The gamma-ray flux at energies above 50 MeV from NP0532 is evaluated for a pulsar model in which the optical and X-radiation is produced by synchrotron effect. Synchrotron radiation and inverse Compton scattering are considered as production mechanisms of the gamma-rays. The theoretical estimates are compared with the experimental values

Treves, A.

English

NASA Technical Reports Server

To be submitted to Nuovo Cimento, December 1970
PRODUCTION OF Y-RAYS IN THE CRAB NEBULA PULSAR
Aldo Treves+
Visiting Scientist
Massachusetts Institute of Technology
Cambridge, Massachusetts
Summary
The Y-ray flux at energies above 50 MeV from NP0532
is evaluated for a Pulsar model in which the optical and
X-radiation is produced by synchrotron effect. Synchro-
tron radiation and inverse Compton scattering are considered
as production mechanisms of the Y-rays. The theoretical
estimates are compared with the experimental values.
Riassunto
Si calcola il flusso di raggi Y di energia maggiore di
50 MeV da NP0532 usando un modello di Pulsar nel quale la
radiazione ottica e la radiazione X sono prodotte per effetto
di sincrotrone. Si considerano come meccanismi di produzione
dei raggi Y, la radiazione di sincrotrone e la diffusione
Compton inversa. Le previsioni teoriche sono confrontate con
i valori sperimentali.
Permanent address: Istituto di Fisica dell'UniversitA,
Milano
N72-13842 (NASA-CR-1248 3 3 ) PRODUCTION OF GANMA RAYS
IN THE CRAB NEBULA PULSAR A. Treves
Unclas (Massachusetts Inst. of Tech.) Dec. 1970
90005 CSCL 03A
90005 \ | A ~ ~ ELZL3
-~~~ 1 -. _______~~~~~I 
https://ntrs.nasa.gov/search.jsp?R=19720006193 2020-03-11T21:49:33+00:00Z
1. Introduction
Recently Vasseur et al.l have reported evidence that
NP0532, the Pulsar in the Crab Nebula, is pulsating at Y-ray
frequencies. The pulsed flux at energies above 50 MeV was
found to be of the order of 10-5Y/cm2sec, which corresponds
to an energy output of -10 35erg/sec.
In this paper the Y-ray observations are related to a
model of the Pulsar in which the optical and X-radiations are
the synchrotron radiation of relativistic electrons moving
around the lines of force of the magnetic field.
2A model of this kind has been discussed by Shklovsky2
From the similarity of the spectral shape of the Pulsar in
the X-ray domain and of the Nebula in the radio domain, he
deduces a value H - 4x103gauss for the perpendicular component
of the magnetic field in the emitting region of the Pulsar.
By the same argument he suggests that the electron spectrum
N(E)=KE
-
has a slope a=1.6 for energies Ee <5x109eV; a=2.4
e
for Ee 5xlO9 eV. Assuming that the emitting region has a
linear dimension -~ 107cm from the observed X-ray flux, the
following values of the spectral index K are obtained:
K = 5xlO erg cm
-
3 for E <5x109 eV;
e
K = 1010 erg-lcm
-
3 for E >5x109eV.K erg c ~e
In the following we shall use these parameters to give a
quantitative estimate of the Y-ray fluxes.
In section 2 the contribution to the Y-ray flux due to
synchrotron effect is evaluated.
-2-
In section 3 the contribution due to the Compton
scattering of relativistic electrons colliding with optical
and X-ray photons (Compton-synchrotron effect) is examined;
the Y-ray flux is given as a function of the scattering angle
because the photon and electron distributions can be highly
anisotropic. The relevant formulae for inverse Compton
scattering at small angles are given in Appendix I.
A discussion of the results in given in section 4.
2. Synchrotron Radiation
Synchrotron radiation is an obvious process for explaining
the Y-ray observations of Vasseur et al. because the observed
flux lies near the extrapolation of the X-ray spectrum. From
fig. 2 of Fishman et al., the following extrapolated value
for the pulsed flux above 50 Mev is obtained:
Iy(>50 MeV) = 4x10-6Y/cm2sec.
