## Problems

12.2 Neutrino observatories

Problem 12.21. (See Section 10.4 for cross sections, etc.) (a) Derive a relationship for the creation rate r (s-1) of37 Ar nuclei in the Homestake tank (ignoring 37 Ar decays) as a function of (i) the solar flux of 8B neutrinos F^ (neutrinos s-1m-2) at the earth in the Homestake energy range, (ii) the average (over all energies) cross section a (m2) for absorption of a neutrino by a single chlorine nucleus, (iii) the mass density p (kg/m3) of the C2Cl4 fluid, (iv) the volume Vof the tank, and (v) the mass mp of the proton. Consider the probability of absorption to be so low that the beam is not significantly attenuated by the fluid. (b) What is the expected number of interactions per second, per day, per 35 d mean life of37 Ar, and per three 37Ar mean lives if F^ = 7 x 1010 s—1 m—2, p = 1600 kg/m3, a = 3 x 10—47 m2, V = 400m3, and mp = 1.7 x 10—27 kg? [Ans. (b) ~1.5 day-1]

Problem 12.22. (a) Show analytically that competing creation and decay processes in the Homestake neutrino tank yield a net number N(t) of 38 Ar atoms that, starting from zero, initially increases rapidly and linearly and then increases more and more slowly toward a maximum equilibrium value. Proceed as follows. Define a creation rate r (atoms/s) and take the mean decay time (mean lifetime) to be t . Write down the appropriate differential equation for a differential time interval dt and integrate it to obtain N (t) in terms of r, t and time t. Let N = 0 at t = 0, and use the fact that t-1 is the probability per unit time for the decay of a single atom to occur. Make a sketch of N(t). (b) What is N(t) in the limits of short times (t << t) and long times (t >> t)? Demonstrate that the latter is a measure of the creation rate r. Would a measurement at t = t provide a value of r ? (c) Discuss the arguments for sweeping the tank every ~100 days (t « 3t). Be quantitative. Consider for example the long term average rate of accumulated events. (d) What is the number N at t = 3t (~100 d) if the creation rate r has the value you obtained in (or saw in the answer to) Problem 21? [Ans. (a) N = r t[1 — exp(—t/t)]; (d) ~50]

412 12 Astronomy beyond photons 12.3 Cosmic ray observatories

Problem 12.31. The principle of relativistic time dilation states that the mean life at rest, t = 2.2 ^s, of a fast moving muon will be extended to t' according to

## Telescopes Mastery

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