# Overview of tides – extra mathematics

This section gives an overview of the mathematics of how the change in the tidal force due to the Moon varies at different places on Earth.

### Gravitational fields and tidal forces

A gravitational field is an example of a vector field. The gravitational field at a point in space is the gravitational force which acts on a one kg mass at that point. From Newton’s law of gravitation, the magnitude of the Moon’s gravitational field at a distance R from its centre is given by the following relationship.

Where

• F(R) is the Moon’s gravitational field at a distance R from its centre.
• Because F(R) is a vector quantity it has a direction as well as a magnitude. As gravity is an attractive force, the direction of F(R) is always towards the centre of the Moon.
• |F(R)| indicates the magnitude or strength of the gravitational field F(R). The two ‘|’ s are mathematical notation for the magnitude of a quantity.
• Mm is the mass of the Moon, 7.346 x 1024
• G is a number known as the gravitational constant and is equal to 6.674 x 10-11m3 kg-1 s-2.  G is always spelt with a capital letter and usually pronounced ‘Big G’ to avoid confusion with g (which is the strength of gravity at the Earth’s surface).

Gravitational fields are often measured in the SI units of Newtons per kilogramme.

The diagram below shows how the magnitude and direction of F(R) varies at a number of locations on the Earth.

The diagram shows a two-dimensional slice through the Earth-Moon system. The point marked with a purple dot is the centre of the Earth. Other locations on the Earth are marked with a black dot. At each location, the direction of the red arrow marks the direction of the Moon’s gravitational field and the length its magnitude.

### Definition of tidal field (due to the Moon)

For any  location, on the Earth,  this the difference between the Moon’s gravitational field at that location and its value at the centre of the Earth.

Tidal field (at location X) = Moon’s gravitational field (at X) – Moon’s gravitational field (at the Earth’s centre)

The tidal field is a vector quantity having both a magnitude and a direction. The diagram below shows how the tidal field varies across the Earth.

The term tidal force is commonly used. The tidal force is simply the force on a object due to the tidal field. It is equal to the mass of the object multiplied by the tidal field

In this diagram a two dimensional slice has been taken through the Earth.

• At the location on the Earth-Moon axis closest to the Moon -marked A, the tidal field is directed upwards away from the Earth’s centre in the direction towards the Moon.
• At the location on the Earth-Moon axis furthest from the Moon -marked B, the tidal field is directed upwards away from the Earth’s centre in the direction away from the Moon.
• At locations at right angles to the direction of the Moon -marked C, the tidal field is directed inwards towards the centre of the Earth.
• At other locations on the Earth, labelled D, the tidal field will be at an angle of of between zero and ninety degrees to the Earth’s surface.

### Working out the magnitude of the tidal field

For simplicity I have calculated this for the two locations which are closest to and furthest from the Moon. At these locations the tidal force is directed away from the Earth’s centre along the Earth-Moon axis which makes the calculation simpler

If we take the point on the Earth’s surface closest to the Moon, then it is at a distance from the centre of the Moon of DEM – RE, where DEM is the  distance between the centre of the Moon and the centre of the Earth and RE is the radius of the Earth.

The magnitude of the Moon’s gravitational field F1 at this point is

The tidal field T1 is given by subtracting the gravitational field due to the Moon at the Earth’s centre Fc, from F1. Because the magnitude of F1 is larger than Fc, T1 points in the direction of the Moon. The magnitude of   T1 is given by:

Because the radius of the Earth (RE) , is significantly smaller than the Earth-Moon distance  (DEM ) then DEM – RE ≈DEM and RE 2 is small compared to 2REDEM and can be neglected. Therefore, the equation simplifies to:

So, the tidal field varies as the inverse cube of the distance from the Moon.

Conversely, if we take the point on the Earth’s surface furthest away from the Moon, then its distance from the centre of the Moon is DEM + RE.

The strength of the Moon’s gravitational field F2 at this point is

The tidal field T2 is given by subtracting the gravitational field due to the Moon at the Earth’s centre Fc, from F2. Because the magnitude of F2 is smaller than Fc,  T2 points away from the Moon. The magnitude of T2 is given by:

As with the previous case, because the radius of the Earth (RE) is significantly smaller than the Earth-Moon distance  (DEM ),   DEM + RE ≈DEM and RE 2 is small compared to 2REDEM and can be neglected. Once again, the equation simplifies to:

So, the magnitude of the tidal forces at the points closest to the Moon and furthest away are approximately equal.

### Some examples of tidal field at different Earth-Moon distances

If we use the value of the Moon’s mean distance from the Earth DEM of 384 400 km, then the tidal force on a 1 kg mass at the two locations are as follows.

• At the location closest to the Moon, the tidal force T1 has a magnitude of 1.13 x 10-6 Newtons towards the Moon.
• At the location farthest from the Moon, the tidal force T2 has a magnitude of 1.07 x 10-6 Newtons away from the Moon.

Although the standard unit of force used by physicists is the Newton, the strength of tidal fields are often expressed in units of g. One g is the average acceleration due to gravity on the Earth’s surface and is equal to 9.81 Newtons per kilogramme. So, to convert from Newtons per kilogramme to g you need to divide by 9.81.

• The magnitude of T1 in units of g is 1.15 x 10-7 g and of T2 is 1.09 x 10-7 g. Compared to the Earth’s gravity the tidal force exerted by the Moon is very weak. For example, a person who weighs 70kg would weigh a mere 8 milligrams lighter when the Moon is directly overhead due to the Moon’s tidal force !
• At the Moon’s closest distance from Earth (known as its perigee) DEM is 363 300 km and the magnitude of T1 is 1.39 x 10-7 g and of T2  1.29 x 10-7 g. The tidal forces, due to the Moon, are 18% stronger than their average value.
• At the Moon’s farthest distance from Earth (known as its apogee) DEM is 405 500 km and the magnitude of T1 is 9.78 x 10-8 g and of T2 is 9.33 x 10-8 g. The tidal forces are 15% weaker than their average value.
• If we go back in time four billion years when life first emerged on Earth, then the Moon was on average only 138 000 km from Earth. In this case T1 was  2.60 x 10-6 g and of T2 is 2.26 x 10-6 g. At this time in the Earth’s early history, the tidal forces were 21.6 times greater than they are today.