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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.

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 [

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 [

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 [

Abnormally high, not giving in to a physical explanation of residuals and second derivative variations are also discussed in the extensive publication of Hobbs, Lyne and Kramer [

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.

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 [

Pulsar time (PT) intervals of any pulsed periodic radiation event N express by the function of its rotation parameters P 0 , P ˙ , P ¨ in the time domain:

P T ( P 0 , P ˙ , P ¨ ) = P 0 N + 1 2 P 0 P ˙ N 2 + 1 6 ( P 0 2 P ¨ − 2 P 0 P ˙ 2 ) N 3 , N = 1 , 2 , 3 , ⋯ (1)

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 ( P 0 ≈ 1 s , P ˙ ≈ 10 − 15 - 10 − 16 s ⋅ s − 1 , P ¨ ≈ 10 − 28 - 10 − 29 s − 1 ) within the two-year span of observations linear component reaches 10^{7} - 10^{8} s, square component is about 10 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 are the observable parameters P 0 ∗ , P ˙ , P ¨ , corresponding to the condition of convergence of the series (1). The equation of the observed intervals PT_{i} in accordance with the relation (1) is:

P T i = ( 1 + α i ) ( P 0 ∗ N + 1 2 P 0 ∗ P ˙ N 2 + 1 6 ( P 0 ∗ 2 P ¨ − 2 P 0 ∗ P ˙ 2 ) N 3 ) i (2)

Here are: PT_{i} is the numerical values of the observed intervals obtained from the planetary ephemerides; P 0 ∗ , P ˙ , P ¨ are the observed pulsar rotation parameters obtained by solving Equation (2); α i 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 P 0 ∗ , P ˙ , P ¨ is unique for each pulsar, wherein convergence of polynomial series (1) is reached. The 3-component series in the second brackets of the Equation (2) is an analytical form P T c a l c of the observed intervals P T o b s , which corresponds to numerical value P Т i / ( 1 + α i ) defining by the parametric approximation of PT_{i} by the calculated intervals with computing precision (inaccuracy less than 1 ns):

P T o b s = P T i ; (3)

P T c a l c = ( P 0 ∗ N + 1 2 P 0 ∗ P ˙ N 2 + 1 6 ( P 0 ∗ 2 P ¨ − 2 P 0 ∗ P ˙ 2 ) N 3 ) i (4)

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:

P T o b s = ( 1 + α P T ) P T c a l c , (5)

The numerical value α P T , as well as rotation parameters, is also the solution of the Equation (2). It characterizes unmodelled timing noise in the sequence of observed intervals P T o b s , it isn’t correlated with an analytical pulsar time scale P T c a l c 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) is form-invariant under coordinate transformations, in which the numerical values of the observed rotation period are coinciding in the barycentric and topocentric coordinate systems at the same epochs of the local time. Left part of the Equation (2) consists the observed topocentric T T o b s or barycentric T B o b s intervals. The right part contains the intervals T T c a l c or T B c a l c , which are calculated by the observed rotation parameters obtained by approximation of T T o b s or T B o b s , respectively.

So, the analytical model of timing (2) determines the unique consistent values P 0 ∗ , P ˙ , P ¨ corresponding to the coherence of periodic radiation of pulsar, on the observed ToAs only. Random variations of ToAs due to imprecise knowledge of physical variables and unmodelled noise of observations have no significant impact on the modelled intervals of coherent periodic radiation of pulsar, which are identified with a machine accuracy and sub-nanosecond resolution by the consistent rotation parameters of the pulsar.

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.

It is evident, that the value of the period P 0 ∗ 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 t = P 0 ∗ N . Included in the Equation (2) the numerical value P 0 ∗ represents the rotation period at the initial epoch of observations, and any current epoch t takes a value consistent with the derivatives P ˙ , P ¨ :

Р ( t ) = P 0 ∗ + Р ˙ 0 ⋅ t + 1 2 P ¨ ⋅ t 2 , t = P 0 ∗ N , 1 < N < ∞ (6)

Relation (6) corresponds to a property of the coherence of pulsar periodic radiation and hence indicates monotonous slowing down of the pulsar rotation ( P ˙ > 0, P ¨ > 0), which is an observational sign of convergence of the polynomial power series (1). Monotonic slowing down of the pulsar caused the negative values of derivatives υ ˙ < 0, υ ¨ < 0 in the frequency domain, respectively.