In the Shklovsky model the contribution due to synchrotron
radiation can be even greater because the ratio of the pulsed
component to the steady component is supposedly increasing
with increasing energy so that above 50 Mev the flux can be
entirely pulsed. The well known formula for synchrotron
radiation (Ginzburg and Syrovatskii4 ) reads:
a+1 a-l 3-a
1.35x1022 a()KVH (6.6x 18) v 2
I xv = (1)V
-2(c.g.s. units) R 2
where a(a)=8.5x10- 2 for a=2.4; V is the volume of the emitting
region,10 21cm3; R is the distance of the source,-3x10 21cm and
v is 5the frequency. We have then:
' j
-* * C 
·'. .
-3-
Iy(>50 MeV) = 10 erg/cm2sec = 10 5 Y/cm2 sec (2)
The energy of the electrons producing the Y-ray photons
is:
I9 Ey(eV) 1/2HEe ~= 7.6xlO H ge , (3)
for Ey>50 MeV it is:
E > 10 1 2 eV.
e ~
3. Inverse Compton Scattering
In the model considered the photons are produced by
relativistic electrons by synchrotron effect. They must then
have the same direction as do the electrons. These electrons
are supposed to move on trajectories characterized by small
-3
pitch angles *. Shklovsky suggests =10 3 a value which has
been criticized by O'Dell and Sartori5 on the basis of the
theory of synchrotron radiation from an anisotropic distribu-
tion 6f electrons. The condition of pulsation5 gives:
l0 -1
* < 10o1
where T is the pulse width and P the period. If the magnetic
field were completely uniform, the scattering angle 0 between
electrons and photons would be = *. The magnetic field in the
region of emission at the moment is unknown. Nevertheless,
it seems reasonable that the effective scattering angle 8eff
be small, although probably greater than p.
The photon spectrum arising from Compton scattering at
small angles is given by formula (A-10).
e+l
IY eff Iy isotropic
-4-
and (Ginzburg and Syrovatskii )
-22 W V 3 - -a 2
Iy(E) = 5.5x10 -2 2 W KeEe Y/cm sec erg
isotropic Eph R/E E
Eph E
-8
where Eph is the mean energy of the X-ray photons, 2x10 erg;
W is the photon energy density, 6x101 0 erg/cm3 as deduced
by the X-ray observations, and
cY 2
Ee M / c ms4 Eph
For Ey>50 MeV, it is:
Iy(>5 0 MeV) - 3x10
-
6 Y/cm2 sec
isotropic
The energy of the electrons producing the Y-ray photons is
given by (A-5):
E ~//2Ey mc 2 -1
Eph
For Ey 50 MeV
Ee ~ 50 elMeV
e
The mean life Tc of electrons for Compton losses is given by:
dE
T
c
= E/ dt
dE
wheredt is the electron energy loss for Compton scattering
dE ~4 dE
dt dt isotropic (see formula A-9)
-5-
and dE 10 2 5 WE2 ev/sec
dt
where W is measured in eV/cm3 , E in eV. We have then:
25e- 4
lT o2 5 sec (6)WE
4. Discussion
In table 1 the results of the previous calculations are
summarized. The first column gives the flux of Y-rays of
energy greater than 50 MeV; the contribution of Compton
scattering ,is calculated from formula (4) for different
scattering angles 8eff. In the second column the energy of
the electrons necessary for production of Y-rays of 50 MeV
is evaluated using formulae (3) and (5); the mean life for
Compton losses of these electrons, obtained by formula (6),
is reported in the third column.
Some consequences can be deduced from the above values.
