R. COWSIK, YASH PAL, and T. N. RENGARAJAN. Tata Institute of Fundamental Research, Bombay, India. (Received 30 September, 1969). Abstract. An attempt ...
A SEARCH
FOR A CONSISTENT
ELECTROMAGNETIC
SPECTRUM
MODEL
FOR THE
OF THE CRAB NEBULA*
R. C O W S I K , Y A S H PAL, and T. N. R E N G A R A J A N Tata Institute of Fundamental Research, Bombay, India
(Received 30 September, 1969)
Abstract. An attempt is made to search for a consistent model to explain the electromagnetic spectrum of the Crab nebula (Tau A). It is assumed that there is a continuous injection of electrons at the centre of the nebula with an energy spectrum E -1.54 as evidenced by radio data. This spectrum must steepen to a slope larger than 2 at some energy E~ in order to ensure that the energy input into electrons remains finite. The spectrum must also steepen beyond an energy Ec depending on the magnetic field because of synchrotron energy losses. Two types of models are considered: Class I, in which the whole nebula is characterised by a uniform magnetic field, and Class II, in which besides the general field H0, small filamentary regions of strong field//8 are postulated. In models of Class I, the best fit to the observed data is obtained when E~> Ec and H0 -~ 5 • 10-4 gauss. However, this predicts a decrease in X-ray source size beyond ~ 40 KeV. There are two possibilities of Class II model depending on the residence time of electrons in strong field regions being small or large. The former case explains the flattening in the optical spectrmn. Experiments to distinguish between the various models are indicated.
1. Introduction There have been m a n y attempts to u n d e r s t a n d the electromagnetic spectrum o f the C r a b N e b u l a (Shklovski, 1968; A p p a r a o , 1967; Tucker, 1967; Burbidge a n d Hoyle, 1969), which has been studied over a wide range o f frequencies, ranging in the r a d i o region f r o m ~ 107 H z to the h a r d X - r a y region ~ 102o H z ; at higher frequencies, up to ~ 1029 Hz, useful u p p e r limits have been placed on the flux. The observations are s u m m a r i s e d in F i g u r e 1. I n this p a p e r we discuss a n u m b e r o f consistent models for interpreting the observed spectrum, a n d indicate future measurements which might distinguish between them. A recent study o f the optical spectra o f the m o v i n g wisps in the C r a b nebula (Scargle, 1969) indicates a c o n t i n u o u s activity near the centre o f the n e b u l a a n d a possible renewal o f the relativistic electrons. A l s o a pulsar, NP0532, has been identified at this site a n d is the p r o b a b l e cause o f this activity. This p u l s a r has a p e r i o d o f ~ 30 ms a n d a characteristic slowing d o w n time o f ~ 2 0 0 0 years. It has been suggested ( G o l d , 1968) t h a t pulsars w o u l d be i m p o r t a n t sources o f cosmic rays a n d t h a t the a b o v e rate o f slowing d o w n could be due to emission o f relativistic particles at a rate 1037-3s ergs/sec. Therefore, in the following discussions we shall assume t h a t there is a continuous injection o f high-energy electrons at the centre o f Crab. The r a d i o spectrum, f r o m 10 to 1000 M H z , has an index o f ~ 0 . 2 7 , indicating a slope o f ~ 1.54 for the electron spectrum. The injection spectrum o f electrons should * Presented at the International Conference on Cosmic Rays, Budapest, 1969. Astrophysics and Space Science 6 (1970) 390-395. All Rights Reserved Copyright 9 1970 by D. Reidel Publishing Company, Dordrecht-Holland
391
A MODEL FOR THE ELECTROMAGNETIC SPECTRUM OF THE CRAB NEBULA 22
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LOG FREQUENCY Hz Fig. 1. The available spectral data on the Crab in the frequency interval 107-1026 Hz are summarised. The optical data, corrected for interstellar absorption (A v = 2; Crab distance- 1.8 Kpc), are taken from Scargle (1969). The X-ray and the 7-ray data are from Clark et al. (1968); Frye and Wang (1969); Grader et al. (1966) and (1969); Boldt et al. (1969); Glass (1969); Peterson et al. (1966) and Haymes et al. (1968). The y-ray spectra arising from Compton scattering of the synchrotron quanta (auto-Compton) and the 3 K and sub-millimeter quanta (Houck and Harwit, 1969) are also plotted for a magnetic field of 5 • 10 4 gauss. An assumption of substantially smaller field would contradict the ?,-ray upper limits (see text, model 3). The dashed curves marked (4) represent the contributions to the spectrum from the general and strong field regions as per model 4.
