Back-of-the-Envelope Estimate of Time Required to Align the GLAST Towers with Ground-Level Cosmic Ray Muons

                                                S.R, 18 October 2000

This is a very rough estimate, done quickly to check whether we can feasibly monitor thermal distortions during environmental testing using ground-level cosmic ray muons.

Inputs and Assumptions

• The towers are principally carbon, so the thermal distortions are completely dominated by the effects of the aluminum grid.  Furthermore, the towers can be treated as individual, completely rigid units that have no significant thermal distortions internally (within each tray, or tray-to-tray).  Thus, we have only 16 objects to co-align.  This very significantly reduces the combinatorics.
• The ground level vertical muon flux is approximately 80 m^-2s^-1sr^-1, with an angular dependence proportional to cos²q.  The flux through a horizontal plane is thus 130 m^-2s^-1.  We use that fraction of the flux that passes through at least 2 or 3 XY planes in two towers.  Note there is an additional soft component, at about 25% this rate, consisting mainly of electrons.  A small (but not needed) bonus if we do thermal testing in an inverted position to allow the heat pipes to be functional is that the calorimeter will nicely attenuate the soft flux.

Estimates

Statistics  For perfectly straight trajectories, the tracking resolution is very roughly given by the hit precision/lever arm, and for each event the tower with the smaller track length will dominate the resolution.  Assuming at least three XY planes for a track results in a limiting resolution of O(1) mrad.  However, multiple scattering is also important.  For 2 GeV muons, the characteristic multiple scattering angle after passing through 3 trays in the Front section is 2.6 mrad (5 mrad in the Back).  Furthermore since the multiple scattering a 1/E, on a differential muon flux spectrum that falls like 1/E² the mean multiple scattering angle in the sample is more like 8 mrad (for a weighted mean energy of 700 MeV).  Thus, for each degree of freedom, a knowledge of better than 0.05 mrad (10 arcsec) will be obtained with a sample of ~25,000 events.  To do a two-tower alignment in 6 degrees of freedom implies a sample size of 150,000 events.

Rates  Given the above fluxes, the whole instrument will see approximately 400 Hz of muons, or roughly about 25 Hz entering each tower from the top.  To do the alignment, we need tracks that cross tower boundaries.  On average, tracks at angles of incidence more than ~10°-15° will have a significant enough path length in two towers.  Given the cos²q dependence, and neglecting track paths through the Back section, we estimate 1 Hz per tower of tracks will satisfy the necessary conditions for a particular tower pair [a more careful calculation is underway].  Since both towers in a pair contribute useful flux by symmetry, the usable rate for a tower pair alignment is 2 Hz.  Note that we have neglected muons entering from the sides of the full instrument.

Time  There are different approaches, but a data-hungry, over-constrained approach would be an iterative pair-wise alignment of a tower with all nearest neighbors, resulting in a statistics requirement of 150,000 useful events per tower pair; which, at 2 Hz, can be accumulated in about 21 hours.  Note that since the same tower is analyzed by several nearest neighbors, a more system-wide coherent calibration can in principle be done with lower statistics.

Conclusion

At first glance, it appears that the flux of ground-level muons will be sufficient to monitor the effects of thermal distortions during environmental testing on the relevant timescales.  These estimates should be checked.

Acknowledgement:  I thank Bob Hartman for a very useful discussion.