Main

The discovery of the bilobate shape of MU69 and its peculiar configuration provided new clues and opened avenues of exploration into the physical processes that sculpt the Solar System. Here we describe an evolutionary channel for the formation of MU69 from an initially wide binary. We consider the initial binary to be a member of a hierarchical triple together with the Sun. Owing to secular evolution induced by the Sun, the inner orbit may experience changes in its eccentricity (e) and mutual inclination (i) on secular timescales much longer than the orbital period, known as Lidov–Kozai (LK) oscillations, which can be modelled using a secular orbit-averaging approach10,11. Large LK oscillations take place when the mutual inclination is large (40° ≱ i ≱ 140°). The highest eccentricities are attained as the binary evolves to the lowest inclinations and vice versa12.

If the eccentricity of the binary exceeds a threshold ecoll, the small pericentre allows binary collisions. Thus, LK evolution could lead to coalescence of individual Kuiper belt binary (KBB) members into a single, probably irregularly shaped, KBO5. However, because the closest approach occurs concurrently with the lowest inclinations, collisions mostly occur near i ≈ 40° and i ≈ 140° (ref. 13). Moreover, tidal effects and the non-spherical structure of KBB components quench LK evolution, which makes collision possible only in a small part of the parameter space5,14. The standard LK mechanism is therefore disfavoured for the origin of the highly oblique MU69, but can explain the origin of highly eccentric KBBs such as WW31 and 2001 QW3225,15,16.

For larger ratios of the inner period to outer period, secular averaging breaks down and the evolution becomes semi-secular. The orbit of the inner binary now evolves considerably on timescales of the outer orbit, and short-term fluctuations arise, making the LK evolution more complex6,7,17. The maximal eccentricity can be calculated analytically, including in domains where it is unconstrained7 and the evolution is non-secular. Figure 1a shows the analytical two-dimensional parameter space for allowed and forbidden domains for collisions in terms of the initial inclination cosi0. The initial separation of the inner binary is normalized to the Hill radius, rH = aout(min/3M☉)1/3, where aout is the outer semi-major axis, M☉ is the mass of the Sun and min is the mass of the inner binary. For an inner semi-major axis a, the (dimensionless) separation α ≡ a/rH cannot exceed the Hill stability limit for highly inclined orbits8, αH = 0.4. We use the outer orbit parameters of MU69: aout = 44.581 au and eccentricity eout = 0.041. We model the lobes as triaxial ellipsoids of dimensions approximately 22 × 20 × 7 and 14 × 14 × 10 km3 (ref. 1), leading to a total radius Rtot = 18 km and inner mass min = (1.61 + 1.03) × 1018 g = 2.64 × 1018 g for a density of ρ = 1 g cm−3 (see Methods for other densities). Secular collisions occur only for sufficiently large critical inclination and beyond a certain initial separation αcoll, which overcomes LK quenching (see equation (12) and Methods). Non-secular collisions will dominate over secular collisions beyond a transitional separation αt:

$${\alpha }_{{\rm{t}}}={3}^{1/3}\,{[\frac{128}{135}\frac{(1-{e}_{{\rm{out}}}^{2})}{{\left(1+\frac{2\sqrt{2}}{3}{e}_{{\rm{out}}}\right)}^{2}}{\left(\frac{{M}_{\odot }}{{m}_{{\rm{in}}}}\right)}^{1/3}\frac{{R}_{{\rm{tot}}}}{{a}_{{\rm{out}}}}]}^{1/4}$$
(1)
Fig. 1: Roadmap to collisions of MU69.
figure 1