also have the same slope, since with a field o f ~ 5 x 10 . 4 gauss, the s p e c t r u m o f electrons at low energies ( ~ 1 G e V ) remains u n a l t e r e d during the lifetime ( ~ 900 yrs) o f Crab. However, in o r d e r that the t o t a l energy o f the injected electrons be finite, the injection s p e c t r u m m u s t steepen at some high energy E i to a l o g a r i t h m i c slope greater t h a n 2. Relativistic electrons suffer energy losses mainly because o f s y n c h r o t r o n r a d i a t i o n a n d b e t a t r o n deceleration in the e x p a n d i n g envelope. W i t h c o n t i n u o u s injection ( K a r d a s h e v , 1962) the electrons at present w o u l d have the same spectral slope as t h a t o f the injected spectrum, u p to an energy E ~ = 4 / b T , T being the age o f the n e b u l a ( ~ 9 0 0 yrs) a n d b~-2e 4 ( H e ) / 3 m 4 c 7 = 1.95 x 10 - 9 ( H 2 ) / m c 2. B e y o n d E c the spectral slope increases by unity. The resulting s p e c t r u m m a y again be injected into localised regions o f stronger m a g n e t i c field. Then, if b S is the characteristic energy loss p a r a m e t e r in this field a n d z the c o n t a i n m e n t time therein, the s p e c t r u m will experience a further steepening by one in the index at energies b e y o n d 1/bsz.
392
R. COWSIK ET AL.
2. The Models We will consider two broad classes of models viz. class I, in which the magnetic field is reasonably uniform over the whole nebular volume and the continuous injection occurs at a point (close to the pulsar) near its centre; class II, in which there is a general magnetic field H o over the volume, but in addition, there are filamentary regions throughout the volume where the value of the magnetic field H s is much higher. Clearly either of these model types is an idealised representation of the actual situation. 2.1.
M O D E L S OF CLASS I
Three possibilities exist, which provide different levels of fit to the experimental data. These models are basically distinguished by the criteria that the steepening (at E~) in the injection spectrum occurs at the same energy, earlier or later than the steepening (at Ec) due to synchrotron energy loss. (1) E~~-Ec: In this case, the slopes of the optical and X-ray spectra will be the same. It may be barely possible to reconcile the experimental data with a single power law spectrum v- 1.1. Then we have to choose Er to correspond to a frequency ~ 5 x 1013 Hz (see Figure 1) and thus obtain Ho"~ 5 x 10 -4 gauss. The injection spectrum beyond E i will have to have a slope of 2.2, which, because of synchrotron-steepening becomes 3.2, giving a slope of ( 3 . 2 - 1 ) / 2 = 1.1 for the synchrotron radiation spectrum. (2) Ei > E c: In this case we have to put E~ at the same place as in the last case, giving a break at ~ 5 x 1013 Hz and a magnetic field H o = 5 x 10 - 4 gauss. The optical spectrum would then have an index (1.54+ 1 - 1)/2=0.77, which is not inconsistent with experiment. If E~ now corresponds to a frequency ~ 1.6 x 1016 H z , the appropriate slope of the X-ray spectrum can then be explained by a suitable choice of the spectral slope of the injected electrons beyond E i. (3) E~
