J. Astrophys. Astr. (1984) 5, 307–316

Braking Index Diagnostics of Pulsars. I. Alignment, Counteralignment and Slowing-down Noise Pranab Ghosh

Tata Institute of Fundamental Research, Homi Bhabha Road,

Bombay 400005 Received 1984 February 19; accepted 1984 May 25

Abstract. We show that the strong correlation observed between the braking indices (n) and the slowing-down ages (τ) of pulsars is inconsistent with counteralignment between their rotation and magnetic axes, but that the data on pulsars with positive braking indices is consistent with alignment. Alternatively, slowing-down noise can quantitatively account for the data on all pulsars except the Crab and the Vela, and so for the apparent |n| ~ τ2 correlation observed for the older pulsars. Key

words:

pulsars—braking index—alignment—noise

1. Introduction Whether the magnetic and rotation axes of pulsars align or counteralign with age has been a much debated but unsettled point. The many theoretical calculations performed to date give widely different results. Davis & Goldstein (1970) and Michel & Goldwire (1970) showed that, for a perfectly conducting spherical star, the two axes align on a braking timescale, a result that can be generalized to a fluid body whose rotation deformation follows the instantaneous rotation axis (Macy 1974). In Goldreich’s (1970) analysis of an imperfectly rigid, rotationally and/or magnetically distorted neutron star, the magnetic axis remains fixed relative to the symmetry axis, while the latter precesses about the instantaneous rotation axis with an amplitude which is increased or decreased by the radiation-reaction torque depending on whether the angle between the magnetic and rotation axes is large or small. Cracking and creeping of the crust, as well as internal friction, can cause alignment or counteralignment (Goldreich 1970; Chau & Henriksen 1971). Macy’s (1974) analysis suggests that alignment or counteralignment can occur when a magnetic distortion moves through the star due to the kind of instability which is believed to cause polar wandering on the Earth, there being alignment or counteralignment according as the external polodial magnetic field of the – pulsar is larger or smaller than l/√3 times the (volume-weighted-averaged) internal torodial magnetic field. Detailed considerations of the properties of neutron-star matter have led to contradictory results. Jones (1975, 1976) argues that alignment occurs after the star has cooled sufficiently to make the temperature-dependent dissipative torque negligible. On the other hand, Flowers & Ruderman (1977) suggest that, through internal fluid motions, the magnetic field relaxes to a minimum energy configuration of counteralignment. The observational situation is uncertain. The presence of a strong interpulse in the Crab was used to argue that rotation and magnetic axes were nearly orthogonal in this

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pulsar (Radhakrishnan & Cooke 1969). Several authors (Lyne, Ritchings & Smith 1975; Jones 1976) have argued that the strong increase in the downward slope of evolutionary · tracks in the P–P plane which is caused by alignment is in better agreement with the · · P–P data than the Standard (Ρ ∝ P–l) fixed-inclination slope. Further, both Jones (1976) and Candy & Blair (1983) have argued that the observed widening of the beam with age argues in favour of alignment. On the other hand, Macy (1974) identifies type D pulsars as those which are aligning and type C pulsars as those which are counteraligning, by relating their general properties to his theory. Flowers & Ruderman (1977) have argued in favour of counteralignment by showing that, among pulsars with drifting subpulses, the drift direction reverses with age, as is predicted by their model. ·· Measurements of the second derivative of the pulsar period, P, (or, equivalently, the ·· 2 ·· frequency second derivative ν, or the braking index, n = vv/v ) provide us with another potential diagnostic tool for testing alignment or counteralignment. In this paper, we · as a introduce the plot of the braking index, n, versus the slowing-down age, τ = Ρ/Ρ, powerful way of using the second derivative data. The available data on 19 pulsars shows that 11 of them have positive braking indices and 8 of them have negative indices and that there are strong, essentially identical, positive correlations between |n| and τ in the two classes. We show that aligning and counteraligning pulsars have distinctly different evolutionary tracks on this plot. For those pulsars which have positive braking indices, the data are consistent with alignment on timescales ~ 103–105 yr, and inconsistent with counteralignment. The data on pulsars with negative braking indices are inconsistent with either alignment or counteralignment. Hence there is no evidence in the present braking index data for a general counteralignment in pulsars, contrary to Nowakowski’s (1983b) suggestion. We then investigate the alternative hypothesis that the observed braking indices are dominated by noise processes. We show that, with the exception of the young Crab and Vela pulsars, the observed correlation between |n| and τ for both classes of older pulsars is consistent with the hypothesis that most of the apparent ν·· arises from a noise in the slowing-down process, and the observed |n| ∝ τ2 correlation for the older (τ ≳ 105 yr) pulsars is thus adequately explained by this hypothesis.

