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The Spin Down of the Radio Pulsars. Bralting Index V.S. BESKIN, A.V. GUREVICH, AND YA.N. ISTOMIN Lebedev Physical Ins citute At present, the value of the retardation dP/dt is well mown for most radio pulsars. It is negative for all cases except one and is of the order of 10-~5. That single case is when the pulsar, which Is located in the star globular system, can have a considerable acceleration leading to the opposite sign of P = dP/dt due to the Doppler effect (Wolszczan et al. 1989~. Careful measurements of the period P also allow one to determine the variation of this retardation with me murse of time P = d2P/dt2. We results of these measurements are usually represented in the form of the dimensionless retardation index n = Q QIQ2 = 2 _ pp~p2 (Q is the angular velociW). The data for 21 pulsars are grven in the table. The parameter n is strongly undetermined both in value and sign In all cases except for four pulsars. Changes of the rotation period P and the inclination angle X, the angle between the axes of rotation and the magnetic moment are caused by two processes: me regular retardation and notation due to deviation from the strict spherical shape of the neutron star. Q = AQa 9(X, Q); X = Q f (X) + [XQ,1 COS Ant ' (1) where ~ and A are constants and Ok and [X are the frequency and amplitude of the notations. The functions g~x,Q) and fix) depend on the retardation mechanism. Id particular, for the magnetodipole losses gkx) = sin2X, f(X) = Age, ~ = 3. Here we consider mainly losses, which are caused by the currents Dowing in the magnetosphere of the neutron star and being closed on the 14

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HIGH-ENERGY ASTROPHYSICS TABLE PSR P P 1015 n = 2 _ ppJp2 0329 + 54 0.714 2.05 (4.81 +0.18) 103 0531 ~ 21 0.033 4.22 lo2 2.515 + 0.005 0540 + 23 0.246 1.54 101 (2.5 + 0.05) 10 0540 - 69 0.050 4.79 102 3.6 + 0.S 0611 + 22 0.335 5.96 10 ~3.5 lo2 0823 + 26 0.531 1.72 -1 104 0833 - 45 0.089 1.25 lo2 (~.2 + 1.~) 10~ 0950 ~ 08 0.253 2.29 10~ ~(-5.2 + 0.~) IO4 1508 + 55 0.740 5.03 3.25 103 1509 - 58 0.150 1.490 103 2.83 + 0.03 1541 + 09 0.748 4.3 lo-l (-2.5 + 0.2) 1 1604 - 00 0.422 3.06 10~1 (1.8 + 0.4) 104 1859 + 03 0.655 7.49 (5 5 + 0.05) ld 1900 + 01 0.729 4.03 (5.0 + 1.2) 103 1gO7 + 00 1.017 551 (-8.7 + 0.~) 103 1907 ~ 02 0.495 2.76 (-9.5 + 3.1) 102 1907 + 10 0.2&t 2.64 (-5.5 + 0.2) 103 1915 ~ 13 0.195 7.20 (4.2 + 0.3) 101 1929 + 10 0.226 1.16 (-2.8 + 0.1) 103 2002 + 31 2.111 7.46 101 (1.2 + 0.05) 102 2020 + 28 0.343 1.89 (1.2 + 0.1) 103 15 star surface. Such losses are ~itical for the neutron stars magnetosphere which is full of dense plasma (the densibr is higher than that of Goldreich- Julian). Since the radioemission is generated in the dense plasma of the polar magnetosphere (Beskin et al. 1988), then pract~cally all radio pulsars are retarded by the current mechanism. l~o cases can be separated here, when the inclination angle X is not too close to 90 and X > ~ (2,rR/cP)~/2, where R ~s the radius of the neutron star. In the first case (Beskin et al. 1983; Beskin e! al. 1984), ~ = 1.93, g~x) = cos2& X, f(X) =-t9X, d = 0.75. (3) For X ~ 90 the retardation dynamics are defined by the asymmetrical current iA which flows out of one half of the polar cap and flows into another ~ = 4~9(Q) = iA(Qj,/(X) = 0 (4) Using expressions (1) we can easily obtain the formula for the braking index

