Converters

Section 2 discussed four different models to scale TCN production rates from SLHL to any other location on Earth. All these models have in common that they require two columns of data in CosmoCalc: ``latitude'' and ``elevation''. They differ in how they quantify these two pieces of information. The scaling factors of Lal (1991) are the only ones that use the actual geographical latitude (in degrees) and elevation (in meters). Stone (2000) also uses the geographical latitude for estimating the latitude effect, but uses atmospheric pressure (in mbar) for modeling the elevation effect. Dunai (2000) uses the geomagnetic inclination (in degrees) instead of latitude, and atmospheric depth (in g/cm$^2$) instead of elevation. Finally, Desilets et al. (2003, 2006) use cut-off rigidity (in GV) for the latitude effect and atmospheric depth for the elevation effect.

All these different measures of ``latitude'' and ``elevation'' are related to each other and can be converted into each other. To facilitate the comparison of the different methods and, for example, reinterpret published literature data, CosmoCalc provides a series of easy-to-use conversion tools.

Converting different measures of ``elevation''

To convert elevation (z, in m) to atmospheric pressure (p, in mbar) (Iribane and Godson, 1992):

$$p = p_0 \left(1 - \frac{\beta_0 z}{T_0}\right)^{\frac{g_0}{R_d \beta_0}}$$ (26)

With $p_0$ the pressure at sea level, $\beta_0$ the adiabatic lapse rate, $T_0$ the temperature at sea level, $g_0$ the gravitational constant and $R_d$ the universal gas constant. In the standard atmospheric model, $\beta_0$ = 6.5 K/km, $g_0$ = 9.80665 m/s$^2$, $p_0$ = 1013.25 mbar and $T_0$ = 288.15 K. However, these values are not valid for Antarctica, where $p_0$ $\approx$ 989.1 mbar and $T_0$ $\approx$ 250 K. The modified Equation 26 can be rewritten as (Stone, 2000):

$$p_{ant} = 989.1 e^{-\frac{z}{7588}}$$ (27)

Atmospheric pressure is converted to atmospheric depth (g/cm$^2$) by:

$$d = 10 \frac{p}{g_0}$$ (28)

The reverse conversions are trivial inversions of these equations.

Converting different measures of ``latitude''

Converting latitude (L, in degrees) to geomagnetic inclination (I, in degrees) and back:

tan I = 2 tan L (29)

Converting latitude to geomagnetic cut-off rigidity (in GV) for a geomagnetic field strength M, compared to the 1945 reference value ($M_0$ = 8.085$\times$10$^2$ A m$^2$):

$$R_c = \sum_{i=1}^{6}\left(e_i + f_i\left(\frac{M}{M_0}\right)\right) L^i$$ (30)

The default value for M/$M_0$ = 1, but can be changed by clicking the ``Option'' button of the CosmoCalc conversion form. $e_1,...,$$e_6$ and $f_1,...,$$f_6$ are defined in Table 8 of Desilets and Zreda (2003). The reverse operation of Equation 30 does not have an analytical solution and is solved iteratively with Newton's method.