V (0) = 0 at x = 0. Find the position x of the maximum V. Make plots of Ex (x) and V(x). What is the significance of the position where Ex = 0? [Ans. xmax = 5.25d ]

6.4 Gamma-ray instruments

Problem 6.41. Each of the 36 EGRET spark-chamber modules yields x and y coordinates of the track of an electron that passes through the module (see text). Two particle tracks are the typical consequence of the conversion of a gamma ray (Fig. 5a). In a given module, these lead to two x coordinates and two y coordinates which intersect at four locations, two that are valid and two that are spurious. (a) In comparing the spark locations from module to module, do the spurious locations form tracks that are discontinuous and thus easily seen to be non-physical? (b) Can you think of a simple modification to the experimental arrangement that would remove this ambiguity? In fact such a feature was incorporated into EGRET. (c) The efficiency for a given spark chamber to produce a spark is only 80-85%. Does this help remove this ambiguity? Consider the cases where a chamber produces one spark or no sparks.

6.5 Statistics of measurements

Problem 6.51. (a) Add three rows to Table 1 for values of the Poisson function for the expected means, m = 0.01,0.1, and 0.3, each for x = 0,1, 2, 3,4. (b) Plot histograms (by hand or with computer) for these values and also those for m = 1 on a single plot. Use an expanded scale; truncate the plot after x = 4. Comment on the behavior of the four histograms.

Problem 6.52. (Makes use of probabilities from previous problem.) (a) How many x rays from a distant point-like star must be detected (in one resolution element or pixel) at the focus of an x-ray telescope for the x-ray detection to be considered real (with fair confidence)? Let the background (averaged over all pixels) yield exactly 1.00 x 10-2 counts per pixel during the exposure. Require that there be less than a 2% chance that a fluctuation in the background rate could give your number. The location of the star in the sky is known, as is the particular pixel where its x-ray image would appear. What kind of statistics are appropriate here? (b) If one is searching for a point-like x-ray image whose position is not known, one must consider all 105 pixels of the image. How many x rays from the point source must be detected in this case? Require that the probability for a background fluctuation up to your value in any one of the 105 pixels be less than 2%. [Ans. very small integer values]

Problem 6.53. (a) Demonstrate that the normal distribution is normalized to unity according to (4). (b) Calculate by numerical integration, say with 20 steps, the probability that a measurement obeying the normal distribution will deviate from the expected mean by more than 4 standard deviations in the positive direction. [Ans. (b) See Table 2]

Problem 6.54. The expression for the significance S/as of a counting measurement derived in the text (16) is valid if the exposure times for the on-source and off-source measurements are equal. Here, let these times be different and designate them ts (= ts+b) and tb respectively. (a) Write the counting rate rs (cts/s) of the source alone in terms of the measured rates rs+b and rb, and find the standard deviation ars of the rate rs in terms of these rates and the times ts and tb. Note that the number of background counts Bs detected "on source" may differ from the number Bb detected "off source". Be careful to distinguish total counts, S + Bs and Bb from the counting rates rs+b, rb and rs. (b) Use your answer to write directly the expression for the significance rs /ars, of the source rate. Find an expression for the time ts that yields the maximum significance if the observing time T = ts + tb is fixed. In other words, how should the time T be apportioned between on-source and background measurements? Hints: substitute tb = T- ts, define g = ts/ T and solve for the optimum value of g while noting that rs+b = rs + rb. (c) Evaluate your expression for three cases: rs ^ rb; rs = rb; rs > rb. [Ans. (b) 1 — g =

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