The observed increase in the Limb B ( arc sec) line widths over those at Limb A ( arc sec) in the mid-transition region are consistent with previous results (e.g., Shine et al. 1976, Mariska et al. 1978). Published results of on-disk measurements of center-to-limb variations is however more limited. For example, Feldman et al. (1976) concluded that the line widths of several optically thin lines were similar at disk center and the limb. Exceptions to this were the strong resonance lines of C IV and Si IV , which were explained in terms of opacity effects. The data presented in Fig. 2a suggests a center-to-limb variation which we believe cannot be explained by optical depth effects.
The maximum difference seems to occur at two different temperature regimes; the first at 20,000 K and the second at 250,000 K, with little or no difference at lower chromospheric temperatures or coronal temperatures. The lines with the highest probability of being affected by opacity are the lower temperature lines, notably C II . In Table 3 we tabulate the line ratios of the various doublets. As can be seen, the C II lines are optically thick everywhere, while the remaining lines are effected by opacity mostly for the Limb A data. Thus the C II data-point plotted in Fig. 2a (i.e., Limb-A) could be affected by opacity although the line ratio from Table 3 indicates a similar effect at disk center and the limb, thus whether the difference (limb minus disk center) is seriously affected is unclear. For the Limb-B data, (Fig. 2b), the data-points for C II , S II and Fe II are all in very good agreement, furthermore Table 3 indicates only a small line opacity effect for S II . Our data at 250,000 K include the weaker component of the strong resonance lines of N V and O VI , the stronger component showing a slightly larger line width suggesting opacity effects. Also included here is the resonance line of O V 630 Å. Thus we do not ascribe all of the excess (limb minus disk center) broadening in Figs. 2a,b to opacity. The important message to note from these figures is the clear excess limb minus disk center broadening.
Table 3. The line ratio for the various doublets observed in this programme
Roussel-Dupré et al. (1979) in a detailed analysis of
Si IV 1393, showed that this line undergoes a
marked broadening from disk center to the limb. Their Fig. 1
indicates additional broadening at compared to
(disk center). At disk center their Si
IV line has a half-width of 0.09
Å compared to 0.105 Å at ,
indicating excess broadening equivalent to km
s-1. This suggested that horizontal velocities were
somewhat greater than the radial velocities, similar to the results
reported here. As in the Feldman et al. results, some of this could be
explained by optical depth effects. Mariska (1992) suggests that the
difference in line width can be accounted for with a small increase in
The solar atmosphere is a highly non-uniform plasma embedded in a magnetic field which is a natural media for MHD waves. MHD waves might play an important role in explaining the observed high temperatures (i.e., K) in the solar corona. These waves can be generated either by the granular motions (e.g., foot-point motions) or excited locally by magnetic reconnection, with part of their kinetic energy being transformed into heating the magnetic loops. In this paper we do not study how MHD waves are excited, we suppose a priori they are present in the solar atmosphere. Furthermore, we suppose that such waves can propagate along the magnetic field lines and carry energy to the higher parts of the atmosphere. We also do not discuss how this energy is dissipated and converted into heat.
We are more concerned with a method of testing their existence via calculating the contribution of wave broadening to the overall line widths. This wave broadening is strongly influenced by the orientation of the magnetic field with respect to the line of sight and the polarisation of the waves.
Below we derive line widths calculated assuming (i) Alfvén wave heating and (ii) magneto-acoustic wave heating for comparison with the observational data. In the analysis which follows we assume an isothermal and uniform plasma.
4.1. An estimate of line widths as a result of Alfvén waves
Observations show that the aspect ratio of coronal loops are much larger than unity, i.e., coronal loops can be approximated by straight cylindrical magnetic flux tubes. The very high electrical conductivity of coronal plasma enables us to describe the induced electrical field due to wave motion by the ideal Ohm's law,
By supposing that there is no background equilibrium motion in the plasma, a travelling Alfvén wave along a magnetic field line B results in a velocity perturbation of a perfectly conducting plasma,
where c is the speed of light, and is
the induced electrical field caused by the Alfvén wave.
The energy flux associated with an Alfvén wave propagating along the magnetic field lines is
where is the perturbation of the magnetic field caused by the Alfvén wave, and denotes the Alfvén speed. In solar applications , and by using
where is the plasma electron density, and denotes the proton mass. Here we assume random polarisation, in which case the mean-square velocity component towards an observer is given by
Let us estimate the line broadening due to a propagating Alfvén wave. For a thermally broadened line produced by an ion of mass , the Doppler temperature is given by
Using the above equations, this implies, that line broadening caused by an Alfvén wave travelling along the magnetic field lines (which is perpendicular to the line of sight) can be estimated as
where is the Doppler width given in Table 1.
When, however, the magnetic field is directed towards the observer, (i.e., the oscillations caused by Alfvén waves are perpendicular to the line of sight), the line width is given by
The second set of parameters within the brackets in Eq. (10) does not contain a temperature dependent term and is equated with the measured non-thermal line broadening parameter (see Eq. 1). For the values of , B, and given in Tables 4 & 5 for the hot and cool loops, km s-1. An alternative way of comparing with the observational results is given in Tables 4 & 5. Here, the estimated excess Doppler line broadening at the limb compared to disk center due to the passage of purely Alfvén waves are compared with those derived from the data given in Table 1.
Table 4. The calculated excess Doppler line broadening in km s-1 at the two limb positions A & B compared to disk center for the ions given in Table 1 for cool loops assuming (i) Alfvén waves and (ii) acoustic waves. Here we used erg cm-2 s-1, G, and cm-3 (see text).
Table 5. The calculated excess Doppler line broadening in km s-1 at the two limb positions A & B compared to disk center for the ions given in Table 1 for hot loops assuming (i) Alfvén waves and (ii) acoustic waves. Here we used erg cm-2 s-1, G, and cm-3 (see text).
4.2. An estimate of the line width as a result of magneto-acoustic waves
In the previous section we have assumed a linear dispersion relation for the Alfvén waves, i.e., the Alfvén waves had a group velocity . Generalising this to any wave mode we can replace the group velocity in Eq. (7) by the group velocity of the wave. Let us suppose that there are magneto-acoustic waves travelling along the magnetic field lines. In a low-beta plasma like the solar corona these waves show the characteristics of sound waves, i.e., the group velocity can be expressed by
where is the ratio of specific heats. These special magneto-acoustic waves have oscillations only along the magnetic field lines, the perturbations perpendicular to the field lines are identically zero. Following the method given in the previous subsection, when the magnetic field lines coincide with the line of sight, one can observe line broadening caused by the magneto-acoustic waves passing along the field lines, e.g.,
However when the magnetic field lines are perpendicular to the line of sight, the magneto-acoustic waves travelling along the magnetic field lines do not cause any additional line broadening, i.e., the line width is described by Eq. (11). This shows that pure Alfvén waves and these type of magneto-acoustic waves show opposite behaviour as regards the line width and the direction of the magnetic field. The estimated line widths due to the passage of these magneto-acoustic waves are shown in Tables 4 & 5.
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998