Discerning the 'Shape' of Mars: Is it 'Pear-shaped', 'Tetrahedroid-shaped' or some other shape?

There exist two descriptions of the ‘figure of the Earth’ in the literature. First, the notion of the ‘Pear-shape of the Earth’ is relatively well-known to the public-at-large since the early days of satellite launches. The southern hemisphere bulges out more than the northern hemisphere on average. Further, the north polar area is 18 m higher from the sea-level whereas the south polar area is 28 m below the sea level. When the altitudes are averaged over circles of latitude, the so-called ‘pear-shape of the Earth’ emerges about the rotational axis of the planet. Second, there exists another description of the figure of the Earth called the ‘Tetrahedroid-shape of the Earth’, which actually predates the ‘pear-shape’ concept by 84 years. In 1875, William Lowthian Green proposed the Tetrahedroid-shape of the Earth in order to explain the distribution of land and water on the planet earth. It was hypothesized that a ‘law of least action’ would solidify a molten spherical Earth into the smallest ‘Platonic solid’ or ‘Tetrahedroid shape’. This concept received strong historical support before being forgotten. In the tetrahedroid model, one vertex is situated at the north pole, and three others on the circle of latitude 19.5oS, separated by 120o. It should be noted that the tetrahedroid model reduces to the pear-shaped model upon averaging over the longitudes, and is therefore more general than the latter model. In this study, the figure of the planet Mars is investigated. The Martian surface has been thoroughly surveyed by the Martian Orbiter Laser Altimeter (MOLA) aboard the Mars Global Surveyor (MGS) spacecraft for a period of nine years. Topographical maps produced by the MOLA measurements are of the finest quality. A Lambert’s equi-angular cylindrical projection map of the Martian surface is used. The coordinate system is defined as a planet centric right-handed spherical polar coordinate system with its z-axis coinciding with the rotational axis of the planet; and the prime meridian passing above the crater Airy-0. The zero elevation is defined as the equipotential surface (formed by gravitational and centrifugal forces) whose average value at the equator is the mean radius of the planet. The results show that the Martian topography was quite similar to that of the Earth, with a major distinction being that its south polar region is an elevated area rather than a depressed one. Consequently, the averaged figure of Mars about its rotational axis is ‘lemon-shaped’ rather that ‘pear-shaped’. Finally, if we ignore the south-polar rise, a ‘tetrahedroid-shape’ of Mars becomes apparent.