Elsevier

Icarus

Volume 370, December 2021, 114624
Icarus

Research Paper
The excited spin state of Dimorphos resulting from the DART impact

https://doi.org/10.1016/j.icarus.2021.114624Get rights and content

Highlights

  • High-fidelity numerical codes are essential for modeling the long-term spin evolution.

  • DART may excite Dimorphos’ spin, leading to attitude instability and chaotic tumbling.

  • Dimorphos is especially prone to unstable rotation about its long axis.

  • A chaotic spin state will affect the system’s BYORP and tidal evolution.

  • ESA’s Hera mission may be able to place constraints on the system’s tidal parameters.

Abstract

The NASA Double Asteroid Redirection Test (DART) mission is a planetary defense-driven test of a kinetic impactor on Dimorphos, the satellite of the binary asteroid 65803 Didymos. DART will intercept Dimorphos at a relative speed of 6.5km s1, perturbing Dimorphos’s orbital velocity and changing the binary orbital period. We present three independent methods (one analytic and two numerical) to investigate the post-impact attitude stability of Dimorphos as a function of its axial ratios, a/b and b/c (abc), and the momentum transfer efficiency β. The first method uses a novel analytic approach in which we assume a circular orbit and a point-mass primary that identifies four fundamental frequencies of motion corresponding to the secondary’s mean motion, libration, precession, and nutation frequencies. At resonance locations among these four frequencies, we find that attitude instabilities are possible. Using two independent numerical codes, we recover many of the resonances predicted by the analytic model and indeed show attitude instability. With one code, we use fast Lyapunov indicators to show that the secondary’s attitude can evolve chaotically near the resonance locations. Then, using a high-fidelity numerical model, we find that Dimorphos enters a chaotic tumbling state near the resonance locations and is especially prone to unstable rotation about its long axis, which can be confirmed by ESA’s Hera mission arriving at Didymos in late 2026. We also show that a fully coupled treatment of the spin and orbital evolution of both bodies is crucial to accurately model the long-term evolution of the secondary’s spin state and libration amplitude. Finally, we discuss the implications of a post-impact tumbling or rolling state, including the possibility of terminating BYORP evolution if Dimorphos is no longer in synchronous rotation.

Introduction

NASA’s Double Asteroid Redirection Test (DART) mission will be the first to demonstrate asteroid deflection by kinetic impact as a realistic assessment for planetary defense. The DART spacecraft will intercept the secondary (Dimorphos) of the near-Earth binary asteroid system 65803 Didymos in the fall of 2022 (Cheng et al., 2018). The European Space Agency’s (ESA) Hera mission will arrive at the binary 4 years later to investigate the resulting dynamical and geophysical changes to the system (Michel et al., 2018). The nominal DART trajectory is an approximate head-on collision with Dimorphos, impulsively reducing its relative orbital speed, and thereby shortening the mutual orbit period and semimajor axis. The binary orbit eccentricity and inclination will also change, depending on the impact circumstances (Cheng et al., 2016). Fig. 1 shows a sketch of the binary system and the geometry of the problem. The change in orbit period will be measured with ground-based observations in order to infer β, the momentum transfer efficiency. The change in velocity of an asteroid in response to a kinetic impact can be written as (Feldhacker et al., 2017, Cheng et al., 2020), Δv=mM(u+(β1)(nˆu)nˆ),where m is the impactor mass, M is the target mass, u is the impactor velocity, and nˆ is the outward surface normal at the impact site. The first term represents the incident momentum of the spacecraft, and the second term is the contribution of escaping momentum, which is assumed to be along the surface normal. β can then be written as the ratio of the total transferred momentum to the momentum delivered by the impactor: β=M(nˆΔv)m(nˆu).In reality, β is a complicated function of the material properties and geometry of both the target and impactor (Stickle et al., 2020). If we assume a head-on impact on a flat surface (allowing us to ignore the impact geometry), we can express β as a simple function of scalars, β=1+pejectapDART,where pDART is the scalar momentum carried by the DART spacecraft, and pejecta is the scalar momentum carried by impact ejecta (which travels in the opposite direction). This expression for β is much simpler than the equation used in practice, as it assumes that pejecta and pDART are perfectly anti-aligned. However, this version is sufficient for describing why β is important: it tells us how much momentum is transferred to the target as a function of the impactor and ejecta momenta. For a more formal description and derivation of β, see Rivkin et al. (2021).

