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OCR for page 9
On the Evolution of Pulsars
V.S. BESKIN, A.V. GUREVICH, YA.N. ISTOMIN
Lebedev Physical Institute
Recently Lyne and Manchester (1988) presented data on the angle X
between the axis of rotation and the magnetic dipole awns, determined from
polarization observations. Such a complete catalogue makes it possible to
carter out a detailed comparison of our theoretical results with the observed
distribution of radio pulsars over the angle X.
Before making such a comparison, we recall the main features of
our theory (for more details see Beskin e' al. 19861. The results of such
calculations of the generation of particles and of directed pulsar radio
emission flux suggest that the properties of pulsars depend substantially on
the parameter
(Is) (10 )
(1)
So, pulsars with Q ~ 1 generate particles practically on the entire surface
of the polar cap whereas pulsars with Q > 1 may generate only within
a thin ring, and the generation itself may be of nonstationary, irregular
character. In other words, pulsars with Q > 1 are located near We death
line on the diagram PP. Indeed, all pulsars exhibiting pulling, subpulse
drifts, and mode switching have Q > 1, whereas pulsars with Q < 1 are
characterized by stable radio emission.
The theory of neutron star evolution associated with current losses is
different for pulsars with Q 1. It is developed more thoroughly
for pulsars with Q < 1. It is just for this reason that in all our preceding
papers we distinguished between pulsars with Q
1 and compared theory with experiment separately for ~em. The same
should be done in the analysis of pulsar distribution over the slope X.
9
OCR for page 10
10
AMERICAN AND SOVIET PERSPECTIVES
Let us first see how the distribution of observed pulsars with Q <
1 over X will loot To this end we shall make use of the distribution
function of pulsars NQ<1 (P. X, B12) over the period P (in seconds), over
the magnetic field B:2 = B/10~2 Gauss and over the slope X Of the axes
(Beskin e! al. 1986~.
N Npv+iB,-io/7(l+B~2)-~-~-§F(x)~[Coso5X-B:2 P ~ (2)
Here N is the total number of observed radio pulsars, ~ = -.1, ~ = 3, ,B =
0~75, F(x) = Amp cos-° sx and the 8-function just separates pulsars with
Q < 1. As shown by Beside et al. (1986), the distribution function (23 is
in agreement with the observed distn~ution of pulsars (with Q < 1!) over
period P and magnetic field B:2.
Integrating now the distribution function NQ < 1 over the magnetic
field B. we can find distn~ution function NQ < 1 (P. X) by which one can
estimate, for example, the mean value of the angle X as a function of period
P:
X(P) = ~ NQ<1(P, x)xdx
(3)
The results are presented in Figure 1. We see that the mean slope X(P) is
in perfect agreement with results, decreases as P increases, in spite of the
fact that for each pulsar the angle X increases with time due to the current
losses. The reason for this is that pulsars having Q < 1 and large periods
P cannot have angles X close to 90°. The region Q < 1 on the diagram P
-sin X is shown in Figure 2. One can see that the generation of particles
and radio emission by pulsars with a large enough period P is only possible
when X are small Thus, the observed decrease of the mean angle X with
increasing P cannot be regarded as an argument in favor of the decrease
of X with time. This is rather an indication of the fact that the observed
radio emission is indeed associated with particle generation near the star
surface.
Similarly, going over in (2) to the variables B12, X and to dynamical
age ED = P/2P, we obtain
NQ<1(~15, B12, X) ~ NT15+1B112-1°/7~+1(1 + B12)-1-~-~8(COSX)1 5(~ 2)
F(X)~[1-ri517B92/49cosx,
(4)
where 715 = {D/15 million years. The analysis of (4) shows mat the
distribution function NQ<1 (~15, X) = ~ NQ<1 (715 ,B12, X)dBl2, for pulsars
OCR for page 11
HIGH-ENERGY ASTROPHYSICS
75
Ox 50
25
~1
OIL -
. , ,
iLy,_
l
.2 .4 .6
P(S)
.8 1.
