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Astron. Astrophys. 358, 347-352 (2000)

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4. Application to solar modulation

To demonstrate the potential and flexibility provided with the solution Eq. (10) we apply it to the problem of ACR and GCR proton modulation assuming for the turbulence the mixed model (M) with [FORMULA]. In order to do so we have to select physical parameters being typical for the heliosphere and specify boundary spectra.

4.1. Turbulence in the heliosphere

The theory of the diffusion tensor in the heliosphere is usually formulated in terms of the diffusion along ([FORMULA]) and perpendicular ([FORMULA] with respect to the local heliospheric magnetic field [FORMULA]. While from quasilinear theory it is found that [FORMULA] so that [FORMULA] in the outer heliosphere, more recent studies, based on comprehensive magnetohydrodynamic turbulence models including turbulence generation by stream interactions as well as by pick-up ions, resulted in more complicated dependencies of the parallel diffusion coefficient on heliocentric distance (see, e.g., Burger & Hattingh 1998; Zank et al. 1998). Combining these new results with the assumption of long standing that the perpendicular diffusion is of the order of [FORMULA] that has been confirmed by recent numerical analyses (e.g., Giacalone & Jokipii 1999) one obtains a radial variation of the spatial diffusion [FORMULA] with [FORMULA] ([FORMULA] denotes the angle between the average magnetic field [FORMULA] and the radial direction, so that [FORMULA] beyond a few astronomical units from the Sun). Also, recent analyses of ACR and GCR data (Steenberg 1998; Moraal et al. 1999) resulted in the finding that [FORMULA].

In view of this knowledge about the spatial diffusion in the heliosphere, we select [FORMULA] and [FORMULA]. Besides being a typical value in the interval suggested by observation and theory, this value has the additional advantage to result in the modified Bessel function [FORMULA] in Eq. (10) when combined with a constant solar wind speed which we assume to be [FORMULA], i.e. [FORMULA] (Eq. (2) above).

4.2. The boundary spectra of ACRs and GCRs

The solution Eq. (10) is formulated in terms of a source function [FORMULA]. So, the task is to determine source functions for ACRs and GCRs such that the correct boundary spectra [FORMULA] and [FORMULA] result.

4.2.1. ACRs

According to the theory of diffusive shock acceleration (for a review see, e.g., Drury 1983), for ACRs one expects the accelerated spectrum at the solar wind termination shock to be related to the scattering centers' compression ratio s (Vainio & Schlickeiser 1999), namely [FORMULA] with [FORMULA]. The spectrum cuts off in the range [FORMULA] due to the finite radius of the termination shock.

Such a spectrum will result if the ACR source function is chosen as:


i.e. we use an exponential cut-off controlled with the constant momentum [FORMULA], have defined a normalisation momentum [FORMULA], and assume the solar wind termination shock to be located at a heliocentric distance [FORMULA]. The factor [FORMULA] allows for suitable normalisation.

4.2.2. GCRs

Many studies have been carried out to derive the interstellar proton spectrum (for a recent compilation see, e.g., Fig. 6 in Mori 1997). We selected the one obtained by Webber et al. (1987) who found [FORMULA]. E and [FORMULA] denote the kinetic and the rest energy of a proton, and c is the speed of light.

In order to obtain such a spectrum from Eq. (10) the GCR source function has to be taken as


Analogously to the ACRs, [FORMULA] is a normalisation factor. The delta function indicates that we assume the solar wind termination to define the modulation barrier for GCRs. This barrier might actually be located farther out, however this is unimportant for the illustration below.

For the illustration we assume that the termination shock is located at a heliocentric distance of [FORMULA] and has a compression ratio of [FORMULA], which appear as reasonable choices according to both observation (Stone et al. 1996) and theory (le Roux & Fichtner 1997).

4.3. Results

The resulting spectra, computed with a numerical integration of Eq. (10), are shown in Fig. 1. Inspection of the figure results, evidently, in the finding that all characteristic features of spherical CR modulation in the heliosphere are clearly visible. At low energies the spectra are dominated by the ACR contribution clearly showing the correct power-law behaviour. At high energies the spectra are dominated by GCRs and the amount of modulation decreases with increasing kinetic energy. The modulated spectra have the expected shape (Christian et al. 1995; le Roux et al. 1996; le Roux & Fichtner 1997). At small heliocentric distances, the spectrum still contains information about the source functions according to the asymptotic limit Eq. (11). Finally, at low kinetic energies the spectra exhibit the expected asymptotic form [FORMULA] according to Eq. (12), reflecting the dominance of adiabatic cooling in this energy range.

[FIGURE] Fig. 1. The modulated spectra of ACRs and GCRs in the heliosphere. The solar wind termination shock marking the position of the sources is located at [FORMULA]. The solid lines are the combined spectra, the dotted and dashed lines indicate the individual contributions from ACRs and GCRs, respectively.

As a further illustration, we study the dependence of the modulation on the underlaying turbulence model. Fig. 2 displays the ACR spectra at a heliocentric distance of [FORMULA] for the three turbulence models introduced in Sect. 2. The importance of the chosen turbulence model at low energies is obvious. The ordering of the curves is a consequence of the choice [FORMULA] (with [FORMULA] being the proton rest mass). For [FORMULA] one has [FORMULA] in Eq. (5), and a greater [FORMULA] yields a lower diffusion coefficient. Thus, the greater [FORMULA], the lower the differential intensities below [FORMULA]. At high energies the flux levels are approaching each other due to the exponential cut-off.

[FIGURE] Fig. 2. The modulated spectra of ACRs at a heliocentric distance of [FORMULA] for the three turbulence models considered: slab Alfvén waves ([FORMULA], dotted line), isotropic fast magnetosonic waves ([FORMULA], dashed line) and the mixed case ([FORMULA], solid line).

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000