A synchronous moon as a possible cause
of Mars’ initial triaxiality  
 

Mars’ triaxiality makes itself most evident through the equatorial ellipticity produced by the Tharsis Rise and by a less prominent elevation located almost diametrically opposite to Tharsis and constituted by Syrtis Major Planum and an adjacent part of Terra Sabaea (Smith and Zuber, 1996) .

Terra Sabatea being more ancient, Syrtis Major’s present form is dated by crater counts to the early Hesperian epoch (Hiesinger and Head, 2004), and is a smoothly sloping shield-volcano dome of an average $2.1$ km height, its peak extending to 2.3 km.

The largest highlands province on Mars and in the solar system, Tharsis is rising, in average, to about $7$ km, excluding the much higher volcanoes. Its formation had begun at least in the Noachian period, and continued through the entire Hesperian, the volcanoes staying active into the Amazonian epoch. As was pointed out by Zuber and Smith (1997), this rise may mask or modify gravitational signatures that contain important information on the Martian gravitational field as a whole. For this reason, those authors produced a “Mars without Tharsis” gravitational field, by subtracting the gravitational signature of Tharsis from the full field of the planet. Even then, Mars retained much of its triaxiality.

Checking on item (b) requires quite a programme of research, but a qualitative argument in support of this item is available readily for a moon whose orbit is synchronous but not circular. A finite eccentricity causes the tidal deformation to oscillate and produce heat, because an elliptic synchronous orbit makes the moon’s position appear to oscillate E-W in the sky over the same region (or to describe an analemma if the inclination is nonzero). The submoon and antimoon zones might have been warmer than the rest of the surface — though this outcome is very structure-dependent and is not warranted (Segatz et al., 1988). Modeling of a young Kelvin-Voigt Moon has demonstrated that the closer the Moon was to the Earth the higher the ratio of near to farside tidal heat flux, the difference being in the dozens of percent (Quillen et al., 2019, Figure 6). This motivates us to extend this observation to Mars, and to expect that the degree-3 tide made the submoon zone warmer than the antimoon zone, with the possibility of a plume emerging in the submoon zone, as Tharsis’ predecessor. It should be added, though, that the cited model contains also an argument against our hypothesis, an increase of the heat flux toward the polar zones — which poses a question why the volcanic activity near the poles was lower than on Tharsis. To answer this question, accurate modeling based on a more realistic rheology is required.

As per advice from Benoît Noyelles, who kindly reviewed this paper and contributed greatly to its improvement, we christen the hypothetical moon Nerio, after a war goddess who was Mars’ partner in ancient cult practices, later to be supplanted by deities adapted from other religions.

Consider a static configuration comprising two mutually synchronised bodies: a planet of mass $M$ and radius $R\nobreak\hskip 1.1pt\nobreak$, and a moon of mass $M^{\,\prime}$. Aiming at crude estimates, we set the eccentricity zero, so the semimajor axis coincides with the distance $r$ between the centres of mass of the partners. This enables us to employ a simple formalism developed for static tides with no lag.

Let $\bf r$ and $\bf R$ be planetocentric vectors pointing, correspondingly, at the moon and at some surface point of the planet. In this surface point, the moon-generated perturbing potential is

$\gamma$ being the angular separation between $\bf R$ and $\bf r$, and $P_{l}(\cos\gamma)$ being the Legendre polynomials. Specifically,

In a static setting, the tidal elevation in the surface point $\bf R$ reads:

where $h_{l}$ are static quadrupole Love numbers, while the surface gravity is

In the submoon point, we have $\gamma=0$ and therefore $P_{2}(\cos\gamma)=P_{3}(\cos\gamma)=1$. In the antimoon one, we get $\gamma=\pi$, whence $P_{2}(\cos\gamma)=1$ and $P_{3}(\cos\gamma)=-1$.

Together, equations (1 - 4) entail the following expression for the elevation in the submoon and antimoon points:

where

the “plus” and “minus” signs corresponding to the sub- and antimoon points.

For a moon on a circular synchronous orbit, $r$ is the synchronous radius:

$\Omega$ being the planet’s ‘initial’ rotation rate. By ‘initial’, we understand the rate established at the time of rotation synchronisation, i.e., equal to the mean motion of the moon at that time. This rotation rate was thereafter sustained by the synchronous moon till its elimination.

Since a part of the centrifugal force mimics the quadrupole component of the tidal force (see Appendix C), a link exists between the then values of the dynamical oblateness $J_{2}$, the Love number $k_{2}$, and the rotation rate $\Omega\;$:

where $J_{2}$ is related to the principal moments of inertia $A<B<C$ through

Combined, formulae (8) and (9) yield a convenient expression for the synchronous radius’ early value:

A subsequent insertion of this expression in equations (6 - 7) results in

where we used the hydrodynamic relations $h_{2}/k_{2}=5/3$ and $h_{3}/k_{3}=7/3$ acceptable for a hot planet (Appendix A).

