The pull of the planets

This piece was entirely rewritten 2017-06-20 in response to a correction by Adam Atkinson.

A friend of a friend stated:

"... the planets exert an enormous influence on the tides..."

... and that set my oh-no-they-don't-o-meter. Let's have a look, shall we?

You might think - as I did when I first wrote this post - that Newton's inverse square law was the key thing in terms of tides, but it's not. At least, not as it stands.

While the force attracting a particle (of mass $m$) to a body (of mass $M$, a distance $d$ away) is indeed $F = \frac {G M m}{d^2}$, where $d$ is the distance between the centres and $G$ the gravitational constant, the important thing is the difference between the force at the top and the force at the bottom - to all intents and purposes, the derivative, with respect to distance, of the force. That is to say, $F' = -2 \frac{G M m}{r^3}$.

This is now in newtons per metre, so strictly we'd need to multiply by the height of the particle to get a force - however, we're just comparing things, so let's not bother. In fact, let's also ignore $G$ and the mass of the particle, which also remain constant.

The only things that change when we're comparing the tidal forces due to planets are the mass of the planet and the distance to them.

$\frac{M_{moon}}{R_{moon}^3}$ turns out to be $1.5 \times 10^{-3}$ kg/m³, which will be the basis for our comparisons.

Let's look at Venus. According to Wolfram|Alpha, Venus has a mass of $4.9 \times 10^{24}$kg, and is currently $1.25 \times 10^{11}$m away. Its tidal effect is therefore about $2.5 \times 10^{-9}$kg/m³.

That's 6 orders of magnitude smaller than the Moon's. A million times, if you prefer numbers with names.

(The Sun, meanwhile, has an effect of $5.7 \times 10^{-4}$kg/m³, about 40% of the moon's effect1 .)

Looking at the planets, we get the following values (as of June 2017):

Planet Effect (kg/m³)
Jupiter $4.2\times 10^{-9}$
Venus $2.5\times 10^{-9}$
Saturn $2.3\times 10^{-10}$
Mercury $4.3\times 10^{-11}$
Mars $1.1\times 10^{-11}$
Uranus $3.1\times 10^{-12}$
Neptune $1.2\times 10^{-12}$

Jupiter exerts the biggest effect out of all of the planets - but it's still 300,000 times smaller than the pull of the moon.

So, no. The planets most certainly do not exert an enormous influence on the tides.

* Further edited 2017-10-15 to correct reference to wrong heavenly body, thanks again to Adam.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. This number is unfamiliar to me; if you feel like checking my work, please do! []


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