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as a continuation of our study of the effects of scattering of x - rays by interstellar grains , we have undertaken an examination of the surface brightness profile ( sbp ) of the crab nebula . toward this end , we have investigated the large - scale structure of the nebula related to the existence of an x - ray halo .
previous observations have offered convincing evidence for such a halo ( ku , et al .
1976 ; toor , et al .
1976 ; charles & culhane 1977 ) , though a decisive explanation as to its existence has , until the present , been lacking .
two competing theories have been advanced .
first , in analogy with other snrs , the halo is attributed to the emission of a strong shock .
this interpretation is troubled by the lack of x - ray emission lines from this region ( schattenburg , et al .
1980 ) , though it can not be excluded by this fact alone , given the featureless x - ray spectrum for the sn 1006 remnant ( becker , et al .  1980 ) .
a second theory which explains the x - ray halo is that it is due to the scattering of x - rays by interstellar grains .
we argue below that this interpretation is the correct one , and discuss the crab s halo as a natural , observable feature of all bright galactic x - ray sources .
the crab nebula was imaged by both the high resolution imager ( hri ) and the imaging proportional counter ( ipc ) aboard the einstein observatory .
a quantitative display of the hri data is presented in figure 1a .
this is the azimuthally summed , radially binned sbp .
the binning neglects the ellipticity of the nebula , and is centered roughly 16 arcseconds northwest of the pulsar at the centroid of the light in the phase - averaged image .
three things are readily apparent : ( 1 ) the surface brightness drops off dramatically beyond about 50 arcseconds , ( 2 ) a bump in this trend kicks in at about @xmath1 arcseconds , and ( 3 ) we run out of the hri s field of view just when things are getting interesting .
this brings us to the ipc data .
we gain with the ipc in two ways : with its wide field of view ( @xmath2 arcminutes ) , and with its lower background .
both of these features allow us to track the sbp of the crab to very large radii . in exactly the same way as the hri sbp , this data is presented in figure 1b .
again , three things are apparent : ( 1 ) due to this instrument s broad spatial resolution , the structure of the nebula is entirely lost , ( 2 ) the flair in the sbp is real , extending in roughly power - law form beyond @xmath3 arcseconds , and ( 3 ) there is no hint of a shell , which would appear as a shelf - like extension of the sbp .
obviously , we could hide a shell under this profile if it were dim and/or small enough that it did nt disturb the power - law tail on the measured sbp .
in this way , we could place an upper limit on the luminosity of a shell from a shock wave surrounding the crab .
we will return to this later .
in order to address the shapes observed for the sbps , we need to consider carefully the contributions of the point response function ( prf ) , which is due to the point response of the mirror and the spatial resolution of each detector .
since the hri has extremely fine ( 0.5 arcsecond ) spatial resolution , its prf is the point response of the mirrors .
the prf of the ipc ( fig .
2 ) is much more complicated due to its broad spatial resolution .
conceptually , however , we expect the ipc s prf to be the convolution of the mirror response ( which is measured by the hri ) , and its own intrinsic spatial resolution .
the prf for both the hri and the ipc were determined during calibration testing prior to the observatory s launch .
an analysis of this data reveals that the prf for each instrument contains an appreciable amount of power at large radii due to the scattering of x - rays from surface imperfections in the mirror elements .
this effect produces nearly power - law prfs for the hri , and adds similar wings onto the ipc s prfs , which are themselves dominated by the broad , nearly gaussian , spatial resolution of this instrument .
this explains to zeroth order the observed sbps for the crab nebula ( fig .  1 ) .
the excess surface brightness beyond @xmath4 arcseconds in the hri profile is due to scattering by the mirror elements of x - rays from the pulsar and nebula .
similarly , the power - law wing on the ipc s sbp is attributable to x - rays scattered by the mirror , while the width of the core is due as much to the spatial resolution of the detector as to the extent of the nebula .
upon closer examination , the sbp of the crab , as measured by the ipc , has too much power at large radii : typically , 10% of the power of the prfs falls beyond 3 arcminutes , whereas this fraction is 15% for the crab s sbp .
note that the @xmath5 arcsecond extent of the nebula , being the order of , or smaller than , the spatial resolution of the detector , and in turn being so much smaller than the scale of the effect we are discussing here , insignificantly impacts this comparison .
the difference of 5% between the measured sbp and the prf is of the order of the halo intensity reported by others for the crab .
a more detailed investigation involves the construction of a model profile for the response of the ipc to the intrinsic sbp of the crab .
this is handled in the following way . to first order
, the hri measures the intrinsic sbp of the crab 
the differences between these being the effect of mirror scattering and the hri s 0.5 arcsecond spatial resolution .
