Determination of Orbital Elements of Near-Earth Asteroids

Six near-earth asteroids were observed by T. Statler and students over several nights with the 2.4m Hiltner telescope at MDM Observatory. Image files were subsequently processed using coordinate determination and orbit fitting software to determine orbital parameters. These parameters were compared to accepted values stored in the JPL small-body database (JPLSBDB). Correlations in the range 4% - 20% were observed, depending upon object.

1. Introduction

On January 1, 1801, Guisseppi Piazzi discovered an object in the sky that was different from most. This object did not travel with the stars but in front of them. It was Ceres, a dwarf planet in the asteroid belt (Fodera Serio et al., 2002). When Piazzi alerted the scientific community, many people attempted to derive a method for determining this object's trajectory based on observations. Only William Herschel's 1781 discovery of Uranus provided any previous experience with fitting a trajectory to a newly discovered object. That particular case was successful mainly because Uranus has such a small eccentricity that it can be assumed to have a circular orbit. Other favorable circumstances included the planet's slow motion and the very small inclination angle of the orbit to the plane of the ecliptic. These characteristics and special mathematical methods not applicable to other situations led to simple calculations in order to determine its orbit. At the time, it was believed that one would need many observations over a longer period of time in order to obtain a precise orbital diagram. It was this rationale that inhibited scientists from truly understanding how to find an elliptical object's trajectory. Carl Friedrich Gauss was unsatisfied with the method used to find Uranus' trajectory as a standard model, so he created his own. He used Kepler's three laws to deduce that the entire orbital motion of an object can be determined from the internal "curvature" of any portion of the orbit. Gauss also hypothesized that the orbit of any object that doesn't pass extremely close to some other body in our solar system must have the form of a simple conic section with the focal point being at the center of the sun (Director & Tennenbaum, 1997).

With this hypothesis came five orbital parameters needed to specify the orbit of an object. Orbital eccentricity is the measure of how much an orbit deviates from a circle, defined for an ellipse as the distance from its center to one focus, divided by the length of the semi-major axis. The major axis of an ellipse is a diameter which passes through its center and both foci, and the semi-major axis is the corresponding radius, half the value of the major axis. An asteroids inclination angle refers to the angle between its orbital plane and the equatorial plane. The longitude of the ascending node is the angle, measured counter-clockwise, between the vernal equinox and the ascending node, where the ascending node is the point at which the asteroid crosses the ecliptic while traveling north. The argument of perihelion is the angle, with reference to the asteroid, between the ascending node line and the perihelion, perihelion being the point in the orbit at which the asteroid is closest to the sun. This angle is measured in the direction of the body's orbit (JPLSBDB, 2008).

2. Observations

3. Reductions

The UTC time stored in each image header was the time at the beginning of the exposure. Because imexam returned each object's position at approximately the middle of each exposure, and the CCD shutter took a finite amount of time to open, we had to modify the stored UTC time using the following equation:

Actual Time = Time Observed + .5 x Exposure Time + .5 seconds   (1)

Having obtained both precise coordinates and a reasonable approximation of observation time, we input these values into the OrbFit Consortium's "OrbFit" software to derive each asteroid's orbital elements: eccentricity, semi-major axis, inclination angle, longitude of ascending node, and argument of perihelion (2008).

4. Results

Table 1. Observational data for asteroids 2006WU29, 2006UQ17, and 2006UN216.

Table 2. Observational data for asteroids 2006PA1, 2006CY10, and 2006CL9.

Table 3. Calculated orbital parameters (top) compared to published values.

Table 4. Uncertainties of published orbital parameters. Data taken from JPL small-body database.

randomn(seed, 1) * 2.35   (2)

The factor of 2.35 in equation (2) corresponds to the curve's full width at half maximum. These random values were then added to the times calculated using equation (1), and this sum was input into OrbFit along with positional data. By taking the standard deviation of the ten resulting values collected in tables 5 & 6, we arrived at our final error approximation, the rows labeled "STDEV".

Table 5. Calculated orbital parameter errors for asteroids 2006WU29, 2006CL9, and 2006CY10.

Table 6. Calculated orbital parameter errors for asteroids 2006PA1, 2006UQ17, and 2006UN216.

Figure 1. Orbits of inner planets and asteroid 2006CL9.
Image reproduced from JPLSBDB.

Figure 2. Orbits of inner planets and asteroid 2006CY10.
Image reproduced from JPLSBDB.

Figure 3. Orbits of inner planets and asteroid 2006PA1.
Image reproduced from JPLSBDB.

Figure 4. Orbits of inner planets and asteroid 2006UN216.
Image reproduced from JPLSBDB.

Figure 5. Orbits of inner planets and asteroid 2006UQ17.
Image reproduced from JPLSBDB.

Figure 6. Orbits of inner planets and asteroid 2006WU29.
Image reproduced from JPLSBDB.

5. Discussion

After comparing our calculated results to those published in the JPLSBD, we generally observe a very tight correlation the two. The greatest differences between accepted and calculated values occurred with asteroid 2006WU29, for which we had only four frames of data, two of which were compromised by bad seeing. Excluding this extreme case, in which the accepted eccentricity was almost 16% larger than its calculated counterpart, many of our results matched published values to within a few percent.

From Kepler's Third Law,

P_yr² = a_AU³   (3)

we can approximate the periods of our objects as being between 1.5 - 5.5 years, which is considerably longer than the two or three days worth of observations that we considered. It is not surprising, however, that good results could be obtained from such a short interval. For Keplerian objects, any arc drawn out between two points along the orbit is unique, except for symmetrical positions at opposite ends of the orbit (Director & Tannenbaum, 1997). Should any parameter describing the orbit change, even slightly, then the observed arc would no longer fit the new orbit at any location. From this, one can conclude that an object's orbit can be determined by as few as three data points, the minimum required, by definition, to specify an arc in coordinate space. Given the achievable accuracy from few observations, it is also unsurprising that a newly discovered NEO's orbit is often made available as quickly as twenty-four hours after initial discovery (Minor Planet Center, 2008).

References