Improved Determination of the Location of the Temperature Maximum in the Corona

Abstract

The most used method to calculate the coronal electron temperature [$T_{\mathrm{e}} (r)$] from a coronal density distribution [$n_{\mathrm{e}} (r)$] is the scale-height method (SHM). We introduce a novel method that is a generalization of a method introduced by Alfvén (Ark. Mat. Astron. Fys. 27, 1, 1941) to calculate $T_{\mathrm{e}}(r)$ for a corona in hydrostatic equilibrium: the “HST” method. All of the methods discussed here require given electron-density distributions [$n_{\mathrm{e}} (r)$] which can be derived from white-light (WL) eclipse observations. The new “DYN” method determines the unique solution of $T_{\mathrm{e}}(r)$ for which $T_{\mathrm{e}}(r \rightarrow \infty) \rightarrow 0$ when the solar corona expands radially as realized in hydrodynamical solar-wind models. The applications of the SHM method and DYN method give comparable distributions for $T_{\mathrm{e}}(r)$. Both have a maximum [$T_{\max}$] whose value ranges between 1 – 3 MK. However, the peak of temperature is located at a different altitude in both cases. Close to the Sun where the expansion velocity is subsonic ($r < 1.3\,\mathrm{R}_{\odot}$) the DYN method gives the same results as the HST method. The effects of the other free parameters on the DYN temperature distribution are presented in the last part of this study. Our DYN method is a new tool to evaluate the range of altitudes where the heating rate is maximum in the solar corona when the electron-density distribution is obtained from WL coronal observations.

Appendix: Derivation of the Differential Equation Giving the DYN Temperature Distribution

Let us consider that the electrons and all ion species have the same average bulk velocities [$\boldsymbol{u}_{j}$], which are all equal to the (single-fluid) plasma bulk speed [$\boldsymbol{u}$]. Note that this postulate is made in single-fluid hydrodynamic models the solar wind: it implies diffusive equilibrium in the multi-component coronal plasma. Although this might be considered as a questionable hypothesis, it will also be adopted here, knowing that it can be relaxed in future more sophisticated formulations.

Furthermore, let us assume that the coronal and solar-wind plasmas are homogeneous, and that $\alpha = n_{\alpha}/n_{\mathrm{p}}$ is a free input parameter (independent of $r$ and time). Under such simplifying assumptions the radial component of the bulk velocity can be determined from the flux conservation equation,

$$ u(r) = u_{\mathrm{E}} \bigl[A_{\mathrm{E}}/A(r)\bigr] \bigl[n_{\mathrm{E}}/n_{\mathrm{e}} (r)\bigr], $$

(18)

where $A(r)$ is the cross-section flow tubes, and $n_{\mathrm{e}}(r)$ is the radial distribution that is obtained from WL eclipse observations; $n_{\mathrm{E}}$ is the electron density measured at 1 AU. The subscript “E” designates values at the Earth’s orbit ($r = 215 \,\mathrm{R}_{\odot}$).

Following Kopp and Holzer (1976), it has become a common practice to approximate $A(r)$ by an exponential function of $r$:

$$\begin{aligned} A(r)/A(1) =& r^{2} f(r) \\ =& r^{2} \bigl\{ f_{\max} \exp\bigl[(r - r_{1})/ \sigma\bigr] + f_{1}\bigr\} / \bigl\{ \exp\bigl[(r - r_{1})/ \sigma\bigr] + 1\bigr\} , \end{aligned}$$

(19)

where $f_{1} = 1 + (1 - f_{\max}) \exp[(1 - r_{1})/\sigma]$; where $f_{\max}$ is the total geometrical expansion rate, $r_{1}$ is the heliocentric distance of most rapid geometrical expansion, and $\sigma$ is the range over which this takes place in the corona. When $f_{\max} =1$, then $f(r) \equiv 1$ and the radial expansion is then recovered, as in early hydrodynamic and kinetic models of the solar wind.

When the average bulk velocities of the electrons and ions are all equal to $u$, and they are determined by Equation (18), the radial component of the steady-state momentum-transport equations for electrons, protons, and $\alpha$-particles are, respectively,

$$\begin{aligned} n_{\mathrm{e}} \mathrm{m}_{\mathrm{e}} u \mathrm{d}u/ \mathrm{d}r + \mathrm{d}p_{\mathrm{e}}/\mathrm{d}r =& - n_{\mathrm{e}} \mathrm{m}_{\mathrm{e}} g - n_{\mathrm{e}} \mathrm{e} E, \end{aligned}$$

(20)

$$\begin{aligned} n_{\mathrm{p}} \mathrm{m}_{\mathrm{H}} u \mathrm{d}u/ \mathrm{d}r + \mathrm{d}p_{\mathrm{p}}/\mathrm{d}r =& - n_{\mathrm{p}} \mathrm{m}_{\mathrm{H}} g + n_{\mathrm{p}} \mathrm{e} E, \end{aligned}$$