The approximation of the observed intervals (2) also detects the random variations of the observed emission period Р ∗ ( t ) . The magnitude of variation is expressed as follows:

Δ Р ( t ) = α ( t ) ( P 0 ∗ + Р ˙ 0 ⋅ t + 1 2 P ¨ ⋅ t 2 ) (7)

By integration of function in an interval {tj, ti}, (here i > j), the deviation of observed intervals is calculated:

Δ P T j , i = 1 P 0 ∗ ∫ t j t i Δ P ( t ) d t (8)

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:

Δ P T j , i = ( α i + α j ) 2 ( P T i − P T j ) . (9)

The main characteristic of the observed slowing down of the pulsar rotation is the braking index n describing dependence of angular speed Ω = 2 π υ = 2 π / P 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:

n = Ω Ω ¨ Ω ˙ 2 = υ υ ¨ υ ˙ 2 , or n = 2 − P P ¨ P ˙ 2 (10)

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 accepted by the authors, there were obtained the measured braking indices for the non-recycled pulsars range from −287,986 to +36,246 with a mean of −1713 and a median of 22. If to restrict the sample of the 30-years data spanning, there were obtained braking indices ranging from −1701 to +36,246 with a mean of +3750 and median of +29. Out of a sample of 366 pulsars, 193 (53 per cent) have a positive υ ¨ value and the remaining 173 (47 per cent) have negative υ ¨ [

Revealed abnormal dispersion of a braking index of pulsars in different sources demonstrate first of all discrepancies exactly of the second derivative challenging additional studying and specification exactly about it.

Parametrical approximation of observed intervals was applied to timing of pulsars В0809+74, В1919+21, В0834+06, J1509+5531, B2217+47, В0329+54, В0531+21 in 2007-2015 on the BSA radio telescope, which operated at frequencies close to 111.3 MHz. A 64-channel radiometer with a channel bandwidth of 20 kHz was used for the pulsar observations. The data were sampled at intervals of 2.56 or 1.28 ms. The BSA radio telescope, made up of a linearly polarized transit antenna with a beam size of about (3.5/cosδ) min, provided the duration of the observing session ranging from 3 to 11 minutes at different pulsar declination δ. During this time, individual pulses were summed synchronously with a predicted topocentric pulsar period to form the mean pulse profile in each 20 kHz channel. After dispersion removal, all the channel profiles were summed to produce a mean pulse profile for the given observing session. The topocentric arrival times of the pulses for each observing session were calculated by cross-correlating the mean pulse profile with a standard low-noise template.

By approximation of observed intervals according to the Equation (2) were obtained the consistency parameters of rotation P , P ˙ , P ¨ which are given in

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

PSR | Р (MJD) [s] | P ˙ [s∙s^{−1}] | P ¨ [s^{−1}] | n |
---|---|---|---|---|

В0809+74 | 1.29224154962235 (56,280.98428) 1.29224137163571 (43,989.63277) | 1.676∙10^{−16} 1.676∙10^{−16} | <10^{−30 } 5.89∙10^{−32 } | Not defined ?0.71 |

В1919+21 | 1.33730279821909 (54,477.39095) | 1.34809∙10^{−15} | 3.99573∙10^{−30} | ?0.94 |

В0834+06 | 1.27377145381349 (54,103.96609) | 6.79918∙10^{−15} | 1.07448∙10^{−28} | ?0.96 |

J1509+5531 | 0.739683736756101 (54,104.2354) | 4.99821∙10^{−15} | 9.82117∙10^{−29} | ?0.91 |

В2217+47 | 0.538470617227171 (54,117.498697) | 2.76421∙10^{−15} | 4.30247∙10^{−29} | ?1.03 |

В0329+54 | 0.714521031744271 (54,104.750676) | 2.04959∙10^{−15} | 1.67089∙10^{−29} | ?0.84 |

B0531+21 | 0.033638877858401 (55,218.787512) 0.033403347409400 (48,743.0) | 4.2096∙10^{−13} | 1.57877∙10^{−23} | ?0.997 ?0.976 |

pulsar B1919+21 ( P ¨ near 10^{−30} s^{−1}) requires at least 7 - 8 years of observations to detect P ¨ . Even after 9 years of observations Pulsar B0809+74 within 2007-2009 and 2012-2017 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.

