The exospheric paradigm and related models

3.3 The exospheric paradigm and related models

As the consequence of rare collisions, the idea has been around since the early days of solar wind modelling to describe coronal expansion as a collisionless process, in which the corona kind of evaporates from an assumed exosphere. This exospheric model is based on the simplest approach that could be thought of to model analytically the solar wind: Consider neither collisions nor waves but only free protons and electrons that move in the gravitational field and interplanetary electric field, while being guided by the magnetic field. The work of Jockers (1970

) and Lemaire and Scherer (1971a ,b ) are classic references for this topic. In these kinetic exospheric models the exobase is defined as the altitude where the mean free paths of the coronal ions and electrons become larger than the barometric scale height. In reality, this transition is of course continuous, and thus the exosphere empirically is badly defined. After Parker’s work (Parker, 1963

), these models ran out of fashion. Yet in the recent past, they became fashionable again, although the importance of collisions and wave-particle interactions in the solar corona and solar wind is out of question more than ever. Certainly, the exospheric model contains some of the basic kinetic physics and only makes a few fundamental assumptions (for example about the VDFs at the exobase of the corona). It works well in predicting a supersonic solar wind. We give a short review of the exospheric paradigm to do justice to the existing literature. The reader who wants to know more is referred to Lemaire and Pierrard (2003 ) for a concise modern account of exospheric theory and the references therein.

In the old models, the exobase was usually located at a distance beyond $5– 10 R⊙$ . However, since the number density is lower in open coronal holes than closed regions of the magnetised solar atmosphere, the exobase has to be lower in coronal holes, more realistically perhaps at about $1.1 –5 R ⊙$ . At such distances, gravitational attraction is still larger than electric repulsion for protons. In the recent exospheric models (Lamy et al., 2003 ; Zouganelis et al., 2003 , 2004 ) a non-monotonic total potential energy for the protons was therefore assumed (as Jockers (1970 ) did already), and by lowering the altitude of the exobase below the maximum of the potential energy, an acceleration of the solar wind to high velocities was obtained. The profile with radial distance, $r$ , of the accelerating mean electric field, (which is to say of the electron partial pressure gradient) is the key ingredient of the models, besides the coronal magnetic field, guiding the particle motion through its Lorentz force. Yet, the essential characteristic of the new exospheric models is that it provides a driving mechanism for the fast solar wind through a strong electric field, which is largely set up by the suprathermal electrons in the VDF, $f (r ,v) e 0$ (assumed at the exobase with radius $r 0$ ), from which the interplanetary VDF can by means of Liouville’s theorem be constructed everywhere. Direct observational evidence for the existence of suprathermal electrons in the corona is still lacking.

Kinetic exospheric models assume that the charged particles move without collisions in the Sun’s gravitational field, the mean electric field, $E(r)$ , and the solar and interplanetary magnetic field, $B (r)$ , along trajectories that are determined by their total energy, $E$ , and pitch angle or magnetic moment, $μ$ . When accounting for a spiral field, the two basic conserved quantities therefore are:

where for simplicity a radial dependence on $r$ only was assumed. Here $Φe (r)$ is the selfconsistent electric potential, $Φg (r) = − GM ⊙ ∕r$ , the Sun’s gravitational potential, and $λ$ the heliographic latitude. The particle mass is $m$ and charge $q$ . The first term contributing to the constant $E$ in Equation (3

) is the kinetic energy in the rotating frame and the second the centrifugal energy. Solar rotation at a frequency $Ω ⊙$ at the equator leads to a co-rotation speed of $Ω ⊙R ⊙ = 2 km s−1$ . Further details of a collisionless solar wind model in a simple spiral magnetic field are discussed by Pierrard et al. (2001 ).

Figure 6:

Left: Electrostatic interplanetary potential from the exobase (at $2 R ⊙$ ) out to $215 R⊙$ in an exospheric model with kappa VDFs of the solar wind consisting of protons and electrons, with base temperature, $Te(r0) = Tp (r0) = 106 K$ and $κe = 2.5$ . Right: Module of the ratio of outward directed electric force, $qE (r)$ , and inward directed gravity, $m g(r) p$ , acting on a proton. This ratio in the solar wind is plotted versus radial distance (after Pierrard et al., 2004

Even including the field curvature does not heal a main problem of exospheric theory, which is that the thermal anisotropies of ions and electrons come out too large, inconsistent with in situ measurements, which for their explanation require ion scattering by waves and collisions (as discussed in subsequent sections). For the magnetic field model a simple Parker spiral of the form

may be assumed. Then the effective potential,

which corresponds to the magnetic mirror force, the gravitational attraction and the attraction (for electrons) or repulsion (for positive ions) due to the ambipolar electrostatic potential, is the key quantity in (steady state) exospheric theory and fully determines the energetics of the coronal expansion and solar wind, together with the VDF at the exobase, $f(r0,v )$ . The model VDF is assumed to be given at the exobase at $r 0$ , and is then according to Liouville’s theorem determined everywhere in corona and heliosphere. In the exospheric paradigm, it is the electric field that drives the expansion and drags out the ions against gravity.

