1 Effect of varying the coronal density.

Figure 1: Flow velocity, Mach number and proton potential energy profiles for four different values of the number of particles $N$ in the system. >From bottom to top the velocity profiles correspond to $N=400, 784, 1600, 6400$, respectively. In all runs $\gamma =4$ and $m_{\rm p}/m_{\rm e}=400$. The normalizing velocity $v_0$ is the proton thermal velocity $v_{\rm p}(r_0)$. The Mach number is defined as the ratio of the radial bulk velocity of the plasma divided by the proton thermal speed $v_{\rm p\parallel}\equiv (2k_{\rm B}T_{\rm p\parallel}/m_{\rm p})^{1/2}$. The normalizing energy $\Psi _0$ is given by $\Psi_0=GMm_{\rm p}/r_0$. Thus, as a reference, if the charge and current neutralizing electric field was the Pannekoek-Rosseland potential $\Psi_{\rm p}/\Psi_0 \approx 0.5$ for $r\gg r_0$.

The number density $n$ and the collisionality of the simulated plasma is dependent on the number of particles $N$ used in the model. Fig. 1 shows the results of 4 simulations which only differ in the number $N$ of simulated particles, i.e. $N=400, 784, 1600, 6400$, the $N=400$ run being the one with the most tenuous (i.e. less collisional) atmosphere. The corresponding number densities at the base of the system are $n_0 [10^8 {\rm cm}^{-3}]=0.8, 1.5, 3.6$ and 13.4, respectively. The differences between the 4 simulations are substantial in many respects. The most evident difference is that the wind acceleration is much more efficient in the high density case, even though the thermal Knudsen number $K_T\equiv \overline\lambda_{\rm ep} \vert \partial \ln T_{\rm e\parallel}(r)/\partial r \vert$ (where $\overline\lambda_{\rm ep}\equiv v_{\rm e\parallel}/ \overline\nu_{\rm ep}$ is the electron-proton collisional mean free path based on the electron-proton rate of momentum exchange $\overline\nu_{\rm ep}$ (Landi & Pantellini, 2001, Appendix B), and where $v_{\rm e\parallel}\equiv (2k_{\rm B}T_{\rm e\parallel}/m_{\rm e})^{1/2}$ is the radial thermal electron velocity) is much smaller than unity for all runs, ranging from $10^{-3}$, for the densest case, to $10^{-2}$ for the most tenuous case. >From the figure it appears that the two more tenuous cases do not even become supersonic with respect to radial proton thermal velocity $v_{\rm p\parallel }\equiv (2k_{\rm B}T_{\rm p\parallel }/m_{\rm p})^{1/2}$ (which coincides with the fluid isothermal sound speed when $T_{\rm p\parallel}=T_{\rm e\parallel}$). The curves on the bottom panel of Fig. 1 illustrate the effect of collisions on the proton potential energy $\Psi_{\rm p}$. The latter results from the sum of the gravitational potential and a charge neutralizing electrostatic potential $\phi$:

$$\Psi_{\rm p} = -{{GMm_{\rm p}}\over{r_0}}\left({{r_0}\over{r}}-1\right) + e\phi(r)$$ (4)

where $e$ is the absolute value of the electron charge and where, because of the finite extent of the simulation domain we choose the level $r=r_0$ to be the reference level for both the gravitational potential and the electrostatic potential, rather than $r=\infty$. Thus $\Psi_j(r_0)=0$ by construction. The decreasing height of the potential barrier the protons have to overcome as the plasma collisionality increases is evident. In the least collisional case (dash-dot profiles in Fig. 1) the potential $\Psi_{\rm p}$ is a monotonically growing function of $r$. However, increasing the system's collisionality beyond a given threshold makes the potential $\Psi_{\rm p}$ become non monotonic, with the peculiar formation of a maximum a few solar radii above the bottom boundary at $r=r_\psi$. As already stated in the introduction the existence of a maximum in the proton potential has been suggested some time ago by Jockers (1970). Scudder (1996a) pushed a step farther by postulating $r_\psi$ to coincide with the position of the isothermal sonic point of Parker's fluid theory. Fig. 1 shows that when a maximum of $\Psi_{\rm p}$ exists, it is located above the sonic point, in agreement with the theoretical predictions (Meyer-Vernet et al., 2002). Fig. 1 also shows that the low density cases do neither produce a maximum in the proton potential nor a supersonic wind, at least if terms of the parallel temperature based Mach number. One may suspect that if the Mach number was defined with respect to the mean temperature $T\equiv (T_\parallel+2 T_\perp)/3$ the flow would more easily become supersonic at large distances because of the $T_\perp \propto r^{-2}$ dependence implied by the conservation of angular momentum in a collisionless plasma with negligible heat flux. In the end, however, given that in the collisionless limit and for $r \rightarrow \infty$, one has $T_\parallel \rightarrow {\rm const.}$ (e.g., Meyer-Vernet & Issautier, 1998), so that $T\rightarrow T_\parallel$, asymptotically. As a consequence, the distant Mach number does not depend on which of the two above definitions has been used.

In summary, the main consequence of increasing the plasma density beyond some threshold appears to be a reduction of the potential barrier the protons have to overcome in order to escape to infinity, accompanied by the formation of a local maximum in the $\Psi_{\rm p}(r)$ profile. As we shall see below the formation of the maximum in the proton potential energy is intimately related to the existence of both an outward directed, and radially decreasing heat flux, and a radially decreasing temperature profile. Both contribute in strengthening the outward directed electric field $\mathcal{E}=-\partial\phi/\partial r$, thus facilitating the extraction of the protons. In this context we shall remember that if the plasma was static (impermeable boundaries) with equal electron and proton temperatures, the charge neutralizing potential $\phi$ would be the celebrated Pannekoek-Rosseland potential (e.g., Rosseland, 1924)

$$\phi(r)=\phi_{\rm PR}(r)\equiv {{GM}\over{r_0}}\,{{m_{\rm ... ...{\rm e}}\over{m_{\rm p}}}\right) \left({r_0\over{r}}-1\right)$$ (5)

and the total potential energy of a proton would be a monotonically increasing function of $r$

$$\Psi_{\rm p}= -{{GM}\over{r_0}}\,{{m_{\rm p}}\over 2} \left(1+{{m... ..._{\rm p}}}\right) \left({{r_0}\over r}-1 \right)\;\;\;\; {\rm (static\; limit)}$$

asymptotically reaching the value $GM m_{\rm p}/(2 r_0)$ which is much higher than the values observed in the simulations (cf bottom panel of Fig. 1).