Astrophysics I HS 2016 Problem sheet 7 Prof. A. Refregier

Due: 8. November 2016

Exercise 7.1 The galactic disk We can obtain a simplified model of the galactic disk by assuming it to be an infinite sheet of constant thickness with constant inner density ρ as shown in Fig. 1.

1. Show that a star displaced by an arbitrary distance z from the mid-plane of the Galaxy in the vertical direction undergoes simple harmonic oscillations around the mid-plane (assuming that the star always remains within the region of constant density).

6 −3 2. Assuming that the density of the galactic disk is about ρ ' 5 × 10 mH m , where mH is the mass of a hydrogen atom, estimate the period of oscillation of a star around the mid-plane of the Galaxy.

3. How does this oscillation period compare with the period of revolution of a star in the solar neighbourhood around the galactic centre (GC)?

Hint: To determine the gravitational force on a particle inside an infinite sheet you can use Gauss’ law for gravity. The period of revolution of a star in the solar neighbourhood −1 around the GC can be determined using v ' 220 km s for the velocity of the Sun around the GC and R = 8 kpc for the Sun’s distance from the GC.

Galactic disk

z Mid-plane

Figure 1: Simplified model of galactic disk

Exercise 7.2 The relations between the Einstein coefficients The Einstein coefficients characterise the emission and absorbtion processes in atoms in the presence of a radiation field with specific energy density Uν. We will consider an atom with two energy levels, which are separated by an energy ∆E = hν. The subscript u denotes the upper while the subscript l denotes the lower . Let the number of atoms per unit volume in the lower and upper levels be nl and nu. There are three processes which result in the transition between the two energy levels: 1. : the spontaneous transition probability per unit time is given by Aul, where Aul is the coefficient of spontaneous emission. 2. : the stimulated transition probability per unit time is given by BulUν, where Bul is the coefficient of stimulated emission and Uν is the energy density of the radiation field at the transition frequency ν.

3. Absorption: the absorption probability per unit time is given by BluUν, where Blu is the absorption coefficient and Uν is the energy density of the radiation field at the transition frequency ν.

We can derive the relations between the Einstein coefficients considering the special situa- tion in which these two-level atoms are in thermodynamic equilibrium with the radiation surrounding them. In this case, the number of upward transitions of the atoms per unit time per unit volume Tl→u has to equal the number of downward transitions per unit time per unit volume Tu→l. Furthermore if the two-level system is in thermodynamic equilibrium, then the occupation numbers of the two energy levels nl, nu will follow the Maxwell-Boltzmann distribution at T , such that their ratio is given by

nl gl hν = e kBT (1) nu gu

where gu and gl denote the statistical weights of the upper and lower states. Using these facts, derive an expression for the energy density of the radiation field at the transition frequency ν. By noting that the latter quantity has to equal the Planck distribution in thermodynamic equilibrium, conclude that the Einstein coefficients have to satisfy the relations

8πhν3 A = B (2) ul c3 ul guBul = glBlu. (3)

Equations 2 and 3 connect fundamental atomic properties. This means that, even though we derived them assuming thermodynamic equilibrium, the relations between the Einstein coefficients will hold even if the system is not in thermodynamic equilibrium.

Hint: You can assume the transition between the two energy levels to exactly occur at frequency ν, i.e. the line profile φ (∆ν) is thin.