Australian Cosmic Ray Modulation Research

M. L. Duldig
, PASA, 18 (1), in press.

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Contents Page: Volume 18, Number 1

Subsections




Cosmic Ray Modulation



The Heliosphere

The heliosphere is the region of space where the solar wind momentum is sufficiently high that it excludes the interstellar medium. This region is thus dominated by the solar magnetic field carried outward by the solar wind plasma. Galactic cosmic rays beyond this region are considered to be temporally and spatially isotropic, at least over timescales of decades to centuries. It is likely that the heliosphere is not spherical but that it interacts with the interstellar medium as shown schematically in Figure 1.

Figure 1: A Schematic view of the heliosphere and its interaction with the interstellar medium. (From Venkatesan & Badruddin 1990).
\begin{figure} \begin{center} \epsfig{file=mld-fig01.eps,height=9cm} \end{center} \end{figure}

Cosmic rays enter the heliosphere due to random motions and diffuse inward toward the Sun, gyrating around the interplanetary magnetic field (IMF) and scattering at irregularities in the field. They will also experience gradient and curvature drifts (Isenberg & Jokipii 1979) and will be convected back toward the boundary by the solar wind and lose energy through adiabatic cooling, although the latter process is only important below a few GeV and does not affect ground based observations. The combined effect of these processes is the modulation of the cosmic ray distribution in the heliosphere (Forman & Gleeson 1975). It should be remembered that the approximately 11-year solar activity cycle is reflected in the strength of the IMF, the frequency of coronal mass ejections (CMEs) and shocks propagating outward and the strength of those shocks. The solar magnetic field reverses at each solar activity maximum resulting in 22-year cycles as well. The field orientation is known as its polarity and is positive when the field is outward from the Sun in the northern hemisphere (e.g. during the 1970's and 1990's) and negative when the field is outward in the southern hemisphere. A positive polarity field is denoted by A>0 and a negative field by A<0.



Heliospheric Neutral Sheet

The expanding solar wind plasma carries with it the IMF. A neutral sheet separates the field into two distinct hemispheres; one above the sheet with the field either emerging from or returning to the Sun and the other below the sheet with the field in the opposite sense. The solar magnetic field is not aligned with the solar rotation axis and is also more complex than a simple dipole. As a result, the neutral sheet is not flat but wavy, rotating with the Sun every 27 days. At solar minimum the waviness of the sheet is limited to about 10o helio-latitude but near solar maximum the extent of the sheet may almost reach the poles. Figure 2 shows an artists impression of the structure of the neutral sheet for relatively quiet solar times.

Figure 2: Artists impression of the structure of the heliospheric neutral sheet. Artist: Werner Heil - 1977, Commissioning Scientist: John M. Wilcox.
\begin{figure} \begin{center} \epsfig{file=mld-fig02.eps,height=8cm} \end{center} \end{figure}

With the rotation of the sheet every 27 days the Earth is alternately above and below the sheet and thus in an alternating regime of magnetic field directed toward or away from the Sun (but at an angle of 45o to the west of the Sun-Earth line). This alternating field orientation at the Earth's orbit is known as the IMF sector structure. The neutral sheet structure is such that there are usually two or four crossings per solar rotation. The example in Figure 2 is for a four sector IMF.



Cosmic Ray Transport

Early work by Parker (1965) and Gleeson & Axford (1967) paved the way for the theoretical formalism developed by Forman & Gleeson (1975) that describes the cosmic ray density distribution throughout the heliosphere. Isenberg & Jokipii (1979) further developed the treatment of the distribution function. Here we briefly summarize the formalism following Hall et al. (1996). If F(x, p, t) describes the distribution of particles such that

\begin{displaymath}\textit{p$^{2}$\ F}(\textbf{x}, \textbf{p}, \textit{t})\; \... ...^{3}\textit{x}\;\textnormal d\textit{p} \;\textnormal d\Omega\end{displaymath}


is the number of particles in a volume d3x and momentum range p to p + dp and centred in the solid angle d$\Omega$ then Isenberg and Jokipii (1979) showed that

\begin{displaymath} \frac{\partial U}{\partial t}+\nabla \cdot \textbf{S}=0 \end{displaymath} (1)

