OBSERVATIONS OF MICROGLITCHES IN HartRAO RADIO PULSARS

Available at: http://publications.ictp.it IC/2008/093 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS OBSERVATIONS OF MICROGLITCHES IN HartRAO RADIO PULSARS A.E. Chukwude 1 Department of Physics and Astronomy, University of Nigeria, Nsukka, Enugu State, Nigeria and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract Timing observations of on 26 radio pulsars, made at Hartebeesthoek Radio Astronomy Observatory (HartRAO) between 1984 and 2001, were analysed in order to probe the microglitch phenomenon in slow radio pulsars. The conventional manual identification method (which relies on visual inspection of plots of phase derivatives) was complimented by a new technique (in which the computer uses the Goodness of fit parameter ) to isolate candidate jumps in pulsar rotation frequency ( ν) and spin-down rate ( ν). Our results show that: (i) only 21 pulsars exhibit significant level of timing activity (with signal-to-noise ratio, SNR > 5) to be considered suitable for microglitch analysis; (ii) the manual and automatic techniques identified 60 and 240 events, respectively, whose sizes are simultaneously significant in rotation frequency (ν) and spin-down rate ( ν); (iii) the observed jumps ( ν, ν) have all possible signatures, with 73, 86, 75 and 66, respectively, having the signs (+, ), (+, +), (, +) and (, ); (iv) irrespective of signs the jumps have amplitudes in the range (0.018 66) 10 9 Hz and (0.00032 8.0) 10 15 Hz/s; (v) an apparent trend in the microglitch data suggests that, irrespective of signs, large jumps in ν correspond to large jumps in ν. Our results are discussed in the context of the current understanding of radio pulsar standard model. MIRAMARE TRIESTE December 2008 1 Junior Associate of ICTP.

1. INTRODUCTION Discrete discontinuities in the rotation of pulsars are broadly of two sorts: macroglitches and microglitches (Cordes, Downs & Krause-Polstorff 1988). Macroglitches (conventionally known as glitches) are characterised by sudden a increase in pulsar rotation frequency, ν, usually accompanied by an increase in magnitude of the spin-down rate, ν, (e.g. Lyne 1987; Lyne & Pritchard 1987; Flanagan, 1990; Lyne et al. 1996; Wang et al. 2001; Urama 2002). To date, about 280 glitches have been detected in 100, mostly young, pulsars (Lyne, Shemar & Smith 2000; Janssen & Stappers 2006; Melatos, Peralta & Wyithe 2008). Macroglitches have been extensively studied and are now associated with some unique features. Firstly, discrete discontinuities in the rotation frequency ( ν) and the spin-down rate ( ν) have definite signature: ( ν, ν) = (+, ) and have magnitudes, respectively, in the range of 10 10 ν 10 6 ν and 10 5 ν 10 1 ν (Janssen & Stappers 2006; Melatos et al. 2008). In some cases, the increases in the parameters are known to relax to their pre-glitch values on a wide range of time scales (Flanagan 1990, 1995; Lyne, 1987; Lyne, & Pritchard 1987; Cordes, Downs & Krause-Polstorff 1988; Lyne et al. 1996). Macroglitches are generally believed to originate from sudden and irregular transfer of angular momentum from the more rapidly rotating inner superfluid components to the relatively slowly spinning crust (Alpar et al. 1984, 1996; Ruderman et al. 1998). On the other hand, microglitches constitute a class of small amplitude, but resolvable, jumps in both, or either of the pulsar rotation frequency and its first time derivatives. Unlike the spectacular glitch events, the study of microglitches have received little attention in the last three dacades of pulsar timing observations and very little is known about this class of rotational discontinuities (Cordes et al. 1988; Chukwude 2002). Presently, only about 90 microglitches have been reported in less than 25 radio pulsars (Cordes & Downs 1985; Cordes et al. 1988; D Alessandro et al. 1995). The amplitudes of the jumps vary over a wide range, but are generally believed to be within ν < 10 10 ν and ν < 10 3 ν. However, unlike macroglitches, where the jumps have definite signature ( ν, ν) = (+, ), microglitches have all possible combination of signs (Cordes et al. 1988). However, developing a comprehensive observational description of the phenomenon of microglitches is particularly important for a number of reasons. It will