The electric potential and the force ratio are plotted versus distance from the Sun in Figure 6 , which was taken from the model of Lamy et al. (2003 ) and Pierrard et al. (2004 ). According to simple fluid theory, when fully neglecting all terms of the order of the electron mass, one finds that $− e0neE (r) = − d∕dr(ne(r)kBTe(r))$ (with the electric charge unit $e0$ ). The ambipolar electric field is given by the electron partial-pressure gradient, which can be made large either by very hot bulk electrons (for which however there is no observational evidence, see David et al., 1998 ), or by sizable tails of suprathermal electrons (for which there is indirect evidence from the deviation from ionization equilibrium see, e.g., Esser and Edgar, 2000 ). The existence of such VDF was first proposed by Scudder (1992a ,b ), to explain merely by collisionless electron kinetics the high temperature of the corona. These extended electron tails are required to set up a sufficiently strong electrostatic potential in the corona.

Such VDFs can be described as $κ$ -functions or generalised Lorentzians, and are a key property of the modern exospheric model VDFs that are assumed to exist at the exobase, which was assumed to be located at $r = 2 R 0 ⊙$ in the recent models, e.g., Pierrard and Lemaire (1996 ). The non-thermal $κ$ -VDF (see the paper of Maksimovic et al., 1997a ) reads as a function of the particle speed $v$ as follows:

with the equivalent thermal speed $v κ$ . The temperature is as usually calculated as the second moment of the VDF (with number density $n$ ) and reads: $2 kBT κ = m < v > ∕3$ . The symbol $Γ (x)$ denotes the gamma function. For $κ → ∞$ one retains a Maxwellian. The value of $κ$ determines the slope of the energy spectrum of the suprathermal particles, and gives the exponent of the power-law tail for $v > > vκ$ , where $f (v) ∼ v−2(k+1)$ . Small $κ$ values (e.g., smaller than $3$ ) mean a hard spectrum.

Figure 7 shows various examples of kappa functions that are plotted in the left frame (and are extracted from the paper by Maksimovic et al., 1997a ). The logarithm of the VDF is given so that a Maxwellian appears as a parabola. One can see the appearance of extended high-energy tails for low values of $κ$ . In the right frame a classical electron VDF as measured in the solar wind is shown after Feldman et al. (1975 ). The diamonds represent the measurements, while the dashed lines represent a double-Maxwellian core/halo model fit, which will be explained in more detail below. The continuous line represents the VDF according to Equation (6 ) for $κ = 4$ . Note, however, that the detailed pitch-angle distributions observed in situ often reveal distinct anisotropies which are not well described by $κ$ -functions. Some typical examples of measured electron VDF are given in Figure 2 in Section 2, where it is also shown that the suprathermal electrons in fast solar wind are best represented by an isotropic halo and a highly anisotropic strahl which is the primary carrier of the heat flux.

The restriction to isotropy is particularly questionable for the weakly collisional coronal electrons that are moving in the non-uniform, mirror-type or flux-tube-like, configurations of the coronal magnetic field. Furthermore, an obvious yet unanswered question is what physical process generates in the low corona or at the exospheric base a $κ$ -function in the first place? Several authors, such as Collier (1993 ), Roberts and Miller (1998 ), Viñas et al. (2000 ), and Leubner (2002 ) have made proposals for kinetic processes that could produce such distributions in the chromosphere and corona.

Figure 7:

Left: Different examples of $κ$ -functions (after Maksimovic et al., 1997a

), all normalised to unity at $v = 0$ . Obviously, in the limit $κ → ∞$ , these functions transform into a Maxwellian or Gaussian (solid line). Right: Measured electron VDF (after Feldman et al., 1975

) in the solar wind (diamonds). The dashed lines correspond to the classical model VDF, being composed of two Maxwellians: a core with $− 3 nc = 30.8 cm$ and $5 Tc = 1.6 × 10 K$ , and a halo with $− 3 nh = 2.2 cm$ and $Th = 8.9 × 105 K$ . The full line represents the $κ$ -VDF model fit with $n = 33.9 cm −3$ , $T κ = 1.9 × 105 K$ and $κ = 4$ .

Taking these functions for granted, exospheric models based on them were in the past years developed. The results presented by Maksimovic et al. (1997a ,b ) indicated that basic features of the fast solar wind can indeed be explained, if the exospheric electron VDFs in coronal holes have enhanced tails, which result in a sufficiently strong electric field accelerating the protons (see Figure 6 again). Quantitatively speaking, for an electron base temperature of $6 Te(r0) = 2 × 10 K$ , the bulk speed at $1 AU$ is about $400$ , $500$ , and $− 1 800 km s$ for assumed kappa-values of $6$ , $3$ , and $2$ , the latter value corresponding to a huge, presumably unrealistic reservoir of suprathermal electron energy in the lower corona.

Pierrard et al. (2004 ) recently investigated also the acceleration of heavy solar ions on the basis of an exospheric Lorentzian model and showed that heavy ions can flow faster than protons if their temperatures in the corona are more than proportional to their masses. The $κ$ -function kinetic exospheric model (Pierrard and Lemaire, 1996 ), initially developed only for electrons and protons, was generalised to the case of a non-monotonic effective potential energy, $Ψ(r)$ , for heavy coronal ions and solar wind ions (see Figure 6 for the relevant radial profiles).