Due to the irregular shapes of both components and their close proximity, the spin and orbit of Dimorphos are highly coupled and non-Keplerian, meaning the dynamics cannot be treated as a simple point-mass 2-body problem. Therefore the use of high-fidelity, full-two-body-problem (F2BP) codes is crucial to understanding the complex dynamics (Agrusa et al., 2020). Further, the shape of Dimorphos is still unknown and could have a major effect on the system’s dynamics. With an assumed triaxial ellipsoid shape for Dimorphos, we explore the post-impact dynamical evolution of the system as a function of the possible axial ratios of the secondary and the momentum transferred by the DART impact (β). In Section 1.1, we give some brief background on the Didymos binary and the DART impact’s implications for the secondary’s libration state. Then Section 2 introduces our novel analytic approach and two numerical methods for studying the spin dynamics of Dimorphos. The results for each of these three methods are presented in Section 3. Finally, we discuss the implications of our results in Section 4.

Although it has not yet been confirmed with observations we nominally assume that Dimorphos is in the 1:1 spin–orbit resonance (i.e., tidally locked)1. Didymos’s spinning-top shape and fast rotation are suggestive of a rubble-pile structure, owing to likely formation scenarios such as spin-up-driven mass loss (followed by gravitational accumulation of the secondary), or gravitational reaccumulation after a catastrophic disruption (Richardson and Walsh, 2006). In addition, its spin rate exceeds the spin barrier at the nominal bulk density of 2.17 g cm3, implying some level of interparticle cohesion and/or higher bulk density (Zhang et al., 2017, Zhang et al., 2018, Zhang et al., 2021). If Dimorphos and Didymos have a common origin, this suggests that Dimorphos is also a rubble pile. The highly dissipative nature of rubble-pile asteroids implies that the system has had sufficient time for Dimorphos to become tidally locked and enter a dynamically relaxed state (Goldreich and Sari, 2009, Jacobson and Scheeres, 2011a). For these reasons we assume the system’s pre-impact dynamical state is relaxed, meaning the mutual orbit is well-circularized with the secondary in the 1:1 spin–orbit resonance and any free libration is minimized. However, it should be noted that observations have not confirmed such a relaxed state, rather it just has not been ruled out (Pravec et al., 2006, Scheirich and Pravec, 2009, Naidu et al., 2020). If, upon arrival at the Didymos system, we find that the mutual orbit and secondary spin are already excited, the DART impact will likely further excite the mutual dynamics. Therefore, the results presented in this work should be interpreted as a conservative estimate of the possible impact outcomes.

The angle between the line-of-centers (LOC) and the secondary’s long axis is commonly referred to as a libration angle. In the classic (uncoupled) spin–orbit problem, there are two distinct libration modes: free and forced (Murray and Dermott, 2000, Naidu and Margot, 2015). Although this paper explores the dynamics of the fully coupled spin and orbital dynamics of the Didymos–Dimorphos system, the insights from the classic spin–orbit problem provide useful intuition for understanding the dynamics when we consider the full problem. For a circular, uncoupled planar orbit, a first-order approximation for the frequency of free libration is given by (see Ch. 5 of Murray and Dermott (2000)), ωlib=n(3(BA)C)1/2,where n is the mean motion, and A, B, and C are the secondary’s three principal moments of inertia (which correspond to the axis lengths abc). For certain combinations of the three moments of inertia, the free libration frequency can become resonant with the forced libration frequency (i.e., the mean motion) and a secondary resonance can occur (Melnikov, 2001, Gkolias et al., 2019). This can lead to an intricate dynamical environment, which only becomes more complicated when we allow for non-zero eccentricity, out-of-plane motion, and a full coupling between the mutual orbit and the spin states of both bodies.

