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Pulsed TeV radiation from Hercules X-1
Nuclear Physics B (Proc . Suppl .) 14A (1990) 200-204 North-Holland
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PULSED TEV RADIATION FROM HERCULES X-1 1,4 P.T. Reynolds,' M.F. Cawley,2 D .J. Fegan,' A.M. Hillas,' P.W. Kwok,4 R.C. Lamb,' M .J. Lang, D.A. Lewis,' D. Macomb5 , G. Vacant',' and T.C. Weekes4 'Physics Department, University College Dublin, Ireland ZPhysics Department, St. Patrick's College, Co . Kildare, Ireland 'Physics Department, University of Leeds, Leeds, UK 4 Whipple Observatory, P.O. Boa 97, Amado, Arizona 856¢5 USA 'Physics Department, Iowa State University, Ames, IA 50011 USA A four-year database (spring 1984 through spring 1987) of observations of Hercules X-1 by the Whipple Observatory Collaboration is being analyzed as a whole for evidence of TeV -y-ray emission . The full distribution of Rayleigh powers near the x-ray pulsar frequency differs significantly from the background distribution, giving evidence for sporadic emission at better than a 99% confidence level.
Air showers associated with brief (< 100 minute) episodes of emission from the binary x-ray source Hercules X-1 have been reported at primary energies of about 1 TeV using atmospheric Cherenkov light detectors1 ,2,3,4 and greater than 100 TeV using arrays of particle detectors .',' With the possible exception of Cygnus X-3, it is the best studied x-ray binary at TeV and greater energies . The purpose of this paper is to report on an analysis of the extensive Whipple Observatory database of observations of Hercules X-1 at TeV energies. We are analyzing the data taken as a complete set rather than examining only the largest Rayleigh powers . The analysis is not complete and, in particular, imaging (using Cherenkov light images to reject cosmic-ray background showers) has not been applied in a systematic way to the full set of observations . Nevertheless the present analysis shows evidence for TeV radiation from the source at better than 99% confidence level. The observations were taken using the Whipple Observatory 10 m-diameter reflector used to focus Cheyx~ov light from individual air showers onto a hexagonal array of 37 photomultipliers (pmts) in the image plane of the reflector . The telescope is then a fast, large-aperture, low-resolution camera.' Each 0920-5632/90/$03 .50 © Elsevier Science Publishers B.V. (Nortli-Ilolland)
pmt subtended a full angle of 0.4° and was spaced by 0.5° from its neighbors . The camera was triggered by the coincidence of signals from a preset number of pmts each exceeding a threshold of 50 photoelectrons . For most ofthe observations described here the trigger requirement was a 10 ns coincidence between any 2 of the inner 19 pmts. This corresponded to an effective energy threshold of 0.7 TeV. Observations were generally conducted on moonless nights under clear-sky conditions. The overall trigger rate for observations near the zenith was approximately 3 Hz. For each shower the digitized image ofthe pattern of Cherenkov light in the camera was recorded as well at its time of arrival . A total of 275 hours of data were measured in the period from spring 1984 through spring 1987. During this same period a number of other candidate sources, including the Crab Nebula, were also observed . Eight episodes of emission have already been reported 2 for the data of 1984-86 . The analyses were based on all events detected, i.e., there was no image selection. The primary evidence for emission was the presence of a periodic modulation in the shower rate at a frequency at or near the known spin frequency of the neutron star. No weight was given to the image
