LUMINOSITY FUNCTION A N D SPATIAL DENSITY OF PULSARS O. Kh. Guseinov,

F. K. Kasumov,

and I. M. Yusifov

The luminosity function of pulsars is investigated in the complete range of observed luminosities. It is shown that it satisfies the power law dN/dL L -Y, where y = 1.76 • 0.06 in the luminosity interval 3o1026 ~ L ~ 2.1030 erg/sec. For L < 3-1026 erg/sec, the luminosity function changes its slope and has the form dN/dL ~ L -Yl, where Y1 = 0.7 • 0.2. This luminosity fUnction is used to estimate the total number and production frequency of pulsars in the Galaxy; these are, respectively, Np = (5-10)'104 and ~p = 0.025 • 0.008 year -I, or one pulsar in 4 0 ~ 0 years.

For a long time after the discovery of pulsars and accumulation of information about their parameters it was impossible to make correct statistical investigations of the spatial distribution of these objects, despite a number of attempts [1-3]. The main cause of this was the u n c e r t a i n t i e s in the estimates of the distances to them and the low sensitivity of the radio searches. The situation has now changed, mainly as a result of the systematic searches for pulsars with the high-sensitivity apparatus commissioned in 1975 at Jodre!l Bank [2, 4]. Up to 1975, 105 pulsars had been discovered. On the basis of the known parameters of these pulsars, Groth [5] obtained for them the form of the luminosity function. Since the observable parameter that characterizes the distance to the pulsars is the dispersion measure DM, he plotted log N against log DM, which is shown by the points in the upper part of Fig. I. Analysis of this curve over the complete range of DM values made it possible to obtain for the luminosity function the expression dN/dL ~ L -T , where L is the pulsar luminosity and y = 1.75. For the given statistical sample, a correlation between the flux and the dispersion measure was not traced and the limits of the range of determination of the luminosity function with given y with respect to DM differed by an order of magnitude, so that the luminosity of the majority of pulsars differs by not more than two orders of magnitude~ The p u b l i c a t i o n of [4], which gave the results of a careful search for pulsars with high sensitivity in the region 4 f < 1 < 6 ~ and ]b I < ~ enabled Roberts [6] to obtain the luminosity function of pulsars in a different way on the basis of the data of [4] (~40 pulsars). His form of the luminosity function is almost the same as in [5]. The range of determination of the form of the luminosity function in [6] covers not more than three orders of magnitude. Since the observed spread in the luminosities of individual pulsars exceeds five orders of magnitude, we consider below in more detail the form of the luminosity function in the complete range of observed luminosities. If the form of the luminosity function found for pulsars in [5, 6] is correct~ then~ integrating it from a certain value L~ to infinity, we can obtain the total number of pulsars with luminosities greater than LI: ~[0.75. Since the values of L were obtained for a!l pulsars in [7], we can plot log N against log L (Fig. 2). As can be seen from Fig. 2, at L 0 = 3-1028 erg/sec there is an inflection. The section of the curve with L > 3.1028 erg/ sac corresponds to the form of the luminosity function given above, whereas for pulsars with L < 3-1028 erg/sec the luminosity function apparently changes its form. However, this is not so, for the following reason. Since the limiting flux from pulsars with L > 3"1028 (i flux unit = 10 -26 W.m-2.Hz-l), they must be detected at right to the boundary of the Galaxy. But for pulsars with effects become important: at the given sensitivity of the stricted volume can be detected. Indeed, let us construct pulsars with L < 3-1028 erg/sec by the method proposed in

erg/sec is S o = 0.01 flux units distances up to 16 kpo~ l . e ~ L ~ 3"1028 erg/sec, selection search, only pulsars ~rom a rethe luminosity function for [5]. In this case, the dependence

Shemakha A s t r o p h y s i c a l O b s e r v a t o r y . Translated from Astrofizika, 351-356, April-June, 1978. Original article s u b m i t t e d J u n e 24, 1977; March 24, 1978.

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V o l . 14, No. 2, p p . revision submitted

Corporation

2

~.0

2.0

IgDM Fig. 1. Distribution of pulsars as a function of the dispersion measure (DM). The c u r v e i n t h e u p p e r p a r t o f the figure is plotted on t h e b a s i s o f the data of [5]; the curve in the lower part, using the data of [4].