2. Alignment and counteralignment The braking index of a pulsar slowing down via magnetic dipole radiation can be written (Macy 1974) as (1) Here I0 and R are respectively the moment of inertia and radius of the (rotationally) undistorted star, Μ is the stellar mass, Ω is the angular frequency, χ is the angle between the rotation and magnetic axes and τB is the timescale for magnetic-field decay. The first term on the right-hand side of Equation (1) is that given by the ‘standard’ model: a perfect sphere with constant magnetic field inclined at a constant angle to the rotation axis. The second term comes from rotational distortion, the effects of which are discussed in detail in Cowsik, Ghosh & Melvin (1983). The first part of the third term

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describes the effects of alignment/counteralignment, and the second part those of magnetic-field decay. Rotational distortion and counteralignment decrease n from its standard value, while field decay and alignment increase it above this value. Since n has been accurately measured only for pulsars with Ρ ≳ 33 ms (indeed, Ρ ~ 1 s for most of the sample), the effects of rotational distortion are negligible for this sample (Cowsik, Ghosh & Melvin 1983), and we shall neglect these effects in what follows. For alignment, we adopt Jones’ model, which gives (2) τa being the alignment timescale. This gives (3) For counteralignment, we adopt (Flowers & Ruderman 1977; Nowakowski 1983b) the form (4) τc being the counteralignment timescale. This gives (Nowakowski 1983b) (5) the relation between the actual age and the slowing-down age being

(6) Here τ0 is the value of τ at t = 0. The variations of na and nc with τ are shown in Fig. 1. We have chosen various values of τB in the range 106–107 yr, in agreement with current understanding of the magnetic-field decay processes, and have tried various values of τa and τc in the range 103–107 yr. We see that, due to alignment and field decay (particularly alignment), n can become orders of magnitude larger than its standard value of 3 and thus explain the observed large positive values of n (and so the large ·· negative values of P: see Gullahorn & Rankin 1982; Manchester et al. 1983; Nowakowski 1983a, b). Counteralignment does reduce n below its standard value (and ·· can make Ρ > 0), but does not lead to negative values of n. The counteralignment curves shown in Fig. 1 correspond to χ0 = 0 (alignment at t = 0), which maximizes the counteralignment effects. At large τ, the field-decay effects dominate, and the counteralignment curves also rise to large positive values of n. For the alignment curves shown in Fig. 1, different values of τB in the 106–107 yr range make no visible difference. Similarly, for the counteralignment curves shown, different values of τc in the 103–107 yr range make little visible difference. Hence, the former curves are labelled by .. values of τa, and the latter, by those of τB. The data on 19 pulsars for which ν has been measured are shown in Fig. 1 (Gullahorn & Rankin 1978a, 1982; Demiański & Prószyński 1979 and the references therein; Downs 1981) and displayed in Table 1. We have not included those pulsars for which only an upper limit to ν·· is available. The data fall into two distinct classes, eleven pulsars with n > 0 and eight with n < 0 (Table 1). In

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Figure 1. Braking index–slowing-down-age correlation. Plotted is the logarithm of the absolute value of the braking index against the logarithm of age. Also shown are the theoretical curves for alignment (solid lines), those for counteralignment (dotted lines) and that given by the Standard n = 3 model (dashed line). The alignment curves are labelled by alignment timescales, τa; different values of the field decay timescale, τΒ, in the range 106 yr–107 yr make no visible difference for these curves. The counteralignment curves are labelled by τΒ; different values of the counteralignment timescale, τc, in the range 103 yr–107 yr make little visible difference for these curves. Filled circles: measurements quoted with error bars as shown. Open circles: measurements quoted without error bars.

each class, there is a strong correlation (indeed an almost obvious linear relation) between log | n | and log τ. The theoretical curves for alignment are consistent with the n > 0 data for τa ~ 103–105 yr (the effect on these curves of varying τB in the range 106–107 yr is negligible). The curves for counteralignment are inconsistent with the n > 0 data. (The case of Crab pulsar, which has n ≃ 2.5, is a special one, and is discussed later). The n < 0 data are in complete disagreement with either alignment or counteralignment. It is thus clear from the braking index data available at the present time that there is no evidence for general counteralignment in pulsars. This is in contradiction with the conclusion of Nowakowski (1983b), who did not attempt a quantitative comparison of theory with observation.

