Astronautics Primer - Your Understanding of Space Starts Here!



Earth's Oblateness - J2

Columbus was wrong! The Earth isn't really round. From space, it looks like a big, blue spherical marble, but if you take a closer look, it's really kind of squashed. Thus, it can't most accurately be treated as a point mass, as it is treated in the two-body assumption. We call this squashed shape oblateness.

What exactly does an oblate Earth look like? Imagine spinning a ball of jello around its axis and you can visualize how the middle (or equator) of the spinning jello would bulge out-the Earth is fatter at the equator than at the poles. This bulge can be modeled by complex mathematics (which we won't do here) and is frequently referred to as the J2 effect.

J2 is a constant describing the size of the bulge in the mathematical formulas used to model the oblate Earth. Why "J2"? This term arises from the mathematical short-hand used to describe Earth's gravitational field. (Gravitational acceleration at any point on Earth is commonly expressed as a geopotential function expressed in terms of Legendre polynomials and dimensionless coefficients Jn-whew!). J2, J3 and J4 are the zonal coefficients that depend on latitude. Of these, J2 is by far the most important; it is roughly 1000 times greater than either J3 or J4.

However, for more precise modeling of the Earth's oblateness, all three of these must be taken into account. In addition, other, higher order terms can be included in the model. These terms serve to slice the Earth into wedges that depend on longitude (sectoral terms) and slice it again into regions of longitude and latitude (tesseral terms).

Let's concentrate on the simplest and most profound case, J2. What effect does J2 have on the orbit? Let's look at the next figure. Here it's shown exaggerated; actually the bulge is only about 22 km thick. That is, the Earth's radius is about 22 km longer along the equator than through the poles.

The Earth's oblateness, shown here as a bulge at the equator (highly exaggerated to demonstrate the concept) causes a twisting force on satellite orbits that change various orbital elements over time.
Let's see if we can reason out how this bulge will affect the orbital elements. The force caused by the equatorial bulge is still gravity. Recall that gravity is a conservative force; therefore, the total mechanical energy in an orbit must be conserved. Total mechanical energy depends on the orbit's semimajor axis. Thus, as long as energy remains constant (i.e., no drag or other forces adding or stealing energy), the semimajor axis also remains constant.

It turns out that the eccentricity, e, also doesn't change, although the explanation for this is beyond the scope of our discussion here. Although you might expect the inclination to change because the bulge pulls on our orbit, it doesn't! However, it does affect the orbit by changing the right ascension of the ascending node, , and moving the argument of perigee, , within the plane. That's not very intuitive, but it's like a force acting on a spinning top. If you stand a nonspinning top on its point, gravity causes it to fall over. If you spin the top first, gravity still tries to make it fall but, because of its angular momentum, it begins to swivel-this motion is called precession. Let's examine the effect of precession on the ascending node and the argument of perigee more closely.



Astronautics Primer - Your Understanding of Space Starts Here!