3.1 The physical basis of the method

3.1.1 Heliospheric modulation of cosmic rays

Before reaching the vicinity of Earth, galactic cosmic rays experience complicated transport in the heliosphere that leads to modulation of their flux. Heliospheric transport of GCR is described by Parker’s theory (Parker, 1965Toptygin, 1985) and includes four basic processes: the diffusion of particles due to their scattering on magnetic inhomogeneities, the convection of particles by out-blowing solar wind, adiabatic energy losses in expanding solar wind, drifts of particles in the magnetic field, including the gradient-curvature drift in the regular heliospheric magnetic field, and the drift along the heliospheric current sheet, which is a thin magnetic interface between the two heliomagnetic hemispheres. Because of variable solar-magnetic activity, CR flux in the vicinity of Earth is strongly modulated (see Figure 2View Image). The most prominent feature in CR modulation is the 11-year cycle, which is in inverse relation to solar activity. The 11-year cycle in CR is delayed (from a month up to two years) with respect to the sunspots (Usoskin et al., 1998). The time profile of cosmic-ray flux, as measured by the longest-running neutron-monitor Climax, is shown in Figure 2View Image (panel b) together with the sunspot numbers (panel a). Besides the inverse relation between them, some other features can also be noted. A 22-year cyclicity manifests itself in cosmic-ray modulation through the alteration of sharp and flat maxima in cosmic-ray data, originated from the charge-dependent drift mechanism. One may also note short-term fluctuations, which are not directly related to sunspot numbers but are driven by interplanetary transients caused by solar eruptive events, e.g., flares or CMEs. For the last 50 years of high and roughly-stable solar activity, no trends have been observed in CR data; however, as will be discussed later, the overall level of CR has changed significantly on the centurial-millennial timescales.

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Figure 2: Cyclic variations. Panel a: Time profiles of sunspot numbers; Panel b: Cosmic-ray flux as the count rate of Climax neutron monitor (NM) (100% NM count rate corresponds to May 1965).

Full solution of the CR transport problems is a complicated task and requires sophisticated 3D time-dependent self-consistent modelling. However, the problem can be essentially simplified for applications at a long-timescale. An assumption on the azimuthal symmetry (requires times longer that the solar-rotation period) and quasi-steady changes reduces it to a 2D quasi-steady problem. Further assumption of the spherical symmetry of the heliosphere reduces the problem to a 1D case. This approximation can be used only for rough estimates, since it neglects the drift effect, but it is useful for long-term studies, when the heliospheric parameters cannot be evaluated independently. Further, but still reasonable, assumptions (constant solar-wind speed, roughly power-law CR energy spectrum, slow spatial changes of the CR density) lead to the force-field approximation (Gleeson and Axford, 1968), which can be solved analytically. The differential intensity Ji of the cosmic-ray nuclei of type i with kinetic energy T at 1 AU is given in this case as

-----(T)(T-+-2Tr-)----- Ji(T, ϕ) = JLIS,i(T + Φi)(T + Φ )(T + Φ + 2T ) , (3 ) i r
where Φi = (Zie∕Ai)ϕ for a cosmic nuclei of i-th type (charge and mass numbers are Zi and Ai), T and ϕ are expressed in MeV/nucleon and in MV, respectively, Tr = 938 MeV. T is the CR particle’s kinetic energy, and ϕ is the modulation potential. The local interstellar spectrum (LIS) JLIS forms the boundary condition for the heliospheric transport problem. Since LIS is not measured directly, i.e., outside the heliosphere, it is not well known in the energy range affected by CR modulation (below 100 GeV). Presently-used approximations for LIS (e.g., Garcia-Muñoz et al., 1975Burger et al., 2000Jump To The Next Citation PointWebber and Higbie, 2003Jump To The Next Citation Point2009)UpdateJump To The Next Update Information agree with each other for energies above 20 GeV but may contain uncertainties of up to a factor of 1.5 around 1 GeV. These uncertainties in the boundary conditions make the results of the modulation theory slightly model-dependent (see discussion in Usoskin et al., 2005aJump To The Next Citation Point) and require the LIS model to be explicitly cited. This approach gives results, which are at least dimensionally consistent with the full theory and can be used for long-term studies1 (Usoskin et al., 2002bJump To The Next Citation PointCaballero-Lopez and Moraal, 2004Jump To The Next Citation Point). Differential CR intensity is described by the only time-variable parameter, called the modulation potential ϕ, which is mathematically interpreted as the averaged rigidity (i.e., the particle’s momentum per unit of charge) loss of a CR particle in the heliosphere. However, it is only a formal spectral index whose physical interpretation is not straightforward, especially on short timescales and during active periods of the sun (Caballero-Lopez and Moraal, 2004Jump To The Next Citation Point). Despite its cloudy physical meaning, this force-field approach provides a very useful and simple single-parametric approximation for the differential spectrum of GCR, since the spectrum of different GCR species directly measured near the Earth can be perfectly fitted by Equation 3View Equation using only the parameter ϕ in a wide range of solar activity level (Usoskin et al., 2005aJump To The Next Citation Point). Therefore, changes in the whole energy spectrum (in the energy range from 100 MeV/nucleon to 100 GeV/nucleon) of cosmic rays due to the solar modulation can be described by this single number within the framework of the adopted LIS. The concept of modulation potential is a key concept for the method of solar-activity reconstruction by cosmogenic isotope proxy.

