Wrong. The paper I cited on NGC 3516 (
http://www.journals.uchicago.edu/doi/abs/10.1086/305779 ) talks about 5 quasars along the minor axis. The object at z = 0.089 is identified this way: "there is a very strong X-ray source that is listed as having a Seyfert spectrum (Veron-Cetty & Veron 1996) with redshift z = 0.089 (about 10 times the redshift of NGC 3516). Optically it is a compact, semistellar object. With its strong X-ray and radio properties, it is closely allied to BL Lac objects and therefore to the transition between quasars and objects with increasing components of stellar populations."
True, but I was trying to simplify the problem by balancing the fact that 3 (of the five) fall into intervals less than 0.10 in width. But since you apparently aren't satisfied with my simplification, let's take another look at the whole problem.
Let's start by matching each observation with it's corresponding Karlsson value: (0.33,0.30) (0.69,0.60) (0.93,0.96) (1.40,1.41) (2.10,1.96). That means the distance from the Karlsson value in each case is 0.03, 0.09, 0.03, 0.01, 0.14 , respectively. To compare with the 0.10 discretization I used in my calculation, we must double those values: 0.06, 0.18, 0.06, 0.02, 0.28 . Note that 3 (of five) fall within a 0.06 discretization but you are correct that 2 don't fall within a 0.10 interval.
Let's redo the calculation for just those three cases and see what we get, assuming again that the quasars randomly came from a population with an equal distribution of probability between 0 and 3.0. There are 50 possible values in that range given an increment of 0.06. Now looking at this again, I don't think I should have used the permutation formula in the previous calculation. This time let's just use the combination formula ... in other words, let's find the number of possible combinations of those those r values from n possible values.
The formula for that is n!/((n-r)!r!). Thus the probability of seeing those 3 values turn up is 1/(n!/((n-r)!r!). In this case, that works out to 1/(50*49*48/3*2) = 5.1 x 10
-5.
And now let's factor in the unlikelihood that we'd find 2 more quasars near that galaxy that are unusually close to the Karlsson (K) values of 0.60, and 1.96. Surely a conservative estimate for that probability would be to simply find the chance of each specific number turning up given an increment appropriate for that case. For the 0.69 case, for example, where the increment is 0.18 (twice the 0.09) value, over the range 0 to 3.0, there are at least 16 increments. So, the probability of finding that number 1/16 = 0.06. For the 1.96 case, the increment needs to be 0.28 and there are 10 possible values. The probability is 1/10 = 0.10. And finding these z's should be relatively independent of finding the others, so the probabilities should simply multiply together to give a final combined probability.
Therefore, I assert that, to a first order, the probability from the 3 number sequence can be adjusted to account for the unlikelihood of the 2 other quasars by multiplying it by 0.06 * 0.1. And that results in a final combined probability of 5.1 x 10
-5 * 0.06 * 0.10 = 3.06 x 10
-7. So to a first order, it appears my initial assumption that an increment of 0.1 used for all of them would balance everything out off by a factor of about 10.
And lest you think the selection of z = 3.0 as the upper bound in my calculation is arbitrary, let me note that I found a mainstream source that said, based on the SDSS study, the number of quasars decreases by a factor of 40 to 50 from z = 2.5 to z = 6.0. Therefore, I think I am justified in using a range of z = 0 - 3.0 in my calculations for quasar z. I will agree that the density of quasars of different z is not uniform over the range. Several of the sources I found indicated that it climbs rather steeply from a value at z = 0 to z = 1.5 and then levels off through z = 3.0. I don't see an easy way to incorporate this fact into the calculation but I don't think it really makes much of a difference since the differences between the Karlsson values and the observed z don't appear to have much of a trend up or down over the range.
However, since you questioned my 0.10 simplification, I'm going to take another look at the rest of the calculation, starting with what I estimated was the total number of quasars in the sky. Recall that I estimated the total number of quasars that can be seen as 1,237,500 ... by multiplying the number of square degrees in the sky (41,250) by 30 quasars per square degree. But is 30 quasars per square degree really a reasonable value to use?