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.

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 [

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 2 and the JPL DE200 ephemeris. The topocentric arrival times collected at the PRAO and the geocentric arrival times obtained from the archival JPL timing data were all corrected to the barycenter of the solar system at infinite frequency. The position and the proper motion required for this correction were taken from [

Further are given the results of timing of 4 pulsars, two of which (B1919+21 and B0809+74) contain both in

At these pulsars, apparently from ^{−1}) on previous data sets of long-term observations of Shabanova and later observations on the BSA FIAN radio telescope.

1) B1919+21.

The results of our timing processing of PSR B1919+21 are presented in Figures 1-4. We get topocentric and by means of TIMAPR [

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

The second derivative was determined according to ratios (3), (4) and (5) direct detecting by observed intervals of PT_{obs}. Considering that parameters P

and Pdot are known, the component of the second derivative has been just subtracted from PT_{obs} of a component of this interval calculated in the known parameters P and Pdot:

P T o b s ( P 2 d o t ) = P T o b s − P T c a l c ( P , P d o t ) . (11)

Detecting of the second derivative was made equal in both systems, observed ToAs were counted is independent of the general initial impulse defining an initial epoch of observations of all subsequent impulses of TTobs and TBobs in each reference system.

The result of detecting of the P2dot component in topocentric and barycentric systems is given in

The numerical value of the second derivative establishes approximation is 3.99573∙10^{−30} s^{−1}. It coincides with the P2dot value obtained in later observations in _{0}(TT) = 1.337302232070039186 s (MJD 49616.69854), in the barycentric reference frame P_{0}(TB) = 1.337302232070342499 s (MJD 49616.70114). The specified distinctions of the period are defined by the difference of an initial epoch of observations (TT-TB)_{0} = -224.4 s on

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

reaches a limit of discrete resolution of the approximating calculator that in this case corresponds to picosecond permission of observed intervals, thus that not correlated noise of timing don’t exert impact on the result of the approximation.

Thus, by means of analytical models of timing (2) the compatibility of observed intervals of radiation in a topocentric and a barycentric reference frame is reached. Observed intervals, due to the coherence of radiation of a pulsar, are expressed through the rotation parameters identical for coinciding epoch and coordinated within any duration of the observations. Therefore, the periods and its derivatives received as a result of the approximation of observed intervals stay mutually coordinated within any duration, irrespective of the choice of an initial epoch of observations. It is only necessary to provide the duration of observations sufficient for sure detecting of the second derivative taking into account the required accuracy of the parametric approximation of observed intervals.

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

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

parameters of rotation, agreed upon at any epoch, no matter what events were chosen for observations.

Compared to the PSR P1919+21, the timing of the PSR B0809+74 reveals the following features:

• Higher level of unmodelled timing noise and about an order of magnitude smaller the value of the detected component of the second derivative, relatively low value of S/N ≈ 10;

• The more accurate installation of the initial MJD epoch of observations in topocentric and barycentric reference frames as a condition for the equivalence of the cubic components of P2dot and the consistency of the initial period Po at the 34-year interval is defined;

• The value of P2dot is approximately 2 orders of magnitude smaller than that of the B1919+21 pulsar and 3 - 4 orders of magnitude lower than that of the pulsars B0834+06, J1509+5531, B2217+47, B0329+54 from the

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 ^{−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

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.

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 ˙ .

