Statistical analysis of the radio luminosity of pulsars

Chinese Astronomy

4 (1980) 202-206

Pergamon Press. Printed in Great Britain 0146-6364/80/0601-0202-$07.50/0

Acta Astr. Sinica 20 (1979) 1 9 9 - 2 0 3

STATISTICAL

Wang

ANALYSIS

Zhen-ru,

OF THE

Qu Q i n - y u e ,

University, and Luo L i a o - f u

RPDIO

Lt~4INOSITY

OF P U L S A R S

Department of Astronomy, Nanjing Department of Physics, Inner-blongolia University

Lu Tan

Received 1978 June 7

ABSTRACT. The r e s u l t s

of statistical

a n a l y s e s o f t h e r a d i o l u m i n s o t i y o f 81

p u l s a r s o b t a i n e d i n t h i s p a p e r a p p e a r t o be i n f a v o u r o f t h e view t h a t t h e radio emission originates statistical

relations

in regions close to the light cylinder.

The v a r i o u s

g i v e n i n TABLES 1 and 2 may p r o v i d e some c l u e s on t h e

radio emission.

i.

INTRODUCTION

Less t h a n two y e a r s from t h e d i s c o v e r y o f p u l s a r s , n e u t r o n s t a r was b a s i c a l l y

established

t o JP 1953, were e x p r e s s e d i n [4]. authors in a previous statistical the theoretical

[1-3].

t h e p h y s i c a l model u s i n g a r o t a t i n g

Doubts on t h i s model, p a r t i c u l a r l y

in regard

An e x p l a n a t i o n i n t h i s c a s e was g i v e n by t h e p r e s e n t analysis

[5].

In t h e Appendix ( F i g . 3 ) , we have g i v e n

c u r v e u s i n g t h e method o f [5] which t a k e s a c c o u n t o f m a g n e t i c d e c a y , and we

s e e t h a t t h e c u r v e r e p r e s e n t s w e l l t h e 81 p u l s a r s o b s e r v e d .

This s t a t i s t i c a l

analysis of

o b s e r v e d d a t a on p u l s a r s p r o v i d e s a s t r o n g s u p p o r t f o r t h e m a g n e t i c d i p o l e model o f a rotating neutron star. Radio e m i s s i o n from p u l s a r s have b e e n o b s e r v e d f o r a l r e a d y 10 y e a r s , b u t t h e q u e s t i o n of its mechanism is far from having been solved, and also opinions differ widely as to the location of the radio emission [6]. The radiating mechanism and region being an obviously important topic in the theory of pulsars, we shall

attempt here to apply further statistical

analyses to the observed data on pulsars, with the aim of finding some clues and evidences on these questions.

2.

STATISTICAL ANALYSES OF OBSERVED DATA

On the great volume of observed data of pulsars given by Taylor and Manchester [13], we have made 13 different statistical analyses.

As ordinate, we always take either the logarithm

of the radio luminosity ig L R or the log of its ratio to the energy decay rate,

Ig(LR/E),

and for the abscissa, we take the various significant quantities such as log of the period,

Radio L u m i n o s i t y o f P u l s a r s

lgR

t i m e p a r a m e t e r T = P/2P, log o f i t s

log o f t h e d e r i v a t i v e o f t h e p e r i o d l g P, l g o f i t s

lg t ,

log o f t h e s q u a r e o f i t s

c y l i n d e r l g Be2 and log o f i t s

surface magnetic f i e l d energy decay r a t e ,

203

l g B02, t h a t o f t h e f i e l d

The r a d i o l u m i n o s i t y v a l u e s LR a r e t a k e n from [7].

