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I have a homework problem:

The Sun emits $ \sim5 x 10^{23}$ photons per second with $hν > 13.6$ $eV$. If the density of hydrogen atoms in interplanetary space is $n =$ $109 m^{-3}$, what is the size of the Stromgren sphere? Assume a recombination coefficient $α = 2.6 x 10^{-19} m^3s^{-1}$.

From Wikipedia, I was able to get to

$$ R_S = \left( \frac{3S_*}{4\pi n^2 \beta_2} \right)^\frac{1}{3} $$

And I know $\beta_2 = \frac{\alpha}{T}$, but I have no idea what the $S_*$ is. Wikipedia describes it as a source of flux, which is obviously the sun, but I cannot figure out anything else about it. I'm actually quite sure that the professor would have given us a different equation, but he never went over it, so I don't know. Anyone know how to solve this?

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In astronomy, S is usually flux. Here is it probably total watts put out by the star. The wikipedia article is sloppy in not defining its terms or linking to a page defining these. –  DarenW Oct 29 '13 at 0:36
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1 Answer 1

The Strömgren sphere is defined as the sphere bounding a region that, if fully ionized, would undergo recombination at a rate equal to the rate of ionization. Calculating the radius is much simpler than Wikipedia makes it out to be, and it's worthwhile to figure this out on your own rather than look up the formula.

Fundamentally, this is just balancing rates to find an equilibrium. On one side, you have the ionization rate, $5\times10^{23}\ \mathrm{s}^{-1}$, often denoted $Q$. On the other side you should write, as a function of $R_\mathrm{s}$, the rate of recombinations in the entire volume given the density of $\mathrm{H}^+$ and $e^-$. No $\beta$'s or $S$'s required.

Some of the tacit assumptions here are that (1) there is nothing else around being ionized and (2) every ionizing photon leads to an average of one ionization in the sphere.

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