Autocorrelation functions for diffuse cosmic X ray

Bleach, R. D.

Boldt, E. A.

Holt, S. S.

Schwartz, D. A.

Serlemitsos, P. J.

English

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7,;!, X-
LIMITS ON THE SMALL SCALE STRUCTURE
OF THE DIFFUSE COSMIC X.-RAYS
if
DANIEL A. SCHWARTZ
ELIHU A. BOLDT
STEPHEN S. HOLT
PETER J. SERLEMITSOS
R. D. BLEACH
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•LIMITS ON THE SMALL SCALE STRUCTURE
OF THE DIFFUSE COSMIC X-RAYS
Daniel A. Schwartz*, Elihu A. Boldt, Stephen S. Holt,
and Peter J. Serlemitsos
Goddard Space Flight Center
Greenbelt, Maryland 20771
and
Richard D. Bleach
University of Maryland
College Park, Marvland 20742
I. INTRODUCTION
In all popular theories, the diffuse cosmic X-rays contain a sub-
stantial contribution from sources at a redshift distance of order
z = 1, thus measurement of their angular distribution provides data
with cosmological relevance 1,2 . Wolfe and Burbidge 2 have constructed
angular autocorrelation functions for X-rays arising directly or indir-
ectly from the observed distribution of galaxies. Available data on the
X-ray isotropy in the 10 to 40 keV range  had been obtained on an angular
scale of 12°. To coripFte with that data, Wolfe and Burbidge had to inte-
grate their autocorrelation functions to define an effective angular
scale of fluctuations, and use beam convolution techniques  to extra-
polate the measured upper limit X-ray fluctuations down to that angular
scale. Here we compare the experimentally measured autocorrelation
fi.nction with the calculations of Wolfe and Burbidge and confirm their
conclusions that the X-ray isotropy is not consistent with the observed
superclustering,and probably not with clustering, of galaxies. If the
bulk of the diffuse cosmic X-rays arise from discrete sources, then the
weakest lower limit to their number is obtained by ignoring possible
clustering and yields a minimum of 5 x 10 5 objects assuming constant
apparent luminosity or 4 x 10 6 objects assuming constant absolute luminosity.
\A:;-NRC Resident Research Associate
-2-
II. EXPERIMENTAL DETAILS
The X-ray detector consisted of a low-background multi-anodo
proportional chamber of 650 cm  effective area collimated 2° x 8°
FWIIM to a 3.2 c1112 ster telescope factor. The counter contained one
atmosphere of xenon-methane with a 25 gm mylar window. Events occur-
ing in one and only one anode, and not vetoed by a simultaneous event
in any immediately adjacent anode or in any of the anodes adjacent to
the counter walls, entered the 128 channel analog to digital convers'_)n.
Our pulse height analyzer accepted events between 2 and 32 keV, giving a
median energy of 6.3 keV for a diffuse X-ray count.
Aerobee flight 13.07 carried this detector to an altitude of 91 miles
above White Sands Missile Range on September 21, 1970. We gathered
spectral data from the Cygnus region and then searched for new sources
between Cygnus and the known source Serpens XR-1. Using photographs
of the star field, we verify that the rocket rolled the 2 0 aperture at
about 0.51 degrees per second, along the galactic plane. During 44 sec.,
as the field of view moved from kII = 69 0 to fII = 46°, we detected no
discrete sources above a 3 	 limit of 0.02 photons (cm 2 sec)- 1 between 2
and 10 keV. We base the discussion in this letter upon the data obtained
during this time.
Figure 1 presents our raw data vs. time sorted into bins of — 2° width.
The count rates shown represent mostly diffuse X-rays, but also contain 	 I
some background due to electrons produced in the collimator and to high
energy y-rays which Compton scatter, depositing only a small fraction
of their energy. The background spectrum is flat, with a small turn-up
6
6-3-
at a few keV. We have three independent estimates of the background flux;
an upper limit of 10 -3 (cm 2keV) -1
 obtained by viewing the earth in a previous
rocket flight; an upper limit of 8 x 10 
4 (cm 2 kev) -1 obtained by assuming
