4 Results

Utilizing the density models defined in Sect. 2 the frequencies $f_{\rm L}(t_i)$ are converted to radial distances R_i(t_i), where $R=r/r_{\odot}$ is the heliocentric distance normalized with respect to the solar radius $r_{\odot}$. The obtained values are used to determine the radial velocities:

$$% v_i\left(\overline t_i\right)= \frac{R(t_{i+1})-R(t_i)}{t_{i+1}-t_i}~r_{\odot}.$$ (6)

In Figs. 3a and b we show graphs v_i(R_i) and X_i(R_i), where R_i=[R(t_i+1)+R(t_i)]/2, and X_i is defined by Eqs. (2) and (5). Using the expressions given in Appendix and applying various combinations of $\theta$ and $\beta$ the values X_i(R_i) are converted to  $M_{{\rm A}i}(R_i)$. The outcome for $\beta =0$ and $\theta =90^{\rm o}$ is presented in Fig. 3c. The inferred shock velocity v decreases with the heliocentric distance, whereas the values of the density jump X (and thus the inferred $M_{\rm A}$) do not show a systematic increase or decrease. Yet it should be noted that at $R\approx 2$, where the shock velocities show a local minimum, the density jumps show a local maximum.

The values v_i(R_i) and $M_{{\rm A}i}(R_i)$ are used to evaluate the Alfvén velocity $v_{{\rm A}i}(R_i)=v_i/M_{{\rm A}i}$ and the corresponding magnetic field (Eq. (3)). The results are shown in Fig. 4 for the two-fold Newkirk density model so that the B(R)dependence can be compared with the results reported by Smerd et al. (1974, 1975) which are shown by gray crosses in Fig. 4b.

The $v_{\rm A}(R)$ dependence shows a local minimum at $R\approx 2$. It is somewhat less exposed if the Saito model is applied (the values based on the five-fold Saito model are shown in the inset of Fig. 4a) due to a less steep slope of n(R) around that distance (see Fig. 8 in the Appendix). However, whatever polynomial fit is used, at least an inflection in the fitted curve is found. We note that the results do not change significantly if the two data points showing a comparatively high Alfvén velocity at $R\approx 1.3$ are removed.

In Figs. 5 and 6 the average values $\overline v_{\rm A}(\overline R)$ and $\overline B(\overline R)$ are shown, where each data point represents the mean of 40 distance-successive data points. This representation is chosen to illustrate more transparently how different choices of $\beta$ (Figs. 5a and 6a), propagation angle $\theta$ (Figs. 5b and 6b), and a density model (Figs. 5c and 6c) affect the results. Clearly, the results are most sensitive on the choice of the density model. Figure 5 indicates again that there is a local minimum, or at least a stagnation in the decrease of $v_{\rm A}$ at $R\approx 2\pm 0.3$.

The B(R) dependence can be approximated well by the power law fit of the form B(R)=a R^-b. The results are summarized in Table 1 where the power-law coefficients a and b are given. In Table 1 we also show the coefficients for the power-law fit expressing the magnetic field as a function of the normalized height $H\equiv R-1$. Inspecting Table 1 one finds that the Saito model gives somewhat steeper decrease of the magnetic field with the distance than the Newkirk model. If the magnetic field decrease is expressed as a function of the height above the solar surface, the slope is in the range $b\approx 1.5$-2, where the former value corresponds to the two-fold Newkirk model. Similar results are obtained if the averaged data shown in Fig. 6 are used, now with the correlation coefficients C>0.98.

Figure 3: Results obtained applying the five-fold Saito density model: a) shock velocities; b) downstream/upstream density jump; c) Alfvén Mach number obtained using $\beta =0$ and assuming the perpendicular propagation ( $\theta =90^{\rm o}$). The 4th degree polynomial least square fits are shown.

Figure 4: a) Alfvén velocity and b) magnetic field, evaluated assuming the perpendicular shock propagation ( $\theta =90^{\rm o}$), the two-fold Newkirk coronal density model, and $\beta =0$. The 5th degree polynomial least squares fit is presented for $v_{\rm A}(R)$. Note the local minimum at $R\approx 2$ and maximum at $R\approx 2.5$. In the inset the $v_{\rm A}(R)$ graph is shown for the five-fold Saito density model, showing somewhat less pronounced minimum and maximum of $v_{\rm A}(R)$ than in the two-fold Newkirk model results. In the bottom graph the power-law fit is presented (bold). The two thin lines represent the mean values shown in Fig. 6c for the two- and ten-fold Saito model. The gray curve represents values of B* defined by two-fold Newkirk model, $T=1.5\times 10^6$ K, and $\beta ^*=1$. The results by Smerd et al. (1974, 1975) are shown by gray crosses.

Figure 5: Alfvén velocity shown as a function of heliocentric distance for: a) five-fold Saito coronal density model and $\theta =90^{\rm o}$, with: $\beta =0$ (gray circles), $\beta =0.1$ (black circles), and $\beta =1$ (triangles); b) five-fold Saito density model and $\beta =0$, with: $\theta =90^{\rm o}$ (perpendicular shock), $\theta =60^{\rm o}$, $\theta =45^{\rm o}$, $\theta =30^{\rm o}$, and $\theta =0^{\rm o}$ (longitudinal shock) c) $\beta =0$, $\theta =90^{\rm o}$, for: two-, five-, and ten-fold Saito density model, as well as two-fold Newkirk model (drawn by squares, black triangles, circles, and gray triangles, respectively.)

Figure 6: Magnetic field shown as a function of heliocentric distance for the same combinations of parameters as used in Fig. 5. The $\beta =0$-0.1 and $\theta =30^{\rm o}$- $90^{\rm o}$ curves are not resolved (the top and middle panel, respectively). In the top panel the curves B*(R) defined by the five-fold Saito model and $T=2\times 10^6$ K are drawn for $\beta ^*=0.01$, $\beta ^*=0.1$, and $\beta ^*=1$ by thick gray lines. In the bottom panel the relationship B=0.5 H-1.5 proposed by Dulk & McLean (1978) is shown by the bold line.