The observations of Kaspi et al. 1994 were performed on observatory in Arecibo at frequencies of 1.4 GHz of PSR B1855+09 ((J1857+0943), and, respectively 1.4 and 2.4 GHz of PSR B1937+21 (J1939+2134). The method of precision timing the moments of observed impulses of PSR B1855+09 with an accuracy of 0.8 mks and PSR B1937+21 with an accuracy of 0.2 mks were measured. In

PSR | В0105+68 | J1214-5814 | J1638-4344 | J1739-1313 | J1852+0305 | J1901+1306 | J2136-1606 |
---|---|---|---|---|---|---|---|

Р,[s] | 1.283 | 0.909 | 1.12 | 1.215 | 1.32 | 1.83 | 1.22 |

P ˙ , [s∙s^{−1}] | 4.8 × 10^{−17} | 5 × 10^{−17} | 2.5 × 10^{−17} | 8.17 × 10^{−17} | 1.0 × 10^{−16} | 1.3 × 10^{−16} | 1.6 × 10^{−17} |

PSR | Р(MJD) [s] | P ˙ [s∙s^{−1}] | P ¨ [s^{−1}] |
---|---|---|---|

В1937+21 | 0.001557806468819794 (47500.000000015133) | 1.051193∙10^{−19} | 2.085∙10^{−35} |

В1855+09 | 0.00536210045404154 (47526.000000067884) | 1.78363∙10^{−20} | 1.745∙10^{−37} |

J1640+2224 | 0.00316331586776034 (55000) | 2.8161∙10^{−21} | 7.431∙10^{−39} |

J2145-0750 | 0.01605242391938130 (55000) | 2.978810^{−20} | 1.625∙10^{−37} |

J0613-0200 | 0.003061844087680279 (55000) | 9.5899∙10^{−21} | 8.830∙10^{−38} |

J1713+0747 | 0.004570136526356975 (53729) | 8.52961∙10^{−21} | 4.680∙10^{−38} |

J1643-1224 | 0.00462164152493627 (54500) | 1.84617∙10^{−20} | 2.168∙10^{−37} |

are specified. At PSR B1937+21, besides, the period format is expanded to 18 decimal signs. For these pulsars the second derivatives in frequency domain are specified:

for PSR B1937+21 F2dot = (13.2 ± 0.3) E-27 s^{−3} or P2dot= -(3.2 ± 0.7) E-32 s^{−1}

for PSR B1855+09 F2dot = - (1.0 ± 0.9) E-27 s^{−3} or P2dot= (2.87 ± 2.58) E-32 s^{−1}.

The calculation for a ratio (10) showed that the numerical values of the second derivative P ¨ of the period of millisecond pulsars В1937+21, В1855+09, J1640+2224, J2145-0750, J0613-0200, J1713+0747, J1643-1224 are in the range 10^{−35} - 10^{−38} s^{−1}. It is at least 7 - 8 orders less, than at second pulsars, and is much lower than a detection threshold that excludes a possibility of measurement P ¨ directly by the timing of millisecond pulsars.

In work of Hobbs et al. (2010) [

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).

N | PSR J | FO [Hz] | Fdot [s^{−2}] | F2dot [s^{−3}] | Epoch [MJD] | F2dot [s^{−3}] (n_{avrg}= ?0.94) | |
---|---|---|---|---|---|---|---|