(21)

$$\begin{aligned} 4 n_{\alpha} \mathrm{m}_{\mathrm{H}} u \mathrm{d}u/ \mathrm{d}r + \mathrm{d}p_{\alpha}/\mathrm{d}r =& - 4 n_{\alpha} \mathrm{m}_{\mathrm{H}} g + 2 n_{\alpha} \mathrm{e} E, \end{aligned}$$

(22)

where $g$ is the gravitational acceleration [$g(r) = g_{\mathrm{o}}/r^{2}$] and $E(r)$ is the outward-directed polarization electric field that accelerates the positive charges to supersonic velocities in the SW. Parker (2010) has called this charge-separation electric field that is generated inside coronal plasma by the charge separation the “Lemaire–Scherer” electric field, since Lemaire and Scherer (1970, 1973) emphasized its role in driving the SW expansion.

Concatenation of Equations (20), (21), and (22) leads to the familiar (single-fluid) hydrodynamic momentum equation adopted in hydrodynamic models,

$$ \rho u \mathrm{d}u/\mathrm{d}r + \mathrm{d}p/\mathrm{d}r = \rho g. $$

(23)

The total electrostatic force density [$\Sigma_{j} \mathrm{Z}_{i} n_{j}\mathrm{e} E$] has disappeared from Equation (23) because of the neutrality condition: $n_{\mathrm{e}} = n_{\mathrm{p}} + 2 n_{\alpha}$, or $n_{\mathrm{e}} = n_{\mathrm{p}} (1 + 2 \alpha)$.

The mass density [$\Sigma_{j} n_{j} \mathrm{m}_{j}$] is equal to $\rho = n_{\mathrm{p}} \mathrm{m}_{\mathrm{H}} + 4 n_{\alpha} \mathrm{m}_{\mathrm{H}} + n_{\mathrm{e}} \mathrm{m}_{\mathrm{e}} \approx (n_{\mathrm{p}} + 4 n_{\alpha} ) \mathrm{m}_{\mathrm{H}} = n_{\mathrm{p}} (1 + 4 \alpha) \mathrm{m}_{\mathrm{H}} = n_{\mathrm{e}} \mu_{\mathrm{e}} \mathrm{m}_{\mathrm{H}}$ where

$$ \mu_{\mathrm{e}} = (1 + 4 \alpha)/(1 + 2 \alpha). $$

(24)

In Equation (23), the total kinetic pressure [$\Sigma_{j} p_{j} = \Sigma_{j} n_{j} \mathrm{k} T_{j}$] is given by $p = p_{\mathrm{e}} + p_{\mathrm{p}} + p_{\alpha} = \nu_{\mathrm{p}} n_{\mathrm{e}} \mathrm{k} T_{\mathrm{e}}$ where

$$ \nu_{\mathrm{p}} = (1+ 2 \alpha + \tau_{\mathrm{p}} + \alpha \tau_{\alpha})/(1 + 2 \alpha) $$

(25)

and where $\tau_{\mathrm{p}(\alpha)} = T_{\mathrm{p}(\alpha)}/T_{\mathrm{e}}$ are the ratios between the proton or $\alpha$ temperature and electron temperature; these dimensionless constants are arbitrary input parameters. The mean molecular mass of the plasma is given by $\mu = \mu_{\mathrm{e}}/\nu_{\mathrm{p}}$.

When the temperature ratios [$\tau_{\mathrm{p}}$ and $\tau_{\alpha}$] are independent of $r$, the values of $\mu_{\mathrm{e}}$ and $\nu_{\mathrm{p}}$ are also independent of $r$. Since the plasma is assumed to be homogeneous, $\alpha$ is independent of $r$. Of course, all of these simplifying assumptions could be relaxed, but at this stage they will serve to investigate the gross influence that these parameters have on the DYN temperature distributions. The results presented here are exploratory steps awaiting more comprehensive observations.

Since the mass of the electron is much smaller than the ion masses, Equation (20) can be approximated by

$$ \mathrm{k} T_{\mathrm{e}} \mathrm{d} \ln n_{\mathrm{e}}/ \mathrm{d}r + \mathrm{d} (\mathrm{k} T_{\mathrm{e}})/\mathrm{d}r \approx - \mathrm{e} E. $$

(26)

This equation allows us to determine the distribution of the Lemaire–Scherer polarization electric field [$E(r)$] since the distribution of $n_{\mathrm{e}}(r)$ is available from WL coronagraphic observations, and $T_{\mathrm{e}}(r)$ can be calculated by using the DYN method introduced in Section 7.