The e n e r g y d e c a y r a t e i s E = ( 2 ~ ) 2 I P / P 3,

t a k e n t o be 10 ~s g c m - 3 . B0 and B a r e t h e f i e l d e and a t t h e l i g h t c y l i n d e r a c c o r d i n g t o t h e d i p o l e model, t h e y a r e g i v e n by

t a k e n t o be 106 cm, and t h e age o f t h e p u l s a r ,

c a l c u l a t e d from Eqn. (2) i n t h e Appendix. coefficient

at the surface

24Ix 4

where R i s t h e r a d i u s o f t h e p u l s a r ,

TABLES 1 and 2 l i s t

at the light

l g E.

I b e i n g t h e moment o f i n e r t i a ,

3I¢~pp

#, i s

CGVSu n i t s a r e u s e d t h r o u g h o u t .

the following results

from t h e s t a t i s t i c a l

y, t h e r e g r e s s i o n e q u a t i o n , t h e s t a n d a r d e r r o r S, s . e .

analyses: the correlation in the g r a d i e n t of the

r e g r e s s i o n l i n e S B , and s . e . i n t h e i n t e r c e p t SA. TABLE 1 r e f e r s t o u s i n g l g ( L R / ~ . two c o r r e l a t i o n diagrams i n v o l v i n g Ig Be a r e shown i n F i g s . 1 and 2.

Table 1 corr.coeff

Irl

l

regression equation

]!

y

-- Bx

5s

+ a

lgP

0.25

IgLe ~- -0.811gP + 27.55

1.10

0.35

0.14

lgib

0.30

lgL• ~ 0.361gib + 32.98

1.08

0.i3

1.91"

lgs.,

0.19

(lgLa ~ 0.221gB~ + 22.35)

1.12

0.13

3.09

0.36

IgLR

1.06

0.06

0.23

lgx

0.36

IgLa ~ --0.4118x + 33.51

1.06

0.12

1.67

Igt

0.35

IgLA ~ --0.651gt + $6.45

1.06

0.20

2,65

0.38

lgLa .L 0.311g~ + 17.69

1.05

0.08

2.73

~- 0 . 2 0 1 g ~

+ 27.01

Table 2 --Icorr.coeff

regression equation

Irl

SD

y ----B~ + a LR

1.~

0,43

0.17

Ig-~8- ~ --0.651g/b -- 14.46

1.30

0.16

2.29

0.20

( l g - ~ - ~ --0.30lIB| + 2.26)

1.41

0.16

3.90

IgB,2

0.67

lg ~

1.06

0.06

0.23

Igr

0.58

lg--~- ~ 0.841gx -- 16.76

1.17

0.13

1.84

lgt

0.62

I g ~ -~. =~ 1,451gt--24.52

1.13

0.21

2.82

0.44

lg T

0.42 lgB~

lIP

age

~ 1.881gp - 4.62

~ --0.491gB~ - - 3.32

The

204

Radio Luminosity of Pulsars

lgL.

4

.:...'.;':... o•

.

28

•

o• •

26

•

•

ooe oe

•

25,

•

•

•

•

°o

o ° •

•

•

• • •°

•

•e

24

I

1

J

I

I

I

i

•

i

I

1

2

$

4

5

6

7

8

9

10

-

I

,i

!

n

12

1~...

Fig. I C o r r e l a t i o n diagram between Ig LB and Ig Bo2

--2

--3 --4 -5

•

o.

• •

--8 -9

--lO

I

I

I

I

l

r

I

I

I

I

I

1

l

2

3

4

f

6

7

8

9

10

.11

i2

Fig. 2 Correlation diagram between Ig(L/E)

3.

and Ig B 2

DISCUSSION

According to the theory of statistical significance, for a sample size of about 80, if IYI>0.22 we have significance at the level of e=0.05, significance at the level of e = 0.01.

and if IYI > 0.28, we have

The values of y given in the tables differ widely

among themselves; the least one, y= .19 (or .20) is between Ig and this value is not significant even at the 0.05 level.

B02 and Ig LR (or ig ( L / ~ ) ,

Thus these two quantities do

not seem to be linearly correlated, and it is pointless in this case to fit a regression line.