It is important to note that the DART impact will excite both free and forced libration modes, even if they have been damped to a minimum prior to the impact. The velocity perturbation from DART will increase the binary eccentricity (Cheng et al., 2016), increasing the forced libration mode, due to the restoring torque that the secondary feels as it becomes misaligned with the LOC as the orbital angular velocity changes throughout the orbit. With a nearly instantaneous perturbation to the orbital velocity of the secondary, DART will also induce free libration modes by creating a difference in its instantaneous orbital and spin angular velocities.

In reality, Dimorphos’s attitude has three degrees of freedom relative to the uniformly rotating orbit frame (roll, pitch, and yaw) and the system could have a nonzero eccentricity and inclination. Therefore, its spin evolution will be more complicated than the two idealized libration modes used here as a conceptual example. Namely, the excited planar libration modes, for particular shapes of the secondary, can induce significant out-of-plane rotation (Kane, 1965, Eapen et al., 2021). Moreover, energy transitions can happen between the planar and out-of-plane rotational degrees of freedom that is attributed to resonant phenomena (Breakwell and Pringle, 1965). We will see that the excitation of Dimorphos’s libration state, primarily due to the excitation of nonplanar rotation, can lead to chaotic motion. Chaotic rotation has been observed for many other bodies in our solar system such as the triple system (47171) Lempo, Saturn’s Hyperion, and Pluto’s outer four satellites, to name a few examples (Correia, 2018, Wisdom et al., 1984, Showalter and Hamilton, 2015).

In this work, we treat the “libration angle” as simply the angle between the long-axis of the secondary and the line-of-centers. As described above, in the classic spin–orbit problem, this angle would be purely within the plane of the orbit. However, we will see that this angle will have nonplanar components if the secondary’s attitude becomes unstable.

Instead of just looking at the libration angle, we can examine all three Euler angles that make up the secondary’s attitude. We use the 1-2-3 Euler angle set (roll-pitch-yaw) shown in the diagram on Fig. 2, where the Euler angles give Dimorphos’s attitude in the frame rotating with the orbit. At each simulation output the rotating frame is defined as follows: the x-axis points along the LOC, the z-axis is the direction of the mutual orbit pole (i.e., the orbital angular momentum vector), and the y-axis completes the right-handed triad. A direction cosine matrix between the secondary’s body-fixed frame to the rotating frame is constructed, from which the three Euler angles are computed. See Appendix B of Schaub et al. (2009) for the precise mathematical derivation of this Euler angle set.

The Euler angles θ2 (pitch) and θ3 (yaw) can be thought of as two libration angles; θ2 is analogous to an out-of-plane (latitude) libration, and θ3 is analogous to the planar (longitude) libration. Due to the ordering sequence of the Euler angles, this is only technically true when θ1 is exactly zero, which we will see is not the case. However, thinking about θ3 and θ2 as the respective planar and non-planar components of the secondary’s libration can be a useful conceptual tool.

Section snippets

Methods

In Section 2.1, we use an analytic approach to investigate the attitude stability and presence of resonant libration frequencies of Dimorphos, under the assumption that the primary is a uniform sphere and that the system is in an equilibrium state (i.e., a circular orbit).2

Analytic model results

We performed a grid search over the solution space of axis ratios, ranging from 1<a/b<1.5 and 1<b/c<1.5. Due to the lack of a well-constrained shape for Dimorphos, the parameter space was instead selected because of an observed upper-limit of binary asteroid satellites with elongations a/b>1.5 in the near-Earth, Mars-crossing, and small main belt populations (Pravec et al., 2016). For each value of a/b and b/c, the inertia tensor is computed for a uniform triaxial ellipsoid and normalized. The

Implications of a post-impact tumbling state

One concern about a post-impact tumbling state is that the periodic (and chaotic) exchange of angular momentum between the mutual orbit and Dimorphos’s spin state could affect the post-impact ground-based measurements of the orbit period. This could cause a portion of the orbit period change to be misattributed to the DART impact, and complicate the estimate of β based on the orbit period change, which is a Level 1 mission requirement. To get a rough idea of how important this effect might be,

Conclusions

We presented three independent methods – one analytic and two numerical – to study the attitude dynamics of Dimorphos. The analytic model found four fundamental periods of motion relating to the mean motion and the free libration, precession, and nutation frequencies of the secondary. At the resonance locations among these various frequencies, we predicted that unstable motion could be possible. Then, using the “simplified 3D model”, we simulated the post-impact attitude evolution of the

Acknowledgments

H.F.A. would like to thank Douglas Hamilton, Kevin Walsh, and Jean-Baptiste Vincent for useful discussions.

