Energy Transport in the Sun: From Models to Observables

1 Introduction

Although sun-like stars only account for 6% of stars in our local 10 pc volume (Beech 2011), understanding and modelling energy transport in the Sun leads to insight into other main sequence stellar types, given the composite nature of stars. All stars must transport energy via radiation, convection, or conduction; producing testable models and theories for the Sun therefore allows theories about all stars to become more accurate and reliable.

The Sun, like all stars of approximately its mass, comprises a core in which energy production occurs predominantly through nucleosynthesis, a radiative region where energy transport by radiation dominates, and finally a convective region where convection is the dominant mechanism of energy transport. Conduction is not a method employed by the Sun and is generally restricted to white dwarfs (LeBlanc 2010, p 166).

Each of these key regions will be discussed either directly in terms of the energy transport mechanism or in terms of the effect of energy production. In particular, physical evidence for these mechanisms will be examined, along with how they fit into current stellar models.

Energy transport due to convection is of particular interest because of the complexities of modelling this hydrodynamical problem in the highly pressurised, notoriously compressed stellar plasma. The mixing-length theory and its relative success will be particularly examined. Overall, the theoretical framework describing energy transport in the Sun will be discussed in relation to producing empirically verifiable models, such as a synthetic solar spectrum.

2 Radiative Energy Transport in the Sun’s Interior

Initially, it is important to arrive at a useful equation for the total Eddington flux at high optical depths, as although the stellar interior is not directly observable, it does determine the features of the stellar atmosphere. Using a Taylor series to approximate the specific intensity (LeBlanc 2010, p 92):

\[\begin{equation} I_\nu(\tau_\nu,u)=\sum_{n=0}^\infty u^n\frac{d^nB_\nu}{d\tau_\nu^n}\bigg|_{\tau_\nu} \end{equation}\]

From this, the average total flux can easily be obtained and then approximated by neglecting the higher-order derivatives:

\[\begin{equation} H_\nu(\tau_\nu)=\frac{1}{2}\int_{-1}^1I_\nu(\tau_\nu,u)udu=\frac{1}{3}\frac{dB_\nu}{d\tau_\nu}+\frac{1}{5}\frac{d^3B_\nu}{d\tau^3_\nu}+\dots\approx\frac{1}{3}\frac{dB_\nu}{d\tau_\nu} \end{equation}\]

This can then be seen to be a function of the temperature gradient, since \(d\tau_\nu=-k_\nu \rho dz\).

\[\begin{equation} H_\nu\approx\frac{1}{3}\frac{dB_\nu}{d\tau_\nu}=-\frac{1}{3k_\nu\rho}\frac{dB_\nu}{dz}=-\frac{1}{3k_\nu\rho}\frac{dB_\nu}{dT}\frac{dT}{dz} \end{equation}\]

This is the diffusion approximation (LeBlanc 2010, p 93), and it can then be seen that the monochromatic flux increases proportionally to the negative temperature gradient. Combining equation 3 with another definition of the integrated Eddington flux as a function of the luminosity, with the Rosseland mean opacity \(k_R\) and \(z\rightarrow r\) for simplicity, it is seen that:

\[\begin{equation} H(r)=\frac{1}{4\pi}\frac{L(r)}{4\pi r^2} \end{equation}\]

Combining this with the integrated Planck function:

\[\begin{equation} B(r)=\frac{\sigma T^4}{\pi} \end{equation}\]

The temperature gradient for radiation is then seen to be:

\[\begin{equation} \frac{dT(r)}{dr}=-\frac{3k_R\rho}{64\pi r^2 \sigma T^3}L(r) \end{equation}\]

Thus the temperature gradient that dictates radiative energy transport depends on a function of the luminosity. It is then important, for the later discussion of convection, that the variable commonly called the temperature gradient is defined; in this case, for radiative transport in hydrostatic equilibrium:

\[\begin{equation} \nabla_{rad}=\frac{d\ln T}{d\ln P}=\frac{3k_R}{64\pi r^2 g}\frac{P}{\sigma T^4}L(r) \end{equation}\]

Although this equation only pertains to radiation, it is clearly directly proportional to the opacity. At large enough opacities, the temperature gradient of radiative energy transport becomes so large that convection becomes an efficient mechanism for transferring energy in such a region of a star. The Sun possesses such a convective region, so determining the exact condition that decides whether a region is convective or radiative in terms of energy transport allows more accurate predictions to be made about the stellar interiors of other stars. Additionally, since this result was derived from the total integrated Eddington flux, the structure of the Sun and other stars must depend on the global properties of the radiation field (LeBlanc 2010, p 160). The next logical step is to examine the monochromatic flux in the stellar interior, although this does not have as large a role to play in the structure as the total integrated Eddington flux does.