3. Slowing-down noise The secular nature of the apparent second derivatives of pulsar periods has been questioned before (see Cordes & Helfand 1980 and references therein). The facts that almost half the known sample shows unexplained large negative braking indices, and that both halves show essentially the same strong correlation between the absolute

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value of the braking index and the slowing-down age, make us reconsider this point. Random walks in phase, frequency, or the first derivative of frequency can give rise to apparent second derivatives. For the last case, that of the so-called slowing-down noise, the apparent second derivative, 〈 v··R 〉, is related to the rms residual, σR(2,Τ), of a leastsquares second-order polynomial fit to the timing data over an interval of length Τ through the inequality (Cordes & Helfand 1980) (7) This immediately gives (8) where (9) is the ‘random-noise age’. τ R is determined by the nature of the noise processes and by the observation interval Τ. σR (2, Τ) and T have been given by Cordes & Helfand (1980) for 16 of the 19 pulsars considered here. In Table 2, we give the values of τR and 〈|nR|〉, and compare 〈|nR|〉 with the observed |n| in Fig. 2 and Table 2. For all but the Crab and Vela pulsars, there is excellent agreement, |n|obs always lying below the upper limit on 〈|nR|〉 given by Equation (8) with two slight exceptions. For Crab and Vela, 〈|nR|〉max are far below the observed values. We thus arrive at the conclusion that for aging pulsars (τ ≳ 105 yr) which show apparent braking indices whose magnitudes are very much larger than 3, the second derivative may very well be dominated by slowingdown noise, whereas for the young Crab and Vela pulsars, the second derivative cannot be so dominated. We stress that Equation (8) does not imply 〈|nR|〉 ∝ τ2, since the random-noise age Table 2. Slowing-down noise.

Data in columns 2 and 3 are taken from Cordes & Helfand (1980)

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Figure 2. Same as Fig. 1, but comparing the observed braking indices (circles) with those expected from slowing-down noise (triangles). Filled circles: measurements quoted with error bars as shown. Open circles: measurements quoted without error bars. For vela, the slowing down noise component itself has an uncertainty as shown (Downs 1981). Also shown are the ‘noise-band’ (the area between the two solid lines which corresponding to the maximum and minimum values of τR in Table 2) and the relation |n| — (τ/105 yr)2 (dashed line) which describes most pulsars adequately.

τR depends on the observation interval T, on the properties of neutron stars and on those of pulsar magnetospheres. However, for most of the pulsar sample considered here, τR ~ 105 yr (see Table 2), so that |n| ≃ (τ/105 yr)2 gives a good description to 16 of the 19 pulsars (the exceptions being Crab, Vela and PSR 0611 + 22) in Fig. 2, where the ‘noise band’ corresponds to the range of τR given in Table 2. On the basis of the slowingdown noise hypothesis, we expect future data on older pulsars to fall in the ‘noise band’ of Fig. 2, and to cluster around the line |n| = (τ/10s yr)2. This point is brought out again by Fig. 3, where the observables |nohs|/τ2 and τR are plotted against each other. The shaded region, which corresponds to slowing-down noise, essentially accounts for all pulsars except Crab and Vela, which stand out far above noise. Also, most points cluster around τR = 105 yr, thus giving the apparent |n| ~ τ2 correlation as before. Plots like Fig. 3 should be useful in quickly separating the slowing-down-noise component in future data on pulsars. 4. Discussion Our main conclusions are: 1. There is no evidence for general counteralignment in the present data on the braking indices of pulsars.

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Figure 3. |n|/τ2 vs τR, both on logarithmic scales, for 16 pulsars (see Table 2). Filled circles: measurements quoted with error bars as shown. Open circles: measurements quoted without error bars. Shaded region: slowing-down noise contribution, bounded by the line |n|τ–2 = τ R–2. Note the particularly large uncertainties for Vela (Downs 1981).

2. Pulsars with n > 0 are consistent with alignment on timescales ~ 103–105 yr and field decay on timescales ~ 106–107 yr. 3. For all but the Crab and Vela pulsars, the data can be accounted for by the hypothesis that slowing-down noise dominates the measured second derivative of frequency. We predict that future data on older pulsars will fall in the ‘noise band’ of Fig. 2 and cluster around the |n| = (τ/105 yr)2 line. In view of the last conclusion we do not give consideration to the short alignment timescales found from braking indices (~ 103–105 yr). However, it is now clear from Equation (3) and Fig. 1 why short field-decay timescales ~ 103 yr were inferred by Nowakowski (1983a) on the hypothesis that the large values of n were entirely caused by field-decay effects. It is also disturbing to note that short alignment timescales imply a near alignment, and therefore a difficulty in the pulse generation, in the older pulsars. Thus, even for n > 0, alignment is not a particularly attractive mechanism for producing large braking indices. Braking-index diagnostics can become most useful in studying the basic processes underlying pulsar braking torques when the random noise age τR can be made greater than the age of the pulsar (see Equation 8), since different physical processes will give characteristically different ‘tracks’ on Fig. I. Increasing the observation interval Τ increases τR (see Equation 9).