P. T. Reynolds et al. /Pulsed TeV radiation from Hercules X-1 characteristics of the showers. The first of the eight episodes (April 4, 1934) !s remarkable because the source was simultaneously seen by both the Whipple and the Durham groups.8 The strongest episode of emission9 occurred in June 1986 at a frequency (0.16 f 0.02)% higher than that of the neutron star. Emission at a frequency virtually identical to that was observed in May 1986 by the Haleakala group3 at TeV energies and in July 1986 by the Los Alamos group at energies above 100 TeV .s In an effort to establish a more rigorous statistical limit to the significance of the Whipple detections, we have reanalyzed the four years of observa tions in the following manner. Inasmuch as 5 of the 8 previously reported episodes had an apparent duration of about 30 minutes, the data were divided into 578 non-overlapping segments of approximately 30minutes duration . The independent Fourier frequency (IFF) spacing for a 30-minute segment is 1/1800 Hz. Each segment was Fourier transformed over the frequency interval 1/3 to 2 Hz. This interval, which corresponds to 3000 IFF's, encompasses both the fundamental, 0.8079 Hz, and the second harmonic of the neutron star's spin frequency. This large number of trial frequencies is used to characterize the statistics of the database and to enable the determination of the significance of any putative signal by direct comparison with background in the same database. A frequency search range for possible signals was decided upon prior to looking at the data, namely ±1 IFF about the fundamental frequency and f2 IFF about the second harmonic. This search range was chosen because it corresponded to that used by Middleditch et al. 1 ° in the detection of optical and infra-red pulsations from Hercules X-1 . In computing the Fourier transforms, the data were oversampled by a factor of 5, i.e., 5 Rayleigh powers were computed within each IFF for a total of 15,000 frequencies . To search for persistent but low-level emission from the source, the Rayleigh powers at each trial frequency were added for all segments. While there is a small excess in the total Rayleigh power near the
neutron star frequency, it is not significant . Two types of tests for sporadic ruiissicn were made. For both types of tests, only peak powers are counted and at most one peak power is associated with each IFF. Since identification of peaks is difficult for powers less than about 3, only peak powers greater than 3 are used in the tests described below. In the first type of test, the segments in which the peak Rayleigh power exceeded arbitrarily chosen values of 4, 6, 8, or 10 are counted at each independent frequency. There are significant excesses at thresholds of 6 (6.4 expected from the background distribution and 15 found) and 8 (0.9 expected and 3 found). For instance, the Poisson probability for a measurement of 15 with a background average of 6.4 is 2.5 - 10-3, indicative of a signal. The 15 segments producing powers greater than 6.0 at the fundamental were examined in detail for evidence of a d.c . excess; it was determined that 7 of the segments had do excess consistent with the Rayleigh power." No attempt has been made to incorporate this result into the overall significance although it does indicate that the observed excess is not inconsistent with emission from the source . In the second type of test, the cumulative peak power distribution for the 4:1 IFF signal search was compared with the measured background cumulative distribution . These are shown in Figure 1. Only peak powers greater than 3 are included for reasons given above. The distributions have been compared using the Kolmogorov, 12 Anderson and Darling, 13 and Fisher 12,14 tests which yield chance-origin probabilities of about 1 - 10-2,2 - 10-3, and 3 - 10-4 , respectively. There is good reason to believe that the Fisher test should be the most sensitive of the three . 15 A spectrum based on the Fisher test and showing the full 0.33 Hz to 2 Hz range analyzed is shown in Figure 2. In order to assign a overall significance to the effect, degrees of freedom must be taken into consideration. A factor of two is included to account for pe riodicity testing at the neutron star second harmonic frequency as well as at the fundamental . An addi-
P.T. Reynolds et al./Pulsed TeV radiation from Hercules X-1
Cumulative Power Distribution +- 1 IFF
Rayleigh power threshold FIGURE 1 The cumulative power distribution within the 2 IFF signal search range is shown and compared with the background distribution in the figure.
Hercules X-1
Spectrum
Sin size is 2 IFF
.® O
a.
u O
Frequency in Hz FIGURE 2 The full spectrum from 0.33 to 2.0 Hz is shown above. The ordinate is -Log e of the probability of obtaining, by chance, a value of the Fisher statistic equal to or greater than that measured . The largest peak occurs exactly at the X-ray pulsar frequency.
