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Astron. Astrophys. 328, 390-398 (1997)

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4. Decimetric continuum

As already mentioned in the Introduction, there are several arguments which lead to the assumption that the solar decimetric continuum is generated by plasma waves at the upper hybrid frequency [FORMULA] (Kuijpers 1974; Zaitsev & Stepanov 1975; Benz & Kuijpers 1976). These waves are generated by trapped energetic electrons having an anisotropic velocity distribution of the loss-cone type.

It was shown in Sect. 3 that the flare-related magnetically trapped fast electrons with a power-law energy distribution are able to generate plasma waves at the upper hybrid frequency, which in turn may lead to the solar decimetric continuum. The close association of the sources of the decimetric continuum with the region of the primary flare energy release follows not only from the near-coincidence of the time profiles of the decimetric continuum and microwave bursts already mentioned in Sect. 1, but also from the following observations:

  • The decimetric continuum is generated in source regions with small sizes comparable with those of microwave bursts (Zheleznyakov 1970).
  • The sources of the decimetric continuum do not show a strong displacement across the solar disk similar as the sources of microwave bursts (Fleisher & Oshima 1961; Krishnan & Mullaly 1961).
  • According to limb observations the source heights of the decimetric continuum are about [FORMULA] cm which is close to the heights of magnetic loops where the gyrosynchrotron radiation of flare electrons, i. e., solar microwave bursts are generated (Kundu & Firor 1961).
  • The high correlation of decimetric bursts with hard X-ray bursts (Aschwanden & Güdel 1992).

We will assume that the decimetric continuum originates in relatively weak, closed magnetic fields directly adjoining the flare loops. The flare originates either at the top or the feet of a magnetic loop and acts as the general source of the energetic electrons for both the microwave burst and the decimetric continuum (Fig. 3).

[FIGURE] Fig. 3a and b. Schematic view of the relative positions of the source regions of the microwave burst (Region I) and the decimetric continuum burst (Region II) invoked in the present paper. The flare occurs either at the top of Region I or at its footpoints and acts as primary source of energetic power-law electrons for both, the microwave burst and the decimetric continuum. The source of the decimetric continuum may be a high-lying relatively cool loop (a) or a loop adjacent to the microwave burst source (b).

In the following the microwave-burst and decimeter-continuum source will be denoted as Region I and Region II, respectively, both strongly differing in their parameters.

On the average, the solar microwave emission has a maximum flux density at, say, the frequency [FORMULA] GHz. Inferring gyrosynchrotron radiation of energetic electrons with a power-law energy spectrum this corresponds roughly to magnetic fields of [FORMULA] G in the source region (Kundu & Vlahos 1982) and a value of the gyrofrequency [FORMULA] GHz. In Region I, on the average, the plasma frequency should obey the relation [FORMULA]. In the case [FORMULA] a strong suppression of the gyrosynchrotron radiation occurs (Razin-Tsytovich effect). For example, for [FORMULA] the brightness temperature of the source of the microwave burst emission decreases by 2-3 orders of magnitude to the level of Coulomb bremsstrahlung of the plasma filling the source volume (cf. Fig. 4). The condition [FORMULA] gives a restriction on the density of the background plasma in Region I to values [FORMULA] cm-3.

[FIGURE] Fig. 4. Theoretical spectra of gyrosynchrotron radiation (GSR) generated by power-law electrons assuming various densities of the thermal background plasma in the (homogeneous) source volume. The numbers on the curves are density values in units of [FORMULA] cm-3. The low-frequency suppression (Razin-Tsytovich Effect) can clearly be recognized for increasing density and will suppress the gyrosynchrotron emission below the level of thermal bremsstrahlung (BS) for [FORMULA] cm-3. For comparison, the resulting spectrum including bremsstrahlung for [FORMULA] cm-3 is also shown in the plot (thin solid line). The other parameters are: [FORMULA] G, [FORMULA] cm-3, [FORMULA] cm, [FORMULA] K, [FORMULA], [FORMULA] keV. In these calculations the power-law distribution was cut at the energy [FORMULA] keV.

