last update: 8 Oct 97
Introduction
Data Set
I have spun through the set of moon files I have at UCI refitting the shower angles using the current online routines (version 2.1?). This data set includes runs from 191 through 330 but does not include all runs from this period (May & June, 1997). I have added to this set runs: 457-505 (not complete) (Sept, 1997).
All events within 15 degrees of the moon are included in the data set. However, upon refitting, about 20% of events are lost near the edge of the region (are > 15 degrees away from the moon after refitting). Because of this, I limit any analysis to the inner 10 degrees. There are approximately 4.6 million events within 10.0 degrees of the moon in this set.
Is this data set large enough to see a shadow? Obviously the answer depends on our angular resolution and on whether we can compensate or limit any effects due to the earth's magnetic field. Ignoring the the mag. field effects, the equation for estimating the significance given an event density (rho) and an angular resolution (sigma) is:
Nsigma = 0.071*0.72*sqrt(rho)/sigma
where Nsigma is the significance, sigma is in degrees, and rho is in events/sq. deg. With the data set described above, rho ~ 14,500 events/sq.deg. If our resolution is indeed 0.5 degrees and there are no magnetic field effects, then with this data set the signficance of the shadow should be 12.3 sigma! If the resolution is 1.0, we will still have 6 sigma from the shadow.
Unfortunitly, there are large deflections due to the earth's magnetic field. It has been estimated that the deflection for a 1.0 TeV proton will be about 1.7 degrees. This effect is energy dependent and thus will not just shift the position of the shadow, but will also modify its shape. There are (at least) two ways to deal with this. If we can determine the energy of each shower accurrately, then we can correct each shower's angles to compensate for the deflection. This obviously requires a pretty good energy resolution which we probably won't have for a while (if ever). Or we could cut out the low energy events using their observable properties. This could be accomplished with a somewhat poorer energy resolution. One could make a very hard cut to insure very little low energy events for the price of losing some of the high energy events.
Analysis Method
My method of fitting the shadow is the same as I used in Cygnus. It is a event-by-event maximum-likelihood procedure with the hypothesis probability being a Gaussian shape for the shadow and a constant event density determined from the data. The content of the Gaussian is fixed to the expected deficit of events. So the only free parameter was the width of the Gaussian (sigma). All events within 5.0 degrees of the target position were used in the fits.
The results which are presented below include the following quantities:
1. True width of the Gaussian (sigma) which is related to the fitted width (sigma1) by:
sigma1 = sigma + exp(-c1-(c2*sigma))
where c1 = 3.23714, and c2 = 1.64457, (determined by Bob Ellsworth which compensates for the fact that the moon is not a point source).
2. The significance of the shadow (Nsigma), given by:
Nsigma = sqrt(2*(w))
where w is sum of -log(P1/P2) and P1 is probability with a shadow and P2 is probability without a shadow.
3. The Chi-Squared per degree of freedom (ChiSq) computed using the function obtained by the maximum-likelihood method and the observed event density distribution. ChiSq is not used in the fitting process but indicates the "goodness" of the fit.
4. The angular distance one has to integrate out to to acheive 72% of the expected deficit (sigma72). This does not use the above fit, but is computed directly from the data assuming a constant density background in R (space-angle difference between event and target).
5. Nsigma72 is the significance of getting 72% of the deficit given the observed level of background.
Search Strategy
Given the recent, preliminary, results presented by Gus Sinnis on Mrk501, I decided to look for the moon at two positions which are the true moon position, and the position of the possible excess seen by Gus, which is a deltaRA of -1.0 and a deltaDEC of -0.8 degrees
For each of these positions, I decided on a set of cuts which were guided by Gaurang's work using Monte Carlo data which shows that one can increase the "high energy" fraction of an event data set by making cuts on the number of pmts hit in the shower (Npmts) and on Gus's work showing that deleo improves with the number of pmts in the fit (NFit).
Results
Below is a table of the results of the fits to the data sets obtained with the above data cuts.
Position #Events NHit NFit sigma Nsigma ChiSq sigma72 Nsigma72 True 1141547 all all --- 0.0 1.53 4.95 1.95 Offset 1193702 all all --- 0.0 1.83 4.95 1.99 True 1006930 all >=20 2.95+?.?-0.9 0.55 1.42 3.35 2.7 Offset 1055798 all >=20 2.63+?.?-0.8 0.33 1.66 4.95 1.9 True 342427 all >=40 1.80+1.0-0.4 1.49 0.68 2.55 2.1 Offset 359660 all >=40 1.29+.34-.21 2.60 1.02 1.85 2.92 True 450580 >=150 all 1.98+.89-.42 1.60 0.91 2.55 2.32 Offset 471526 >=150 all 1.53+.76-.33 1.56 1.41 2.75 2.25 True 439711 >=150 >=20 1.94+.90-.41 1.58 0.88 2.55 2.34 Offset 460278 >=150 >=20 1.63+1.1-.39 1.32 1.41 2.75 2.22 True 318486 >=150 >=40 1.66+.71-.41 1.80 0.77 2.45 2.08 Offset 333936 >=150 >=40 1.23+.28-.21 2.93 1.11 1.75 2.98 True 146987 >=200 all 2.60+?.?-1.1 0.43 1.27 3.15 1.10 Offset 153207 >=200 all 1.50+?.?-.46 0.67 1.03 2.75 1.28 True 145264 >=200 >=20 2.60+?.?-1.1 0.46 1.18 3.15 1.09 Offset 151362 >=200 >=20 1.68+?.?-.57 0.41 1.02 2.75 1.27 True 139195 >=200 >=40 2.13+?.?-1.0 0.72 1.21 3.25 1.03 Offset 145022 >=200 >=40 1.44+?.?-.41 0.83 0.97 2.75 1.25
Discussion
Very little in the above results is striking. Certainly, none of the results are significant. Possible explainations for the lack of a significant shadow are that:
You can choose your own poison.