where

\begin{displaymath}U(\textbf{x}, p, t)=p^{2} \int _{4\pi} F(\textbf{x}, \textbf{p}, t)\;\textnormal d\Omega\end{displaymath}


and S is the streaming vector:

\begin{displaymath} \textbf{S}(\textbf{x}, p, t)=CU\textbf{V}-\kappa(\nabla U)... ...1+(\omega\tau)^{2}}\;(\nabla U\times \widehat{\textbf{B}}) \end{displaymath} (2)

and $\omega$, gyro-frequency of the particle's orbit; $\tau$, mean time between scattering; $\kappa$, diffusion coefficient (isotropic); C, Compton-Getting coefficient (Compton & Getting 1935, Forman 1970);

$\widehat{\textbf{B}}$, unit vector in the direction of the IMF; r, the radial direction in a heliocentric coordinate system; V, solar wind velocity; and U, number density of cosmic ray particles. As already noted, adiabatic cooling is relatively unimportant at the energies observed by ground based systems and so it has not been included in Equation 1. Equation 2 may be considered in several parts. The first term describes the convection of the cosmic ray particles away from the Sun by the solar wind. The second and third terms represent diffusion of the particles in the heliosphere parallel to and perpendicular to the IMF respectively. The last term describes the gradient and curvature drifts. Jokipii (1967, 1971) expressed Equation 2 in terms of a diffusion tensor

\begin{displaymath} S=CU\textbf{V}-\underline{\underline{\kappa}}\cdot (\nab... ...t & 0 \\ 0 & 0 & \kappa_{\vert\vert} \end{array} \right) \end{displaymath} (3)

where $\kappa _{\vert\vert}$, $\kappa _\bot$ are the parallel and perpendicular diffusion coefficients and the off-diagonal elements $\kappa _T$ are related to gradient and curvature drifts (see Equation 5 below, Isenberg & Jokipii 1979). Then

\begin{displaymath} \frac{\partial U}{\partial t}= -\nabla\cdot(CU\textbf{V}-\underline{\underline{\kappa}}\cdot\nabla U) \end{displaymath} (4)

Equation 4 is a time dependent diffusion equation known as the transport equation. If we note that

\begin{displaymath} \begin {array}{lll} \left(\displaystyle\frac{\partial U}... ...^S\cdot\nabla U)+\textbf{V}_D\cdot\nabla U \end{array} \end{displaymath} (5)

where

$(\partial U/\partial t)^D$ refers only to the non-convective terms in Equation 4 and $\kappa^S$ and $\kappa^A$ refer to

$\underline{\underline{\kappa}}$ being split into symmetric and anti-symmetric tensors, we find that

$\nabla\cdot\kappa^A$ is the drift velocity, VD, of a charged particle in a magnetic field with a gradient and curvature. Thus Equation 4 is an equation explicitly representing the transport of cosmic rays in the heliosphere by convection, diffusion and drift.



Modulation Model Predictions

The diffusion and convection components of Equation 4 are independent of the solar polarity and will only vary with the solar activity cycle. Conversely, the drift components will have opposite effects in each activity cycle following the field reversals. Jokipii et al. (1977) and Isenberg & Jokipii (1978) investigated the effects of this polarity dependence by numerically solving the transport equation. They showed that the cosmic rays would essentially enter the heliosphere along the helio-equator and exit via the poles in the A<0 polarity state. In the A>0 polarity state the flow would be reversed with particles entering over the poles and exiting along the equator. This is shown schematically in Figure 3 (Duldig 2000). Kota (1979) and Jokipii & Thomas (1981) showed that the neutral sheet would play a more prominent role in the A<0 state when cosmic rays entered the heliosphere along the helio-equator and would interact with the sheet. Because particles enter over the poles in the A>0 state they rarely encounter the neutral sheet on their inward journey and the density is thus relatively unaffected by the neutral sheet in this state.

Figure 3: Global cosmic ray transport predicted by modern modulation models. (From Duldig 2000).
\begin{figure} \begin{center} \epsfig{file=mld-fig03.eps,height=6cm} \end{center} \end{figure}

It was clear from the models that there would be a radial gradient in the cosmic ray density and that the gradient would vary with solar activity. Thus the cosmic ray density would exhibit the 11-year solar cycle variation with maximum cosmic ray density at times of solar minimum and minimum cosmic ray density at times of solar maximum activity (and field reversal). Figure 4 shows this anti-correlation from the long record of the Climax neutron monitor (for the data source see http://ulysses.uchicago.edu/NeutronMonitor/Misc/neutron2.html).