2.1 Monochromatic Flux

Making the assumption that the luminosity is constant and equal to \(L(r)=4\pi R^2\sigma T_{eff}^4\) outside of the core where energy production due to nucleosynthesis occurs, the monochromatic flux in the stellar interior is given by (LeBlanc 2010, p 164):

\[\begin{equation} H_\nu=\frac{1}{16}\frac{k_R}{k_\nu}\frac{T^4_{eff}}{T^3}\bigg(\frac{R}{r}\bigg)^2\frac{dB_\nu}{dT} \end{equation}\]

Equation 8 is useful because, after using the substitution \(u=\frac{h\nu}{kT}\), the maximum of the total monochromatic Eddington flux can be determined independently of knowledge of the temperature \(T\) and the frequency \(\nu\). Since stellar models normally contain all the physical variables required, predominantly the Rosseland mean opacity \(k_R\), it is then possible to calculate the monochromatic opacity spectrum, and thus the total integrated Eddington flux. This is an extremely useful tool for describing radiative energy transport in the stellar interior; it allows verification of the accuracy and predictive power of a solar model, as it can be easily compared to the observed \(H_\nu\) of our Sun.

3 Solar Energy Production

As seen in equation 3, an important aspect of the global integrated flux is the temperature gradient, which in turn depends on the luminosity as a function of \(r\). The region of the Sun in which energy production occurs therefore needs to be examined. The luminosity at a given radius is given by (LeBlanc 2010, p 161):

\[\begin{equation} L(r)=\int_0^r 4\pi r^2 \rho(r)\varepsilon(r)dr \end{equation}\]

Here \(\rho(r)\) is the local density and \(\varepsilon(r)\) is the local nuclear energy generation rate. Outside the solar thermonuclear core, however, the nuclear energy generation rate goes to zero and the luminosity function remains constant, since only the core of the star is dense enough for nuclear fusion to occur. For the Sun, this region lies between its centre at \(r=0\) and approximately \(r=0.25R_\odot\) (LeBlanc 2010, p 161). Understanding energy production in the Sun therefore has a large bearing on the luminosity function, which in turn affects the temperature gradient and the total integrated Eddington flux of the model. Thus, to effectively model the Sun such that the observed flux is close to that predicted by the model, the energy production process at work must be understood. If the model does match the observed integrated Eddington flux, this shows that energy generation and energy transport in the Sun are well understood.

3.1 Proton-Proton Chains

In the thermonuclear cores of stars with masses approximately less than \(1.5M_\odot\) (such as our Sun), the dominant form of energy generation on the main sequence is the proton-proton chain. These thermonuclear reactions all begin with two protons reacting to form deuterium. There are three chains, all of which release approximately \(27.0\) MeV upon completion (Kippenhahn et al. 2012, p 193) through the formation of helium, equivalent to the mass defect between four hydrogen nuclei and a helium nucleus.

The first chain, PPI, dominates, making up 69% of proton-proton chain energy generation in the Sun. All three thermonuclear reaction chains produce neutrinos. Because neutrinos only interact via the weak interaction, they act as an energy sink: they do not play a role in energy transport in the Sun, since they simply do not interact with stellar plasma. On average (LeBlanc 2010, p 220), PPI loses 2.0%, PPII loses 4.0%, and PPIII loses 27.9% of their energy due to neutrinos simply not interacting with the star and escaping to space. The detection of solar neutrinos on Earth (Collaboration et al. 2011) has confirmed that this is the form of energy generation occurring in the Sun.

This gives further accuracy and precision to the radiative energy transport model seen in earlier sections, which depends on the energy generation rate in the Sun’s core. Because the probability of each chain occurring is known for the Sun, it is possible to determine an energy generation rate for all three chains by taking \(\varepsilon_{PP}\propto\varepsilon_{PPI}\) (Kippenhahn et al. 2012, p 194).

\[\begin{equation} \begin{split} \varepsilon_{PP}&=\frac{2.57\times10^4\psi f_{11}g_{11}\rho X_1^2}{T_9^{\frac{2}{3}}}e^{-\frac{3.381}{T_9^\frac{1}{3}}}\\ g_{11}&=1+3.82T_9+1.51T_9^2+0.144T_9^3-0.0114T_9^4 \end{split} \end{equation}\]

Here \(T_9=T\times10^9 \ \text{K}\), and \(f_{11}\) is the shielding factor for this reaction. The factor \(\psi\) corrects for additional energy generation in the PPII and PPIII branches, and is a function of \(T\). \(X_1\) is, of course, the mass fraction of hydrogen. This therefore gives a reliable model for energy generation in the Sun’s core, allowing models of radiative energy transport in the Sun to be very accurate. Using data from the study of nuclear physics found in (Angulo et al. 1999), a very accurate description of the nuclear production rate in the Sun’s thermonuclear core can be formulated and then tested by observing whether the spectrum produced is what we would expect.

