X-axis attitude error: 206.7 marcs rms
Y-axis attitude error: 285.6 marcs rms

For the sake of simplicity, when describing GP-B’s mission, we show its polar orbit as a circle around a spherical Earth. But, as several astute readers have pointed out, this is not truly the case. The Earth is not perfectly spherical in two ways: one, it has a slight bulge around its waistline, called “oblateness”. Two, its gravitational field differs from place to place around the globe due to varying distributions of the Earth’s mass.

How then do these factors affect GP-B’s orbit, and more importantly, affect the geodetic and frame-dragging predictions? If GP-B’s predictions for the drift of a gyroscope (6.614 arcsecs - geodetic; 0.0409 arcsecs – frame-dragging) assumed a perfectly spherical Earth and a perfectly circular orbit, wouldn’t these numbers be wrong?

The answer is, ‘Yes…and no.’ Leonard Schiff’s original “relativity gyroscope” predictions did have to take into account these two aspects of the Earth – but not as much as you might think. Schiff’s original calculations, based on a spherical Earth and a corresponding circular orbit, came within 0.1% of the final calculation for gyroscopic drift.

OBLATENESS
By far, the oblateness, or bulging, of the Earth is the more significant factor of the two that had to be taken into account. The Earth’s rotation causes its center, or equator, to bulge out slightly due to rotational momentum. At present, the equatorial radius is about 21 km longer than the Earth’s polar radius (a difference of 0.33%), meaning the Earth has a slightly oblong shape when viewed from space. Consequently, a polar orbiting satellite will not follow a perfectly circular path; its orbit will be slightly elliptical as it goes from pole to equator to pole. In regards to Schiff’s calculations and the predictions for GP-B, the Earth’s oblateness altered the final number by about 0.1%. See this NASA article for more info on the Earth’s changing shape.

GRAVITATIONAL VARIATIONS
The second aspect, variations in Earth’s gravitational field, has an even smaller effect on an orbiting satellite. The strength of Earth’s gravitational field varies with the amount of mass in the Earth at a given location. Mountain ranges like the Himalayas, which comprise of an enormous amount of mass in a given area, exert a stronger gravitational pull than do areas like the Indian Ocean, where most of the mass is in the form of water (less dense, so less mass in a given area).

In theory, these variations in gravity could effect the GP-B satellite. A stronger or weaker gravitational area could change the position of the satellite, ever so slightly. In reality, however, these variations are far too miniscule to affect the GP-B gyroscopes and satellite. Any possible changes in the satellite’s orbit have been calculated to be well below the tolerances of the SQUID (device that monitors gyroscopic drift) and of the experiment as a whole.

For more information about mapping the Earth’s gravity field, check out the GRACE experiment, including its incredible models of Earth’s gravity!

So the answer to the initial question is both ‘Yes’ and ‘No’. GP-B scientists did have to account for the Earth’s oblateness to make an accurate prediction of a gyroscope’s drift. But since the gravitational variations around the Earth are so small, GP-B scientists did not need to take them into account. Just in case, GP-B is carefully monitoring the orbit of the satellite and its gyroscopes and will compare these orbit paths with the predicted path after the completion of the mission.

FACTOID OF THE WEEK

Did you know that the Earth has been experiencing “post-glacial rebound”? During the most recent ice age, tens of thousands of years ago, the massive weight of the ice caps was flattening the Earth. Since then, as the ice caps melt, the Earth has been regaining its spherical figure slowly but surely. As of today, the Earth is nearly spherical – less 0.5% from perfectly round!