definitely improve on the current statistics of microglitches, throwing more light on their incidence and amplitudes. Such description is required in order to ascertain the applicability of models involving the neutron star interior and magnetospheric torque fluctuations (e.g. Alpar et al. 1984; Cordes & Greenstein 1981; Arons 1981). This is necessary to constrain the possible origin of the phenomenon and crucial in understanding the relationship, if any, between glitches and microglitches. For instance, it is still unclear whether microglitches are simply the scaled-down version of macroglitches or are entirely different phenomenon, having different origin and requiring different physics (Cordes et al. 1988). Previous efforts to study microglitches relied only on the manual technique, in which phase residuals were numerically differentiated using different methods, to obtain frequency and 2

spin-down rate residuals, δν and δ ν, respectively, (Cordes & Downs 1985; Cordes et al. 1988; D Alessandro et al. 1995). Plots of δν and δ ν against time were visually inspected for scatters in excess of measurement uncertainties, indicating microglitch events. Save for the well sampled Vela pulsar data (Cordes et al. 1988), all other samples have long (> 30 d) data sampling intervals. This could introduce large uncertainties in the epochs of events, which ultimately limits their amplitude resolution of the events. In addition, there is a problem of completely averaging out some candidate microglitches in the process of numerical differentiation, given that appropriate data segments could span more than 200 days. In this paper, we present a more complete observational description of microglitches in a sample of 26 radio pulsars, whose data span about 16 years. Our data have a much improved sampling rate of 1 0.07 d 1, yielding a mean data frequency of 0.35 d 1. The short data sampling intervals and resultant high data frequency offer us an unprecedented flexibility in the microglitch analysis, allowing us to combine the manual search method with a more objective automatic technique. 2. OBSERVATIONS Regular timing observations of all objects in the current sample of 26 radio pulsars commenced at Hartebeesthoek Radio Astronomy Observatory between 1984 January and 1987 May and is still ongoing. However, a major interruption in HartRAO pulsar timing program occurred between 1999 June and 2000 August during a major hardware upgrade. Save for pulsars B0833 45 and B1641 45, which are on real time glitch monitoring program, no pulsar was observed during this period. However, these two pulsars were excluded from the current analysis. In HartRAO, pulse times of arrival (TOAs) were measured regularly at intervals 1 14 days using the observatory 26-m parabolic radio telescope. Radio pulses were recorded by a single 10 MHz bandwidth receiver centred near either 13 or 18 cm and no pre-detection dedispersion hardware was implemented. For each pulsar, detected pulses are smoothed with an appropriate filter-time constant, and folded over N p consecutive rotation periods to beat down the background noise. N p is different for different pulsars, but generally vary between 500 and 5000 for current sample, corresponding to integration time of 0.8 32 mins. Integration was usually started at a particular second by synchronization to the station clock, which is derived from a hydrogen maser and is referenced to UTC via the Global Position satellite (GPS). Three such on-line integrations were made during an observing session. Details of data acquisition and reduction at HartRAO have been described elsewhere (Flanagan 1995). The resulting topocentric arrival times were transformed to infinite observing frequency at the Solar System Barycentre (SSB) using the Jet Propulsion Laboratory DE200 solar system ephemeris (Standish 1982) using TEMPO software package (http://pulsar.princeton.edu/tempo). 3