On the average the spectral maximum of the decimetric continuum lies in the frequency range [FORMULA] GHz (Isliker & Benz 1994). Hence we obtain a plasma density in the source region of [FORMULA] cm-3 for [FORMULA] or [FORMULA] cm-3 for [FORMULA]. Here we assume that in Region II the gyrofrequency is much smaller than the plasma frequency. This condition [FORMULA] is necessary in order to exclude strong gyroresonance absorption at the levels [FORMULA] and [FORMULA] at the escape of the radiation from Region II outwards.

The temperature of the plasma inside (hot) flare loops is typically [FORMULA] K (Tsuneta 1996). We will assume that outside these flare loops the temperature is lower. For Region II (as sketched in Fig. 3 b) we assume [FORMULA] K although a heating of this region during the flare cannot be excluded (Aschwanden & Benz 1995; Tsuneta 1996)). Hence, summarizing the above arguments we use the following estimates for the characteristic plasma parameters in the Regions I and II taking into account that they may slightly differ towards higher or lower values:

  • Region I (microwave burst source):
    [FORMULA] G
    [FORMULA] K
    [FORMULA] cm-3.
  • Region II (source of decimetric continuum):
    [FORMULA] G
    [FORMULA] K
    [FORMULA] cm-3.

4.1. Brightness temperature of the radiation in the source region of the decimetric continuum

Analyzing the flare-energy support of the source of the decimetric continuum, the absorption of the radiation by electron-ion collisions at the wave propagation from the source through the solar atmosphere is important. Since this absorption is rather high, the effective temperatures of the decimetric continuum up to [FORMULA] K (Kundu 1961; Krishnan & Mullaly 1961) can be ensured only as a result of a sufficiently high number of fast electrons injected from the flare region into the source region of the decimetric continuum comparable with the number of fast electrons in the microwave burst source.

The optical depth [FORMULA] of the corona due to electron-ion collisions for radiation propagating from the source region of the decimetric continuum to the observer is given by

[EQUATION]

where [FORMULA] is determined by Eq. (20) and [FORMULA] is the angle between the line of sight and the direction of the density gradient. Furthermore,

[EQUATION]

is the scale height of plasma density variation in the source region of the decimetric continuum. Using Eqs. (20) and (35), Eq. (34) can be rewritten:

[EQUATION]

assuming that [FORMULA] is given in GHz. Taking [FORMULA] GHz, [FORMULA] K, [FORMULA], we obtain from Eq. (36) [FORMULA]. We can see that even in the most favourable case ([FORMULA]) the optical depth becomes [FORMULA] and the radiation at the second harmonic is weakened by [FORMULA] at the transit through the source of the decimetric continuum while the radiation of the first harmonic is weakened by a factor [FORMULA] 1014!

The necessity of a consideration of a significant absorption in the corona for the study of the decimetric continuum was already early mentioned by Kuijpers (1974). Our estimations show that, due to the rather strong absorption of the first harmonic, the most favourable mechanism for the decimetric continuum is the generation of the second harmonic as a result of coupling of strong plasma waves excited by the loss-cone instability. Here the observed brightness temperatures [FORMULA] K should correspond to brightness temperatures inside the source of the decimetric continuum which are about three orders higher, i. e.,

[EQUATION]

This brightness temperature is less than the temperature of the upper hybrid Langmuir waves

[EQUATION]

which for the case given here is of the order [FORMULA] - [FORMULA] K.

4.2. Generation of the second harmonic

The emission at the second harmonic of the plasma frequency [FORMULA] is generated by nonlinear coalescence of two plama waves (combination scattering) if the resonance condition

[EQUATION]

is fulfilled. The transfer equation for the brightness temperature of the emission has the form

[EQUATION]

Here [FORMULA] is the emission coefficient, [FORMULA] is the absorption coefficient related to the decay of an electromagnetic wave of the frequency [FORMULA] into two plasma waves, [FORMULA] is the absorption coefficient due to the absorption of electromagnetic waves by electron-ion collisions inside the source of radiation, and l is the coordinate along the ray propagation.

