What is the shape of Earth? Sounds like an easy question but it's not at all. The dispute over the shape of the Earth has been going on for millennia, because it is a problem that is far from trivial. Despite what the flat-earthers say, the hypothesis of the "Spherical Earth" was already known in Antiquity: using a very ingenious geometric method, the mathematician Eratosthenes calculated with astonishing precision the size of the Earth [1], wrong a few hundred kilometers only. A trifle, considering the method. All this in spite of Columbus, who several centuries later blatantly misjudged his estimates, he tried to reach the Indies by sea and was saved from certain death by the fortuitous presence of the American continent on his route.

Actually, to the brilliant intuition of the Greek philosophers on the sphericity of the Earth (put down by Aristotle [2]) a fundamental ingredient was missing: rotation. Finally, in the sixteenth century Galileo and Copernicus sensed that the Earth is not at the center of the Universe, but it orbits the Sun and rotates on itself. Starting from this, Newton came to a better estimate of the shape of our planet: thanks to an elaborate balance between gravity and centripetal force, Newton defined the Earth as a rotation ellipsoid (also called oblate spheroid).

We're not there yet though. The Earth is not a homogeneous body, because inside there are materials with very different properties. The complex density distribution of the Earth's interior makes its shape very difficult to determine., due to gravity changes; therefore, complicated mathematical functions are needed to calculate it (spherical harmonics). The advent of supercomputers and the use of satellites has allowed us to arrive at the most accurate possible representation of the shape of the Earth. Ladies and gentlemen, This is a geoid.

Anomalies of the Earth's gravity field, measured by the satellite network GRACE. In red are indicated the areas where the gravity is stronger than the reference, in blue where it is weaker. Credit: NASA/JPL/University of Texas Center for Space Research.)

Quoting Full Metal Jacket's Sergeant Major Hartman, "it's so ugly it could be a modern art masterpiece". Aesthetic taste aside, let's try to better understand what we are looking at. This animation shows the mathematical representation of the Earth, that is, the shape that our planet would have if its surface were an equipotential surface. To observe an equipotential surface is very easy: take a closed bottle of water and observe how the water level always remains horizontal, no matter how you move of the bottle. This is because any liquid tends to move in order to minimize its potential energy; its surface is in fact always perpendicular to the direction of the force of gravity. So would the ocean, if it were not for the action of the wind, of sea currents and tides. The Earth's surface therefore also tries to be an equipotential surface, but without succeeding (the forces to which it is subjected are too great).

Years of research were needed to measure the Earth's gravity field and its variations with great precision. The mission GRACE of NASA, started in 2002 and ended in 2017, allowed you to draw the animation that we showed you earlier. It is made up of a network of satellites, "coupled" two by two to measure the Earth's gravity anomalies. In 2009 it was ESA's turn with the mission JOY, consisting of a single satellite, which has allowed us to achieve even greater precision. The mission ended in 2013, when the satellite ran out of fuel and re-entered the atmosphere.

The geoid, however, represents the time-mediated shape of our planet. Its real form can be very different. In fact, our planet changes shape continuously due to the tides: the best known are the lunar ones, given the Earth's proximity to our only natural satellite. There are others less known, like those of the Sun and Jupiter [3].

But why all this expenditure of time and energy? Knowing with great precision the shape of the geoid and the Earth's gravity field is one of the tasks of the geodesy [4], a branch of physics that has many practical applications, from geo-localization to oceanography, from seismology to the search for underground resources. Not to mention the fact that the earth's inner motions are constantly changing its mass distribution., causing continuous adjustments of the Earth's gravity that must be carefully measured. Impossible to experience such small changes, unless you drink one glass too many.

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