We observe that the relative asymmetry $H_{3}/H_{2}$ is proportional to the factors $(3\nobreak\hskip 1.1pt\nobreak J_{2})^{1/3}$ and $\nobreak\hskip 1.1pt\nobreak{k_{3}}\nobreak\hskip 1.1pt\nobreak{k_{2}^{\nobreak\hskip 0.77pt\nobreak-\nobreak\hskip 0.77pt\nobreak 4/3}}\nobreak\hskip 1.1pt\nobreak$. As will be explained in Section 4, the former of these factors has undergone a very limited time evolution and may be attributed its present-day value: $\nobreak\hskip 1.1pt\nobreak(3\nobreak\hskip 1.1pt\nobreak J_{2})^{1/3}=0.18\,$. By distinction, $\nobreak\hskip 1.1pt\nobreak{k_{3}}\nobreak\hskip 1.1pt\nobreak{k_{2}^{\nobreak\hskip 0.77pt\nobreak-\nobreak\hskip 0.77pt\nobreak 4/3}}$ dropped at some point. Thence, very naturally, the earlier the triaxial figure was formed the larger its seed asymmetry was. We shall dwell on this in Section 6 in more detail.

Provided in Table 1, the present values of the Martian mass and mean radius,

include the late veneer carried out mostly by the planetesimals left-over from the terrestrial planet formation period. According to Nesvorný et al. (2023, Fig. 17a) it added 1.6% to Mars’s mass, almost a third of this addition having arrived during the LHB. Hence the estimates for a young Mars:

Now, the dynamical oblateness and triaxiality, whose current values are given in Table 1. With Tharsis “removed”, $J_{2}$ is reduced by approximately $5\%$ for a fully hydrostatic $J_{2}$ to approximately $10\%$ for $J_{2}$ that is $20\%$ nonhydrostatic (Zuber and Smith, 1997). In our case, however, the reduction of $J_{2}$ would have been much less, because we would ignore not the entire Tharsis but only its highest part caused by later uplifts and eruptions, and would leave the tidal contribution. Another change in $J_{2}$ might have come from the accretion of exterior material, especially during the LHB. While the arrival rate of deposits may have, in principle, depended on the latitude, any such nonuniformity of mass influx was smeared by the equinoctial precession of Mars and large variations of Mars’ obliquity over its history. This averages out the influence of bombardment on $J_{2}$. These considerations legitimise our use of the present oblateness in calculations pertaining to the figure-formation stage (especially in formulae (LABEL:H3) and (24) for the seed triaxiality, where $J_{2}$ is raised to the power of $1/3$):

Related to the principal moments $\nobreak\hskip 1.1pt\nobreak A<B<C\nobreak\hskip 1.1pt\nobreak$ through

the dynamical triaxiality may have been more sensitive to later uplifts and volcanism, especially at Tharsis and opposite to it. (Recall that within our hypothesis the areas around the frozen tidal bulges are supposed to have been more prone to these processes.) So the relative increase of $J_{22}$ may have been more noticeable than that of $J_{2}$. Therefore the present triaxiality $J_{22}^{\rm(present)}$ serves as the upper bound for the early value $J_{22}\,$:

For the purpose of estimate, we assume the seed equatorial asymmetry to be small, $\nobreak\hskip 1.1pt\nobreak|\nobreak\hskip 1.1pt\nobreak H_{3}\nobreak\hskip 1.1pt\nobreak|\nobreak\hskip 1.1pt\nobreak\ll\nobreak\hskip 1.1pt\nobreak H_{2}\nobreak\hskip 1.1pt\nobreak$, and model the soft young planet with a homogeneous ellipsoid having the principal axes $a>b>c$, whence $\nobreak\hskip 1.1pt\nobreak A\nobreak\hskip 1.1pt\nobreak=\nobreak\hskip 1.1pt\nobreak\frac{\textstyle M}{\textstyle 5}\nobreak\hskip 1.1pt\nobreak(b^{2}+c^{2})\nobreak\hskip 1.1pt\nobreak$ and $\nobreak\hskip 1.1pt\nobreak B\nobreak\hskip 1.1pt\nobreak=\nobreak\hskip 1.1pt\nobreak\frac{\textstyle M}{\textstyle 5}\nobreak\hskip 1.1pt\nobreak(c^{2}+a^{2})\nobreak\hskip 1.1pt\nobreak$. Insertion of these formulae in equation (17) produces

Identifying the tidal elevation $H_{2}$ with $\frac{\textstyle a-b}{\textstyle 2}$, we obtain:

which is about $1.1$ km.