Figure 4: Long term Climax neutron monitor observations and smoothed sunspot numbers. Solar magnetic reversals for each pole are indicated.
\begin{figure} \begin{center} \epsfig{file=mld-fig04.eps,height=6cm} \end{center} \end{figure}

Jokipii & Kopriva (1979) extended the analysis and showed that the A<0 polarity would have larger radial gradients of particles. It is also apparent from Figure 4 that the cosmic ray peaks at solar minimum alternate from sharply peaked in the A<0 polarity state to flat topped in the A>0 state. This is not well fitted by modulation models but is clearly related to the polarity differences and probably to the effects of the neutral sheet on the cosmic ray transport shown in Figure 3. Jokipii & Kopriva (1979) also found that the transport of cosmic rays would result in a minimum in the cosmic ray density at the neutral sheet during A>0 polarity states and a maximum at the neutral sheet in the A<0 state. There would therefore be a bi-directional latitudinal (or vertical) gradient symmetrical about the neutral sheet and reversing sign with each solar polarity reversal. Jokipii & Davila (1981) and Kota & Jokipii (1983) further developed the numerical solutions with more realistic models and more dimensions to the models. They found that the minimum density at the neutral sheet predicted for the A>0 state would be slightly offset from the neutral sheet as shown in Figure 5 (Jokipii & Kota 1983). Independently, Potgieter & Moraal (1985) made the same predictions using a model with a single set of diffusion coefficients.

Figure 5: The predicted latitudinal distribution of cosmic rays near the heliospheric neutral sheet in the A>0 polarity state. (From Kota & Jokipii 1983).
\begin{figure} \begin{center} \epsfig{file=mld-fig05.eps,height=7cm} \end{center} \end{figure}

More recent models have included polar fields that are less radial than previously thought but the predictions of the models remain generally the same (Jokipii & Kota 1989; Jokipii 1989; Moraal 1990; Potgieter & Le Roux 1992). It is worth noting that the Ulysses spacecraft found that the magnetic field at heliolatitudes up to  50o was well represented by the Parker spiral field but that there was a large amount of variance in the transverse component of the IMF (Smith et al. 1995a, 1995b).



Solar Diurnal Anisotropy

Forman & Gleeson (1975) showed that the cosmic ray particles would co-rotate with the IMF. At 1 AU this represents a speed of order 400 km s-1 in the same direction as the Earth's orbital motion (at 30 km s-1). Thus the cosmic rays will overtake the Earth from the direction of 18 hours local time as shown in Figure 6.

Figure 6: The solar diurnal anisotropy resulting from co-rotational streaming of particles past the Earth. This view from above the ecliptic plane shows local solar times. (From Hall et al. 1996; Duldig 2000).
\begin{figure} \begin{center} \epsfig{file=mld-fig06.eps,height=7cm} \end{center} \end{figure}

Drift terms were neglected by Forman & Gleeson (1975) and their results indicated that the anisotropy should have an amplitude of 0.6%. Later models by Levy (1976) and Erdös & Kota (1979) that included drifts showed that the anisotropy should have an amplitude given by:

\begin{displaymath} 0.6\times\frac{1-\alpha}{1+\alpha}\;\% \end{displaymath}


where

$\alpha=\kappa_{\bot}/\kappa_{\vert\vert}=\lambda_{\bot}/\lambda_{\vert\vert}$ is ratio of perpendicular to parallel diffusion coefficients that can be shown to be equal to the ratio of perpendicular to parallel mean free paths of the particles. The arrival direction of the anisotropy is also affected by drifts shifting from 18 hours local time in the A<0 polarity state to 15 hours local time in the A>0 state. In Figure 7 we see observations from a number of underground telescopes of the anisotropy. These observations are not corrected for geomagnetic bending so the absolute phases do not generally represent those of the anisotropy in free space. The changes in phase of the anisotropy are, however, readily apparent at the times of solar field reversal. It should also be noted that the Mawson underground north telescope views along the local magnetic field and is not subject to geomagnetic deflections (see Section 3.2 above) and shows the expected free space phases.