FIGURE 1 The mean value of the slope angle X as a function of period P for pulsars with
Q < 1. Ibe curve corresponds to formula (33, the points correspond to the observations
reported by Lyne and Manchester (1988~.
vvith ED < 15 million years tie., practically for all pulsars with Q < 1)
must depend wealthy on the AD value, and the distribution over X must be
approximately equiprobable. Indeed, if we consider only pulsars with Q <
1, then, as shown in Figure 3, their distn~udon over X for ED < 2 million
years and word > 2 million years practically coincide, the mean value of X
even somewhat increasing with time.
X(~< 2mln.yr) = 43° ~ 24°
X(~TD> 2mln.yr) = 48° ~ 29°
(~5')
But this increase cannot be regarded as statistically significant. Hence, here
too, we see close agreement between our theory and experiment.
Let us now consider pulsars with Q > 1. The distribution function of
such pulsars over the magnetic field B and the angle X, according to Beskin
et al. (1986), has the form
NQ>1.(BI2, X) = NQ<~EP(X, Bt2), X, Bt2iPr
(6)
where P = 10-~5B'2~°/7cos: 5x is the rotation deceleration velocity in the
mechanism of current losses, ~ = 3 Unix B~2-~°/7 minion years is the
characteristic life-time of radio pulsars at the stage Q < 1, and the relation
P = P(x, B:2) corresponds to the condition Q = 1 which describes transition
of pulsars from ache region Q 1 as in formula (4~.
As a result, for the distribution function over the angle X for pulsars with
Q ~ 1, we obtain
OCR for page 12
12
AMERICAN AND SOVIET PERSPECTIVES
if
3~10 \
\3,1042
\ \1012
Q 1 \
.5
1.
1.5
P(s)
FIGURE 2 The region Q < 1 on the diagram P - sin X. The boundary is shown for
three characteristic magnetic fields 31011 Gauss, 1012 Gauss, 31012 Gauss.
NQ>1(X) = 2 Xcos1 5X
(7)
As shown In Figure 4, here we also see agreement between our theory and
the experimental results reported by Lyne and Manchester (1988~.
Concluding, we would like to emphasize that an unquestionable con-
clusion of the modern theory is that for rotating pulsar magnetosphere
filled with plasma, the law of vacuum deceleration
dW 1B2Q4R6 . 2
-= 6 ,~ sin X
(8)
is not obeyed. The presence of plasma in pulsar magnetosphere follows
from the very fact of the existence of observed powerful coherent radio
emission of pulsars, and the rotation deceleration is determined by the
ponderomotive action of current leading to an increase with time of the
slope of axes for each individual pulsar. We can see that the conclusions
of this theory are in satisfactory agreement with observations.
Thus, the observational data presented by Lyne and Manchester are
OCR for page 13
HIGH-ENERGY ASTROPHYSICS
ED ~ 2 million years
3
a)
ILL
I 1 1
30 60 90
At
x°
13
~ b)
3
ED > 2 million years
-I
l l
30 60 90
x°
FIGURE 3 Distnbution of pulsars with Q ~ 1 over the slope angle X for a) ED < 2
million years; by TD > 2 million years.
of
20
10
. ~
_ ~
. 1 1 1
90
x°
FIGURE 4 Distribution of pulsars with Q > 1 over the slope angle X. The come
corresponds to formula (73.
in good agreement with our theory of neutron star evolution. New ob-
se~vations which would promote direct estimation of the value dX/dt for
individual radio pulsars would be, of course, of great interest.
REFERENCES
Beskin, V.S., NV. Gurevith, and Ya.N. Istomin. 1986. Sov. Phys. Usp. 29:946.
Lyne, JOG., and RN. Manchester. 1988. MADRAS. 234:477.
Representative terms from entire chapter:
distribution function
