how are quasar redshifts distributed, in [0,3]? If they are not distributed equally (to within some bound), then probability calculations need to reflect that non-equal distribution.
I addressed that question. As I said, the frequency of redshifts is not constant over the entire range. Based on recent mainstream sources it looks like it rises from a finite value (about 1/8th of the max) near z=0 to a max at about z=1 to 1.5, then levels off till around z=2.5 to 3.0, where it precipitously drops reaching a value of only 1/40th to 1/50th the maximum at z=0.6. Thus, the low z data points (say below z = 1) are overweighted in my calculation compared to the higher z values in the range 1 to 3. That would affect the overall calculation more significantly if we observed that the separation between observed redshifts and Karlsson values as a percentage of their spacing between Karlsson values remained constant or trended upward or downward. But it doesn't. That spacing as a percentage goes up and down from point to point. Finding this effect is fairly involved but we can at least gage whether ignoring this in the calculation is conservative.
In the case of NGC 3516, observed z = 0.33, 0.69, 0.93, 1.40, 2.10. The Karlsson z = 0.3, 0.6, 0.96, 1.41, 1.96, 2.64. Therefore, the spacings are +0.03, +0.09, -0.03, -0.01, +0.14 which, as a percentage of the distance between the two nearest Karlsson values are 10%, 25%, 8%, 2%, 20%. Thus, the first two values with separations from the Karlsson value of 10% and 25% are overweighted, compared to the ones that have separations of 8%, 2% and 20%. That means in the calculation involving the three quasars with the lowest separations, the two lowest separations are underweighted compared to the highest separation of the three. Meaning that the corrected probability from that calculation
would be lower than was calculated. And in the two separate calculations to account for effect of the other two quasars, one of the two is somewhat overweighted but the other may be slightly underweighted. So I assert that incorporating this factor into this particular calculation would LOWER the final probability from the value I determined.
In looking this over for the NGC 5985 case, I find I made a mistake in the previous calculation. The observed z = 0.35, 0.59, 0.81, 1.97, 2.13. The spacings are therefore +0.05, -0.01, -0.15, +0.01, +0.17. In the previous calculation, I used a separation of +0.03 for the last data point instead of +0.17. That effects the calculation in a number of ways, so I'm going to redo the whole calculation before addressing the z distribution evenness issue.
Now we could do the same as before and simply calculate the combinatorial probability of finding the lowest three spacing data points, z= 0.35 (+0.05), 0.59 (-0.01) and 1.97 (+0.01). In that case, the required increment would be 0.10 and the probability would be 1/((30 * 29 * 28)/(3*2*1)) = 2 x 10
-4.
But that might significantly overestimate the probability since two of the data points are within an increment of only 0.02. So what's the combinatorial probability of finding 2 data points with a increment of 0.02? The answer is 1/((150 * 149 )/(2*1)) = 9 x 10
-5. Which is less than the above estimate so let's use it.
Now we add in the effect of the 0.35 (+0.05), 2.13 (+0.17) and 0.81 (-0.15) values. The probability of seeing the 0.35 data point with a increment of 0.10 is about 1/30 = 0.033; the probability of seeing the 0.81 data point with a increment of 0.30 is about 1/10 = 0.1; and the probability of seeing the 2.13 data point with an increment of 0.34 is about 1/8.8 = 0.11. Combined, these would reduce the 9 x 10
-5 probability estimate to only 3 x 10
-8.
Next, we must account for the actual number of quasars that might be seen near galaxies in groups of the size needed to do the above calculations. Previously, I found that the mainstream estimates there should be a total of about 410,000 quasars in the sky. And I then assumed (very conservatively, I think) that only half are near low redshift galaxies. That left us with 205,000 quasars to distribute. Then I assumed (again, very conservatively) that half of these would be distributed in quantities less than five to all the galaxies available, leaving 103,000 that are in groups of 5 near low redshift galaxies. Dividing by 5, the final result was 20,600 galaxies with at least 5 nearby quasars. Multiplying the above probability by 20,600 yields a probability of 5.4 x 10
-5.
It's at this point, however, that I notice another possible complication in my previous procedure. Since about half of the 3 x 10
-8 probability comes from only 2 quasars being together near a galaxy, the number of galaxies that might have 2 quasars is larger (by 2.5 times). Thus, the importance of those 2 quasars could be improperly diminished if I assume 20,600 as the total number of galaxies. Thus, we can expect an UPPER BOUND of 5/2 * 5.4 x 10
-5 = 0.000135 for the probability at this stage of the calculation. Let's conservatively use that.
Finally, we add in the fact that all 5 of these objects are aligned with the minor axis. As before, the alignment probability reduces the likelihood by 0.08, giving a final probability value of 0.000135 * 0.08 = 1.1 x 10
-5 for the NGC 5985 case.
(By the way, accounting for an increase in galaxy sample size in the NGC 3516 case because much of the probability only depends on 3 quasars, one can estimate an upper bound probability of 5/3 * 0.0005 (the original probability in that case) = 0.00083 ... a very, very small likelihood of that case turning up at all if we were somehow able to check every single quasar thought to be visible in the sky.
Now let's examine your concern about the z distribution in the NGC 5985 case. The percentage of distance between the two nearest Karlsson z values for the z = 0.35, 0.59, 0.81, 1.97, 2.13 observations are 16%, 3%, 42%, 1%, 25%, respectively. In this case, the 16%, 3% and 42% data points are overweighted, while the 1% and 25% data points are underweighted. In the two quasar calculation (which used z=0.59 and 1.97), the 3% value is somewhat overweighted. This would increase the probability at least a bit ... perhaps a factor of 2? But counteracting this is that fact that the z = 0.35 data point with a very large increment is underweighted. Likewise the z = 0.81 data point with an even bigger increment is also underweighted. But if you like, I'll still give you that factor of 2. In which case, the final probability of seeing NGC 5985, if one could check every single quasar out there, would only be 2 x 10
-5 ... again a VERY small number.
Any way you cut it, DeiRenDopa, this calculation proves that the mainstream's theory about quasars is on shaky ground. They need to reexamine that theory in light of this data or come up with an explanation why redshifts seem to be quantized around certain values and show up with a higher than expected frequency around galaxies along their minor axes. Or one has to illogically believe that Arp was REALLY LUCKY in turning up 2 cases with likelihoods of only 0.0008 and 0.0002 even if all the galaxies in the sky with quasars could be examined (which he didn't come close to doing).