Here's a 2005 study
http://www.iop.org/EJ/abstract/1538-3881/129/5/204 that indicates an average density of 8.25 deg
-2 based on the SDSS survey then argues it should be corrected upward to 10.2 deg
-2 to make it complete. And if you go to the SDSS website (
http://www.sdss.jhu.edu/ ) you find they say the effort will observe 100,000 quasars over a 10,000 deg
2 area. That also works out to about 10 quasars deg
-2. So it looks like I used a number that was 3 times too large in my earlier calculation. In this revised calculation, I will only assume the average quasar density is 10 deg
-2. That means the total number of quasars than can be seen from earth is around 410,000.
So now we come to the question of how those 410,000 quasars are distributed, or more precisely, how many galaxies with 5 or more quasars near them can be expected in the total population of galaxies that can be seen. Now recall that in my previous calculation I initially assumed that the all the quasars are located near galaxies and distributed 5 per galaxy until the number of quasars available is exhausted. That resulted in an estimate of 250,000 (~ 1,237,500 /5) galaxies with 5 quasars each. Doing that maximized the total number of galaxies assumed to have 5 quasars which was a conservative approach from the standpoint of not wanting to over estimate the improbability of the observation of NGC 3516.
But the truth is that most quasars do not lie close to galaxies at all (certainly not galaxies where we can discern any detail as is the case in all three examples of interest here) so that's why I later multiplied the calculated probability by 0.10 to account for the assumption that only 10% of quasars lie next to a galaxy. I still think that's probably a reasonable number. But for this calculation, I'm going to give your side the benefit of the doubt and assume that fully half of all quasars are near galaxies. That has to be very conservative. Wouldn't you agree. So now there are 205,000 in the population that we need to distribute amongst galaxies.
It's also apparent that most galaxies that have nearby quasars only have a few quasars ... not 5 or more. I didn't find any source actually quantifying this but we can observe that in Arp's catalog of anomalous quasar/galaxy associations, relatively few of the examples have 5 or more quasars in the field. Therefore, I think it's conservative to assume that only half the quasars are in groups of 5 or more near galaxies. You would agree, right? In fact, I think this is very conservative assumption, otherwise Arp's list of anomalous quasar/galaxy associations would have likely contained far more examples with large numbers of quasars. In any case, I'm going to reduce the number of quasars available to comprise the population of galaxies that have 5 quasars by half ... to 103,000. Now if you divide that number by 5, that means there are at most 20,600 galaxies visible that have 5 quasars in their vicinity.
Therefore where previously I effectively multiplied the probability of any given galaxy having 5 quasars with Karlsson redshifts by 25,000 (1,237,500 / 5 * 0.10), now I'm going to multiply the new probability calculated for NGC 3516 by only 20,600. Doing so produces a probability (for finding the 5 quasars with the specific z's near NGC 3516 amongst the total population of quasars/galaxy associations) of 3.06 x 10
-7 * 20,600 = .0063 .
Now let's complete the calculation by again adding in the fact that all 5 objects are aligned rather narrowly along the minor axis. I'll just use the dart board example I used previously, where I found that the probability of throwing 5 darts in a row that land within a 15 degree zone extending from opposite sides of the center of the dart board as being 3.9 x 10
-6 per galaxy. And again, we have to multiply by the number of galaxies with 5 quasars that can be aligned. With only 20,600 such cases possible (conservatively), the probability of finding 5 quasars aligned along the minor axis is therefore 3.9 x 10
-6 * 20,600 = 0.08 which makes the total likelihood of encountering this one case if one carefully studied the entire quasar population equal to 0.08 * 0.0063 = ~0.0005 .
That's a very small probability. Yet Arp found such a case after looking, not at all the galaxies that have quasars near them, but by looking at only a tiny fraction of those galaxies. Which makes his observation even more improbable. Perhaps significantly so.
And he not only found that case, he found two others that have large numbers of quasars with values close to the Karlsson values aligned with the minor axis of galaxies. Recall that NGC 5985 had 5 that are lined up along its minor axis with redshifts of 2.13, 1.97, 0.59, 0.81 and 0.35. The corresponding delta to the nearest Karlsson values are 0.03, 0.01, 0.01, 0.15, 0.05. Let's ignore the 0.81 and 0.35 values for the moment and find the probability of encountering the first three values, on a combinatorial basis. With an increment equal to twice the largest delta (i.e., 0.06), that probability is 1/((50 * 49 * 48)/(3*2*1)) = 1/19600 = 5.1 * 10
-5.