If the electron spectrum maintains the slope a=2.4 up to
1210 eV, synchrotron radiation is the dominant effect in
producing Y-rays above 50 MeV. Compton scattering is an
important effect only if the interaction between electrons
and photons can be considered isotropic, i.e., 0 effl. In
this case the energy required for the electrons to
produce Y-rays would be Ee~100 MeV. We observe that an
isotropic distribution of electrons implies a very short life
for Compton losses. For instance, the electrons which produce
X-rays of 10 keV, whose energy is EelO 1 0 eV, would live
-6-
only for 10 8sec in comparison with the pulse width T=10 6sec.
The mechanisms examined for producing Y-rays, i.e.,
synchrotron radiation and Compton scattering, will be active
even if the structure of the source is different from that
of the model we have considered. Nevertheless, it is obvious
that, were the values of the physical parameters much different
from those proposed by Shklovsky, the estimates of the Y-ray
flux should be changed. On the other hand, if new experiments
confirm the existence of a Y-ray flux Iy~10-
5
- 10 6Y/cm sec
above 50 MeV, an indirect test for the model will be achieved.
-7-
Appendix - Inverse Compton Scattering at Small Angles
a. General Formulae
7Baylis, Schmidt and Lusher 7(B.S.L. in the following text)
have studied inverse Compton effect for these scattering
angles: 8=0; 80 =2 ; 08=. We are interested in the case 0<6<<1.2
The treatment by B.S.L. will be followed closely.
Let (W,K), (W',K')be the energy and momentum of the
photons before and after the collision; (Y,p), (Y',p'), the
energy and momentum of the electrons; X, X', the angles defined
by:
X=KP, X''P; Y=(l-v)1/2; re= 2.82x10 3 cm
We put h = c = me 1.
duThe scattering differential cross section dU, can be
obtained by the Klein Nishina formula by application of a
Lorentz transformation. For w, w', 1<< Y, it is (B.S.L.,formula
24):
do 2Tre W+W 2++6w'2 3ww 3l(.+2') 2
_. = [1 - + +dw'YK YK 2Y2 K 2 3Y2 K3 4Y 4 K4
where K= YW(1-vcosx).
The electron energy loss per unit time is given by:
dY da 2 n
dY = -dwd' IdQW(-w) dal ana- (l-vcosx) (A-1)
and the spectrum of the created photons by:
dR dFdYd do a n
dw'" ',·idw dQ- ~'avc' sx) (A-2)
-8-
where d is the spectrum of the electrons and aE;, the
spectrum of the ambient photons per unit solid angle. For
a discussion of formulae (A-1) and (A-2), we refer to B.S.L.,
section IV.
We suppose that the shape of the spectrum of the ambient
photons be independent of the direction, i.e.,
a2n dn(w)
We can then define a function
I(w,w',Y) = JdQd'"T(Q)(1-vcosx)
so that (A-l) and (A-2) can be written in the form:
dY _ IId (d'IdnI ,w,Y)(w-w') (A-3)
dt -ff dd
dR dF dn
d' = dYdR d=d I(w',w,Y) (A-4)
For the case of interest to us the integrals (A-3) and (A-4)
must be calculated for:
Y(Q) = 2- 6(cosx - cose)
1 2
-2 6(cosx - 1 + -2) for O<<1
and correspondingly
2
I( ,w',Y) = [1 - 2 +
+2+ 2- 22 2 2 w 4 4
Yw Y1e Y 6 Y e
w' 1 ,w 2 1 w' 1
+ 12- + 2- 12
- 12 2
124 - + 12 2
W,2 a6 2w Y8 wy81
-9-
The range of variation of w' is a complicated function of
X, X', Y, w(see Felten and Morrison , formula 5); if the
supplementary hypothesis Yw<<l is made, we obtain:
Y 2w(l - ScosO)(l - B)< w' < 2Y 2w( - vcos0)
and for 6<<1,
2
4- < < ' 2 2 w. (A-5)4
b. The Energy Loss of the Electrons
For 0<<1 we have:
722
9 I(w,w',Y)w'dw' = 4rrer 2 ,
and hence:
dt 2 3 
which indicates that the electrons will gain energy when
202<2 and lose energy in the opposite case. For 22>>2:
dY 2 2 2Y4 ddnndY 2 2 2 (A-7)
trreY 6wL (A-7)dt dw
which is the same expression given by Woltjer .
