Issue 
A&A
Volume 684, April 2024



Article Number  L20  
Number of page(s)  6  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/202449716  
Published online  19 April 2024 
Letter to the Editor
Age of (152830) Dinkinesh I Selam constrained by secular tidalBYORP theory
^{1}
Cornell University, Ithaca, NY 14853, USA
email: ccm242@cornell.edu
^{2}
University of Colorado, Boulder, CO 80303, USA
^{3}
University of Bern, Bern, Switzerland
Received:
23
February
2024
Accepted:
5
April
2024
We constrained the age of the main belt binary asteroid system, (152830) Dinkinesh, through secular dynamics and assuming the secondary, Selam, is at equilibrium. We reproduced Selam’s current semimajor axis and rotation period and Dinkinesh’s current rotation period, starting from the initial conditions of the spinup fission event. The method presented here includes the secular effects of YORP, BYORP, and tidal interactions, while also allowing for the strengths of these effects to be constrained. Based on this method, we performed a Monte Carlo simulation, which indicates that Selam’s age is likely to be ∼1–10 Ma. Derivations for improved equations that describe tidalBYORP equilibria in binary asteroid systems are also presented in this work. In particular, we derived: (1) a tidalBYORP equilibrium equation that scales appropriately with the secondary mass and accounts for differences in the primary and secondary densities; and (2) an equation for YORPdriven primary spin stability, with the secondary being at equilibrium.
Key words: celestial mechanics / minor planets / asteroids: general / planets and satellites: dynamical evolution and stability
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
Binary asteroid systems account for 15% of the nearEarth asteroid population between 200 m and 10 km in diameter (Pravec 2006; Margot et al. 2002). Nearly all asteroids in this size range are expected to be rubble piles (see review in Walsh 2018) and the dominant theory for binary system formation is the rotational fission of a rubble pile (Walsh et al. 2008; Rubincam 2000). This fission occurs when the YarkovskyO’KeefeRadzievskiiPaddack (YORP) effect increases the spin rate of the parent body (or primary) until it reaches a critical spin rate where the centrifugal acceleration on the surface material supersedes the body’s gravitational acceleration (Rubincam 2000; Vokrouhlický & Capek 2002). Material is then lofted from the surface and a satellite (or secondary) is formed by accretion beyond the Roche limit of the primary (Walsh et al. 2008, 2012). After forming, the secondary will synchronize with the primary due to tidal effects (typically on the order of 10^{5} years, Goldreich & Sari 2009). The secondary’s longterm orbital evolution will then be dominated by the binaryYORP (BYORP) effect and tides (Ćuk & Burns 2005). Both the BYORP effect and tides can either expand or contract the secondary’s semimajor axis but tides tend to be expansive, as the spin rate of the primary is almost always greater than the orbit rate of the secondary (McMahon & Scheeres 2010b; Pravec 2006). This follows naturally from the rotational fission theory for binary systems: the primary’s fast spin is typically preserved as the secondary forms; thus, many binary systems feature fastrotating primaries (e.g., Didymos’s period is 2.2600 ± 0.0001 h Scheirich & Pravec 2022). Tides and BYORP change the energy and angular momentum in the system and may cause the secondary to migrate and reach the Hill radius or impact the primary (Ćuk 2007; McMahon & Scheeres 2010a).
