Monday, March 14, 2011

The physics of the "Supermoon", part 2

In the last post, we used Newton's Law of Gravitation to calculate the relative strengths of the gravitational force on the surface of the Earth from (1) the Earth, (2) the Sun, (3) the Moon at closest approach, and (4) the Moon at its farthest distance from Earth. We also looked at the difference between (3) and (4), and argued that this is the important quantity when considering the impact of a "supermoon" compared with other times. The calculations were in the previous post, and summarized in a table, reproduced below:



In this post, we'll look at why the relative position of the Sun, Earth, and Moon influence tides, but the change in distance between the Earth and Moon is not as significant.

The "supermoon" on or about March 18 is occurring when the moon is in full phase, and at closest approach. This is shown in the figure below.

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There are two configurations that create fairly strong tides. Spring tides, the strongest tides, occur when the Sun-Moon-and Earth are in a line, and in that order. 

The configuration above, which will occur around March 18, produces weaker tides, known as neap tides.

Tides occur because the Earth takes up a 3-dimensional space. Because one side of the Earth is a little bit closer to the Moon than the other side, that side experiences a slightly larger gravitational force. 

The force on the closer side of Earth is 


while the force on the far side of Earth is



Now, I'm using the mass of the Earth and treating it as if the entire Earth's mass is concentrated either in the near or far parts. This is incorrect. But since I'll be interested in only the relative strengths of these tidal forces, this is okay for now.

I first factor out the distance between the Earth and Moon of the denominator, and I remember that the radius of the Earth is much, much smaller than the distance between the Earth and the Moon. (The radius of the Earth is about 2% of the distance between the Earth and Moon). Therefore, I can approximate it using a Taylor (MacLauran) expansion (precalculus/calculus).






The tidal force is defined as the difference between the near side and far side forces. Therefore,


Simplifying, I get



Notice that it depends upon the distance to the third power. Therefore, tidal forces from far objects are much weaker than from nearby objects. 

Using a similar process, I can also get the tidal force due to the Sun:



Using the numbers summarized in the first table, I get the following values for the tidal force due to the Sun and the Moon:


I can now calculate the tidal force from the Sun and the Moon on the Earth in various configurations.

Let's go back to spring tides and neap tides. 

The strongest spring tide will occur when the Moon is closest to the Earth, 



The total tidal force in this case would be


The weakest spring tide will occur when the moon is farthest from the Earth.
In this case, the total tidal force would be



The strongest neap tide will occur when the moon is on the opposite side of the Earth relative to the Sun, but it is at the closest point in its orbit. (This is the configuration anticipated on Friday.)
In this case, the net tidal force is


Note: the minus sign only means that the net force is heading in the direction of the Moon, not the Sun.

The weakest neap tide occurs when the Moon is farthest from the Earth and on the opposite side of the Earth relative to the Sun:




In this case, the net force is


Here's a table summarizing the total tidal force in the four configurations of interest.

I scaled it to the weakest spring tide. Why did I do that? Because, every month, the Earth experiences a spring tide at least that strong. There is no reason, then, to freak out about a tide that is going to be less than or equal to half as strong as the tide experienced every new moon. 

In other words, don't freak out too much. There's nothing out of the ordinary with this configuration, either based on gravitational strength or on tides.

Additional notes:

Simplifications: I did not account for the relative inclination of the Moon relative to the Sun-Earth plane. Because the inclination is about 5 degrees, it will affect the results by at most a factor of 1-cos(5 degrees) ~ 0.004 = 0.4% for values involving the Moon. 

1 comment:

Bert said...

You are wrong about neap tides. Neap tides are produced at first and third quarter moon. Spring tides are produced at full moon and new moon. Whether there is a difference between a new moon spring tide and a full moon spring tide (Earth to sun distance being the same and Earth to moon distance being the same) I don't know.