The Relationships between Emission Geometry and Jitter Noise in Millisecond Pulsars

The relationships between several parameters of an emission geometry and jitter noise in 17 millisecond pulsars (MSPs) are investigated. By assuming the jitter noise is due only to a pulse variation in phase, the former can be modeled as changes in the plasma flow rate leading to variation in the measured pulse arrival time relative to the predicted time. In the model for pulsar magnetospheres with multiple emission states, the plasma flow is associated with the emission states, and a change in the emission state corresponds to a change in the plasma flow causing variation in the pulse arrival time. These can be specified in an emission geometry defined by the obliquity and viewing angles, measured from the rotation axis to the magnetic axis and to the line of sight, respectively. We calculate the maximum change in the emission state based on the reported jitter noise for each of the MSPs. Using the results, we show that the MSPs possess relatively large obliquity angles, which is consistent with observation, and the jitter noise exhibits dependency on frequency. We find that the jitter noise in our sample displays an exponential decay as a function that combines the obliquity angle and the rotation period, revealing the correlation among the three parameters. This suggests that the magnitude of the jitter noise is likely specific to an MSP. We discuss how jitter noise may be related to the evolution of an MSP.


Introduction
In an ideal world, pulse signals from a pulsar would arrive at only one particular and exact frequency, which is also stable in time.However, they often contain "noise," which exists in the form of fluctuations in the frequency and amplitude.This introduces an uncertainty in the measurement of the pulse arrival times (ToAs) in the form of timing variations in the single-pulse signals.Known as jitter noise, it is present in every millisecond pulsar (MSP; Cognard & Backer 2004;McKee et al. 2016).MSPs are renowned for their high spin frequency and unique formation history.In addition, their steadiness in rotation and emission properties make them ideal for use in applications that require a high degree of stability in those properties (Cordes et al. 2004;Kramer 2004;Demorest et al. 2010;Antoniadis et al. 2013).This includes the identification of low-mass primordial black holes (Seto & Cooray 2007) and the possible detection of gravitational waves (Hobbs 2008;Hobbs & Dai 2017;Arzoumanian et al. 2018).The latter also relies on the accumulation of long data sets with highly accurate ToA data from arrays of MSPs, such that a reliable analysis can be performed to reveal the presence of weak gravitational-wave signals.It is then apparent that the presence of jitter noise poses a significant limitation on such detections as well as the performance of any accurate systems of timing using MSPs.Since the noise becomes more prominent as the signal-to-noise ratio increases, and it scales roughly inversely to the square root of the number of the individual pulses (Rathnasree & Rankin 1995), an effective means to combat the noise is by increasing the observation time in order to obtain a stable averaged profile.However, little is known about the relationships between jitter noise and the pulsar parameters and how a similar radio emission mechanism can give rise to the different levels of jitter noise in different MSPs.
Pulse jitter arises from variation in the single-pulse emission, which manifests as changes in (i) the pulse shape and amplitude and (ii) the arrival phase in a stream of single pulses.Point (i) involves pulsar electrodynamics and generation of the radio emission.Models for pulsar radio generation are usually proposed assuming a corotating plasma-filled magnetosphere with small regions of charge starvation known as the vacuum gap (Ruderman & Sutherland 1975).Charge starvation in a corotating magnetosphere arose in models for an aligned rotator, first introduced by Goldreich & Julian (1969), and subsequent development extended the aligned scenario to oblique rotators by assuming corotation (Fawley et al. 1977;Scharlemann et al. 1978).The model is used for analysis of a wide range of pulsar data (Deshpande & Rankin 1999, 2001;Gil & Sendyk 2003;Bhattacharyya et al. 2007;van Leeuwen & Timokhin 2012;Szary & van Leeuwen 2017).There are also different suggestions for the gap location (Bai & Spitkovsky 2010;Yuki & Shibata 2012) and even alternative models for pair production without a gap (Coroniti 1990;Wada & Shibata 2007;Philippov & Spitkovsky 2014), meaning that at present, there is no clear identification of the specific radio emission mechanism (Melrose & Yuen 2016).Point (ii) relates to the pulse phase jitter, which is suggested to be dominated by the rotation rate of the magnetosphere and its fluctuation (Cordes & Shannon 2010).The variation can reach up to an amount equivalent to the width of an individual pulse in phase (McLaughlin et al. 2002).Observations (Deshpande & Rankin 1999;Gil et al. 2003;Janssen & van Leeuwen 2004;Esamdin et al. 2005;Mitra & Rankin 2008) suggested that pulsar emission originates from discrete regions of the favored emission location that are flowing relative to corotation.In addition, it is widely accepted that the flow rate of the emitting plasma is determined by the electric drift velocity v dr = E × B/B 2 , or ω dr = v dr /r when expressed in drift frequency at a radial distance r.Assuming a uniform distribution of m discrete emission regions around the magnetic pole, and treating the emission regions as fixed to the plasma, the flow rate of the emission regions is given by ω dr .This implies that the uncertainty due to pulse jitter arises from the continuing variation in the motion of the radio emission regions around a magnetosphere of oblique rotation.It follows that the detected arrival phase of the pulses is a reflection of the rotation rate of the emitting plasma.