On November 1, 2023, NASA’s Lucy spacecraft flew by the main belt asteroid, (152830) Dinkinesh, which it observed to be a binary system (Levison et al. 2024). The only other small binary asteroid system visited by a spacecraft is the nearEarth system, (65803) Didymos (Daly et al. 2023; Chabot et al. 2024). The Dinkinesh and Didymos systems share remarkable similarities, which help to highlight their differences and makes their comparison particularly enriching. Although the primary body (hereafter, Dinkinesh) has a similar size and spectral type to Didymos, it differs in terms of its morphological features, as Dinkinesh has large troughs and steep walls that appear to be absent on Didymos (Levison et al. 2024; Barnouin et al. 2024). The secondary of the Dinkinesh system, Selam, differs from Dimorphos in many aspects, including its size, shape, and mass (Chabot et al. 2024). In Table 1, we summarize the key comparisons between the Didymos and Dinkinesh systems. Selam is a definitively unique secondary and is the first contact binary observed as the secondary in a binary system (Levison et al. 2024). Contact binaries have been observed in the nearEarth population, the main belt, the Jupiterfamily comets, and the Kuiper belt. They likely make up between 14% and 30% of small bodies larger than 200 meters (Virkki et al. 2022). Some of these bodies may have been formed via reaccumulation after disruption (presented as a mechanism to form Itokawa in Michel & Richardson 2013) while others may be primordial bodies (e.g., Arrokoth, McKinnon et al. 2020; Marohnic et al. 2021) or formed via lowspeed collisions (Jutzi & Asphaug 2015). The creation of contact binaries in binary systems has been discussed in the context of secondary fission as a possible mechanism (Jacobson & Scheeres 2011a). Because of the diversity of these objects and lack of formation observations, they have also been the subject of recent mission designs (e.g., Merrill et al. 2024). Despite the extensive literature that focuses on contact binaries, the bilobate shape of Selam remains a puzzle. In this paper, we aim to contribute to the solution of this puzzle by constraining the age of the system through dynamical simulations. The results from this study can be compared to ages derived by crater sizefrequency analyses and, when combined with these analyses, could provide insight into the dynamical history of the system (e.g., Barnouin et al. 2024).
Characteristics of the Didymos and Dinkinesh binary systems with 1σ uncertanties.
2. Methods
2.1. YORP theory
The YORP effect is a torque on asteroid bodies driven by solar radiation pressure. An extensive derivation of YORP torques is given in Scheeres (2007), but we only consider the leadingorder form of this effect, outlined in Rossi et al. (2009). The rate of change of the spin rate of the primary from YORP is
where R_{p} is the radius of a volumeequivalent sphere for the primary, ρ_{p} is the density of the primary, C_{Y} is the normalized YORP coefficient,
F_{⊙} = 10^{17} N, a_{⊙} is the heliocentric semimajor axis, and e_{⊙} is the heliocentric eccentricity. Rossi et al. (2009) found that applying a Gaussian distribution of C_{Y} = 0 ± 0.0125 (1σ) successfully reproduced the spin rates of bodies in the nearEarth population.
2.2. BYORP theory
We refer to McMahon & Scheeres (2010a,b) and Cueva et al. (2024) for extensive BYORP theory development and applications, as we only offer a summary of the effect here. We use a leadingorder BYORP theory based on the yearaveraged equations from McMahon & Scheeres (2010a). The rate of change to the semimajor axis due to the BYORP effect is
where μ = G(M_{p} + M_{s}) is the standard gravitational parameter of the system, G is the gravitational constant, M_{p} is the mass of the primary, M_{s} is the mass of the secondary, a is the semimajor axis of the secondary, e is the eccentricity of the secondary, R_{s} is the radius of a volumeequivalent sphere for the secondary, ρ_{s} is the density of the secondary, and B is the normalized BYORP coefficient given by
Here, A_{0}(2) is the alongtrack component of the zerothorder Fourier coefficient in the secondary’s bodyfixed frame presented in McMahon & Scheeres (2010a,b) and Scheeres (2007). In particular, B is a difficult parameter to predict for asteroids and has a strong dependence on the body’s shape and surface topography. A nearlysynchronous secondary is necessary for the BYORP effect (Ćuk & Burns 2005). Nonsynchronous motion will effectively “turn off” this effect, as there will no longer be a consistent portion of the body interacting with the solar radiation pressure and the torques exerted on the body will net 0 (Quillen et al. 2022). It appears that Selam is tidally locked and so, we can assume that BYORP is active (Levison et al. 2024).
