Compatibility of the Observed Rotation Parameters of Radio Pulsars on Long Time Scales

We present the results of an analysis of the rotation regularities of pulsars from the current observations and retrospective data on the timing of pulsars at Pushchino Observatory, obtained since 1978 to 2017. Parametric approximation of numerical series of Times of Arrival (ToAs) the convergence polynomial series defined the combinations of numerical values, unique for each pulsar, identical in any reference system for coinciding epochs, irrespective of duration of observations that corresponds to the coherence of periodic radiation of pulsars. The braking index at all observed pulsars is n = –(0.9 ± 0.2) that corresponds monotony of slowing down of rotation (>0). The polynomial power series of the intervals calculated in observed parameters of rotation represent an analytical time scale on which variations of observed intervals, uncorrelated with rotation, are counted, as at pulsars B1919+21 and B0809+74, or the unpredictable variations of intervals defining the movement of a pulsar in the supernova remnants, as at PSR B0531+21 or at PSR B1822-09 which spontaneous movement also doesn’t exclude its belonging to the supernova remnants.


Introduction
The most significant results in studying the physical properties of pulsars are connected with periodic regularity of a radio emission of the rotating magnetized neutron star. The energy losses accompanying star radiation lead to the gradual slowing down of its rotation, so the numerical value of the period of rotation specified in the catalog is fixed together with the relating observation epoch.
The rotation period, being observed parameter, represents the measured value, which is determined directly by the moments of arrival of impulses of radiation on the radio telescope. Already from the beginning of the 70th years, near only several years after opening and fast addition of the list of new pulsars, the numerical values of the period and its derivative were accurately measured for all the pulsars included in the first fundamental catalog of pulsars [1], which have not lost meaning so far.
Many of results are obtained by the pulsar timing based on the precise measuring the topocentric Times-of-Arrival (TоAs) of the pulsar signals at the radio telescopes, denominated in the UTC time scale [2]. The process of pulsar timing is fundamentally dependent on an accurate description of everything that affects the ToAs of the pulsed radiation at the telescope. In addition to a local time standard and the Solar-System ephemerides pulsar timing requires knowledge of the pulsar's pulse period and spin-down, its astrometric position and proper motion, its distance, parallax (where detectable) dispersion measure in the interstellar medium and (on the binary system) any orbital parameters. All of these parameters are included in a timing model, which can be used to predict the phase of the pulsar's periodic signal at any point in time. The differences between the measured ToAs and those predicted by the timing are the "timing residuals", which are the unmodelled difference between the observations and the theory [3].
The continuing research of pulsars, especially in the process of accumulation long-term, within several decades, archives of timing, showed that braking of pulsars is not limited only to the first derivative. An analysis of timing noise has shown that these deviations are due to a random, continuous wandering of the pulse phase that produces long-term cubic polynomial components in the timing residuals, which are correlated with the period derivative and are quantified by a parameter based on the non-zero second derivative. The timing residuals cannot be explained by the models that are based on noise-like processes because such models are not consistent with observations over long time scales.
Crucial is an exact definition of the second derivative which contribution usually doesn't exceed several microseconds within 2 -3 years of observations that is several orders lower not than a threshold of detection against the background of unmodelled timing noise. As Shabanova notes in the generalizing paper [4]: "For ordinary pulsars, the values of P2dot due to slowdown are very small in comparison to the measurement uncertainties. In actuality, the long-term cubic polynomial components in the timing residuals of most ordinary pulsars lead to the large observed P2dot values and accordingly to anomalously high values of the braking indices ( Hobbs, Lyne and Kramer [5]: "For a large number of pulsars, the timing residuals deviate by more than one pulse period…The observed ν2dot values for the majority of pulsars are not caused by magnetic dipole radiation or by any other systematic loss of rotational energy, but are dominated by the amount of timing noise present in the residuals and the data span". Therefore, the accepted timing models quite often lead to considerable distortions in estimates of physical properties of the pulsars, which are strongly differing for the same objects on different sources. Unpredictable, abnormally high variations of residuals are interpreted as sudden failures of the period of rotation (glitches). In this consideration, especially significant is a reliable extraction of physical characteristics of rotation of pulsars, including the second derivative, on directly observed data of, in view of basic restrictions of reliability of statistical estimates on residuals.
Our approach, in general, is to find the analytical relation of the pulsar time intervals and the physical parameters so that the numerical values of these parameters should be determined and best matched with measured values of the observed intervals. The comprehensive fitting should be enough only those parameters whose exact values can be derived directly from observation.

