CHANCE COINCIDENCES AND THE SO-CALLED REDSHIFT SYSTEMS IN THE ABSORPTION SPECTRUM OF PKS 0237-23

Varshni,Y.P.: 1981, Atrophys.Space Sci., 74, 3.

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Abstract. It is shown that the number of redshift systems based on C IV doublets, proposed by Boronson et al (1978) in the absorption spectrum of the quasar PKS 0237-23, is insignificantly different from that that would be expected from chance coincidences. Consequently, these systems and their z-values appear to be devoid of any physical significance.

1. Introduction

The object PKS 0237-23 was identified as a quasar by Arp et al. (1967) who also discovered that there are many lines in its absorption spectrum. Subsequent investigations on its absorption spectrum were those of Greenstein and Schmidt (1967), Burbidge et al. (1968) and Boksenberg and Sargent (1975).

Recently, Boroson et al. (1978, hereafter BSBC) have reported on the absorption spectrum of PKS 0237-23. Six observations made over 3 years were combined to yield wavelengths and equivalent widths of 193 absorption lines in the spectral region 3715-4290 . These authors claim that many of these absorption lines are due to C IV 1548.20, 1550.77 at various redshifts. A total of 45 redshifts have been claimed.

We have proposed a theory of quasars (Varshni, 1973, 1974a, 1975a, 1977a, b, 1978, 1979. Varshni and Lam, 1976), based on sound physical principles, which does not need the artificial assumption of redshifts and provides satisfactory explanations of the various phenomena associated with quasars. All the observational evidence on quasars either supports our theory, or is consistent with it (Varshni, 1979). Clearly, the claims of BSBC require examination.

2. The 'Redshift' Systems of BSBC

BSBC carried out their search for redshifted C IV lines as follows: "Line pairs identified as C IV doublets were required to have the correct separation to 0.3 ngstroms. The ratio of the equivalent widths W(1548 ): W(1551 ) should lie between 2 and 1. This requirement was made with appropriate account taken of the accuracy of the equivalent widths. Occasional exceptions were made to this constraint if one of the lines was obviously an unresolved blend." Forty-five redshift systems thus found are given in their Table 3. A check on these systems showed that the discrepancy allowed for the separation was more accurately 0.33 . To clarify the constraints imposed on the ratio W(1548 )/W(1551 ), we searched for all such pairs of lines whose separation would be the same as that of redshifted C IV lines, within 0.33 . A total of 52 such pairs were found and are shown in Table I:

Table I

'Redshifted' C IV doublets in PKS 0237-23


              C IV 1548.20    C IV 1550.77
              ____________    ____________

System              Equivalent       Equiv.          Credibility
number  z          width()        width     R(EW) rating