2.3. Tidal theory
To account for tidal effects, we use the classical tidal theory presented in Chapt. 4.9 of Murray & Dermott (1999). The magnitude of the tidal torque of the secondary on the primary is given by
where k is the primary’s tidal Love number and Q is the primary’s tidal quality factor. The tidal torque can be related to the rate of change in mechanical energy with
where n is the mean motion of the secondary. This can be related to the rate of change in the semimajor axis due to tides with
Rearranging the equation for ȧ_{T} and substituting in Eq. (6) results in
which is the change in semimajor axis due to tidal torque. The tidal torque’s effect on the primary’s spin is given by
from Murray & Dermott (1999), where we assume α = 2/5. In this work, we adopted the convention derived in Nimmo & Matsuyama (2019), where tidal dissipation is primarily driven by the surface layer of regolith on the asteroid. Pou & Nimmo (2024) combined this work with that of Goldreich & Sari (2009), which solves for k/Q via Eqs. (16) and (12) in Pou & Nimmo (2024) to obtain
where t is the regolith thickness and the leading constant has units of m^{5} kg^{−2} s^{−2}. The constant is computed with a yield strain of ϵ_{Y} = 10^{−2} and rigidity μ^{*} = 10^{10} Pa for individual boulders, which are the values used in Nimmo & Matsuyama (2019) and Pou & Nimmo (2024). These values were selected because they are typical for rubble pile bodies and this allows us to maintain consistency between this study and Nimmo & Matsuyama (2019) and Pou & Nimmo (2024). This approach is adopted solely to constrain k/Q bounds and discussed to explain where uncertainty in k/Q can arise; we highlight that it is not meant to constrain the surface properties of Dinkinesh.
2.4. TidalBYORP equilibrium
There is an equilibrium semimajor axis where expansive tides and contractive BYORP (i.e., B < 0) are equal and opposite (i.e., ȧ_{B} + ȧ_{T} = 0, Jacobson & Scheeres 2011b). In typical binary systems, where the primary’s spin rate is greater than the secondary’s mean motion (i.e., ω_{p} > n), this equilibrium semimajor axis, a_{eq}, is stable. For such systems, when a < a_{eq}, tidal effects are stronger than BYORP and increase the semimajor axis back to a_{eq}. When a > a_{eq}, BYORP effects are stronger than tidal effects and shrink the semimajor axis to a_{eq} and so the equilibrium is stable. This ratio of B, Q, and k can then be found as
where a_{eq} is the equilibrium semimajor axis that the secondary will asymptotically converge to for B < 0 and ω_{p} > n. We solve for a_{eq} to obtain
These equations are qualitatively similar to those in Jacobson & Scheeres (2011b) but are more accurate for more massive secondaries, as they scale with the mass of the secondary and account for differences in the primary and secondary densities. It appears that Eq. (3) in Jacobson & Scheeres (2011b) mistakenly uses the reduced mass of the system rather than the secondary mass, which leads to an error in computation. Assuming the secondary and primary densities are equal and the secondary is of negligible mass, Eq. (11) will be equal to Eq. (11) from Jacobson & Scheeres (2011b). However, we recommend the use of this work’s equations in future studies to ensure accuracy.
2.5. Primary spin stability
The tidalBYORP equilibrium we discuss here is a stable equilibrium, with the assumption that the primary’s spin does not evolve substantially. If the primary’s spin is reduced such that ω_{p} = n, a doublysynchronous system is achieved which may destabilize small secondaries (Jacobson et al. 2014). If the primary’s spin is increased, it may further undergo fission, which could also destabilize the system. Thus, the tidalBYORP equilibrium, alone, is not necessarily a final state for binary systems due to the primary’s changing spin state. However, if the spin of the primary is kept constant, then the system can theoretically exist in the stable equilibrium for perpetuity. This stable equilibrium requires that the torque on the primary from tides is equivalent to the torque provided by the YORP effect (i.e., , Golubov et al. 2018). We set Eq. (1) equal to Eq. (9) to solve for the semimajor axis:
where the torque on the primary from YORP and tides are equal and opposite. When a > a_{Y}, the torque from YORP will dominate the spin evolution of the primary and when a < a_{Y}, the torque from tides will dominate the spin evolution of the primary. There is a special case where a_{Y} = a_{eq} and this is when:
Here, the primary’s spin will not evolve when the secondary’s semimajor axis is a_{eq}. We note that C_{Y} is actually a function of the primary and Eq. (14) should only be used to estimate C_{Y} if a system is found to be particularly longlived and stable. The stability for this condition is as follows: if BYORP is contractive and tides are expansive (i.e., ω_{p} > n and B < 0), then a secondary at a_{eq} will be stable (Jacobson & Scheeres 2011b). If C_{Y} is equivalent to Eq. (14) and tidal torques are negative (i.e., C_{Y} > 0 and ω_{p} > n), then the primary spin will also be stabilized (Golubov et al. 2018). If any of these three conditions are not fulfilled (ω_{p} > n, B < 0, and C_{Y} > 0), the secondary’s orbit and primary’s spin will not be stabilized simultaneously. It is possible that the system will leave this state when either body experiences a significant impact or the system has a close encounter with a planet. However, there appears to be no effect internal to the system that could destabilize it.
