Estimate the emission measure and the ionized density from H alpha brightness of LDN 1616

[will-henney]

Using the brightness measured in #3 we can calculate the emission measure of the ionized gas at the surface of the globule:

For this, we will need the following steps:

• [x] Find the units of the Finkbeiner map and convert them to cgs
• [x] Find values for the factors $\alphaB$ (recombination coefficient), $f\mathrm{H\alpha}$ (fraction of recombinations that produce an H alpha photon), $E_\mathrm{H\alpha}$ (energy of H alpha photon). Although it is possible that the last of these might not be needed).
• [x] Convert the H alpha brightness value to an emission measure (EM)

After that, we can convert it to a density, assuming $n_0^2 r_0 \approx \mathrm{EM}$

[will-henney]

This is how we convert from Rayleighs to emission measure:

We use the definition of Rayleigh from Wikipedia, but converted to cgs

We also use an effective H alpha recombination rate of 1.17e-13, which is the value given in the Draine textbook for a temperature of 10,000 K

The final result is EM = 2.76 times the surface brightness in Rayleighs, where EM is in units of pc per cm^6

[will-henney]

The temperature dependence of the factor 2.76 is approximately linear. For instance, at a temperature of 8000 K, it would be 2.2

We should use 2.2 for the conversion factor, since a lower temperature is more realistic.

For comparison, Finkbeiner gave a number that implies 2.56 for this factor

Also, Pon et al (2014) have the following:

So, all these estimates are in agreement to within about 10%

[will-henney]

@RobeReyes please repeat the calculation of the density to document it here. We can assume that the line of sight depth is related to the globule radius as $\ell = (2 / \sqrt{10}) r_0$.

Then use the fact that $\mathrm{EM} = n_0^2 \ell$

[RobeReyes]

For the density we have that $n_0=\sqrt{\frac{EM}{l}}$ and if we suppose a brightness of 30, we have that $n_0=\sqrt{\frac{2.76 I/Ry}{(2/\sqrt{10})r_0}}\approx 9.88 cm^{-3}$