3.1 The returns and the impacts

Our analysis of the orbital evolutions propagated from the catalog of 1201 multiple solutions shows that there are four possible collisions associated to this asteroid in the next half century: collisions are possible in January of 2014, 2038, 2044, and 2046. The existence of each of these impacting solutions has been confirmed by totally independent orbit computations (P. Chodas, private communication, 1999). Table 1 provides details for the corresponding VIs, including the MOID with gravitational focusing effects included (MOID _GF), the stretching on the MTP ( $\Gamma_{\mathrm{MTP}}$), and the probabilities of encounter within the lunar distance and of Earth collision (P_LD and $P_{R_\oplus}$, respectively). The $\sigma$ values listed in the table refer to the increase in $\xi$, the RMS of the observational residuals, relative to $\xi_0=0.528$, the RMS for the nominal orbit, according to the relation $\sigma^2=2N_{obs}\left( \xi^2-\xi_0^2\right)$, where N_obs=21 is the number of astrometric positions used in the fit. The keplerian orbital elements for each VI are also given. The J2000 reference frame is used throughout this paper.

 
 

Table 1: 1998 OX₄ Virtual Impactor Data.

Year of Impact	2014	2038	2044	2046
Date (UT)	Jan 20.539	Jan 20.203	Jan 20.686	Jan 20.264
$\sigma$	2.422	0.746	0.760	0.779
H mag	21.2	21.3	21.3	21.3
MOID GF ($R_\oplus$)	$0.480\pm 0.126$	$0.373\pm 0.174$	$0.710\pm 0.188$	$0.990\pm 0.189$
$\Gamma_{\mathrm{MTP}}$ (AU/$\sigma$)	397	95	47	18
PLD	$2\times 10^{-6}$	$6\times 10^{-6}$	$20\times 10^{-6}$	$40\times 10^{-6}$
$P_{R_\oplus}$	$0.3\times 10^{-7}$	$1\times 10^{-7}$	$2\times 10^{-7}$	$2\times 10^{-7}$
Keplerian orbital elements at epoch MJD 51024.0 TDT
a (AU)	1.626809266912	1.606833539990	1.607099946443	1.607352219135
e	0.506326452475	0.497502331734	0.497622010073	0.497735281195
I (deg)	4.767779650096	4.664179526644	4.665577804419	4.666901397937
$\Omega$ (deg)	300.4692804569	300.3776807433	300.3789300531	300.3801125065
$\omega$ (deg)	116.5952500167	116.5727153899	116.5730145568	116.5732965881
M (deg)	310.6465675773	309.6526746252	309.6661466286	309.6788984818

These impact possibilities result from resonant returns after other close approaches closer to the present time. The 2014 VI is comparatively far from the other ones in the confidence region of element space, and undergoes a very close approach in January 2001. The three VIs for 2038, 2044, 2046 belong to virtual asteroids close together along the line of variations; all three have a shallow encounter in January 2001, and they all have a quite close approach in January 2003, after which a high order resonant return leads to the possible collision. A complete description of the cascade of returns would be more complex, because there are many shallow encounters and even non-resonant returns: this asteroid has a moderate inclination $(4^\circ.5)$ and approaches within 0.1 AU can occur quite far from the descending node, and even near the ascending node.

It needs to be stressed that the use of the discrete set of 1201 virtual asteroids to represent the line of variations implies a resolution limit, thus it cannot be guaranteed that the impact possibilities listed in Table 1 are the only ones; however, the ones which might have been missed must have very small probabilities. Another possible reason of incomplete exploration of the confidence region is due to its large size, resulting in a shape significantly different from the ellipsoid described by the quadratic Eq. 1. More refined scans of the confidence region could be done, but a decision needs to be taken on the level of probability at which an impact possibility deserves a negative observation campaign.