Quantification of my post on tides

I was asked a quite reasonable question on my post about tides. Azmat Hussain posted the question to my blog and on LinkedIn. Here is what he had to say on LinkedIn:

"I liked your writing style, and but it could have used some mathematics and some quantification. Like how much is the force on the body of water on this side vs the other side, and is the difference significant?"

I like it when people try to keep me honest!  

I looked on Wikipedia to get some rough numbers. The perigee and apogee of the Moon (closest and farthest points from the Earth) are 362,570 km and 405,410 km. Let's just say that the Moon is 380,000 km away from the Earth when we happen to be looking at the tide. The Earth has a radius of 6,371 km. I'm going to round that to 6,000 km just for ease of reading.

So, here are the numbers that I am going to play with: 374,000 km from the Moon to the closest point on Earth, 380,000 km to the center of the Earth, and 386,000 km to the far side of the Earth. [1]

Next, we throw in out good friend the inverse square law. Someone wrote a very entertaining blog that had something to do with that. It says that the pull of gravity falls off as the square of the distance between two objects. Based on that, I have assembled this table that shows the relative strength of the Moon's pull.

Location	Distance	Relative pull
Side closest to Moon	374,000 km	103.23%
Center of Earth	380,000 km	100%
Side farthest from Moon	386,000 km	96.92%

So... we have a total swing in gravitational pull of 6%. Wow. I was actually expecting a smaller number! I think this proves this is a huge effect, right? [2] And it also explains the whole monthly weight gain problem that women complain about. It's not bloating, ladies. It's the gravitational effect of the Moon!

But, then again, this isn't the whole story. I am talking about the gravitational pull of the Moon. Isn't that tiny? Who cares if it is varies, if it is too small to measure?

Hmmm... If I am standing on the Earth, how does the gravitational pull of the Moon compare to the gravitational pull of the Earth? The gravitational pull goes as the square of the distance. I am 6,000 km from the center of the Earth and 380,000 km from the Moon, so we are talking (6000 / 380,000) squared, which is about 0.025%.

Wait!! That's only a part of the answer, since gravitational pull also goes as the product of the masses, that is, the product of my mass with that of the Earth, and the product of my mass with that of the Moon. My mass is the same, so relatively speaking, we can just look at the relative masses of the Moon and Earth.

The Moon is about 7 X 10^22 kg, and the Earth is about 6 X 10^24 kg. The ratio here is about 1.2%. 

So, I conclude that the gravitational pull of the Moon is on the order of 0.025% X 1.2% of the pull of the Earth. If I am using my calculator correctly, I cipher this out to about 3 parts in a million.

I have gone through the whole calculation below.

Location	Distance	Relative pull	Pull relative to the Earth
Side closest to Moon	374,000 km	103.23%	3.00 X 10-6
Center of Earth	380,000 km	100%	2.91 X 10-6
Side farthest from Moon	386,000 km	96.92%	2.82 X 10-6

Wow. I was actually expecting a larger number! I think this proves that the Moon has a negligible effect, right? 

Then again, this effect is tiny, but we are talking about tides of a few meters... just a tiny distance compared with 380,000 km, about 5 parts per billion.

Azmat has asked a very reasonable question. At this point, I am afraid I must shrug my shoulders and say "flies walk on the ceiling". I say this whenever I am confronted with a problem that is orders of magnitude outside of where I normally live.When I make this comment about flies, I mean that the glue on the foot of a fly that holds it to the ceiling is tiny in my experience, and big for a fly. Alternately, gravity is a pretty big deal for me, but flies seem to be able to take it or leave it.  

My next inclination would be to put together an experiment to test the hypothesis. I would measure some tides with the Moon in place where it is, and then remove the Moon and measure the tides again. Simple experiment, really. Maybe I would repeat it a few times. I have applied for a grant to the NSF. Right now, it's hung up on the "romantic impact assessment". Some silly folks are concerned that the removal of the Moon might have a negative impact on the mating habits of homo sapiens. Darn tree huggers! I'll let you know when I hear back from the NSF.

Seriously, my next approach would be a computer simulation in which I modeled the Earth as a thousand little balls. Each ball would be given an initial position and direction vector. They would move through space under the effects of momentum and gravity, and subject to the constraint of non-compenetrability [they can't occupy the same space). Simply enough to write the code, right? (John rolls his eyes.)

In the mean time, I think that Azmat has provided a significant question about insignificance.