1 | J0034-0534 | 532.713 | −1.4093E−15 | −0.043(10)E−24 | 51096.0 | −3.50E−33 | |

2 | J0218+4232 | 430.461 | −1.4340E−14 | −0.007(8)E−24 | 51268.0 | −4.49E−31 | |

3 | J0613-0200 | 326.601 | −1.02E−15 | −0.009(4)E−24 | 51240.0 | −2.99E−33 | |

4 | J0751+1807 | 287.458 | −0.643E−15 | −0.002(4)E−24 | 51389.0 | −1.35E−31 | |

5 | J1012+5307 | 190.268 | −0.620E−16 | 0.0031(12)E−24 | 51331.0 | −1.90E−33 | |

6 | J1024-0719 | 193.716 | −6.9523E−16 | 0.007(6) E−24 | 51422.0 | −2.35E−33 | |

7 | J1300+1240 | 160.810 | −2.957E−15 | −0.002(3)E−24 | 50788.0 | −5.11E−32 | |

8 | J1455-3330 | 125.200 | −3.81E−16 | 0.008(4)E−24 | 51190.0 | −1.09E−33 | |

9 | J1643-1224 | 216.373 | −8.65E−16 | −0.0043(19)E−24 | 51251.0 | −3.25E−33 | |

10 | J1713+0747 | 218.812 | −4.09E−16 | 0.0019(13)E−24 | 51318.0 | −7.16E−34 | |

11 | J1730-2304 | 123.110 | −3.06E−16 | −0.0053(13)E−24 | 51240.0 | −7.15E−34 | |

12 | J1744-1134 | 245.426 | −5.389E−16 | −0.0076(12)E−24 | 51490.0 | −1.11E−33 | |

13 | J1804-2717 | 107.032 | −4.68E−16 | −0.008(5)E−24 | 51440.0 | −1.93E−33 | |

14 | J1823-3021A | 183.823 | −1.14336E−13 | 0.522(4)E−24 | 50719.0 | −6.68E−29 | |

15 | J1824-2452 | 327.406 | −173.519E−15 | 0.047(7)E−24 | 50238.0 | −8.64E−33 | |

16 | J1857+0943 | 186.494 | −0.620E−15 | 0.15(7)E−26 | 50027.0 | −1.94E−33 | |

17 | J1911-1114 | 275.805 | −1.065E−15 | 0.016(9)E−24 | 51513.0 | −3.87E−33 | |

18 | J1939+2134 | 641.928 | −4.3314E−14 | 0.0174(8)E−24 | 49389.0 | −2.75E−30 | |

19 | J1955+2908 | 163.048 | −0.789E−15 | −0.002(4)E−24 | 49456.0 | −3.59E−33 | |

20 | J2019+2425 | 254.160 | −4.54E−16 | −0.032(16)E−24 | 51351.0 | −7.61E−34 | |

21 | J2051-0827 | 221.796 | −6.26E−16 | 0.011(11)E−24 | 51481.0 | −1.66E−33 | |

22 | J2124-3358 | 202.794 | −8.46E−16 | 0.011(8)E−24 | 51284.0 | −3.32E−33 | |

23 | J2229+2643 | 335.816 | −1.71E−16 | −0.011(6)E−24 | 51625.0 | −8.19E−35 | |

24 | J2317+1439 | 290.255 | −2.04E−16 | −0.005(7)E−24 | 51656.0 | −1.35E−34 | |

25 | J2322+2057 | 207.968 | −4.18E−16 | 0.007(11)E−24 | 51335.0 | −795E−34 |

Such computing procedure was performed in relation to all millisecond pulsars in _{avrg} = -0.94 are placed. Thereby the coordinated values of the second derivative in according to the sign, corresponding to monotonous slowing down, are obtained.

The specific features of pulsars associated with the supernova remnants, are larger magnetic fields, shorter spin periods, than the typical pulsars. These pulsars differ much bigger, than at usual second ones, in energy losses therefore the 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 [

V. Kaspi selected about 30 pulsars, which can be associated with the supernova remnants [

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. [

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} (^{−6} [

These spontaneous deviations of the observed emission period that is not associated with changes in the pulsar rotation parameters, can be interpreted as a Doppler shift due to the motion of the pulsar because of the effects of unbalanced forces in radial directions. The radial velocity is then determined by the deviation α i of the period P obtained from Equation (2) and the light velocity c: V i = α i ⋅ c (

Parameters of the movement―the speed, acceleration and movement of a pulsar, as well as rotation parameters, are defined by the Equation (2). Only,

unlike other observed pulsars from

α i = Δ P i / P 0 ∗ .

The beam velocity of the movement of a pulsar is defined on an epoch of observation as follows:

V i = α i ⋅ c , here сis light speed.

Displacement L i of a pulsar within some interval counted from an initial epoch in an assumption of constant speed is defined by the following expression:

L i = V i ⋅ P T i = α i ⋅ c ⋅ P T i .