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Of the two pulsars (Crab and Vela) in which slowing-down noise cannot account for most of the second derivative, Crab is easily understandable in terms of the standard model, since magnetospheric (Roberts & Sturrock 1972) or other effects, including those of counteralignment, can easily account for the small ( ≃ 0.5) deviation from the standard model value of n = 3. Vela remains a problem, since none of the host of processes which are ordinarily thought to be able to affect the braking index (see Manchester & Taylor 1977 for a list of these processes and their effects) will increase it to n 3. These processes include multipole electromagnetic radiation, gravitational quadrupole radiation, magnetospheric effects, rotational distortion, and pulsar proper motion (for a discussion of the inadequacy of the last process, see Gullahorn & Rankin 1982 and references therein). Alignment on a short timescale (see Fig. 1) is implausible in view of Vela’s age (see above). We note that other explanations for large braking indices suggested so far are tentative. Gullahorn & Rankin’s (1978b) braking torque variations are formally similar to slowing-down noise. Doppler effects due to planets with long (Pb ~ 50 yr) orbital periods (Demiański & Prószyński 1979) will produce an apparent correlation |n| ~ τ2 over observation intervals Τ ≪ Pb for a collection of pulsars with similar planetary masses, orbital periods and inclination angles, but n will eventually show periodic variations. Finally, even if a neutron star could sustain triaxial distortions to the moment of inertia (due, for example, to misaligned rotation and magnetic axes), to achieve distortions of sufficient magnitude to explain the observed braking indices (Tademaru 1981, personal communication to Gullahorn & Rankin) is problematic, particularly for the older pulsars, in view of the known results for spheroidal distortions (see above and Cowsik, Ghosh & Melvin 1983). What other evidence is available on the alignment/counteralignment question of our pulsar sample? As pointed out by Lyne, Ritchings & Smith (1975), a traditional · · ‘alignment’ plot of PP vs P/P would be unreliable because of the narrow range of Ρ values considered in our sample. However, we do note that the two classes (n > 0 and n < 0) of pulsars seem to be very similar in other properties, including their pulse-widths, W(Manchester & Taylor 1981). It is known that WP29/42 goes like cosec χ and so should increase with age (Jones 1976) for aligning pulsars, and that W itself should show a characteristic rise with age for old aligning pulsars (Candy & Blair 1983). We have performed these tests on our sample, from which we draw the tentative conclusions that the two classes behave essentially identically in these tests, and that both seem to be undergoing alignment on the canonical timescale ~ 106–107 yr.

Acknowledgements It is a pleasure to thank Professor R. Cowsik for stimulating discussions and Mr. P. K. Ghosh for vital encouragement. An anonymous referee called my attention to an instructive plot in Lyne, Ritchings & Smith (1975).

References Candy, B. N., Blair, D. G. 1983, Μ on. Not. R. astr. Soc, 205, 281. Chau, W. Y., Henriksen, R. N. 1971, Astrophys. Lett., 8, 49. J.A.A.– 9

