We learned in Sections 24.2 and 24.3 that the Sun is currently located 30 pc north of the Galactic midplane and moving away from it with a velocity w⊙=7.2 km s−1. The z component of the gravitational acceleration vector is directed toward the midplane, so the Sun's peculiar velocity in the z direction must be decreasing. Eventually the direction of motion will reverse and the Sun will pass through the midplane heading in the opposite direction. At that time the direction of the z component of the gravitational acceleration vector will also reverse, ultimately causing the Sun to move northward again. This oscillatory behavior above and below the midplane has a well-defined period and amplitude that we will estimate in this problem.
Assume that the disk of the Milky Way has a radius that is much larger than its thickness. In this case, as long as we confine ourselves to regions near the midplane, the disk appears to be infinite in the z=0 plane. Consequently, the gravitational acceleration vector is always oriented in the ±z direction. We will neglect the radial acceleration component in this problem.
(a) By constructing an appropriate Gaussian surface and using Eq. (24.56), derive an expression for the gravitational acceleration vector at a height z above the midplane, assuming that the Sun always remains inside the disk of constant density ρ.
(b) Using Newton's second law, show that the motion of the Sun in the z direction can be described by a differential equation of the form
dt2d2z+kz=0.
Express k in terms of ρ and G. This is just the familiar equation for simple harmonic motion.
(c) Find general expressions for z and w as functions of time.
(d) If the total mass density in the solar neighborhood (including stars, gas, dust, and dark matter) is 0.15 M⊙ pc−3, estimate the oscillation period.
(e) By combining the current determinations of z⊙ and w⊙, estimate the amplitude of the solar oscillation and compare your answer with the vertical scale height of the thin disk.
(f) Approximately how many vertical oscillations does the Sun execute during one orbital period around the Galactic center?