Displacement L j , i of a pulsar between randomly chosen observed events in an interval {РТj,РТi} is calculated on the ratio (9).

Over the span of nearly 6-year observations of pulsar B0531+21 within 2010-2015, its velocity changed in range, from 0.1 m/s or less up to 500 m/s or more; displacement of pulsar along the line of sight changed from a few hundred meters up to 100 million km and more. Calculated from deviation in velocity within the intervals of 1 - 7 days between observations, acceleration changed in range, from 10^{−8} m∙s^{−2} and less up to 10^{−4} m∙s^{−2} or more.

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 [

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) PSR B1822-09

This pulsar from a list of 27 PSR of Shabanova et al. (2013) [^{−1} which 1 - 2 orders more than value, usual for second pulsars, and there is closer to Pdot = 4.096E−13 s∙s^{−1} of a pulsar B0531+21 which is located in the supernova remnant.

Shabanova, as she notes, was the first to detect slow glitches of the pulsar B1822-09. Slow glitches, according to Shabanova et al. [

We carried out the processing of observation data of a pulsar of B1822-09 on the analytical model of time processing, similar to PSR B0531+21, having taken as initial the topocentric ToAs PSR B1822-09 of Shabanova within an interval of 1994-2005. Besides, to this series, we added data of later observations, performed on BSA FIAN in 2015-2016. Results of processing are given in the Figures 13-15.

On the coordinated values of parameters of rotation P_{0} = 0.6897223949 s (MJD 46800.0) and Pdot = 5.432E−14 s∙s^{−1} from the PSR Catalogue [^{−1} that corresponds to an braking index n = - 0.94 and to monotonous slowing down of rotation were found and measured the spontaneous deviations of the observed emission period which are interpreted in this case by analogy with the PSR B0531+21 as a Doppler shift due to the motion of the pulsar. Observed spontaneous deviations of the period fall just on an interval 1994-2005

to which the estimated glitch or noise of timing specified by the author’s of this pulsar belongs.

Further observations of a pulsar of B1822-09 in 2015-2016 showed that the beam speed of a pulsar remained without significant changes and after a 10-year break in observations. Respectively acceleration became several orders less in comparison with an interval of 1994-2000 (

The analytical timing model, based on the parametric approximation of the numerical series ToAs by the convergent power series (2), is an effective and reliable tool for direct extraction and testing of the consistent physical parameters of the pulsar rotation. For each pulsar, there is a single combination of numerical values P , P ˙ , P ¨ ( υ , υ ˙ , υ ¨ ) that are the same in any arbitrarily chosen observing reference system within an unlimited duration of observations, both current and retrospective.

The consistency of the rotation parameters does not depend on the choice of the initial epoch; in any case, the initial period value corresponding to the chosen sample of the initial epoch, taking into account the derivatives of the period (6), is determined by approximating the intervals PTobs.

A prior knowledge of the parameters of pulsar rotation from previous observations, confirmed by subsequent observations, indicates the coherence of periodic pulsar radiation and the necessary condition for the convergence of the observed PTobs series.

The main problems in approximating the observed intervals were related to the exact determination of the second derivative, since the cubic component of the polynomial power series of intervals PTcalc, due to a very small, only a few microseconds or less, reaches the threshold of reliable detection only after several tens of years of observations.

A direct consequence of the consistency of parameters P 0 * , P ˙ , P ¨ of coherent radiation is the monotonicity of the deceleration of pulsar rotation, expressed 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.

Authors are thankful to the technical and scientific staff of the Pushchino Radio Astronomy Observatory for the active collaboration and comprehensive supporting in caring out of the regular long-term observations of the pulsars.

Avramenko, A.E., Losovsky, B.Y., Pugachev, V.D. and Shabanova, T.V. (2018) Compatibility of the Observed Rotation Parameters of Radio Pulsars on Long Time Scales. International Journal of Astronomy and Astrophysics, 8, 24-45. https://doi.org/10.4236/ijaa.2018.81003