# How tides work

Although the Moon is much smaller and less massive than the Earth its gravitational field still has significant effects on the Earth. The most noticeable of these are tides, the periodic rise and fall of sea levels. High and low tides- Images from Wikimedia Commons

Causes of Tides

The average Earth- Moon distance is 384 400 km. This, however, is the distance of the centre of the Moon from the centre of the Earth and the Earth itself is 12 740 km in diameter. So, the point on the Earth’s surface which is closest to the Moon is on average a distance of 378 030 km away and the point on the Earth’s surface farthest away a distance of 390 770 km. The principle cause of tides is that the pull of the Moon’s gravity is stronger at the area of the Earth closest to the Moon and weaker at the area facing away. The average strength of the Moon’s gravitational field at the location on Earth closest to the Moon is 0.000 003 497g, where g is the acceleration due to gravity on the Earth’s surface. The strength of the Moon’s gravitational field at the location farthest from the Moon is significantly weaker at 0.000 003 272g.

Definition of tidal force (due to the Moon)

The tidal force at a location, on the Earth, is the difference between the Moon’s gravitational field at that location and its value at the centre of the Earth. For readers wanting any more detail on the mathematics behind this see the notes at the end of this post.

The tidal force due to the Moon tends to stretch the Earth slightly along the line connecting the two bodies. The solid Earth can deform a little, but ocean water, being fluid, is free to move much more in response to the tidal force. This causes a tidal bulge in the area closest to the Moon, shown as A in the diagram below. Another tidal bulge also occurs in the area of the Earth farthest away from the Moon, where the Moon’s gravity is weaker than its average value. This is shown as C in the diagram. As the Earth rotates, high tides occur at A and C and low tides at B and D

Taken together these two tidal bulges mean that at a given location there are normally two high tides every 24 hours 50 minutes.  This period of time, which is sometimes called a ‘tidal lunar day’ is the interval of time between successive occasions when the Moon is at its highest in the sky. However, in reality, tides are a little more complicated than this – as described later in this post.

Spring and Neap Tides

The Sun also contributes to tides but, because the Sun is much farther away than the Moon, the difference between the pull of the Sun’s gravity at the location on Earth closest to the Sun and the location farthest away is smaller than compared to the Moon. The average strength of the Sun’s gravitational field at the area of the Earth closest to the Sun is 0.000 604 3g. The average strength of the Sun’s gravitational field at the area of the Earth fathest away from the Sun is slightly weaker at 0.000 604 2g. Because the difference between the two values is smaller, the tidal force due to the Sun is only 46% of the tidal force due to the Moon.

When the Earth, Sun and Moon are in a line, which happens at full moon and new moon the tidal force of the Sun adds to the tidal force of the Moon and the total tidal force is larger than average. On these occasions, which are called spring tides, high tides will be higher than average and low tides will be lower.  The word spring in this case has nothing to do with the season, instead it comes from the verb to move or jump suddenly or rapidly upwards or forwards. Spring tides

Conversely, when the Earth, Sun and Moon are at ninety degrees to each other, which happens at first and last quarter, the tidal force of the Sun subtracts from the tidal force of the Moon and the total tidal forces are lower than average. On these occasions, which are called neap tides, the tidal range is smaller. High tides are less high than average and low tides are not so low. The word neap is derived from the Middle English word neep which means scant or lacking. Tidal Friction

The Earth rotates on its axis in just under 24 hours, whereas the Moon takes 27.5 days to complete an orbit of the Earth.  Because the Earth rotates on its axis faster than the Moon revolves around the Earth, the tidal bulge is always a little bit ahead of the Moon. The tidal bulge is always ahead of the Moon’s orbital position. This ‘pulls the Moon along’ in its orbit.

This causes two separate effects: one on the Moon and one on the Earth.

• The pull of the tidal bulge ahead of the Moon causes the Moon to accelerate very slightly.  In effect the Moon saps the Earth’s rotational energy, causing it to gradually spiral away from the Earth.
• As the Earth’s rotational energy is sapped, it rotates more slowly. This causes the length of the day to get very slightly longer, at the rate of approximately 0.0023 seconds per century.

The sapping of the Earth’s rotational energy by the Moon is not 100% efficient. Rather than all of the extracted energy going to accelerate the Moon away from the Earth, some of it is dissipated as heat – warming up the oceans slightly.

These effects are discussed in a previous post The Days are getting Longer.

Could the Moon be responsible for the Origins of Life

Because the Moon has been getting farther away from the Earth, in the distant past the Moon was much closer than it is today. When the Moon was first formed about 4.5 billion years ago it was only 25 000 km away. The Moon’s proximity to Earth meant tidal forces were much stronger and when the first primitive single celled life forms emerged, about four billion years ago, the Moon was already around 138,000 km away from Earth, 36% of its current value. At this time, the Earth rotated faster and a day was around 18 hours in length. It would have taken only eight of these 18-hour days for the Moon to complete one orbit around the Earth.