The idea of emission from discrete regions that flows at a rate defined by the electric drift of the magnetospheric plasma is also useful for explaining another phenomenon known as subpulse drifting.This common phenomenon appears as systematic movement of subpulses (discrete regions) across the profile window (Drake & Craft 1968;Manchester & Taylor 1977;Weltevrede et al. 2006Weltevrede et al. , 2007)).The deviation in the plasma flow from corotation results in the observable emission regions drifting around the magnetic axis.This implies that the difference between subpulse drifting and jitter noise lies in the stability of such deviation.For drifting subpulses in a pulsar, the deviation is generally consistent in both magnitude and direction, resulting in a stable and repeating pattern of drift bands (Drake & Craft 1968;Rankin 1986).Such consistency allows quantifications of the phenomenon to be made in terms of two parameters described by P 2 , which measures the interval between successive drift bands, and P 3 , which denotes the vertical spacing between two consecutive drift bands (Manchester & Taylor 1977).From the observations of changes in subpulse drifting between different drift rates (e.g., Smits et al. 2005;Yan et al. 2023;Xu et al. 2024), it is evident that sudden changes can occur in a pulsar magnetosphere on different timescales and with different magnitudes.The cause of these changes is not likely due to the electric drift but is probably from a deeper mechanism (Timokhin 2010).The mechanism alters the plasma flow and manifests in the electric drift as different deviations.It is suggested that the characteristic time of change due to the mechanism may be random and specific to the pulsar (Timokhin 2010).For some pulsars, the drift is more consistent, giving rise to drifting subpulses, while for others, it may be more irregular, giving rise to jitter noise.In jitter noise, the deviation fluctuates with varying magnitudes and directions, giving rise to different "shifts" of the pulse profile from its predicted location every pulsar rotation.As mentioned above, such deviation is likely random (Parthasarathy et al. 2021), and hence the associated jitter noise would appear timeuncorrelated.Since v dr varies with r, if assuming radius-tofrequency mapping, the jitter noise of an MSP is expected to show dependency on observing frequency.
The goal of this paper is to examine the properties of jitter noise as revealed by MSPs and their connections with several pulsar parameters.Recognizing such links allows predictions to be made for the jitter noise in an MSP and an estimation of its overall impact on a specific application.There have been proposals that jitter noise is related to several characteristics of an MSP, such as the profile width and the rotation period.Furthermore, variation in the pulse phase is likely related to the obliquity angle, α, and the viewing angle, ζ, of the MSP assuming such variation is dominated by the plasma flow in the magnetosphere (Cordes & Shannon 2010).However, the relationships between jitter noise and the two angles are uncertain.As we show in Section 2, the plasma flow rate is related to the emission state in a pulsar magnetosphere that possesses multiple emission states.In this model, changes in the plasma flow rate correspond to changes in the emission state that can be associated with the variation in pulse phase.The inferred changes in the emission state from the jitter noise may be used to infer several properties of the MSP in relation to ζ and α.Recently, Parthasarathy et al. (2021) used MeerKAT to measure the jitter noise in 29 MSPs, with the results achieved at the nanosecond level.For our investigation, we will focus on the results from that work in order to obtain more quantitative results on the yet-unclear roles of the emission geometry in the observed level of jitter noise in different MSPs.Only MSPs with measured jitter noise values and quoted uncertainty are selected.For simplicity, we attribute the jitter noise as variation in the pulse arrival phase, or jitter in the pulse phase, without considering changes in the pulse shape.We assume that the reported values represent the discrepancy of a pulse from the ideal position or the maximum deviation in the plasma flow from corotation.The yetunidentified link between jitter noise and the radio emission mechanism, with the latter being responsible for the formation of each individual pulse and the unstable shape, implies that there may be unknown factors missing from the traditional models.In addition, the large differences in the jitter noise level across different MSPs suggest that the properties of these factors are likely MSP-specific and distinctive to jitter noise.Since the relationships between these factors and jitter noise are still uncertain, before taking such components into account, it is useful to inspect the implications of the idealized model.As pulsar radio emission is suggested to be coming from lower heights in the magnetosphere, we also assume that the magnetic field lines around the radio emission region can be approximated by a dipolar structure.Recent modeling of the X-ray waveform from MSP PSR J0030+0451 using the Neutron Star Interior Composition Explorer (NICER; Gendreau et al. 2016) suggests a mulitpolar magnetic field structure (Bilous et al. 2019).Later studies show that a consistent emission geometry can also be obtained from a joint fitting using both γ-ray and radio light curves (Benli et al. 2021;Pétri et al. 2023).This supports the idea that an almost dipolar magnetic field can apply equally well to MSPs, even if a multipolar field may have effects on the surface emission.
The paper is organized as follows.In Section 2, we describe the model for jitter in phase.Here, we outline a model that relates the change in ω dr to a change of the emission state in an obliquely rotating pulsar magnetosphere and the conditions for detection of such changes.Section 3 gives the details of the samples used in our investigation, and the results of the simulation are given in Section 4. Certain properties of the jitter noise as revealed by its correlation with several pulsar parameters are examined in Section 5.The limitations in our modeling are also addressed in this section.We discuss and summarize the paper in Section 6.The electromagnetic fields related to this paper are described in Appendix A, and the matrices needed for transforming between different coordinate systems employed in the emission geometry are given in Appendices B and C, respectively.

