Tidal Effects: The Planets vs. the Pebbles in Your Yard

Do the planets have tidal effects on human beings?

In the August 2015 issue of Sky & Telescope, in my Focal Point column about astrology and my friends who cast horoscopes for pay, I wrote

They’ve learned better than to try to give me lame physics woo, such as the stuff you’ll hear about our bodies being mostly water and celestial objects raising tides within us. They’ve been well instructed that each of the millions of pebbles in your yard has a greater tidal effect on your body (water or otherwise) than Jupiter, Mars, or Venus do.

. . . and I promised to give you the math at this link. So here we go.

To keep this as simple as possible, here's the key: Tidal effects decrease as the cube of the distance from the object doing the pulling.

Plain old gravity decreases as the square of the distance. But tides are a step up in complexity. "Tidal force" is the difference in the gravitational pull that something exerts on a body's near side and far side.

So, for instance, the Moon's tidal force on the Earth is the difference between the Moon's pull on Earth's near side and its far side. The difference isn't much, considering how far away the Moon is compared to Earth's diameter. But the difference makes the oceans slosh around enough for someone by the seaside to notice. The reason it's enough to notice is because the oceans have several thousand miles of width to slosh in. Unlike a lake, or a dish of water, or your body, in which the tides are fantastically miniscule (decreasing as the inverse cube of width).

Now, what else changes as the cube of distance? The volume of any two bodies whose angular size appears the same.

For instance, picture two round stones 1 inch and 2 inches in diameter. The latter will have 8 times the volume, and hence 8 times the mass, of the former: 2 cubed. If you hold the 2-inch ball twice as far away as the 1-inch ball, they will have the same apparent angular size.

And since twice as far means one-eighth the tidal effect, the 8-times-as-massive ball twice as far away will have the same tidal effect on your head as the smaller, closer one.

Any objects you see with the same angular diameter have the same tidal effect on you, assuming they have the same density (like rock).

An example: the Moon and Sun appear nearly the same angular diameter as seen from Earth, although the Sun is 400 times farther away. You may have read that the Sun exerts 40% of the tidal force that the Moon does. (That's why we have spring tides and neap tides depending on whether the Sun and Moon are aligned or crosswise.) If the Moon and Sun had the same density, their tidal effects would be the same. This fact alone tells you that the Sun is made of stuff that's 40% as dense as the Moon's stuff — without knowing a thing about the Sun being gaseous, or its interior compression, or anything else!

And in fact, look it up and you'll see that the Moon's average density is 3.34 grams per cubic centimeter, while the Sun's is only 1.41 grams per cc. Forty percent as much. Cool, huh?

Now let's apply this to pebbles in your yard compared to planets.

Mars, for instance, never appears wider than 25 arcseconds. It's made of rock. A pebble of rock at the back of your yard 20 meters (66 feet) away from you appears 25 arcseconds across if it's just 2.4 millimeters wide — a tenth of an inch! That's more like a large sand grain than a pebble. It exerts the same tidal force on you that Mars does.

Which is fantastically tiny in any case, in part because your body is way smaller than the ocean.

Millions of pebbles larger than that are on and under your yard. Move a single one of them, and it changes the tidal force on your body more than the moving of Mars does.

So when astrologers talk about tides linking us to the planets, they're babbling pathetic woo.