The Bill O'Reilly Problem

 “Tide goes in, tide goes out. Never a miscommunication. You can’t explain that. You can’t explain why the tide goes in.”

—Bill O'Reilly

 
Political pundit Bill O'Reilly elevated himself to meme status this past month because of his assertion that you can't explain the rise and fall of the tides.  Most people probably don't know the exact mechanism, but even in our Are You Smarter Than a 5th Grader? society, people should at least recall that there was some explanation learned in middle school science.  After all, it was over 300 years ago that Newton figured out the tides were caused by the Moon's gravitational field.  How does the Moon produce the tides and can we use physics to estimate just how big they should be?

We live in a gravitational field.  At the surface of the Earth, the field causes all masses to accelerate to the ground at a rate of 9.8 m/s².  But the Earth is not the only object that has a gravitational field: all masses are attracted to each other by gravity.  Right now, the mass of your body is exerting a very slight pull on the pyramids of Egypt due to gravity.  While your gravitational field is very weak, massive things like the Moon can create very large gravitational fields.  We can show the gravitational fields due to the Moon and the Earth schematically like this:

Gravitational field lines around the Earth (blue) and Moon (orange).

At each point in space, an arrow tells you what direction an object placed there would be pulled.  As you can see, the more massive (i.e attractive) Earth has more arrows pointing towards it.  Only arrows very close to the Moon point towards it.  The reason is that gravitational fields get weaker as you get further away, so if you're very close to the Moon you'll be pulled more towards it than the Earth.  This decrease in strength as you get further away is the reason there are two tides per day.  The Moon pulls very hard on the part of the Earth that is facing it.  The water there will shift a little closer to the Moon creating high tide.  Why is there a second tide 12 hours later?  In addition to pulling on the water, the Moon also pulls on the center of the Earth.  It pulls a little harder on the center of the Earth than it does on the water at the opposite end of the Earth because the center is closer.  This effect elongates the surface of the water giving it an oval shape as seen below:

The gravitational pull of the Moon effectively stretches the Earth.  (Not to scale.)

This is a nice qualitative description, but is there a way we can compute the height of the tides?  Yes!  It turns out that fields are not the only useful way to describe gravity.  There's also something called "gravitational potential".  Masses tend to move from areas of higher potential to areas of lower potential.  The potential V at some point due to a mass M is given by

V(r) = -G M / r,

where r is the distance between the point and the center of the mass and G = 6.67×10^-11 N·m²/kg² is the fundamental gravitational constant of the universe. If water in the oceans can flow to a lower potential, it will.  For this reason, we know that the gravitational potential should be the same everywhere on the surface of the ocean, because if it wasn't water would flow until it was.  We can write the potential due to the Earth and the Moon as

V = -G ( M_Earth/ r_Earth + M_Moon/ r_Moon),

where M_Earth/Moon and r_Earth/Moon are the mass and distance from the Earth/Moon, respectively.  We can plot lines of equal potential much like you can plot lines of equal height on a topographic map: 

Contour plot showing lines of equal gravitational potential.  The white circle on the left represents the Earth while the dark speck in the center represents the Moon. 

Notice the the potential lines are slightly warped by the Moon giving the characteristic oval shape one observes in the tides.  We can look up the mass of the Earth (M_Earth=6.0×10²⁴ kg) and the mass of the Moon (M_Moon=7.4×10²² kg).  When the Moon is directly overhead, r_Moon is roughly 3.7×10⁸ m and r_Earth=6.4×10⁶ m.  From this, we can calculate the gravitational potential at the surface of ocean to be  V=-6.25445×10⁷J/kg.  Halfway around the Earth, the height of the water will be lower but the potential must be the same.  If you calculate this height using the values listed and a little geometry, you find the difference between high tide and low tide is about 22 m.  In the Bay of Fundy, which has the largest tides in the world, the water rises and drops about 16 m, so our estimate is very good considering we didn't consider land masses or other complicating factors.