2. Relative motion of the nuclei
A total of 124 positions of nucleus A with respect to B has been collected from the available astrometric data, covering a period of time from May 5, 1997 to Jan. 23, 1998, or 8 months. The major contributors were Gajdo et al. (1997, 1998), Sugie (1997), Yamanishi et al. (1997, 1998), and Nakamura (1997, 1998). Positional data were likewise reported by Kojima (1997), by Kobayashi (1997), by Manca and Cavagna (1997), by Holvorcem (1997), by Hergenrother and Spahr (1997), and by Pravec and arounová (1997). The maximum angular separation between the two components, about 3 arcmin, occurred in October 1997.
Analysis of the positional offsets has followed the standard technique for split comets. As described by Sekanina (1982a), this iterative least-squares differential-correction orbital procedure solves for up to five separation parameters: the time of splitting, ; three components of the separation velocity, , in the cardinal directions; and the companion's relative radial nongravitational deceleration, . The separation-velocity components in the directions referred to the plane of the parent comet's heliocentric orbit are, respectively, the radial (away from the Sun), transverse, and normal velocities, , , and , in the right-handed RTN coordinate system. The deceleration is assumed to vary inversely as a square of heliocentric distance and is usually expressed in units of 10the solar gravitational acceleration. The mutual gravitational attraction of the fragment nuclei is ignored.
Unless the comet experiences a grazing approach to a planet, the planetary perturbations can safely be neglected. The orbital elements by Nakano (1998), calculated for an osculating epoch of Dec. 23, 1996, have been used below, after they were precessed to equinox B1950.0. Because of the diffuse nature of the nuclear condensations, their astrometric positions are measured with a nontrivial uncertainty, usually a fraction of 1 arcsec. It is therefore necessary to prescribe a rejection cutoff for the residuals of the offsets in right ascension and declination. In this investigation, the separation parameters were computed for six assumed rejection cutoffs that vary from 1.2 arcsec down to 0.2 arcsec in steps of 0.2 arcsec.
The sets of separation parameters from the orbital solutions constrained by each of the six rejection cutoffs are listed in Table 1. The apparent, expected decrease in the nominal mean residual with decreasing rejection cutoff is diagnostically meaningless. In fact, an excessively tight rejection cutoff requires that most observations be discarded (e.g., 90% for a cutoff of 0.2 arcsec), including many at either end of the time span covered, thus shortening the orbital arc to be used in the computations and leading necessarily to relatively inferior solutions. In truth, the most constraining rejection cutoff is yielded by its minimum value that is expressed in terms of the standard deviation of a fitted Gaussian distribution law.
Table 1. Separation parameters from solutions for varying rejection cutoffs (equinox B1950.0).
A discriminating search criterion can appropriately be formulated on the basis of a simple consideration that follows. Let - c be the residual between the observed value of a offset of nucleus A from nucleus B and its value calculated from the chosen solution. Let be the absolute value of an intrinsic rejection cutoff. Since the offset residuals for each solution's output are given to 0.01 arcsec, the intrinsic rejection cutoff equals its nominal value (as listed in Table 1) + 0.005 arcsec. Similarly, the intrinsic number of the offset residuals equals to , where N is the number of pairs of offsets. The unity is the most likely correction that, for a continuous distribution, expresses the fact that N pairs of offsets do satisfy the rejection cutoff, but do not. Since, for a Gaussian distribution, and since , then
where C is a constant of proportionality and is given by
Since, furthermore, the squares of residuals summed up over the N pairs of offsets can be expressed as
a ratio (o - c) can be written in the form
It can be shown that F is a monotonically nonincreasing function of , with and . Since, for any solution, F is expressible in terms of the quality-of-fit data, Eqs. (2) and (5) can be used to find , the quantity of interest here, from this information.
Table 2 compares the six solutions in terms of . It is apparent that the condition is satisfied by each of the tabulated values of F and that the most constraining rejection cutoffs are near 0.6 arcsec. The standard deviation of the Gaussian function amounts then to 0.57 arcsec, or about 1.7 times the mean residual listed in Table 1. Inspection of the actual distribution of the relevant offset residuals has confirmed that within 0.6 arcsec they have indeed been fitted most satisfactorily by the Gaussian law; for larger residuals, the distribution rapidly becomes distinctly non-Gaussian. The author is satisfied that the solution with the rejection cutoff of 0.6 arcsec offers a high-quality set of parametric values, probably the best achievable one under the circumstances.
Table 2. Comparison of the solutions
The magnitude of the nongravitational deceleration classifies nucleus A as a typical short-lived companion, even though its calculated endurance (cf. Sekanina 1982a for the definition) of 109 equivalent days is more than twice the expected value for its minimum lifetime.
The excellent match to the measured offsets of nucleus A is apparent from Fig. 1, which also shows the computed motion of the companion prior to its discovery. Unfortunately, the comet's appearance at the time of splitting will never be known, as the object was then only from the Sun, heading for conjunction with it on Feb. 10, 1997, 6 weeks after perihelion. In fact, the comet was not at all observed between July 18, 1996 and May 5, 1997. Even though its perihelion distance was 1.3 AU, it was never seen at heliocentric distances under 2.2 AU!
© European Southern Observatory (ESO) 1998
Online publication: September 30, 1998