## 4. DiscussionCan the observed time delay between modes be the effect of different group velocities of the propagating radiation? Relative delays of the two modes result in a magnetized plasma, where the indices of refraction differ for the two modes. If the wave frequency is well above the cutoff and (where is the propagation angle relative to the magnetic field, and is the electron gyrofrequency), the quasi-longitudinal approximation of the index of refraction is valid. Thus where ( being the plasma frequency) and . The upper and lower sign in Eq. (21) correspond to the ordinary and extraordinary mode, respectively. The approximation (21) holds in most of the corona since generally , except near sources of plasma emission and regions of nearby perpendicular propagation (e.g. Benz 1993). Since , the group velocity can be derived from Eq. (21). The time delay between modes is the integral along the ray path from the source to the observer We have evaluated Eq. (22) for a simple model, assuming a constant ratio , where Assuming a constant angle , a constant
implies both constant Alfvén velocity
and constant in the plasma where the difference
in the integral of Eq. (22) is significant. The constant ratio also
implies the same scale height for the magnetic field and the square
root of the electron density. Therefore the delay can be easily
evaluated as a function of and
, the starting value of is the angle between the vertical and the ray path. The result is shown in Fig. 5 for a density scale height corresponding to a temperature of K. At the cutoff frequency of the extraordinary mode, the delay becomes infinity since the group velocity of the extraordinary wave vanishes. A decreasing lets the polarization originate at a higher frequency relative to the local plasma frequency, reducing the delay.
Comparing Fig. 5 with the results of Table 2 immediately
demonstrates that the observed delays are many orders of magnitude
smaller than may be expected from spike sources. Either the spike
polarization originates at extremely small values of
and (requiring small
) or small density scale heights or both. For
and , Eqs. (21) and (22)
can be approximated to lowest order in It corresponds to the lower left corner of Fig. 5 and small . Fig. 5 and Eq. (25) indicate the dependence of the delay on the coronal model: At large it is dominated by , and at small by and in addition.
Model A assumes that the polarization originates in the spike
source. Lower limits on and
can be derived from proposed emission
mechanisms. For illustration we use the model of Willes & Robinson
(1996). Based on observations of harmonic spikes they propose as
typical source parameters and
. Thus an observed delay of 100
As an alternative to extremely small density scale lengths it is conceivable that the spike emission originates as highly polarized radiation, but is transformed into a mixture of modes in the higher corona, where the local plasma frequency and electron gyrofrequency are much lower than the observing frequency, . The values of and can then be considerably lower than in the source. Two such processes have been proposed: (i) The crossing of a quasi-transverse region and (ii) scattering on lower-hybrid waves. (i) In an inhomogeneous medium of propagation with a quasi-transverse magnetic field, the two modes are coupled. If the coupling is intermediate, i.e. where an incoming circularly polarized wave becomes fully linearly polarized and subsequently gets completely depolarized by Faraday rotation. For in the quasi-transverse region and , it follows from Eq. (27) that Using a conventional upper limit on the magnetic scale height of cm, Eq. (25) yields s. Thus the depolarization in a quasi-transverse region is compatible with the observed delays for . (ii) The deflection of radio waves on lower-hybrid waves has been proposed by Wentzel et al.(1986) for the production of two modes (equivalent to depolarization) in type I bursts. It may be noted here that the depolarizing lower-hybrid waves would have to occur on similar heights in the corona for spikes and type I bursts (Wentzel 1997). The conservation conditions require The deflection changes the degree of polarization depending on the angle of incidence between the radio wave and the lower-hybrid wave. Since the frequency of the lower-hybrid wave is small compared to the radio wave, the frequency remains practically unchanged for the deflected wave. The deflection can take place at much lower and than in the source of emission. Thus the delay between modes can be as small as observed, if the combination of deflected and direct waves does not introduce additional delays. Both processes for reduced polarization in spikes have starting values and at higher altitude and lower and than in the source. Using a conventional density scale height of cos cm and an observed s, Eq. (25) yields . They are consistent with the emission model quoted above if the polarization originates at a site where e.g. the density and magnetic field are both reduced by more than an order of magnitude compared to the source. © European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |