Hydrogen Ionization Inside the Sun

Abstract

Hydrogen is the main chemical component of the solar plasma, and H-ionization determines basic properties of the first adiabatic exponent ${\Gamma _{1}}$. Its ionization significantly differs from the ionization of other chemicals. Due to the large number concentration, H-ionization causes a pronounced lowering of ${\Gamma _{1}}$, with a strongly asymmetric and extending across almost the entire solar convective zone. The excited states in the hydrogen atom are modeled using a partition function, which accounts for the internal degrees of freedom of the composite particle. A temperature-dependent partition function with an asymptotic cut-off tail is derived from the quantum mechanical solution for the hydrogen atom in the plasma. We present numerical simulations of hydrogen ionization, calculated using two partition function models: Planck-Larkin (PL) and Starostin-Roerich (SR). In the SR model, the hydrogen ionization shifts to higher temperatures than in the PL model. Different models for excited states of the hydrogen atom may change ${\Gamma _{1}}$ by as much as $10^{-2}$. The ${\Gamma _{1}}$ profiles for pure hydrogen exhibit a “twisted rope” structure for the two models, significantly affecting the helium ionization and the position of the helium hump. This entanglement of H and He effect provides a valuable opportunity to investigate the role of excited states in the solar plasma.

Appendices

Appendix A: Hydrogen Ionization in Other Equations of State

The ionization of hydrogen is generally different in various equations of state. Comparison of SAHA-S EOS with selected equations of state was performed previously by Gryaznov and Iosilevskiy (2016) along isochores. In contrast, we show a comparison at the points of the solar model. We examine equations of state widely used in astrophysics, i.e., OPAL (Rogers and Nayfonov 2002) and FreeEOS (Irwin 2012).

Figure 7 shows differences of $\Gamma _{1}$ for hydrogen plasma. FreeEOS and OPAL curves lie between the SR and PL ones, closer to the SR, in the region of hydrogen ionization ($\log T\sim 4-5$). Their maximal deviation from PL curve is about $6\cdot 10^{-3}$. The deviations in other regions are smaller $10^{-3}$.

Figure 7

Deviations of $\Gamma _{1}$ for hydrogen plasma computed in various EOS from $\Gamma _{1}$ in SAHA-S with the PL partition function.

Appendix B: Cut-Off Factors and Partition Functions for Several States

Figure 8 shows the values of the weights of excited states for a fixed temperature. A sufficiently low temperature $T = 1.5 \cdot {10^{4}}K$ was chosen as an example. The weight factor for the ground state of hydrogen turns out to be close to unity. The values of other excited states were calculated for the same temperature using Expression 9. The ionization potentials for excited states are significantly lower than those of the ground state. Therefore, the calculation results turn out to be located at lower values of ${\alpha _{n}}$, i.e., shifted to the right on the graph. The values of the weight factors of excited states quickly fall into the “cut-off” section of the weight function, even if the ground state continues to be equal to unity.

Figure 8

Cut-off factors ${w_{n}}$ according expressions 7 and 9. An argument $\alpha =I_{n}/kT$, where $I_{n}=I_{\mathrm {H}}/n^{2}$ is ionization energy for state $n$. The curves are temperature-independent, but the values $w_{n}$ depend on temperature and are shown for $T=10^{4}\text{ K}$.

Figure 9 shows the contribution of each subsequent excited state to the sum $Q$ compared to the sum calculated with a smaller number of states. For the SR model, the contribution of excited states is quite significant compared to the sum over the ground state, reaching 30% in the region of the maximum sum $\log T = 4.8$. Excited states in the case of PL make a noticeable contribution, but it does not exceed 10% of the ground state, and the sum itself does not exceed 1.04.

Figure 9

Partition function $Q$ calculated for several limit numbers $n$.

Appendix C: Fraction of Hydrogen Atoms in the Central Part of the Sun

The fraction of neutral hydrogen below the convection zone is plotted in Figure 10 for the PL and SR models. A part of neutral hydrogen is small as $10^{-4}$.

Figure 10

Fraction of hydrogen atoms in the central part of the Sun.

Appendix D: Discussion on Mott Condition

The functions in the PL and SR models are temperature-dependent and independent of density. In this section, we evaluate the validity of this approximation under the conditions of the convective zone of the Sun. At higher densities, when the average distance between atoms is smaller than their sizes, the atoms are destroyed. This condition is called the Mott condition (see, e.g., Ebeling, Kraeft, and Röpke 2012). The radius of a hydrogen atom in an excited state with number $n$ can be estimated by the formula

$$ a_{n}=n^{2}a_{\mathrm{B}} \, , $$

(13)

where $a_{\mathrm{B}}$ is the Bohr radius. This equality allows us to estimate the critical density $\rho _{n}$, above which atoms with the level $n$ and higher do not exist:

$$ \rho _{n}=\frac{m_{\mathrm{H}}}{(4/3)\pi a_{n}^{3}} \, . $$

(14)

Here, $m_{\mathrm{H}}$ is the mass of the hydrogen atom. In Figure 11, the critical densities for $n$ from 1 to 20 are shown by black horizontal lines. The difference between the densities $\rho _{n}$ also decreases with increasing $n$.

Figure 11

Densities corresponding to the Mott condition for $n=1-20$ (black horizontal lines), $n=40$ (blue line), as well as the temperature and density in the solar model (red curve).

This figure also shows the temperature and density in the solar model (red curve). According to the Mott condition, there are no neutral atoms at $\log T>6.58$; they are destroyed due to the high density. At lower temperatures, $\log T=6.45-6.58$, only atoms in the ground state ($n=1$) can exist. At $\log T=6.16-6.45$, atoms exist in states $n=1$ and 2, and so on.

We calculate the partition function SR, leaving only a limited number of levels. The result is shown in Figure 12 by blue circles. Taking into account the bound states, using the Mott conditions does not significantly affect the value of the partition function $Q(H)$.

Figure 12

Partition function SR calculated for different numbers of levels (curves), and also taking into account the Mott condition (blue circles).

Appendix E: Influence of Partition Function on Pressure

Figure 13 shows pressure differences between the PL and SR cases for pure hydrogen and hydrogen-helium mixture. The pressure in the SR case is always lower than in the PL. The difference between the pressures in the ionization region of hydrogen and helium reaches one percent.

Figure 13

Difference between pressures computed in SR and PL approaches.

Appendix F: Sensitivity to Solar Models

In this section, we examine if our result depends on points $(T,\rho )$. There are many standard and nonstandard solar models in up-to-date literature (see, e.g., review by Christensen-Dalsgaard 2021). Our main result is obtained for the standard solar model computed with the SAHA-S equation of state for high-Z solar abundances, i.e., mass fraction of elements heavier than helium is $Z=0.018$ (Ayukov and Baturin 2017). Now, we consider a solar model computed for low-Z abundances $Z=0.0136$ (Ayukov and Baturin 2017). We also consider Model S by Christensen-Dalsgaard et al. (1996), a high-Z model computed with OPAL equation of state, other nuclear reactions etc.

Figure 14 shows the difference between $\Gamma _{1}$ for hydrogen plasma computed using the SR and PL partition functions at points $(T,\rho )$ of these three models. The curves are not distinguishable at the scale of the figure. Thus, the effect of partition function on adiabatic exponent is the same at points of various solar models.

Figure 14

Difference between $\Gamma _{1}$ for hydrogen plasma computed using SR and PL partition functions at points $(T,\rho )$ of various models.