On the other hand, the correlation coefficient between ig Be2 and Ig LR ~or Ig ~ L / ~ )

is highly signi£icant - at the 0.01 level.

Therefore, results of our analysis favour the

view that the radio emission originates in the vicinity of the light cylinder and not on the surface.

The correlation between Ig P and Ig

LR is significant at the 0.05 level, while

all the other correlations not mentioned so far are significant even at the 0.01 level.

Radio L u m i n o s i t y o f P u l s a r s

Among t h e l a t t e r ,

there is still

205

a s p r e a d i n t h e a c t u a l v a l u e s , t h e l a r g e s t one b e i n g

between l g Bc2 and l g ( L £ E ) . B e s i d e s , t h e above r e g r e s s i o n a n a l y s e s have l e d t o t h e f o l l o w i n g r e s u l t s : (i) rate

Because LR~E°. 31, we have L B / E ~ - 0 . 6 9

is,

t h a t i s to say, the l a r g e r the energy decay

t h e l a r g e r w i l l be t h e r a d i o l u m i n o s i t y LR, b u t t h e r a t i o

l u m i n o s i t y and t h e e n e r g y d e c a y r a t e w i l l be s m a l l e r . the radio emission is,

between t h e r a d i o

Or , we can say t h a t ,

the larger

t h e s m a l l e r i s i t s p r o p o r t i o n a t e s h a r e i n t h e e n e r g y decay.

t h a t LR~BoO. 4, which means t h a t t h e s t r o n g e r t h e f i e l d

at the light cylinder is,

Noting the greater

w i l l be t h e r a d i o e m i s s i o n , b u t t h e f r a c t i o n o f e n e r g y t h a t i s t r a n s f o r m e d i n t o t h e r a d i o p a r t w i l l be p r o p o r t i o n a t e l y l e s s .

For example, t h e two p u l s a r s ,

NP0532 and PSR0835 have

t h e l a r g e s t known Be2 and t h e i r LR a r e a l s o l a r g e , b u t t h e i r L~/E a r e t h e l e a s t .

This i s

p r o b a b l y caused by t h e r a p i d i n c r e a s e u n d e r a s t r o n g f i e l d

in the p r o p o r t i o n o f r a d i a t i o n

e m i t t e d i n t h e v e r y s h o r t w a v e l e n g t h r a n g e and as p a r t i c l e

radiation.

T h i s view i s

s u p p o r t e d by t h e d i s c o v e r y o f s t r o n g y r a d i a t i o n from t h e s e two o b j e c t s . (ii)

Any r e a s o n a b l e r a d i a t i n g mechanism must a c c o u n t f o r t h e c o r r e l a t i o n s

TABLES 1 and 2. radiation

and have g i v e n t h e o r e t i c a l

radiation

and t h e p e r i o d (~ p-m).

statistical

listed

in

R e c e n t l y , many p e o p l e [8] have p r o p o s e d t h e mechanism o f c u r v a t u r e

relation

relation

between t h e r a t e o f d o i n g work by c u r v a t u r e

There i s s t i l l

some d i s t a n c e between t h i s and t h e

£ R ~ P - 0 " 8 1 g i v e n i n TABLE 1, and i t r e m a i n s an i n t r i g u i n g problem t o

f i n d a r a d i a t i n g mechanism t h a t can a c c o u n t f o r t h e v a r i o u s c o r r e l a t i o n s

ennumerated above.

APPENDIX Repeating the statistical analysis in [S] for 81 pulsars, we have found the statistical curve shown in Fig. 3 and the equations --/=(7.8-I-1.3)XI0-", v~

~=(3.54+1.85)X10-"7 s e e

where I i s t h e moment o f i n e r t i a

of the pulsar,

(1)

~0 i s t h e i n i t i a l

m a g n e t i c moment, ~ i s

t h e m a g n e t i c d e c a y p a r a m e t e r , t h e v a l u e g i v e n h e r e i s t h e median v a l u e . about t h e c u r v e i s about 0 . 6 0 . t

= 2/~ = 1.8 × 106 y r .

