5 Discussion

5.1 Starting assumptions

Figures 5 and 6 show that the inferred values of the Alfvén velocity and magnetic field do not significantly depend on the choice of the plasma parameter $\beta$ and the propagation angle $\theta$. Comparing Figs. 5a and 5b with Fig 4a, as well as Figs. 6a and 6b with Fig. 4b, one finds that the effect is much smaller than the data scatter.

 

 

Table 1: Power-law coefficients a and band correlation coefficients C for the B(R) and B(H)dependence.

model		B(R)=a R-b				B(H)=a H-b
		a	b	C		a	b	C
2$\times$Saito		7.9	4.44	0.82		0.41	1.49	0.81
5$\times$Saito		17.6	4.22	0.83		0.89	1.85	0.83
10$\times$Saito		29.2	3.94	0.82		1.77	2.00	0.83
2$\times$Newkirk		9.6	3.40	0.76		0.86	1.50	0.78

The thick gray curve in Fig. 4b represents the magnetic field $B^*=2\mu nkT/\beta^*$ defined by the two-fold Newkirk density model, the temperature $T=1.5\times 10^6$ K, and $\beta ^*=1$. Only ten data points fall below the curve, i.e. into the $\beta^*>1$ region. Note that these data points are well isolated from the rest of the sample and could be treated as exceptions. In the case of the five-fold Saito model only five data points show $\beta^*>1$.

The magnetic field B^*(R) defined by the five-fold Saito density model, the temperature $T=2\times 10^6$ K, and three values of $\beta^*$ (1, 0.1, and 0.01) is compared with the measurements in Fig. 6a. The measured values $\overline B(\overline R)$match closely the curve B^*(R) defined by $\beta^*\approx 0.1$. This justifies the applied approximation $\beta =0$ since the difference in the results obtained by taking $\beta =0.1$ and $\beta =0$ is negligible. We note that the slope of $\overline B(\overline R)$ is steeper than B^*(R) which is consistent with the expected increase of the parameter $\beta$ with the height (see, e.g., Gary 2001 and references therein).

Figures 5b and 6b show that the variation of the propagation angle $\theta$ affects the results even less than the choice of $\beta$. Inspecting Fig. 9a one finds that at BDW=0.35 the difference in $M_{\rm A}$ for $\theta =90^{\rm o}$ and $\theta =0^{\rm o}$ amounts to $\approx$20%. Since only 2% of events are characterized by BDW>0.35 the average values of $v_{\rm A}$and B are not significantly influenced by different choices of $\theta$.

Figures 5 and 6 show that the inferred values of the Alfvén velocity and the magnetic field are most sensitive on the density model used. When applying a higher density model, the values of $v_{\rm A}$ and B become larger. At the same time the corresponding radial distances increase. The slopes of  $v_{\rm A}(R)$and B(R) relationships at a given R depend directly on the slope of n(R) which is different in each model (see Fig. 8 in the Appendix).

The ambiguity in choosing the density model results in a range of possible values of Alfvén velocity. For example, presuming that two- to ten-fold Saito model embraces most of the situations involved (see Sect. 2), one finds that at $R\approx 1.6$ the Alfvén velocity spans between 450 and 1300 km s^-1. The corresponding magnetic field is in the range 1-7 G.

Such a span of $\overline B(\overline R)$ values is comparable with the scatter of individual data points. This can be seen in Fig. 4b where the two thin lines that represent $\overline B(\overline R)$ values obtained by using two- and ten-fold Saito model embrace a large majority of the data points. Since the data scatter is much larger than the errors of measurements, the scatter has to be attributed primarily to the diversity of the coronal conditions and to different angles between the direction of the shock propagation and the density gradient.

Finally it should be noted that the magnetic field strengths obtained by Smerd et al. (1974, 1975) are about two times larger than found herein (Fig. 4b). We emphasize that this cannot be explained by the hydrodynamic approximation ( $\beta=\infty$) used therein, since the corresponding difference would not be larger than 10-20% for the considered range of values of BDW (see Fig. 9b in Appendix). Unfortunately in Smerd et al. (1974, 1975) the procedure of measurement is not described in detail, so we cannot resolve the origin of this discrepancy.

5.2 Radial dependence of the magnetic field

The range of magnetic field strengths exposed in Fig. 6c is generally consistent with the values often found at these heights (cf. Dulk & McLean 1978; Krüger & Hildebrandt 1993). In Fig. 7 our results, represented by the shaded gray area that covers the range of values defined by Fig. 6c, are exposed in a wider context. A number of estimates based on different methods are shown, mostly reported after the paper by Dulk & McLean (1978). The data are clearly clustered along the B(H)=5 H^-1.5 relationship proposed by Dulk & McLean (1978) for 1<R<10 (thick gray line in Fig 7).