Figure 7: Underground observations of the solar diurnal variation uncorrected for geomagnetic bending. The change of phase after each IMF reversal is clearly seen. The years of reversal are shown. It should be noted that the Mawson underground north telescope is unaffected by geomagnetic bending and shows the phases expected. Top left: Hobart vertical; Top right: Hobart north; Centre: Mawson north; Bottom left: Embudo north; and Bottom right: Embudo vertical. (From Duldig 2000).
\begin{figure} \begin{center} \epsfig{file=mld-fig07.eps,height=15cm} \end{center} \end{figure}

Rao et al. (1963) analysed the solar diurnal anisotropy, $\xi_{SD}$, as observed by neutron monitors and concluded that it arose from a streaming of particles from somewhere close to 90o east of the Sun-Earth line (i.e. 18 hours). The spectrum was assumed to be a power law in rigidity ($\vert\xi_{SD}\vert$=$\eta$P$^{\gamma}$, where $\eta$ is an amplitude constant, $\gamma$ the spectral exponent and P is rigidity) with some cut-off to the rigidity of particles that were responsible for the anisotropy. This cut-off became known as the Upper Limiting Rigidity (Pu) of $\xi_{SD}$. Although Pu is generally employed as a sharp spectral cut-off it is in reality the rigidity at which the anisotropy ceases to contribute significantly to a telescope response. Rao et al. (1963) found that the anisotropy was independent of rigidity ($\gamma$=0) and Pu was 200 GV. Further analysis by Jacklyn & Humble (1965) found that Pu was not constant. This was confirmed by Peacock & Thambyahpillai (1967) and Peacock et al. (1968) who showed Pu varying from 130 GV during 1960-1964 to 70 GV in 1965. Duggal et al. (1967) showed that the amplitude was not constant. Jacklyn et al. (1969) were able to show that these changes were not due to a change in the spectrum but that Pu did vary in the manner described by Peacock et al. (1968) and that the amplitude also varied as described by Duggal et al. (1967). Furthermore they showed that the spectral exponent was slightly negative ($\gamma$=-0.2). Ahluwalia & Erickson (1969) and Humble (1971) also found Pu varied but did not agree about the spectral index finding that it was 0 and slightly positive respectively. Concurrently with these studies, Forbush (1967) showed that there was an $\sim $20 year cycle in variation in data recorded by ionization chambers from 1937-1965. Duggal & Pomerantz (1975) subsequently verified conclusively that there is a 22-year variation in the anisotropy that is directly related to the solar polarity. Forbush (1967) had suggested that the long term variation was due to two components. Duggal et al. (1969) investigated the two components and determined that they had the same spectrum. Ahluwalia (1988a, b) disagreed that there were two independent components always present but conceded that there were two components during the A>0 polarity state - one radial and the other aligned in the east-west (18 hours local time) direction, termed the E-W anisotropy. He argued that the radial component disappeared during the A<0 polarity state. This could explain the 22-year phase variation in the anisotropy. Swinson et al. (1990) showed that the radial component of the anisotropy was correlated with the square of the IMF magnitude indicating that the radial component must be related to the convection of particles away from the Sun. This convection is generated by IMF irregularities carried radially outward by the solar wind. The correlation found by Swinson et al. (1990) was greater during A>0 polarity states in agreement with Ahluwalia (1988a, b). Ahluwalia (1991) and Ahluwalia & Sabbah (1993) discovered a correlation between Pu and the magnitude of the IMF. Unexpectedly high values of Pu were observed after the solar maximum of 1979, increasing to 180 GV in 1983. These results were confirmed by Hall et al. (1993). The most recent analysis of the anisotropy was carried out by Hall (1995) and Hall et al. (1997). These results are reproduced in Figure 8 and are derived from a study using 7 neutron monitors, 4 underground telescopes and 1 surface telescope, covering a rigidity range of 17 GV to 195 GV from 1957 to 1990. The 11-year variation in the amplitude is clear and there is some evidence for an 11-year period in Pu. The four very large values of Pu are probably unreliable as the $\chi^{2}$ contours of the fit indicated a large range of possible solutions. The spectrum also appears to depend on the solar polarity with the A>0 polarity state (1970's) showing positive spectral indices for much of the time.

