Tidal force, or The Moon and the Mosquito revisited

As Ξ noted in an earlier post, the claim that the gravitational pull of a mosquito is stronger than the gravitational pull of the moon is off by a fair bit … roughly five orders of magnitude.

The original author of the claim, George Abell, was an astronomer.  Aren’t astronomers supposed to be good at working with large numbers?  Wondering if he had been misquoted in Scientific American, I set out to find out what I could about this claim.

A google search turned up a fair number of sources that describe Abell’s claims; most of their accounts are similar to that of Lilienfeld and Arkowitz.  One of the more widely read accounts appears in The Skeptics Dictionary (by Robert T. Carroll):

Astronomer George O. Abell claims that a mosquito would exert more gravitational pull on your arm than the moon would (Abell 1979).

The secondary literature is pretty much in agreement: Abell claimed that the mosquito exerts a stronger gravitational pull than the moon.  (And clearly that claim is false.)

A quick stop at the library of a nearby college turned up Abell’s original article.  He had written a piece for the Skeptical Inquirer, a review of a book on the putative effects of the moon on human behavior.  The book’s author had suggested a plausible mechanism for such influence:  the body is largely made up of water, and we all know the moon is a primary cause of tides on the Earth.

Abell’s discussion notes the source of the Moon’s influence on terrestrial tides:  not the gravitational pull of the moon, but rather the difference in that force between the nearest and farthest points on the Earth.  Because of that difference, the Earth is (very slightly) distorted, with its fluid surface in motion attempting to achieve equilibrium.

Abell notes that while the Sun’s gravitational pull on the Earth is more than 100 times stronger than the Moon’s, its tidal force — the difference in the Sun’s pull over the diameter of the Earth — is less than half that of the tidal force of the Moon.

If the Moon’s influence on human behavior were tidal (acting on the fluids in the body), then that tidal effect would  be due to differences in the Moon’s gravitational pull on the fluid in different parts of your body — differences due to the fact that (for a 6 foot tall person) part of their body could be six feet closer to the center of mass of the Moon than other parts of their body.   Abell quotes this difference as being “about one part in (or 30 trillion) of the weight of that fluid”, and states that the copy of a magazine in the reader’s hand is tens of thousands of times more significant in producing tidal forces.

If we take the weight of the magazine to be roughly 1/2 lb [based on weighing a stack of AMS Notices], or 2.2 Newtons, leading to a mass of 0.2 kg, then the tidal force due to the magazine (over distances ranging from .2 m to .5 m — assuming I hold the book about 20 cm from my torso, and my torso is roughly 30 cm deep) will be

(where G is the gravitational constant, and M is the mass of the fluids in your body), which works out to approximately newtons.  A similar calculation for the lunar tidal force acting on my body when I am standing (taking my height to be 2 meters):

which is roughly newtons.

So indeed, as Abell claimed, it is reasonable to conclude that a small book or magazine in your hand exerts a tidal force on the order of thousands of times stronger than the moon on your body.

But wait!  Where are the mosquitoes?

Abell described a second potential source of lunar influence on the human body.  He writes:

What might matter is the difference between your weight in the presence of the moon’s gravitational effect and what it would be if there were no moon.  At the most, that difference amounts to only 0.01 gram, or about 0.0003 ounces, less than the effect of a mosquito on your shoulder.

In the paragraph leading up to that passage, Abell noted that this isn’t the same as the gravitational pull of the Moon on your body, since the Moon is also pulling on the Earth (so you have a slightly larger acceleration toward the Moon than the center of the Earth does — in effect, this is the tidal force calculation in a different guise).

It isn’t quite clear what Abell means by “the effect of a mosquito on your shoulder”:  is he referring to a gravitational (tidal) effect due to the mosquito, or is he just referring to the weight of the mosquito — the amount by which it increases your weight when it lands?

I’ll leave the analysis of these two interpretations as exercises for our readers.  But the conclusions I found:  if we compute the tidal force of the mosquito on your body, it is far greater than the tidal force of the moon on your body (because of its proximity versus the great distance to the Moon).  If instead we believe that Abell is claiming that the weight (mass) of the mosquito is greater than the 0.01 g that he had computed, then he is wrong:   Ξ had generously estimated the mass of a mosquito at 5 mg; Abell’s estimate of the Moon’s influence on a person’s weight is about twice that.

Abell’s review of Arnold Leiber’s The Lunar Effect: Biological Tides and Human Emotions (1st ed, 1978) appeared in the Skeptical Inquirer 3 (1979) pp 68 – 73.