A question of waves

A tide, earlier
Every now and then someone sends me an email with a question in it, and I try to answer.*

Sometimes these questions are rather silly - and that's fine. That's how we learn, by asking silly questions. (I do it myself all the time with real scientists.) Sometimes the questions are pretty straightforward, or mind-bogglingly wacky. But just occasionally you get one that's really interesting - and I had such a one the other day about the tide.

As we all know, the tides are primarily influenced by the Moon, though there is also some input from the Sun. But my questioner wondered why this would be the case, as the Sun has a much bigger gravitational pull on the water than the Moon does.

Two questions, then. Was he right, and if so, why does the Sun's influence appear so understated? This is one of this areas where a few back-of-an-envelope calculations can give you a useful feel for what's going on. Thanks to Mr Newton (we don't need general relativity for this, thankfully), we know that gravitational force is proportional to the mass of the body producing it divided by the square of the distance. Plug in the numbers for the Sun and the Moon and you'll find that the Sun does indeed out-pull the Moon by a factor of 160 or so. (This shouldn't be surprising - the Earth's orbit would be distinctly scary if the Moon beat the Sun.)

However, tides are not about absolute gravitational pull. Working out the actual details of tides is messy indeed, but we can get a feel by thinking that the key factor in producing them is the difference between the pull the water feels on the surface of the Earth and the pull it would feel if it were at the centre of the planet. (There are other factors, including the spin of the Earth and the fluid nature of water than add the horrendous complexity of the real calculations, but this gives us a feel.) That distance, the Earth's radius, is significant in terms of the distance to the Moon, but makes very little difference compared to the distance the Sun. It's the amount of variation of gravitational pull that matters for tides, not the absolute value. This results in the tidal effect being approximately dependent on the inverse of the distance cubed, not the distance squared as in the usual gravitational calculation. And hence the Moon becomes the big cheese. As it were.

* If it's about something in one of my books. I reserve the right not to answer questions about, say, One Direction or fashion or many other subjects.