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[1] A thought occurred to me when I was figgering those numbers. I have made the assumption that perigee and apogee mean the distance between the centers of the Earth and whatever satellite  we are talking about. I wasn't sure about that, so I went to look it up on Wikipedia:

"An apsis ... is the point of greatest or least distance of a body from one of the foci of its elliptical orbit. In modern celestial mechanics this focus is also the center of attraction, which is usually the center of mass of the system. Historically, in geocentric systems, apsides were measured from the center of the Earth."

http://en.wikipedia.org/wiki/Apogee

Uh-oh. Things are getting complicated. I'm gonna stick with the geocentric model.

[2] Rule number 1 about reading John the Math Guy blogs. When I end a statement with "right?" it means that it's not right.

[3] This footnote is not referred to in the main text, but I thought I should put it in for completeness.

A Tidey Question

First question - Why are there tides?

The connection between the Moon and Sun to tides has been known at least since 150 BC. Seleucus was the ancient Greek go-to guy on tides. He knew that the Moon was in control of tides, but he thought the interaction was through "pneuma". Pneuma is something that doesn't exist but is all over the place. Pneuma was invented to allow the ancient Greeks to pretend that they understood. In this way, pneuma is akin to aether, phlogiston, electromagnetic waves, dark matter, and my non-existent buddy Horace. I pretend to go out for a beer with him when I don't want to tell my wife where I really am. 

Where am I when I don't want my wife to know where I am? Probably singing an old Righteous Brothers song at a karaoke bar. She is trying to cure me of this addiction. [1]

One of my "go to" karaoke songs when I am in a tidal mood

Posidonius (who lived around 100 BC) was another ancient Greek who was interested in tides. He said that the Moon's effect was because the Moon heated the water enough to make it expand, but not enough to make it boil. The effect of the Moon on tides was, in his mind, proof of astrology. If the effect of the Moon on the entire ocean is so large, then why can't a star that is a zillion miles away have enough of a selective force over my life so as to predestine who I would fall in love with and that I should be a karaoke star? Damn the stars for sentencing me to this unresolvable conflict!

It wasn't until Newton's Philosophiæ Naturalis Principia Mathematica that gravitational forces were singled out as the way the Moon and Sun controlled the tides. That's not such a big surprise, since Newton invented gravity. By the way, gravity didn't exist before July 5, 1687, and neither did tides.

Tides are caused by the gravitational pull of the Moon and the Sun on the water on the water. Gravity is pulling on the solid parts of Earth as well, but the water is free to slosh about. With respect to a point on Earth, the Sun rotates around every 24 hours, and the Moon every 24 hours and 50 minutes. This gives lots of opportunity for them to combine efforts and for them to cancel each other out as the two forces get into and out of phase with each other.

Fun fact #273 - Tidal waves are caused by earthquakes, not tides

Second question - Why are there two tides a day?

Now we have the more difficult question! The simple answer is that the oceans bow out on both sides of the Earth. The side facing the Moon or Sun has high tide, and the side facing away from the Moon or Sun also has high tide. Note that if the Moon is making high tides for me in Milwaukee and for my good friends in India, the Moon is doing nothing for my buddies in England.

Actual unretouched photo of the Earth with some way big tides

I have used a technique here called "answering a hard question with a vague and incomplete answer that really doesn't address the real question." This never worked on my mother, either.

Third question - Why is there a tidal bulge on the opposite side of the Earth?

This third question is the tough question. It almost looks like the Moon's gravitational force is pushing the water on the other side of the planet away. That don't make no sense. I went googling for the answer to this enigma. I found lots of answers. I have tried to arrange them according to similar explanations:

Crazy quirky

Additionally, by a crazy quirk of physics, it also causes the water to dome on the opposite side of the earth.
http://answers.yahoo.com/question/index?qid=20110330220751AAkEDdr

I will add "crazy quirk" to my list including pneuma, aether, and my non-existent buddy Horace.

Effect of the Sun

BUT, the sun is also pulling at the same time in the opposite direction halfway around the world. This produces two bulges, one near the sun, and one near the moon.

http://answers.yahoo.com/question/index?qid=20100131114142AA71hNp

Ummm... so the Sun is always directly opposite the Moon?

Centrifugal force

I found a lot of explanations that call on centrifugal force to explain the bulge on the opposite side.

The centrifugal force produced by the Earth's rotation cause water to pile up on the opposite side as well.

http://answers.yahoo.com/question/index?qid=20100320143833AADcbFf

On the side of the earth directly opposite the moon, the net tide-producing force is in the direction of the greater centrifugal force, or away from the moon.

http://co-ops.nos.noaa.gov/restles2.html

At the centre of the earth the two forces acting: gravity towards the moon and a rotational force away from the moon are perfectly in balance. … On the opposite side of the earth, gravity is less as it is further from the moon, so the rotational force is dominant.

http://www.pol.ac.uk/home/insight/tidefaq.html#4

…on the near side the direct pull dominates and causes the oceans to bulge in the direction of the moon; on the far side the centrifugal effect dominates and causes the oceans to bulge in the direction away from the moon.

http://scubageek.com/articles/wwwtides.html

These explanations are cool, and obvious, right? But, they kinda miss the point. Isn't the centrifugal force pretty much the same all the way around the globe? Why does the water bulge just at the two ends?