Using formulae (12) and (20), we arrive at a remarkably simple relation:

In this expression, both $J_{2}$ and $J_{22}$ are parameters of a young Mars. As was explained in Section 4, the present value of the oblateness is still a tolerable estimate for the early oblateness, while the present triaxiality is only an upper bound for the early value. Combined with equality (16) and inequality (18), the above relation becomes:

Insertion of the numerical values of $J_{22}^{\rm(present)}$, $J_{2}^{\rm(present)}$,  and $M$ gives us:

which is less than one third of the lunar mass. It could have been even less than that, depending on how $J_{22}$ has been boosted by the later volcanic activity.

Had we assumed that the elevation $H_{2}$ is comparable to the height of Syrtis Major Planum (on average, $2.1$ km) or of Tharsis (on average, about $7$ km above the reference datum), we would have raised the value of $H_{2}$ by a factor of 1.9 to 6.4, which would accordingly require, through equation (12), a moon of mass $(4.3~{}-~{}17)\times 10^{22}$ kg, which would be between $0.6$ and $2.3$ lunar masses. Geophysical evidence, however, is indicating that both Syrtis Major and Tharsis owe much of their elevation to processes subsequent to the initial tidal distortion (Syrtis assumed its present shape in the early Hesperian; Tharsis in the late Hesperian, if not later). So a third of the lunar mass, equation (23), is a trustworthy bound for $M^{\,\prime}$.

Owing to equations (LABEL:H3), (16), and (23), the seed elevations $H_{2}+H_{3}$ in the submoon and antimoon points were

with the “plus” sign for the submoon point, “minus” for antimoon.

While addressing the timing issue would require modeling of the solidification of an early magma ocean, and of the formation and thickening of Mars’ stagnant lid, a simplified analytical approach is nonetheless possible. The insertion of expression (32) for $k_{l}$ into the second term of equation (24) renders the relative triaxiality, $H_{3}/H_{2}$, as a function of $\mu$. This effectively would be a function of the figure formation time, had we known the history of solidification expressed as $\mu(t)$. Establishing of this time-dependence would constitute a separate, heavily numerical project. So for now we are using the evolving $\mu$ as a parameter, instead of time. Along these lines, we find in Appendix A that the values of ${k_{3}}{k_{2}^{-4/3}}$ are residing in the interval

The upper bound corresponds to a figure that was formed at the magma-ocean stage and sustained by the moon into the solidification epoch. This scenario implies that the moon showed up on the then synchronous orbit when the Martian mean shear modulus was not very different from that of a near-liquid lava, $\nobreak\hskip 1.1pt\nobreak\mu\nobreak\hskip 1.1pt\nobreak\simeq\nobreak\hskip 1.1pt\nobreak 0.2\times 10^{9}$ Pa. The lower bound in equation (25) pertains to a scenario wherein the moon got synchronised when the solidification was already going on, with the mean viscosity about $\mu\simeq 18\times 10^{9}$ Pa.

We observe that an early formation renders a higher seed asymmetry, with an elevation of $\simeq 1.14\,H_{2}$ in the submoon point, and $\simeq 0.86\,H_{2}$ in the antimoon one. A later formation produces a lower asymmetry, with elevations of $\simeq 1.03\,H_{2}$ and $\simeq 0.97\,H_{2}$ in the submoon and antimoon points, correspondingly. This may be regarded as an argument in favour of the moon having emerged and synchronised its mean motion during the magna-ocean period already. On the other hand, if part (b) of our hypothesis in Section 1 is correct, and the tidally elevated areas were most prone to geophysical processes, then even a small initial tidal asymmetry could later entail a larger asymmetry of shape — an issue requiring numerical treatment.

It is proposed in this work that an initial, ‘seed’ triaxiality of Mars was produced by a massive moon that was orbiting a young Mars on a synchronous orbit. Depending on whether the moon’s orbit was circular or not, the tide was static or near-static (slightly librating about the mean direction toward the synchronous moon). After the seed triaxiality was created, the tidally elevated areas on the two opposite tips became, hypothetically, more prone to geophysical processes than the rest of the Martian surface — which led to these tips’ further rise and a resulting triaxiality increase.

The author wishes to acknowledge helpful consultations provided to him by Amirhossein Bagheri, Andrew Dombard, Amir Khan, Alessandro Morbidelli, Anthony L. Piro, Robert Schulmann, Francis Nimmo and Michaela Walterova, none of which colleagues shares responsibility for the hypothesis proposed in this work.

The author was also fortunate in having had the assistance of two reviewers who donated generously of their time, helping to shape and improve the contours of the manuscript. Their advice is greatly appreciated. Particular thanks are owed to Benoît Noyelles for suggesting the name Nerio for the putative moon.

The shear rigidity values used in this paper are borrowed from James et al. (2004). The value of Mars’ present dynamical oblateness $J_{2}^{(present)}$ is provided in Konopliv et al. (2011). The value of Mars’ present dynamical triaxiality $J_{22}^{(present)}$ is given in Konopliv et al. (2020). The value $M^{(present)}$ of Mars’ present mass is borrowed from the PDS file for the Mars gravity field MRO120D used in Konopliv et al. (2016).