Modeling of Phase Jitter Noise
An increasing number of observations reveal that sudden changes in some pulsar magnetospheres can alter the observed radio emission features (e.g., Smits et al. 2005;Kramer et al. 2006;Lyne et al. 2017), with the timescale and the characteristics of change being different in different pulsars.Since the observed features are strongly linked to emission properties in the emission region (Ruderman & Sutherland 1975), these observations suggest that multiple emission states, each with different emission properties, are allowed in a pulsar magnetosphere.A pulsar can switch abruptly between different emission states, resulting in changes in the observed emission features.In this section, we outline the model for pulsar magnetospheres of multiple emission states (Melrose & Yuen 2014) and a change in the emission state corresponding to a change in the plasma flow rate in relation to phase jitter noise.An important feature of the model, as compared to an aligned model, is the incorporation of obliquity and the presence of an inductive field (Cruz et al. 2021;Tolman et al. 2022).

Changes in the Plasma Flow
Our model is based on an interpolation of the plasma motions between two cases of (i) the vacuum-dipole model and (ii) the corotation for the pulsar magnetosphere.There is no plasma in (i), and the inductive electric field, E ind , is subjected to the rotating magnetic dipole (see Appendix A).In this scenario, any test charge has an electric drift across the magnetic field lines at a velocity, say v ind .The v ind is interpreted as the velocity of plasma flow in the limit where the plasma density is arbitrarily small to provide the potential field implied by corotation.The plasma velocity in (ii), denoted by v cor⊥ , is the perpendicular component of the corotation velocity, which is the electric drift induced by the corotation electric field, E cor , defined by Equation (A5).The two plasma flows may then be regarded as two limiting conditions as they originate from two limiting cases for the magnetospheric plasma.We then consider a class of synthesized models for emission states described by a single parameter y, such that the actual plasma flow velocity is given by (Melrose & Yuen 2014) y takes a value between 0 and 1, corresponding to the two cases of (ii) and (i), respectively.The model identifies the plasma flow intermediate between the two cases, and a change in the velocity of the plasma is parameterized by a change in the value of y.Dividing Equation (1) by r gives the drift frequency as a function of y defined by Figure 1 shows the variations in the magnitude of ω dr in units of ω cor for four different values of y and a pair of ζ and α.In general, the value of ω dr /ω cor at a given rotational phase, ψ, is different for different y values, and a change in the y value will result in a change of the ω dr value.In addition, the plasma flow varies across a magnetosphere with the variation depending on α.In the ideal case for α = 0 (not shown) and all y, the ω dr /ω cor is a constant irrespective of ψ giving a straight and horizontal line, with the value decreasing with increasing y.For oblique rotators, the value of the plasma flow varies as a function of y and ψ, such that changes of ω dr /ω cor result in a curve with the maxima and minima occurring at either ψ = 0°or 180°on different curves.The figure also shows that some curves (in this case, brown and gray) intersect at specific points.These points correspond to parameters for which the plasma flows between the different y values are equal.Of particular physical interest is that the difference between ω cor and ω dr varies as functions of y and ψ.Since the value of ω dr depends on {ζ, α} and y, a situation may arise in which two MSPs have different {ζ, α}, as well as different y values, but in such a way that the average differences between ω cor and ω dr along the two trajectories are similar.
The interpretation of y is in relation to the plasma flow as shown in Equation (2).The plasma flow is also related to the gap model for an obliquely rotating magnetosphere.In such a model, the screening of the parallel electric field, E ∥ , cannot occur everywhere (Melrose & Yuen 2012).This leads to the existence of a region with E ∥ ≠ 0 called the vacuum gap.The field lines that pass through it are not in equipotential, which gives rise to a potential difference causing variation in the plasma flow through the gap.In the Ruderman & Sutherland (1975) model, the plasma flow is different below and above the gap, with the latter being at a lower angular speed than corotation.When coupled with Equation (2), the plasma flow deviates from corotation below the gap, where ω dr = ω cor (y = 0), to <  above the gap.The deviation is roughly proportional to y, such that the former increases as the value of y increases.Therefore, the value of y may be considered as representing the height, with a larger y value corresponding to a higher height.