Analytical Model of Pulsar Timing
The main point of the model is to convert the numerical series of the observed ToAs, measured with respect to the some initial event of pulsar radiation, into the analytical form of intervals, calculated from the observed rotation parameters that most closely correspond to the numerical series of the ToAs [6].
Pulsar time (PT) intervals of any pulsed periodic radiation event N express by the function of its rotation parameters 0 , , P P P   in the time domain: 2  2  3  0  0  0  0  0   1  1  , ,  2  ,  1, 2, 3,  2 6 PT P P P P N P PN P P P P N N The right part of the Equation (1) represents a power series expansion Maclaurin of intervals PT, counted from the initial emitted pulsar event. The structure of intervals PT contains linear, square and cubic components, which are defined by fixed rotation period P 0 on the initial epoch and its derivatives , P P   . The power series on the right side (1) is limited to only 3 parametric components, sufficient for its convergence. Indeed, for a typical pulsar (   15  16  1  28  29  1  0 1 , 10 -10 , 10 -10 P s P ss P s s, cubic component is not more than a few microseconds. Further continuation of the series shows that estimates the contribution of the third derivative is less than 1ns, and therefore cannot be detected on noise background, so higher derivatives of order 2 in a power series (1) cannot be considered at all.
In the relation (1) the intervals РТ are calculated from the parameters of rotation, which are known merely suspected a priori. However, by linking these parameters with the observed intervals PTi, we obtain an equation whose solutions Here are: PT i is the numerical values of the observed intervals obtained from the planetary ephemerides; 0 , , P P P *   are the observed pulsar rotation parameters obtained by solving Equation (2); i α is the divergence measure of series (2) of the PT i approximated by the rotation parameters of pulsar.
By parametric approximation of the intervals, PT i (2) counted from the initial observed pulsar event, the fixed rotation period and its derivatives on the initial epoch in accordance with the power series expansion Maclaurin (1), are defined.
The combination of parameters 0 , , P P P *   is unique for each pulsar, wherein convergence of polynomial series (1) In a generalized view of a ratio (3) and (4) for all events of radiation of a pulsar observed within any interval are expressed as follows: The numerical value PT α , as well as rotation parameters, is also the solution of the Equation (2). It characterizes unmodelled timing noise in the sequence of observed intervals obs PT , it isn't correlated with an analytical pulsar time scale calc PT which is defined by the observed parameters of rotation only and, therefore, doesn't depend on that unmodelled timing noise.
It has been shown that the Equation (2)  As a result of dividing the observed intervals, the random variations are not contained detectable rotation components. They can be analyzed additionally to identify with affecting factors, for example, the distortion of an atomic time scale of the radio telescope, for possible physical interpretation. It is assumed also, that the pulsar rotation parameters are independent of any physical factors, which are involved in a parametrical fitting.

Consistency of the Observed Rotation Parameters of Pulsars
It is evident, that the value of the period 0 P * depends on any choice of the epoch of the initial event, taking into account the nonzero derivatives P  , P  . The corresponding settings of the rotation parameters also satisfy the convergence of the series expansion (2) for any extension in the vicinity the variable specified by ( ) Relation (6) By integration of function in an interval {tj, ti}, (here i > j), the deviation of observed intervals is calculated: Replacing continuous function with its discrete approximation on the Equation (2), and assuming that change of the observed period in an interval {tj, ti} happens evenly, we obtain deviations of observed intervals in the form of the final differences between any two observed events of radiation of a pulsar within this interval: A. E. Avramenko et al. International Journal of Astronomy and Astrophysics The main characteristic of the observed slowing down of the pulsar rotation is the braking index n describing dependence of angular speed 2 2 P πυ π Ω = = of rotation of a neutron star on time. The braking index is the only direct test connected with a slowing down mechanism of radio pulsars now. On the braking index it is possible to compare pulsars, the period and which derivatives differ on several orders, but being observed parameters, they are available to comparison according to catalogs and other sources. They it is equivalent are expressed in a frequency and time domain: Considering that the numerical values of the period P and its derivative P  were correctly measured for all the pulsars contained in the fundamental catalogs, there are followed the evident discrepancy for a pulsar slowing down evaluation. Instead of inspecting n = 3 for the magnetic dipole braking model ac- By approximation of observed intervals according to the Equation (2) were obtained the consistency parameters of rotation , , P P P   which are given in Table 1 together with corresponding braking indexes n [7].
As a component of the second derivative P  in the PT intervals, it does not exceed a few microseconds, even at the pulsars with relatively large values P  = 10 −28 -10 −29 s −1 , it is detected not earlier than after two years of observations. The   Calculated from the observed parameters , , P P P   the value of braking index is a numerical invariant n = -(0.9 ± 0.2), corresponding to coherent radiation at monotonous slowing down of these pulsars.