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Cordes, J. Μ., Helfand, D. J. 1980, Astrophys. J., 239, 640. Cowsik, R., Ghosh, P., Melvin, Μ. Α. 1983, Nature, 303, 308. Davis, L., Goldstein, Μ. 1970, Astrophys. J., 159, L81. Demiański, Μ., Prószyński, Μ. 1979, Nature, 282, 383. Downs, G. S. 1981, Astrophys. J., 249, 687. Flowers, Ε., Ruderman, Μ. Α. 1977, Astrophys. J., 215, 302. Goldreich, P. 1970, Astrophys. J., 160, L11. Groth, Ε. J. 1975, Astrophys. J. Suppl. Ser., 29, 453. Gullahorn, G. E., Rankin, J. M. 1978a, Astrophys. J., 83, 1219. Gullahorn, G. E., Rankin, J. M. 1978b, Bull. am. astr. Soc., 9, 562. Gullahorn, G. E., Rankin, J. M. 1982, Astrophys. J., 260, 520. Helfand, D. J., Taylor, J. H., Backus, P. R., Cordes, J. Μ. 1980, Astrophys. J., 237, 206. Jones, P. Β. 1975, Astrophys. Space Sci., 33, 215. Jones, P. B. 1976, Nature, 262, 120. Lyne, A. G., Ritchings, R. T, Smith, F. G. 1975, Mon. Not. R. astr. Soc, 171, 579. Macy, W. W. 1974, Astrophys. J., 190, 153. Manchester, R. Ν., Newton, L. Μ., Hamilton, P. Α., Goss, W. Μ. 1983, Mon. Not. R. astr. Soc, 202, 269. Manchester, R. Ν., Taylor, J. Η. 1977, Pulsars, W. Η. Freeman, San Francisco, pp. 121, 188. Manchester, R. N., Taylor, J. Η. 1981, Astrophys. J., 86, 1953. Michel, F. C, Goldwire, H. C. 1970, Astrophys. Lett., 5, 21. Nowakowski, L. A. 1983a, Astr. Astrophys., 118, 29. Nowakowski, L. A. 1983b, Astr. Astrophys., 127, 259. Radhakrishnan, V., Cooke, D. J. 1969, Astrophys. Lett., 3, 225. Roberts, D. H., Sturrock, P. A. 1972, Astrophys. J., 173, L33.

Braking Index Diagnostics of Pulsars. I. Alignment, Counteralignment and Slowing-down Noise Pranab Ghosh

Tata Institute of Fundamental Research, Homi Bhabha Road,

Bombay 400005 Received 1984 February 19; accepted 1984 May 25

Abstract. We show that the strong correlation observed between the braking indices (n) and the slowing-down ages (τ) of pulsars is inconsistent with counteralignment between their rotation and magnetic axes, but that the data on pulsars with positive braking indices is consistent with alignment. Alternatively, slowing-down noise can quantitatively account for the data on all pulsars except the Crab and the Vela, and so for the apparent |n| ~ τ2 correlation observed for the older pulsars. Key

words:

pulsars—braking index—alignment—noise

1. Introduction Whether the magnetic and rotation axes of pulsars align or counteralign with age has been a much debated but unsettled point. The many theoretical calculations performed to date give widely different results. Davis & Goldstein (1970) and Michel & Goldwire (1970) showed that, for a perfectly conducting spherical star, the two axes align on a braking timescale, a result that can be generalized to a fluid body whose rotation deformation follows the instantaneous rotation axis (Macy 1974). In Goldreich’s (1970) analysis of an imperfectly rigid, rotationally and/or magnetically distorted neutron star, the magnetic axis remains fixed relative to the symmetry axis, while the latter precesses about the instantaneous rotation axis with an amplitude which is increased or decreased by the radiation-reaction torque depending on whether the angle between the magnetic and rotation axes is large or small. Cracking and creeping of the crust, as well as internal friction, can cause alignment or counteralignment (Goldreich 1970; Chau & Henriksen 1971). Macy’s (1974) analysis suggests that alignment or counteralignment can occur when a magnetic distortion moves through the star due to the kind of instability which is believed to cause polar wandering on the Earth, there being alignment or counteralignment according as the external polodial magnetic field of the – pulsar is larger or smaller than l/√3 times the (volume-weighted-averaged) internal torodial magnetic field. Detailed considerations of the properties of neutron-star matter have led to contradictory results. Jones (1975, 1976) argues that alignment occurs after the star has cooled sufficiently to make the temperature-dependent dissipative torque negligible. On the other hand, Flowers & Ruderman (1977) suggest that, through internal fluid motions, the magnetic field relaxes to a minimum energy configuration of counteralignment. The observational situation is uncertain. The presence of a strong interpulse in the Crab was used to argue that rotation and magnetic axes were nearly orthogonal in this

