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OCR for page 14

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

OCR for page 14

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 =

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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.