Observations and Timing Analysis
Now, in view of the results of the pulsar timing expressed as analytical power series, we will expand a scope of this model, having extended it also to other observation data and objects for comparison of results and their physical interpretations. For this purpose, we will address the list from 27 pulsars, observed from 1978 to 2012 years on the BSA FIAN radio telescope by an identical technique in the same conditions [4]. All pulsars from Table 1, except PSR B0531+21 contain in the list 27 PSR of Shabanova as well. In turn, all pulsars from Shabanova's list contain in the extensive list from 366 pulsars observed over the past 36 years using the 76-m Lovell radio telescope at the Jodrell Bank observatory, which are investigated in details regarding the existence of irregularities of rotation in a well-known work of Hobbs et al. [5]. Observations on the BSA and Lovell radio telescopes were carried out almost in parallel at the same years, authors were rather well mutually informed and actively discussed results, paying special attention to the available divergences, analyzed their possible reasons.
Timing parameters of 27 pulsars were determined by the initial topocentric ToAs performed on regular observations on the BSA radio telescope in the automatic mode. Timing parameters were determined using the program TEMPO The position and the proper motion required for this correction were taken from [5] and were held constant during the fitting procedure. The positions of most of the pulsars from our sample do not require correction since their residuals (defined as the observed minus predicted arrival times) obtained after the fitting procedure did not show a sinusoid with a period of 1 yr within the precision of the measurements. Position corrections were needed for pulsars B1508+55, B2020+28, and B2224+65, which are fast moving pulsars, but a timing solution did not provide improved positions for these pulsars because of the large timing noise. As the timing behavior of these pulsars was analyzed over a long interval, the presence of a weak annual sine wave at the end of this interval does not affect the large-scale structure of the timing residuals.
Further are given the results of timing of 4 pulsars, two of which (B1919+21 and B0809+74) contain both in Table 1, and in the list of 27 pulsars of Shabanova, a pulsar B0531+21 is only in Table 1, pulsar B1822-09 is given in the list of Shabanova only. Timing of all listed pulsars is performed on analytical model and observed rotation parameters of pulsars corresponding to it.

Pulsars B1919+21 and В0809+74
At these pulsars, apparently from Table 1, there are the smallest values of the second derivative, at PSR B0809+74 it doesn't reach a detecting threshold even for several years of observations. Therefore, it is possible to hope that retrospective archives of observations of Shabanova lasting several decades will allow to extract the reliable value of the second derivative. On the example of PSR B1919+21 we will compare results of detecting of second derivatives (P2dot near 10 -30 s −1 ) on previous data sets of long-term observations of Shabanova and later observations on the BSA FIAN radio telescope.  Table 1. We set a task to compare values of the second derivative coordinated with other parameters of rotation on both of these selections.
On analytical model we looked for equivalent decisions for the second derivative both in topocentric and in the barycentric coordinates. As duration of both selections is sufficient for sure detection of a cubic component, we can directly compare the results of detecting of the second derivative and test its consistency with other parameters of rotation. When detecting the second derivative we operated with absolute values of ToAs both in the topocentric system (TT), and in the barycentric system (TB). On graphics of Figure 1 their differences are shown.
The second derivative was determined according to ratios (3), (4) and (5)   It is easy to notice that the value of the period is defined much more precisely, than it is specified in the catalog. And the longer duration of the observations, the higher the accuracy corresponding to an approximation of observed intervals. With a duration of observations in several decades the inaccuracy of approximation