Figure 8: Solar diurnal variation, annual average best-fit parameters. Typical 1$\sigma $ errors are shown. (From Hall et al. 1997).
\begin{figure} \begin{center} \epsfig{file=mld-fig08.eps,height=9cm} \end{center} \end{figure}

Hall et al. (1997) summarized the results of analyzing the anisotropy for the period 1957-1990. They concluded that: 1. The anisotropy had a spectral index of -0.1$\pm$0.2 and an upper limiting rigidity of 100$\pm$25 GV; 2. The rigidity spectrum may be polarity dependent; 3. The spectral index is relatively constant within a polarity state but the upper limiting rigidity varies roughly in phase with the solar cycle; and 4. The amplitude of the anisotropy varies with an 11-year solar cycle variation that is not due to spectral variations.



North-South Anisotropy

Compton & Getting (1935) analysed ionization chamber data for a sidereal variation and found that the peak of the variation had a phase of about 20 hours local sidereal time. The observations were all made in the northern hemisphere. Clearly an anisotropy existed with a direction fixed relative to the background stars and not to the Sun-Earth line as for the solar diurnal anisotropy. Subsequently, Elliot & Dolbear (1951) analysed southern hemisphere data and found a sidereal diurnal variation 12 hours out of phase from the result of Compton & Getting (1935). Jacklyn (1966) studied the sidereal diurnal variation in underground data collected at Cambridge in Tasmania. He employed two telescopes, one viewing north (into the northern heliospheric hemisphere) and the other vertically (into the southern heliospheric hemisphere). The southern view produced a maximum response at a phase of 6 hours local sidereal time whilst the northern view gave a maximum response phase at 18 hours. Jacklyn (1966) attributed this to a bi-directional streaming (or pitch-angle anisotropy) along the local galactic magnetic field. Swinson (1969) disagreed proposing instead that the anisotropy responsible for the sidereal diurnal variation was IMF sector polarity dependent and directed perpendicular to the ecliptic plane. The streaming of particles perpendicular to the ecliptic had been described by Berkovitch (1970) and Swinson realised that the anisotropy would have a component in the equatorial plane. Figure 9 shows how this North-South anisotropy arises from the gyro-orbits of cosmic ray particles about the IMF. When the Earth is on one side of the neutral sheet (in one sector) there will be a component of the field parallel to the Earth's orbit as shown in the top part of Figure 9. As the neutral sheet rotates, the Earth passes into the next solar sector and this component of the field reverses as in the bottom part of Figure 9. The direction of gyration of cosmic ray particles about the field reverses with the field reversal. Because a radial gradient is present there is a higher density of cosmic rays farther from the Sun. Thus the region of higher density alternately feeds in from the south (top of Figure 9) and then the north (bottom of Figure 9). The lower density of particles on the sunward side of the figure similarly reverse giving rise to a lower flux at the opposite pole. The anisotropy simply arises from a B

x Gr flow where Gr represents the radial gradient of the particles. The flow of particles perpendicular to the helio-equator is not aligned with the Earth's rotation axis.

Figure 9: Schematic representation of the North-South anisotropy and its dependence on the IMF direction. (From Duldig 2000).
\begin{figure} \begin{center} \epsfig{file=mld-fig09.eps,height=6cm} \end{center} \end{figure}

Figure 10 shows the components of the anisotropy as viewed from the Earth and Figure 11 shows the geometry of the Earth's orbit which must also be included in an analysis of the anisotropy.

Figure 10: Geometric components of the North-South anisotropy. (From Duldig 2000).
\begin{figure} \begin{center} \epsfig{file=mld-fig10.eps,height=7cm} \end{center} \end{figure}

Figure 11: Orientation of the Earth's rotation axis to the heliographic equator. (From Duldig 2000).
\begin{figure} \begin{center} \epsfig{file=mld-fig11.eps,height=5cm} \end{center} \end{figure}

Nagashima et al. (1985) demonstrated that the solar semi-diurnal anisotropy (a pitch-angle or bi-directional anisotropy) is annually modulated leading to a spurious sidereal variation that contaminates the real sidereal diurnal variation. In the case of the North-South anisotropy this can be removed by appropriate analysis against the solar sectorization and removal of the spurious response by the method of Nagashima et al. (1985). It turned out that both Jacklyn (1966) and Swinson (1969) were correct and that a bi-directional sidereal anisotropy and a sidereal anisotropy resulting from the North-South anisotropy co-existed in the 1950's and 1960's. It would appear that the amplitude of the bi-directional anisotropy diminished greatly in the early 1970's (Jacklyn & Duldig 1985; Jacklyn 1986) and has not recovered, a result that remains unexplained.