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16 AMERICAN AND SOVIET PERSPECTIVES n = ot + 9 aft + f (X) .'X/9 + [xQn Q COS tint' (5) This consists of two portions: the first has the constant sign and is defined by the regular part of star braldng; and the second which is changeable over time and caused by the notations. Values of n presented in the table do not, in most cases, correspond to the regular values, which should be of the order of several units. This means that the last member of the expression (5) is dommeer~g. The exception includes four pulsars 0531 ~ 21 (Crab), 0540-69, 083~45 (Vela), and 1509-58, which agree with the picture of the regular brnlang n = 2 (~21rR/cP) (7) The dependence of the assymetrical current on the rotation frequency has the power form (Beskin et al. 1983), so that for current losses at % = 9oo n ~ 2.6, X ~ 90 We characteristic dependence of the braking index n on the angle X (7) is shown In the figure. The observed values of n for four pulsars included in the table are also shown here. Values Of X are taken from the observations of X-ray radiation for pulsars 054~69 and 1509-58; and for the two other pulsars 0531 + 21 and 0833~5 from the tangential condition for the line of observation of the polar cap edge. We can see that the theory of current losses correctly reflects the character of the dependence of the braldng index n on the angle X. It should be noted that a slight discrepancy between the theory and observations can be explained by the usage of a sumple approximation in which the longitudinal current is considered as constant ~ the whole area of open magnetic field lines. More accurate estimations, when the effect of the electrical current flowing near the internal surface of the hollow cone was taken into consideration, were made by Besldn et aL (1986~. The result was a decrease of the value n. The angle X = 52 ~ 2 was also determined from the observed value of n = 2.83 ~ 0.03 (Manchester et al. 1985) for

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HIGH-ENERGY ASTROPHYSICS 40 20 10 _ 8 4 2 1 17 0540 69 1509~59 0531 -21 , , , _ 30 60 90 Xo FIGURE 1 The braking index n versus angle X. The curve corresponds to formulae (7) for the current spin down mechanism. The measurements are presented for the four pulsam with regular baking. the pulsar 1509-58. This result agrees with the value of angle X, which was defined from the modulation of the observed pulsar X-ray radiation (Seward and Harnden 1982~. For the remaining pulsars in the table the characteristic values are of the order of ~ (102 - 104~. Then from the expression (5) it follows that Pn /[X < 10 2 p/ p. For characteristic values P ~ 10-i5, P ~ 1 see we have Pn/iX < 10~3sec. Note that this value of Pn corresponds to the star asymmetry which is caused by the magnetic field (Goldreich 1970~. Actually, in this case B2R n 87rGMp X (~8) where B is the magnitude of magnetic field, M is the neutron star mass, p is the density. Putting characteristic parameters into (8) we get Pn = 3 1Ol2PB~-22R6 4(M/M<:,)2 COS~i X. (9) Here BE = B 10-~2 G-i, R6 = R 10-6 cm~~. Thus, we can see that uncertainty of values n for most pulsars can be caused by slow notations (Pn < 10~2 see, {X = ~ - X) due to the magnetic field. For the quickly decelerated pulsars

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18 AMERICAN AND SOVIET PERSPECTIVES P > [Xp > 10-l~xcosx n (10) the notations can become negligible and the braking index n is determined by the regular star braking processes (6). The inequality (10) takes place for four indicated pulsars, when either P ~ 10-15 or X ~ 90 (or these both). It should be especially noted that for the pulsar 0531 ~ 21 (Crab) it was possible to measure the third derivative Q. It gave us the bral~g parameter of the second order 2 n = `.~3 It was equal to n(2) = 10 ~ 1 (correspondingly, Q = ~ 10-3i sec~4) (Blandford and Romani 1988; Lyne et al. 1988~. The determination of n(2) also gives us ache possibility to clarity ache character of the neutron stars evolution Indeed, neglecting notations (as in the case of Crab) from expression (1) we get n(2) = n(~2n-1~) Jr f (`X`) AX (11) Since the pulsar 0531+21 is an interpulse one (% ~ 90), then fix) = 0, and we have n(2) = n(2n-1~. (12) Expression (12) agrees with the measurement results because n = 2.509. This proves the current mechanism of losses which we proposed (Beskin et al. 1983; Beskin et al. 1984~. At X ~ 90 current losses only lead to expression (12~. In this case X = 0, be. X = 90, and this is the stationary value which the angle X between axes approaches due to the evolution of the neuron star rotation. REFERENCES Beskin, VS., AU Gurevich, and Ya.N. Istomin. 1983. Soviet Phys. JEEP 58:235. Besldn, V.S., AV. Gurevith, and Ya.N. Istomin. 1984. Astrophys. Space Sci. 102:301. Besldn, V.S., AV. Gurevich, and Ya.N. Istomin. 1986. Page 361. Proceedings of the Joint Varenna-Abastumani School. Besl~, V.S., AV. Gurevich, and Ya.N. Istomin. 1988. Astrophys. Space Sci. 146:2Q5. Blandford, RD., and R.W. Romani. 1988 M.N.RAS. Z34:37. Goldreich, P. 1970. Astrophys. J. l9O:L11. Lyne, AG., R.S. Pritchard, and F.G. Smith. 1988. M.N.RNS. 233:667. Manchester, R., J.M. Durdin, and L~M. Newton. 1985. Nature 313: 374. Seward, F.D., and F.R. Harden, Jr. 1982. Astrophys. J. 256: L45. Wolsz~an, A, S.R Kulkarni, J. Middleditch, D.C Backer, AS. Fruchter, and RJ. Dewey. 1989. Nature 337:531.