a second order approximation to the crab s intrinsic sbp would result if we could deconvolve the hri sbp with the hri s prf .
we settle with truncating the hri profile at @xmath1 arcseconds : this removes the power - law wing obviously associated with mirror scattering , while neglecting the broadening of the profile due to scattering interior to this radius .
this approximation to the true profile suffices if we are not interested in , nor sensitive to , the small - scale structure of the nebula when observed with the ipc .
we construct a model ipc sbp for the crab by convolving the ipc s prf with the crab s intrinsic sbp , which is approximated in the manner just described as the truncated hri sbp .
the prf we use is a linear combination of nine monoenergetic calibration prfs , weighted is such a way as to approximate the response of the ipc to the crab s continuous spectrum 
careful account being made of the gain of the detector and the modification of the crab s intrinsic spectrum by the effective area of the mirror / detector combination .
this prf is convolved with the crab s ` true ' sbp to produce a model ipc sbp .
this is compared to the measured profile in figure 3a : a significant difference is seen to exist between these two profiles beyond @xmath6 arcseconds .
the difference between the measured and model profiles reveals the size , shape , and intensity of the excess emission beyond the borders of the nebula . the magnitude of this excess , defined as the fractional difference between these two profiles is @xmath7 .
while this figure agrees with the magnitude of the excess emission reported by other observers , we are able for the first time to choose unambiguously between the two theories advanced to explain it . in particular , this emission has the character of a halo produced by the scattering of x - rays from interstellar grains and not that of a shock . a theoretical explanation as to why the former interpretation is correct would take us too far afield ;
the interested reader is left to pursue this topic in the literature ( overbeck 1965 ; hayakawa 1970 ; mauche , et al .
we simply assert two things : ( 1 ) the halo around the crab nebula has ` exactly ' the same shape as the halo around the compact galactic x - ray source gx13 + 1 , shown in figure 3b , and ( 2 ) the halo around gx13 + 1 is due to scattering from dust .
gx13 + 1 is but one of the compact sources for which we have detected halos .
the strength of these have been found to correlate well with two quantities which measure the column density of grains to a given source : ( 1 ) the amount of visual extinction , and ( 2 ) the distance through the galaxy s dust layer .
the data points for the crab are shown plotted amongst the points for the compact sources in figure 4 : the crab s relatively weak halo is consistent with its extinction as well as with its location with respect to the galactic plane .
these two observations  the similarity between figures 3a and 3b , and the correlation in figure 4  unmistakably identifies the mechanism for the crab s halo as scattering from dust
. however , we need to return to what the halo is not : emission from a shell associated with a shock wave .
an upper limit on the luminosity of such a shell is dependent on its angular size , due to the obscuring effects of mirror and dust scattering on the ipc sbp , in such a way that the upper limit is smaller ( larger ) for a shell of larger ( smaller ) size .
we choose to calculate the upper limit for a size of 9 arcminutes , consistent with the prediction of a sedov solution with nominal parameters , as well as with the ( age and distance ) scaled size of cas a. our estimate of the maximum surface brightness for a shell of this size is shown in figure 3a .
the associated 0.54 kev luminosity is about 1% of the crab s , or about @xmath0 .
significantly , this upper limit is a factor of @xmath8 larger than the luminosity ( though only a factor of three in central surface brightness ) of the sn 1006 remnant , which is of approximately the same age as the crab .
consequently , we can not place a strong upper limit on the shell s luminosity
. the important result of this paper is however that at ` soft ' x - ray energies the crab s dust - scattering halo completely obscures its putative shell . due to the @xmath9 dependence in the size of the halo , and the @xmath10 dependence in the halo intensity
, the prospects for detecting the shell would be much better at higher energies .
we have searched the region surrounding the crab nebula for the existence of a shock wave with the imaging instruments of the einstein observatory . 
 the search is complicated by the scattering of nebula and pulsar x - rays from the imperfectly polished surfaces of the telescope mirror , as well as from interstellar grains along the line of sight . 
 both of these effects lead to the appearance of x - ray emission , in the form of an x - ray halo , beyond the boundaries of the nebula filaments .    
 we show that the size , shape , and intensity of the halo around the crab nebula , above the contribution of mirror scattering , is consistent with what is expected from the scattering from interstellar grains . 
 the upper limit on the x - ray emission from a shock wave is about 1% of the total 0.54 kev luminosity of the crab or about @xmath0 ( assuming a distance of 2.2 kpc ) . 
 this figure applies to a shell whose angular radius is 9 arcminutes . 
 the upper limit is smaller ( larger ) for a shell of larger ( smaller ) size . 
 this upper limit is an order of magnitude or more below the flux of cas a , tycho , and kepler snrs , which are 2 to 3 times younger , but it is still above that of sn 1006 .