu }_{{\rm{d}}2}|$	⋯	⋯	⋯	⋯	1	0.86	0.92	−0.96	0.97	−0.4
${\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}2}$	⋯	⋯	⋯	⋯	⋯	1	0.60	−0.69	0.74	0.02
Δνp	⋯	⋯	⋯	⋯	⋯	⋯	1	−0.98	0.95	−0.67
${\rm{\Delta }}{\dot{\nu }}_{p}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1	−0.99	0.52
Δνg	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1	−0.42
$\hat{Q}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1

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Table 3.  Spearman Correlations Between the Parameters for G1–G5

ρ	τ1	$| {\rm{\Delta }}{\nu }_{{\rm{d}}1}|$	${\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}1}$	τ2	Δνd2	$| {\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}2}|$	Δνp	${\rm{\Delta }}{\dot{\nu }}_{p}$	Δνg	$\hat{Q}$
τ1	1	0.6	−0.6	0.9	0.9	0.8	0.9	−0.7	0.9	−0.3
$| {\rm{\Delta }}{\nu }_{{\rm{d}}1}|$	⋯	1	0.2	0.7	0.7	0.4	0.3	−0.5	0.7	−0.1
${\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}1}$	⋯	⋯	1	−0.3	−0.3	−0.5	−0.7	0.1	−0.3	0.1
τ2	⋯	⋯	⋯	1	1.0	0.9	0.7	−0.9	1.0	−0.1
$| {\rm{\Delta }}{\nu }_{{\rm{d}}2}|$	⋯	⋯	⋯	⋯	1	0.9	0.7	−0.9	1.0	−0.1
${\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}2}$	⋯	⋯	⋯	⋯	⋯	1	0.6	−0.8	0.9	0.2
Δνp	⋯	⋯	⋯	⋯	⋯	⋯	1	−0.6	0.7	−0.6
${\rm{\Delta }}{\dot{\nu }}_{p}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1	−0.9	0.2
Δνg	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1	−0.1
$\hat{Q}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1

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We also find that the correlations between ${\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}1}$ and $| {\rm{\Delta }}{\dot{\nu }}_{{\rm{d}}2}|$, $| {\rm{\Delta }}{\dot{\nu }}_{{\rm{p}}}|$, τ₁, τ₂, $\hat{Q}$ are weak listed in Tables 2 and 2.

It is generally believed that a NS has the following interior structure: the outer crust made by degenerated electrons and an ion crystal lattice; the inner crust composed of nucleus, superfluid neutrons, probably superfluid protons and leptons; the outer core that contains superfluid neutrons, superfluid protons and electrons; and the inner core (Anderson & Itoh 1975; Alpar et al. 1984b). The angular momentum transfer from the inner superfluid component to the outer normal component can explain the observed frequency jumps (glitches) of pulsars (Anderson & Itoh 1975; Alpar et al. 1981). The response to the glitch of the thermal vortex creep process in the pinned superfluids are suggested to be responsible for the post-glitch behaviors (Alpar et al. 1984a, 1984b; Larson & Link 2002). A quick rise of the spin rate in crust, resulting from the initial energy deposition, could be followed by a slower rise due to the thermal wave dissipation in the effective crust with thickness 200 m, depending on the crust equation of state (Link & Epstein 1996; Larson & Link 2002). Another possible scenario is that vortex accumulation in strong pinning regions leads to differential rotation and the propagation of vortex fronts, which naturally produces a slower component of the rise after the initial fast step in frequency jump (Haskell & Antonopoulou 2014; Haskell et al. 2018; Khomenko & Haskell 2018). Recently, the combination of crust-quake vortex and unpinning models is proposed to explain the whole glitch behavior as suggested by Gügercinõglu & Alpar (2019). Haskell et al. (2018) estimated the rising timescale of rapid initial spin-up of the largest glitch G5 is ∼0.1 hr, which is consistent with upper limit of 0.48 hr for G5. We hope that the positive correlations between the amplitudes and time scales of C2 and C3 can be also used to constrain the properties of NS structures.

Figure 5 shows the relation between Δν and $| {\rm{\Delta }}\dot{\nu }|$ for the Crab pulsar (Espinoza et al. 2011). The Crab pulsar, PSR J0537−6910, and the Vela pulsar have relatively large glitches as characterized by both the large Δν and $| {\rm{\Delta }}\dot{\nu }|$ values. However, the glitch properties of the Crab pulsar are very different from those of PSR J0537−6910 and the Vela pulsar. Δν and $| {\rm{\Delta }}\dot{\nu }|$ of Crab's glitches have a strong positive correlation (Lyne et al. 2015), in contrast to the other two pulsars without such correlation (Espinoza et al. 2011; Antonopoulou et al. 2018). Five spin-up events are found for the Crab pulsar; however, no similar spin-up phenomenon has been reported for either PSR J0537–6910 or the Vela pulsar. The glitch around MJD 57,734 from the Vela pulsar is observed with the rising timescale less than 12.6 s (Ashton et al. 2019) and does not show any evidence for delayed spin-up process. Given the different ages of these three pulsars, we speculate that the states of their crusts and interiors are different, and so the conditions of the physical processes involved in glitches are different for pulsars with different ages.

In summary, in this work we have studied the glitches of the Crab pulsar, which have delayed spin-up processes. First, in addition to the three glitches that occurred in 1989, 1996, and 2017, we also found that the second and fourth largest glitches of the Crab pulsar detected in 2004 and 2011 have delayed spin-up processes, with the second and fourth largest glitches detected in 2004 and 2011 are analyzed and these two glitches are found also with delayed spin-up processes of τ₁ = 1.7 ± 0.8 days and τ₁ = 1.6 ± 0.4 days, respectively. Using observations from Insight-HXMT, Radio Telescopes in Xinjiang and Kunming China and Fermi, we studied the largest glitch in 2017 and obtained similar results with Shaw et al. (2018a) and Zhan et al. (2018), and further constrained its rising time to less than 0.48 hr.

We obtained the correlations among the parameters of the delayed spin-up processes and the parameters of the exponential decay processes: the amplitudes of the delayed spin-up ($| {\rm{\Delta }}{\nu }_{{\rm{d}}1}|$) and the recovery process (Δν_d2), their respective time scales (τ₁, τ₂), and permanent changes of spin frequency (Δν_p) have strong positive correlations, while the rest parameters do not show any correlation with each other. From the positive correlations, we suggest that further analysis of the existing data for smaller glitches are needed to search for any evidence of delayed spin-up processes with possibly shorter spin-up time scales, and more high cadence observations of the Crab pulsar in the future are also critical in understanding delayed spin-up processes and the interior structure of NSs.

This work is supported by the National Key R&D Program of China (2016YFA0400800) and the National Natural Science Foundation of China under grants U1838201, U1838202, U1938109, U1838104, and U1838101. This work made use of the data from the HXMT mission, a project funded by China National Space Administration (CNSA) and the Chinese Academy of Sciences (CAS). This work also made use of the radio observations from Yunnan Observatory. The Nanshan 25 m radio telescope is jointly operated and administrated by Xinjiang Astronomical Observatory and Center for Astronomical Mega-Science, Chinese Academy of Sciences. We acknowledge the use of the public data from the RXTE, INTEGRAL, and Fermi data archive.