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The Double Asteroid Redirection Test (DART) mission was the first planetary defense demonstration mission to test the effectiveness of a kinetic impactor to deflect an asteroid (Cheng et al. 2016). The spacecraft was launched on 2021 November 24 UTC from the Vandenberg Space Force Base in California, and after a roughly 10 month cruise, it impacted Dimorphos, the satellite of binary near-Earth asteroid Didymos, on 2022 September 26 at 23:14 UTC (Daly et al. 2023). Goldstone radar observations about 12 hr after impact provided the first estimate of the orbital period change of Dimorphos of −36 ± 15 minutes (Thomas et al. 2023). Subsequent optical and radar observations over a timespan of about 2 weeks refined the period change estimate to −33.0 ± 1.0 minutes (Thomas et al. 2023). This orbital period change corresponds to a momentum transfer efficiency, β, between 2.2 and 4.9 (Cheng et al. 2023). Using images taken by the Didymos Reconnaissance and Asteroid Camera for Optical navigation (DRACO) on board the DART spacecraft, Daly et al. (2024) estimate the axes of the best-fitting Dimorphos ellipsoid to be 173 × 170 × 113 m prior to the DART impact. Elsewhere, Barnouin et al. (2023) used DRACO and the Light Italian Cubesat for Imaging of Asteroids (LICIACube) imagery to obtain the best-fitting ellipsoid of Didymos as 818 × 796 × 590 m.The impact and the aftermath were observed by several ground-based observatories as well as spacecraft, including the Lucy spacecraft (H. Weaver et al. 2023, in preparation) and the LICIACube, which was released 15 days prior to impact (Dotto & Zinzi 2023). Using the Hubble Space Telescope, Li et al. (2023) monitored the complex evolution of the ejecta under the influence of gravity and solar radiation pressure starting about 15 minutes after impact for about 19 days. The observations not only provided insight into the kinetic impactor technique of asteroid deflection but also shed light on the formation and evolution of active asteroids.Moskovitz et al. (2024) continued photometric observations of the system until 2023 February, extending the observational data arc by about 4 months beyond that used in Thomas et al. (2023). Here we show that the additional data reveal more complex dynamical effects and enable estimation of higher-order terms in the gravitational fields of Didymos and Dimorphos. In this work, we used the same general estimation approach as described in Naidu et al. (2022) and Thomas et al. (2023), although fitting the extended data set required a higher-fidelity dynamical model (Section 2) and more precise measurements of the mutual event times (Section 3).Our approach differs from Scheirich et al. (2024) in terms of the observables, models, and fitting algorithm. The method of Scheirich et al. (2024) fits the entire primary-subtracted lightcurve, including portions that are outside the mutual events, whereas we model only mutual event times observed in the primary-subtracted lightcurves. In addition to lightcurves, we also fit radar delay and Doppler measurements as well as DART spacecraft observations that were taken on approach to the Didymos system. Our orbital model contains additional parameters such as the absolute dimensions of Dimorphos and Didymos, the J2 of Didymos, and the ΔV of Dimorphos due to the DART impact.

For the preimpact orbit, we used the same dynamical model as that described in Naidu et al. (2022) and Thomas et al. (2023). We assumed Dimorphos is a point mass on a modified circular Keplerian orbit about Didymos. The sole non-Keplerian effect was an additional term to capture the drift in mean motion of Dimorphos due to nongravitational forces such as binary YORP. The mean anomaly (M) and mean motion (n) of Dimorphos at time t < timp (time of DART impact) are given by where M0 and n0 are the mean anomaly and mean motion at the epoch t0 and is the rate of change of mean motion.We treat the effect of DART on the Dimorphos orbit as an instantaneous, impact-induced velocity change (ΔV) in the radial (ΔVR ) and transverse (ΔVT ) directions. Since the DART velocity vector was almost aligned to the plane of the mutual orbit (Daly et al. 2023), we assumed no velocity change in the normal direction (ΔVN ), thus keeping the problem planar and simplifying the postimpact orbit integration.We obtain the state (i.e., position and velocity) of Dimorphos immediately after impact by adding the impulse (ΔV) to the velocity at impact from the preimpact orbit. We used this state as the initial condition for integrating the postimpact orbit. This approach maintains position continuity across the DART impact and guarantees self-consistency between the preimpact and postimpact positions of Dimorphos, which is not necessarily the case when fitting separate preimpact and postimpact orbits as was done in Thomas et al. (2023).To integrate the postimpact orbit, we assumed Didymos to be an oblate spheroid and Dimorphos to be a triaxial ellipsoid. With these assumptions, we derived the equations of motion following the planar full two-body problem formulation described by Scheeres (2009).The potential energy for the system shown in Figure 1 is defined as Here mA and mB are the masses of the primary and secondary, and are the mass-normalized inertia tensors of the primary and the secondary, ϕA and ϕB are the orientation angles of the primary and secondary with respect to the inertial frame, and r is the distance between the two objects. For a triaxial ellipsoid, and , where a, b, and c are the semiaxes of the ellipsoid from longest to shortest. Since we assume Didymos to be an oblate spheroid, we have , and the orientation angle of Didymos is ϕA = 0. We can also replace the moments of inertia with spherical harmonics coefficients, C20 and C22, using the following relations: where R is the longest semiaxis of the body (circumscribing radius). Zoom In Zoom Out Reset image size Figure 1. The planar full two-body problem from Scheeres (2009).Download figure: Standard image High-resolution image Using Equations (2) and (3) and the equations in Section 2.3 in Scheeres (2009) and introducing a time-dependent, draglike transverse acceleration (described in Section 5), we can derive the following equations of motion: where m = mA mB /(mA + mB ) is the reduced mass of the system, μ = G(mA + mB ), RA and RB are the circumscribing radii of the primary and the secondary, , and .We tested the importance of solar tides on the mutual orbit and found them to be negligible due to the proximity of the two bodies compared to the distance of the Sun. Solar tides arise from small differences in the solar gravitational force on the two bodies.

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