P.T. Reynolds et al . /Pulsed TeV radiation from Hercules X-1
tional factor of two might be included to account for the failure of the incoherent power summation to produce a significant effect, although there is some excess power at the pulsar frequency. Including a factor of 4, the Poisson test for powers greater than 6, the A&D test and the Fisher test all yield a chance, background noise origin probability of less than 1%. In a modelis developed after the reports of emission from Hercules X-1, emission at a blue-shift of 0.1% is expected. Some clustering of high Rayleigh powers at adjoining blue shifted Rayleigh frequencies was also noted in the above analysis (17 on an expectation of 9.6 in the range [+2 to +4] IFF from the pulsar frequency), but as the initial hypothesis did not specify this search range it is impossible to assign a probability to this occurrence unambiguously. In summary, it appears from a comprehensive analysis of the dataset that there is evidence for TeV emission from Hercules X-1 at better than the 99% confidence level, although we are still investigating the statistical properties of the dataset . A parallel analysis in which only events which have y-ray Cherenkov light image characteristics is in progress. ACKNOWLEDGEMENTS We acknowledge the contributions of P.W. Gorham, K.G. Gibbs, D.F. Leibing, V.J. Stenger, N.A, Porter in earlier phases of this work and the continu ing assistance of K. Harris in taking the observations. This program is supported by the U.S. Department of Energy, the Smithsonian Scholarly Studies Fund, and the National Board of Science and Technology of Ireland. AMH and TCW acknowledge the assistance from a NATO grant. REFERENCES 1. J.C. Dowthwaite, A.B . Harrison, I.W . Kirkman, H .J. MacRae, K.J. Orford, K.E. Turver and M. Walmsley, Nature 309 (1984) 691 . 2. P.W. Gorham, M.F. Cawley, D .J . Fegtsn, K .G. Gibbs, S. Kenny, R.C. Lamb, D.F. Leibing, N.A. Porter, V.J. Stenger and T.C. Weekes, Astrophys. Journ. 309 (1986) 114 ; P.W. Gorham, M.F. Cawley, D.J. Fegtsn, K .G. Gibbs, R.C. Lamb,
203
D.F. Leibing, N.A. Porter, V.J. Stenger and T.C. Weekes, Astrophys . Journ. (Letters) 3 (1986) L11; P.W. Gorham, M.F. Cawley, D.J. Fegan, K.G. Gibbs, R.C. Lamb, D.F. Leibing, N .A. Porter, V.J. Stenger and T.C. Weekes, Proc. NATO Workshop on Very High Energy Gamma Ray Astronomy (Durham, 1986) 125 . 3. L.K. Resvanis, A. Szentgyorgyi, J. Hudson, L. Kelley, J.G. Learned, C. Sinnis, D.D. Weeks, J. Gaidos, M. Kertzmtsn, F. Loeffler, T. Palfrey, G. Sembrowski, C. Wilson ., U. Ctsmerini, J.P. Finley, W. Fry, M. Jaworski, J. Jennings, A. Kenter, M. Lomperski, R. Loveless, R. March, J. Matthews, R. Morse, D. Reeder, and P. Slane, Astrophys . Journ. (Letters) 328 (1988) L9. 