4 Solar Convective Energy Transport

Unlike radiative energy transport, which relies solely on photons transporting energy, convective energy transport relies on cells of hotter mass in dynamically unstable regions rising and exchanging their excess thermal energy with the cooler surroundings. Convective transport occurs when the opacity is high, when the radiative flux is high, or both (LeBlanc 2010, p 168), all of which (as seen in equation 7) lead to a high temperature gradient. Under these conditions, convection becomes a more efficient energy transport mechanism than radiation. The Sun possesses a convective region, and therefore convective turbulence characterised by eddies of all sizes; this region is extremely dense, so the fluid is very compressible. This is an active area of research, and no complete theoretical framework for convective transport exists. There has been considerable effort to apply the Reynolds Stress models (Kippenhahn et al. 2012, p 61), which have had success in oceanography and atmospheric sciences, but these models are not ready to be easily used in stellar interior calculations because they are computationally very expensive. A useful approximation is therefore applied: the mixing-length theory, which has had great success in many fields of science and is applied to a surprisingly high degree of success in stars. The mixing-length theory approximates the eddies of all sizes as a single large eddy, and treats the fluid as incompressible (Canuto and Mazzitelli 1991).

By assuming that convection is carried out by rising and falling cells which do not exchange energy with their surroundings, except at the final point where they stop rising, it can be reasonably assumed that for a cell to rise, it must be more buoyant, and therefore less dense, than the surrounding medium. The process of the cell rising is adiabatic, so the mean molecular weight \(\mu\) does not change (LeBlanc 2010, p 169). As this is an adiabatic expansion, the pressure inside the cell is proportional to a density power law \(P\propto\rho^\gamma\), where \(\gamma\) is the ratio of the specific heats at constant pressure and volume. This relates the natural logarithm of the density in the adiabatic cell to the density in the radiative surrounding medium, which itself follows \(P\propto\rho T\).

\[\begin{equation} \bigg(\frac{d\ln\rho}{dr}\bigg)_{adi}=\frac{1}{\gamma}\bigg(\frac{d\ln\rho}{dr}\bigg)_{adi}<\bigg(\frac{d\ln P}{dr}\bigg)_{rad}-\bigg(\frac{d\ln T}{dr}\bigg)_{rad}=\bigg(\frac{d\ln\rho}{dr}\bigg)_{rad} \end{equation}\]

The rise of the cell is assumed to be adiabatic, so the gradients of pressure in the cell and the radiative medium are the same. It is therefore found that:

\[\begin{equation} \frac{\gamma-1}{\gamma}<\nabla_{rad} \end{equation}\]

Making the ideal gas assumption in the cell, it is seen that \(T\propto P^{\frac{\gamma-1}{\gamma}}\), resulting in:

\[\begin{equation} \frac{\gamma-1}{\gamma}=\nabla_{adi} \end{equation}\]

Therefore, the condition known as the Schwarzschild criterion, for which convective energy transport will most likely occur, is given by:

\[\begin{equation} \nabla_{rad}=\frac{3k_R}{64\pi r^2 g}\frac{P}{\sigma T^4}L(r)>\frac{\gamma-1}{\gamma}=\nabla_{adi} \end{equation}\]

The condition for convective transport to dominate is therefore clearly when there is a large flux or opacity.

It is certain that there is a convective energy transport region in the Sun, since subsurface convective motions are visible on the surface of the Sun in a phenomenon called solar granulation (Leighton 1963). This convective zone, which begins approximately 300 km below the visible photosphere, is due to the high opacity and resulting large temperature gradient caused by the ionisation of hydrogen (Leighton 1963). Ionisation raises the opacity because, when hydrogen is partially ionised, the bound-bound and bound-free radiative opacity of the medium is increased, since the excited atomic levels of the less charged ion become highly populated (LeBlanc 2010, p 168). Because the boundaries of these convective cells are observable on the solar surface, the cells used to model convective energy transport are themselves directly observable.

