Well I guess I’m just all confused and going to have to give up on this thread.
from RM Mentock"
Egg-shaped, pear-shaped, whatever. The point was the shape is slightly unsymmetrical such that more volume (as well as mass) is above the equator than below. Maybe it’s the moon that’s egg-shaped and the earth is pear-shaped. *I’m * not playing so picky on the cross sectional geometry.
OK, I’ll accept egg-shaped as synonymous with pear-shape–I’m don’t want to be picky.
As for the point–it’s not true. Egg-shape or pear-shape, there is not more mass above the equator than below. The top of the “pear” is taller, but skinnier. The net effect is no difference between the two hemispheres.
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My dictionary says the equator is everywhere equidistant from the two poles. That would mean it can’t be taller, because that moves the equator higher. Thus, the lump is big on one “half” and small on the other “half”. 
Of course the distinction between mass and volume could be relevant, because the mass contained in the liquid core if off-center, I believe.
But now I’m just making things up. So while I’m at it, I’ll make up a new answer for the original question.
Down is soft and fluffy, and some people are allergic.
Here is an physical analogy that may help with understanding the concept of a geoid.
Fill a water container to the one liter mark. What if the water contains ice? The surface of the water would seem to be ambiguous, because some of the ice protrudes above the water level. However, if the ice were to melt, the water level would not change. So, it is reasonable to take that level as the actual measurement.
For the earth, 70% of the earth is covered with water. The surface of that water is known as the geoid. The continents “float” in the material of the earth’s mantle much like ice cubes do–the effect is called “isostasy.” Mountains are sticking up because their roots are less dense–just like ice cubes in water. If a mountain and its root were to “melt”, the result would be reduced to the geoid level.
When we say, a mountain is 2000m above sea level, we mean that the top of it is 2000m above the geoid. Such measurements have increased in accuracy over the years, but there are refinements still being made.
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It’s been bothering me that we’ve been neglecting, or at least glossing over, the effect of the Earth’s rotation. I get this effect to be 11.05 km at the Earth’s center at 45 degrees latitude; fully half the difference between down and the direction to the Earth’s center is due to the Earth’s rotation. Presumably, the other half comes from the Earth’s mass distribution.
For the effect of the Moon, I get less than a meter difference at the Earth’s center, and even this amount would tend to average to zero over time, so it can be neglected. Since the Moon causes the biggest tide, the other solar system masses are even more negligible.
If you’re not interested in the math, skip the rest of this message:
gmass = 980 cm/s^2 radially
grot = 3.39 cm/s^2cos(theta) away from axis of rotation (rhodially? 
)
The angle gmass + grot pulls is atan(gmass sin(theta) / ((gmass+grot)cos(theta))
= atan(1.00347 tan(theta))
= 45.09927 degrees at theta = 45 degrees
So the difference of 0.09927 degrees is the effect on “down” due to the Earth’s rotation. Converting to the difference at the Earth’s center gives 6378 * tan(0.09927) = 11.05 km.
The effect of the moon’s gravity will be smaller than the effect of the Earth by the factor
(Mmoon/MEarth) * (Rearth/Dmoon)^2 * (2 * REarth / Dmoon)
where Mmoon, MEarth are the moon and Earth masses, REarth is the Earth Radius, and Dmoon is the distance to the moon. The last factor
comes from the acceleration we care about being the difference between the acceleration at the Earth center and our location on the surface. This is the worst case number.
Using Mmoon/Mearth = 0.013, Dmoon/Rearth = 60, the Moon’s acceleration relative to Earth is smaller by 1.2E-7, less than a meter difference at the Earth’s center.
It is too clear, and so it is hard to see.
RM:
Of course the sea isn’t level. Wind and currents stir up waves, altering the coastal height at different locations. Thus the Panama Canal having a long series of locks.
Irishman writes:
No, no, no. As I wrote in Triumph of the Straight Dope, p. 46, the difference in height between the Pacific and Atlantic oceans at the entrances to the Panama Canal is only about eight inches. The locks are needed because the land is 85 feet higher in the middle.
ZenBeam
That is a very good point.
We haven’t necessarily been neglecting the effect of the rotation of the Earth–that it was important was the gist of my original post.
I did have a little trouble following your calculations, but it makes sense to me, if the plus sign in (gmass+grot) is changed to a minus.
We have relied upon Karen’s original calculation, but it seems accurate enough. If you were to take the equation for an ellipse, x^2/a^2 + z^2/c^2 = 1, where a is the Earth’s equatorial radius, and c is the polar radius, and find the point at which the slope of the normal is 45 degrees (latitude 45 degrees), you’d find that the normal at that point misses the center of that ellipse by about 21 kilometers. That appears to be what Karen calculated.
Cecil
Welcome to the party! Good to see you up and about after the Y2K thing.
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It is too clear, and so it is hard to see.
from Cecil Adams:
Administrator
[sheepishly]Oops. Guess I missed that one.[/sheepishly]
Fear not, Irishman, being corrected by Cecil is like being corrected by God – good for the soul, and not many mortals are so favoured.
An amended Mailbag item on this topic will be published shortly, and our thanks to all those who contributed thoughts.
RM Mentock… isn’t that the Chandler Wobble guy?
Hi Jill
Just thought I’d bug Dex for awhile, instead.
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Way to go, Mentock!
I would’ve said “down” was toward the center of gravity of a planet or star or other heavenly body also. And “up” would be the opposite direction.
Feel free to correct me at any time. But don’t be surprised if I try to correct you.
jab1
Actually, that’s something that I should have included in my OP, but I didn’t. ZenBeam is the one who pushed on that one: the pull of gravity that you experience is not necessarily towards the center of gravity. That only works in approximation (when you’re far away from the mass), or for symmetric spheres.
A good example of this is our solar system. Where is the center of gravity? Somewhere close to the Sun. But we here on earth don’t perceive a force towards that center of gravity. We wouldn’t even if we were not revolving or rotating.
rocks