4. P.R. Vishwanath, P.N. Bhat, P.V. Ramants murthy and B.V. Sreekantau, Astrophys . Journ. (1989) in press. 5. R.M. Baltrusatis, G .L. Cassiday, R. Cooper, J.W. Elbert, P.R. Gerhardy, E .C. Loh, Y. Mizumota, P. Sokolsky, P. Sommers and D. Steck, Astrophys. Journ. (Letters) 293 (1985) L69. 6. B.L. Dingus, D .E. Alexandreas, R.C . Allen, R.L. Burman, K.B . Butterfield, R. Cady, C.Y. Chang, R.W. Ellsworth, J.A. Goodman, S.K. Gupta, T.J. Haines, D.A. Krakauer, J. Lloyd-Evans, D.E. Nagle, M. Potter, V.D. Sandberg, R.L. Talaga, C.A. Wilkinson, and G.B . Yodh, Phys. Rev . Lett . 61 (1988) 1906 . 7. M.F. Cawley, D.J. Fegan, K .G. Gibbs, P.W. Gorham, R.C. Lamb, D.F. Leibing, P.K. MacKeown, N.A . Porter, V.J . Stenger and T.C. Weekes, Proc. Intern. Cosmic Ray Conf. (La Jolla, 1985) 3 (1985) 453 . 8. P.M. Chadwick, N.A. Dipper, I.W.Kirkman, T.J.L . McComb, K.J. Orford, and K.E. Turver, Proc. NATO Workshop on Very High Energy Gamma-Ray Astronomy (Durham, 1986) 121 . 9. R.C. Lamb, M.F. Cawley, D.J. Fegern, K.G. Gibbs, P.W. Gorham, A.M. Hillas, D.A. Lewis, N.A. Porter, P.T. Reynolds, and T.C. Weekes, Astrophys . Journ. (Letters) 328 (1988) L13. 10. J . Middleditch and J. Nelson, Astrophys . Journ. 208 (1976) 567; J. Middleditch, C.R. Pennypacker, and M.S. Burns, Astrophys . Journ. 274
204
P.T. Reynolds et al . /Pulsed TeV radiation from Hercules X-1 (1983) 313; J. Middleditch, R.C. Pentter and C.R . Pennypacker, Astrophys. Journ. 292 (1985) 267.
11 . D.A . Lewis, Astron . Astrophys., (1989) in press. 12 . W.T . Eadie, D. Drijard, F.E. James, J. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971). 13. T.W . Anderson and D.A . Darling, Ann. Math . Statist. 23 . (1952) 193.
14 . R.A . Fisher, Statistical Methods for Research Workers (Oliver and Boyd, Edinburgh and London, 1958). 15 . D.A . Lewis, in this volume. 16 . K.S . Cheng and M. Ruderman, Astrophys. Journ. (Letters) 337 L77.
200
PULSED TEV RADIATION FROM HERCULES X-1 1,4 P.T. Reynolds,' M.F. Cawley,2 D .J. Fegan,' A.M. Hillas,' P.W. Kwok,4 R.C. Lamb,' M .J. Lang, D.A. Lewis,' D. Macomb5 , G. Vacant',' and T.C. Weekes4 'Physics Department, University College Dublin, Ireland ZPhysics Department, St. Patrick's College, Co . Kildare, Ireland 'Physics Department, University of Leeds, Leeds, UK 4 Whipple Observatory, P.O. Boa 97, Amado, Arizona 856¢5 USA 'Physics Department, Iowa State University, Ames, IA 50011 USA A four-year database (spring 1984 through spring 1987) of observations of Hercules X-1 by the Whipple Observatory Collaboration is being analyzed as a whole for evidence of TeV -y-ray emission . The full distribution of Rayleigh powers near the x-ray pulsar frequency differs significantly from the background distribution, giving evidence for sporadic emission at better than a 99% confidence level.