2) В0809+74
As it was shown upper, even after 9 years of observations of pulsar B0809+74 on the BSA FIAN radio telescope it has not been found the meaningful sign of the second derivative, i.e. P  < 10 -30 s -1 , which does not exceed a specified detection threshold, as noted in Table 1. The absence of a significant second derivative at relatively short intervals of 2 -3 years simplified the processing of the timing data when only the first derivative could be limited. However, the increasing demands on the accuracy of timing, raises the question of the exact definition of the second derivative, especially sharply, taking into account the anomalous variations of its existing estimates. Therefore, the unique 34-year observations pulsar B0809+74, performed by Shabanova et al. [4], are of particular interest as a chance to overcome the limiting detection threshold and to obtain the actual measured value of the cubic component of the second derivative by using long-term archival data.
The method of processing observational data is completely analogous to that used for the pulsar B1919+21. In Figures 5-8 shows the same structure of graphs, calculations and estimates as in Figure 1 for the pulsar B1919+21. In the graph on Figure 5 shows the difference between the initial topocentric and barycentric ToAs within 34 years of observation. We note that two breaks in observations of 1 -2 or more years in the initial stage had no effect on the timing results, since, firstly, all the intervals of the observed events are counted from the general beginning and, secondly, the intervals are approximated observable   Table 1; These features of the timing of the B0809+74 pulsar, which are directly related to its physical properties, are manifested in the conditions of data processing in the accepted parametric time processing model. It should be specially noted that the braking index of the PSR B0809+74 is n = -0.71 and, unlike the other pulsars in Table 1, is close to the lower limit of the range n = -(0.9 ± 0.2) established by observation. So far, according to observations of only one pulsar, it is unclear whether this quantity is random or it reflects the A. E. Avramenko et al. International Journal of Astronomy and Astrophysics general pattern characteristic of pulsars with a small deceleration, that is, pulsars with relatively small derivatives similar to PSR B0809+74. In general, such pulsars with a period of about a second, in which P  ≤ 10 -16 c•с −1 , are not known so much. So, in the ATNF catalog, containing more than 2600 pulsars, there are only about two dozen of them. Several such pulsars are given in Table 2.
From the observational data on the timing of these pulsars, a direct measurement of the cubic component of the observed intervals can be used to measure the exact value of the second derivative to refine the lower limit of the braking index. It should only take into account that for a reliable detection of P2dot observations will be required for 40 years or more.

On the Consistency of the Rotation Parameters of Millisecond Pulsars
For millisecond pulsar, whose periodic radiation is also coherent, the derivative of the period P  usually does not exceed 10 −18 -10 −21 s•s −1 . The expected magnitude of the second derivative P  , by the consistency of the rotation parameters according to the criterion of monotony slowing down (6), has to be within 10 −35 -10 −39 s −1 . This is a lot of order below the detection threshold, and P  can be evaluated theoretically only, by analogy with calculation of the braking index of second pulsars. Therefore, proceeding from the numerical invariance of the braking index with an average value of the braking index n avrg = -0.94, the numerical values of P  as their upper limit for millisecond pulsars have been calculated in the known rotation parameters , P P  . Table 3 summarizes the consistent rotation parameters of 7 millisecond pulsars. The parameters , P P  of the PSRs J1640+2224, J2145-0750, J0613-0200, J1713+0747, J1643-1224 were taken from the catalog ATNF [8], these parameters of the PSRs B1937+21 and B1855+09 were published in the fundamental research Kaspi et al. [9], special devoted to these pulsars, together with ToAs, which were kindly provided via the Internet by the authors.  Table 3 at these pulsars the format of an epoch differs in a large number of decimal signs of a fractional part whereas at all other pulsars of Table 3 the integer MJD only    Table 4 together with their frequency parameters of rotation FO, Fdot, F2dot. Some of pulsars, namely J1939+2134, J1857+0943, J0613-0200 contain in Table 3 in which parameters of rotation it is equivalent in a time domain are expressed.
It is obvious that for millisecond pulsars at which the second derivative is a priori 7 -8 orders less, than at second pulsars to find its numerical value in the observation way, even a parametric approximation of observed intervals, it is impossible in principle. The abnormal discrepancies in the definition of the second derivative are especially inherent in millisecond pulsars at which the relation of a cubic component in observed intervals to unmodelled variance of residuals objective is several orders less, than at ordinary pulsars.
Therefore the only opportunity to define the second derivative at millisecond pulsars is to establish it in the settlement way as the top limit coordinated with known for the period (frequency) of rotation and the first derivative on a numerical invariant of an braking index of n = -(0.9 ± 0.2) according to a ratio (10).  period derivative at them is more at least on 1 -2 orders. Even before opening and the beginning of active observations of pulsars, F. Pacini [10] predicted that after explosion supernova has to be a very dense star kernel. Developing these representations, F. Pacini [11] noted that the rotation of neutron stars, which can be formed during the explosion of supernova, has to be connected with a surrounding gaseous cover. V. Kaspi selected about 30 pulsars, which can be associated with the supernova remnants [12]. Important criterion of accessory are speeds of the movement of a pulsar, is emphasized that appropriate measurements of the movement are extremely important for confirmation of authenticity of many associations of pulsars listed in the table with supernova. Proper motion measurements for young pulsars are crucial for deciding whether many of the associations are genuine. The most striking conclusion following examination of the associations between neutron stars and supernova remnants is that the paradigm that every young neutron star is like the Crab pulsar requires reconsideration.
Meanwhile, it should be noted that the pulsar B0531+21, in particular, was not included in the number of pulsars 366 analyzed by Hobbs et al. [5]: "Although PSRs B0531+21, <B1758-23 and B1757-24 > have been regularly observed, we cannot obtain a useful data set for the analysis described in this paper. These pulsars are therefore not included in our sample". Really, in this work the movement of pulsars as the factor of influence on the results of timing wasn't considered in common.

