# Problem Set 4

## Due Week February 12, 2001

1. Relativistic Kinematics The cosmic rays have been observed with energies as high as 10^20 to 10^21 eV.

Assuming (1) the cosmic ray is a proton (mc^2 = 938 MeV) or (2) an iron nucleus (Z = 26, A = 56),

(a) What are the beta = v/c and gamma for such a cosmic ray?

(b) Assuming a mean, ordered (means directionally aligned in the Galactic plane) magnetic field of 3 microGauss, what is the radius of curvature of such a cosmic ray travelling perpendicular to the plane? How does that compare to the Galactic mean half-height of three thousand light years?

(c) How accurately would such a cosmic ray proton point back to its source?

(d) Show that such an energetic, extragalactic cosmic ray cannot be an iron nucleus, if it comes across intergalactic space. Hint: intergalactic space is filled with cosmic microwave background photons, about 413 per cc, each with a mean energy of kT with T = 3K in our frame and the iron nucleus has a mean binding energy of about 8 MeV per nucleon in its frame.

2. The Doppler Shift: The brightest member of a binary star pair moves in a circle of radius R about the system center of mass with angular velocity omega. The bright star's spectrum includes a bright line of natural frequency f. Find the frequency observed at a great distance D (which remains constant with respect to the binary's center of mass) in the plane of the circular orbit. Find it as a function of time, keeping terms through second order in (omega R /c) if expanded. Does the observed frequency oscillate in time? If not, is some other variation present? What is the time averaged observed frequency?

3. Harmonic Motion A particle at the origin executes simple harmonic motion in the y direction with frequency f and amplitude a. Determine its position and velocity as a function of time in a reference frame S' moving with velocity v in the x direction. Calculate the time and distance between successive maxima of y'.

4. Derive the equations for transformation of the components of acceleration a_x, a_y, and a_z in system S to a_x', a_y', and a_z ' in system S' moving with velocity v into the x direction, proceeding directly from the Einstein velocity transformation equations and the Lorentz transformation.

5. Use of Invariants A proton moving with energy gamma times its rest mass collides with an identical proton moving, also with the same energy gamma times its rest mass, but at right angles to the first proton in the lab. Derive a formula and evaluate for gamma = 10.

(a) What is the center of momentum system total energy of the ensemble?

(b) What is the momentum and energy of one proton in the frame of the other?