From t h e median v a l u e o f ~, we f i n d t h e m a g n e t i c d e c a y t i m e ,

I f we t a k e i n t o a c c o u n t t h e s c a t t e r

t o have t h e p r o b a b l e bounds ( 1 . 2 - 3.7) × 106 y r . calculate

The mean s c a t t e r

i n ~, we f i n d t h e decay time

Using t h e v a l u e s o f ~ g i v e n h e r e , we can

t h e t r u e age o f t h e p u l s a r a c c o r d i n g t o t h e e q u a t i o n

, = ~11 ~ ( ~ + D.

(2)*

For those pulsars with known P but unknown P we can also use the following formula, deriveable from the theory in [S], for calculating their ages:

where I'.

~V/, • = e(t-~oo) = ~6,e k"~'~ l~l

* The f o r m u l a (2) was i n d e p e n d e n t l y d e r i v e d i n [5] i n [9]

(4)

206

Radio Luminosity of Pulsars

--8 --9 --10

--12 Ig.-..-r.

,,, " • .. %e "eO •

--14

" "

--15

• " °:~ "," " ° "

--16

..

. ~ - .953

--17 I 11

1o

1 12

Fig. 3 Statistical

I

13

relation

. i '14

~ 15

l 16

t 17

~_

b e t w e e n l g ( P / P 3) a n d l g T

REFERENCES [1 ] [ 2] [3] [ 4] [5 ]

[6] [7] [ 8] [9]

Gold, T . Nature, |18(1968), 731; ~ 1 ( 1 9 6 9 ) , 25. Oold~eieh, P. and Julian, W. ]K., Ap. •., 157(1969), 869. Ostriker, g. P. and Gunn, J. E., Ap. g., 157(L969), 1395. Phys. Today, 28(1975), N~. 3, 19. Qu Q i n - y u e , ~Jang Z h e n - r u , Lun T a n , Luo L i a o - f u , Ke~ue Yongbao 2] ( 1 9 7 6 ) Olnzburg, V. L. aufl Zheleznyakov, V. V., A~t. Rev. Astron. a.d Ap., 15(1975), 511. Taylor, J. H. and Manchester, R. N., A. g., 80(1975), 794. ~ord. G. and Buseh~uero R., M. iV., 179(1977)o 99. Lyne, A. G., Ritchlng, R. T. amd Smith, F. G., M. N., 171(1975), 579.

ON GAS DYNAMICAL Cai

MODELS

176.

OF G A L A X I E S *

Department of Mechanics, University of Science and Technology

Shu-tang

of China ABSTRACT,

This paper discussed

used in gas dynamical models. collapse,

the

the reaction state

quantities

still

t h e n we m u s t ,

(2)

out that

and the equation The p r e s e n c e

of the gas dynamical equations

(1),

when d e a l i n g

of continuity

of turbulence

simplifying

assumptions,

holds

coworkers even if

in general,

are precisely there

is

equations

turbulence,

take account

for

with gravitational are inapplicable

only affects

m o t i o n i s g i v e n a n d i f we n e g l e c t

small

for

The e q u a t i o n s

small perturbations,

if

the basic-

higher-order

then the equations

a r e o f t h e same f o r m a s i n t h e a b s e n c e o f t u r b u l e n c e .

C. C. L i n and h i s theory

are large,

If the basic-state

a n d i f we make c e r t a i n

perturbations

is pointed

law o f m a s s c o n s e r v a t i o n energies

motion.

the range of applicability It

small used by

hence Lin's

I f we w i s h t o go t o t h e n e x t a p p r o x i m a t i o n ,

of turbulence.

* Translator's Note: Because of the non-specific ences to Chinese sources, only the abstract will

character of the text be translated here.

and its

heavy refer-