Shukie said:
Yeah, I know it's an old topic, but something bugged me about this answer. I have never heard this explanation for the second water bulge. I thought the second one was created due to the centrifugal force that arises from the Earth and moon revolving around their centre of mass?
That is a common but completely erroneous explanation. First off, what centrifugal force?
Secondly, and more importantly, suppose an object of the Moon's mass is falling straight toward the Earth with zero tangential velocity. There is no centrifugal force in this case; the object is falling straight into the Earth. At the point in time that this object reaches the Moon's orbital distance the tidal forces will be exactly equal to those exerted by the Moon. The tidal forces exerted by the Moon has nothing
per se to do with Moon's orbit. It is solely a function of where the Moon is.
So, how to explain that there are two bulges? The answer is simple. The Earth as a whole is accelerating gravitationally toward the Moon:
a_e = \frac {GM_m}{{R_m}^2}
where
ae is the acceleration of the Earth toward the Moon,
Mm is the mass of the Moon, and
Rm is the distance between the centers of the Earth and the Moon.
The distances between the center of the Moon and the points on the surface of the Earth directly opposite the Moon and directly between the Earth and Moon are
Rm+re and
Rm-re, respectively. Here
re[/i] is the radius of the Earth
Because these points are a bit further from/closer to the Moon than the center of the Earth, the acceleration toward the Moon at these points will be slightly different than that of the Earth as a whole. In particular,
a_p = \frac {GM_m}{(R_m\pm r_e)^2}
It is the difference in acceleration that is important here. Calculating this,
a_{p,\text{rel}} = \frac {GM_m}{(R_m\pm r_e)^2} - \frac {GM_m}{{R_m}^2}<br />
\approx \mp\, 2\,\frac{GM_mr_e}{R_m^3}
The point on the surface of the Earth directly between the centers of the Earth and the Moon experiences a bit more acceleration toward the Moon that does the Earth as a whole, so this differential acceleration is directed toward the Moon -- that is, away from the center of the Earth. The point on the surface of the Earth directly opposite the Moon experiences a bit less acceleration toward the Moon that does the Earth as a whole, so this differential acceleration is directed away from the Moon -- which is, once more, away from the center of the Earth. The Moon pulls the submoon point and its antipode apart, like a piece of taffy. For the points on the surface of the Earth where the Moon is at the horizon, the action is to squeeze those points inward a tiny bit.
For the Moon these tidal forces are very, very small. For a neutron star or black hole they are not. There is a cool technical term for how black holes pull the extremes of some object apart and squeeze in at the middle: http://en.wikipedia.org/wiki/Spaghettification" .