As in the other case, we still have to add in the effect of the other data two points. Following the same approach as before, the probability of seeing the 0.35 value with an increment of 0.1 is 1/30 = 0.033. The probability of seeing the 0.81 data point with a increment of 0.30 is 1/10 = 0.1.
Therefore, the combined probability for NGC 5985 is 5.1 * 10
-5 * 0.033 * 0.1 = 1.683 x 10
-7. Accounting for the actual number of quasars that might be seen near galaxies in groups of 5 and the fact that all these objects are aligned with the minor axis gives a final probability of 1.683 x 10
-7 * 20,600 * 0.08 = ~0.0003 .
That's another very small probability. And finding two such improbable associations when Arp didn't look at all that many galaxies/quasars in order to find these cases, is makes this even more improbable.
Any way you look at it, DRD, this finding does not bode well for the theory that quasar redshifts are not quantized and have nothing to do with the galaxies that they are near. And I draw your attention to the use of Bayes' Theorem that I outlined in my earlier post to David (post #151).
I can update that case for the new probabilities calculated above as follows.
Suppose apriori we are really sure that the mainstream theory about quasars and redshift is correct. Let's say Pr
0(A) = 0.999, leaving just a little room for doubt. That means Pr
0(B) = 0.001. Fair enough?
Next, we "measure" that sequence of 5 redshift values from NGC 3516 that are all aligned with the minor axis of the galaxy. And based on the calculation I did above, the probability of that sequence of values and alignment occurring under the assumption that hypothesis A is correct (P
A(x
i)) is calculated to be no better than 0.0005. At the same time, we can say that P
B(x
i) = 0.9995.
Now let's compute Pr
1(A) and Pr
1(B).
Pr
1(A) = (0.999 * 0.0005) / (0.999 * 0.0005 + 0.001 * 0.9995) = 0.33
Pr
1(B) = 0.67
In other words, based on that single observation, the probability that your hypothesis is correct has dropped from 99.9% to 33% and the probability that the quasars' redshifts and positions aren't just a matter of random chance has risen from 0.1% to 67%.
That theorem shows that finding these cases significantly reduces the probability that the mainstream hypothesis about quasars is correct. At least enough that it would behoove the mainstream to take a closer look rather than just try to dismiss this out of hand as they have done, and you and David are now trying to do.
That's an interesting comment but I don't think it's correct because the theory has it that z is a function of the age of the quasar. There is no requirement in any given case that the galaxy have been producing quasars the entire time, including up till recently. There is therefore no requirement that there be quasars corresponding to each Karlsson value that is possible. Values can be skipped because for some reason the galaxy stopped producing quasars for a time. Or there may be more than one value at a given z because the galaxy was producing more for a time. Or there may not be any high ones because the galaxy stopped producing them at some point. Or perhaps we don't see any higher z quasars because they tend to be much closer to the site where they were produced and are therefore lost in the glare or opacity of the parent galaxy.
I think the logic was clear enough in what I wrote earlier but see the revised calculation above. I tried to make it even clearer.
Obviously, I would predict they'd tend to be near Karlsson values. Do you have some data to suggest they are not? If not, I don't think this concern has much merit at all and just lengthens the thread further.
There is apparently a limit to the total number of quasars. The methodology used in SDSS was designed to produce a relatively complete list of the surveyed region and one of the papers I found concluded that it had succeeded ... with well above 90% completeness in that region. So I don't anticipate new observations that will increase the estimated total quasar count much higher than it already is ... provided the regions that were already surveyed are representative of the whole. Sure, individual quasars will be found in those regions of the sky that weren't previously surveyed but that shouldn't increase total quasar counts.
And for the record, I have no idea whether NGC 3516 lies in an already surveyed region or not. Do you? If so, then by all means tell us the latest data. No need to be coy. That can only serve to make the thread longer and we know how you dislike that.
Again, I shall just quote some excerpts that certainly seem to indicate VCVcat data being used for statistical analysis.
