Calculating the density of a HII region

[bcx]

Hey,

Been stuck on a problem for over a day now, can't seem to figure it out!
The question reads as follows:

A single high-mass star is surrounded by an HII region that
subtends an angle of 10 arcmin on the sky. The optical colours
of the star suggest it has a bolometric luminosity of 5∙10^(31) W and
it is observed to have a bolometric flux of 4.4∙10(-7) W m-2.
Assuming most of this luminosity is emitted as 10 eV photons, what
is the density of hydrogen gas around the star?

To solve this, I have intially used the formula

to calculate the distance of the high mass star. I have then
calculated the circumference at that distance, and using the
knowledge that the HII subtends an angle of 10 arcminutes, I have
converted the overall circumference into the radius of the HII
region (first by dividing the circumference by (360*6), then
by 2). From this I have determined the volume of the HII region
using 4/3*pi*r^3. Then, I have converted the luminosity of the star
into electron volts (by dividing by 1.6*10^-19), and used the Ionisation formula as follows (the one typically used to find the stromgren radius)
[PLAIN]http://rpmedia.ask.com/ts?u=/math/5/1/2/5128062b3e1ef80caa7cf0ac4f1bb3c2.png

Where I have taken S* as the total luminosity divided by the energy of a single 10eV photon (as stated in the problem).

Then from this I have a value of n^2*beta. where beta is the recombination constant and n is the number desity of hydrogen.
I take an value of beta as 3.1*10^-19 (assuming star is at 8000K), and from this calculated n.

The value is supposed to be 6000/cm^3, but my values always seem to lie in a higher region. I've tried working backwards but all I seem to get is silly values for the temperature of the star. I can't seem to see where I am going wrong. Any light on the situation would be appreciated!

 

[cepheid]

Welcome to PF bcx!

To solve this, I have intially used the formula

to calculate the distance of the high mass star. I have then
calculated the circumference at that distance, and using the
knowledge that the HII subtends an angle of 10 arcminutes, I have
converted the overall circumference into the radius of the HII
region (first by dividing the circumference by (360*6), then
by 2).

You don't need the "circumference" or anything (it doesn't even make sense to talk about a circumference, since you're modelling the HII region as a sphere). It would be much more direct to just use the definition of an angle:

angular diameter (in radians) = (physical diameter)/(distance)

Once you know the physical diameter, you know the radius.

From this I have determined the volume of the HII region
using 4/3*pi*r^3.

You don't need to explicitly compute the volume. You can just assume that the radius obtained using the method above is the Stromgren radius and enter that into your equation below.

Then, I have converted the luminosity of the star
into electron volts (by dividing by 1.6*10^-19), and used the Ionisation formula as follows (the one typically used to find the stromgren radius)
http://rpmedia.ask.com/ts?u=/math/5/1/2/5128062b3e1ef80caa7cf0ac4f1bb3c2.png

Where I have taken S* as the total luminosity divided by the energy of a single 10eV photon (as stated in the problem).

Right, well S* is the number of ionizing photons emitted per unit time (in s^-1). You need 13.6 eV to ionize a hydrogen atom, so it's a little confusing that you are asked to assume that all of the photons are at 10 eV. Maybe you are just meant to use 10 eV as a ballpark figure, and assume that these photons are all ionizing. In that case, your method is fine.

Then from this I have a value of n^2*beta. where beta is the recombination constant and n is the number desity of hydrogen.
I take an value of beta as 3.1*10^-19 (assuming star is at 8000K), and from this calculated n.

Well, the star in question has a luminosity of 130,242 solar luminosities. Assuming that it is a main sequence star, and looking at the main sequence on a typical HR diagram (for example the one given at the top of the Wikipedia article for "Main Sequence") it looks like stars with luminosities of 100,000 times that of the sun are only the hottest ones, having O spectral type. This means that their surface temperatures would lie somewhere in the range of 30,000 K - 50,000 K. Do you have a value of $\beta_2$ for this surface temperature?

 

[bcx]

Hey Cephid, Cheers for the help!

I recalculated it with the method you described for calculating the diameter of the HII region (which makes a lot more sense than my circumference method).
I also thought it was a bit unusual that it used 10eV instead of 13.6 as standard, but it's what the question asked.

No value of b2 was given in the question, which made me doubt whether my methodology was correct in the first place. I calculated a value of b2 to be about 7.07*10^(-20) using

. Where Te is the temperature of the electrons, which is apparently the same as the temperature of the star.
But with the value of 30,000-50,000k which seems more fitting, the value is still incorrect. In fact with this method all my answers seem miles off. I'm starting to think my method may be completely off tbh.