all the measured counts above 25 keV were due to background; and similar
numbers derived by comparing the measured count rates with rates predicted
using the known detector response and the measured diffuse spectrum of
15.2E -1.7 above 8 keV 5 and 8.4 -1.35 below 8 keV6 . From the last con-
sideration, we consider the "noise" contribution as ` 25% for the data
in Figure 1.
:'rom the original results on an unresolved galactic canponent of
X-ray emission 7 9 we would have expected 8 counts sec -1
 due to galactic
X-rays. Subtracting 20 seconds of data with the detector off the galac-
tic plane-' gives an estimate of only 4 +4 counts per second, a nebligible
contribution to the total diffuse count rate. Note, however, that Cooke
and Pounds  report the resolution of the previously diffuse galactic
II
component at 1	 300° into a few discrete X-ray sources which would
lie above our present sensitivity. Thus we have no reason to expect
r
such an extended galactic plane flux in the inter-arm region surveyed
In this experiment.
iii. LIMITS ON DISCRETE SOURCLS
Let us assume that the diffuse X-rays arise as the integrated effect
of discrete sources of constant apparent luminosity j(photons/sect dis-
tributed uniformly randomly over the sky. Then in a set of bins containing
-;The detector actually pointed t th C nus Lpop during th'
	 ime. T
obviate our limit on galactic panel- rya s, this source wou^a iave to grave
a spectrum flatter than the diffuse component, whereas a previous measure-
ment(8) shows this to have the softest known X-ray spectrum.
6=4-
an average number of background counts C b , we should observe a mean
number of counts N = j n + C b
 and a variance in the number of counts
S2 = j .1 (j + 1) + Cb.10 From these relations we can calculate the
2
average number of discrete sources in one bin on the sky: n = (N'b)
S - Iv
(If one considers that only a fraction f of the diffuse component is due
to discrete sources then n = f 2 (N - C b )
2
	 Lumping all the counts into
SZ	 N
M = 11 bins each 2° wide, we find N = 110, C b = 27, and the measured
_	 2
variance estimate S 2 = ^^ l 	 N) = 62. The data thus all S2 - N = 0,
10
n	 The variance in the estimate of S 2 is 11 var S 2 = 2	 2 2	 2) 	 (28)
Adding 3	 var S2 to S 2 , we estimate (S2 - N) s 36, or n = 190 to about
80% confidence. Since it takes 2600 of the 2° x 8° resolution elements
to fill the sky, it would take at least 5 x 10 5 discrete sources to be
consistent with the measured isotropy.
This compares with previous lower limits of 10 5 fbr the 10 to 40 keV
range  and a limit (private communication from P. Sanford) of about 1.5
X 106 (to the same statistical significance) 12 obtained with an argon
counter having a median energy of about 4 keV. Because the diffuse
component has a complicated spectral shape, the predominant sources
might differ as a function of energy and thus it is important to consider
All the above limits as independent information concerning an appropriate
range of energies.
Note that the limit obtai.ned above applies only to sources of con-
stant apparent luminosity. Sources of constant absolute luminosity dis-
tributed throughout a volume will add a term n var j to the expected
variance, due to the dispersion in apparent emissivity proportional to j-5/2.
I
-5-
We now expect S2 = j n (j + 1) + n var j + Cb . The total number of
sources is still only proportional to the inverse square of the ob-
served fluctuations, ii = q (N - C b ) 2 AS 2 - N) but is now multiplied by
1 r
q = < j2 >/< j > 2 ^r 3 mo , for a model of sources distributed within
1
some maximum luminosity distance r o , and where sources less distant
than r l could be recognized individually. Our discrete source sensi-
tivity stated previously gives us r  5 4.3 x 10 26 cm, and taking ro = 1028 cm
gives q = 7.8, or a lower limit of 4 x 10 0 sources of constant absolute
luminosity in the sky. (If one asks how many of the discrete sources
were recognized individually above a sensitivity F photons cm-2 sec-1,