Figure 12: Derived amplitude of the North-South anisotropy as determined from observations by northern and southern polar neutron monitors. (From Bieber & Pomerantz 1986).
\begin{figure} \begin{center} \epsfig{file=mld-fig12.eps,height=5cm} \end{center} \end{figure}

There are several ways that the North-South anisotropy may be derived from observations. The differences between northern and southern viewing telescopes at a single site, taking into account the expected responses, can be used. Similarly, the differences between the responses of northern and southern polar neutron monitors that have appropriate cones of view (see Section 6.1.1 below) may be employed (Chen & Bieber 1993). Figure 12 shows the results of such an analysis by Bieber & Pomerantz (1986). Finally, analysis of the difference between the response to the sidereal diurnal anisotropy in the toward and away sectors of the IMF can be employed to derive the anisotropy (Hall 1995; Hall et al. 1994a; Hall et al. 1995a).

Figure 13: Observed annual average toward-away sidereal diurnal vectors from a sample of stations. The circles represent 1$\sigma $ errors for individual years. (From Duldig 2000; Hall 1995; Hall et al. 1994a, b, 1995a, b).
\begin{figure} \begin{center} \epsfig{file=mld-fig13.eps,height=6.5cm} \end{center} \end{figure}

A number of recent studies of the anisotropy have been undertaken by Australian researchers (Hall 1995; Hall et al. 1994a, b, 1995a, b). Figure 13 shows some of the results from these studies that involved almost 200 detector years of observation from twelve telescope systems at eight locations around the globe.



Deriving Modulation Parameters from Observations

Yasue (1980) and Hall et al. (1994a) have presented a complete description of the derivation of the anisotropy, $\xi_{NS}$, the rigidity spectrum and, as a result, the radial density gradient from multiple telescope and neutron monitor measurements of the sidereal variation. Assuming that there is little anisotropy arising from perpendicular diffusion compared with that caused by drifts, they showed that the radial density gradient as a function of rigidity, Gr(P) is

\begin{displaymath} G_r(P)\approx-\frac{\xi^{T-A}_{NS}(P)}{\rho\sin\chi} \end{displaymath} (6)

where $\rho$ is the gyro-radius of a particle at rigidity P and $\chi$ is the angle of the IMF to the Sun-Earth line (typically 45o).

$\xi^{T-A}_{NS}(P)$ is a measurement of half the difference between $\xi_{NS}$ averaged over periods when the Earth is in toward IMF sectors and when the Earth is in away IMF sectors. So it is possible to obtain a measure of the radial gradient at 1 AU directly from measurements of the sector dependent sidereal diurnal variation. In a benchmark paper Bieber & Chen (1991) further developed the cosmic ray modulation theory and showed that

\begin{displaymath} \overline{\lambda_{\vert\vert}G_{r}}=\frac{1}{\cos\chi}\le... ...{1}_{1}\Big)+ \eta_{ODV}\sin\chi+\eta_{c}\cos\chi\right] \end{displaymath} (7)

where ASD and tSD are the annual average amplitude and phase of the solar diurnal anisotropy ($\xi_{SD}$),

$\delta A^{1}_{1}$ and

$\delta t^{1}_{1}$ are the coupling coefficients that correct the amplitude and phase respectively to the free space values of the anisotropy beyond the effect of the Earth's magnetic field(Yasue et al. 1982; Fujimoto et al. 1984), G(P) is the rigidity spectrum of the anisotropy, $\eta_{ODV}$ (=0.045%) is the orbital doppler effect arising from the motion of the Earth around its orbit, $\eta_{c}$ is the Compton-Getting effect arising from the convection of cosmic rays by the solar wind and $\chi$ is the angle of the IMF at the Earth. Forman (1970) showed that $\eta_{c}$=1.5 assuming a solar wind speed of 400 km s-1 whilst Chen & Bieber (1993) used in situ solar wind measurements and found that there was no significant difference from the Forman (1970) approximation. The parameters ASD and tSD are directly derived from observations whilst the spectrum can be deduced from observations by a number of telescopes with differing median rigidities of response. The remaining parameters may be considered constants. It is therefore possible to determine the average annual product of the radial gradient, Gr, and the parallel mean free path $\lambda _{\vert\vert}$. Figures 14 and 15 show determinations of the product for neutron monitors and muon telescopes respectively.