Air showers associated with brief (< 100 minute) episodes of emission from the binary x-ray source Hercules X-1 have been reported at primary energies of about 1 TeV using atmospheric Cherenkov light detectors1 ,2,3,4 and greater than 100 TeV using arrays of particle detectors .',' With the possible exception of Cygnus X-3, it is the best studied x-ray binary at TeV and greater energies . The purpose of this paper is to report on an analysis of the extensive Whipple Observatory database of observations of Hercules X-1 at TeV energies. We are analyzing the data taken as a complete set rather than examining only the largest Rayleigh powers . The analysis is not complete and, in particular, imaging (using Cherenkov light images to reject cosmic-ray background showers) has not been applied in a systematic way to the full set of observations . Nevertheless the present analysis shows evidence for TeV radiation from the source at better than 99% confidence level. The observations were taken using the Whipple Observatory 10 m-diameter reflector used to focus Cheyx~ov light from individual air showers onto a hexagonal array of 37 photomultipliers (pmts) in the image plane of the reflector . The telescope is then a fast, large-aperture, low-resolution camera.' Each 0920-5632/90/$03 .50 © Elsevier Science Publishers B.V. (Nortli-Ilolland)
pmt subtended a full angle of 0.4° and was spaced by 0.5° from its neighbors . The camera was triggered by the coincidence of signals from a preset number of pmts each exceeding a threshold of 50 photoelectrons . For most ofthe observations described here the trigger requirement was a 10 ns coincidence between any 2 of the inner 19 pmts. This corresponded to an effective energy threshold of 0.7 TeV. Observations were generally conducted on moonless nights under clear-sky conditions. The overall trigger rate for observations near the zenith was approximately 3 Hz. For each shower the digitized image ofthe pattern of Cherenkov light in the camera was recorded as well at its time of arrival . A total of 275 hours of data were measured in the period from spring 1984 through spring 1987. During this same period a number of other candidate sources, including the Crab Nebula, were also observed . Eight episodes of emission have already been reported 2 for the data of 1984-86 . The analyses were based on all events detected, i.e., there was no image selection. The primary evidence for emission was the presence of a periodic modulation in the shower rate at a frequency at or near the known spin frequency of the neutron star. No weight was given to the image
P. T. Reynolds et al. /Pulsed TeV radiation from Hercules X-1 characteristics of the showers. The first of the eight episodes (April 4, 1934) !s remarkable because the source was simultaneously seen by both the Whipple and the Durham groups.8 The strongest episode of emission9 occurred in June 1986 at a frequency (0.16 f 0.02)% higher than that of the neutron star. Emission at a frequency virtually identical to that was observed in May 1986 by the Haleakala group3 at TeV energies and in July 1986 by the Los Alamos group at energies above 100 TeV .s In an effort to establish a more rigorous statistical limit to the significance of the Whipple detections, we have reanalyzed the four years of observa tions in the following manner. Inasmuch as 5 of the 8 previously reported episodes had an apparent duration of about 30 minutes, the data were divided into 578 non-overlapping segments of approximately 30minutes duration . The independent Fourier frequency (IFF) spacing for a 30-minute segment is 1/1800 Hz. Each segment was Fourier transformed over the frequency interval 1/3 to 2 Hz. This interval, which corresponds to 3000 IFF's, encompasses both the fundamental, 0.8079 Hz, and the second harmonic of the neutron star's spin frequency. This large number of trial frequencies is used to characterize the statistics of the database and to enable the determination of the significance of any putative signal by direct comparison with background in the same