2.6. Monte Carlo analysis
We produce a Monte Carlo analysis to obtain statistical data on the age of Selam. Here, the “age” is the time starting from the formation of Selam to the current system’s state. Table 2 lists the variables used in the tests and describes how they are varied. It is assumed in these tests that the current semimajor axis of Selam is equivalent to a_{eq}, which is a reasonable assumption given that Selam appears to be in synchronous rotation and has no measurable eccentricity (Levison et al. 2024). The coefficient of YORP, C_{Y}, has a welldefined normal distribution (Rossi et al. 2009) but we only accept values of C_{Y} that are less than the value in Eq. (14). Because the spin of the primary is initially faster than the current spin rate, we require the primary to slow down (i.e., C_{Y} must not be greater than the value given by Eq. (14) or else the primary will tend to speed up in time and never reach its current spin rate). To constrain the k/Q values, we use Eq. (10), with a regolith thickness of 10 m < t < 100 m, which results in 10^{−5} < k/Q < 10^{−3} ; these are typical k/Q values for rubble piles (Nimmo & Matsuyama 2019; Pou & Nimmo 2024). With the assumption that the system is at a_{eq}, this results in 10^{−4} < B< 10^{−2}, which is also a typical range for B magnitudes (Jacobson & Scheeres 2011b). We only considered contractive BYORP, as this is a necessary condition for our a_{eq} assumption to hold.
Variables and distributions used for the Monte Carlo analysis.
We uniformly distributed Selam’s initial semimajor axis between the Roche limits for bodies with strength and zero cohesion (1.5R_{p} < a_{i} < 2.5R_{p}, Holsapple & Michel 2006, 2008). Dinkinesh’s initial spin rate derived via angular momentum conservation is
where is the disruption spin rate of a sphere and M^{*} = M_{p}M_{s}/(M_{p} + M_{s}) is the reduced mass of the system (Pravec et al. 2010). We note that we effectively treat Selam as a single body that formed immediately after the disruption of Dinkinesh. Despite the morphological complexity of Selam, this is a reasonable assumption given that Agrusa et al. (2024) found that ∼8% of fission events create more than a single secondary within the first 100 days and an early impact is much more likely than a persisting multibody system Ferrari et al. (2024). We assume the synchronization timescale is small compared to the timescale for orbit evolution. This is a fitting assumption given that the synchronization timescale should be no greater than ∼10^{5} years (Goldreich & Sari 2009; Levison et al. 2024); however, could be as low as ∼1 year, as some secondaries form with nearly synchronous spin (Agrusa et al. 2024). The maximum age we allowed our simulations to produce is 1 Ga (i.e., the collisional lifetime of Dinkineshsized bodies, Richardson et al. 2022; Bottke et al. 2005; Raducan et al. 2024). The simulations were concluded when two conditions are simultaneously satisfied: (1) the period of the primary reaches 3.7387 hours and (2) the semimajor axis of the secondary is within 0.1% of a_{eq} (and, thus, the secondary has reached its equilibrium position). If either of these two criteria are not satisfied, the run is not counted, as it has not accurately modeled the Dinkinesh system’s current state. We ran the simulation until we produced 10^{6} cases that satisfy the two previously mentioned conditions. To offer some model validation, our model outputs are ρ = 2420 ± 260 kg m^{−3} and μ = 33.1 ± 1.6 m^{3}/s^{2}, which are comparable to the current estimates for Dinkinesh (ρ = 2400 ± 350 kg m^{−3}; μ = 33.0 ± 1.6 m^{3}/s^{2}; Levison et al. 2024).
3. Results and discussion
3.1. Selam’s age
Although Selam’s age spans four orders of magnitude in our Monte Carlo simulations, we find that ∼2/3 of runs conclude between 1–10 Ma. Figure 1 plots the resulting age and first, second, and third quartiles of 1.38, 2.97, and 7.18 Ma, respectively. The mode of the data (for 100 bins per order of magnitude) falls at 2.00–2.04 Ma. Ages of > 10^{8} years (∼1% of cases) indicate that C_{Y} is close to the equilibrium C_{Y} from Eq. (14) and are slowevolving systems. We compare the age of Selam to that of Dimorphos (< 0.3 Ma; Barnouin et al. 2024), reasoning that Selam is likely to be an older secondary than Dimorphos would be. The short lifetimes of these objects indicate that binary systems are continuously created.