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pulsar (Radhakrishnan & Cooke 1969). Several authors (Lyne, Ritchings & Smith 1975; Jones 1976) have argued that the strong increase in the downward slope of evolutionary · tracks in the P–P plane which is caused by alignment is in better agreement with the · · P–P data than the Standard (Ρ ∝ P–l) fixed-inclination slope. Further, both Jones (1976) and Candy & Blair (1983) have argued that the observed widening of the beam with age argues in favour of alignment. On the other hand, Macy (1974) identifies type D pulsars as those which are aligning and type C pulsars as those which are counteraligning, by relating their general properties to his theory. Flowers & Ruderman (1977) have argued in favour of counteralignment by showing that, among pulsars with drifting subpulses, the drift direction reverses with age, as is predicted by their model. ·· Measurements of the second derivative of the pulsar period, P, (or, equivalently, the ·· 2 ·· frequency second derivative ν, or the braking index, n = vv/v ) provide us with another potential diagnostic tool for testing alignment or counteralignment. In this paper, we · as a introduce the plot of the braking index, n, versus the slowing-down age, τ = Ρ/Ρ, powerful way of using the second derivative data. The available data on 19 pulsars shows that 11 of them have positive braking indices and 8 of them have negative indices and that there are strong, essentially identical, positive correlations between |n| and τ in the two classes. We show that aligning and counteraligning pulsars have distinctly different evolutionary tracks on this plot. For those pulsars which have positive braking indices, the data are consistent with alignment on timescales ~ 103–105 yr, and inconsistent with counteralignment. The data on pulsars with negative braking indices are inconsistent with either alignment or counteralignment. Hence there is no evidence in the present braking index data for a general counteralignment in pulsars, contrary to Nowakowski’s (1983b) suggestion. We then investigate the alternative hypothesis that the observed braking indices are dominated by noise processes. We show that, with the exception of the young Crab and Vela pulsars, the observed correlation between |n| and τ for both classes of older pulsars is consistent with the hypothesis that most of the apparent ν·· arises from a noise in the slowing-down process, and the observed |n| ∝ τ2 correlation for the older (τ ≳ 105 yr) pulsars is thus adequately explained by this hypothesis.

2. Alignment and counteralignment The braking index of a pulsar slowing down via magnetic dipole radiation can be written (Macy 1974) as (1) Here I0 and R are respectively the moment of inertia and radius of the (rotationally) undistorted star, Μ is the stellar mass, Ω is the angular frequency, χ is the angle between the rotation and magnetic axes and τB is the timescale for magnetic-field decay. The first term on the right-hand side of Equation (1) is that given by the ‘standard’ model: a perfect sphere with constant magnetic field inclined at a constant angle to the rotation axis. The second term comes from rotational distortion, the effects of which are discussed in detail in Cowsik, Ghosh & Melvin (1983). The first part of the third term

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describes the effects of alignment/counteralignment, and the second part those of magnetic-field decay. Rotational distortion and counteralignment decrease n from its standard value, while field decay and alignment increase it above this value. Since n has been accurately measured only for pulsars with Ρ ≳ 33 ms (indeed, Ρ ~ 1 s for most of the sample), the effects of rotational distortion are negligible for this sample (Cowsik, Ghosh & Melvin 1983), and we shall neglect these effects in what follows. For alignment, we adopt Jones’ model, which gives (2) τa being the alignment timescale. This gives (3) For counteralignment, we adopt (Flowers & Ruderman 1977; Nowakowski 1983b) the form (4) τc being the counteralignment timescale. This gives (Nowakowski 1983b) (5) the relation between the actual age and the slowing-down age being

(6) Here τ0 is the value of τ at t = 0. The variations of na and nc with τ are shown in Fig. 1. We have chosen various values of τB in the range 106–107 yr, in agreement with current understanding of the magnetic-field decay processes, and have tried various values of τa and τc in the range 103–107 yr. We see that, due to alignment and field decay (particularly alignment), n can become orders of magnitude larger than its standard value of 3 and thus explain the observed large positive values of n (and so the large ·· negative values of P: see Gullahorn & Rankin 1982; Manchester et al. 1983; Nowakowski 1983a, b). Counteralignment does reduce n below its standard value (and ·· can make Ρ > 0), but does not lead to negative values of n. The counteralignment curves shown in Fig. 1 correspond to χ0 = 0 (alignment at t = 0), which maximizes the counteralignment effects. At large τ, the field-decay effects dominate, and the counteralignment curves also rise to large positive values of n. For the alignment curves shown in Fig. 1, different values of τB in the 106–107 yr range make no visible difference. Similarly, for the counteralignment curves shown, different values of τc in the 103–107 yr range make little visible difference. Hence, the former curves are labelled by .. values of τa, and the latter, by those of τB. The data on 19 pulsars for which ν has been measured are shown in Fig. 1 (Gullahorn & Rankin 1978a, 1982; Demiański & Prószyński 1979 and the references therein; Downs 1981) and displayed in Table 1. We have not included those pulsars for which only an upper limit to ν·· is available. The data fall into two distinct classes, eleven pulsars with n > 0 and eight with n < 0 (Table 1). In