database. A frequency search range for possible signals was decided upon prior to looking at the data, namely ±1 IFF about the fundamental frequency and f2 IFF about the second harmonic. This search range was chosen because it corresponded to that used by Middleditch et al. 1 ° in the detection of optical and infra-red pulsations from Hercules X-1 . In computing the Fourier transforms, the data were oversampled by a factor of 5, i.e., 5 Rayleigh powers were computed within each IFF for a total of 15,000 frequencies . To search for persistent but low-level emission from the source, the Rayleigh powers at each trial frequency were added for all segments. While there is a small excess in the total Rayleigh power near the
neutron star frequency, it is not significant . Two types of tests for sporadic ruiissicn were made. For both types of tests, only peak powers are counted and at most one peak power is associated with each IFF. Since identification of peaks is difficult for powers less than about 3, only peak powers greater than 3 are used in the tests described below. In the first type of test, the segments in which the peak Rayleigh power exceeded arbitrarily chosen values of 4, 6, 8, or 10 are counted at each independent frequency. There are significant excesses at thresholds of 6 (6.4 expected from the background distribution and 15 found) and 8 (0.9 expected and 3 found). For instance, the Poisson probability for a measurement of 15 with a background average of 6.4 is 2.5 - 10-3, indicative of a signal. The 15 segments producing powers greater than 6.0 at the fundamental were examined in detail for evidence of a d.c . excess; it was determined that 7 of the segments had do excess consistent with the Rayleigh power." No attempt has been made to incorporate this result into the overall significance although it does indicate that the observed excess is not inconsistent with emission from the source . In the second type of test, the cumulative peak power distribution for the 4:1 IFF signal search was compared with the measured background cumulative distribution . These are shown in Figure 1. Only peak powers greater than 3 are included for reasons given above. The distributions have been compared using the Kolmogorov, 12 Anderson and Darling, 13 and Fisher 12,14 tests which yield chance-origin probabilities of about 1 - 10-2,2 - 10-3, and 3 - 10-4 , respectively. There is good reason to believe that the Fisher test should be the most sensitive of the three . 15 A spectrum based on the Fisher test and showing the full 0.33 Hz to 2 Hz range analyzed is shown in Figure 2. In order to assign a overall significance to the effect, degrees of freedom must be taken into consideration. A factor of two is included to account for pe riodicity testing at the neutron star second harmonic frequency as well as at the fundamental . An addi-
P.T. Reynolds et al./Pulsed TeV radiation from Hercules X-1
Cumulative Power Distribution +- 1 IFF
Rayleigh power threshold FIGURE 1 The cumulative power distribution within the 2 IFF signal search range is shown and compared with the background distribution in the figure.
Hercules X-1
Spectrum
Sin size is 2 IFF
.® O
a.
u O
Frequency in Hz FIGURE 2 The full spectrum from 0.33 to 2.0 Hz is shown above. The ordinate is -Log e of the probability of obtaining, by chance, a value of the Fisher statistic equal to or greater than that measured . The largest peak occurs exactly at the X-ray pulsar frequency.
P.T. Reynolds et al . /Pulsed TeV radiation from Hercules X-1