Detectable Emission
Assuming the discrete emission regions each fixed to the magnetospheric plasma implies that their flow rate is given by Equation (2).We assume that the locations of the m discrete emission regions are distributed uniformly in azimuthal angle, f b , around the magnetic pole forming a periodic pattern that varies ( ) m cos b f µ (Clemens & Rosen 2004;Godoberidze et al. 2005).The subscript b signifies quantities in the magnetic frame.From a pure magnetospheric perspective, the pattern can be interpreted as due to a standing wave structure at a specific spherical harmonic.This leads to a repeating pattern of overdense and underdense plasma regions forming a carrousel-type model that consisted of m antinodes (overdense) and nodes (underdense), with emission being assumed coming from the antinodes.The model of discrete emission regions on a rotating carrousel is successful for describing the characteristics of drifting subpulses (Deshpande & Rankin 1999), although alternatives have also been proposed (Bilous 2018).For MSPs, their multicomponent profiles may be thought of as a superposition of several emission components, each of a Gaussian shape (Krishnamohan & Downs 1983;Kramer et al. 1994;Wu et al. 1998), coming from different emission regions.However, it is uncertain as to how the m regions are organized in height, r, and in polar angle, θ b .In the following, we assume that the diameters for all m regions are identical, and each is equidistant from its neighbors (Gil & Sendyk 2000).We further treat their distribution as independent of θ b and locally independent of r.This results in alignment of the regions along the radial direction producing a structure of radial spokes when projected onto a surface of constant r, as demonstrated in Figure 2.
We adopt the assumption that emission from an emission region is visible if it originates from a source point where the emission is tangential to the local dipolar field line (Cordes 1978;Hibschman & Arons 2001;Kijak & Gil 2003) and the emission direction is parallel to the line of sight (Yuen & Melrose 2014), as given by the geometry in Appendix C. For a given pair of ζ and α, specific locations for the visible point as the pulsar rotates can be expressed either in the polar, θ bV , and azimuthal, f bV , angles in the magnetic frame, as given by Equation (C1), or in θ V and f V in the observer's frame using the equations in Appendix B. In general, the position of the visible point is not a constant but changes with ψ, such that it traces a closed trajectory after one complete pulsar rotation.r r arcsin 0 for 0 f b < 2π and r 0 = r L , where r L = c/ω å , with c and ω å representing light speed and the angular frequency of the star, respectively.Hence, the boundary is dependent on both ψ and the height r, and emission can only be seen from r within a pulse window that is bounded between the pulsar phases when the trajectory cuts this boundary.For emission occurring only on the last closed field lines, the pulse window reduces to a point (if one magnetic pole is visible) at a particular ψ and from a minimum height r min , where the trajectory is tangent to the boundary.This implies the dependence of the visible point on height in the form of the requirement that r r

Measuring Jitter in Phase
We define jitter as a variation in the time when a pulse crosses a designated rotational phase or a variation in the period of the pulse relative to that of a reference profile.In our model, detection of a pulse profile occurs when the visible point traverses an emission spot at a certain phase, giving its ToA.Treating the emission spots as fixed to the magnetospheric plasma implies that the flow rate of the former can be described by Equation (2).Since the average difference between the jitter noise and the rotation period of an MSP in our sample is a factor of about 10 −5 (see Table 1), we assume that the same emission spot is observed every pulsar rotation.We also assume that the timing model for an MSP represents the average case (the reference profile) for corotating plasma flow in an emission state, in which the pulses arrive at the same time as each pulsar rotation giving zero jitter noise.A deviation in the arrival phase will then correspond to the emission spot being intersected by the trajectory of the visible point at a different rotational phase other than the predicted phase.We suggest that it arises from a different plasma flow due to a change in emission state that causes shifting of the observable emission spot where it crosses the designated phase when comparing with the timing model.In our model, a phase jitter is represented by a deviation of the plasma flow (ω dr ) from corotation (ω cor ).Consider the special case in which the trajectory of the visible point lies entirely inside the open-field region, such as the one in yellow in Figure 2. Emission is potentially observable from the whole trajectory, and hence the calculation of ω dr is along the trajectory.For other cases, emission is observable only within a small range of rotational phase (pulse window), as indicated by the blue part of the middle trajectory in Figure 2.This means that any variations in the plasma flow along other parts of the trajectory (in brown) are nondetectable, and the observed ToA is the result of averaging the overall effects along the trajectory.Hence, When the value of y changes, dr w also changes, giving rise to fluctuation in ω δ , which leads to variation in the ToA every pulsar rotation.The jitter noise, when expressed in units of seconds, is then given by δ ω = ω δ P 2 /2π for an MSP possessing a rotation period of P. It follows that δ ω = 0 for perfect corotation, and the emission spot returns exactly to its original location in ψ, resulting in zero jitter noise as measured from the pulse window.Depending on the magnitudes of the jitter noise, the deviation of plasma flow from corotation, as specified by the value of y, will be different.

The Samples
Recently, Parthasarathy et al. (2021) reported observations of 29 MSPs using the latest MeerKAT radio telescope of the Square Kilometre Array (SKA) precursor.For accurate results, most of the jitter noise in this investigation is taken from MSPs in the list of that paper.Only MSPs with designated values for the jitter noise and uncertainty are considered.For an MSP whose jitter noise is not accompanied by an uncertainty in the original results, the corresponding value obtained from the previous study is used (shown in the last column in Table 1 of Parthasarathy et al. 2021), provided that it comes with an uncertainty.In the following, we refer to these MSPs as simply from "previous results."We obtained a total of 17 MSPs, whose details are listed in the second to fifth columns of Table 1.The profile widths are mainly obtained from the ATNF Pulsar Catalogue (Manchester et al. 2005), and some are from the European Pulsar Network.We treat the observed jitter noise, δ j , as representing the average magnitude of the discrepancy in phase.

Simulation and Results
The jitter noise for each of the MSPs is simulated using Equation (3).Doing this requires determining the plasma flow along the trajectory of the visible point for a given pair of {ζ, α}.For an MSP, different combinations of ζ and α will result in different trajectories.Study of the polarization position angles from a large pulsar population (Rankin 1993;Tauris & Manchester 1998) shows that the obliquity angle is distributed within the range of α  90°and leans heavily toward small values (Rankin 1990), which is also supported by simulations (Zhang et al. 2003;Kolonko et al. 2004).Our simulation considers different values of ζ from 0°to 90°in steps of 0°.5 and different values of α from 1°to 90°, also in steps of 0°.5.This gives 32,399 different combinations of {ζ, α}.Then, the restriction for the impact parameter |β| = |ζ − α| 15°is imposed.The restriction on β is chosen to be consistent with the results obtained from analysis of the shape of the radio beams and the associated polarizations (Lyne & Manchester 1988).In addition, radio emission is suggested to come from a height of around 0.1r L or above (Johnston & Weisberg 2006), and we assume a maximum height of 0.2r L .Subsequently, a maximum height of 0.2r L is fixed for the openfield region.For a given {ζ, α}, the boundary of the region is determined based on the value of α.Next, the values of {θ V , f V } along the trajectory of the visible point for the {ζ, α} are derived from Equation (C1) with the transformation matrices in Appendix B. The intersections of the trajectory and w is determined at y = 0, and dr w is computed for each y value from 0 to 1 (the two limiting cases for pulsar magnetospheres) in steps of 0.01.At each y value, the δ ω is also determined using Equation (3) followed by transforming the result into seconds.As discussed in Section 2.1, different combinations of {ζ, α} and y may give similar jitter noise values.This may lead to large scatter in the results, especially for jitter noise with large uncertainty.Since simulated data points closer to an observed value are more likely to have a greater significance, we first evaluate a weight based on the observational uncertainty in δ j .For a measured δ j , weights are calculated for data points according to their distance from δ j using a Gaussian function, such that higher (or lower) weights are assigned to points that are closer to (or farther away from) δ j .Then, each simulated value from the same calculation will receive identical weight.Finally, a weighted average is determined for each of the parameters in the simulation for the MSP.
The results of our simulation for the jitter noise and the corresponding y value together with the values for α and β are shown, respectively, in the fifth through eighth columns in Table 1.The simulated jitter noise agrees with the observed value within the uncertainty for each MSP.Treating the reported jitter noise values as the maximum discrepancies in the ToAs gives the y values with the largest deviation from corotation.We obtain an average y value of 0.37, and none of the y values in any of the MSPs is 0, which suggests that the plasma flow is not in corotation.Our results show that a large δ j does not necessarily correspond to a large y value.For example, PSRs J1603−7202 and J2145−0750 possess the largest δ j in the sample, but the corresponding y values are different by nearly a factor of 2. Furthermore, MSPs with similar δ j could possess similar y values, such as that for PSRs J0030−0451 and J0437−4715, whereas others have different y values, as seen in PSRs J0711−6830 and J2010−1323.From Section 2, the y value is a function of both δ j and α, with δ j being broadly proportional to P (Parthasarathy et al. 2021).In addition, the potential α value is restricted by the profile width of an MSP, which means that the value of y is also related to Δψ.It follows that the same y value does not necessarily give the same δ j .The result is that the final y value is dependent on δ j (or P) and Δψ, which are different for different MSPs, resulting in similar y values for some MSPs and different values for others.This indicates that jitter noise and the associated change in the emission condition in the magnetosphere are dependent on the MSP.In general, the values of α obtained for our samples tend to be high, with an average of 48°. 1.The values of β are relatively small, with an average of about −1°.30.Our predicted values for α and β are consistent with the recent estimation using MSPs with time-aligned radio and γ-ray pulse profiles (Benli et al. 2021), which gives α and ζ both larger than approximately 45°.For instance, PSR J0030+0451 is shown with large α and negative β values (Benli et al. 2021), which agree with our prediction.Furthermore, investigation of the MSP using the NICER data suggested that ζ ≈ 53°.9 (+6°.3,−5°.7) (Riley et al. 2019;Pétri et al. 2023), which is also consistent with our result.Our predicted α is smaller, but it lies within the distribution of the reduced chi-squared values.The difference is likely due to the uncertainty in the locations of the radio and γ-ray sites leading to variation in the parameter (Benli et al. 2021).From geometry alone, a relatively broad profile width would require that β be small.Similarly, the values of α would also tend to be small for a fixed emission height (Esamdin et al. 2005).The large estimated values of α in MSPs from observations may be related to the small size of their light cylinders and the impact of the recycling process on the emission configuration (Manchester 2017).

Properties of Jitter Noise
In this section, we examine several characteristics of jitter noise in the MSPs based on the results in Section 4.

