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where E is the total energy (rest energy + kinetic energy), and mc2 = 106 MeV is the muon rest energy. What is the kinetic energy Ek of a muon (in GeV) that will just make it to sea level at the end of its (extended) mean life? Assume the muon is created at a depth of 1000 kg/m2 in the atmosphere, or about 18 km above sea level. You may approximate its speed of travel as about equal to the speed of light c and neglect any energy loss due to ionization during passage through the atmosphere. To what extent are these assumptions warranted? [Ans. ~3 GeV]

Problem 12.32. A relativistic muon passes vertically through you while you are standing up. (a) About how much energy is dissipated by it inside your body, in units of MeV? The energy deposition rate per kg/m2 is about the same as in the atmosphere; see text. (b) Roughly, how much energy is deposited in your brain by cosmic ray muons in one second? (See text for muon flux.) How many molecules (mostly water) of your brain are likely to suffer an atomic ionization in that one second? (It takes 13 eV to ionize either an H or an O atom.) By what age might you expect to have all the molecules in your brain ionized, if recombination is neglected? [Ans. ~200 MeV; 1012 yr]

Problem 12.33. A fairly large EAS due to an incident gamma ray reaches its maximum size of 1 x 107 electrons, positrons, and gamma rays as it reaches sea level (atmospheric depth 10 300 kg/m2). Consider all of the created particles to still be present at the maximum. What approximately is the mean interaction length in the atmosphere, in kg/m2, for either the pair production or the bremsstrahlung process; assume they are equal. This is known as the "radiation length" in air. Hint: assume that each interaction doubles the number of particles. [Ans. ~ 400 kg/m2, or ~1/25 of the atmosphere]

Problem 12.34. (a) What is the approximate time delay expected between the arrival time of the electrons at the detectors under the right edge of the EAS shown in Fig. 4 compared to those at the left edge? The arrival direction of the primary proton is tilted 15° from the vertical. (b) How accurate must the timing be to attain a precision of A0 in the (left-right) arrival direction of an EAS of width D arriving from a zenith angle 0 ? Evaluate your expression for A0 = 1° for arrival directions 0 = 0°, 15°, and 30°. (c) Do you think these accuracies are attainable? Discuss. [Ans. ~200 ns; 10-15 ns]

Problem 12.41. Construct from (8) the quadrupole tensor for several elementary distributions of discrete masses. That is, evaluate (8) for each of the 9 components Q11, Q12, Q13, Q21,... ,Q33. It is convenient to present the 9 values in a matrix of 3 rows and 3 columns. The goal is to give a feel for the meaning of "quadrupole moment" and the tensor that describes it. Comment on what you learn; e.g., what does the Q33 term represent, what components are systematically zero and what kind of mass distribution would give non-zero values, and do the diagonal elements have a systematic relationship? The mass distributions consist of discrete masses, each of mass m placed on the 3 axes (x, y, z) as follows: (a) Two masses on z axis at z = ±r0 (symmetry along one axis). (b) Four masses, two as in (a) and two on y axis at y = ±r0 (symmetry along two axes). (c) Six masses, four as in (b) and two on x axis at x = ±r0 (symmetry along all three axes). (d) Same as (c) except the two on the z axis are closer to origin at z = ±r0/2 (oblate distribution). (e) Same as (c) except the two on the z axis are further out at z = ±2r0 (prolate distribution). (f) Six masses, same as oblate distribution (d) except the pattern is rotated 45° about the z axis, so the four masses in the x, y plane are no longer on the axes and have both x and y components. [Ans. Q33 values are (in units of mr^): 4/3; 2/3; 0; -1 ; 4; -1]

Problem 12.42. (a) What is the approximate orbital period of two neutron stars orbiting each other just barely in contact? Let each star have mass m = 1.4M0 and radius R = 10km. Consider all the mass of each star to be at a point at its center. Ignore effects of general relativity and tidal forces. Use selected laws by Newton or Kepler. (b) Construct from the general quadrupole tensor (8) the quadrupole tensor (10) as a function of time t for a binary system consisting of two neutron stars of equal masses m, each in a circular orbit in the xy plane with orbital angular frequency m. The masses rotate about the z axis at radius r and are aligned along the x axis at t = 0 (see Fig. 8a). Again assume point masses. (c) Find the strain tensor (11) for a binary from the general expression (9) and your answer to (b). (d) Find the units of the tensor component hjk from (9). Is your answer what you expect? [Ans. —1 ms; Eq. (10); Eq. (11); —]

Problem 12.43. Find, from (13), the amplitudes and frequency v (Hz) of each strain polarization (h+ and hx) at the earth due to gravitational radiation from the Hulse-Taylor binary pulsar at its present period (7.75 h), if its orbit were circular. The masses of the two neutron stars are —1.4 M0, and the earth is distant —16 000 LY in the direction 47° from the orbital pole. Do you think this could be detected with the currently operating detection systems, e.g., LIGO? [Ans. —10-4 Hz, —5 x 10-23 for each polarization]

Problem 12.44. (a) Calculate from (6) the numerical value of dE/dt for a binary pulsar containing two neutron stars, each of mass m = 1.4 M0, in circular orbits of period the same as the H-T pulsar, P = 7.75 h. (Use selected laws of Newton or Kepler to obtain the needed parameter a.) Compare to the solar luminosity. (b) Find an expression for the characteristic decay time t = E/(dE/dt) of the system in terms of the variables given in (6) where E, the total energy (kinetic plus potential), is taken from Newtonian mechanics. What is the physical significance of the characteristic time for such a binary system? Evaluate your expression for our binary and compare your answer to the age of the universe, ~ 15 Gyr. (c) From the rate of energy loss, estimate the advance of the time of periastron in 25 years. Make simplifying assumptions as needed, e.g., that the rate of energy loss does not change appreciably in 25 years. Hints: find a relation between AP and Aa and also one between AE and Aa to find the period change after 1/2 the 25 yr. (A fun aside: how much does the separation decrease in 12.5 yr?) Compare to the 26 s actually measured for the H-T pulsar. How can you explain the difference? (d) Recalculate the phase advance for a circular orbit that is at the closer periastron distance of the H-T pulsar, a' = 0.383a. Again compare to the H-T pulsar phase advance and comment. [Ans. ~ 1024W; < 1010 yr; 2 s; ~-25 s]

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