It's obvious

This is one of my favorite explanations. Wrap a hard problem in fancy words and then slip in an "it's obvious".

If every particle of the earth and ocean were being urged by equal and parallel forces, there would be no cause for relative motion between the ocean and the earth. Hence it is the departure of the force acting on any particle from the average which constitutes the tide-generating force. Now it is obvious that on the side of the earth towards the moon the departure from the average is a small force directed towards the moon; and on the side of the earth away from the moon the departure is a small force directed away from the moon.

http://www.1902encyclopedia.com/T/TID/tides-03.html

I was with ya until the part about "every particle"

Gravity differential, correct but confusing

Now we come to the real reason for two tides. The pull of gravity is slightly greater on the side of the Earth that faces the Moon as compared with the pull on the side that is opposite the Moon. I'll give my explanation of why this should cause tides in a bit, but first I want to acknowledge some explanations that are correct, but still confusing.

The bulge on the side of the Earth opposite the moon is caused by the moon "pulling the Earth away" from the water on that side.

http://science.howstuffworks.com/environmental/earth/geophysics/tide-cause.htm

What???

The not so obvious part is that the water on the far side is getting left behind because the earth is getting pulled away from it.

http://wiki.answers.com/Q/Why_are_there_two_high_tides_in_one_day

This sounds like that last explanation that I couldn't understand!

Owing to the differences of distance of the moon from various portions of the earth, the amount of attractive force will be different in different places and tend to produce a deformation.

Van Nostrand's Scientific Encyclopedia, Fourth Edition, 1968, entry on Tides

Ok, so why will it tend to produce a deformation???

And the water which is closer to the moon is pulled more strongly and so it’s pulled up into a tide. The [water] on the opposite side is pulled slightly less strongly and so it’s pulled down less strongly towards the surface of the Earth and so you get a second bulge on the far side of the Earth.

http://www.thenakedscientists.com/forum/index.php?topic=33978.0

Ummm... it is pulled less strongly... why does that cause a bulge on the opposite side? [2]

The Moon exerts a force on the Earth, and Earth responds by accelerating toward the Moon; however, the waters on the side facing then Moon, being closer to the Moon, accelerate more and fall ahead of Earth. Similarly, Earth itself accelerates more than the waters on the far side and falls ahead of these waters. Thus two aqueous bulges are produced, one on the side of Earth facing the Moon, and one on the side facing away from the Moon.

http://www.thehighesttides.com/what-causes-the-highest-tides.shtml

The Moon is falling! The Moon is falling! We must run and tell the king!

Trophy for best explanation I could find

I won this trophy for my karaoke rendition of Fly Me to the Moon

Here is the explanation that came closest to explaining it for me. I still think it needs work, but this got me thinking along the right track.

The pull is greater on the side facing the Moon, pulling the water there closer to the Moon, while the pull is weaker on the side away from the Moon, making the water there lag behind. This stretches out the Earth and the water on it, creating two bulges.

http://www.math.nus.edu.sg/aslaksen/teaching/tides.html

My own answer

First, gravitational pull decreases with distance. If I am standing directly beneath the Moon, it's tug on me will be larger than the tug I would get if I were on the opposite side of the Earth. This has nothing to do with the Earth getting in the way. It is all about distance.

Now, if I pull really hard on one side of a ball, and pull not quite so hard on the middle, and still a little less on the opposite side, the ball will deform a bit. 

That's the layman's explanation. The answer for a graduate physics exam would probably be a bit more involved. There would be some stuff about the inverse square law [3], and how the Moon and Earth are star-crossed lovers, destined to never meet [4]. Momentum and some crazy stuff about adding vectors of motion together would probably jump out.

But in the end, it's all about stretching.

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[1]  I know that, as a blogger, I have a solemn obligation to always tell the truth, but... I lied when I said that my wife is trying to cure me of my karaoke addiction. She has been known to grab the mic herself.

[2] This answer was from the Naked Scientists podcast. Very entertaining. I recommend it. Except they have this thing about trying to make science entertaining. Come on. Science is serious business. Stop making it fun.

[3] This law is on the statutes for Rhode Island. It says that you are not allowed to invert a square on any holidays that involve eating.

[4] The term "star-crossed lovers" comes from Shakespeare in reference to Romeo and Juliet. This is an astrological reference. It means that the stars have thwarted the romance. I think it was very clever of me to slip in another reference to astrology.