then in surveying a fraction of the skyot one would expect on the average
3/2
to detect n = N 1/2(	 r	 sources, where I is the average diffuse
/	 13,3
intensity, and N the total number of sources in the sky, 	 (cf. ref. 14,
where 61 = 3 F). We observe no discrete sources, therefore to 98 percent
confidence n !(. 3. We have a = 0.004, giving us a lower limit of only 7000
sources using this technique, and we feel a previously published lower
limit 
12 
of 109 sources should be corrected to a similarly small number.)
A much stronger lower limit is obtained if one considers the actual
clustered distribution of galaxies instead of a random distribution. We
compare with the calculations of Wolfe and Burbidge 2 , who used the correla-
tion technique developed by Chandrasekhar and Munch 15 . We divide our data
6
into ^i = 220 bins of 0.205 seconds during the scan and construct the observed
-6-
autocorrela;ion function F (A) - f (8)/f(0),
1	 M-k
f (B)	 M - k	 "	 (N i	N) (N 1.+k N)
where A = 0.102 k degrees, and X represents the average number of counts 	 r
in one bin. We present the results in Figure 2, along with the previous
theoretical calculation. Our data are consistent with random scatter
about zero intrinsic correlation. The valid range of these measurements
extend from the collimator resolution of 2° up to about $°, where scatter
increases die to the 23° limit of our scan. The data present a clear
contradiction to supercluster and some cluster models for the origin of
the diffuse X-rays. Independently, the data stand as a constraint to
other theoretical calculations which might employ different cosmological
models or connect the X-rays to galaxies in a manner different than the
direct proportionality assumed by Wolfe and Burbidge.
6
References
1. Wolfe, A., Astrophys. J. (Letters) 159, L61 (1970).
2. Wolfe, A.M., and Burbidge, G.R., Nature 228, 1170 (1970).
3. Schwartz, D.A., Astrophys. J. 162, 439 (1970).
4. Conklin, E.K., and Bracewell, R.N., Nature 216, 777 (1967).
5. Schwartz, D.A., Hudson, H.S., and Peterson, L.E., Astrophys. J. 162,
431 (1970).
6. Boldt, E.A., Desai, U.D., Bolt, S.S., and Serlemitsos, P.J., Nature
224, 677 (1969).
7. Cooke, B.A., Griffiths, R.E., and Pounds, K.A., Nature 224, 134 (1969).
8. Grader, R.J., Hill, K.IJ	 and Stoering, J.P., Astrophys. J. (Letters),
161, L45 (1970).
9. Cooke,	 B.A.,	 and Pounds, K.A., Nature 229,	 144 (1971).
10. Schwartz,	 D.A., thesis, Univ. of California at San Diego	 (1969).
11. Guest, P.G., Numerical Methods of Curve Fitting (Cambridge; Cambridge
University Press)(1961).
12. Fabian, A.C., and Sanford, P.W., Nature 231, 52 (1971).
13. Bunner, A.M., Coleman, P.C., Kraush aar, W.L., McCammon, D., Palmieri, T.IM.,
Shilepsky, A., and Ulme1-, M., Nature 223, 1222 (1969).
14. Sill., J., Space Sci. Rev. 11, 671 (1970).
15^ Chandrasekhar, S., and Munch, G., Astrophys. J. 115, 103 (1952).
16. 3e%laucouleurs, G., Science 167, 1203 (1970).
4
FIGURE CAPTIONS
Figure 1. Total counts per 2.07° bin in galactic longitude. Fluctuations
are consistent with counting statistics and allow no less than
an average of 190 discrete sources to produce the total diffuse
counting rate in each bin.
Figure 2. Autocorrelation functions	 Solid lines: Theoretical calcula-
tions of Wolfe and Burbidge for superclusters (upper pair) or
clusters (lower pair) of galaxies. The two lines in each pair
are distinguished by the assumed total numiuer of member galaxies.
Triangles: Possible correlation modeled for our detector based
on a 23° cut throug' , contours of smoothed galaxy counts (to
magnitude 19.4) presented by de Vaucouleurs 16 , Circles:
Measured correlation, this experiment.
01
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GALACTIC SCAN
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69.0	 60.7	 52.4	 46 11
64.9	 56.6	 48.3
GALACTIC LONGITUDE
FIGURE 1
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FIGURE 2


Limits on the small scale structure of the diffuse cosmic X-rays

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