 1    1.4007  3716.9  0.6     3722.8  0.9      0.67  PO
 2    1.4174  3742.6  0.3     3748.8  0.2      1.50  PO
 3    1.4575  3804.7  0.5     3810.9  0.3      1.67  PO
 3a   1.4794  3838.5  1.2     3845.2  0.1     12.00
 4    1.5023  3874.2  0.7     3880.3  0.7      1.00  C
 4a   1.5063  3880.3  0.7     3886.7  1.9      0.37
 5    1.5144  3892.8  0.8     3899.3  0.3      2.67  PR
 6    1.5153  3894.1  0.4     3900.7  0.3      1.33  PR
 6a   1.5185  3899.3  0.3     3905.5  0.9      0.33
 7    1.5260  3910.8  0.2     3917.1  0.2      1.00  PO
 8    1.5348  3924.6  0.3     3930.8  0.1      3.00  PR
 9    1.5389  3930.8  0.1     3937.2  0.1      1.00  PO
 9a   1.5517  3950.6  1.0     3957.0  0.1     10.00
 10   1.5546  3954.9  0.1     3961.7  0.2      0.50  PO
 11   1.5559  3957.0  0.1     3963.7  0.2      0.50  PR
 12   1.5603  3963.7  0.2     3970.6  0.1      2.00  PR
 12a  1.5787  3992.3  0.1     3999.0  0.1      1.00
 13   1.5950  4017.6  0.7     4024.3  0.3      2.33  C
 13a  1.5956  4018.6  0.4     4025.1  1.2      0.33
 14   1.5964  4019.8  1.2     4026.3  0.8      1.50  C
 15   1.6102  4041.0  0.6     4047.8  0.4      1.50  C
 16   1.6146  4047.8  0.4     4054.8  0.1      4.00  PO
 17   1.6164  4050.6  0.1     4057.6  0.1      1.00  PO
 18   1.6209  4057.6  0.1     4064.4  0.1      1.00  PO
 19   1.6311  4073.4  0.1     4080.4  0.3      0.33  PO
 20   1.6356  4080.4  0.3     4087.2  0.2      1.50  PR
 21   1.6365  4081.7  0.2     4088.7  0.1      2.00  PR
 22   1.6399  4087.2  0.2     4093.9  0.1      2.00  PR
 23   1.6410  4088.7  0.1     4095.7  0.2      0.50  PO
 24   1.6429  4091.7  0.1     4098.5  0.1      1.00  PO
 25   1.6443  4093.9  0.1     4100.7  0.1      1.00  PO
 26   1.6512  4104.5  0.1     4111.4  0.9      0.11  PO
 27   1.6556  4111.4  0.9     4118.2  0.7      1.29  C
 28   1.6568  4113.2  0.9     4120.2  1.1      0.82  C
 29   1.6581  4115.2  0.4     4122.2  0.2      2.00  C
 30   1.6591  4116.9  0.2     4123.7  0.2      1.00  PO
 31   1.6598  4117.7  0.2     4124.8  0.2      1.00  PR
 32   1.6599  4118.2  0.7     4124.8  0.2      3.50  PR
 33   1.6636  4123.7  0.2     4130.7  0.1      2.00  PO
 33a  1.6689  4132.0  0.1     4138.7  0.8      0.13
 34   1.6703  4134.2  0.9     4141.0  1.1      0.82  C
 35   1.6714  4135.9  0.9     4142.7  1.4      0.64  C
 36   1.6732  4138.7  0.8     4145.5  0.4      2.00  C
 37   1.6747  4141.0  1.1     4147.7  0.5      2.20  C
 38   1.6775  4145.5  0.4     4152.1  0.1      4.00  PO
 39   1.6820  4152.1  0.1     4159.3  0.2      0.50  PR
 40   1.6907  4165.6  0.3     4172.8  0.1      3.00  PR
 41   1.7188  4209.3  0.2     4216.1  0.2      1.00  PO
 42   1.7240  4217.4  0.1     4224.1  0.1      1.00  PO
 43   1.7292  4225.3  0.1     4232.1  0.2      0.50  PO
 44   1.7455  4250.6  0.1     4257.6  0.1      1.00  PO
 45   1.7533  4262.6  0.1     4269.7  0.2      0.50  PO

The forty-five pairs of BSBC are labelled 1 to 45. The seven pairs rejected by BSBC are indicated by 3a, 4a, 6a, 9a, 12a, 13a, and 33a. The ratio of the equivalent widths, R(EW)=W(1548 )/W(1551 ), is shown in column 7. BSBC have rated the credibility of the systems as follows: C-Certain, PR-Probable, and PO-Possible. These ratings are shown in the last column of Table I. An inspection of the R(EW) values shows that considerable latitude was allowed. The accepted systems have R(EW) lying between 0.11 to 4.0. As R(EW)=0.11 has been considered acceptable, very likely R(EW)= 5 or 6 would have been accepted, had such a value occurred in the actual data. There are twelve cases with R(EW) < 1 and eight cases with R(EW) > 2.