o f l o g N o n l o g DM m u s t h a v e t h e f o r m sh o w n i n t h e u p p e r p a r t o f F i g . 1 ( o p e n c i r c l e s ) . The p a r t o f t h e c u r v e i n t h e r e g i o n DM < 16 p c / c m 3 h a s r e m a i n e d u n c h a n g e d , b u t i n t h e r e g i o n 16 ~ DM ~ 158 p c / c m 3 t h e s l o p e h a s c h a n g e d v e r y s l i g h t l y . The change in the exponent of the luminosity function corresponding t o t h i s i s 47 = 0 . 0 5 , i . e . , the relative c h a n g e i n "the f o r m of the luminosity function i s l e s s t h a n 3%. Since the upper luminosity limit of this section is ~3-1028 erg/sec, it must be assumed that the luminosity function preserves its form r i g h t down t o L = 3 . 1 0 2 6 e r g / s e c . Similarly, one can obtain the luminosity function on the basis of the data of [4]. I n [4] t h e s e a r c h s e n s i t i v i t y was a l m o s t an o r d e r o f m a g n i t u d e h i g h e r t h a n i n t h e p r e c e d i n g searches, and therefore the inflection i n t h e c u r v e i s o b s e r v e d i n t h i s c a s e a t L0 : 5 . 1 0 2 7 erg/sec. The plot of log N against l o g DM i n t h i s c a s e i s s h o w n i n F i g . 1 ( l o w e r c u r v e ) . Over the section 50 ~ DM ~ 160 p c / c m 3 t h e s l o p e i s 1 . 3 . Since the pulsar distribution satisfies a nearly spherical law in this region, we o b t a i n y = 1 . 8 2 , i . e . , once more the

i

0.15

i

?5

2;

2'8

~ log

L

2'9

~'o

(erg/sec)

Fig. 2. P u l s a r d i s t r i b u t i o n function. The c u r v e is c o n s t r u c t e d on the b a s i s of a s t a t i s t i c a l sample i n c l u d i n g all the p u l s a r s k n o w n up to 1975.

201

relative change in the form of the luminosity function is very slight (~4%)~ In this case, on the section of the curve with DM ~ 50 pc/cm 3 there are only four pulsars, so that the slope 0.6 need not worry us (the statistics is clearly too low). Therefore, from the data of this statistical sample too it follows that the form of the luminosity function is pre, served right down to L ~ 5-1026 erg/sec since the upper limit is L~ = 5-1027 erg/sec, and the region of determination of the luminosity function is ~DM 2 ~ (i@0/50) 2, i.e.~ an order of magnitude. Thus, we must regard as valid the pulsar luminosity function given by power y = 1.76 • 0.06 in almost the complete range of observed luminosities from 3"1026 erg/sec. In order to find the behavior proceed as follows.

of the luminosity

function

for L < 3"1026

law with to 241030

erg/sec~

we

If the luminosity function has the form dN/dL ~ L "1"76• and if the luminosity intervals (LI, L 2) and (L2, L 3) are chosen in such a way that L3/L 2 = L2/L 1 = 3o16, then the ratio of the numbers of pulsars in these intervals in unit volume is N12/N23 = 3'160~76• = 2.41 • 0.17. If pulsars with L' (lower limit of the interval) can be detected at distance r', then all the remaining pulsars with luminosities in this given interval must necessarily be observed (under otherwise equal conditions) at distances r < r'. The ratio of the number of pulsars with L < 1026 erg/sec is in accordance with the data of [7] equal to 0.5, w h i l e for L > 1026 erg/sec it is greater than I. This indicates that the lumin0sity function changes sign within the interval 5-1026-3.1026 erg/see. Knowing the ratio NI2/N23 from ob ~ servations and taking into account the selection effect, we can assume that yl = 0.7 • 0.2. It should however be noted that this result is based on a low statistics (only !4) and is preliminary. In the generally adopted model of pulsars, which regards them as inclined magnetic rotators, the luminosity must depend on the age. In aBnumber of investigations [8-10], this dependence is expressed by the power law L ~ Clt- . To determine the values of ~he parameters C 1 and B, we use the form of the luminoslty function determined above, Since

N = C~ ~ t - ~ d t expressing

L in terms

of t,

= (~ - - I ) - l

C~t <1-~,

(I)

we o b t a i n

N = ('~ -- I)-'C~C~I-T! t -'3(I-~,

(2)

with age less than t. On the other hand, if the pulsar production frequency is ~ p , then the number of pulsars with age 3.1026 erg/sec in the Galaxy: Np = (5'i0)'i04 and for mean age T p = 3.106 years [3, 12] the production frequency ~p = 0.025 • 0,008 year -I. Using this value of ~p, we also determine the value of the parameter Ci = (4,3 • 2.6).1034 Thus, the change in the luminosity with age can be represented by the power law L = (4.3 • 2 . 6 ) . I 0 ~ 1 - 1 ' 3 2 - ~

where L is expressed

in erg/sec

(3)

and t in years.

