ASTRONOMY 402T – Spring 2018
Problems for Tutorial Week #5

Consider a pure hydrogen cloud with uniform density surrounding a hot star. This star emits Q ultraviolet photons per second beyond the Lyman limit, i.e., photons capable of ionizing hydrogen from the ground state. Assume that each photon ionizes one and only one hydrogen atom.

Let R be the number of recombinations per cm³ per second. In a steady state, the number of recombinations will equal the number of ionizations inside the resulting spherical ionized region, called a Strömgren sphere, whose Strömgren radius is given by:

          R_S = (3Q/4πα_Bn_e²)^1/3

The recombination rate involves a two-body process: the two bodies are the electron and the proton. The rate must be proportional to the product of their number densities, n_e and n_p. Overall charge neutrality requires that n_e = n_p and for a pure hydrogen nebula = n_H. The total recombination rate per cm³ per second is α_Bn_e², where α_B is the recombination coefficient for Case B, where recombinations to the ground state are ignored. (Why can we say that?)

For temperatures characteristic of H II regions, α_B is approximately 3 x 10^-13 cm³sec^-1. Assume n_e = 10 cm^-3.

a. Using these values, write down a parameterized expression for the Strömgren radius in cm.

b. Compute r in cm and in parsecs when Q = 3 x 10⁴⁹ sec^-1 (an O5 V star with T~40,000K and R~20 R_☉).

c. Compute r in cm and in parsecs when Q = 4 x 10⁴⁶ sec^-1 (a BO V star with T~30,000K and R~5 R_☉).

d. Compute r in cm and in parsecs when Q = 1 x 10³⁹ sec^-1 (a G2 V star with T~6000K and R~R_☉).

e. What kinds of main-sequence stars create significant H II regions around them?

f. O stars are often born in clusters like the Trapezium in Orion. The Orion Nebula is 450 pc away, and the portion known as M42 has an angular diameter of about 1 degree. Assume that M42 is a spherical pure hydrogen nebula with an average density of 200 cm^-3. How many equivalent O5 stars are required to ionize M42? Compare this with the number of the brightest Trapezium stars.

g. Now consider a planetary nebula central star with a temperature of 150,000K and a radius ~0.1 R_☉. It has Q = 1 x 10⁴⁷ sec^-1. Compute r in cm and in parsecs for this case, using the same value for α_B, but with a density characteristic of planetary nebulae, about 10³ cm^-3. Compare with the O  star in part (b) above. Also comment qualitatively on the relative ionization levels you would expect for elements like nitrogen, oxygen, and neon, if they were present in both the planetary nebula and the H II region around the O star in part (b) above.

h. Calculate the mass, in solar masses, contained in the Strömgren sphere of the planetary nebula above. Compare this with the typical  planetary nebula mass you learned about in Astronomy 111 or in your own studying.

i. Calculate the mass, in solar masses, contained in the Strömgren sphere produced by the O5 star in part b. Compare this value with the mass of  the planetary nebula, and comment on the relative size of these values.

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