The shape of the Earth

What shape is the Earth? It sounds like a pretty basic question, but it’s trickier to answer than you might think.

For starters, let’s define what we mean by ‘the Earth’. You might not often consider this, but there’s a whole bunch of atmosphere that is really part of our planet. The fact that the portion of our planet above us is gaseous and the stuff below us is solid is just a happenstance of our density. If we were made of Helium we’d all be floating around and rarely bothering with the solid stuff way beneath us. Everyone agrees Jupiter is a massive planet, even though most of it is gas. So don’t be gas haters. That said, it is not easy to pick a satisfying edge of the atmosphere from which to define the planet’s shape; we will go back to ignoring it. It’s also not much good using the ground surface (elevation/seafloor depth) because it changes every time there there is a landslide, volcanic eruption, or Russian oligarch’s mining project. 

Instead, we’ll choose a more intuitive surface from which to explore the Earth’s shape – the sea level. This is a nice reference because water flows so that its surface is ‘flat’ with respect to the direction of gravity. The liquid in your teacup cannot all bunch up at one side because gravity will pull it back down until no point is any higher than another point. While this makes things look flat on a small scale (since the force of gravity on either side of your cup points in almost exactly the same direction) on a large scale, the surface is obviously curved. What is really going on here is that the Earth’s attraction is producing an “equipotential surface” - a surface of equal gravitational potential. The fluid will flow in order to minimise its potential at all points. If it were to be pushed to higher potential (e.g. when your grumpy uncle Joseph petulantly splashes the dolphin with whom he is supposed to be communing on a ‘relaxing’ holiday) it will immediately flow back down again. The sea surface therefore defines an equipotential surface that we call the “Geoid”: the notional shape of the Earth. 

Okay, so what shape is the geoid? 

If you guessed ‘sphere’ you get half a point. For a static, uniform body, gravity would indeed pull it into a sphere. Cf. astronauts playing with water droplets on the International Space Station

But the Earth is also spinning on its axis (we have days, nights, etc.). This means the inward force of gravity is somewhat balanced by the outward centrifugal force (a fake force, but let’s not be judgemental) at the equator and so the geoid bulges there. At the poles, the gravitational force is un-challenged, so it pulls in the geoid there. The result is a squashed sphere, known more technically as an oblate spheroid. Other things that look like oblate spheroids are: hamburgers and oranges that you have sat on. I can’t think of better examples. 

It turns out that the Earth’s equatorial radius is about 21 km greater than its polar radius. This has some interesting implications. Devotees of the excellent show QI will recall that the ‘highest’ mountain on Earth (measured from the very centre of the Earth) is not Mt. Everest, but Chimborazo, Ecuador. Even though Everest stands 8,848 m above sea level, and Chimborazo is just 6,268 m above sea level, the sea level isn’t level

In fact, the beaches in Ecuador (sitting on the equatorial bulge) come about half way up the full height of Everest, a little shy of Base Camp. Take that, Tenzing Norgay

But the geoid’s interesting features do not stop there. By precisely measuring sea surface heights(1) we find that it varies a good deal from a perfect oblate spheroid. The geoid has some large valleys and hills. The largest of these is a really odd zone just south of  Sri Lanka, in the Indian ocean. Measuring about 3500 km across, the geoid here is up to 100m deeper than the average. Thinking back to the definition of the geoid, this means that if you set out in a boat going SE from India you will gradually sail(2) down a valley 100m deep and then back up the other side, never once changing your gravitational potential energy(3). How cool is that? A whole valley, up and down, at no cost to you. You’re welcome. 

These variations in the geoid at the large-scale level tell us about large-scale features in the Earth, like thinned or thickened crust as a result of continents breaking up or squelching together. That deep geoid valley next to India has been linked to a graveyard of sunken dense material in the deep mantle. On a much finer scale (which we’ve only been able to attain since the advent of satellite measurements) geoid anomalies inform us about subsurface mineral deposits or water reservoirs. I just learned that we now have atomic clocks so accurate that (using some general relativity badass-ery) we can measure the geoid from the comfort of our homes.

Now you know what it is, you can look up the height of the geoid where you are and call your friends to brag about being ‘higher’ than them. Fun for all the family.

(1) The waves are averaged out, obviously. 
(2) Why you are in a sailboat, no one knows. 
(3) That’s like going all the way down the stairs of a high-rise apartment block and then all the way back up, without expending any energy.

A rather exaggerated geoid surface. Source

A cartoon explaining the difference between topography (land height), reference ellipsoid (that’s the oblate spheroid) and the geoid. Source.