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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.

Volatile Mass Balance

The mass balance of a given volatile in the interior 1 is:

\[X_v^s M^s + X_v^l M^l + X_v^g M^g + m_v^e + m_v^o + m_v^r = X_v^{\rm init} M^m\]

where superscripts \(s\), \(l\), \(g\), \(e\), \(o\), \(t\) denote solid, liquid (melt), gas, escaped, ocean, and total. Only solid, liquid, and atmosphere are physical reservoirs.

The partition coefficient relates volatile concentrations:

\[k_v = \frac{X_v^s}{X_v}\]

Atmospheric mass

The total atmospheric mass of \(q\) species composes as:

\[m_t^g = \frac{4 \pi R_p^2}{g} P_s\]

where \(R_p\) is planetary radius and \(P_s\) is surface pressure.

The mass of a given volatile species is:

\[m_v^g = 4 \pi R_p^2 \left( \frac{\mu_v^g}{\bar{\mu}} \right) \frac{p_v}{g}\]

Partial pressure follows a modified (power-law) Henry's law:

\[p_v ( X_v ) = \left( \frac{X_v}{\alpha_v} \right)^{\beta_v}\]

In SPIDER we use scaled mass (omitting the \(4 \pi\) factor):

\[X_v (k_v M^s + M^l) + \frac{R_p^2}{g} \left( \frac{\mu_v^g}{\bar{\mu}} \right) p_v + m_v^e + m_v^o + m_v^r = X_v^{\rm init} M^m\]

We solve for volatile mass fraction in the liquid phase, from which we can compute volatile mass in solid and gas phases.

Non-dimensionalisation

Mass

Masses are non-dimensionalised as:

\[M = \rho_0 R_0^3 \hat{M}\]

Volatile Concentration

Volatile concentration is expressed as scaled mass fraction:

\[X_v = V_0 \hat{X}_v\]

where \(V_0=10^{-6}\) gives parts-per-million (ppm), \(V_0=10^{-2}\) gives weight percent (wt%), and \(V_0=1\) gives mass fraction.

Power Law Solubility

Non-dimensional partial pressure:

\[\hat{p}_v ( \hat{X}_v ) = \left( \frac{\hat{X}_v}{\hat{\alpha}_v} \right)^{\beta_v}\]

where:

\[\hat{\alpha}_v = \frac{\alpha_v^{{\rm ppm/Pa}^{1/\beta_v}}}{10^6} \frac{P_0^\frac{1}{\beta_v}}{V_0}\]

Sossi Solubility

For H\(_2\)O 2:

\[X_v = A{f_{H_2O}}^\frac{1}{2}+B G f_{H_2O}\]

where fugacities are constrained by oxygen buffer, and \(A=534\) ppm/bar\(^{0.5}\) and \(B=723\) ppm/bar.

Initial Volatile Concentration

For an initial condition, we prescribe the total volatile concentration and solve the mass balance to obtain initial partial pressure consistent with chemical equilibrium criteria.

Chemical Reactions

Reactions transfer mass between volatile species. For example:

\[[\rm{H}_2O]\leftrightarrow \frac{1}{2} [\rm{O}_2] + [\rm{H}_2]\]

with equilibrium constant:

\[K=\frac{p_{\rm H_2} f_{\rm O_2}^{1/2}}{p_{\rm H_2\rm O}}\]

Mass is conserved through stoichiometry:

\[m_{H_2O} = m_{O2} + m_{H_2}\]

Atmospheric Escape

Jeans escape

\[\frac{d m_v^e}{dt} = \left( \frac{d m_{\rm v}^{\rm g}}{dt} \right) \mathcal{R} (1 + \lambda_s) \exp(-\lambda_s) + \frac{\Phi}{4 \pi}\]

where Jeans parameter:

\[\lambda_s = \frac{g R_p \mu_{\rm v}}{k_b T_s N_A}\]

Zahnle escape model

For H\(_2\) 3:

\[\phi_{H_2} \approx \Gamma \frac{(1 \times 10^{12}) f_{H_2} S}{\sqrt{1+0.006S^2}}\]

where \(S\) is non-dimensional and \(\Gamma\) is a scaling constant.

Grey atmosphere model

Optical depth

\[\tau^\ast = \frac{3 \kappa^\prime p(\tau^\ast)}{2g}\]

Effective emissivity

Optical depths for each volatile are combined:

\[\epsilon = \frac{2}{\sum_j \tau_j^\ast +2}\]

Atmosphere temperature structure

Temperature as function of optical depth 4:

\[T(\tau^\ast) = \left( T_0^4 \frac{(\tau^\ast+1)}{2} +T_\infty^4 \right)^\frac{1}{4}\]

where:

\[T_0 = \left( \frac{F_{atm}}{\sigma} \right)^\frac{1}{4}\]

Stellar flux

\[F_{sun} = \sigma T_{eqm}^4 = (1-\alpha) \frac{F_0^\prime}{D^2}\]

where \(\alpha\) is bolometric albedo, \(F_0'\) is averaged solar constant, and \(D\) is planet-star distance.

  1. Lebrun, T.; Massol, H.; Chassefi`ere, E.; Davaille, A.; Marcq, E.; Sarda, P.; Leblanc, F.; Brandeis, G., Thermal evolution of an early magma ocean in interaction with the atmosphere, J. Geophys. Res.-Planet., 2013. ↩

  2. Laura Schaefer; Bruce Fegley, Redox States of Initial Atmospheres Outgassed on Rocky Planets and Planetesimals, Astrophys. J., 2017. ↩

  3. Kevin J. Zahnle; Marko Gacesa; David C. Catling, Strange messenger: A new history of hydrogen on Earth, as told by Xenon, Geochim. Cosmochim. Acta, 2019. ↩

  4. Abe, Yutaka; Matsui, Takafumi, The formation of an impact-generated H2O atmosphere and its implications for the early thermal history of the Earth, J. Geophys. Res. Solid Earth, 1985. ↩