Astron. Astrophys. 348, 614-620 (1999)

3. Discussion

Now, the heliospheric density model derived in the previous section and presented in Fig. 1 is compared with observations in the solar corona and the interplanetary space.

In Fig. 3 the density model (full line) is represented in the corona, i. e., in the range or above the photosphere, in comparison with the fourfold Newkirk (1961) model, i. e., with . The Newkirk (1961) model resulted from measurements of white light scattering in the corona during a solar minimum period. The inspection shows, that the model agrees well with the fourfold Newkirk model within a mean error of 15%. The deviation between the Newkirk model and the model derived in this paper is growing beyond . Recently, Koutchmy (1994) reported on optical ground based measurements during the eclipse on July 11, 1991. The resulting radial behaviour of the electron density was presented in Fig. 4 of the paper by Koutchmy (1994) for coronal loops, streamers, quiet equatorial and polar regions. The density in these regions differ by three orders of magnitude. For example, a density of , , and is found in coronal streamers, quiet equatorial and polar regions at a distance of (cf. Fig. 4 in Koutchmy (1994)), respectively. At the same distance the model provides a density of , which is the mean value of the density in the corona at this height.

Fig. 3. Comparison of the particle number density (full line) according to Fig. 1 and the fourfold Newkirk (1961) model (dashed line) in the range

The density behaviour in the outer corona and the interplanetary space was recently studied by the coronal radio sounding experiment at the ULYSSES spacecraft. Figs. 4 and 5 show the radial behaviour of the particle number density in the range according to the model (full line) and the radio sounding measurements (dashed line) during the ingress phase (cf. Fig. 4) and egress phase (cf. Fig. 5) (Bird et al., 1994). The radio sounding measurements were perfomed by Bird et al. (1994) during the 1991 solar conjunction of the ULYSSES spacecraft. The ingress and egress phase cover a range and , respectively. The inspection of the Figs. 4 and 5 reveals, that the model derived in Sect. 2 agrees well with the observations by ULYSSES in the range between and . The deviation between the model and the measurements has a mean value of 18%.

Fig. 4. Comparison of the density model (full line) (cf. Fig. 1) with the density measurements (dashed line) by the coronal radio soundig experiment (Bird et al. (1994)) of the ULYSSES mission during the ingress phase

Fig. 5. The same format as displayed in Fig. 4 for the ULYSSES data during the egress phase (cf. Bird et al. (1994))

Fig. 6 represents the radial density behaviour according to the model (full line) (cf. Sect. 2) in comparison with in-situ density measurements (dashed line) by the HELIOS 1 and 2 satellites (Bougeret et al., 1984) and the so-called RAE model (long dashed line) (Fainberg and Stone, 1971) in the range . Bougeret et al. (1984) derived a radial density model of the heliosphere by employing the in-situ density measurements during the period 1974 - 1980. These observations are made in the range . The best fit of the data are obtained by in for solar minimum conditions. The RAE model (Fainberg and Stone 1971) results from the investigation of type III bursts and type III radio storm bursts in the frequency range below 1 MHz by interplanetary radio measurements. Type III radio bursts appear as rapidly drifting emission stripes in dynamic radio spectra in the range . They are interpretated as the radio signature of sub-relativistic electron beams travelling from the solar corona along open magnetic field lines into the interplanetary space (cf. Suzuki and Dulk (1985) as a review). The model derived in Sect. 2 agrees well with the averaged data of the HELIOS in-situ measurements as demonstrated in Fig. 6. The deviation between the model by Bougeret et al. (1984) and our model has a mean value of . On the other hand there is a great difference with the RAE model. This difference is not too surprising, since the RAE model (Fainberg and Stone 1971) is a indirectly derived model, i. e., it results from type III storm radio bursts measurements, while the model by Bougeret et al. (1984) uses the HELIOS in-situ measurements. Similar differences have been noted for other radio source locations (cf. e. g. Steinberg et al. (1984, 1985)). Robinson (1992) deduced a radial density behaviour of from studying the radial variation of interplanetary type III burst source parameters. His result agrees with our model.