3. TIMING ANALYSIS AND RESULTS Subsequent modelling of the barycentric times of pulse arrival (hereafter referred to as BTOAs) was accomplished with the HartRAO in-house timing analysis software (Flanagan 1995), which is based on the standard pulsar timing technique (e.g. Manchester & Taylor (1977). The BTOAs were fitted with a simple Taylor series of the form (1) φ(t) = φ 0 + ν(t t 0 ) + 1 2 ν(t t 0) 2, where φ 0 is the phase at an arbitrary time t 0, ν and ν are, respectively, the rotation frequency and its first time derivative. Basically, equation 1 was used to model the BTOAs to account for the deterministic pulsar spin down (Lorimer & Kramer 2005). The difference between the observed BTOAs and the best-fit model (the phase residuals) were subsequently used to obtain improved estimates of the pulsar spin-down parameters (ν and ν). The phase residuals, defined in the sense of model-predicted minus observed arrival times, for a selection of 6 HartRAO pulsars are shown in Fig. 1. The observed timing activity in five pulsars (B0450 18, B1133+16, B1426 66, B1451 68 and B2045 16) was found to be exceptionally weak to allow for meaningful microglitch analysis. For these pulsars, the rms timing noise, σ TN (σ TN = σr 2 σ2 W ; where σ R is the root-mean-squares phase residuals from 2nd-order polynomial fit to the entire data span and σ W is the rms white noise) is typically < 5 mp. These objects are characterised by signal-to-noise ratio (SNR = σ R /σ W ) < 3 and are excluded from further analysis in this paper. For the remaining 21 pulsars, the observed timing activity vary over a wide range, with SNR in the range of 5 700. The phase residuals of some of the objects, e.g. B0740 28, B1323 62, B1356 60 and B1749 28, are dominated by piecewise linear segments, suggesting enhanced discrete discontinuities (Cordes & Downs 1985). 3.1. BTOA Error Assignment. Assignment of correct errors to the pulse times of arrival data is very crucial in microglitch analysis. Since such analysis involves calculation of relatively small event sizes and the associated errors, correct BTOA error assignment will not only improve the sensitivity of the method, but also the reliability of the results. Correct BTOA error assignment is particularly useful in the new automatic microglitch detection technique, which uses the goodness of fit parameter (Q) as a tracer of microglitch event. In principle, correct BTOA errors will minimise the sensitivity of Q values to mere large scatter in BTOA data. The formal BTOA errors returned by the routine that extracts arrival times from observations are grossly underestimated (Flanagan 1995). In HartRAO, realistic BTOA errors are obtained from the real scatter in the data (Flanagan 1995). Short segments of data, spanning between 50 and 300 days, are modelled with 2nd-order polynomial plus a dispersion measure (DM) term. The lengths of the segments depended on the level of timing activity in the pulsar and the data sampling frequency, but are generally such that the resulting phase resdiduals are statistically equal to zero. The residuals are manually examined for possible outliers. Finally, BTOA errors are calculated from the real scatter in the phase residuals. Our estimator (following Flanagan 4

1995) is the two-sample variance averaged over 5 10 days, depending on the data sampling rates. The error obtained from each data segment was assigned to all BTOAs within the segment. The 13 and 18 cm observations were separately analysed, since the amplitudes of the observed scatter are different at 13 and 18 cm. BTOA errors calculated using this method are found to be larger than the those returned by the fitting routine, on average, by factors of 2 and 4 at 18 and 13 cm, respectively, and have been shown to represent more realistic error estimates(chukwude 2002; Urama 2002). (Chukwude 2002). 3.2. Manual identification of microglitches. The manual technique for microglitch detection relies on a visual inspection of plots of carefully calculated residuals of pulse spin frequency ( ν) and/or its first derivative ( ν) for scatters in excess of measurement uncertainty. The frequency and spin-down rate residuals can be calculated either directly, using coefficients of local fits to short spans of BTOAs (Cordes et al. 1988), or indirectly, by numerically differentiating phase residuals (e.g. Cordes & Downs 1985; D Alessandro et al. 1995). The high data frequency and the regularity of observation allowed us to implement the former method, which appears to be more sensitive to these small amplitude discontinuities. For each of the 21 pulsars, a time series of the pulsar rotation frequency (ν(t)) was calculated by performing weighted