It follows from (3) that the mean age of pulsars with luminosities L > 3"1026 erg/sec is Tp = (2 • 1)'106 years, which agrees well with the estimate obtained by the standard procedure [13]. For faint pulsars with L < 3-1026 erg/sec this expression does not apply since, like (i), it is valid for 7 > 1.

202

LITERATURE CITED 1. J 2. R 3. 0 4. R 5. E P 6. D 7. J 8. A 9. J 10. O 11. O 12. K

Gunn a n d P . O s t r i k e r , A s t r o p h y s . J . , 1 6 0 , 979 ( 1 9 7 0 ) . A. R u l s e a n d J . H. T a y l o r , A s t r o p h y s . J . , 1 9 1 , L 59 ( 1 9 7 4 ) . Kh. G u s e i n o v , F . K. K a s u m o v , a n d I . M. Y u s i f o v , T r . I I I EAK, T b i l i s i (1975). A. H u l s e a n d J . H. T a y l o r , A s t r o p h y s . J . , 2 0 1 , L 55 ( 1 9 7 5 ) . J . G r o t h , N e u t r o n S t a r s , B l a c k H o l e s and X-Ray S o u r c e s , R e i d e l , D o r d r e c h t ( 1 9 7 5 ) , 143. H. R o b e r t s , A s t r o p h y s . J . , 2 0 5 , L 29 ( 1 9 7 6 ) . H. T a y l o r a n d M. M a n c h e s t o r , A s t r o n . J . , 8 0 , 794 ( 1 9 7 5 ) . G. L y n e , R. T. R i t c h i n g s , a n d F . G. S m i t h , M . N . , 1 7 1 , 579 ( 1 9 7 5 ) . Gunn and P. O s t r i k e r , Astrophys. J., 158, 3 (1969). Kh. G u s e i n o v a n d F . K. K a s u m o v , T s i r k . S h e m a k h i n s k o i O b s . , No. 4 , 11 ( 1 9 7 3 ) . H. G u s e i n o v a n d F . K. K a s u m o v , A s t r o p h y s . S p a c e S c i . , (1978) (in press). L a n d e a n d W. E. S t e p h e n s , A s t r o p h y s . S p a c e S c i . , 4 9 , 169 ( 1 9 7 7 ) .

BRIE F COMMUNICATIONS ENERGY LOSS OF C O S M I C RAYS IN EXPANDING

REGIONS

OF THE GALAXY V. G. K r i v d i k In all models of the propagation of cosmic rays in of the cosmic rays lose energy basically as a result of gas [1, 2 ] . To c a l c u l a t e the loss, a very simple model It is supposed that the interstellar medium c o n s i s t s of homogeneously and isotropically in space.

the Galaxy it is assumed that nuclei nuclear interactions with interstellar of the interstellar medium i s a s s u m e d . gas in a stationary state distributed

In reality, there are in the Galaxy a large number of rather extended regions (supernova remnants, regions of ionized hydrogen, star winds), in which gas expands radially and in which there are inhomogeneities of the magnetic field [3, 4 ] . When t h e y t r a v e l t h r o u g h these regions, the cosmic rays lose energy as a result of scattering on the radially expanding inhomogeneities of the magnetic field [5]. I n t h i s p a p e r , we show t h a t t h e e n e r g y l o s s e s of the cosmic rays in these regions of the Galaxy are in order of magnitude comparable with the energy losses due to nuclear interactions of the cosmic rays with the interstellar gas. region

The e n e r g y of radius

loss for one relativistic r that is expanding with

particle velocity

0

with energy E that passes through v is given by the expression [5]

a

r

The total energy losses for all the cosmic-ray particles in all expanding regions of t h e G a l a x y i s t h e sum o f t h e e n e r g y l o s s e s o f t h e p a r t i c l e s in each expanding region of the Galaxy. winds)

In each individual type of expanding the particles lose the energy

region

(supernova

_

~

~ k-

remnants,

H II

regions,

stellar

_

(2)

- - M k ----- V~n ( E ) E, r k

w_here k i _ n d i c a t e s t h e t y p e o f r e g i o n ; Mk i s t h e n u m b e r o f r e g i o n s o f t h e r k , a n d Vk a r e , r e s p e c t i v e l y , the mean values of the expansion velocity, the volume for the region of the given type; and n(E) is the density of particles (it is assumed that the distribution of the cosmic rays in the

given type; Vk, the radius, and the cosmic-ray G a l a x y i s homo-

Institute of G e o p h y s i c s , A c a d e m y of S c i e n c e s of t h e U k r a i n i a n SSR. Translated from Astrofizika, V o l . 1 4 , No. 2, p p . 3 5 7 - 3 5 9 , A p r i l - J u n e , 1978. Original article submitted November 29, 1977.

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