Replacing $\rho$ and $p$ by the expressions given above, and assuming that $\mu_{\mathrm{e}}$ and $\nu_{\mathrm{p}}$ are constants independent of $r$, Equation (23) becomes

$$ n_{\mathrm{e}} \mu_{\mathrm{e}} \mathrm{m}_{\mathrm{H}} u \mathrm{d}u/\mathrm{d}r + \nu_{\mathrm{p}} \mathrm{k} T_{\mathrm{e}} \mathrm{d}n_{\mathrm{e}}/\mathrm{d}r + \nu_{\mathrm{p}} n_{\mathrm{e}} \mathrm{k} \mathrm{d}T_{\mathrm{e}}/\mathrm{d}r = - n_{\mathrm{e}} \mathrm{m}_{\mathrm{H}} \mu_{\mathrm{e}} g_{\mathrm{o}}/r^{2}. $$

(27)

Using Equation (24), (25), and $\mu = \mu_{\mathrm{e}}/\nu_{\mathrm{p}}$, Alfvén’s “normalization temperature” is determined by

$$ T^{*} = \mathrm{G} \mathrm{M}_{\odot} \mathrm{m}_{\mathrm{H}} \mu_{\mathrm{e}}/\nu_{\mathrm{p}} \mathrm{k}\, \mathrm{R}_{\odot} = g_{\mathrm{o}} \mathrm{R}_{\odot} \mathrm{m}_{\mathrm{H}} \mu/ \mathrm{k}. $$

(28)

This expression depends on the concentration of $\mbox{He}^{++}$ ions, and on the ratios between the temperatures of the ions and the electron temperature: $\tau_{\mathrm{p}(\alpha)} = T_{\mathrm{p}(\alpha)}/T_{\mathrm{e}}$.

Using the solar radius as the unit for $r$, and $y = T_{\mathrm{e}}/T^{*}$, Equation (27) becomes

$$ \mathrm{d}y/\mathrm{d}r + y \mathrm{d}\ln n_{\mathrm{e}}/ \mathrm{d}r = - \bigl(1/r^{2}\bigr) + (\mathrm{m}_{\mathrm{H}} \mu/ \mathrm{k} T^{*}) u \mathrm{d}u/\mathrm{d}r. $$

(29)

The left-hand side of Equation (29) corresponds to the hydrostatic-pressure gradient. The first term in the right-hand side corresponds to the downward gravitational force, and the last term corresponds to the inertia force, which determines the outward acceleration of coronal plasma.

The general solutions of this first-order differential equation diverge when $r \rightarrow \infty$. The solution of this equation for which $y(r \rightarrow \infty) = 0$ is determined by

$$ y = \bigl[1/n_{\mathrm{e}} (r)\bigr] \int_{\infty}^{r} \bigl\{ n_{\mathrm{e}} \bigl(r '\bigr)/r^{\prime2} \bigr\} \bigl[1 + F\bigl(r '\bigr)\bigr] \mathrm{d}r ', $$

(30)

where $F(r)$ is the ratio of the radial components of the inertial acceleration and the gravitational deceleration

$$ F(r) = (\mathrm{m}_{\mathrm{H}} \mu_{\mathrm{e}}/ \nu_{\mathrm{p}} \mathrm{k} T^{*}) r^{2} u \mathrm{d}u/\mathrm{d}r = r^{2} (u \mathrm{d}u/\mathrm{d}r)/g_{\mathrm{o}} \mathrm{R}_{\odot}. $$

(31)

The function $F(r)$ is determined by Equations (18) and (19), because the distribution of $n_{\mathrm{e}}(r)$ is determined as a function of $r$ by Equation (24). Since $r^{2} (u \mathrm{d}u/\mathrm{d}r)$ tends to zero for $r \rightarrow \infty$, $\lim F(r \rightarrow \infty) = 0$.

It can be seen that Equation (30) is a generalization of the hydrostatic solution; indeed, when $u(r) \equiv 0$, $F(r) \equiv 0$, and Equation (30) is then identical to Equation (28) of the main text.

Thus the solution in Equation (30) introduced in this article should be used to determine electron-temperature distributions when the solar corona expands radially, instead of being in hydrostatic equilibrium as was assumed before 1958. Of course, close enough to the base of the solar corona, where the expansion velocity is small (subsonic), the HST temperatures are fully appropriate, while the SHM temperatures are not.

Further generalizations of Equation (30) can be envisaged by assuming anisotropic pressures and temperatures for the coronal ions and electrons, as well as non-uniform distributions for the relative concentrations of the minor ions [$\alpha = n_{\alpha} (r)/n_{\mathrm{p}}(r)$] or of the temperature ratios [$\tau_{\mathrm{p}(\alpha)} = T_{\mathrm{p}(\alpha)}/T_{\mathrm{e}}$].