tional factor of two might be included to account for the failure of the incoherent power summation to produce a significant effect, although there is some excess power at the pulsar frequency. Including a factor of 4, the Poisson test for powers greater than 6, the A&D test and the Fisher test all yield a chance, background noise origin probability of less than 1%. In a modelis developed after the reports of emission from Hercules X-1, emission at a blue-shift of 0.1% is expected. Some clustering of high Rayleigh powers at adjoining blue shifted Rayleigh frequencies was also noted in the above analysis (17 on an expectation of 9.6 in the range [+2 to +4] IFF from the pulsar frequency), but as the initial hypothesis did not specify this search range it is impossible to assign a probability to this occurrence unambiguously. In summary, it appears from a comprehensive analysis of the dataset that there is evidence for TeV emission from Hercules X-1 at better than the 99% confidence level, although we are still investigating the statistical properties of the dataset . A parallel analysis in which only events which have y-ray Cherenkov light image characteristics is in progress. ACKNOWLEDGEMENTS We acknowledge the contributions of P.W. Gorham, K.G. Gibbs, D.F. Leibing, V.J. Stenger, N.A, Porter in earlier phases of this work and the continu ing assistance of K. Harris in taking the observations. This program is supported by the U.S. Department of Energy, the Smithsonian Scholarly Studies Fund, and the National Board of Science and Technology of Ireland. AMH and TCW acknowledge the assistance from a NATO grant. REFERENCES 1. J.C. Dowthwaite, A.B . Harrison, I.W . Kirkman, H .J. MacRae, K.J. Orford, K.E. Turver and M. Walmsley, Nature 309 (1984) 691 . 2. P.W. Gorham, M.F. Cawley, D .J . Fegtsn, K .G. Gibbs, S. Kenny, R.C. Lamb, D.F. Leibing, N.A. Porter, V.J. Stenger and T.C. Weekes, Astrophys. Journ. 309 (1986) 114 ; P.W. Gorham, M.F. Cawley, D.J. Fegtsn, K .G. Gibbs, R.C. Lamb,
203
D.F. Leibing, N.A. Porter, V.J. Stenger and T.C. Weekes, Astrophys . Journ. (Letters) 3 (1986) L11; P.W. Gorham, M.F. Cawley, D.J. Fegan, K.G. Gibbs, R.C. Lamb, D.F. Leibing, N .A. Porter, V.J. Stenger and T.C. Weekes, Proc. NATO Workshop on Very High Energy Gamma Ray Astronomy (Durham, 1986) 125 . 3. L.K. Resvanis, A. Szentgyorgyi, J. Hudson, L. Kelley, J.G. Learned, C. Sinnis, D.D. Weeks, J. Gaidos, M. Kertzmtsn, F. Loeffler, T. Palfrey, G. Sembrowski, C. Wilson ., U. Ctsmerini, J.P. Finley, W. Fry, M. Jaworski, J. Jennings, A. Kenter, M. Lomperski, R. Loveless, R. March, J. Matthews, R. Morse, D. Reeder, and P. Slane, Astrophys . Journ. (Letters) 328 (1988) L9. 4. P.R. Vishwanath, P.N. Bhat, P.V. Ramants murthy and B.V. Sreekantau, Astrophys . Journ. (1989) in press. 5. R.M. Baltrusatis, G .L. Cassiday, R. Cooper, J.W. Elbert, P.R. Gerhardy, E .C. Loh, Y. Mizumota, P. Sokolsky, P. Sommers and D. Steck, Astrophys. Journ. (Letters) 293 (1985) L69. 6. B.L. Dingus, D .E. Alexandreas, R.C . Allen, R.L. Burman, K.B . Butterfield, R. Cady, C.Y. Chang, R.W. Ellsworth, J.A. Goodman, S.K. Gupta, T.J. Haines, D.A. Krakauer, J. Lloyd-Evans, D.E. Nagle, M. Potter, V.D. Sandberg, R.L. Talaga, C.A. Wilkinson, and G.B . Yodh, Phys. Rev . Lett . 61 (1988) 1906 . 7. M.F. Cawley, D.J. Fegan, K .G. Gibbs, P.W. Gorham, R.C. Lamb, D.F. Leibing, P.K. MacKeown, N.A . Porter, V.J . Stenger and T.C. Weekes, Proc. Intern. Cosmic Ray Conf. (La Jolla, 1985) 3 (1985) 453 . 8. P.M. Chadwick, N.A. Dipper, I.W.Kirkman, T.J.L . McComb, K.J. Orford, and K.E. Turver, Proc. NATO Workshop on Very High Energy Gamma-Ray Astronomy (Durham, 1986) 121 . 9. R.C. Lamb, M.F. Cawley, D.J. Fegern, K.G. Gibbs, P.W. Gorham, A.M. Hillas, D.A. Lewis, N.A. Porter, P.T. Reynolds, and T.C. Weekes, Astrophys . Journ. (Letters) 328 (1988) L13. 10. J . Middleditch and J. Nelson, Astrophys . Journ. 208 (1976) 567; J. Middleditch, C.R. Pennypacker, and M.S. Burns, Astrophys . Journ. 274
204
P.T. Reynolds et al . /Pulsed TeV radiation from Hercules X-1 (1983) 313; J. Middleditch, R.C. Pentter and C.R . Pennypacker, Astrophys. Journ. 292 (1985) 267.
11 . D.A . Lewis, Astron . Astrophys., (1989) in press. 12 . W.T . Eadie, D. Drijard, F.E. James, J. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics (North-Holland, Amsterdam, 1971). 13. T.W . Anderson and D.A . Darling, Ann. Math . Statist. 23 . (1952) 193.
14 . R.A . Fisher, Statistical Methods for Research Workers (Oliver and Boyd, Edinburgh and London, 1958). 15 . D.A . Lewis, in this volume. 16 . K.S . Cheng and M. Ruderman, Astrophys. Journ. (Letters) 337 L77.