least-squares fits of a 2nd-order polynomial to short independent blocks of BTOA data. The block lengths were determined by the data sampling rates, the intrinsic scatter in the BTOAs and our desire to keep the formal errors in the ν to 1 part in 10 9. These conditions resulted in segment lengths, generally, in the range of 35 120 days. For each pulsar, ( ν) were obtained by subtracting a model of ν and ν fitted to the entire data span length from the ν data. The frequency residuals of 4 of the 21 pulsars, used in the manual identification of microglitches, are shown in Fig. 2. The epochs of candidate events are marked with arrows. Once a candidate event has been identified in this manner, short data segments spanning 200 400-d, and bracketing the candidate microglitch event were modelled with a 2nd-order polynomial. The resulting phase residuals were carefully examined for a sudden slope change, possibly, caused by the microglitch event. Fig. 3 shows typical examples of the phase residuals from short segments known to habour one or more candidate events, first identified with the ν plots. Once an improved epoch of the candidate event has been obtained in this manner, the relevant jump parameters (the epoch, the jumps in ν and ν, ν and ν, respectively, are estimated by modelling BTOA data extending between 200 and 400 d on both sides of an identified event with equation 1. This method resulted in two sets of ephemerides for each event. The amplitudes of events in ν and ν are estimated (following Cordes & Downs 1985; D Alessandro et al. 1995), respectively, from ν = ν post ν pre and ν = ν post ν pre, where the parameters ν pre and ν pre refer to the pre-event values extrapolated to the event epoch, while ν post and ν post are the corresponding post-event values, also extrapolated to the event epoch. 5

3.3. Automatic method for microglitches. The automatic search method employs a statistic, the goodness of fit parameter (Q), to determine when the model fit to BTOAs suddenly becomes too bad, probably due to discrete rotation discontinuity (Flanagan 1995). In this method, weighted least-squares fits are performed. About 50-d block of BTOAs was modelled with equation 1 plus a DM term. The block length was subsequently incremented in steps of 7 or 14 d, correponding roughly to the data sampling interval, until the value of Q fell below 0.1. Such low Q value was, generally, interpreted to mean a poor description of the data by the model due to a sudden jump in the pulsar rotation. The resulting ephemerides were written to a file. The procedure was repeated, starting from the epoch of this presumed event, until the end of the data set was reached (hereafter referred to as forward pass). The entire process identification of candidate events, writing of ephemerides to a file and restarting the model-fit is automated. Hence, once the parameters are properly set up, a complete pass through the data was achieved, with minimum manual intervention. To ensure that very small amplitude events were not averaged out, the 50-d start block was staggered by 14 d, at a time, and whole process was repeated. Finally, the process was repeated starting from the end of a data set going backwards until the start of the data is reached (hereafter referred to as backward pass). The result of these procedures were sets of ephemeris files containing ephemerides extending on both sides of the candidate microglitch events. The epoch of an event was taken as the two arrival times bracketing the event and the jump parameters, ν and ν calculated as the difference between consecutive ephemerides after extrapolating to the event epoch. 3.4. Testing the significance of identified microglitch events. Firstly, the amplitude of all identified candidate events in ν and/or ν was tested against the null hypothesis that they are mere accumulations of random walks of preferred signs. Following Cordes & Downs (1985), D Alessandro et al. (1995), the significance test was accomplished by comparing the amplitude of the observed events in ν and ν with the expected standard deviations of idealized large-rate random walk processes. Events are considered real, that is their amplitudes are too large to be due to random walk fluctuations, if ν σ ν N and ν σ ν N, where σ ν = S 1 t and σ ν = S 2 t