BSBC recognized that many systems could arise by chance, and made a statistical assessment of the overall significance of the identifications. Earlier Varshni (1974b, c, 1975b) had emphasized the importance of taking into account the density distribution of lines for determining chance coincidence "redshifts". Since, in the case of PKS 0237-23, the distribution of lines in wavelength is not uniform, BSBC divided the line list into six 100 regions and calculated the expected number of chance C IV doublets in each region by the following formula E(C IV)=[N(N-1)/L](2deltaLambda)(f), where N is the number of lines in the region, L is the extent of the region, deltaLambda is the required accuracy of the separation, and f is the probability of fulfilling the relative equivalent-width criterion by chance. The quantity f was arbitrarily given a value 0.75 by BSBC. We have noted earlier that in the spectrum of PKS 0237-23 there are 52 systems, of which, BSBC have accepted 45. It would be more logical to assume f=45/52=0.87. If the results of BSBC are taken at face value, these indicate that in two wavelength intervals, namely, 4000-4100 and 4100-4200 , the number of observed systems is much larger than that expected by chance. Taken together, in the two intervals, there are 28 observed systems, whereas 14.13.5 are expected by chance.

3. The Error in the Work of BSBC

There is a basic error in BSBC's calculation of the number of chance C IV doublets. They have not taken into account the variations in the density distribution of lines in the absorption spectrum of PKS 0237-23 in a proper way. The separation between the two C IV lines varies from 6.17 at z=1.40, to 7.09 at z=2.76. Thus, it is seen that at any given redshift only an interval of about 10 of the spectrum is sampled. In Figure 1 we show a histogram of the number of lines in 10 intervals as a function of the wavelength. It will be noticed that at several places there are large fluctuations in the density of lines.

Plot of Line Density of Spectrum

Fig.1. Number of lines in 10 intervals as a function of wavelength for PKS 0237-23. The data are due to BSBC. The vertical bars at the top indicate the positions of the "walls" used in generating the ghost spectra.

The density of lines plays a very important role in determining chance coincidences (Russell and Bowen, 1929. Russell et al., 1944; Varshni (1974b, c, 1975b). The fluctuations in the density of lines in the spectrum of PKS 0237-23 must be properly taken into account in any calculation of the number of chance coincidences.

4. Chance Coincidences

Let us calculate the chance coincidence probability of finding a C IV doublet at a certain redshift. Suppose there are n observed lines in a wavelength interval X, and x is the wavelength coincidence tolerance (0.16 in our case). Then the probability, p1, of one of the observed lines falling within x of one of the C IV lines is p1=2nx/X. Similarly, p2=2(n-1)x/(X-2x). Thus, the probability of both the occurrences is P=4n(n-1)x^2/X(X-2x).

For the mathematically-minded reader, we note that the BSBC procedure is equivalent to calculating

[(Sum from i=1 to m of ai)^2 - (Sum from i=1 to m of ai)]/m

as compared to the correct expression:

Sum from i=1 to m of ai(ai-1),

where ai represents a sequence of numbers. Only if the dispersion in the ai values is very small, the two expressions will be close to each other. If the dispersion in the ai's is large, the former expression can be substantially smaller than the latter one.

The calculated values of P in steps of deltaz=0.002, as a function of z, are shown in Figure 2(a). These values were averaged over deltaz=0.02 and the resulting histogram is shown in Figure 2(b), which may be compared with the results of BSBC, shown in Figure 2(c). The empty squares in Figure 2(c) represent the systems rejected by BSBC. It will be noticed that the chance-coincidence redshift distribution is very similar to that of the actual systems found for PKS 0237-23. We conclude that the proposed systems are consistent with the hypothesis that these are just chance-coincidences without any physical significance.

Redshift Histograms

Fig.2. (a) P as a function of z, in steps of deltaz=0.002. (b) Histogram showing averaged P over deltaz=0.02, as function of z. (c) Histogram showing the distribution of absorption redshifts. The credibility rating of the 45 systems of BSBC is indicated by C,PR or PO.

To obtain the actual number of chance-coincidence redshift systems, two methods were used.