3.1.2 Geomagnetic shielding

Cosmic rays are charged particles and therefore are affected by the Earth’s magnetic field. Thus the geomagnetic field puts an additional shielding on the incoming flux of cosmic rays. The shielding effect of the geomagnetic field is usually expressed in terms of the cutoff rigidity P c, which is the minimum rigidity a CR particle must posses (on average) in order to reach the ground at a given location and time (Cooke et al., 1991). Neglecting such effects as the East-West asymmetry, which is roughly averaged out for the isotropic particle flux, or nondipole magnetic momenta, which decay rapidly with distance, one can come to a simple approximation, called the Störmer’s equation, that describes the vertical geomagnetic cutoff rigidity Pc:

2 Pc ≈ 1.9M (Ro ∕R ) cos4λG [GV ], (4 )
where M is the geomagnetic dipole moment (in 1025 G cm3), Ro is the Earth’s mean radius, R is the distance from the given location to the dipole center, and λG is the geomagnetic latitude. This approximation provides a good compromise between simplicity and reality, especially when using the eccentric dipole description of the geomagnetic field (Fraser-Smith, 1987). The eccentric dipole has the same dipole moment and orientation as the centered dipole, but the dipole’s center and consequently the poles, defined as crossings of the axis with the surface, are shifted with respect to geographical ones.

The shielding effect is the strongest at the geomagnetic equator, where the present-day value of Pc may reach up to 17 GV in the region of India. There is almost no cutoff in the geomagnetic polar regions (λG ≥ 60∘). However, even in the latter case the atmospheric cutoff becomes important, i.e., particles must have rigidity above 0.5 GV in order to initiate the atmospheric cascade (see Section 3.1.3).

The geomagnetic field is seemingly stable on the short-term scale, but it changes essentially on centurial-to-millennial timescales (e.g., Korte and Constable, 2006). Such past changes can be evaluated based on measurements of the residual magnetization of independently-dated samples. These can be paleo (i.e., natural stratified archives such as lake or marine sediments or volcanic lava) or archaeological (e.g., clay bricks that preserve magnetization upon baking) samples. Most paleo-magnetic data preserve not only the magnetic field intensity but also the direction of the local field, while archeo-magnetic samples provide information on the intensity only. Using a large database of such samples, it is possible to reconstruct (under reasonable assumptions) the large-scale magnetic field of the Earth. Data available provides good global coverage for the last 3 millennia, allowing for a reliable paleomagnetic reconstruction of the true dipole moment (DM) or virtual dipole moment2 (VDM) and its orientation (the ArcheoInt collaboration – Genevey et al., 2008Jump To The Next Citation Point). Less precise, but still reliable reconstructions of the DM and its orientation are possible for the last seven millennia (the CALS7K.2 model by Korte and Constable, 2005Jump To The Next Citation Point), however they may slightly underestimate the dipole moment, especially in the earlier part of the period (Korte and Constable, 2008Jump To The Next Citation Point). No reliable directional paleomagnetic reconstruction is presently possible on a longer timescale, because of the spatial sparseness of the paleo/archeo-magnetic samples in the earlier part of the Holocene. Therefore, only the virtual axial dipole moment3 (VADM) can be estimated before ca. 5000 BC (Yang et al., 2000Jump To The Next Citation PointKnudsen et al., 2008Jump To The Next Citation Point).UpdateJump To The Next Update Information Note that the strong assumption of the coincidence of magnetic and geographical axes may lead to an overestimate of the true dipole moment, if the magnetic axis is actually tilted. Accordingly, we assume that the two models of Y00 and CALS7K.2 bind the true dipole moment variations. These paleomagnetic reconstructions are shown in Figure 3View Image. All paleomagnetic models depict a similar long-term trend – an enhanced intensity during the period between 1500 BC and 500 AD and a significantly lower field before that.