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Figure 1. Braking index–slowing-down-age correlation. Plotted is the logarithm of the absolute value of the braking index against the logarithm of age. Also shown are the theoretical curves for alignment (solid lines), those for counteralignment (dotted lines) and that given by the Standard n = 3 model (dashed line). The alignment curves are labelled by alignment timescales, τa; different values of the field decay timescale, τΒ, in the range 106 yr–107 yr make no visible difference for these curves. The counteralignment curves are labelled by τΒ; different values of the counteralignment timescale, τc, in the range 103 yr–107 yr make little visible difference for these curves. Filled circles: measurements quoted with error bars as shown. Open circles: measurements quoted without error bars.

each class, there is a strong correlation (indeed an almost obvious linear relation) between log | n | and log τ. The theoretical curves for alignment are consistent with the n > 0 data for τa ~ 103–105 yr (the effect on these curves of varying τB in the range 106–107 yr is negligible). The curves for counteralignment are inconsistent with the n > 0 data. (The case of Crab pulsar, which has n ≃ 2.5, is a special one, and is discussed later). The n < 0 data are in complete disagreement with either alignment or counteralignment. It is thus clear from the braking index data available at the present time that there is no evidence for general counteralignment in pulsars. This is in contradiction with the conclusion of Nowakowski (1983b), who did not attempt a quantitative comparison of theory with observation.

3. Slowing-down noise The secular nature of the apparent second derivatives of pulsar periods has been questioned before (see Cordes & Helfand 1980 and references therein). The facts that almost half the known sample shows unexplained large negative braking indices, and that both halves show essentially the same strong correlation between the absolute

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value of the braking index and the slowing-down age, make us reconsider this point. Random walks in phase, frequency, or the first derivative of frequency can give rise to apparent second derivatives. For the last case, that of the so-called slowing-down noise, the apparent second derivative, 〈 v··R 〉, is related to the rms residual, σR(2,Τ), of a leastsquares second-order polynomial fit to the timing data over an interval of length Τ through the inequality (Cordes & Helfand 1980) (7) This immediately gives (8) where (9) is the ‘random-noise age’. τ R is determined by the nature of the noise processes and by the observation interval Τ. σR (2, Τ) and T have been given by Cordes & Helfand (1980) for 16 of the 19 pulsars considered here. In Table 2, we give the values of τR and 〈|nR|〉, and compare 〈|nR|〉 with the observed |n| in Fig. 2 and Table 2. For all but the Crab and Vela pulsars, there is excellent agreement, |n|obs always lying below the upper limit on 〈|nR|〉 given by Equation (8) with two slight exceptions. For Crab and Vela, 〈|nR|〉max are far below the observed values. We thus arrive at the conclusion that for aging pulsars (τ ≳ 105 yr) which show apparent braking indices whose magnitudes are very much larger than 3, the second derivative may very well be dominated by slowingdown noise, whereas for the young Crab and Vela pulsars, the second derivative cannot be so dominated. We stress that Equation (8) does not imply 〈|nR|〉 ∝ τ2, since the random-noise age Table 2. Slowing-down noise.

Data in columns 2 and 3 are taken from Cordes & Helfand (1980)

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Figure 2. Same as Fig. 1, but comparing the observed braking indices (circles) with those expected from slowing-down noise (triangles). Filled circles: measurements quoted with error bars as shown. Open circles: measurements quoted without error bars. For vela, the slowing down noise component itself has an uncertainty as shown (Downs 1981). Also shown are the ‘noise-band’ (the area between the two solid lines which corresponding to the maximum and minimum values of τR in Table 2) and the relation |n| — (τ/105 yr)2 (dashed line) which describes most pulsars adequately.

τR depends on the observation interval T, on the properties of neutron stars and on those of pulsar magnetospheres. However, for most of the pulsar sample considered here, τR ~ 105 yr (see Table 2), so that |n| ≃ (τ/105 yr)2 gives a good description to 16 of the 19 pulsars (the exceptions being Crab, Vela and PSR 0611 + 22) in Fig. 2, where the ‘noise band’ corresponds to the range of τR given in Table 2. On the basis of the slowingdown noise hypothesis, we expect future data on older pulsars to fall in the ‘noise band’ of Fig. 2, and to cluster around the line |n| = (τ/10s yr)2. This point is brought out again by Fig. 3, where the observables |nohs|/τ2 and τR are plotted against each other. The shaded region, which corresponds to slowing-down noise, essentially accounts for all pulsars except Crab and Vela, which stand out far above noise. Also, most points cluster around τR = 105 yr, thus giving the apparent |n| ~ τ2 correlation as before. Plots like Fig. 3 should be useful in quickly separating the slowing-down-noise component in future data on pulsars. 4. Discussion Our main conclusions are: 1. There is no evidence for general counteralignment in the present data on the braking indices of pulsars.

