Stas pointed out an error in my memo. In Table 1, I took the deficits
from the wrong figure. I should have put in the deficit (MC minus
data) for the whole attico running period, runs 7473-13120, with a
total deficit of 105 events. Instead I put in the deficit for the
second attico running period, runs 11513-13120, with a total deficit
of 52 events. (I used Figure 28b of the Montaruli/Ronga memo on
systematics studies, when I should have used Figure 29.) This turns
out to have an even greater effect than the factor of two in total
events. It turns out that for the running period I used to calculate
the deficits, the deficits were distributed more uniformly than usual
as a function of nadir angle. When that data was transformed to
geomagnetic coordinates, it was quite flat. However, for the full
data set the deficit is relatively more concentrated at the near
vertical. When it is transformed to geomagnetic coordinates, it
"predicts" greater deficits in the southern hemisphere than in the
northern, and "predicts" a deficit of 9.5 events in the sourthernmost
bin, as opposed to 3.5 events in my memo.
Here is the new data that should have appeared in Tables 1 and 2:
INPUT: observed deficit as a function of nadir angle:
1<cos(theta_macro)<0.9 35.5
0.9<cos(theta_macro)<0.8 33
0.8<cos(theta_macro)<0.7 11
0.7<cos(theta_macro)<0.6 -6
0.6<cos(theta_macro)<0.5 18
0.5<cos(theta_macro)<0.4 11
0.4<cos(theta_macro)<0.3 3
0.3<cos(theta_macro)<0.2 -3.5
0.2<cos(theta_macro)<0.1 2
0.1<cos(theta_macro)<0 1
OUTPUT: "predicted" deficits as a function of geomagnetic latitude
1<cos(theta_geo)<0.9 (N pole) 3.949
0.9<cos(theta_geo)<0.8 2.543
0.8<cos(theta_geo)<0.7 2.1565
0.7<cos(theta_geo)<0.6 2.7155
0.6<cos(theta_geo)<0.5 2.8455
0.5<cos(theta_geo)<0.4 3.6625
0.4<cos(theta_geo)<0.3 3.954
0.3<cos(theta_geo)<0.2 4.198
0.2<cos(theta_geo)<0.1 5.1575
0.1<cos(theta_geo)<0 5.64
0<cos(theta_geo)<-0.1 6.259
-0.1<cos(theta_geo)<-0.2 6.3295
-0.2<cos(theta_geo)<-0.3 6.677
-0.3<cos(theta_geo)<-0.4 6.762
-0.4<cos(theta_geo)<-0.5 5.7865
-0.5<cos(theta_geo)<-0.6 5.6755
-0.6<cos(theta_geo)<-0.7 5.5575
-0.7<cos(theta_geo)<-0.8 7.496
-0.8<cos(theta_geo)<-0.9 8.165
-0.9<cos(theta_geo)<-1 (S pole) 9.4705
Note that this does not affect Figure 5 of my memo, data versus Monte
Carlo. However, it does affect the question "Is there information in
the geomagnetic distribution additional to that in the nadir angle
distribution?" If we take MC minus predicted deficit as the "modified
expectation" for the first bin, we have 15 - 9.5 = 5.5 events
expected. The Poisson probability for 5.5 to fluctuate down to 1 or
fewer is about 2.5%, not unexpected in a plot with 20 bins.
We have almost one year of data collected since the dataset I used in
the memo, so if it's a fluctuation we should know soon.
Of course, if we had never made a nadir angle distribution plot and
had only made geomagnetic distribution plots, we would be really
scratching our head about the 15 expected, 1 observed problem in this
bin. It is only because we are so familiar with the nadir angle
distribution that we are comfortable using it to "predict" a deviation
from Monte Carlo.
I did one more piece of work today -- I asked the reverse question,
"How well does the observed geomagnetic distribution `predict' the
nadir angle distribution?" One could imagine a world where the
mechanism that causes the deficit is a smooth, monotonic function of
geomagnetic latitude. If so, the observed nadir angle distribution
would imperfectly predict the geomagnetic distribution but the
observed geomagnetic distribution would perfectly predict the nadir
angle distribution. So I followed the same procedure, to distribute
the observed deficits from the geomagnetic bins throughout the solid
angle represented by the bin, and transform to nadir angle. The
"predicted" distribution of deficits is:
predicted observed
1>cos(theta_macro)>0.9 21.1555 35.5
0.9>cos(theta_macro)>0.8 17.2055 33
0.8>cos(theta_macro)>0.7 11.325 11
0.7>cos(theta_macro)>0.6 9.7865 -6
0.6>cos(theta_macro)>0.5 9.117 18
0.5>cos(theta_macro)>0.4 7.823 11
0.4>cos(theta_macro)>0.3 6.3395 3
0.3>cos(theta_macro)>0.2 3.358 -3.5
0.2>cos(theta_macro)>0.1 2.022 2
0.1>cos(theta_macro)>0 0.868 1
(The two column sums differ by 17 events because the observed column is
based on background-subtracted distributions from the Montaruli/Ronga
analysis and the "predicted" column has no background subtraction.)
So the observed deficits in nadir angle are not well-predicted by
geomagnetic deficits -- the near-vertical deficit is considerably
worse than predicted, and the predicted distribution of deficits is
smoother than observed. In fact, the geomagnetic prediction of the
nadir angle distribution is probably *worse* than the nadir angle
prediction of the geomagnetic distribution.
Bob