H. J. Fahr - K. Scherer
Institut für Astrophysik und Extraterrestrische Forschung der Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
Received 29 March 2004 / Accepted 12 May 2004
For a long time it has been suggested that at larger solar distances the interplanetary magnetic field may not be behaving as predicted by Parker`s Archimedian spiral field model. This phenomenon, partly identified as the "magnetic flux deficit'' paradox, appears as deficits in the azimuthal magnetic field components connected with "underwound'' magnetic fields, though the pick-up ion decelerated solar wind should rather lead to excesses. Up to now no satisfactory explanation for this phenomenon has been presented. In this paper we study for the first time the diamagnetic effect of pick-up ions which systematically load the solar wind with suprathermal ions at its expansion to larger distances and diamagnetize the plasma. We shall demonstrate that such ions mainly reduce the azimuthal magnetic field component by their diamagnetic action. As we can show the field deficits can easily theoretically be explained by this effect. Also some new light is now shed on the problem of the magnetic field jump to be expected at the termination shock.
Key words: magnetic fields - plasma
In the paper by Winterhalter et al. (1990) it was shown that, on the basis of PIONEER-11 magnetic field data a systematic flux deficit appears in the outer heliosphere with respect to field predictions made by the Parker model (Parker 1958). The resulting deficits amount to about 20 percent at distances of 24 AU. This phenomenon was confirmed by ongoing measurements as documented by Smith et al. (1997,2004) based on ULYSSES magnetic field measurements. However, other authors came to the conclusion, that no such deficit exists (e.g. Burlaga et al. 2002) and there might be still no sufficient observational data to decide on this phenomena. Here we argue that, on a purely theoretical basis, the flux deficit reported by (Smith et al. 1997) should be expected. The latter authors clearly revealed that the magnetic flux invariant behaves normal and interestingly enough, is revealed as latitude-independent. In contrast to the azimuthal component clearly violates Parker's expected field invariant , showing that especially at latitudes with (i.e. near the ecliptic) an azimuthal field deficit by about 10 to 15 percent is recognizable. As these authors state, this latitudinally dependent deficit can hardly be explained by unknown latitudinal variations of the solar rotation period , i.e. effects of differential solar rotation. Also, earlier explanations of an out-of-ecliptic transport of magnetic fields due to meridional solar wind flow components as offered by Pizzo & Goldstein (1987) or Suess et al. (1985) in view of the latitude-independent invariant are obviously ruled out. Hence it still remains to explain why azimuthal field components in the outer heliosphere appear to be reduced instead of enhanced due to the pile-up in the pick-up ion decelerated solar wind when compared with the expectations from Parker`s field model. It is interesting to notice that the well-known pick-up ion effect to decelerate the solar wind at larger distances (see Fahr & Rucinski 2001; Isenberg 1997; Fahr & Rucinski 1999) inducing a magnetic field pile-up should tend to produce flux excesses rather than deficits. Here, we will restrict ourselves only to the derivation of the diamagnetic effect of pick-up ions and its possible implications.
It is well known vacuum electric and magnetic fields, and , are converted into different fields, and , in dielectric and magnetically susceptible media like solid, liquid or gaseous materials or plasmas. In such media not only free electric charges and currents may be present, but also "bound'' charges in form of electrically polarizable materials or in form of "bound'' electrical currents which are connected with motions of "bound'' electric charges. In a plasma for instance charged particles may be magnetically bound to gyro-orbits and thus cause microscopic electric circle currents . One can ask the question what relations may exist between and as well as and in a space plasma like the solar wind. In a fully ionized plasma the polarizability vanishes for stationary cases (i.e. for the frequency ) and the dielectric coefficient is equal to 1, meaning that . The effective magnetic induction , measurable in a plasma background is nevertheless different from the vacuum field since the plasma matter develops bound electric currents connected with gyrating electric charges which by themselves produce magnetic fields.