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Figure 3. |n|/τ2 vs τR, both on logarithmic scales, for 16 pulsars (see Table 2). Filled circles: measurements quoted with error bars as shown. Open circles: measurements quoted without error bars. Shaded region: slowing-down noise contribution, bounded by the line |n|τ–2 = τ R–2. Note the particularly large uncertainties for Vela (Downs 1981).

2. Pulsars with n > 0 are consistent with alignment on timescales ~ 103–105 yr and field decay on timescales ~ 106–107 yr. 3. For all but the Crab and Vela pulsars, the data can be accounted for by the hypothesis that slowing-down noise dominates the measured second derivative of frequency. We predict that future data on older pulsars will fall in the ‘noise band’ of Fig. 2 and cluster around the |n| = (τ/105 yr)2 line. In view of the last conclusion we do not give consideration to the short alignment timescales found from braking indices (~ 103–105 yr). However, it is now clear from Equation (3) and Fig. 1 why short field-decay timescales ~ 103 yr were inferred by Nowakowski (1983a) on the hypothesis that the large values of n were entirely caused by field-decay effects. It is also disturbing to note that short alignment timescales imply a near alignment, and therefore a difficulty in the pulse generation, in the older pulsars. Thus, even for n > 0, alignment is not a particularly attractive mechanism for producing large braking indices. Braking-index diagnostics can become most useful in studying the basic processes underlying pulsar braking torques when the random noise age τR can be made greater than the age of the pulsar (see Equation 8), since different physical processes will give characteristically different ‘tracks’ on Fig. I. Increasing the observation interval Τ increases τR (see Equation 9).

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Of the two pulsars (Crab and Vela) in which slowing-down noise cannot account for most of the second derivative, Crab is easily understandable in terms of the standard model, since magnetospheric (Roberts & Sturrock 1972) or other effects, including those of counteralignment, can easily account for the small ( ≃ 0.5) deviation from the standard model value of n = 3. Vela remains a problem, since none of the host of processes which are ordinarily thought to be able to affect the braking index (see Manchester & Taylor 1977 for a list of these processes and their effects) will increase it to n 3. These processes include multipole electromagnetic radiation, gravitational quadrupole radiation, magnetospheric effects, rotational distortion, and pulsar proper motion (for a discussion of the inadequacy of the last process, see Gullahorn & Rankin 1982 and references therein). Alignment on a short timescale (see Fig. 1) is implausible in view of Vela’s age (see above). We note that other explanations for large braking indices suggested so far are tentative. Gullahorn & Rankin’s (1978b) braking torque variations are formally similar to slowing-down noise. Doppler effects due to planets with long (Pb ~ 50 yr) orbital periods (Demiański & Prószyński 1979) will produce an apparent correlation |n| ~ τ2 over observation intervals Τ ≪ Pb for a collection of pulsars with similar planetary masses, orbital periods and inclination angles, but n will eventually show periodic variations. Finally, even if a neutron star could sustain triaxial distortions to the moment of inertia (due, for example, to misaligned rotation and magnetic axes), to achieve distortions of sufficient magnitude to explain the observed braking indices (Tademaru 1981, personal communication to Gullahorn & Rankin) is problematic, particularly for the older pulsars, in view of the known results for spheroidal distortions (see above and Cowsik, Ghosh & Melvin 1983). What other evidence is available on the alignment/counteralignment question of our pulsar sample? As pointed out by Lyne, Ritchings & Smith (1975), a traditional · · ‘alignment’ plot of PP vs P/P would be unreliable because of the narrow range of Ρ values considered in our sample. However, we do note that the two classes (n > 0 and n < 0) of pulsars seem to be very similar in other properties, including their pulse-widths, W(Manchester & Taylor 1981). It is known that WP29/42 goes like cosec χ and so should increase with age (Jones 1976) for aligning pulsars, and that W itself should show a characteristic rise with age for old aligning pulsars (Candy & Blair 1983). We have performed these tests on our sample, from which we draw the tentative conclusions that the two classes behave essentially identically in these tests, and that both seem to be undergoing alignment on the canonical timescale ~ 106–107 yr.

Acknowledgements It is a pleasure to thank Professor R. Cowsik for stimulating discussions and Mr. P. K. Ghosh for vital encouragement. An anonymous referee called my attention to an instructive plot in Lyne, Ritchings & Smith (1975).

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