Inferring Mimas' spatial distribution of tidal heating from its longwavelength topography
Data files
Apr 19, 2024 version files 10.85 MB

Mimas_AGU2022_conv

Mimas_best_fits_table_revision_from_d_Airy_eccentricity_regweightFalse_pressureFalse_20220705_por0.0.txt

Mimas_best_fits_table_revision_from_d_Airy_eccentricity_regweightFalse_pressureFalse_20220705_por0.1.txt

Mimas_best_fits_table_revision_from_d_Airy_eccentricity_regweightFalse_pressureFalse_20220705_por0.2.txt

Mimas_best_fits_table_revision_from_d_Airy_eccentricity_regweightFalse_pressureFalse_20220705_por0.3.txt

Mimas_best_fits_table_revision_from_d_Airy_eccentricity_regweightFalse_pressureTrue_20220705_por0.0.txt

Mimas_best_fits_table_revision_from_d_Airy_eccentricity_regweightFalse_pressureTrue_20220705_por0.1.txt

Mimas_best_fits_table_revision_from_d_Airy_eccentricity_regweightFalse_pressureTrue_20220705_por0.2.txt

Mimas_best_fits_table_revision_from_d_Airy_eccentricity_regweightFalse_pressureTrue_20220705_por0.3.txt

Mimas_best_fits_table_revision_from_d_Airy_obliquity_regweightFalse_pressureFalse_20220705_por0.0.txt

Mimas_best_fits_table_revision_from_d_Airy_obliquity_regweightFalse_pressureFalse_20220705_por0.1.txt

Mimas_best_fits_table_revision_from_d_Airy_obliquity_regweightFalse_pressureFalse_20220705_por0.2.txt

Mimas_best_fits_table_revision_from_d_Airy_obliquity_regweightFalse_pressureFalse_20220705_por0.3.txt

Mimas_best_fits_table_revision_from_d_Airy_obliquity_regweightFalse_pressureTrue_20220705_por0.0.txt

Mimas_best_fits_table_revision_from_d_Airy_obliquity_regweightFalse_pressureTrue_20220705_por0.1.txt

Mimas_best_fits_table_revision_from_d_Airy_obliquity_regweightFalse_pressureTrue_20220705_por0.2.txt

Mimas_best_fits_table_revision_from_d_Airy_obliquity_regweightFalse_pressureTrue_20220705_por0.3.txt

Mimas_best_fits_table_revision_from_d_Pratt_eccentricity_regweightFalse_pressureFalse_20220705_por0.0.txt

Mimas_best_fits_table_revision_from_d_Pratt_eccentricity_regweightFalse_pressureFalse_20220705_por0.1.txt

Mimas_best_fits_table_revision_from_d_Pratt_eccentricity_regweightFalse_pressureFalse_20220705_por0.2.txt

Mimas_best_fits_table_revision_from_d_Pratt_eccentricity_regweightFalse_pressureFalse_20220705_por0.3.txt

Mimas_best_fits_table_revision_from_d_Pratt_eccentricity_regweightFalse_pressureTrue_20220705_por0.0.txt

Mimas_best_fits_table_revision_from_d_Pratt_eccentricity_regweightFalse_pressureTrue_20220705_por0.1.txt

Mimas_best_fits_table_revision_from_d_Pratt_eccentricity_regweightFalse_pressureTrue_20220705_por0.2.txt

Mimas_best_fits_table_revision_from_d_Pratt_eccentricity_regweightFalse_pressureTrue_20220705_por0.3.txt

Mimas_best_fits_table_revision_from_d_Pratt_obliquity_regweightFalse_pressureFalse_20220705_por0.0.txt

Mimas_best_fits_table_revision_from_d_Pratt_obliquity_regweightFalse_pressureFalse_20220705_por0.1.txt

Mimas_best_fits_table_revision_from_d_Pratt_obliquity_regweightFalse_pressureFalse_20220705_por0.2.txt

Mimas_best_fits_table_revision_from_d_Pratt_obliquity_regweightFalse_pressureFalse_20220705_por0.3.txt

Mimas_best_fits_table_revision_from_d_Pratt_obliquity_regweightFalse_pressureTrue_20220705_por0.0.txt

Mimas_best_fits_table_revision_from_d_Pratt_obliquity_regweightFalse_pressureTrue_20220705_por0.1.txt

Mimas_best_fits_table_revision_from_d_Pratt_obliquity_regweightFalse_pressureTrue_20220705_por0.2.txt

Mimas_best_fits_table_revision_from_d_Pratt_obliquity_regweightFalse_pressureTrue_20220705_por0.3.txt

README.md
Abstract
In new work (Gyalay et al., 2023), we infer the interior of Mimas from its global shape (longwavelength 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 (equalmass vs. equalpressure 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 <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 selfconsistency checks. In the associated paper, we analyze the dataset we produced to make conclusions on Mimas' interior structure and orbital dynamic history.
README: 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 wellfitting 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 solidbody
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 forwardmodeled
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 [isostasytype] isostasy and [tidetype] tides upon Tethys,
[True/False] weighted regression [states whether the multilinear regression was
weighted] and [True/False] pressure isostasy [did we use equalpressure or
equalmass 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<=l: normalized spherical harmonic
weights of topography from our bestfit interior.
z_Clm: zscore of the normClm we calculate vs. those observed.
This is (normClmnormClm_observed)/SD_normClm_observed
where SD is the standard deviation
Methods
This dataset was produced by a model using the methods described in Gyalay & Nimmo (2023a; JGR: Planets 128(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 bestfit 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.
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.
Usage notes
The header of each dataoutput 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 forwardmodeled 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; Icarus 194(2), doi: 10.1016/j.icarus.2007.11.010). Usage of the input file is included in the file README.md