SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 323, 250-258 (1997)

Previous Section Next Section Title Page Table of Contents

4. Discussion

Can 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 [FORMULA] is well above the cutoff and [FORMULA] (where [FORMULA] is the propagation angle relative to the magnetic field, and [FORMULA] is the electron gyrofrequency), the quasi-longitudinal approximation of the index of refraction is valid. Thus

[EQUATION]

where [FORMULA] ([FORMULA] being the plasma frequency) and [FORMULA]. 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 [FORMULA], except near sources of plasma emission and regions of nearby perpendicular propagation (e.g. Benz 1993). Since [FORMULA], the group velocity [FORMULA] can be derived from Eq. (21). The time delay [FORMULA] between modes is the integral along the ray path from the source to the observer

[EQUATION]

We have evaluated Eq. (22) for a simple model, assuming a constant ratio [FORMULA], where

[EQUATION]

Assuming a constant angle [FORMULA], a constant [FORMULA] implies both constant Alfvén velocity and constant [FORMULA] 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 [FORMULA] and [FORMULA], the starting value of X at the site where the polarization originates. A barometric density model, [FORMULA] exp [FORMULA], is assumed. Hence Eq. (22) can be transformed to

[EQUATION]

[FORMULA] 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 [FORMULA] K. At the cutoff frequency of the extraordinary mode, the delay becomes infinity since the group velocity of the extraordinary wave vanishes. A decreasing [FORMULA] lets the polarization originate at a higher frequency relative to the local plasma frequency, reducing the delay.

[FIGURE] Fig. 5. Time delay between circular modes as a function of [FORMULA] for various starting values [FORMULA] indicated in the figure ([FORMULA]). A density scale height of [FORMULA] cm has been assumed.

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 [FORMULA] and [FORMULA] (requiring small [FORMULA]) or small density scale heights or both. For [FORMULA] and [FORMULA], Eqs. (21) and (22) can be approximated to lowest order in X and Y, yielding

[EQUATION]

It corresponds to the lower left corner of Fig. 5 and small [FORMULA]. Fig. 5 and Eq. (25) indicate the dependence of the delay on the coronal model: At large [FORMULA] it is dominated by [FORMULA], and at small [FORMULA] by [FORMULA] and [FORMULA] in addition.

Model A

Model A assumes that the polarization originates in the spike source. Lower limits on [FORMULA] and [FORMULA] 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 [FORMULA] and [FORMULA]. Thus an observed delay of 100 µs can only be explained if the scale length, [FORMULA] cm. This dimension is consistent w typical diameters of coronal loops as seen in soft X-rays (e.g. Golub etal. 1990). However, to avoid a large delay outside the dense structure, the density must decrease by more than 6 of these small scale lengths, i.e. the density in the source must be higher by more than a factor of [FORMULA] than in the ambient medium. Although observations may not exclude this,model A is not supported by observations. We may note that model A does not allow for maser emission requiring [FORMULA] and [FORMULA] close to unity or larger.

Model B

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, [FORMULA]. The values of [FORMULA] and [FORMULA] 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.

[EQUATION]

where

[EQUATION]

an incoming circularly polarized wave becomes fully linearly polarized and subsequently gets completely depolarized by Faraday rotation. For [FORMULA] in the quasi-transverse region and [FORMULA], it follows from Eq. (27) that

[EQUATION]

Using a conventional upper limit on the magnetic scale height of [FORMULA] cm, Eq. (25) yields [FORMULA] s. Thus the depolarization in a quasi-transverse region is compatible with the observed delays for [FORMULA].

(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

[EQUATION]

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 [FORMULA] and [FORMULA] 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 [FORMULA] and [FORMULA] at higher altitude and lower [FORMULA] and [FORMULA] than in the source. Using a conventional density scale height of [FORMULA] cos [FORMULA] cm and an observed [FORMULA] s, Eq. (25) yields [FORMULA]. 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.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

helpdesk.link@springer.de