Relationships with the Emission Geometry
An important parameter for pulsar emission is the obliquity angle, but the relationship between the latter and jitter noise is unclear.The variation in the observed jitter noise (δ j ) as a function of α is shown in Figure 3.It shows that the values of α from our simulation are distributed over a relatively large range between about 20°and 65°, with the majority leaning toward larger values.Although the MSPs with low jitter noise seem to have low obliquity angles, the value of α increases as the jitter noise spreads over an increasingly broader range.Furthermore, MSPs with similar α can exhibit different degrees of jitter noise, as seen for PSRs J1757−5322 and J2145−0750 at α ∼ 38°, and PSRs J1730−2304 and J1744−1134 both have α ∼ 55°.On the contrary, a similar level of jitter noise can be found in pulsars of different α, such as PSRs J0125−2327 and J0437−4715.However, this demonstrates that MSPs in our samples are oblique rotators, and investigation of their emission mechanism in relation to jitter noise must take α ≠ 0 into account.A difference for pulsars with α ≠ 0 relates to the component of the inductive electric field in the electric drift velocity being increasingly important as α increases, which will affect the systematic motion of the plasma, as shown in Equation (2).
Our results suggest that the MSPs in our samples tend to have small impact parameters for the range of |β| 15°c onsidered in the simulation.The simulated β values are significantly smaller in magnitude than the limit of the range

Trend of Jitter Noise
Another significant parameter for MSPs is their rotation periods.Traditional models suggest that the potential difference that accelerates charged particles in the vacuum gap is related to the rotation period (Goldreich & Julian 1969;Ruderman & Sutherland 1975;Arons & Scharlemann 1979).Study of the polarization profiles in MSPs (Yan et al. 2011) demonstrates that a similar radiation process also applies.The magnitude of the accelerating potential varies with P, such that the former is small when the latter is large.Furthermore, the change in the plasma flow, which leads to jitter noise, is related to α in our model as shown in Section 2. Assuming that pulsar energy loss is due to magnetic dipole radiation (Lyne et al. 2015), the value of α is anticorrelated with P. It follows that the two parameters are likely important for describing the jitter noise.Figure 5 shows the variation of δ j plotted against α/P, with both P and δ j being the observed values, whereas α is obtained from our simulation.It demonstrates a continuous decreasing trend in δ j as α/P increases.The decreasing trend can be fitted with an exponential function that contains a term of the form given by 0.88 exp ( ) P a -, with δ j expressed in milliseconds.From Figure 5, a small δ j can be obtained provided that α/P is large.The latter can be achieved with either a very large α or a very small P. On the contrary, an MSP with small α and large P is predicted to exhibit large δ j .It follows that the jitter noise tends to be large for large P, which is consistent with observations (Helfand et al. 1975;Cordes & Downs 1985).Therefore, it makes sense to search for more fast-rotating MSPs for the benefit of timing applications.In general, δ j and P are measurable parameters, but α is difficult to determine.Since δ j and α/P are correlated, it is possible to estimate α for an MSP once δ j is known.
An ideal case relates to the zero jitter noise.In reality, jitter noise below a certain threshold level is desirable for timing applications, such as the detection of gravitational waves.As is apparent from the line of best fit in Figure 5, the lowest jitter noise values are achieved for small P.This is consistent with observations (Parthasarathy et al. 2021).From models for the structure and evolution of neutron stars, it was predicted that pulsars with 1500 rotations s -1 (∼0.67 ms) would disintegrate (Cook et al. 1994;Haensel et al. 1999).Even at 1000 rotations s -1 (1 ms), the rotation deceleration due to energy lost through gravitational radiation would be faster than the speeding up owing to the accretion process (Chakrabarty et al. 2003).From Figure 5, the limit of 1 ms for P will eventually be reached, where the jitter noise is also small.Such fast rotation would provide strong constraints on the equation of state of the matter inside the stars, thus bridging the gap between theory and observation in detecting fast-rotating MSPs (Demorest et al. 2010;Fonseca et al. 2016;Linares et al. 2018).

Dependence on y
The relationship between the values of y and δ j is shown in Figure 6.All the MSPs considered are found to have y values   between 0.25 and 0.6, where both the smallest and the largest δ j reside.An increasing trend in δ j is noticeable as y increases.
Performing the Pearson's r correlation test gives a coefficient of +0.44, suggesting a moderate positive association between the two variables.From the discussion in Section 2.1, treating the parameter y as representing the height, and assuming emission at low frequency is from a higher height, and vice versa, in the radius-to-frequency mapping, the moderate correlation between δ j and y is an indication of the dependence of jitter noise with observing frequency.

Limitations in the Model
We discuss some of the simplifying assumptions made in our modeling of jitter noise in MSPs.First, jitter noise consists of two components, and our assumption of it as represented only by phase jitter is obviously simplified.Furthermore, the description of the jitter noise through changes in the plasma flow is based on a single parameter (y) without referring to the underlying mechanism.Although such a mechanism is still unclear, it will allow inclusion of both components in the jitter noise, which will undoubtedly involve a larger set of parameters in the simulation.Second, pulse profiles of MSPs are complex and highly non-Gaussian.This means that selecting either W 10 or W 50 for making a comparison with the simulation will be biased.Ideally, an analysis should consider the full shape of the pulse profile.However, this requires knowing the arrangement of the emitting plasma in the emission region and its relation to the underlying mechanism.Since we focus on phase jitter, the profile shape is ignored.To cover the maximum emission range, which comes with a reported measured value, we select the W 10 of an averaged profile for constraining the values of {ζ, α}.Third, our simulation is based on a small MSP sample but with highquality jitter noise measurements.A small data set results in unequal representation of some observed parameters, which may obscure certain features causing biases in certain interpretations.A related example concerns the determination for the range of β, which is assumed to be between a minimum of −15°and a maximum of 15°in our simulation.An issue may be raised that while this is consistent with observations (Lyne & Manchester 1988), relaxing the restrictions could lead to a broader range of simulated β values.As mentioned in Section 5.1, the simulated β values are well within this restricted range, which makes the latter a valid assumption.

