## 崱廡偺僯儏乕僗乮Jun丂 11乯

*2000擭6寧11擔庴晅

#### 俀丏徻嵶

http://near.jhuapl.edu/news/sci_updates/000505.html
2000擭5寧5擔乮 May 5, 2000 乯

As of the first week in May, NEAR Shoemaker has reached its prime mission orbit at about 50 km from the center of Eros. In this orbit, NEAR Shoemaker is close enough to the asteroid to measure composition, search for a magnetic field, and study internal structure. The spacecraft is also close enough now for the orbit to be affected strongly by the irregularity of the Eros gravity field. It is by studying the orbit that we will learn about the mass distribution within Eros. In some ways the orbit of NEAR Shoemaker is not very different from what is familiar, but in other ways it is quite strange.

The small size of Eros, and its correspondingly weak gravity compared to that of Earth (for example), mean that the spacecraft orbital velocity is much lower than we are accustomed to. In the 50 km orbit, the orbital velocity is about 3 m/s [7 mph] whereas in low Earth orbit the circular velocity is about 7700 m/s [18000 mph]. However, this is not a fair comparison, because the low Earth orbit is just skimming the surface of the almost spherical Earth, whereas the 50 km orbit around Eros is far above the elongated asteroid surface. If we found the radius of a sphere with the same volume as Eros, the radius of that sphere would be 8.5 km. That is, if Eros were fluid or a strengthless gravel pile, and it were not spinning, it would collapse into a sphere of radius 8.5 km because of its own gravity. Incidentally, we can infer from the shape of Eros that it must have some strength, but the required strength is very low even compared to that of ordinary soils on Earth, let alone rocks - more on that another time. Returning to our orbit around Eros, we should ask how fast would the orbital speed be for a circular orbit just skimming the surface of our (hypothetical) 8.5 km radius asteroid: the orbital speed would now be 7 m/s.

Why did we insist on comparing the orbital speeds this way? Because if we now compare the orbital periods of the low Earth orbit and the 8.5 km orbit around the hypothetical asteroid, we find they are no longer very different - the orbital period is 89 minutes around Earth and 120 minutes around the asteroid. It turns out that if the mean densities of the Earth and the hypothetical asteroid were the same, then the periods of the Earth orbit and the asteroid orbit would be equal. The period of an orbit skimming the surface of a spherical mass is inversely proportional to the square root of the mean density. Actually, the mean density of Eros is about the same as that of Earth's crust, which is only about half the overall mean density of Earth, because Earth's iron-rich core is somewhat denser than its crust. Earth's higher density than Eros means that the period of a surface-skimming orbit is smaller, but not by very much, because the square-root-of-density is never very different when comparing any ordinary materials, whether rocky or icy . At most this square-root-of-density factor differs by something like a factor of two, whereas the size, for example, differs by many orders of magnitude between Earth and asteroids. Hence for any asteroid, the orbital period of a low, surface-skimming circular orbit will always be about the same, regardless of the size of the asteroid. That means the speed in such an orbit is directly proportional to the radius of the asteroid - a spacecraft would orbit at about twice the speed around an asteroid of twice the radius, so as to complete an orbit in about the same time.

What about orbits that are not surface-skimming? We must now scale the orbital radius in terms of the body radius to make a fair comparison. That is, we compare the orbital period at a distance measured in Earth radii from the center, with the orbital period at the same number of asteroid radii, and we find that the orbital periods are the same (except for the inverse dependence on square root of mean density). The orbital period at six Earth radii, or 38000 km from the center of Earth, is about 22 hours, not very different from the 30 hour period at six asteroid radii, or 6 x 8.5 = 51 km from the center of Eros. This is remarkable considering the great disparities in size, mass, and strength of gravity between Earth and an asteroid.

Hence, this aspect of orbiting an asteroid is not terribly different from orbiting a planet like Earth, but that is because we have considered so far only the effect of small size. We have not yet discussed effects of the irregular shape. We know from Kepler's laws that orbits around a spherical body are conic sections, or in our case ellipses (with circles included as a special type of ellipse). However, as soon as we introduce a nonspherical gravity field, such as formed by a mass with a quadrupole moment, the orbits are no longer conic sections. In fact, they are no longer closed curves of any kind at all, but trace out a fantastic three-dimensional filigree, without ever returning to where they started and without even remaining in any fixed plane. We say that such orbits precess. At present, NEAR Shoemaker is trying to stay in an orbit that is close to circular at 50 km radius, but because of Eros's irregular shape, the distance from the center actually varies from 48 to 52 km - an incredibly bumpy ride. NEAR Shoemaker is navigating the most irregular, non-spherical gravity field that any spacecraft has ever experienced.