Hence for instance at some heliocentric distance r in interplanetary
space the fields
taking into account the
effect of these bound currents
are thus connected by
(see e.g. Chen 1977):
At larger solar distances the PUI distribution function due to rapid
pitch-angle scattering can reasonably well be approximated by a
distribution which is essentially gyrotropic and isotropic with respect to
or its cosine
e.g. Chalov & Fahr 1998). Thus the above expression transforms to:
A fairly realistic expression for
can be derived from
results obtained for
by Chalov et al. (1995,1997) as solutions of a complete PUI phase-space
transport equation. As was shown by Fahr & Lay (2000) these
numerical results for
can fairly adequately be
represented by the following analytical formula:
Coming back now to the magnetic moment given by Eq. (5) one thus finds:
Since fields are not measured in the solar wind rest frame, but in a
spacecraft frame which is nearly identical with the heliocentric rest frame, we
have to transform Eq. (16) into the heliocentric rest frame. Therefore, we
use the special relativistic transformation of the fields from the solar wind
rest frame to the heliocentric rest frame and, reminding that the Lorentz
and in the comoving
we then find:
The Archimedian Parker field is assumed to behave as frozen-in the solar wind
flow. This is derived from the well-known frozen-in field condition:
Of interest in the ecliptic of the outer heliosphere is only
the -component of the magnetic field. With the help of
Eq. (17), we then find for the magnetic induction :
Perhaps in a more detailed consideration it also needs to be studied that pick-up ions not only represent bound currents , but they also contribute to "free'' electric currents for the period of time, , before they become fully isotropized by pitch angle scattering, i.e. as long as they are slipping along the local field lines. These currents can be estimated by and they are mainly oriented in radial direction and thus cause a latitudinal gradient of the azimuthal field component by means of the relation: .
The Rankine-Hugoniot field relation for the MHD shock given by the Poisson bracket reads (see e.g. Diver 2001; McKenzie & Westphal 1969; Baumjohann & Treumann 1996): , where is the unit vector of the shock normal.
For the measurable magnetic field
in a magnetized background plasma
the above relation thus translates to:
Especially for the case of the nose region of the solar wind termination shock
which represents a perpendicular shock, i.e. with
one hence derives with the above relation:
>From the magneto-hydrodynamic Rankine-Hugoniot shock relations (see e.g. Diver 2001) one obtains:
The functional dependence for the ratio RB/S is shown in Fig. 1 for five different values of . It can be seen that the compression ratio S can be smaller or larger than the magnetic compression ratio RB, while in the non-diamagnetic case they are identical. Furthermore, for very large compression ratios S the standard magnetohydrodynamic Rankine-Hugoniot relations are no longer valid, since depends on for , leading to negative values of .
Nevertheless, we can state here that without the knowledge of the Alfvén Mach number and the plasma Mach number the magnetic compression ratio RB cannot be fixed and not be connected with the standard compression ratio S.
|Figure 1: The ratio RB/S as function of the standard compression ratio S. The labels at the curves are the ratio of the Alfvén Mach number to the plasma Mach number .|
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We have shown, that the diamagnetic effect of PUIs has to be taken into account, because it reduces dramatically the magnetic induction in the solar wind with increasing distances. In order to get an analytic expression, we had to make some moderate simplifications to the problem. Therefore, we derived only upper limits of the diamagnetic effect, but we clearly have shown, that the decrease in the magnetic induction due to the diamagnetism of the pickup ion plasma has to be taken into account.
Furthermore we have shown, that the magnetic compression ratio RB and the standard compression ratio S are related in a complicated way and cannot be determined from each other without the knowledge of the sound speed and Alfvén speed at the shock.
Similar complications should also arise at cometary shocks, however, here the pick-up ion diamagnetism is harder to estimate since the resulting distribution functions are not well known.
The authors are grateful for financial support granted to them by the Deutsche Forschungsgemeinschaft in the frame of the project "Heliotrigger'' (Fa 97/28-1).