Flights into Fantasy
Having said all that, let me heap some more trials onto an already bleak data set. Several suggestions have been made to me. The two main ones (besides get a REAL job) are to look at different offsets and to look at the deltaDEC distributions for various deltaRA bands in the data. The idea for the second one is that the magnetic deflections should mainly effect RA and not DEC. To do this, I chose the data set with Npmts > 150 and NFit> 40, centered on the true lunar position. I made ten slices in deltaRA (moonRA-eventRA) from -5.0 to +5.0 degrees. The deltaDEC (moonDEC-eventDEC) distributions are shown in Figure 1a, Figure 1b, and Figure 1c. There is an apparent dip seen in the sixth plot near deltaDEC of 2.5 degrees (note: the offset is in the same direction as the offset used in the results section). The sixth plot is a deltaRA slice from 0.0 to 1.0 degrees. Figure 2 shows a fit to this deltaDEC distribution with a linear background and Gaussian sink. The number of background events in this 1.0 degree region is 3680 as determined by the fitted line (dashed, without a Gaussian). The number of "on-sink" events is 3433. There are roughly 10 "bins" used to compute the background and by Li-Ma this gives Nsigma = 3.9, pre-trial of course. I positioned my maximum-likelihood fitter at this offset and obtained the following results (event density is shown in Figure 3):
# events sigma Nsigma ChiSq sigma72 Nsigma72 94549 0.76+.14-.20 3.87 1.22 1.45 3.49
Further Flights
In my earlier work last spring it was suggested that I divide up the moon's orbit around the earth into 4 quadrants relative to the sun-moon angular separation in RA. I used the variable beta = MoonRA-SunRA+45. I then divided beta into four quadrants: 0-90, 90-180, 180-270, 270-360. I used my maximum-likelihood fitter with the offset determined above (-.06, -2.7) and fit the data from each of this quadrants. Below are the results:
Beta # events sigma Nsigma ChiSq sigma72 Nsigma72 <theta> 000-090 94549 2.44+?.?-1.0 0.57 1.0 2.45 1.1 34.7 090-180 80725 0.78+.27-.16 2.45 0.87 1.15 2.2 35.0 180-270 36664 0.76+.36-.22 2.96 1.21 0.95 1.8 46.9 270-360 103996 0.39+.07-.05 3.97 1.50 0.85 3.4 34.9
Figure 4, Figure 5, and Figure 6 show the deltaDEC, deltaRA, and event density distributions for the fourth quadrant, including fits. What is interesting to note is that the shapes in the deltaRA an deltaDEC distributions are very similar. If I make a slice in deltaRA and see a dip with a given width in the deltaDEC distribution, there is no gaurantee to see the same shape in deltaRA.
Back to Reality
Obviously there are several points in this analysis where there are an unknown number of trails taken. For example, looking at the deltaDEC distributions for 10 slices in deltaRA. So, conservatively, no real post-trail significance can be put on the above results!
However, (you knew it was coming) I would like to check an independent data set of earlier runs when the moon was in the fourth quadrant (270<Beta<360) and see if there is a dip at the given offset. The set should contain about 100,000 events within 5 degrees of the target position with cuts of NPMTs>=150 and NFit>=40.
Please contact me via email or phone with your feedback on this, Tony...
Appendix A
After writing the above, I have studied the aledged moon shadow wrt NFit. For this, I used the data set obtained from make the following cuts:
The results are summarized in the table below:
#events NFit sigma Nsigma ChiSq sigma72 Nsigma72 <theta> 316943 all 0.98+.88-.22 1.79 1.12 3.25 1.56 37.6 308116 >=20 0.98+.97-.22 1.76 1.12 3.25 1.54 37.6 220063 >=40 0.57+.11-.08 4.72 1.15 1.05 4.03 36.9 127694 >=60 0.58+.15-.10 4.08 1.28 0.95 3.39 36.3 75384 >=80 0.43+.14-.08 3.62 1.17 0.95 2.61 35.9 42987 >=100 0.33+.10-.05 3.19 0.95 0.95 1.97 35.6 20684 >=120 0.89+.73-.36 1.66 0.79 0.95 1.37 35.4
Note (10/8/97): As pointed out by Todd and Gaurang, the y distribution obtained by Stefan seems to indicate that an offset in y (and thus DEC???) increases with increasing theta. Using very crude estimates of the mean theta for mrk501 and the moon, we get that if the offset in the aledged mrk501 signal and the offset in the aledged moon shadow are related to the offset in y, then offset in the moon should be ~ 2-3 times that of mrk501 which is what is seen in the data.