Fig. 6. Comparison of the density model (full line) (cf. Fig. 1) with the model by Bougeret et al. (1984) (dashed line) and Fainberg and Stone (1971) (long dashed line)

The radio instrument aboard the HELIOS satellite was able to localize the radio source in the interplanetary space. Kayser and Stone (1984) studied type III radio bursts and determined the source location of the different frequencies emitted during the movement of the electron beam in the interplanetary space. The result is presented in Table 2. Here, the radio waves are assumed to be predominantly emitted at the harmonic of the plasma frequency Thus, the particle number densities in the second column of Table 2 are calculated from the frequencies (left column of Table 2) by the assumption of harmonic emission. The corresponding radial source location obtained from the radio instrument aboard HELIOS is given in the third column. Thus, the frequency range covers a range between up to () in the heliosphere. The source location according to our model (cf. Sect. 2) is presented in the fourth column. The inspection of third and fourth column in Table 2 shows, that the model derived in Sect. 2 agrees very well with these observations over a great range in the heliosphere. The deviations between the observations and the model have a maximum and mean value of 15.2% and 11% (cf. right column in Table 2), respectively.

Table 2. Comparison of the radial source location of different plasma frequency levels (first column) as deduced by the radio measurements aboard HELIOS (third column) (cf. Kayser and Stone (1984)) and the model (fourth column) (cf. Sect. 2). The errors between the measurements and the model are given in the fifth column. The particle number densities (second column) are calculated by the assumption of harmonic emission.

At 5 AU our model provides a particle number density of , an electron plasma frequency of 4.15 kHz and a solar wind speed of 533 km s^-1. These values were also approximately found at 5 AU in the ecliptic plane by the ULYSSES satellite (Bame et al., 1992; MacDowall et al., 1996).

As already mentioned the radial density behaviour can be approximated by a barometric height formula (cf. (10)) in the corona. The analytical expression (10) is valid with and and represents an appropriate approximation of the solutions of (7) and (9) with a temperature of in the range within an error of 2%. Thus, the electron plasma frequency behaves according to

with . A plasma frequency of 6.4 MHz is calculated at . The radial distance r of a level with the plasma frequency is deduced to be

from (11). Then, a relationship between the drift rate in dynamic radio spectra and the radial velocity of the associated radio source

can be found with by means of (1), (10), (11), and (12). On the other hand, the radial behaviour of the density can be approximated by

with , , and beyound 0.2 AU. In the range (14) reflects the behaviour of the derived density model within an error of . This result agrees roughly with the density model by Bougeret et al. (1984) (cf. also Fig. 6). Using (1) and (14) the relationship between the drift rate and the radial source velocity

is found with . In (13) and (15) and should be used for the fundamental and harmonic emission, respectively.

In order to demonstrate the use of the density model derived in Sect. 2, it is applied for estimating the source velocity of solar and interplanetary type III radio bursts. As already mentioned type III radio bursts represent the radio signature of electron beams produced by solar flares and, subsequently, propagating along open magnetic field lines through the corona into the interplanetary space (cf. Suzuki and Dulk (1985) and Gurnett (1995) as a review). On December 27, 1994 a group of solar type III radio bursts have been observed by the radiospectrometer (40-800 MHz) (cf. Fig. 7) (Mann et al. 1992) of the Astrophysikalisches Institut Potsdam. They started at 170 MHz on 10:42:15 UT (cf. Fig. 7). The associated interplanetary type III burst extended to lower frequencies, i.e. up to 20 kHz, as recorded by the WAVES instrument (cf. Fig. 8) (Bougeret et al. 1995) and the URAP instrument (cf. Fig. 9) (Stone et al. 1992) aboard the WIND and ULYSSES spacecraft, respectively. The comparison of the Figs. 7, 8 and 9 suggests that the group of solar type III radio bursts was merging to a single interplanetary type III burst, i. e., the single electron beams produced in the low corona were merging to a giant electron beam in the interplanetary space. The measurements of the drift rates of these type III bursts in the dynamic radio spectra (cf. Figs. 7, 8, 9) reveals a relationship