are, respectively, the standard deviations of idealized random walks in ν and ν with strength parameters S 1 and S 2, respectively, N is an integer denoting the significance level of the test and t is the rise time of the event (e.g. Cordes & Helfand 1980; Cordes et al. 1988). Several authors (e.g. Cordes & Downs 1985, Cordes et al. 1988; D Alessandro et al. 1995) have demonstrated that N = 5 is an appropriate signficant level, as it correctly rejects spurious random walk-induced events which could mimic real microglitches. Given the similarity between the HartRAO and JPL pulsar data, in terms of observing frequencies and BTOA signal-to-noise, we also adopted N = 5 for the current test. Following Cordes & Downs (1985), D Alessandro et al. (1995), Chukwude (2002), we calculated strength parameters (S 1 and S 2, respectively) from BTOAs bracketed by any two identified 6

Figure 1. The phase residuals for a selection of 6 HartRAO radio pulsars, obtained as described in text, over the period between 1984 and 1999. candidate events (hereafter referred to as method A) and from entire BTAOs available for a pulsar (hereafter referred to as method B). Our presumption is that data set used in method A, probably, contains no discontinuity and hence has very low timing activity. On the other hand, the data set employed in method B incorporates all the rotational discontinuities, including all the microglitches. In principle, methods A and B are expected to yield, respectively the lower and upper limiting estimates for the strength parameters. The implication is that, though the strength parameter is variable, its true value should lie within the values obtained from the two methods, if random walks are indeed present in the data. The observed rms phase residuals are known to be dominated at short and long data span lengths, respectively, by events in ν and ν (Chukwude 2002 and references therein). Following Cordes & Downs (1985), D Alessandro et al. (1995), methods A and B were weighted heavily for testing the significance of events in ν and ν, respectively. In addition, a candidate event was considered real only if its amplitudes exceed the formal standard error by at least a factor of two. In principle, the necessary condition that must be satisfied by jumps in ν and ν is that ν > 2ǫ ν and ν > 2ǫ ν, respectively, where ǫ ν and ǫ ν are, the formal standard errors in ν and ν, respectively. This condition is believed 7

Figure 2. Time evolution of the frequency residuals ( ν(t), calculated as described in the text, for a selection of 4 HartRAO pulsars over the period between 1984 and 2001. Arrows indicate the points where scatters in ν are signficantly large (in excess of the measurement uncertainty) suggestive of the occurence of real jump in the pulsar rotation. Error bars are 1-σ formal standard errors. to be stringent enough to minimize the leakage of spurious events into the final sample of real microglitch events (Chukwude 2002). A combination of the manual and automatic techniques yielded, respectively, 60 and 240 events, whose amplitudes are simultaneously significant in both the rotation frequency and the spin-down rate. Additional 20 and 14 events were found to be significant in only ν and ν, respectively. The important features of the identified events are summarized as follows: (i) 160 and 140 jumps in rotation frequency ( ν) are positive and negative, respectively; (ii) the change in spin-down rate is ( ν) is positive for 153 and negative for 147 events; (iii) the amplitudes of the events in ν and ν vary over a wide range: 4.0 10 8 < ν < +6.6 10 8 Hz and 7.0 10 15 < ν < +8.0 10 15 Hz/s, respectively; (iv) irrespective of signs, the observed microjumps in the pulsar rotation frequency and spin down rate are characterised by mean values of 3.4 10 9 Hz and 4.0 10 16, respectively. 8