This study was supported in part by the DART mission, NASA Contract #NNN06AA01C to JHU/APL. M.C. acknowledges support from the NASA Solar System Workings program (80NSSC21K0145). I.G., K.T., P.M., and Y.Z. acknowledge funding support from ESA and the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 870377 (project NEO-MAPP). O.K. acknowledges

References (55)

  • FroeschléC. et al.

    The fast Lyapunov indicator: a simple tool to detect weak chaos. application to the structure of the main asteroidal belt

    Planet. Space Sci.

    (1997)
  • GkoliasI. et al.

    Accurate modelling of the low-order secondary resonances in the spin-orbit problem

    Commun. Nonlinear Sci. Numer. Simul.

    (2019)
  • GoldreichP. et al.

    Q in the solar system

    Icarus

    (1966)
  • HarrisA.W. et al.

    On the shapes and spins of “rubble pile” asteroids

    Icarus

    (2009)
  • HirabayashiM. et al.

    Assessing possible mutual orbit period change by shape deformation of didymos after a kinetic impact in the NASA-led double asteroid redirection test

    Adv. Space Res.

    (2019)
  • JacobsonS.A. et al.

    Dynamics of rotationally fissioned asteroids: Source of observed small asteroid systems

    Icarus

    (2011)
  • McMahonJ. et al.

    Detailed prediction for the BYORP effect on binary near-earth asteroid (66391) 1999 KW4 and implications for the binary population

    Icarus

    (2010)
  • MeyerA.J. et al.

    The effect of planetary flybys on singly synchronous binary asteroids

    Icarus

    (2021)
  • MichelP. et al.

    European Component of the AIDA mission to a binary asteroid: Characterization and interpretation of the impact of the DART mission

    Adv. Space Res.

    (2018)
  • NaiduS. et al.

    Radar observations and a physical model of binary near-earth asteroid 65803 didymos, target of the DART mission

    Icarus

    (2020)
  • NimmoF. et al.

    Tidal dissipation in rubble-pile asteroids

    Icarus

    (2019)
  • PravecP. et al.

    Binary asteroid population. 3. Secondary rotations and elongations

    Icarus

    (2016)
  • PravecP. et al.

    Photometric survey of binary near-earth asteroids

    Icarus

    (2006)
  • RubincamD.P.

    Radiative spin-up and spin-down of small asteroids

    Icarus

    (2000)
  • ScheirichP. et al.

    Modeling of lightcurves of binary asteroids

    Icarus

    (2009)
  • ScheirichP. et al.

    The binary near-earth asteroid (175706) 1996 FG3 - an observational constraint on its orbital evolution

    Icarus

    (2015)
  • ScheirichP. et al.

    A satellite orbit drift in binary near-earth asteroids (66391) 1999 KW4 and (88710) 2001 SL9 - indication of the BYORP effect

    Icarus

    (2021)
  • Cited by (47)

    • Energy dissipation in synchronous binary asteroids

      2023, Icarus
      Citation Excerpt :

      Through this strong coupling, the bodies’ spins and mutual orbit will evolve concurrently while energy dissipation occurs. Additionally, spin–orbit coupling can lead to attitude instabilities as a result of orbit perturbations such as the DART impact (Agrusa et al., 2021). There are two main mechanisms of energy dissipation we will consider in this work, both stemming from the deformation of the bodies.

    • Attitude instability of the secondary in the synchronous binary asteroid

      2023, Icarus
      Citation Excerpt :

      Assuming that the sphere–ellipsoid model is restricted to the planar configuration, that is, the spin-axes of the two components are aligned and normal to the orbital plane, there are two well-known libration modes of the 1:1 spin–orbit resonance: forced and free (Agrusa et al., 2021 see also section 2.3).

    View all citing articles on Scopus
    View full text