last revised 3/14/08
ASTRONOMY 402
Question 6
due in class, Thursday 3/13

Consider a pure hydrogen cloud with unform 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, r is given by:

R(4πr3/3)=Q

The recombination rate R 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 (in cm-3), ne and np. Overall charge neutrality requires that ne = np. So we have the total recombination rate per cm3 per second:

R = αne2

where α is the "recombination coefficient," representing the constant of proportionality in the recombination equation.

a. Write down an expressions for the Strömgren radius in cm.

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

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 05 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 R~0.1 R☉.
It has Q = 1 x 1047 sec-1. Compute r in meters and in parsecs for this case, using the same value for α, but with a density characteristic of planetary nebulae, about 103 cm-3. Compare with the O star above. Also comment on the ionization level in the Strömgren sphere compared with that in the H II region for the O star above.

h. Calculate the mass, in solar masses, contained in the Strömgren sphere of the planetary nebulae above. How does this compare this with the typical mass of a planetary nebula you learned about in Astronomy 111, or in your 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

Back to Astronomy 402 syllabus.