Figure 3. The phase residuals around some candidate microglitch events in the pulsar B0740 28, whose observed timing activity show strong evidence for enhanced microglitch activity. (v) a plot of the ν against ν, on a log log scales (Fig. 5) shows an apparent trend, in which large jumps in ν, on average, correspond to large jumps in ν, irrespective of signs. 4. DISCUSSIONS Only twenty one of the current sample of 26 HartRAO pulsars show fluctutaions in their arrival times at significant levels to be considered for meaningful microglitch analysis. For the remaining 5 objects, the timing activity over > 13 yr is almost indistinguishable from measurement uncertainty. The instrinsic scatter in the BTOAs of these objects have amplitudes in excess of 20 ms (see Chukwude 2007) and could certainly swamp low level timing activity in the pulsars. The unusually large scatter in the BTOAs could be attributed to hardware configuration: high observing frequencies (1.6 and 2.3 GHz) and the narrow receiver band of 10 MHz. Hence the received signals are expected to be extremely weak, even after a long integration time (Chukwude 2002). However, three pusars (1426 66, 1451 68 and 2045 16) observed for 7 years at much lower frequencies (650 and 800 MHz) with about 40 MHz receiver bandwith did not show appreciable level of timing activity, suggesting that they may be intrinsically weak activity objects, at least from our locations in the southern hemisphere. Previous independent analysis of the pulsar 1133+16 (Cordes & Down 1985) also reveals dominant instrinsic BTOA scatter. 9

The sub-sample of 21 pulsars, considered suitable for current analysis yielded unprecedented results in terms number and sizes of the identified microglitch events. The manual identification technique isolated a total 60 jumps that are simultaneously significant in both rotation frequency and spin-down rate. The automatic search method detected 240 events, whose sizes are significant in which was ν and ν. A breakdown of the event signatures shows that 73 have signatures ( ν, ν) = (+, ), corresponding to those of classical glitches (Lyne & Graham-Smith 1998). For 87, 75 and 66 events, ( ν, ν) = (+, +), (, ) and (, +), respectively. Our results also reveal large dispersions, up to 4 orders of magnitude, in sizes of the observed jumps both ν and ν. Irrespective of sign, the resolved ν and ν are in the range of 2 10 11 5 10 8 and 3 10 19 7 10 16, respectively, with mean values of 3.3 10 9 Hz and 4.2 10 16 s 2. On avearge, the manually identified microglitches have relatively larger sizes, however, they showed no preferred signs. In terms of amplitude and sign of events, our result is, generally, in good agreement with those of previous independent studies (Cordes & Downs 1985; Cordes et al. 1988; D Alessandro et al. 1995). Current analysis, nonetheless, produced remarkable improvements on the overall statistics of pulsar microglitches. The number of known microglitch events has increased by more than a factor of 5, from about 60 to 300, while extending the minimum detectable size of microglitch in ν down to 10 19 Hz. Current improvements in the statistics of microglitches is as expected, given some unique features of the current HartRAO pulsar data. These include long time, effectively 13 16 yrs, coverage. Short sampling intervals that vary between 1 and 14 days, in sharp contrast with > 30 d characteristic of JPL and Tasmanian data (Cordes & Downs 1985; D Alessandro et al. 1993, 1995). Shorter sampling intervals lead to improved event epoch and, ultimately, to better resolution of the jump parameters (Flanagan 1995; Chukwude 2002). Short sampling intervals and long data span length would imply a high data frequency of BTOAs. The high sampling frequency of current HartRAO data, with a mean value of 0.35 d 1, allowed us some flexibility in the analysis. For manual search method, ν was directly calculated from the coefficient of local fits to short data spans, yielding unambiguous error estimates. Moreover, the automatic technique could not have been effective if not for short sampling intervals. One fundamental problem associated with microglitch observation is the difficulty of distinguishing between real and spurious event. Previous studies (Cordes & Downs 1985; Cordes et al. 1988; D Alessandro et al. 1995) have relied largely on the signficance tests, using timing noise strength parameters, to determine real events. However, there is strong dependence of this method on the rise time of the event t (Cordes & Downs 1985). This implies that the standard deviations could possibly be overestimated or underestimated for very small or very large t, respectively. This could cause leakage of spurious events with large amplitudes or rejection of very small amplitude candidate events. Our data appear to favour the former scenario, given the short sampling intervals and the resulting short rise times. With a mean t 5 d, it is 10

