DOI: 10.7291/d1zm3n
title: Inferring Mimas' spatial distribution of tidal heating from its long-wavelength topography

description: In new work (Gyalay et al., 2023), we infer the interior of Mimas from its
 global shape (long-wavelength topography). To do so, we have to make
 various assumptions on how the ice shell of Mimas operates. This includes
 the temperature at the base of the ice shell, the thickness of the ice
 shell, what mode of isostasy it operates under (equal-mass vs.
 equal-pressure and Airy vs. Pratt), whether tidal tidal heating is due to
 eccentricity vs. obliquity, and how porous the region of the ice shell
 with a temperature &lt;140 K may be. Further, as it has not yet been
 measured for Mimas, we must make an assumption on its moment of inertia.
 We vary through these assumptions and calculate how well an inferred heat
 distribution matches with a tidal heating distribution, among other
 physical self-consistency checks. In the associated paper, we analyze the
 dataset we produced to make conclusions on Mimas' interior structure
 and orbital dynamic history.
description: This dataset was produced by a model using the methods described
 in Gyalay & Nimmo (2023a; <em>JGR: Planets
 128</em>(2), doi: 10.1029/2022JE007550). In that paper, we
 established the mathematics behind how we used assumed parameters (upper
 ice shell porosity, total ice shell thickness, moment of inertia, basal
 temperature at the base of the ice shell) to infer the average basal heat
 flux and fit for spatial patterns of tidal heating (Beuthe, 2013, Icarus).
 Using this best-fit tidal heating for each set of parameters, we forward
 model the topography (also described in that paper) and calculate its
 spherical harmonic weights as well as compare them to the originally
 observed topography. <br><br>The code used to generate this
 dataset and as well as the dataset associated with Gyalay & Nimmo
 (2023a) are included in that paper's associated repository (Gyalay
 & Nimmo, 2023b; Dryad, dataset, doi: doi.org/10.7291/D11969). This
 repository contains only the produced model output for Mimas.
description: The header of each data-output file should describe what each
 column refers to. MoI is the moment of inertia. From these data, one can
 compute density profiles for each model of Tethys (or Enceladus) and judge
 whether it is consistent with the inferred heating pattern weight. Values
 were not printed to file if the calculated average heat flux was NaN, if
 any of chi_A,B,C were not between 0 and 1, or if any of the spherical
 harmonic weights of forward-modeled topography were NaN. Further
 descriptions of parameters and their uses are described in Gyalay
 & Nimmo (2023a) and Gyalay et al. (2023). We
 also include an input file for the solid body tidal heating code of
 Roberts & Nimmo (2008; <em>Icarus 194</em>(2), doi:
 10.1016/j.icarus.2007.11.010). Usage of the input file is included
 in the file README.md
description: # Inferring Mimas' spatial distribution of tidal heating from its
 long- ## wavelength topography In this repository there is a series of
 data files for outputs of Mimas modeled under different assumptions. The
 biggest indicators of well-fitting models are the r_sq, which is the
 coefficient of determination that shows how well the inferred heating
 pattern beneath the ice shell can be fit by spatial patterns of tidal
 heating, and the RMS, which is the root mean square difference between the
 observed topography of Mimas and the topography forward modeled from the
 best fit tidal heating pattern weights. In the associated paper, we
 conclude there was a past epoch of strong obliquity tides in a solid
 Mimas. Additionally, there is one last file that is input for the TIRADE
 solid-body tidal heating code of Roberts & Nimmo 2008. That code is
 not ours to provide, but we can at least provide the input. Further, the
 Roberts & Nimmo code needs to be updated to calculate the tidal
 dissipation and potential due to obliquity tides. The tidal dissipation is
 equation 42 of Beuthe 2013, while the tidal potential is equation 88 of
 Beuthe 2013. These updates must be made to to the file
 "tidal_module.c" at about lines 139 (the variable
 "Ediss") and 979 (the variable "potential"). The
 derivatives of Equation 88 must also be calculated to update variable
 "dpot". The input file is "Mimas_AGU2022_conv" and
 requires the creation of a directory "mimas_agu2022_conv_dir" in
 the same directory as the input file. Then the command
 "./tidal_cond.x Mimas_AGU2022_conv" will place all outputs
 within "mimas_agu2022_conv_dir", assuming all the TIRADE code is
 within the same directory as the input file. ## Description of the data
 and file structure The header of each file should describe what each
 column refers to. MoI is the moment of inertia. From these data, one can
 compute density profiles for each model of Tethys (or Enceladus) and judge
 whether it is consistent with the inferred heating pattern weight. Values
 were not printed to file if the calculated average heat flux was NaN, if
 any of chi_A,B,C were not between 0 and 1, or if any of the spherical
 harmonic weights of forward-modeled topography were NaN. Further
 descriptions of parameters and their uses are described in the paper for
 which this dataset was produced. Further, we include files for Enceladus.
 While the Tethys data include "no_odysseus" in the filename, the
 spherical harmonic coefficients utilized were derived from limb profiles
 that *do* include those that pass over Odysseus crater (Nimmo et al.,
 2011\) The headers contained appear like so: Assuming [isostasy-type]
 isostasy and [tide-type] tides upon Tethys, \[True/False] weighted
 regression [states whether the multilinear regression was weighted] and
 [True/False] pressure isostasy [did we use equal-pressure or equal-mass
 isostasy], each given porosity, MoI [Moment of Inertia], basal temperature
 T_B, and total shell thickness d we calculate necessary basal heat flux
 (F_B). We then calculate the following values: chi_a, chi_b, chi_c:
 heating pattern weights r_sq: coefficient of determination rms: Square
 root of the weighted average of the square of modeled topography minus
 observed. Weighted by area of degree bin. CF20/CF22: Spherical harmonic
 coefficients of flux ratioed. normClm for l=2,4;m=0,2,4&lt;=l: normalized
 spherical harmonic weights of topography from our best-fit interior.
 z_Clm: z-score of the normClm we calculate vs. those observed. This is
 (normClm-normClm_observed)/SD_normClm_observed where SD is the standard
 deviation