Solving for Dactyl's Orbit and Ida's Density

Galileo's recent discovery that Ida has a satellite (now known as Dactyl) suggests that satellites orbiting asteroids may be a commonplace occurrence. In the Solid-State Imaging (SSI) camera data returned from the Ida encounter, Dactyl and Ida appear in 47 images. Their locations in these images were used to estimate Dactyl's orbit and Ida's bulk density, which is of great interest because it may indicate whether Ida is composed of rocks that have been thermally processed deep within a collisionally destroyed planetesimal. Density calculations were based on an Ida volume of 16,100 cubic kilometers (112 percent), which was determined from an accurate model of the shape of Ida. While Dactyl's orbit is of interest, its greatest significance is in providing the first accurate estimate of an S-type asteroid's density--another achievement by Galileo!

Initial attempts to apply classical astronomical orbit-fitting methods to estimate Dactyl's orbit, assuming a reasonable value for Ida's density, suffered from numerical problems caused by the Galileo-to-Ida line of sight being nearly in the plane of Dactyl's orbit for most of the images. Mike Belton, SSI Team leader, then asked the Navigation Team to apply their orbit-determination methods to the problem, which led to this challenging and enjoyable search.

Since our objective was to determine preliminary estimates for Dactyl's orbit and Ida's density, the analysis was simplified by assuming that Dactyl's orbit was affected only by Ida's gravity acting as a point mass. The problem was to find an orbit for Dactyl that was consistent with the locations of Ida and Dactyl in the SSI images. Much of the analysis involved reducing the raw data associated with the images (exposure time, camera-pointing direction, positions of Ida, etc.) to a form usable by a new computer program that could estimate Dactyl's orbit. Another large part of the task involved actually writing and debugging this new program, which is constructed of QUICK commands. (QUICK is an easy-to-use, versatile processor from JPL's Multimission Analysis Software Library.)

It became clear almost immediately that the mass/density of Ida could not be solved at the same time as Dactyl's orbit. Instead, a series of Dactyl orbits were generated for a range of Ida mass/density values--from 1.5 to 4.0 grams per cubic centimeter. For each density value, there is a unique orbit; over this range of densities, these orbits differ greatly. For Ida densities less than about 2.1 grams per cubic centimeter, the orbits are just barely hyperbolic. For higher Ida densities, the orbits are elliptical with a large apoapsis (farthest point from Ida), a periapsis (nearest point to Ida) of around 80-85 km, and periods that range from just over a day to many tens of days. At a density of about 2.8 grams per cubic centimeter,, the orbit is nearly circular (about 82 by 98 km) with a period of about 27 hours. For even higher densities, the elliptical orbits have apoapses of about 95-100 km, with periapses that decrease with increasing density. For an Ida density greater than about 2.9 grams per cubic centimeter, the periapsis is less than about 75 km and the period is less than 24 hours. The geometry for a range of orbit solutions is shown in the accompanying figure. Since this view is from the spin pole of Ida, the motion of Dactyl and Ida's rotation are both counterclockwise.

The figure shows that when points recorded at the same time on each Dactyl orbit are connected, they are parallel to the center line through Ida that points to the spacecraft. All of the images but the very last were taken when Galileo was thousands--or even hundreds of thousands--of kilometers from Ida and nearly in its equatorial plane, so that the spacecraft was viewing the Ida-Dactyl system from the lower right part of the figure. For scale, the long axis of Ida is 58 km, and Ida is shown as it would be oriented at the time of Galileo's closest approach. Thus, the figure only covers a few hundred kilometers around Ida.

The last image mosaic was taken when Galileo was almost at its closest approach to Ida and included parts of both Ida and Dactyl in separate images. At that point, Galileo was essentially looking down on Dactyl's orbit plane (essentially the plane of the figure), and Dactyl was at the point where all possible orbits cross. The lowest parallel line connects the points on each orbit at 5 hours prior to closest approach, or about the time of the earliest image. Since Dactyl was viewed for only a fraction of its orbit and from a nearly edge-on vantage point, all of the orbits shown fit the observations equally well. If one imagines being on the Galileo spacecraft looking at Ida and Dactyl, then all of the orbit solutions would have appeared the same during the 5-hour approach, since the differences between them are all along the line of sight (the parallel lines in the figure).

Thus, for a given mass/density of Ida, a unique and well-determined two-body conic orbit can be found. However, this alone does not help us find the unknown density of Ida. Only by applying the dynamics of motion about a non-point-mass Ida and using our knowledge of the general distribution of asteroidal material in the entire asteroid belt can the range of possible mass/density values for Ida be reduced.

Dynamical studies show that orbits with periapses less than about 75 km from Ida are unstable and either collide with, or escape from, Ida--thus, orbit solutions are not physically possible that correspond to an Ida density of about 2.9 grams per cubic centimeter or greater. At the other extreme, hyperbolic and even highly elliptical orbits around Ida are very unlikely. The observed speed of Dactyl around Ida for any of the orbit solutions is no more than about 10 m/s, about the speed of a fast run or a slowly thrown baseball. Calculations indicate that the chance of a random piece of asteroidal material the size of Dactyl passing by Ida at that speed, just when Galileo was observing it, are about 1 in 1019. In addition, if Dactyl were in a hyperbolic or highly elliptical orbit, it should have been seen by the Hubble Space Telescope (HST) when it observed the region around Ida over an 8-hour period on April 26, 1994 . HST would have easily seen Dactyl had it been more than about 700 km from Ida. Combining these two restrictions gives a preliminary estimate for Ida's density of 2.1 to 2.9 grams per cubic centimeter. Allowing for a 12-percent uncertainty in the modeled volume of Ida increases the range to 1.9 to 3.2 grams per cubic centimeter.

This density range is surprisingly well constrained and suggests that Ida is fairly porous and/or made of fairly light rocks. This result already excludes several classes of dense igneous rocks that had previously been suggested as the primary components of Ida's composition.

Further work on the long-term stability of orbits that fit these observations, as well as a more precise analysis of the SSI images themselves, may lead to a better determination of both the density of Ida and the orbit of Dactyl. These, combined with other ongoing work involving the color, spectral properties, and geology of Ida's surface are expected to lead to major advances in our knowledge of the nature of asteroids and what they can tell us about the birth of the planets.