If particularly the B(H) dependence found from our measurements is considered, it shows a slope between H^-1.5 and H^-2, the former value being obtained by using the Newkirk density model. The result can be considered as compatible with the Dulk & McLean (1978) relationship since our measurements cover only the 1.3<R<2.9 range.

Figure 7: a) Comparison of our results (shaded area covers the results based on two- to ten-fold Saito density model) with some other estimates: 1-3 - Brosius et al. (1993, 1997); 4 - Ramaty & Petrosian (1972); 5 - Kakinuma & Swarup (1962); 6 - Nakariakov & Ofman (2001); 7 - Aurass et al. (1987); 8 - Lin et al. (2000); 9 - Bogod et al. (1993); 10 - Ledenev et al. (2002); 11 - Yu & Yao (2002); 12 - Mann & Baumgärtl (1988); 13 - Gopalswamy et al. (1986); 14 - Mollowo (1988); 15 - Dulk et al. (1976); 16 - Patzold et al. (1987). The results obtained by Smerd et al. (1974, 1975) are framed by dashed line and denoted by S. b) The same data (limiting values) presented in the B(R) graph, provisionally divided into two groups showing different slopes (R<1.3 - circles, R>1.3 - crosses). The bold and thin lines show $B=2\times R^{-2}$ and $B=75\times R^{-10}$, respectively.

As already noted by Dulk & McLean (1978) the $B(H)\propto H^{-1.5}$ relationship, except at low heights, behaves similar to the R^-2 curve, revealing a dominance of the radial field. In Fig. 7b we represent the data from Fig. 7a in the B(R) graph, splitting the sample into the H>0.3 and H<0.3data sets. Each data point represents the limiting values of different estimates shown in Fig. 7a. The B(H)=5 H^-1.5 relationship is drawn by thick gray line. The other two lines show the functions $B=75\times R^{-10}$ and $B=2\times R^{-2}$ (thin and bold, respectively). The former relationship matches roughly the data below R<1.3. The second one is chosen since at 1 AU it gives $B\approx 5$ nT, i.e. the value usually measured at the Earth. The R>1.3 data show a lower limit lying closely by the R^-2 curve. Except the shaded data point, representing the upper-right corner of the "S"-box in Fig. 7a (the data by Smerd et al. 1974, 1975), the upper boundary also approaches the R^-2curve beyond $R\approx 2$. This indicates that the magnetic field is dominated by active region fields below $H\approx 0.3$ and the radial field above H=1. Such a result is consistent with the polarimetric measurements in the Fe  XIII 10747 Å line reported recently by Habbal et al. (2001).

5.3 Radial dependence of the Alfvén speed

The dependence of the Alfvén velocity on the height does not depend qualitatively on the density model used (Fig. 5b). For the two- to ten-fold Saito model the Alfvén velocity shows a local minimum of $v_{\rm A}\approx 400$- 500 km s^-1 at $H\approx 0.7$-1.2, and a maximum of $v_{\rm A}\approx 450$-700 km s^-1 at $H\approx 0.8$-1.5 .

Such a behaviour of the Alfvén speed was anticipated by Mann et al. (1999, 2002): outside of active regions, where the radial magnetic field dominates $v_{\rm A}$ should attain maximum at the height of 2-3 solar radii (Mann et al. 1999); If the bipolar magnetic field of an active region is added, a local minimum appears between the solar surface and the region of the Alfvén velocity maximum (Mann et al. 2002).

Our results find the minimum at somewhat larger height than calculated by Mann et al. (2002), whereas the maximum is for about 1 solar radius lower. Nevertheless the qualitative behaviour is the same, bearing in mind that we have a comparatively small number of data points at R>2.5(Fig. 4a) so the height of the maximum is ambiguous.

The described height dependence of the Alfvén velocity is important for the comprehension of the formation and evolution of MHD shocks in the solar corona. Vrsnak & Lulic (2000) analysed steepening of the large amplitude perturbation (simple wave) into the shock in the uniform Alfvén velocity environment. The frontal part of the perturbation is steepening because later segments (having larger amplitude) are faster than the earlier ones. The effect is obviously enhanced if the Alfvén velocity decreases in the direction of the propagation of the disturbance. Thus, the shock formation is expected to occur in the R<2 range where the Alfvén speed decreases. This is compatible with typical type II burst starting frequencies of $\approx$100 MHz (cf. Nelson & Melrose 1985). Analogously, the shock weakens in the opposite situation. In an environment characterized by increasing Alfvén velocity the amplitude of the shock decreases and eventually becomes too weak to generate the type II burst emission. This is most likely to happen in the region before the $v_{\rm A}$ maximum, i.e. around $R\approx 2.5$, compatible with the usual dm-m type II burst ending frequencies of about 20 MHz (cf. Nelson & Melrose 1985).