Figure 14: Three year running average of

$\overline {\lambda _{\vert\vert}G_{r}}$ for selected neutron monitor observations. Vertical lines indicate years of solar minimum. (From Bieber & Chen 1991).

\begin{figure} \begin{center} \epsfig{file=mld-fig14.eps,height=8cm} \end{center} \end{figure}

Figure 15:

$\overline {\lambda _{\vert\vert}G_{r}}$ derived from muon telescope observations for particles between 50 and 195 GV. (From Hall et al. 1994b).

\begin{figure} \begin{center} \epsfig{file=mld-fig15.eps,height=10cm} \end{center} \end{figure}

Bieber & Chen (1991) also showed that

\begin{displaymath} G_{\vert z\vert}=\frac{\mathrm{sgn(I)}}{\rho} \left[\alp... ...{1}_{1}\Big)+ \eta_{ODV}\cos\chi-\eta_{c}\cos\chi\right] \end{displaymath} (8)

where

\begin{displaymath} \mathrm{sgn(I)}=\Bigg\{ \begin{array}{l} +1, \mathrm{A... ... -1, \mathrm{A<0\quad IMF\; polarity\; state} \end{array} \end{displaymath}


All the parameters are directly measured or known except for

$\alpha(=\lambda_{\bot}/\lambda_{\vert\vert})$. The correct value of $\alpha$ has been strongly debated in the literature. Palmer (1982) estimated consensus values of the mean free paths from earlier studies. From his conclusions $\alpha$ ranged between about 0.08 at 0.001 GV and 0.02 at 4 GV. Ip et al. (1978) derived a value of 0.26$\pm$0.08 at 0.3 GV and Ahluwalia & Sabbah (1993) estimated it must be <0.09. Bieber & Chen (1991) assumed a value of 0.01 for their study. Hall et al. (1995b) studied the effect of varying $\alpha$ on derived modulation parameters. They found that the results were relatively insensitive to values of $\alpha$ between 0.01 and 0.1. They also derived upper limits to the value of $\alpha$ at various rigidities for both polarity states. In the A<0 state the upper limit was 0.3 for rigidities between 17 GV and 185 GV. In the A>0 state the situation was quite different with an upper limit of about 0.15 at 17 GV and increasing with rigidity to very high values (>0.8) at 185 GV. There appeared to be a strong dependence of the maximum value on the rigidity although this does not guarantee that the actual value is similarly dependent. It would appear that a general consensus would be a value of $^{<}_{\sim}$0.1 for neutron monitors but that higher rigidity values require further study.



Separating Gr and $\lambda _{\vert\vert}$

In the previous section we saw how the radial gradient, Gr, and the average product of the radial gradient and the parallel mean free path,

$\overline{G_{r}\lambda_{\vert\vert}}$ could be independently determined from observations of the North-South anisotropy and the solar diurnal anisotropy respectively. If we assume that

$\overline{G_{r}\lambda_{\vert\vert}}=\overline{G_{r}}\cdot\overline{\lambda_{\vert\vert}}$ then we are able to separate out the parallel mean free paths of cosmic rays near 1 AU with Equations 6 and 7. Chen and Bieber (1993) extended their formalism to show that

\begin{displaymath} \overline{G_{r}}=\frac{\xi_{NS}\pm\sqrt{\xi_{NS}^{2}+4\alp... ...xi_{\bot}-\alpha\tan\chi\xi_{\vert\vert})}}{2\rho\sin\chi} \end{displaymath} (9)

Hall (1995) derived a different but equivalent form of the equation

\begin{displaymath} \overline{G_{r}}=\frac{-\xi_{NS}\pm\sqrt{\xi_{NS}^{2}+4\rh... ...\lambda_{\vert\vert}})G_{\vert z\vert}}} {2\rho\sin\chi} \end{displaymath} (10)

Either form of the equation is more accurate than the approximation given in Equation 6 although they introduce the parameter $\alpha$ discussed above in relation to the vertical gradient, G|z| (see Section 5.7). The most recent analyses of this type were undertaken by Hall et al. (1995b, 1997). Their results are reproduced in Figures 16 and 17, for 17 GV particles from neutron monitor observations and 185 GV particles from Hobart underground observations respectively.