Method 1.
A detailed examination of the data of BSBC and Figure 1 shows that the absorption spectrum of PKS 0237-23 can be conveniently divided into 15 intervals, to allow for the varying density of lines, with `walls' at 3715, 3740, 3760, 3810, 3860, 3880, 3900, 4080, 4110, 4120, 4150, 4170, 4205, 4270, 4285, and 4290 . Ten ghost spectra (Varshni, 1975b) were generated on a computer (IBM 360/65). The fifteen intervals were considered separately; inside an interval, the ghost wavelengths were generated by the formula

Li(ghost)=Lmin+Ri(Lmax-Lmin),

where Lmin and Lmax represent the lower and upper limits, respectively, of the interval, and Ri is a uniformly distributed random number between zero and one (generated using the RANDU subroutine of IBM's Scientific Subroutine Package). The minimum separation between two lines in the spectrum reported by BSBC is 0.4 ; it was constrained that the minimum separation between two wavelengths in a ghost spectrum will be the same. The equivalent width of the ith observed line was assigned to Li(ghost). Thus in any given interval, a ghost spectrum has the same number of lines of various equivalent widths as in the observed spectrum. Each ghost spectrum was analysed for `redshifted' C IV doublets such that the separation between the lines is within 0.33 of the correct separation. The initiating number used for RANDU subroutine for generating the ghost spectrum and the total number of C IV doublets found in each of the ten cases are shown in Table II. We have noted earlier that BSBC have not used any well-defined rule for R(EW) for a system to be `acceptable'. For the ghost spectra, we find that if we use the criterion that R(EW) should lie between 0.11 and 5, approximately 85 percent of the systems are `acceptable'. The average total number of chance C IV doublets is 48.52.1 and about 85 percent of these are in the `acceptable' category.


Table II

Number of C IV doublets in ghost spectra


Ghost     Initiating  Total
spectrum  number      number of
number    for RANDU   C IV doublets

 1        28471       53
 2        83123       47
 3        26687       45
 4        42003       48
 5        62893       50
 6        66981       49
 7        10737       47
 8        78241       50
 9          137       49
10         8873       47
        Average       48.52.1

Method 2.
The expression for E(C IV) with f=1 was used to calculate the expected number of chance C IV doublets in each of the fifteen regions which are identified in Method 1. The total number of such doublets is found to be 486.9. If we assume that f=0.87, the number of "acceptable" C IV doublets comes out to be 41.86. We may note here that this method does not take into account doublets formed such that the two lines lie in two separate, adjacent, intervals. However, the agreement of the result obtained by this method with that obtained from Method 1 indicates that the error arising because of this is small. The results presented here conclusively show that the number of C IV redshift systems proposed by BSBC is insignificantly different from that that would be expected from chance coincidences. Consequently, these systems and their z values are devoid of any physical significance. These z values are merely a futile exercise in empty numerology.

The next, and more interesting, problem is what are these lines due to. We have succeeded in identifying a good many of the lines recorded by BSBC on the basis of the theory of quasars that we have proposed. We shall present these identifications and the arguments to support them in due course. (in Varshni, 1988)

Acknowledgment. This work was supported in part by a research grant from the Natural Sciences and Engineering Research Council of Canada

References and Related Papers

  1. Arp,H.C., Bolton,J.G., Kinman,T.D.: 1967, Astrophys.J., 147, 840.
  2. Boksenberg,A., Sargent,W.L.W.: 1975, Astrophys.J., 198, 31.
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  4. Burbidge,E.M., Lynds,C.R., Stockton,A.N.: 1968, Astrophys.J., 152, 1077.
  5. Greenstein,J.L., Schmidt,M.: 1967, Astrophys.J., 148, L13.
  6. Russell,H.N., Bowen,I.S.: 1929, Astrophys.J., 69, 196.
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  10. Varshni,Y.P.: 1974b, Astrophys.J., 193, L5.
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  12. Varshni,Y.P.: 1975a, Astrophys.Space Sci., 37, L1.
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The average scientist is good for at most one revolution. Even if he has the power to make one change in his category system and carry others along, success will make him a recognised leader, with little to gain from another revolution.
-Gerald M Weinberg, Introduction to General Systems Thinking, 1975.

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