Four billion years ago the tidal forces would have been 22 times larger than they are today. There would have been a difference of hundreds of metres between the water levels at low and high tides and a large number of tidal pools  These would have filled and evaporated on a regular basis to produce higher concentrations of amino acids than found in the seas and oceans, which facilitated their combination into large complex molecules.  These complex molecules could well have been the origin of the first single celled lifeforms.

As discussed earlier, the Moon’s gravity is the main cause of tidal forces, and most locations on the Earth have two high tides every 25 hours. However, the water levels at the two high tides are not always the same and for many areas the time when high water occurs is out of step with what would be expected by only considering the Moon’s gravitational field. There are other effects which come in to play as well.

Flow of water, shape of coastline, large landmasses

Water has to flow from an area of low tide to an area of high tide and there may be large landmasses in the way preventing or delaying this flow. If we consider the UK as an example, the shape of the coastline and the water depth results in different tide times around its coast. When the mass movement of water caused by tidal forces crosses into shallow seas, its speed decreases. Also, the outline of the coast prevents the tidal wave from moving in a uniform direction. For example, St Mary’s in the Isles of Scilly (A) experiences high tide whilst at the other end of the south coast, Dover (B) is experiencing low tide. The tidal range, which is the difference in water level between high tide and low tide, varies widely around the UK coast. In general, the tidal range is larger when water is forced through a narrow channel and smaller in flat open coastline. The Bristol Channel(C), a narrow strip of sea 120 km long which separates South West England from South Wales, experiences the third highest range of anywhere in the world, with a mean spring tidal range of 12.3 m. On the east coast Lowestoft (D) experiences a mean spring tidal range of only 1.9 m.  As a general rule, the shapes of the shoreline and ocean floor affect the way that tides propagate to such a degree that there is no simple formula to predict the time of high water from the Moon’s position in the sky.

Inclination of the Moon’s orbit

Another factor is the inclination of the Moon’s orbit to the Earth’s equator. This means, for many locations, one of the daily high tides is significantly higher than the other. For example, if we take a location in the Southern Hemisphere, marked as A in the diagram, when the Moon is directly overhead A lies at the centre of the tidal bulge caused by the Moon. However, twelve and a half hours later, after the Earth has rotated, location A is no longer centred in the tidal bulge (its new location is shown as A’) and the tidal force is significantly weaker. The tidal bulge is now centred at location B, which lies in the Northern Hemisphere.

Shape of the Moon’s orbit

Another factor to consider is that because the Moon moves in an elliptical rather than a circular orbit and its distance from the Earth varies, the maximum strength of the tidal force will vary as well. For example, it will be especially strong if there is a full moon, which is also a supermoon, directly overhead.

And finally

I hope you have enjoyed this post and are staying safe in these difficult times. For any readers wanting further mathematical detail on tidal forces, please see the additional notes.

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

The gravitational field at a point 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 the centre of the Moon. As 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 size 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).

The units gravitational fields are measured in are 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.

The tidal force due to the Moon, at any given location on Earth , is the difference between the Moon’s gravitational field at that location and the  gravitational field due to the Moon at the Earth’s centre. The diagram below shows the tidal force due to the Moon at various locations on the Earth. Direction of the tidal force -The diagram shows a two-dimensional slice through the Earth-Moon system. At each location, the direction of the black arrow marks the direction of the tidal force due to the  Moon’s gravitational field and the length its magnitude.

As you can see from the diagram the tidal force is at its strongest at the location on Earth closest to the Moon (B) where its direction is towards the Moon  and also at the location on Earth  farthest from the Moon(D) where its direction is directly away from the Moon.  At some locations (e.g.  A and C) the tidal force is directed inwards towards the centre of the Earth.

Working out the magnitude of the tidal force

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 force 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 force varies as the inverse cube of the distance from the Moon.

Conversely, if we take the point on the Earth’s surface farthest 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 force 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 ) then   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 farthest away are approximately equal but act in the opposite direction.

Some examples of tidal force at different Earth-Moon distances

If we put the value of the Moon’s mean distance from the Earth DEM, 384 400 km, into the equations then the tidal forces 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 strengths of gravitational forces 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.