Fig. 1. Age of Selam from the Monte Carlo simulation (1 million cases). The vertical dashed gray lines are placed at the first, second, and third quartiles of 1.38, 2.97, and 7.18 Ma, respectively. The bins are logarithmically spaced such that there are 50 bins for each order of magnitude. 
Selam’s rate of orbit expansion is sensitive to the magnitudes of k/Q and B. For a given k/Q, the time it takes Selam to reach its current semimajor axis is the minimum age in Fig. 2. This minimum age can be reduced if Selam formed further from the Roche limit or if Selam is not at a_{eq}. As previously mentioned, the age of Selam should be independently derived via crater analysis. Given that age, constraints may be placed on k/Q and C_{Y} (e.g., if Selam’s age is the mode of our data, 2 Ma, then C_{Y} > −1.0e−2 and k/Q > 6.0e−5 from Fig. 2). We note that Selam being at equilibrium does not exclude periods of nonsynchronous rotation (e.g., tumbling or barrel instability), but those periods should be short compared to Selam’s age (Ćuk et al. 2021; Meyer et al. 2023b). In shorttimescale simulations, the irregular shapes of Dinkinesh and Selam are important to model because of their spinorbit coupling (e.g., Meyer et al. 2023a), but these shortperiod processes should not alter our results.
Fig. 2. Effects of YORP and tides on Selam’s age (1 million cases). 
3.2. BQ/k for Dinkinesh and Didymos
Assuming Selam’s current semimajor axis is a_{eq}, Dinkinesh’s BQ/k magnitude is 10.1 ± 3.8, which is more than an order of magnitude lower than BQ/k for preDART Didymos (358 ± 108 found using the data in Richardson et al. 2024). A potential explanation for Dinkinesh’s lower BQ/k magnitude is Selam’s bilobate shape, which could cause significant selfshadowing and weaken the BYORP effect. If selfshadowing is indeed the primary reason for the lower BQ/k value, this will have implications for other bilobate bodies and contact binaries in general. Because of the constraint placed on k/Q and the derived BQ/k relation, we may also constrain B (e.g., k/Q > 6e−5 indicates B> 6.1e−4 ± 2.3e−4).
4. Conclusions
We derived an improved version of the tidalBYORP equilibrium equation that scales appropriately with the mass of the secondary and accounts for the difference in density between the primary and secondary. We defined the necessary conditions for a coefficient of YORP that will stabilize the spin of the primary when the secondary is at a_{eq}. A system in this state has a stronger stability condition than one strictly characterized by BYORP and tides. Ultimately, we find that only external factors could destabilize such a system.
We reproduced the orbital state of Dinkinesh and Selam starting from spinfission conditions using tidal, BYORP, and YORP effects. We find that the Dinkinesh system has a much lower BQ/k magnitude than the Didymos system, which may be the result of selfshadowing but this requires further investigation. Our Monte Carlo simulation indicates that Selam has an age of 1–10 Ma, with a median age of 3 Ma and it is likely to be older than Dimorphos. If Selam’s absolute age is measured by crater analysis with a higher level of accuracy than that achieved this work, further constraints may be placed on C_{Y}, k/Q, B, and the system’s dynamical history.
Acknowledgments
We thank both Rachel Cueva and our reviewer, Seth Jacobson, whom improved the quality of this paper. We also thank Phil Nicholson, Dmitry Savransky, Charlie Detelich, Veronica Hegelein, and the H.M. folks for their discussions and support.
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All Tables
Characteristics of the Didymos and Dinkinesh binary systems with 1σ uncertanties.
All Figures
Fig. 1. Age of Selam from the Monte Carlo simulation (1 million cases). The vertical dashed gray lines are placed at the first, second, and third quartiles of 1.38, 2.97, and 7.18 Ma, respectively. The bins are logarithmically spaced such that there are 50 bins for each order of magnitude. 

In the text 
Fig. 2. Effects of YORP and tides on Selam’s age (1 million cases). 

In the text 
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