3) PSR B0531+21
The observations of pulsar B0531+21 in the radio range 111 MHz detect an anomalous deviation the observation period P and intervals PT that do not fit into a convergent power series (2) of parameterized components observed PT intervals (Figures 9-12). However, the observed rotation parameters obtained from the Equation (2), matched with the criteria of coherence and correspond to the typical monotone spin down with braking index n = -(0.9 ± 0.2). Meanwhile, the relative magnitude of the observed deviations of the period P during the two-year observations reached the order of 10 −9 (Figure 9(a)), which is 5 -7 orders of magnitude greater than the other pulsars in Table 1 Table 1, for PSR B0531+21 it is necessary to consider that the dimensionless parameter α of approximation of intervals (2) is associated with the movement of a pulsar here and is interpreted as the Doppler shift of the observed period of radiation owing to the movement of a pulsar under the influence of unbalanced forces in the radial directions. The relative shift of the period is expressed as follows: Note that the implied pulsar transverse velocity V t , which is determined by the angular pulsar displacement from the SNR Centre, is specified by Kaspi [12] as a constant value 125 km/s. This is several orders of magnitude greater than the observed value, and does not consider it significant and unpredictable variations.
The observed deviation of the emission period in the pulsar-driven Crab Nebula shows that part of the rotational energy of PSR B0531+21 is converted into energy of the observed accelerated motion, without any violating the monotony its slowing down and coherence of the periodic radiation. 4 We therefore suggest that slow glitches are not a unique phenomenon, but are caused by the same process as the timing noise". We

Summary
The analytical timing model, based on the parametric approximation of the numerical series ToAs by the convergent power series (2) pressed by the coincidence of the sign of the derivatives-positive in the time domain ( , P P   > 0) and negative in the frequency domain ( , υ υ   < 0). Accordingly, the braking index (10) for the observed pulsars is close to the negative unit: n = -(0.9 ± 0.2) and varies in a rather narrow range of values, while the rotation parameters they differ by several orders of magnitude. In essence, this is a numerical invariant for all pulsars, and propagates to those about them, in which the second derivative does not reach the detection threshold in observations and is calculated from the condition of consistency of the three rotation parameters in the time or frequency domain. The analytical model (2) excludes the anomalous distortions of the second derivative, inherent in its estimates from the barycentric residual deviations, The initial numerical series ToAs in the polynomial power series of intervals PTcalc, transformed with machine accuracy, is an analytical time scale from which the random, unmodified variations of the observed intervals PTobs are measured, as in the pulsars B1919+21 and B0809+74, as well as the pronounced spontaneous variations of the intervals defining the movement of PSR B0531+21 pulsar in the supernova remnant and PSR B1822-09, the spontaneous movement of which also suggests that it belongs to the supernova remnant. All observation data of 27 pulsars kindly allowed T.V. Shabanova for the staff of Pushchino Observatory became available for analysis in scientific research after considerable work on their systematization, which is executed by her colleague V.D. Pugachev.