Figure 16: Determinations of Gr, top,

$\overline {\lambda _{\vert\vert}G_{r}}$, middle, and $\lambda _{\vert\vert}$ for 17 GV particles, derived from neutron monitor observations. The solid lines are three point running averages. Error bars are 1$\sigma $. (From Hall et al. 1997).

\begin{figure} \begin{center} \epsfig{file=mld-fig16.eps,height=11cm} \end{center} \end{figure}

Figure 17: Determinations of $\lambda _{\vert\vert}$ for 185 GV particles, derived from the Cambridge underground muon telescope observations. Error bars are 1$\sigma $. (From Hall et al. 1997).
\begin{figure} \begin{center} \epsfig{file=mld-fig17.eps,height=6cm} \end{center} \end{figure}

Hall et al. (1994a) concluded that there was an 11-year variation in the radial gradient that was rigidity dependent as expected from modelling. In their extended analysis Hall et al. (1997) found that the 11-year variation was less convincing when the analysis was extended to the end of the 1980's although the rigidity dependence remained. They also showed that

$\overline {\lambda _{\vert\vert}G_{r}}$ had a 22-year variation with a smaller 11-year variation superposed, both variations being in phase, resulting in smaller values in the A>0 polarity state. They also found that

$\overline {\lambda _{\vert\vert}G_{r}}$ had a greater rigidity dependence in the A>0 polarity state. Finally they found that $\lambda _{\vert\vert}$ in the range 17-195 GV may be polarity dependent with higher values in the A<0 polarity state and that the polarity dependence was larger at higher rigidities.

Figure 18: Determination of the symmetric latitudinal gradient, G|z| from neutron monitor observations of the solar diurnal anisotropy for particles with rigidities between 17 and 37 GV. (From Bieber & Chen 1991).
\begin{figure} \begin{center} \epsfig{file=mld-fig18.eps,height=7cm} \end{center} \end{figure}

Figure 19: Determination of the symmetric latitudinal gradient, G|z| from observations of the solar diurnal anisotropy as recorded by: (a) Mawson neutron monitor ($\sim $17 GV) 1957-1990; (b) Mt Wellington neutron monitor ($\sim $17 GV) 1965-1988; (c) Embudo underground muon telescope ($\sim $135 GV) 1966-1985; and (d) Hobart underground muon telescope ($\sim $185 GV) 1957-1989. Error bars are 1$\sigma $. (From Hall et al. 1997).
\begin{figure} \begin{center} \epsfig{file=mld-fig19.eps,height=7cm} \end{center} \end{figure}



The Symmetric Latitude Gradient, G|z|

We have seen that the symmetric latitude gradient G|z| can be deduced from observations of the solar diurnal variation through the application of Equation 8. Bieber & Chen (1991) undertook the first such analysis and assumed a value of $\alpha$ = 0.01. The appropriate value of $\alpha$ has already been discussed in Section 5.7 above. A positive value of G|z| describes a local maximum in the cosmic ray density at the neutral sheet whilst a negative value represents a local minimum. The results of Bieber & Chen (1991) are reproduced in Figure 18 and clearly show the dependence of G|z| on the polarity state. The bi-directional symmetric latitude gradient does reverse at each solar polarity reversal. We must ignore the shaded periods which are the times when the field was undergoing reversal and was highly disordered. Hall et al. (1997) confirmed Bieber & Chen's results and studied the gradient at higher rigidities, finding the same dependence extended up to at least 185 GV as shown in Figure 19. In fact Ahluwalia (1993, 1994) reports a significant observation of the gradient at 300 GV. The reversal of the gradient is in accordance with drift models. The magnitude of the gradient appears to have its largest values around times of solar minimum activity and may also have slightly lower values during the A>0 polarity state. It should be noted that the results presented in Figures 17 and 18 assumed a constant spectrum for the solar diurnal anisotropy. If the spectrum is allowed to vary then the gradient reversal cannot be confirmed at rigidities above about 50 GV.
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