2 The method and assumptions

Adopting the interpretation in terms of the plasma emission from the upstream and downstream shock regions, the band-split frequencies map the electron densities behind and ahead of the shock front. In front of the shock (upstream region) the plasma is characterized by the electron density n₁ and emits radio waves at the frequency $f_{\rm L}$ (lower frequency branch; LFB). The plasma behind the shock (downstream region) is compressed to the density n₂ > n₁ corresponding to the frequency  $f_{\rm U} > f_{\rm L}$ (upper frequency branch; UFB). Defining the relative band-split (Fig. 1, see also Paper I) as:

$$% BDW=(f_{\rm U}-f_{\rm L})/f_{\rm L}$$ (1)

and bearing in mind $f\propto\sqrt n$, the density jump at the shock can be expressed as:

$$% X \equiv \frac{n_2}{n_1} = \left(\frac{f_{\rm U}}{f_{\rm L}}\right)^2=(BDW+1)^2.$$ (2)

Once the compression X is known, the Alfvén Mach number $M_{\rm A}$can be estimated (see Appendix). Note that in MHD shocks the condition X<4 is always satisfied (cf. Priest 1982), implying $f_{\rm U}/f_{\rm L}<2$ and BDW<1.

On the other hand, the emission frequency f can be transformed into the radial distance r by assuming some density distribution function n(r). Consequently, the shock propagation speed $v=\partial r/\partial t$ can be inferred from the frequency drift $\partial f/\partial t$, providing evaluation of the Alfvén velocity $v_{\rm A}=v/M_{\rm A}$. Finally, the magnetic field can be found using (m.k.s.): $B= v_{\rm A}\sqrt{\mu\rho}= 0.51\times10^{-17}fv_{\rm A}$, where the coronal plasma density is approximated as $\rho\approx m_{\rm p} n$, and $\mu=4\pi\times 10^{-7}$ H m^-1. A more practical expression reads:

$$% B~[{\rm gauss}]= 5.1\times 10^{-5}\times f~[{\rm MHz}]\times v_{\rm A}~\left[{\rm km~s^{-1}}\right].$$ (3)

The procedure is obviously model dependent: i) The inferred height and the velocity of the radio source depend on the coronal density model and on the angle $\phi$ between the direction of the source motion and the density gradient; ii) The relationship between the Alfvén Mach number $M_{\rm A}$ and the density jump X is a parametric function of the plasma-to-magnetic pressure ratio $\beta$ in the upstream region and the angle $\theta$ between the shock normal and the upstream magnetic field (see Appendix).

The emission frequency depends on the local electron density which can be a complicated function of space coordinates, especially in active regions. We note that most of the measurements are performed at frequencies corresponding to relatively large heights, well outside of the active region core. Therefore we assume that the plasma density depends only on the height. In particular, the Saito (1970) and Newkirk (1961) density models with various base densities are applied (see Fig. 8 in Appendix).

Radioheliographic observations indicate that the type II burst sources frequently do not propagate radially, especially at the onset of the burst (see, e.g., Nelson & Robinson 1975; Klassen et al. 1999; Klein et al. 1999). This means that the shock velocity v' is generally underestimated if the radial propagation is assumed. Accordingly, smaller values of the Alfvén speed and the magnetic field are obtained. For a given angle $\phi$ between the (radial) density gradient and the direction of the source motion, the true source velocity is found using $v=v'/{\rm cos}\phi$. Therefore, if $\phi ={\rm const.}$ the evaluation of true velocity is reduced to a simple multiplication of the model densities by factor $k=\cos\phi<1$. For example, the same speed is obtained by using the five-fold Saito density model and $\phi=60^{\rm o}$, or by applying the ten-fold Saito model and $\phi=0^{\rm o}$.

In a statistical sense the problem can be solved by introducing the mean angle $\overline\phi$ at which an "average source" moves through an "average corona". Assuming that most often the corona can be described by the two- to five-fold Saito model, we will present also the outcome for the ten-fold Saito model with $\overline\phi=0$ to represent the propagation in the five-fold Saito model corona under assumption of a strong, $\overline\phi =60^{\rm o}$, deviation from the radial propagation.

The MHD relationship between the density jump and the Mach number depends on the angle $\theta$ between the shock normal and the magnetic field direction. In the quasi-perpendicular regime between $\theta =90^{\rm o}$ and, say, $\theta =60^{\rm o}$ the outcome is only weakly dependent on the value of $\theta$ (see Fig. 9a in Appendix). Comparing the longitudinal, $\theta =0^{\rm o}$, and the perpendicular, $\theta =90^{\rm o}$, propagation one finds that the calculated values of $M_{\rm A}$ are 10-25% lower in the longitudinal case, the difference being larger for a larger band-split.

Another important parameter is the plasma-to-magnetic pressure ratio $\beta = 2\mu p/B^2$ in the upstream region (see Fig. 9b in Appendix). Highly structured patterns observed in the EUV wavelength range indicate that the coronal plasma is controlled by the magnetic field, i.e., $\beta\ll 1$ (for a discussion see Gary 2001). Since the relationship $M_{\rm A}(BDW)$ only weakly depends on the value of $\beta$ when it is smaller than $\approx$0.2 (see Fig. 9b in Appendix) we apply $\beta =0$. This approximation will be checked finally: after the B(r) dependence is established using some density model n(r), the assumption $\beta=2\mu nkT/B^2\ll 1$ can be verified taking for the coronal temperature T= 1-2  $\times\;10^6$ K (Chae et al. 2002).

In order to check how much various approximations affect the results, the analysis is carried out using several different values for all relevant parameters.

Figure 1: Measurements of $f_{\rm U}(t_i)$ and $f_{\rm L}(t_i)$ at the upper and lower frequency branch (UFB and LFB, respectively) of the harmonic band in the type II burst of 21 February 1999. LFB and UFB of the band-splitting are indicated by the lines following the emission ridges and defining the frequencies $f_{\rm L}$ and $f_{\rm U}$. Vertical lines show the times ti at which $f_{\rm L}$ and $f_{\rm U}$ are measured.