between the drift rate (in MHz s^-1) and the frequency f (in MHz) (cf. Fig. 10). The drift rates have been determined at the leading edge of the individual type III bursts. Thus, a mean drift rate of -18.3 MHz s^-1 and -0.0255 kHz s^-1 has been observed at 85 MHz and 40 kHz, respectively. In the corona the radio emission of type III bursts can take place near the fundamental or harmonic of the electron plasma frequency (Melrose 1985). Then, a radial velocity of 43000 km s^-1 or 59000 km s^-1 is found for the type III related electrons in the corona by (13) in the case of fundamental or harmonic emission, respectively. Furthermore, the radio radiation of interplanetary type III bursts is generally assumed to be emitted at the harmonic of the electron plasma frequency (Reiner et al. 1992). Then, the radial velocity of the type III burst related electrons is found to be about 100000 km s^-1 (cf. (15)) at the 40 kHz level corresponding a radial distance of 1.14 AU from the Sun. On the other hand, Lin et al. (1996) measured the energy spectrum of these type III electrons by the 3D plasma instrument (Lin et al. 1995) aboard the WIND satellite on December 27, 1994. These electrons have energies in the range (cf. Lin et al. (1996)), which correspond to radial velocities of 22000 km s^-1  km s^-1. Thus, the beam, which is responsible for the type III burst on December 27, 1994 (cf. Figs. 7, 8, 9), has a broad energy spectrum. The radial velocities of the type III burst related electrons show that the slower electrons of the beam, i. e.,  km s^-1, generated the type III burst in the MHz range, while the faster electrons with  km s^-1 are producing the interplanetary type III burst at 40 kHz. This can be explained in the following manner: As already mentioned the radio radiation is generated by Langmuir waves or upper hybrid waves (Melrose 1985). These high frequency electrostatic waves are produced by energetic electron beams "via a beam-plasma" instability. Such an instabiltity occur if the distribution function of the electrons has a region with a positive slope, i. e., , (cf. Krall and Trivelpiece (1973)). (Here, V denotes the velocity of the electrons.) Initially, electrons with a broad energy spectrum are produced by a flare in the corona. The slower part of this electron ensemble is able to fulfill the above condition of instability. Consequently, these slow electrons are producing the solar type III burst. However these electrons are propagating along open magnetic field lines into the interplanetary space. Thus, the faster part of these electrons is running away, i. e., the fastest electrons are first to reach the interplanetary space (e. g. the 40 kHz level at 1.14 AU), where they produce the interplanetary type III burst. This scenario agrees well with the radio measurements presented in this paper and the measurements by Lin et al. (1996) (cf. Figs. 1 and 3 in Lin et al. (1996)).

Fig. 7. Dynamic radio spectrum of the solar type III burst group on December 27, 1994 as recorded by the radiospectrometer (40-800 MHz) of the Astrophysikalisches Institut Potsdam

Fig. 8. Dynamic radio spectrum of the interplanetary type III burst on December 27, 1994 measured by the WAVES instrument aboard WIND. In the RAD2 plot, signals in the 8-10 MHz range are man-made terrestrial signals, broadened by the interpolation used to provide a continuous spectrum. In the RAD1 plot, the blotchy signals from 200-400 kHz are terrestrial kilometric radiation (TKR)

Fig. 9. Dynamic radio spectrum of the interplanetary type III burst on December 27, 1994 measured by the URAP instrument aboard ULYSSES

Fig. 10. The drift rate versus the frequency F of the solar and associated interplanetary type III radio bursts on December 27, 1994 as derived from the dynamic radio spectra presented in Figs. 7, 8, and 9

The comparison between the density model (cf. Figs. 3-6) and different density measurements in the heliosphere, i. e., from the corona up to a distance of 5 AU in the interplanetary space, demonstrates that the model representing a special solution of Parker 's (1958) wind equation (9) reflects very well the radial density behaviour in the heliosphere, in particular, in the region near the ecliptic plane. The density model derived in Sect. 2 and illustrated in Figs. 1 and 2 should be regarded as a good approximation of the radial behaviour of the density in the heliosphere, although the heliosphere is spatially and temporally varying with respect to the density (cf. Schwenn (1990)). Thus, this model is a useful tool for the interpretation of solar and interplanetary radio data, especially the determining of radial source velocities from drift rates (cf. (1) in dynamic radio spectra.

© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999
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