Appendix

In the presented analysis we have used different approximations and assumptions. In Fig. 8 the density models used are shown. The two-fold Newkirk, two-, five-, and ten-fold Saito models are denoted in the legend of Fig. 8 as 2xNewk., 2xSaito, 5xSaito and 10xSaito.

Figure 8: The coronal density models used.

The relationship between the downstream/upstream density jump X(compression) and the Alfvén Mach number $M_{\rm A}$ depends on the plasma-to-magnetic pressure ratio $\beta$ and the angle $\theta$between the shock normal and the upstream magnetic field. For the oblique shock the Alfvén Mach number $M_{\rm A}$ and the density jump X are related (taking for the adiabatic index $\gamma=5/3$) as:

$$% \begin{array}{l} \left(M_{{\rm A}x}^2 - X\right)^2~\left[5\... ...heta~\left[(5+X)M_{{\rm A}x}^2 + 2X(X-4)\right]=0. \end{array}$$ (7)

Here $M_{{\rm A}x}=v/v_{{\rm A}x}$ is the Alfvén Mach number based on the $v_{{\rm A}x}=B_x/\sqrt{\mu\rho}$ component of the Alfvén velocity, where B_x=Bcos$\theta$ is the magnetic field component normal to the shock front (e.g., Mann et al. 1995). Since the sound speed and the Alfvén velocity are related as $c_{\rm s}=v_{\rm A}\sqrt{\gamma\beta /2}$ the sound Mach number can be also determined:

$$% M_{\rm s}=M_{\rm A}\sqrt{\frac{2}{\gamma\beta}}=M_{\rm A}\sqrt{\frac{1.2}{\beta}}\cdot$$ (8)

In the case of the perpendicular shock ( $\theta =90^{\rm o}$) Eq. (7) becomes:

$$% M_{\rm A}=\sqrt{ \frac{X(X+5+5\beta)}{2(4-X)}}\cdot$$ (9)

In the limiting case, for $\beta\rightarrow 0$ one finds:

$$% M_{\rm A}=\sqrt{ \frac{X(X+5)}{2(4-X)} }\cdot$$ (10)

On the other side, for $B\rightarrow 0$ (i.e. $\beta\rightarrow\infty$) one comes to the hydrodynamic approximation (sound shock wave), and the Mach number is:

$$% M_{\rm s}=\sqrt{ \frac{3X}{4-X} },$$ (11)

which was used by Smerd et al. (1974, 1975).

Figure 9: Relationship between the Alfvén Mach number $M_{\rm A}$ and the relative band-split BDW obtained from $M_{\rm A}(X)$ relationship by substituting $BDW=\sqrt {X-1}$ (see Eq. (2)). Results are presented for a) different angles $\theta$ between the shock normal and the magnetic field, with $\beta =0$; b) different values of plasma-to-magnetic field pressure ratio $\beta$ for the perpendicular shock, as well as for the hydrodynamic shock (gray line).

As Eqs. (7)-(11) show, the compression is limited to X<4, corresponding to $f_{\rm U}/f_{\rm L}<2$, i.e., BDW<1, regardless on the value of $\beta$.

In the case of the longitudinal shock ( $\theta =0^{\rm o}$):

$$% M=\sqrt X,$$ (12)

but there is an upper limit on the compression which depends on the value of $\beta$:

$$% X_{{\rm max}}=4-\frac{5}{2}\beta$$ (13)

(Priest 1982).

The relationships given by Eqs. (7)-(12) are illustrated in Fig. 9 for different combinations of $\theta$ and $\beta$.