# about H$_\alpha$ and H$_\beta$ lines in astronomical objects' spectra

Generally there may be some line ratio between the two lines.

If H$\alpha$ is strong, it is quite possible we can see H$_\beta$ too.

Is it possible we can see H$_\beta$, but can not see H$_\alpha$? In what kind of environments can it happen?

In a normal stellar atmosphere, where the temperature decreases with height, I think this is not possible.

H-alpha absorption arises from the $n=2$ to $n=3$ transition; H-beta from the $n=2$ to $n=4$ transition. Thus both transitions are governed by the number of atoms in the $n=2$ lower level.

The absorption coefficient for a transition can be written in proportionality terms by (in local thermodynamic equilibrium) $$\alpha_\nu \propto \nu B_{lu} n_l ( 1 - \exp[-h\nu/kT]),$$ where $B_{ul}$ is the Einstein absorption coefficient, $n_l$ is the population of the lower energy level and $\nu$ is the photon frequency corresponding to the transition.

Using the well-known relationship between the Einstein emission and absorption coefficients and values for the emission coefficients found here, we can estimate a ratio of absorption coefficients for a given temperature.

$$\frac{\alpha_{H\beta}}{\alpha_{H\alpha}} = \frac{g_{n=4}}{g_{n=3}} \frac{A_{H\beta}}{A_{H\alpha}} \left(\frac{\nu_{\alpha}}{\nu_{\beta}}\right)^2 \left(\frac{1 - \exp(-h\nu_{\beta}/kT)}{1-\exp(-h\nu_{\alpha}/kT}\right),$$ where $A_X$ are the Einstein emission coefficients and the statistical weights $g_n$ are given by $2n^2$.

Thus if we take $\nu_\alpha = 4.57\times10^{14}$ Hs, $\nu_{\beta}=6.17\times10^{14}$ Hz, $A_{H\alpha} \simeq 10^{8}$ s$^{-1}$, $A_{H\beta} \simeq 3\times 10^{7}$ s$^{-1}$, then $$\frac{\alpha_{H\beta}}{\alpha_{H\alpha}} = 0.29 \left( \frac{1 - \exp(-29642/T)}{1 - \exp(-21956/T)}\right)$$

So when $T$ (in Kelvin) is small, the ratio is about 0.3. When $T$ becomes large then the ratio increases, but above $T \sim 12,000$ K, all the Hydrogen is ionised.

Thus in thermodynamic equilibrium is it very difficult to manufacture a circumstance where the optical depth in the H$\beta$ line is larger than that in the H$\alpha$ line.

[I'd be grateful if someone can tell me whether the ratio I calculated above is correct, since I could not find it in any reference and need to work it out from scratch].

• scratch? That is awesome job! Jul 7, 2015 at 9:54
• I revised the title. It is not necessarily a stellar spectrum. Jul 7, 2015 at 12:43

As a small addendum to Rob Jeffries' excellent answer, I'll note that you probably can't get a scenario of "H-beta, but no H-alpha" for emission lines, either. The emission lines are the reverse of the absorption lines, so to have H-beta emission without H-alpha, you'd have to have $n = 4$ to $n = 2$ transitions without also having $n = 3$ to $n = 2$ transitions. But even if you could somehow magically arrange to have all your H atoms in the $n = 4$ level, some of them would transition to $n = 3$, and then to $n = 2$, so you really can't avoid H-alpha emission.

Furthermore, dust extinction is worse at shorter wavelengths, so dust extinction could in principle give you the appearance of H-alpha emission without H-beta emission, but not the reverse.