### Tide-Causing Forces

Friday, November 11th, 2005For thousands of years, humans have noted a relationship between the patterns of the tides and the phases of the moon. Sir Issac Newton was the first to formalize this relationship into a mathematical “model” based on his equation for the gravitational attraction of planetary bodies (a function of the products of their mass divided by the square of their distance). Yet teaching of Newton’s equilibrium model of the tides (and perhaps even understanding of it) often gets jumbled, especially where explanations of tidal bulges directed away from the moon (or sun) are concerned. Here’s my attempt to get it right.

First, we know that the gravitational attraction between the sun and the Earth is greater than the gravitational attraction between the moon and the Earth. Earth orbits the sun, not the moon, eh? Yet the moon has twice the effect on tides as the Earth. Why? It turns out that the tide-causing forces come from the gradient of differences in the gravitational forces at various points on the Earth. In other words, the change or difference in the gravitational force of the moon (or sun) from point to point on the Earth is what is responsible for the tides. More formally, it is the deriviative of the gravitational force, dF, across the radius (r) of the Earth, dr, that gives rise to the tides. If we do the calculus and simplify, we find that the tidal forces are roughly proportional to the product of the masses of the attracting bodies divided by the cube of the distance between them. You can easily look up the masses and the distances and do the math to convince yourself that the tidal force of the sun is about half that of the moon (because distance cubed for the sun is such a large number).

The mathematics get a bit complicated but if you calculate the differential tidal forces across the sphere of the Earth, you will find forces directed towards the attracting body and away from the attracting body. One way to think about this is to imagine a hollow rubber ball (or a balloon) sitting on a table that you press down upon. As you press down, the ball elongates in two directions. Of course, the moon (or sun) doesn’t push down on the top of the Earth, but the tidal forces do include vertical components directed inwards. As a result, tidal forces compress and elongate the Earth. The Earth actually experiences daily tides as a result of tidal forces. moving ever so slightly (imperceptibly but measurably). When we add water to the Earth’s surface (assuming no continents), the lack of tensile strength of water (as opposed to Earth) makes the vertical components (the pushing and pulling) very weak. But the horizontal or tangential components of the tidal forces do move water and they move water towards the location beneath the attracting body and away from the attracting body (on the other side of the Earth). To summarize, it is the differential tidal forces-the gradient of variations in the moon’s or sun’s gravity at different points on Earth-that cause horizontal movements of water to locations beneath and away from the attracting body. The water piled up in these two locations is referred to as the tidal bulges (often terribly exagerrated in illustrations).

Note that we have described the causes of the tides with no mention of a moving Earth. That’s because tides would occur even if the Earth, moon and sun were static (admitedly impossible but significant, nonetheless). An explanation of tides does not require the Earth pulling away from the water, it does not require a centrifugal force and it certainly does not require the moon (or sun) to pull water upwards off the surface of the Earth. Tides occur simply because gravitional attraction varies over Earth’s surface.

Generation of the tidal bulges combined with Earth’s rotation give rise to the semidiurnal patterns of tides observed across much of the Earth. Throw in declination, the degree of “elevation” of the moon or sun relative to the equator, and you can derive an explanation for mixed and diurnal tides, too. Of course, the orbit of the moon around the Earth every 27.3 days (29.5 to return to a position in line with the sun, i.e., a tidal month) and the orbit of the Earth around the sun every 365 days generate monthly, annual, decadal and multidecadal patterns to the tides. In fact, a complete orbital cycle of the moon (ending where it started) takes 18.6 years, which is why NOAA bases its National Tidal Datum on 19 years, the National Tidal Datum Epoch.

Tides sound complicated but they really teach us a lot about Earth and its relation to the sun and the moon. You can learn more about this topic in today’s podcast. Click here. See also http://www.lhup.edu/~dsimanek/scenario/tides.htm.