Modern Physics

# What is a geoid

The geoid or Earth figure is the theoretical surface of our planet, determined by the average level of the oceans and with a very irregular shape. Mathematically, it is defined as the equipotential surface of the Earth’s effective gravitational potential at sea level.

As an imaginary (non-material) surface, it traverses continents and mountains, as if all oceans were connected by water channels that pass through land masses.

The Earth is not a perfect sphere, as the rotation around its axis makes it a kind of balloon flattened by the poles, with valleys and mountains. Therefore, the spheroid shape is still imprecise.

This same rotation adds to the Earth’s gravity force a centrifugal force, whose net or effective force does not point towards the Earth’s center but has a certain gravitational potential associated with it.

Also, the relief shapes create irregularities in density and therefore the gravitational pull in some areas is definitely no longer central.

So scientists, starting with CF Gauss, who invented the original geoid in 1828, created a geometric and mathematical model to more accurately represent the Earth’s surface.

For this, it is assumed an ocean at rest, without tides or ocean currents and with constant density, whose height serves as a reference. Next, it is considered that the Earth’s surface ripples gently rising in places where the local gravity is greatest and when this decreases in sinking.

Under these conditions, allow the effective acceleration of gravity to always be perpendicular to the surface whose points have the same potential, and the result is the geoid, which is irregular because the equipotential is not symmetrical.

## Geoid’s physical base

To determine the geoid’s shape, which has improved over time, scientists performed many measurements, taking into account two factors:

– The first is that the value of g, the Earth’s gravitational field equivalent to the acceleration of gravity , depends on latitude: it is maximum at the poles and minimum at the equator.

– The second is that, as we said before, the Earth’s density is not homogeneous. There are places where it increases because the rocks are denser, there is accumulation of magma or there is a lot of ground on the surface, like a mountain, for example.

Where density is greater, is also greater. Note that g is a vector and is therefore indicated in bold.

### Earth’s gravitational potential

To define the geoid, the potential due to gravity is needed, for which the gravitational field must be defined as the gravitational force per unit of mass.

If a test mass m is placed in this field, the force exerted by the Earth on it is its weight P = mg, so the magnitude of the field is:

Force / mass = P / m = g

We already know its average value: 9.8 m / s 2 and, if the Earth were spherical, it would be directed towards its center. Likewise, according to Newton’s law of universal gravitation:

P = Gm M / r 2

Where M is the Earth’s mass and G is the universal gravitational constant. So the magnitude of the gravitational field g is:

g = gm / r 2

It looks a lot like an electrostatic field, so you can define a gravitational potential analogous to electrostatic:

V = -GM / r

The constant G is the universal gravitational constant. Well, the surfaces where the gravitational potential always has the same value are called the potential surfaces and g is always perpendicular to them, as I said before.

For this specific potential class, equipotential surfaces are concentric spheres. The work required to move a mass over them is nil, because the force is always perpendicular to any path in the equipotential.

#### Lateral component of gravity acceleration

Since the Earth is not spherical, the acceleration due to gravity must have a lateral component, g l , due to the centrifugal acceleration caused by the planet’s rotation around its axis.

The following figure shows this component in green, whose magnitude is:

l = ω 2 a

In this equation ω is the Earth’s angular rotation speed and a is the distance between the point on Earth at a certain latitude and the axis.

And in red is the component that is due to planetary gravitational attraction:

o = GM / r 2

As a result, adding the vector or + l , gives a resultant acceleration g (in blue) , which is the true acceleration of Earth’s gravity (or effective acceleration) and which, as we can see, does not point exactly to the center. .

Furthermore, the lateral component depends on latitude: it is null at the poles and therefore the gravitational field is maximum there. At the equator, it opposes gravitational attraction, reducing the effective gravity, whose magnitude remains:

g = GM / r 2 – ω 2 R

With R = the Earth’s equatorial radius.

It is now understood that the equipotential surfaces of the Earth are not spherical but assume a shape that g is always perpendicular to them at all points.

## Differences between geoid and ellipsoid

Here is the second factor that affects the variation of the Earth’s gravitational field: local variations in gravity. There are places where gravity increases because there is more mass, for example on the hill in figure a).

Or there is an accumulation or excess of mass below the surface, as in b). In both cases, there is an elevation in the geoid because the greater the mass, the greater the intensity of the gravitational field.

On the other hand, over the ocean, the density is lower and, as a consequence, the geoid sinks, as we see on the left of figure a), above the ocean.

In figure b) it is also noted that the local gravity, indicated by arrows, is always perpendicular to the surface of the geoid, as we said. This is not always the case with the reference ellipsoid.

### Geoid Ripples

The figure also indicates, with a bidirectional arrow, the difference in height between the geoid and the ellipsoid, called undulation and called N. Positive undulations are related to excess mass and negative undulations are related to defects.

The swells almost never exceed 200 m. In fact, the values ​​depend on how the reference sea level is chosen, as some countries choose differently according to their regional characteristics.

## Advantages of representing Earth as a geoid

-About the geoid, the effective potential, resulting from the potential due to gravity and the centrifugal potential, is constant.

-The force of gravity always acts perpendicular to the geoid and the horizon is always tangential to it.

-The geoid provides a reference for high-precision mapping applications.

– Using geoid seismologists, it is possible to detect the depth at which earthquakes occur.

-The GPS positioning depends on the geoid to be used as a reference.

-The ocean surface also runs parallel to the geoid.

-The elevations and decreases of the geoid indicate mass excesses or defects, which are gravimetric anomalies . When an anomaly is detected and, depending on its value, it is possible to infer the geological structure of the subsoil, at least at certain depths.

This is the basis of gravimetric methods in geophysics. A gravimetric anomaly can indicate accumulations of certain minerals, structures buried underground or also voids. Underground salt domes, detectable by gravimetric methods, are indicative in some cases of the presence of oil.