Discussion and Summary
We have investigated the correlations between the jitter noise and several pulsar parameters in 17 MSPs based on published results obtained from the observation using Meer-KAT.We assumed that the reported jitter noise represents the average magnitude of deviation in the ToA of the pulse profile relative to the predicted time.We modeled the pulse ToA as related to the rate of the plasma flow under the electric drift in a magnetosphere of multiple emission states.In this model, changes in the emission state lead to changes in the plasma flow resulting in the observed fluctuation in the ToA every rotation.For a magnetosphere in perfect corotation, the ToA of the pulses is identical to the predicted time, giving zero jitter noise.For the MSPs in our sample, we found that the magnetospheric plasma flow is not in corotation, and that their obliquity angles tend to be large.The latter indicates the importance of including the effects of oblique rotation for examination of jitter noise.In our results, the jitter noise exhibits a moderate correlation with frequency.
The jitter noise in our sample MSPs does not show notable relationships with the obliquity and the viewing angles.The two angles define an emission geometry for visible radio radiation, which implies that jitter noise in the MSPs is not likely related to the emission geometry alone.This is further confirmed by their dependence on frequency, or emission height if the radius-to-frequency mapping is assumed.However, a strong correlation is revealed when the obliquity angle and the rotation period are considered together, as demonstrated in Figure 5.As mentioned earlier, the conventional model of pulsar radio emission relates an accelerating potential to the rotation period of a pulsar, such that the former is low when the latter is high, and eventually leads to the "death" of the pulsar (Arons 2000;Zhang et al. 2000;Harding & Muslimov 2002).Therefore, the rotation period is related to the radio emission generation.However, the unique formation of MSPs, which includes a "recycling" process, makes it uncertain whether the conventional description of their magnetospheres and radio emission applies (Xilouris et al. 1998;Luo et al. 2000).As discussed in Section 5.2, a short rotation period is preferable for low jitter noise given similar α values.A short rotation period would imply a more efficient production of the pair particles resulting in ample charged particles in the magnetosphere, which provide the potential field implied by corotation, and the plasma flow approaches corotation.It follows that faster-rotating MSPs would have smaller jitter noise.It also suggests that the conventional models for radio emission are likely to play a significant role in MSPs.
It is generally accepted that the unique properties of MSPs are gained in the "recycling" scenario, during which an MSP is spun up through accretion from the binary stellar companion (Smarr & Blandford 1976;Radhakrishnan & Srinivasan 1982;Manchester 2004).However, recent discoveries of many MSPs without binary companions in globular clusters (Verbunt et al. 1987;Kremer et al. 2022) have defied the formation process described in the recycling model.In particular, the accretion process may be different for different MSPs.The accretion efficiency is dependent on the formation scenario, and different scenarios, such as conventional accretion (Alpar et al. 1982;Radhakrishnan & Srinivasan 1982), stellar collision and tidal disruption (Mukherjee et al. 2015;Parfrey et al. 2017), and a binary involving a "strange" star (Jiang et al. 2020), may lead to different final values for the observed MSP parameters.For example, the luminosities of some isolated MSPs are found to be less luminous (Bailes et al. 1997;Kramer et al. 1998).An intrinsic feature of pulsar radio luminosity is its relation to the properties of the magnetospheric plasma and the spin parameters.Furthermore, the interaction between the accretion disk and the magnetosphere will cause stress and changes in the magnetosphere and its field-line structure, which may also be different in different formation scenarios.It is thought that MSPs originate from ordinary pulsars of (usually) large age.The large age is gained by assuming energy loss through magnetic dipole radiation (Lyne et al. 2015), in which α evolves from large values toward 0. The suggestion that MSPs possess large α (Parthasarathy et al. 2021) implies that the changes to the magnetosphere may also lead to alteration of the original α value.In our model, the values of y are closely related to the magnetospheric plasma flow and α, and hence it is reasonable to assume that its values are also dependent on the formation scenario.It follows that the emission properties of an MSP, including the final P and α, the y values, and the jitter noise level, are likely formed with the star.Furthermore, the correlation shown in Figure 5 indicates that the jitter noise, rotation period, and obliquity angle of an MSP are related.This implies that the course of formation for an MSP may not be random, but rather specific, so that the resulting MSP would display such a correlation.It follows that the companion star and the pulsar in the original system are unique, and not all binary systems can give birth to MSPs.Therefore, the emission properties of MSPs are related to their formation history, and eventually the stellar evolution, with the relevant "aftermath" information of the process captured in the jitter noise.To accurately obtain this information requires high-precision measurements from highly advanced observing instruments.With the contributions from the Five-hundred Aperture Spherical radio Telescope (Li et al. 2018) in pulsar research and the plans for the future Qitai radio Telescope (Wang 2014;Xie et al. 2022) and SKA, it is hopeful that new MSPs, with even shorter rotation periods, will be discovered, which will be beneficial to unlock the secret of jitter noise and to shed new light on the formation of such beautiful pulsar systems.where α is the obliquity angle between the rotation and magnetic axes, ψ is the rotational phase, and ω is the angular velocity of the dipole.We note that E ind is proportional to sin a and so it is absent for an aligned rotator.The magnetic field equation is given by where the terms ∝1/r 2 and ∝1/r are radiative terms, and the term ∝1/r 3 is dipolar (subscript "dip").Since μ is a function of α in the observer's frame, all B, E pot , and E ind are also functions of α.
In the comoving frame of a plasma-filled magnetosphere with negligible particle inertia and infinite conductivity, the electric field vanishes, giving where E ind = −∂A/∂t, and grad cor -F characterizes the electric field in relation to the corotation charge density.In spherical coordinates, E cor has only components along the radial and polar directions and perpendicular to the dipolar field lines.

