The fundamental theories of reconnection outlined previously rely on 2D models of reconnection, which are an idealised version of the magnetic field in the solar corona. 3D reconnecting fields are more representative of the mechanisms taking place, and so are a field of increasing interest, with clear distinctions being made between 2D and 3D reconnection theory (e.g., Birn & Priest, 2007). As such, we will now outline the background behind models of acceleration during reconnec- tion at a 3D null point as described by Dalla & Browning (2005) and Browning
et al. (2010), which is of particular relevance to the work presented in Chapter 7. In these models, a single–particle approach is taken. In order to determine the primary forces acting on a charged particle, we begin again with our equation of motion (Equation 2.15), excluding all forces but the Lorentz force, and assuming a negligible electric field such that m(dv/dt) = q(E+v×B). By taking the dot
into these two components also: dvk dt = 0 (2.46) dv⊥ dt = q m (v⊥×B) (2.47)
The former equation demonstrates that the velocity along the magnetic field is unaffected by this force, while the latter equation describes an acceleration that forces particles to orbit a guiding centre. As a result, charged particles can be highly magnetised, and therefore are treated with the guiding–centre ap- proximation, which approximates particle motion as directly along these centres, neglecting gyration.
A force which works against the magnetisation of particles is the electric drift force, which imparts a velocity of the particle perpendicular to the guiding centre in an electric fieldE and magnetic field B field of
vE =c
E×B
This results in a radius of gyration, or Larmor radius of
rL =
mc2E
qB2 . (2.49)
The degree of magnetisation, which determines whether the drift force is enough to discount the guiding centre approximation is therefore
= rL L =
mc2E
qB2L. (2.50)
In the solar corona, on global length scales, it is often the case that << 1, and therefore particles can be safely treated to move along their guiding centres. However, this approximation is shown not to hold close to magnetic null points.
A simple 3D magnetic null point can be defined by the magnetic field B = (Bx, By, Bz), where Bx =B0 x L (2.51) By =B0 y L (2.52) Bz =−2B0 z L. (2.53)
where B0 is the magnetic field strength near the null, andL is the characteristic length scale. A visual example of this geometry is given in Figure 2.8 Here, the null point is located at (0,0,0), where each component of the magnetic field linearly approaches zero. Here, the degree of magnetisation, asymptotically approaches infinity, and so the guiding centre approximation rapidly becomes insufficient, and the full equation of motion must be used to determine particle
Figure 2.8: Geometry of a magnetic null point. a) Sampled magnetic field lines based on Equations 2.51 - 2.53. b)During reconnection, inflow and outflow respec- tive to the null point are expected, as highlighted in the 3D shaded regions here. (Dalla & Browning, 2005).
motion.
Models using this setup have shown, for a random distribution of test par- ticles, that electrons are efficiently accelerated to energies as high as 100 keV. Furthermore, the steady–state distribution of particles was found to be that of a power–law (Browning et al., 2010), as shown in Figure 2.9. For a magnetic field strength of both 100 G and 20 G, the distribution rapidly evolves from a thermal distribution to a nonthermal power–law, as indicated by flare HXR inversions.
2.2.2
Density Models and Nonuniform Ionisation
Regardless of the initial acceleration mechanism or location primary energy re- lease, the density, temperature and ionisation fraction of the corona and chromo-
Figure 2.9: Model evolution of a proton spectrum accelerated at a null point. On the left, evolution of the spectrum is shown (from black to blue to green to orange to red) for a magnetic field of 100 G, with the initial (black) spectrum representing the chosen initial distribution. On the right, the evolution is shown with the same colour evolution in the case of a magnetic field of 20 G (Browning et al., 2010).
sphere are still crucial in understanding their role as a target in the HXR and radio emission mechanisms outlined in the following section. As such, we outline the current solar atmospheric models here, for quiet–sun conditions as well as in flaring active regions.
The most commonly used models of solar density and temperature are the VAL (Vernazza et al., 1981) and FAL (Fontenla et al., 1990) models. The VAL model is produced using the full radiative transfer and hydrodynamic modelling of Fraunhofer lines detected in the Sun by the Skylab instrument. As a result of fitting emission from H, H−, C, Si, Fe, Mg, Al, He, He II, Ca II, Mg II among others, a model distribution over height of plasma temperature and density were produced, as shown in Figure 2.10. This model provides a basis for the target of a nonthermal electron beam impacting on a quiet–Sun plasma. However, to account for the enhanced and redistributed density in active and flaring regions, more specific models have been developed.
Figure 2.10: The VAL model of the solar atmosphere. Shown are the temperature (solid curve) and mass density (dashed curve) variation with height above the solar surface, which is on the right of the plot. Notable are the sharp drop in temperature and rise in density at the transition region, followed by the roughly exponential rise in density with depth into the chromosphere, where the temperature reaches its minimum (Vernazzaet al., 1981).
As shown in Section 2.3.1, and expanded further in Chapter 4, the collisional thick target model can be used to predict HXR source heights from HXR spectra when using an input density model such as those outlined here. Conversely, HXR spectra can be informed by images, which provide source locations, to produce empirical density models (e.g., Liu et al., 2006). Saint-Hilaire et al.
Figure 2.11: Statistically derived density structure of the flaring solar atmo- sphere. The black solid line represents the density structure based on a fit to results of a statistical RHESSI study based on HXR spectra and locations (Saint- Hilaireet al., 2010).
(2010) outline a survey, performing this technique on 838 flares in order to produce a model density structure of an ‘average flaring chromosphere’, shown in Figure 2.11. In this study it was found that the atmosphere is well described by a double–exponential density structure, with a scale–height of 131 ± 16 km in the chromosphere, and 5.4 ± 0.6 Mm in the corona.
These target models have been used as starting points in modelling the solar atmospheric response to an injection of nonthermal electrons. Allredet al.(2005) use the RADYN (Carlsson & Stein, 1992, 1997) suite of radiative and hydrody- namic models to determine how the temperature, density, and ionisation fraction
tra (Suet al., 2009). The effect of the local NUI presented in the models of Allred
et al.(2005) on the size and structure of RHESSI sources is explored in detail in Chapter 5.