Modeling the earth

[mooeypoo]

The math for doing this is far beyond most undergraduate students. The University of Texas has an ongoing modern pyramid building project involving a large number of slaves graduate students. In this project, called the Gravity Recovery and Climate Experiment, the slaves graduate students struggle valiantly to build very detailed, non-point mass models of the Earth using some rather hairy mathematics. You can read more about GRACE here.

 

BTW, D H, I know I'm a puny undergrad, but this is really interesting; we just learned in class something about distribution of mass (and charge, they seem to have the same general concept). The professor didn't get into it too much but he did say that dipole moment (quadruple moment, etc) define the general shape and 'distortion' of the mass. Is that what you are talking about (simply, of course, I'm sure its a lot more complicated than that) ?

 

btw.. hairy mathematics or hairy mathematicians? .. both can.. be.. uh.. challenging.

[D H]

The professor didn't get into it too much but he did say that dipole moment (quadruple moment, etc) define the general shape and 'distortion' of the mass. Is that what you are talking about (simply, of course, I'm sure its a lot more complicated than that) ?

That's the end result. The zeroth moment is simply the total mass. Since there is no such thing as negative mass, the dipole moment of any mass distribution about the distribution's center of mass is necessarily zero. So the first non-trivial components of a multipole expansion of a mass distribution are the quadrupole moments (2nd degree). This model of the Earth's gravity field goes up to degree and order 2159 (plus a partial expansion to degree 2190).

 

Using these models is relatively easy. All it takes is a good spherical harmonics algorithm. The hairy mathematics is coming up with the models in the first place. The Earth is not a rigid body. There are short and long term variations in the Earth's mass distribution. Over shorter periods of time, earth tides (the entire earth, not just the oceans, heaves and buckles do to the moon and sun, by about 0.35-0.5 meters) and ocean tides really make a mess of things. Over longish periods of time, mass northward during northern hemisphere winter and back toward the equator in northern hemisphere summer. One of the goals of GRACE is to uncover climate changes based on subtle variations in how a couple of satellites orbit the Earth.

[mooeypoo]

That's the end result. The zeroth moment is simply the total mass. Since there is no such thing as negative mass, the dipole moment of any mass distribution about the distribution's center of mass is necessarily zero. So the first non-trivial components of a multipole expansion of a mass distribution are the quadrupole moments (2nd degree). This model of the Earth's gravity field goes up to degree and order 2159 (plus a partial expansion to degree 2190).

wow. Neat. Is that because of the mountains/vallies of the surface of the Earth (like, does it go this deep in terms of accuracy)?

 

Using these models is relatively easy. All it takes is a good spherical harmonics algorithm. The hairy mathematics is coming up with the models in the first place. The Earth is not a rigid body. There are short and long term variations in the Earth's mass distribution. Over shorter periods of time, earth tides (the entire earth, not just the oceans, heaves and buckles do to the moon and sun, by about 0.35-0.5 meters) and ocean tides really make a mess of things. Over longish periods of time, mass northward during northern hemisphere winter and back toward the equator in northern hemisphere summer. One of the goals of GRACE is to uncover climate changes based on subtle variations in how a couple of satellites orbit the Earth.

So, if I got this right, this distribution is a function of time, too? (again, I'm reeeeeeeeally simplifying things, I'm sure, but just conceptually it's really interesting).

And.. if we're on the subject - are those tides oscillating in some frequency (in relation to how close we are to the Sun, or our positions with the other planets) or is it completely varied due to the complexity of all the planets orbits compared to ours and the Sun's mass?

 

This is really interesting, if there are any mods awake, I'd appreciate cutting this part under a new thread

[D H]

The Earth's mass distribution is obviously different when expressed in different frames. The obvious frame of choice is an Earth-centered, Earth-fixed frame, but no matter what reference frame you choose the mass distribution will vary with time because the Earth is not a rigid body. People who use the gravity models to look for oil, for example, don't want to be bothered with these temporal variations. They want a static view of the planet. One of the things the model developers have to do then is to come up with this static view.

 

Regarding your question on tides: while gravity is an inverse square relationship, tidal forces are more-or-less an inverse cube relationship. This means the Moon plays a greater role than the Sun, and the planets are extreme bit players at best. Jupiter has 1/1000 of the Sun's mass and orbits at ~5 AU, so the magnitude of the tides induced by Jupiter are about 1/100,000 of those induced by the Sun.

 

The theoretical models of the tides was developed in the late 19th century / early 20th century by George Airy, George Darwin (Charles' son), and A.E.H Love. Love's formalism involves a lot of different factors called Love numbers (a search for which can return some incredibly unrelated and embarrassing results). If you want to avoid the annoying "what is your love number" hits, I suggest you google something more specific such as "Love number formalism". A good reference on earth tides: http://www.agu.org/reference/5.5_wahr.pdf

 

The different terms in the Love number formalism represent different frequency responses of the Earth to the Moon (and the Sun) induced primarily by the Earth's rotation. The responses are grouped according to period: roughly half a day (semi-diurnal), roughly one day (diurnal), and smaller long term variations driven by the month, the year, and the 18.6 year lunisolar cycle.

 

The developers of a model of the Earth's mass distribution must account for these tidal variations. To do that, they need good models of the Earth's rotation and of the motions of the bodies in the solar system. To do that, they need good models of time and a good inertial reference system. The people who develop models of the Earth's mass distribution, the Earth's rotation, the solar system, time, and reference systems work together to improve their models because the errors in these models are interconnected.