Changes in the dipole moment M inversely modulate the flux of CR at Earth, with strong effects in tropical regions and globally. The migration of the geomagnetic axis, which changes the geomagnetic latitude λG of a given geographical location is also important; while not affecting the global flux of CR, it can dramatically change the CR effect regionally, especially at middle and high latitude (Kovaltsov and Usoskin, 2007). These changes affect the flux of CR impinging on the Earth’s atmosphere both locally and globally and must be taken into account when reconstructing solar activity from terrestrial proxy data (Usoskin et al., 2008).UpdateJump To The Next Update Information Accounting for these effects is quite straightforward provided the geomagnetic changes in the past are known independently, e.g., from archeo and paleo-magnetic studies. However, because of progressively increasing uncertainties of paleomagnetic reconstructions back in time, it presently forms the main difficulty for the proxy method on the long-term scale (Snowball and Muscheler, 2007Jump To The Next Citation Point), especially in the early part of the Holocene. On the other hand, the geomagnetic field variations are relatively well known for the last few millennia (Genevey et al., 2008Jump To The Next Citation PointKorte and Constable, 2008).

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Figure 3: Geomagnetic field intensity over millennia: VADM compilation by Yang et al. (2000Jump To The Next Citation Point, – Y00 curve with 1σ statistical errors of the sample distribution); dipole moment according to Hongre et al. (1998, – HBI red dots) since 800 AD, dipole moment according to CALS7K.2 model (Korte and Constable, 2005Jump To The Next Citation Point, – K05 magenta curve with 1 σ error band) as well as a recent ArcheoInt compilations of VADM/VDM (Genevey et al., 2008, – A08 azure diamonds) and (Knudsen et al., 2008, – K08 blue stars).
UpdateJump To The Next Update Information

3.1.3 Cosmic-ray–induced atmospheric cascade

When an energetic CR particle enters the atmosphere, it first moves straight in the upper layers, suffering mostly from ionization energy losses that lead to the ionization of the ambient rarefied air. However, after traversing some amount of matter (the nuclear interaction mean-free path is on the order of 100 g/cm2 for a proton in the air) the CR particle may collide with a nucleus in the atmosphere, producing a number of secondaries. These secondaries have their own fate in the atmosphere, in particular they may suffer further collisions and interactions forming an atmospheric cascade (e.g., Dorman, 2004). Because of the thickness of the Earth’s atmosphere (1033 g/cm2 at sea level) the number of subsequent interactions can be large, leading to a fully-developed cascade (also called an air shower) consisting of secondary rather than primary particles. Three main components can be separated in the cascade:

The development of the cascade depends mostly on the amount of matter traversed and is usually linked to residual atmospheric depth rather than to the actual altitude, that may vary depending on the exact atmospheric density profile.

Cosmogenic isotopes are a by-product of the hadronic branch of the cascade (details are given below). Accordingly, in order to evaluate cosmic-ray flux from the cosmogenic isotope data, one needs to know the physics of cascade development. Several models have been developed for this cascade, in particular its hadronic branch with emphasis on the generation of cosmogenic isotope production. The first models were simplified quasi-analytical (e.g., Lingenfelter, 1963O’Brien and Burke, 1973) or semi-empirical models (e.g., Castagnoli and Lal, 1980Jump To The Next Citation Point). With the fast advance of computing facilities it became possible to exploit the best numerical method suitable for such problems – Monte-Carlo (e.g., Masarik and Beer, 1999Jump To The Next Citation Point2009Jump To The Next Citation PointWebber and Higbie, 2003Jump To The Next Citation PointWebber et al., 2007Jump To The Next Citation PointUsoskin and Kovaltsov, 2008Jump To The Next Citation PointKovaltsov and Usoskin, 2010Jump To The Next Citation Point). The fact that models, based on different independent Monte-Carlo packages, namely, a general GEANT tool and a specific CORSIKA code, yield similar results provides additional verification of the approach. UpdateJump To The Next Update Information

3.1.4 Transport and deposition

A scheme for the transport and redistribution of the two most useful cosmogenic isotopes, 14C and 10Be, is shown in Figure 4View Image. After a more-or-less similar production, the two isotopes follow quite different fates, as discussed in detail in Sections 3.2.3 and 3.3.3. Therefore, expected terrestrial effects are quite different for the isotopes and comparing them with each other can help in disentangling solar and climatic effects (see Section 3.6.3).

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Figure 4: Schematic representation of 14C (left) and 10Be (right) production chains. The flux of cosmic rays impinging on the Earth is affected by both heliospheric modulation and geomagnetic field changes. The climate may affect the redistribution of the isotopes between different reservoirs. Dashed line denotes a possible influence of solar activity on climate.

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