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\begin{document}

\title{Computational Modeling of Volcanism on Earth-like Planets}



\author[1]{Edward Kim }%
\author[1]{Matthew J.S. Beach}%
\author[1]{Ethan Buchman}%
\affil[1]{Affiliation not available}%


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  \date{May 18, 2024}


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\begin{abstract}
The aim of this paper is to model the level of volcanic activity on
Earth-like planets over time. We consider the level of volcanic activity
on a planet to be a function of the planet's thermodynamic state. A
planet is considered to become volcanically inactive once it has reached
a thermodynamic steady state. We first model the heat flow in the
planet's interior by the heat equation, which we reduce to a
one-dimensional Laplace equation. We then calculate the temperature
field of the planet by numerically solving a spherically symmetric
boundary value problem. Finally, we relate discrete time-steps in our
simulation to real-time via an empirically-informed mapping. Our results
indicate that Earth will remain volcanically active for a total of
~6.0\textbackslash{}sim 6.0 billion years since its formation, while
Earth-like planets of 0.50.5 and 2.02.0 Earth masses will be active for
~5.5\textbackslash{}sim 5.5 and ~7.7\textbackslash{}sim 7.7 billion
years respectively. Our model incorrectly predicts that Earth-like
planets above 2.02.0 Earth masses continually increase their internal
temperatures if their conductivity and density profiles are assumed to
be identical to that of Earth, which suggests some limitations of the
model.%
\end{abstract}%



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