ASTR 3730 PROBLEM SET #7

1. If you were to take a spectrum of a reflection nebula, would you see absorption lines, emission lines, or no spectral lines? How would this help to show that the illumination is by reflection from the central star?

2. To calculate the size of a Stromgren sphere, idealize the problem by considering a pure hydrogen gas of uniform density which surrounds a single hot star. Let N_* be the number of ultraviolet photons beyond the Lyman limit which leave per unit time from the central star. Assume that each such photon will ultimately ionize one and only one hydrogen atom. Let R be the number or recombinations of protons and electrons into hydrogen atoms per unit volume per unit time. In a steady state, the total number of recombinations in the Stromgren sphere of radius r must balance the total number of ionizations:

R(4pr³/3)=N_*

Given R and N_*, this equation would allow us to find r. To obtain R, let us note that recombination at interstellar densities is a two-body process (involving for each recombination one proton and electron). Thus the number of recombinations per unit volume R must be proportional to the product of the number densities of protons and electrons, n_pn_e. The proportionality factor is denoted by a, and it is called the "recombination coefficient." Thus

R=an_pn_e=an_e²

Where we require n_p=n_e for overall charge neutrality. Show now that the Stromgren radius r is given by the formula

r = (3N_*/4pan_e²)^1/3

The "recombination coefficient" a is a function of the temperature of the hydrogen plasma. For temperatures characteristic of Galactic HII regions, a has the approximate value 3x10^-13 cm³sec^-1. Assume n_e=10cm^-3; compute r when N_*= 3x10⁴⁹ sec^-1 (O5 V star), N_* = 4x10⁴⁶sec^-1 (B0 V star) and N_* = 1x10³⁹ sec^-1 (G2V star). Convert your answer to light-years. What types of main-sequence stars have appreciable HII regions?

3. Suppose a spherical shell identified as a supernova remnant is observed with radius r and with outward expansion speed v. Assume the mass density of the ambient medium to have the uniform value r₀; then the supernova remnant must have swept up mass M = r₀4pr³/3. Let the original mass M₀ be ejected at speed v₀. If we ignore communication between different parts of the shell (via the thermal pressure of the hot interior), and suppose that each piece of the shell preserves its outward linear momentum as it sweeps up more material initially at rest, we have the snowplow model. Show that the snowplow model implies

(M+M₀)v=M₀v₀

The original kinetic energy E₀ of the ejected material equals M₀v₀²/2. The present kinetic energy E of the shell equals (M+M₀)v²/2. Show that the ratios E/E₀ and v/v₀ are given by

E/E₀=v/v₀=M₀/(M+M₀)

In a typical supernova explosion, M₀=4M_¤ worth of matter might be ejected from the central star at speeds v₀=5000km/s. What is the original kinetic energy E₀? How much mass M would have to be swept up to bring the supernova shell speed v to a value of 10km/s, which is typical of the random velocities of t interstellar clouds? In the process, by what factor is E reduced from E₀? What must happen to the lost energy in the snowplow model?

What is the radius r when v =10km/s if r₀=2x10^-24g/cm³? Convert your answer to parsecs.

For M>> M₀, show that v is inversely proportional to r³. With the time-rate change of radius proportional to r^-3, show the age t of the remnant will four times shorter than had it expanded to radius r at a constant speed. Compute t for our hypothetical supernova remnant.

4. a) At what velocity must a hydrogen atom hit an ice cube to sputter off a water molecule, given that the boiling point of water is 400K?
b) How far from a one solar mass star does the orbital velocity equal the velocity of part a?
c) The temperature of Saturn is about 100K. How massive would an accretion core have to be (at 100K) to hold a hydrogen atom against thermal escape? Assume a core density of 3g/cc. How does this mass compare to Saturn's mass?