If the source has a steady inhomogeneous distribution of the plasma density n with a characteristic scale height [FORMULA], the integration of Eq. (40) should be carried out through a thin layer [FORMULA] in which the frequency of the elctromagnetic wave is approximately constant, i. e., [FORMULA]. The depth of this layer is (Zaitsev & Stepanov 1983):

[EQUATION]

where [FORMULA] and [FORMULA] are the maximum and minimum values of the wave number in the excited wave spectrum, respectively. In the second part of of Eq. (41) we took [FORMULA], where k is the average wave number of the plasma waves.

The coefficients [FORMULA] and [FORMULA] for the process obeying Eq. (39) have the following form (Zheleznyakov 1996):

[EQUATION]

[EQUATION]

where [FORMULA] characterizes the width of the spectrum of the excited plasma waves, [FORMULA], and

[EQUATION]

In our case we have [FORMULA]. Taking [FORMULA], we obtain [FORMULA]. Maximum growth of the plasma waves occurs at the wave number [FORMULA] ; therefore we have

[EQUATION]

which yields [FORMULA] for [FORMULA] cm s [FORMULA] keV) and [FORMULA].

It can easily be seen that collisional absorption at the second harmonic with the absorption coefficient [FORMULA] within the limit of a layer [FORMULA] as given by formula (41) does not give a significant contribution to the general absorption coefficient. Therefore, integrating Eq. (40) inside a layer [FORMULA] under the assumption [FORMULA], we obtain at [FORMULA]

[EQUATION]

where

[EQUATION]

is the optical depth of the layer [FORMULA] for the decay of electromagnetic waves of the frequeny [FORMULA] into two plasma waves under the assumption that the scale height is [FORMULA] cm. Eq. (46) yields, together with (45) and (47), for [FORMULA] GHz, [FORMULA] cm-3, and [FORMULA] K the required brightness temperature in the source region [FORMULA] K if the energy density of the waves is [FORMULA].

As it was shown in Sect. 3 [cf. Eq. (33)], the generation of plasma waves by fast electrons with a power-law energy spectrum leads to an energy level [FORMULA]. Hence the required brightness temperature in the source can be explained if [FORMULA]. For [FORMULA] cm-3 this corresponds to a density of fast electrons of [FORMULA] cm-3. If the source volume is [FORMULA] cm3 (by taking [FORMULA] cm and [FORMULA] cm) the total number of fast electrons injected into the source region of the decimetric continuum amounts to [FORMULA].

4.3. Polarization of the decimetric continuum

Commonly the polarization of the decimetric continuum is believed to correspond to the ordinary wave mode (cf., e.g., Zheleznyakov 1970, Kuijpers 1980). However, broadband decimetric pulsations were found to be polarized in the extraordinary sense according to the leading spot hypothesis (Aschwanden 1986).

We will suppose that the plasma waves are generated within a certain cone of angles with a half-width [FORMULA], where the direction of the cone differs slightly from the direction perpendicular to the magnetic field:

[EQUATION]

Then, transforming the results of Zlotnik (1981) to our case, we obtain the following expression for the degree of polarization:

[EQUATION]

where [FORMULA] and [FORMULA] are the observed fluxes of the extraordinary and ordinary wave modes, respectively, and [FORMULA] is the angle between the direction of wave propagation and the magnetic field. Thus, the cases [FORMULA] and [FORMULA] correspond to the polarization of the extraordinary and ordinary waves, respectively. For our selected plasma-wave spectrum given in Eq. (48), the function [FORMULA] in Eq. (49) has the following form:

[EQUATION]

with

[EQUATION]

[EQUATION]

In the simplest case of a narrow plasma-wave spectrum, where [FORMULA], the degree of polarization becomes

[EQUATION]

We find that the sense of polarization corresponds to the ordinary wave ([FORMULA]) for angles [FORMULA]. Here, for [FORMULA], the degree of polarization becomes [FORMULA]. For angles [FORMULA] the polarization corresponds to the extraordinary mode. However, for sources of radio emission concentrated in magnetic traps, as in our case, the observation of a major part of the source under angles [FORMULA] near [FORMULA] is very likely. This can mean that, regardless of a part of the source (at the roots of magnetic loops) seen at angles [FORMULA] and hence yielding extraordinary polarization, the main polarization of the decimetric continuum corresponds to the ordinary wave mode. Significant extaordinary polarization requires strongly asymmetric source locations inside the trapping loop close to one footpoint (Aschwanden 1986).

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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