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SPIDER: model overview

Here you can find a detailed overview of the SPIDER formulation.

Note

This model overview is taken from the notes and contains an extended description of the equations and derivations related to the SPIDER code. It is still work in progress.

Radioactive heating

The concentration \(X_i\) of a given isotope \(i\) is 1:

\[X_i(t_\mathrm{age})= X_{i0} \exp{\left( \frac{t_\mathrm{age} \ln2}{T_{i1/2}} \right)}\]

where \(X_{i0}\) is present-day concentration, \(t_\mathrm{age}\) is age (time before present), and \(T_{i1/2}\) is half-life.

With \(t_{i0}\) as the time at which the concentration is known (e.g., 4.54 Gyrs for present-day):

\[X_i(t) = X_{i0} \exp{\left( \frac{(t_{i0}-t) \ln2}{T_{i1/2}} \right)}\]

The heat production rate for isotope \(i\) is:

\[H_i(t)=H_i x_{i0} C_0 \exp{\left( \frac{(t_{i0}-t) \ln2}{T_{i1/2}} \right)}\]

where \(x_{i0}\) is the fractional isotopic abundance and \(C_0\) is the elemental concentration.

Radionuclides

Key radionuclides relevant for planetary heating:

Isotope T\(_{1/2}\) (Myr) \(x_{i0}\) \(H_i\) (W/kg)
\(^{26}\)Al 0.717 0 0.3583
\(^{40}\)K 1248 \(1.1668\times10^{-4}\) \(2.8761\times10^{-5}\)
\(^{60}\)Fe 2.62 0 \(3.6579\times10^{-2}\)
\(^{232}\)Th 14000 1 \(2.6368\times10^{-5}\)
\(^{235}\)U 704 0.0072045 \(5.68402\times10^{-4}\)
\(^{238}\)U 4468 0.9927955 \(9.4946\times10^{-5}\)

Total heating rate is the sum:

\[H(t) = \sum_i H_i(t)\]

Note that 2 computes bulk element power ensuring internal consistency.

  1. Turcotte, D.; Schubert, G., Geodynamics, Cambridge University Press, 2014. ↩

  2. Thomas Ruedas, Radioactive heat production of six geologically important nuclides, Geochem. Geophy. Geosys., 2017. ↩