Figure 4. The log log scatter plot of the magnitude of the 300 identified microglitches in the rotation frequency ν and its first time derivative ν. Key: + and denote microglitches identified, respectively, with manual and automatic search techniques. feared that both σ ν and σ ν could have been underestimated, causing significant leakage of spurious events into the real events (e.g. Chukwude 2002). However, we believe the above 2-σ standard formal error condition imposed on the size identified microglitches is severe enough to remove the false events. This is particularly true given the robust and direct method employed in estimating the parameter errors. Further analysis on the identified microglitches shows that, irrespective of sign, large discontinuities in ν, on average, correspond to large jumps in ν. However, the amplitude scatter in ν ν plot appears to be larger at the lower left hand corner, corresponding to extremely small amplitude jumps in both parameters. At that level, contribution of rotational activity, like timing noise, are expected to be very significant. 5. SUMMARY The phenomenon of microglitches in slow radio pulsars have been probed deeply using the manual and automatic search techniques. An extensive analysis of a sample of 21 pulsars, whose timing data are characterised by long time spans ( 16 yrs) and high sampling frequency ( 0.35 d 1, yielded a phenomenal 300 microglitches in both rotation frequency and its first derivative. The jumps show no preferred signs and have amplitudes that span about 8 and 10 orders of magnitude in ν and ν, respectively. The microglitch data reveal an apparent relationship between the magnitude of the events in in ν and ν, which lends further support to our claim that these events consitute real discontinuities in the rotation of these pulsars. 11

Acknowledgments This work was completed during my visit to the Abdus Salam International Centre for Theoretical for Physics, Trieste, Italy, as a Junior Associate. I am grateful to the Swedish International Development Agency (SIDA) for supporting my visit to ICTP with a travel grant. The author wishes to acknowledge the Director of HartRAO and Dr. C. S. Flanagan for giving him access to the Observatory pulsar data. References [1] Alpar, M.A., Anderson, P.W., Pines, D., Shaham, J.,1984, ApJ, 278, 533 [2] Alpar, M.A., Chau, H.F., Cheng, K.S., Pines, D. 1996, ApJ, 459, 706 [3] Arons, J., in IAU Symposium 95, Pulsars, ed. W. Sieber and R. Wielebinski (Dordrecht: Reidel), 69 [4] Chukwude, A.E., 2002, Ph.D. Thesis, University of Nigeria, Nsukka [5] Chukwude, A.E., 2007, Chin. J. Astron. Astrophys. 7, 521 [6] Cordes, J.M., Greenstein, G., 1981, ApJ, 245, 1060 [7] Cordes, J.M., Helfand, D.J., 1980, ApJ, 239, 640 [8] Cordes, J.M., Downs, G.S., 1985, ApJS, 59, 343 [9] Cordes, J.M., Downs, G.S., Krause-Polstorff, J., 1988, ApJ, 330, 847 [10] D Alessandro, F., McCulloch, P. M., King, E.A., Hamilton, P. A., McConnell, D., 1993, MNRAS, 261, 883 [11] D Alessandro, F., McCulloch, P. M., Hamilton, P. A., Deshpande, A. A., 1995, MNRAS, 277, 1046 [12] Flanagan, C. S., 1995, PhD thesis, Rhodes University, Grahamstown, South Africa [13] Flanagan, C.S., 1990, Nat, 345, 416 [14] Janssen, G.H., Stappers, B.W., 2006, A&A, 457, 611 [15] Lorimer, D. & Kramer, M. 2005, Handbook of Pulsar Astronomy (Cambridge University Press) [16] Lyne, A.G., 1987, Nat, 326, 569 [17] Lyne, A. G., Pritchard, R. S., 1987, MNRAS, 229, 223 [18] Lyne, A. G., Graham-Smith, F., 1998, Pulsar Astronomy (Cambridge University Press) [19] Lyne, A.G., Pritchard, R.S., Kaspi, V.M., Bailes, M., Manchester, R.N., Taylor, H., Arzoumanian, Z., 1996, MNRAS, 281, 14 [20] Lyne, A.G., Shemar, S.L., Smith, F.G., 2000, MNRAS, 315, 534 [21] Melatos, A., Peralta, C., Wyithe, S.B., 2008, ApJ, 672, 1103 [22] Manchester, R. N., Taylor, J. H., 1977, Pulsars (San Francisco: Freeman) [23] Standish, E. M., 1982, Celestial Mechanics, 26, 181 [24] Ruderman, M., Zhu, T., Chen, K., 1998, ApJ, 492, 267 [25] Urama, J.O., 2002, MNRAS, 330, 58 [26] Wang, N., Wu, X., Manchester, R.N., Zhang, J., Lyne, A.G., Yusup, A., 2001, Chin. J. Astron. Astrophys., 1, 195 12