Production rate scaling factors

In the simplest case (no shielding or burial), only three pieces of information are needed to calculate an exposure age or erosion rate: TCN concentration, half-life and production rate. Production rates are the ``achilles heel'' of the TCN method. There exist only a few calibration sites where TCN production rates are accurately known thanks to the availability of independent age constraints (e.g., Nishiizumi et al., 1989; Niedermann et al., 1994; Kubik et al., 1998). These production rates are only valid for the specific conditions (latitude, elevation, age) of each particular calibration site. To apply the TCN method to other field settings, the production rates must be scaled to a common reference at sea level and high latitude (SLHL). Up to 20% uncertainty is associated with this scaling, constituting the bulk of TCN age uncertainty.

Although several efforts have been made to directly measure TCN production rate scaling with latitude and elevation using artificial $H_2O$ and $SiO_2$ targets (Nishiizumi et al., 1996; Brown et al., 2000; Graham et al., 2007), all currently used scaling models are based on neutron monitor surveys. The oldest and still most widely used scaling model is that of Lal (1991). This model is a simple set of polynomial equations giving the (spallogenic + muogenic) production rate relative to SLHL as a function of geographic latitude and elevation. In CosmoCalc, Lal's scaling factors can be calculated by simply selecting two columns of latitude (in degrees) and elevation (in meters) data and clicking ``OK''.

Lal's scaling factors use elevation as a proxy for atmospheric depth, assuming a standard atmosphere approximation. Stone (2000) noted that this approximation is not valid in certain areas, such as Antarctica and Iceland. To avoid the systematic errors caused by the standard atmosphere model, Stone (2000) recast the polynomial equations of Lal (1991) in terms of air pressure instead of elevation. A second improvement of the Stone (2000) model is the independent scaling of TCN production by slow (negative) muons (Heisinger et al., 2002). In spite of this added complexity, the CosmoCalc interface for Stone (2000) scaling factors is identical to that for Lal (1991) scaling: the user simply needs to provide two columns of data, one with latitude and one with air pressure (in mbar). The scaling factors of Stone (2000) can be different for different nuclides, because the importance of muons depends on the nuclide of interest. Because most TCN production rate calibration sites are not located at SLHL, it is crucially important to scale the production rates using the same method as the unknown sample. This is exactly what CosmoCalc does when the user selects a nuclide from the scroll-down menu of the scaling-form. Thus, the program ``forces'' the user to be consistent. The program comes with a set of default calibration sites, but these can be changed. Also the relative importance of the production pathways (neutrons, slow and fast muons) can be changed (Section 7).

Behind the scaling models of both Lal (1991) and Stone (2000) lies an extensive database of neutron monitor measurements, ordered according to geomagnetic latitude. This ordering implies that Earth's magnetic field can be accurately approximated by a simple dipole. To avoid this approximation, Dunai (2000) ordered the neutron monitor data according to geomagnetic inclination, which also represents the non-dipole field. Just like Stone (2000), also Dunai (2000) incorporates separate muon scaling and atmospheric effects. However, atmospheric depth (g/cm$^2$) is used instead of air pressure. Using CosmoCalc it is very easy to convert air pressure to elevation or atmospheric depth and back (Section 6).

Ultimately, both geomagnetic latitude and inclination are merely proxies for a more fundamental physical quantity: the geomagnetic cutoff rigidity ($R_c$), which is the minimum momentum per unit charge (in GV), required for a primary cosmic ray to reach the atmosphere. Ordering the neutron monitor data according to this parameter results in yet another set of scaling factors. Unfortunately, at least three different methods for calculating $R_c$ exist in the literature. Dunai (2001) used a database of horizontal magnetic field intensities and inclinations to estimate the cutoff-rigidity of an equivalent axial dipole field, for which an approximate analytical solution exists. Desilets and Zreda (2003) used a model based on trajectory tracing of an axial dipole field, which is done by numerically testing the feasibility of vertically incident anti-protons to travel from the top of the atmosphere back into space. Finally, Lifton et al. (2005) fit a cosine function to a database of geomagnetic latitudes versus trajectory traced cutoff rigidities for the 1955 magnetic field. These authors consider the scatter around their fit to be a realistic estimator of the natural variability of $R_c$ at any given geomagnetic latitude.

In order to avoid confusion, CosmoCalc currently only implements one of these three methods, namely that of Desilets and Zreda (2003). The scaling model of Desilets and Zreda (2003) makes a distinction between slow and fast muons, each of which scales differently. In an update of their model, Desilets et al. (2006) did a neutron monitor survey at low latitudes, which were undersampled by previous surveys. This resulted in a slightly different set of attenuation length polynomials for spallogenic reactions. The method of Desilets et al. (2003, 2006) is probably the most accurate of all the scaling models implemented in CosmoCalc. It is also the most complex model, but this did not change the user interface. Using the scaling models of Desilets et al. (2003, 2006) is just as easy as that of Lal (1991) in CosmoCalc. Default values for the relative SLHL production rate contributions of neutrons, slow and fast muons can be changed in the Settings menu (Section 7).

The scaling models of Dunai (2001), Desilets et al. (2003, 2006), Pigati and Lifton (2004), and Lifton et al. (2005) are a sensitive function of magnetic field intensity and solar activity, both of which are poorly constrained over geologic time. On the one hand, this temporal variability is the biggest downside to $R_c$-based models for long exposures $(>$ 20 ka; Dunai, 2001). On the other hand, such models also offer the possibility to correct for secular variation of TCN production rates for short exposures. Instructions for doing so are given by Dunai (2001) and Desilets et al. (2003), provided a local record of paleomagnetic intensity is available. Compiling such a record is something for advanced users and falls beyond the scope of CosmoCalc. The scaling models of Pigati and Lifton (2004) and Lifton (2005) are accompanied by global datasets of magnetic field intensity, polar wander and solar activity and in principle, it would be possible to incorporate these datasets into CosmoCalc. However, because they are very large (4 and 7Mb, respectively) in comparison with CosmoCalc ($\sim$ 500kb), the cost of including these scaling models was considered too high. Therefore, researchers working with $^{14}$C, where secular variation of the magnetic field is really crucial, should use CosmoCalc only as an exploratory tool, and use the spreadsheets of Pigati and Lifton (2004) and Lifton et al. (2005) for final calculations.