Enhanced tidal dissipation in rubble pile binary secondaries revealed via numerical studies

1. Binary asteroid modeling in GRAINS

Tidal theory for binary asteroid systems is in a low state of development in comparison to the theory for planetary systems. However, massive N-body simulations present an opportunity to directly simulate and observe relevant physics for binary asteroid systems. The software GRAINS is unique in its N-body formulation, with non-spherical mass elements subject to friction and contact dynamics modeled via the Chrono physics engine [1]. We focus on the spin dynamics of the secondary, modeling a super-synchronously rotating 3000 particle rubble pile secondary orbiting around a point mass primary. The system considered is the Didymos-Dimorphos system, but with Dimorphos spun up to an initial rotation of twice the orbital mean motion, with the axis of rotation perpendicular to the initial orbit plane. This yields a mainly planar problem, simplifying an already complex analysis. This scenario could be loosely representative of a primordial Didymos-Dimorphos before the tidal locking of the secondary, but it also has relevance because of the possibility that the DART impact significantly changed Dimorphos’ initially tidally locked spin state [2]. A depiction of the initial configuration is given in Figure 1 – note that all lengths shown are in non-dimensional units inherent to GRAINS.

Figure 1. Spherical Didymos and rubble pile Dimorphos modeled in GRAINS (non-dimensional units L).

2. Estimating Q/k₂

The fraction Q/k₂ gives the ratio of the quality factor Q to tidal love number k₂, where smaller ratios indicate a more dissipative nature. We opt for a direct measurement of Q/k₂ via rearrangement of the classical MacDonald tidal torque expression [3], and then averaging over full rotations of the super-synchronously rotating Dimorphos:

where L_z gives just the z component of the torque of Didymos on Dimorphos, and this analysis should only be performed for planar super-synchronous rotation. Using this equation and the numerically estimated tidal torques, we obtain estimates for Q/k₂ ranging from 79.4 to 10.7 depending on the averaging window and time in the simulation, and we note that this result is significantly more dissipative than expected [4]. Moreover, we observe significant slow-down of Dimorphos’ spin of ~13% in the ~72 hours simulated. Note that BYORP effects are not included, but the timescale is short, and we are more interested in the spin dynamics than the orbital evolution.

3. Rock movement and tidal motion

Because the location of every rock in the rubble pile is tracked, we can perform complex studies of rock movement. We observe a tendency for a few rocks close to the surface to migrate longitudinally from their original positions with respect to their neighbors. We also observe that in the body frame which diagonalizes the inertia tensor of our rubble pile Dimorphos, there is still some small residual longitudinal oscillation of the topography. Indeed we would only expect the topography to appear static in such a principal axis frame if the body is perfectly rigid. To better isolate relative rock movement patterns, we also define an alternate body frame – a “topographic” frame. This is done via basis vectors computed from the barycenter-relative positions of a few select rocks deep in the interior, wherein we hypothesize that rock movement is mitigated by overburden pressure in comparison to at the surface. At t=0, the first and third basis vectors of this topographic frame are collinear with those of a Hill frame parameterized by the radial direction from Didymos and the direction of the orbital angular momentum vector.

Figure 2. Equatorial cross-section showing rock movement in a body-fixed topographic frame (length units L).

While this study is conducted for rubble piles, which are governed by granular (discretized) physics in lieu of rheological (continuum) physics, we identify global patterns in rock displacement that defy expectations from the unimodal MacDonald model, which predicts a static lag offset between the ground track of the perturber and the raised tide. Figure 2 depicts the relative displacement of rocks at three distinct times in an equatorial (i.e. z-axis) cross-section of Dimorphos. The axis collinear with the peak of a longitudinally traveling tidal “wave” is clearly visible, traveling clockwise, catching up with and overtaking the ground track of Didymos (black arrow), which also travels clockwise in the topographic/body frames. This behavior is somewhat reminiscent of the non-stationary tidal lag predicted from classical Darwin-Kaula theory and other recent related works [5].

4. Conclusions

This work directly explores binary asteroid tidal physics using the GRAINS N-body model. We compute lower-than-expected Q/k₂ for Dimorphos, ostensibly due to the new consideration of high-fidelity contact and friction effects. If accurate, a real-world implication of this could be the revelation of a surprisingly settled rotational state of post-DART impact Dimorphos upon the arrival of the Hera mission. With this model we can also directly observe rock movements and tidal distortion, presenting GRAINS as a useful tool for new developments in rubble pile tidal physics.

5. Disclaimer

Funded by the European Union. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

6. Acknowledgements

This work was performed as part of an MSCA postdoctoral fellowship, grant agreement ID: 101063274.

References

[1] Ferrari, F., Tasora, A., Masarati, P. et al. (2017). N-body gravitational and contact dynamics for asteroid aggregation. Multibody Syst Dyn, 39, 3-20.

[2] Agrusa, H. F., Gkolias, I., Tsiganis, K. et al. (2021). The excited spin state of Dimorphos resulting from the DART impact. Icarus, 370, 114624.

[3] Murray, C. D., Dermott, S. F. (1999). Solar System Dynamics. Cambridge University Press.

[4] Pou, L., Nimmo, F. (2024). Tidal dissipation of binaries in asteroid pairs. Icarus, 411, 115919.

[5] Efroimsky, Michael. (2012). Bodily tides near spin-orbit resonances. Celestial Mechanics and Dynamical Astronomy, 112, 283-330.