Different formulations of the mixing-length theory exist, and this is a very active field of research. Many groups are trying to produce more accurate and useful versions of this theory for stellar convection, as done by Canuto and Mazzitelli, who used observational data from the Sun to assess and improve their model (Canuto and Mazzitelli 1991). They also applied their reformulation of the mixing-length theory to the evolution of solar mass stars, so it is possible to extend the observational data to stars of a similar mass to our Sun and to see whether the model predicts the stellar evolution. A model for the convective energy transport within the Sun therefore helps predict the future evolution of the Sun.

By assuming that the total integrated Eddington flux (as seen in equation 4) is constructed from a radiative and a convective term, a function for the convective flux can be derived using the mixing-length theory. Using the temperature difference between the rising cell and the medium with respect to \(r\), the equation for the energy transported by the cell \(E=\rho c_P \Delta T\), and the knowledge that the convective flux is the energy density times the rate (i.e. the velocity at which this energy moves through a surface), it is seen that:

\[\begin{equation} F_{conv}=4\pi H_{conv}=\bar{V}E \end{equation}\]

It is then possible to derive (LeBlanc 2010, p 174):

\[\begin{equation} H_{conv}=\frac{\rho c_P \bar{V}T}{8\pi}\bigg(\frac{l}{\mathcal{H}}\bigg)(\nabla_{med}-\nabla_{cell}) \end{equation}\]

Here the variable \(\mathcal{H}\) is the pressure scale height in an isothermal gas, and is equal to the distance over which the pressure varies by a factor \(e\). The mixing-length \(l\) is the average distance travelled by one of the cells before it dissipates its energy and dissolves. Calculation of the parameter \(\frac{l}{\mathcal{H}}\) is an ongoing field of research, and the Sun is often the star used to help fit refinements of this theory (Canuto and Mazzitelli 1991). In a paper by Mazzitelli, several mixing-length theory models are evaluated based on data from the Sun. The author finds that the He abundance used in all these models is slightly higher than what is now suggested (Mazzitelli 1979), suggesting that for the Sun the helium abundance is in the range \(0.22\leq Y\leq0.24\), as opposed to the range \(0.25\leq Y\leq0.28\) used by two of the previous models that have been evaluated. The helium abundance found in Mazzitelli’s model is corroborated by Dziembowski et al., who used the p-modes of helioseismology to determine that the helium abundance in the convective envelope lies between \(0.22\leq Y\leq0.24\) (Dziembowski et al. 1991). Therefore, even given the limitations of the mixing-length theory, strikingly good predictions can be made about convective energy transport, which can be verified with observational solar data obtained by other methods.

Mazzitelli also suggests that a mixing-length larger than the commonly used value of unity (LeBlanc 2010, p 175) is numerically favourable, and that a range between \(1.5 \mathcal{H}\rho \leq l \leq2.0 \mathcal{H}\rho\) provides results that fit the Sun well. He does, however, highlight that such large factors for the mixing-length in superadiabatic convection are unphysical, and may not extend well to non-main-sequence stars (Mazzitelli 1979).

5 Conclusions

The Sun presents a nearby and useful object of study; the mechanisms that drive energy transport within it produce a number of directly observable quantities from Earth that we are simply unable to measure for other stars. Even given the assumptions and approximations used to determine the overall mechanisms of radiative and convective transport within the Sun, it is particularly useful to see that the global properties of the radiation field affect the global structure of the stellar interior. The ability to improve the precision of our knowledge of the total integrated Eddington flux by examining the monochromatic flux is also extremely useful.

As noted, the thermonuclear energy generation in the Sun’s core is clearly very important to energy transport, as seen by the role of the luminosity function in the temperature gradient. Knowledge of the energy generation, thanks to the advances of nuclear physics in the 20th century, means that theories about energy transport have extremely small uncertainties in relation to the energy generation component. This is in no small part due to the measurement of solar neutrinos, which has given observers direct knowledge about processes occurring in the Sun’s thermonuclear core.

Finally, convective energy transport, due to its complexity as a hydrodynamical process, warrants an essay in itself. The success of different formulations of the mixing-length theory in making predictions, such as the helium abundance in the convective layer, which have then been supported by observational data, is deeply impressive for such an approximation. A future paper would review the current state of the art in mixing-length theories whilst also examining newer hydrodynamical models of the Sun’s convective zone that rely on detailed numerical analysis and the development and rise of supercomputers. Specifically, it would assess whether the new models created in the Reynolds stress form are actually currently superior to sophisticated mixing-length theories, both in terms of fitting the observed solar data and in terms of computational expenditure.

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