Pierrard et al. (2004 ) showed that the ion velocity filtration effect can lead to very hot ions in the solar corona, given the ion VDF had enhanced suprathermal tails in the low corona. For sufficiently high ion temperatures at the exobase, located at $2 R ⊙$ , their exospheric model could account for the high bulk speeds of the heavy ions in fast solar wind ions at $1 AU$ . The assumed $κ$ -value was $2.1$ , resulting in very high coronal kinetic temperatures, ranging in $MK$ units between from about $60$ for $2+ He$ , to $250$ for $6+ O$ , and up to $950$ for $12+ Fe$ . These are very high ion temperatures, way beyond what was typically inferred from spectroscopic data obtained by SOHO. Inspection of the simple energy constraint (2 ) shows that (for $γ = 5∕3$ , as it should be for a monoatomic gas) a heavy ion of species $i$ must have an effective coronal temperature scaling, such that $T ∼ m C i$ , because only then its coronal thermal speed is of the order of or larger than the escape speed of $−1 618 km s$ from the solar surface.

The new exospheric kinetic models claim to predict the fast solar wind without assuming an unreasonably large $TC$ , and without additional heating of the outer region of the corona, as it is needed in hydrodynamic models to achieve the same solar wind speed through pressure gradient forces. However, the problem of coronal heating has only been circumvented, and it severely reemerges at the exospheric boundary in the new guise of the unknown origin of the crucial suprathermal particles. In their recent collisionless transonic model of the solar wind Zouganelis et al. (2003 , 2004 ) presented parametric studies showing that a high terminal wind speed does not depend on the details of the non-thermal electron VDF, but is claimed to be a robust outcome of a sufficient number of suprathermal electrons. For example, a core-halo electron VDF as occurring in interplanetary space (shown previously in the right frame of Figure 7 ) was assumed to exist in the low corona.

Figure 8:

Left: Electron temperature profile in the corona in an exospheric model with kappa function tails of the VDF. The kappa values are indicated at the different lines. The dashed vertical line corresponds to $1 AU$ . Only $κ$ larger than about $5$ is compatible with empirical constraints. Right: Contours of the terminal (at $1 AU$ ) solar wind speed in $km s−1$ for an electron VDF composed of core and halo Maxwellians. The contours are shown as a function of the relative density, $α0 = nh(r0)∕nc(r0)$ , and the temperature ratio, $τ0 = Th(r0)∕Tc(r0)$ , of the electron core and halo at the exobase, being located on the solar surface, $r0 = R ⊙$ (after Zouganelis et al., 2004

The left frame of Figure 8 from the paper of Zouganelis et al. (2004 ) shows that low values of $κ$ yield enhanced coronal electron temperatures reaching a maximum of $7 × 106 K$ within a few solar radii. This maximum gets smaller for larger values of $κ$ . This temperature increase is a direct consequence of the velocity filtration effect (Scudder, 1992a ,b ), but is not observed, which suggests that $κ$ -VDFs with strong suprathermal tails are not adequate for the corona. One may take double-Maxwellian instead, as was done by Zouganelis et al. (2004 ), with the exobase being located at the Sun’s surface, $r0 = R ⊙$ . The right frame of Figure 8 shows their model results for a sum of two Maxwellians, core and halo. The diagram gives contours of the terminal solar wind speed as function of $α = n (r )∕n (r ) 0 h 0 c 0$ and $τ = T (r )∕T (r ) 0 h 0 c 0$ . One can see that this kind of non-thermal VDF can explain the fast solar wind ( $−1 700– 800 km s$ ) for a dense and hot enough halo, corresponding to substantial number of suprathermal coronal electrons. If the empirical in situ values of $α = 0.04$ and $τ = 7$ , as observed by Ulysses (McComas et al., 1992 ), also prevail in the lower corona, one must come to the conclusion that the resulting electron tails are too weak to accelerate the solar wind to the observed high speeds.

Finally, we would like to mention the recent work by Zouganelis et al. (2005 ) who demonstrated that both approaches, the exospheric and collisional (modelled by direct particle simulation), yield a similar variation of the wind speed with the basic model parameters. In other words, a proper inclusion of collisions in an exospheric approach does not change the effects that suprathermal particles may have on solar wind acceleration.

The new exospheric models, implying the acceleration of coronal ions through the average ambipolar electric field set up by suprathermal electrons, work in principle but seem to fail in providing the high terminal speeds observed in fast streams. To achieve this end requires, as is illustrated in Figure 8 , unrealistically large tails in the VDFs and/or high electron coronal temperatures, which are not consistent with known empirical constraints (Marsch, 1999 ). The consequence therefore has been to look for a direct acceleration of the ions through either their partial pressure gradient force (set up, e.g., by wave heating) when speaking in fluid terms, or kinetically and directly through wave-particle interaction forces, i.e., ultimately by micro-turbulent or coherent-wave electromagnetic fields, yielding the proper Lorentz force in the particles’ frame. This way seems to be the most promising remaining alternative.