Saturday, April 2, 2011

The Earth's geoid is Potato shaped

This video is no April's fool joke: Earth really is shaped like a potato. However, the shape that you see here is, um, slightly exaggerated to highlight its irregularities.

Another caveat: what is depicted here is not the shape of the planet, but rather the shape of an idealized sea-level surface extending around the entire globe—a surface that Earth scientists call the geoid.

The video is the most accurate reconstruction of the geoid to date, and was released by the European Space Agency at a scientific meeting in Munich on March 31 (see a video of the related press conference here).

It is based on data collected by ESA's Gravity Field and Steady-State Ocean Circulation Explorer (GOCE). The five-meter long, arrowhead-shaped probe has been in a low orbit for almost exactly two years, collecting painstakingly accurate measurements of the gravitational field of Earth.

Alas, many of the Web sites and blogs that reported on GOCE's results so far got it wrong. Contrary to what was stated in these reports, the geoid is not a surface on which the strength of gravity is the same at every point.

The geoid is, in the words of an oceanographer on the GOCE team, a surface such that if you placed a marble anywhere on it, it would stay there rather than rolling in any direction. Another way of saying it is, imagine you were an engineer traveling around the world with a level.

Then wherever you go, the level would be exactly parallel to the geoid at that place. Yet another equivalent definition: it is a surface that's everywhere perpendicular to the direction of a plumb-bob, or in other words, to the gravitational field.

Gravity need not have the same strength everywhere on the geoid. In other words, if you could walk on the geoid you would see gravity always pointing exactly downwards, but your weight could slightly change from one region to another.

The misunderstanding may have stemmed from the confusion of two concepts from multivariable calculus: a vector field and its potential. The vector field in this case is the gravity field, and the potential is the gravitational potential (which is essentially the gravitational energy of a unit mass).

At any point in space, the gravity field can be seen as the direction in which the gravitational potential rises fastest. Its magnitude, or length, is the rate of change of the potential. The field defined this way is called the gradient vector field of the potential.

In general, the gradient of a quantity defined in space is the 3-D version of the derivative of a quantity in the sense of single-variable calculus. (To be pedantic: for historical reasons, the gravity field is defined to be the opposite vector to the gradient of the potential.)

If you were to follow a gravitational field line—a curve which is at every point tangent to the vector field at that point—you would follow a curve of steepest ascent on the gravitational potential.

That's why it is called the gradient. (Note that because of inertia, the field lines of the gravitational field are not necessarily the trajectories of a body in free fall. The field won't tell you in which direction you'll be moving—only in which direction you will accelerate.)

Gradient vector fields are difficult to grasp in 3-D, because picturing the potential (to visualize the ascent or descent) would require a fourth dimension. But a 2-D analogy might help.

Think of how water trickles down a hilly surface. Neglecting inertia, a droplet of water would follow a curve of steepest descent, which is a vector field made of vectors along the surface.

In this analogy, the elevation of the droplet is the analogue of the potential at that point. And the analogue of the geoid is an elevation contour on a topographical map: the potential—the height—is constant along the contour. Obviously, some parts of this contour can be on steep terrain, whereas others can be on mild gradients. (On a topographical map, the slope is steeper where many different contour lines are crowded together.)

The magnitude of the gradient vector field—which represents the steepness of a slope—is not the same at all points of the contour.

Thus, gravity does change in strength along the geoid, and so the geoid is not "a shape where the gravity is the same no matter where you stood on it," as one blogger put it.

Nor is it "how the Earth would look like if its shape were distorted to make gravity the same everywhere on its surface," as reported by a major news site, which went on to add that "areas of strongest gravity are in yellow and weakest in blue," apparently not noticing the contradiction with the previous sentence.

More information here at Scientific America

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