How much of the Solar convection zone is completely ionized? I was reading about the energy transportation in stars here, and I found this: "The outer portion of solar mass stars is cool enough that hydrogen is neutral and thus opaque to ultraviolet photons, so convection dominates".
I always thought that almost all hydrogen and helium inside the Sun (and even in the convection zone) are completely ionized, so this doesn't seem to be the correct explanation of why convection dominates there for me.
I know that the solar convection zone is 200,000 kilometers deep from the photosphere, and temperature at this depth is 2 million °K which keeps decreasing until it reaches the 5700 °K of the photosphere.
So my question here is, at what depth from the photosphere is the temperature low enough for hydrogen to exist as atoms and not ions ?
And is this explanation of neutral hydrogen correct ?
EDIT: I added a picture of a relation between the temperature and the radius inside the Sun. Hope it helps. 
 A: The boundary between the radiative and convective zones is at a temperature of around $10^6$ K. Hydrogen atoms will not survive temperatures much above 20,000 K (to see why, study the Saha equation). Therefore the explanation that convection begins because of the increased opacity brought about by hydrogen atoms is incorrect and is there is very little un-ionised hydrogen in the convection zone. The same goes for helium, although that obviously survives to slightly higher temperatures - it starts to be ionised at about 12,000 K and is almost completely ionised above 30,000 K. Thus almost none of the convection zone (in terms of a mass fraction, or even as a fraction of the radius) contains neutral hydrogen and helium atoms.
Convective heat transport takes over when the temperature gradient in the interior becomes steeper than the "adiabatic temperature gradient".
$$\left|\frac{dT}{dr}\right| > \frac{T}{P}\left|\frac{dP}{dr}\right|\left(1 - \frac{1}{\gamma}\right),$$
where $\gamma$ is the adiabatic index of the gas (e.g. 5/3 for a monatomic ideal gas). If we have hydrostatic equilibrium and a perfect gas of mean atomic weight $\mu$ then
$$\left|\frac{dT}{dr}\right| > \frac{\mu}{k_B}\frac{GM(r)}{r^2}\left(1 - \frac{1}{\gamma}\right),$$
where $M(r)$ is the mass interior to $r$.
In the core and deep interior, the energy transport is radiative and the temperature gradient is given by radiative diffusion:
$$\left|\frac{dT}{dr}\right| = \frac{3\rho \kappa L}{64\pi \sigma T^3 r^2},$$
where $\rho$ is the density, $\kappa$ is the opacity, $L$ is the luminosity (generated interior to $r$) and
$\sigma$ is the Stefan-Boltzmann constant.
Now in the outer parts of the star $M(r)$ changes slowly, so comparing the radiative gradient with the adiabatic gradient, we see that convection can be triggered by a combination of increasing opacity and decreasing temperature, but that the gradient is decreased by decreasing density.
At temperatures of a million K, the opacity is given approximately by Kramer's opacity
$\kappa \propto \rho T^{-3.5}$, which is dominated by bound-free absorption by abundant highly ionised elements such as oxygen and iron. So the radiative temperature gradient increases as $\rho^2 T^{-6.5}$ (the decreasing temperature and stronger temperature dependence wins out over the decreasing density).
In the end then it is a combination of decreasing temperature and increasing opacity which triggers convective instability in the Sun.
A: 
I always thought that almost all hydrogen and helium inside the Sun (and even in the convection zone) are completely ionized...

This question has been bothering me ever since it was asked.  I had a similar, apparently naive, assumption until I bugged a solar physicist down the hall.
It turns out that not only does the sun become partially ionized low in the chromosphere and through much of the photosphere into the convection zone.  The interesting part is not only are these regions not totally ionized, the ions that do exist were unexpected.  For instance, there are several heavy elements/metals (e.g., silicon, sodium, etc.) that will freely give up electrons even at the low temperatures of the photosphere and there are corresponding quasi-stable H- and H2+ states in addition to the usual H+, He+, and He2+.

...so this doesn't seem to be the correct explanation of why convection dominates there for me.

As you might imagine, the ionization states are altitude-/depth-dependent and not trivial [e.g., Fontenla et al., 1990, 1991, 1993, 2002].  The level of complexity increases even further when including even heavier elements and the nuances of their electron orbitals, which have different frequency-dependent opacities.
Further, H- and H2+ can dominate the opacity in some places that should be too cold to support H+ [e.g., Fontenla et al., 1990, 1991, 1993, 2002].

So my question here is, at what depth from the photosphere is the temperature low enough for hydrogen to exist as atoms and not ions?

There is apparently neutral hydrogen well above the photospheric surface but it does not become a significant fraction of the charged density until until within a few thousand kilometers of the surface [e.g., Avrett et al., 1976].  The fraction of neutral hydrogen depends upon the rates of recombination and ionization.
Update
As Rob (Jeffries) pointed out, I did not discuss the Saha equation.  It turns out that the regions of interest are not in local thermodynamic equilibrium or LTE, which is required for the use of the Saha equation.  The Fontenla et al. group has been actively creating more and more complex solar atmosphere models for over 20 years and are considered the experts on this topic.  If you scan through the references I provided, they do not mention the Saha equation, which was surprising to me the first time I read them.
However, Avrett et al. [1976] does discuss the continuum source function, $S_{\nu}^{c}$, for non-LTE systems.  In the limit of LTE, the Saha equation can be used to transform $S_{\nu}^{c}$ to recover Planck's law.
Update 2
Rob (Jeffries) made several more arguments as to why the Saha equation is relevant and convinced me it is important for the convection zone.  The models to which I refer only go ~1% into the convection zone and below that it starts to become more and more difficult to argue against using something like the Saha equation (i.e., the fluid goes to LTE rather quickly).
References


*

*Avrett, E.H., et al. "Excitation and ionization of helium in the solar atmosphere," Astrophys. J. 207, pp. L199-L204, 1976.

*Fontenla, J.M., et al. "Energy balance in the solar transition region. I. Hydrostatic thermal models with ambipolar diffusion," Astrophys. J. 355, pp. 700-718, 1990.

*Fontenla, J.M., et al. "Energy balance in the solar transition region. II. Effects of pressure and energy input on hydrostatic models," Astrophys. J. 377, pp. 712-725, 1991.

*Fontenla, J.M., et al. "Energy balance in the solar transition region. III. Helium emission in hydrostatic, constant-abundance models with diffusion," Astrophys. J. 406, pp. 319-345, 1993.

*Fontenla, J.M., et al. "Energy balance in the solar transition region. IV. Hydrogen and helium mass flows with diffusion," Astrophys. J. 572, pp. 636-662, 2002.