Appendix B The Coordinate System
We adopt the arrangement of the rotation and magnetic axes of a pulsar in Cartesian coordinates in such the way that ˆẑ q a q a q f y f q f y a q f y a q = + -= ---
Figure 2 demonstrates three trajectories of the visible point with the same α and different ζ.Conventional assumption states that emission occurs only within the open-field region.The boundary of an open-field region can be defined by the locus of the last closed field lines, which satisfy b q = field line, where r V is given by Equation (C2).In the figure, the boundary of the open-field region at 0.2r L is obtained from Equation (C2) based on locations on the last closed field lines.Emission is visible from the point in phase, ψ 1 , where the trajectory enters the open-field region, to the point ψ 2 , where the trajectory exits the region, as indicated by the blue part of the middle trajectory.The profile width (or pulse window) is then defined by Δψ = ψ 2 − ψ 1 .It is apparent that Δψ is dependent on ζ and α for an open-field region at a fixed height.The location on a radial spoke, where it is cut by the trajectory, is referred to as an emission spot.

Figure 2 .
Figure 2. Plot showing the arrangement of 20 radial spokes in the magnetic frame, where the magnetic pole is located at the origin.The boundary of an open-field region at height 0.2r L is shown by the dotted dark circle.Three trajectories of the visible point with α = 15°and ζ = 20°(yellow), ζ = 49°( blue and brown), and 60°(green) are also shown.The former two trajectories traverse the open-field region at different rotational phases cutting various spokes in the process, whereas the last trajectory does not traverse the openfield region at all.

Figure 3 .
Figure 3. Plot showing the variation of the observed jitter noise as a function of the obliquity angle obtained in the simulation.The MSPs from previous results are indicated in gray.

Figure 4 .
Figure 4. Similar to Figure 3 but for variation of jitter noise as a function of β.Figure 5. Variation in the jitter noise as a function of a combined quantity of α (in degrees) and P (in milliseconds).The line of best fit is shown by a dotted black line.The MSPs from previous results are indicated in gray.

Figure 5 .
Figure 4. Similar to Figure 3 but for variation of jitter noise as a function of β.Figure 5. Variation in the jitter noise as a function of a combined quantity of α (in degrees) and P (in milliseconds).The line of best fit is shown by a dotted black line.The MSPs from previous results are indicated in gray.

Figure 6 .
Figure 6.Variation in the jitter noise as a function of y.The MSPs from previous results are indicated in gray.

x
An obliquely rotating time-dependent magnetic dipole, μ, produces a vector potential, A, of the form given by (Melrose & is the position vector from the stellar center, and r = |x|.The corresponding inductive electric field is given by represents the corotation electric field (Goldreich & Julian 1969), and ω å is the rotation frequency of the star.For an obliquely rotating magnetosphere, Equation (A5) may be written as(Hones & Bergeson 1965;Melrose 1967) subscript b signifies the quantity is expressed in the magnetic frame.Transformation between the unit vectors is given by ψ is the rotational phase and R T is the transpose of R. The equivalent unit vectors for radial, polar, and azimuthal in spherical coordinates are described by ˆˆr , , q f and ˆˆr , ,

Table 1
Details of the MSPs and the Simulated Results The 17 MSPs used in this paper are shown in the second column, together with their rotation periods (in milliseconds) listed in the third column.The profile width at W 10 expressed in milliseconds at about 1.4 GHz is given in the fourth column.The detected jitter noise (δ j ) and the corresponding values obtained from our simulation (δ ω ) are given in the fifth and sixth columns, respectively, in units of nanoseconds.The y values are shown in the seventh column, and the estimated values for α and β for each pulsar are given in the last two columns.The asterisk signifies an MSP whose jitter noise and associated uncertainty are taken from previous results.theboundary of the open field give ψ 1 and ψ 2 and hence Δψ.Since Δψ is related to ζ and α, as shown in Figure2, restrictions on the possible combinations of {ζ, α} can be achieved by requiring the Δψ generated from a {ζ, α} pair to match with the observed profile width at W 10 of the MSP.The next step is to evaluate the values of cor w and dr w for each trajectory of the {ζ, α} pairs that give a matching profile width.Here, cor