## 13 thoughts on “How tides work”

1. Hi Steve, I am afraid the explanation of the tides that you give here is simply wrong, but it is a very common misconception which is perpetuated on a number of respected websites and text books. You state that the tides are raised by the vertical component of the differential gravitational field of the Moon and go on to calculate this as about one ten millionth of a ‘g’. Newton himself did this calculation and knew that this was an impossibly small effect to account for the tides. It was Euler in the mid-eighteenth century who first correctly realised that it is the lateral or tangential components of the off-axis differential forces that are the horizontal tractive forces that raise the tides. These tractive forces are zero along the Earth-Moon line because the gravitational differential is exactly perpendicular to the surface, but away from this line and over all the rest of the planet, the tidal differential has a tangential or horizontal component. Although these tractive forces are also tiny, they are, crucially, cumulative and the Earth’s oceans are vast, and sea water is incompressible, which all leads to the raising of the tides. The tidal bulges do indeed peak along the line of the Earth-Moon axis, giving the false impression that the Moon has lifted them up, when it hasn’t. Instead they have been gathered up around the sides if you like, which is a much more subtle effect. Laplace used Euler’s insights to develop his eponymous Tidal Equations which are a set of differential equations which are still used today and which take the vertical component correctly to be zero and the tide-raising force to be all horizontal. I write this to you in all good faith as I share your enthusiasm for a better understanding of science and the world around us.
Best wishes,
Richard Jowitt

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2. Dear Steve,
An Interesting article and it is easy to see that the Earth and the sea would be pulled as one object towards the moon if they were a point mass at the Earth’s centre and because they are not, and the Moon’s gravity varies across the Earth’s diameter, then the near-side sea is pulled slightly more and the far-side sea is pulled slightly less, leading to the two opposite tidal bulges.
The Moon’s near-side and far-side gravity strengths are 3.497 micro-g and 3.272 micro-g, which is a very small difference and I was toying with the difference between the heights of the opposite bulges. The near-side tidal force is 115 nano-g and the far-side is 109 nano-g, which would seem to indicate a 5.5% difference.
To what extent, if any, is the impact of the difference in centripetal force at the Earth’s surface necessary to cause circular motion about the Earth-Moon barycentre important ?
Assuming a radius for the Earth at 6371 km, a barycentre radius of 4671 km, and a rotation of 27.32 days, the far-side centripetal field required is 7.978 micro-g and the near-side field required is 1.228 micro-g. These are very different values and far exceeding the tidal field force.
I presume that it makes little difference, but I see some references elsewhere to inertia in explaining the opposite bulge. How to put this to bed ?

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3. Hi Steve,
I wonder if you can explain tidal friction in a little more detail. In particular your explanation implies that the moon arrives (at a particular point above the surface of the Earth) after the high tide has already occured at that point. I have read elsewhere that it is the other way around which seems more intuitive. I have also read that the delay is only generally a matter of about 12 minutes. Is this correct?
Many thanks!
Ben

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4. An interesting read. I can see why there is a high tide on the seas facing the moon but why is there also a high tide at the point furthest away from the moon?

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1. The bulge D may be understood as the moon’s tidal force pulling the planet (not the ocean) toward it. (source: National Geographic)

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2. Hi Graham, Thanks for your comment,
As stated in the post,
The tidal force due to the Moon at a given point on the Earth is the difference between the Moon’s gravitational field at that point and the Moon’s gravitational filed Earth’s centre of mass (which is the same as the Earth’s centre, because the Earth is spherical.

Tidal Force at location x = gravitational field at x – gravitational field at the Earth’s centre.

At the point on Earth closest to the Moon, the Moon’s gravitational field is stronger than it is at the Earth’s centre, so the tidal force points towards the Moon.
At the point on Earth farthest from the Moon the Moon’s gravitational field is weaker than it is at the Earth’s centre, so the tidal force points away from the Moon.
At the Earth’s centre the tidal force is always zero.
At another places on the Moon tidal force is not zero and depending on the result of the subtraction, it may point towards the Moon away from the Moon, or indeed at any direction to the Moon. Some examples of the tidal force at different locations on the Earth are shown in the diagram near the bottom of the post labelled ‘Direction of the Tidal Force’

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5. Interesting part of nautical science as well. One of my favourite books is A History of Marine Navigation by the astronomer Per Collinder.

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6. Fascinating. I grew up by the sea and thought I understood tides. Now I realise that I only half understood them! Thank you.

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7. Hi Steve,
A very good explanation, & very well-worded so that your intended readership, once they’ve started, are happy to keep on reading. I suspect you’ll need to be prepared for questions about the Sun’s influence, additive and subtractive – and what would have happened in the Moon’s absence.

I’d imagine that nearby aliens – who no doubt have somewhat better instrumentation than ours – could measure changes in Earth’s shape & diameter, and work out how much of the surface is water-covered; probably how deep the oceans are at various points and observe the much smaller “tidal” bulges of solid land-masses. They’d no doubt go on to deduce quite a few facts about Earth’s interior.

You mentioned the emergence of life: If Darwin’s “small warm rock pool” origin scenario is right, then I’m sure* that tides in a previously-sterile ocean would have been instrumental in its origin. If it originated in deep ocean, e.g. hydothermal vents, then it might have already had time to evolve to a high degree, and the contribution of tides would have been to allow it to spread (rather reluctantly!) onto land.

* well, almost sure; disregarding panspermia etc.
Regards, David.

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1. Thank you David for your comment. Yes it is interesting to speculate what Earth would be like without the Moon, but I strongly suspect that we wouldn’t be here !
Steve

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