-10-
c. The Photon Spectrum
Let the electron spectrum be:
dF -a
dy = Key
and the ambient photon spectrum:
Y 1<Y<2
dn = K W-B
dU w
The integration of (A-4) gives the following:
dR' = dY a I(w,w',Y)d w' dY y )
(w/e 2 w) 1/2
-a-l a+l-2B
2 -+1 -2 2
= 2nr K K +iW W 2
e w0 e
2 w 2 ( 2
+ a+3-2 B(-)f2 (a) + a+5-2B()' f3 (a)
2 W1"2
(1) (a+l-2B) 1
1
where
a+l a+3 a+5
f2(a)
a+3 a+5 a+7
f3(a) 2 12 + 12
a+5 a+7 a+9
For 0<<1 and w'>>2
2 -a-l
4rrr K KdR e  e a+l , 2
dw'e w f1(a) [w2
a+1-2a
a+1-2a
2
a+l-2B
2
1
1< <w 2
·
I  .- W 
X ta--2 fl ( )
x +l--2B t[fa
-11-
Expressions (A-6) and (A-8) indicate that the energy loss
dY
per unit time, dY , depends on the fourth power of the
dR cz+l
scattering angle,8, and the photon spectrum, dw'' on 0
Disregarding small numerical factors we can then write:
dY 4 d¥
dt ()dt isotropic (A-9)
dR a~+l dR
dR' a (dR isotropic (A-10)
Acknowledgements
It is a pleasure to thank Drs. A. Cavaliere, B. Coppi,
M. Rees, D. Sciama for helpful discussions. During the course
of this work the author was supported by an ESRO-NASA
international fellowship.
Table 1
Iy(>50 MeV) is the flux of y-rays above 50 MeV measured at Earth; Ee, the
energy of the electrons which produce photons of 50 MeV; c , the mean
life of the electrons for inverse Compton scattering.
Iy (>50 MeV) Ee c
y/cm2 sec (eV) (sec)
--5
measured flux ' 10
extrapolation
from X-ray ' 4 x 10- 6
observations
[isotropic] 10- 1 0
synchrotron 1-5 12ll 101 6
a -"I 2.4 E 10 _ eff1 310-62.4 -e10-3j 102
sotropic % 3 x 10- 6 \ 108 U 10- 6
Compton eff 10 1 1.2 x 
- 8 109 10 -2
ef1-3 -14 l 610
.~eff ° 0 5 x 10 ____ _ __ 10
i
References
(1) J. Vasseur, J. Paul, B. Parlier, J.P. Leray, M. Forichon, B. Agrinier,
G. Boella, L. Maraschi, A. Treves, R. Buccheri and L. Scarsi, Nature
226, 537 (1970).
(2) I.S. Shklovsky, Astrophys. J. 159, L77 (1970).
(3) G.I, Fishman, F.R. Harnden Jr. and R.C. Haymes, Astrophys. J. 156
L107 (1969).
(4) V.L. Ginzburg and S.I. Syrovatskii, Ann. Rev. Astron. Astrophys. 3,
297 (1965).
(5) S.L. O'Dell and L. Sartori, Astrophys. J. 161, L63 (1970).
(6) V.L. Ginzburg and S.I. Syrovatskii, The Origin of Cosmic Rays,
Pergamon Press (1964).
(7) W.F. Baylis, W.M. Schmidt and E. Lusher, Zeit. Astrophys. 66, 271
(1967).
(8) J. Felten and P. Morrison, Astrophys. J. 146, 686 (1966).
(9) L. Woltjer, Astrophys. J. 146, 599 (1966).


Production of gamma rays in the crab nebula pulsar

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