a, The orbital evolution of MU69 in initial separation–initial inclination space. The initial eccentricity, e, is 0 (blue), 0.4 (green) and 0.8 (red). Solid lines show the condition for non-secular collisions (equation (14)) with unbound eccentricity. Dashed lines show the conditions for a secular collision (equation (4)) with deterministic eccentricity. The different domains are as follows: white, LK oscillations are completely quenched and the eccentricity is constant, α < αL (equation (9)); green, the eccentricity is excited but below ecoll, and collisions are avoided, αL < α < αcoll (equation (12)); grey, secular evolution can lead to a collision, αcoll < α < αt (equation (1)); blue, non-secular perturbations dominate and lead to a collision; red, the initial inclination is too low to induce a collision. b–d, Time evolution of the instantaneous distance, inclination and eccentricity of an individual orbit with initial Keplerian elements: semi-major axis a = 0.3rH, eccentricity e = 0.1, inclination i = 86°, argument of periapse ω = 0, argument of ascending node Ω = π/4 and mean anomaly \( {\mathcal M} =0\). The outer binary is set at ωout = Ωout = 0 and \({{\mathcal{M}}}_{{\rm{o}}{\rm{u}}{\rm{t}}}=-{\rm{\pi }}/4\).

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In our case, αt ≈ 0.174. Figure 1b demonstrates the separation in the non-secular regime before the collision. During the high-eccentricity phase, there are about 10 cycles where the instantaneous separation drops below 103 km. A collision occurs during the third LK cycle after about 4,600 yr. The mutual inclination flips its orientation during the high-eccentricity peak of the LK cycle (Fig. 1c). The eccentricity is essentially unbound and a collision eventually occurs (Fig. 1d).

To explore in detail the overall evolution and statistics of KBBs in the chaotic non-secular regime, we defer to detailed N-body simulations, which provide us with the probability for collisions and the post-collision characteristics. We use the publicly available code REBOUND18 with the IAS1519 integrator (see Methods for details and stopping conditions). We integrate four sets of initial conditions in the non-secular regime. The first three sets have initial separations of α = 0.2, 0.3 and 0.4, and the fourth set has uniformly sampled separations in α = [0.2, 0.4]. The orbital angles are sampled uniformly. The mutual inclination of observed binaries is cosine-uniform9, and thus we follow cosine-uniform sampling with a cut-off at |cosi| ≤ 0.4 (lower inclinations cannot lead to a collision). For each case, we run 250 simulations (except for α = 0.2, for which we run 200 simulations and use |cosi| ≤ 0.3), each up to 5 × 104 yr.

Figure 2 shows the cumulative distribution function of various parameters of the colliding orbits. Both the closest-approach distance q = a(1 − e)/Rtot (Fig. 2a) and the final inclinations (Fig. 2c) at collision are consistent with a uniform distribution (in cosi) between 40° and 140°, suggesting that the orbits are indeed chaotic and in the non-secular regime, as expected. Most orbits induce collisions after about a few thousand years (Fig. 2b). The mean collision time increases with increasing separation. The velocity at impact is comparable to the escape velocity with a very small dispersion, consistent with a gentle collision20 (Fig. 2d).

Fig. 2: Cumulative distributions of the impact characteristics.
figure 2

The cumulative distribution functions (CDFs) are for α = 0.4 (solid blue), 0.3 (dashed red), 0.2 (dotted green) and uniform in (0.2, 0.4) (dash-dotted black). a, Pericentre q/Rtot. b, Time of collision. c, Final inclination at impact. The shaded grey line is a uniform cumulative distribution in cosi in the range 30°−150°. d, Velocity at impact. The vertical black dashed line is the escape velocity, vesc.

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We find the overall merger fractions of wide binaries to be around 12%−18% (see Extended Data Table 1), which is roughly consistent with the observed 10%−25% occurrence of contact binaries for the cold classical belt21. Most mergers occur for initially high inclinations, as expected. About 1%−3% of all wide binaries produce highly oblique contact binaries (i = 80°−100°), consistent with the observed high obliquity of MU69 and providing predictions that can be verified by future KBO observations. There is little dependence on the underlying distribution of α, and merger rates are bounded between minimal and maximal values of 12% (for α = 0.2) and 18% (for α = 0.4). Moreover, in a collisional environment22 the binary orbits can be perturbed such that originally low-inclination orbits become highly inclined and become subject to semi-secular evolution, forming contact binaries; the quoted formation rates are thus lower limits to the total fraction of contact binaries formed through this process.

The non-merging systems continue to evolve quasi-periodically. On longer timescales, three-body encounters are expected to shape the populations of KBBs3,23. Exchange interactions can drive the binaries into equal masses24, and the loose nature of the binaries can result in evaporation (Heggie’s law)25. There are only a handful of KBBs beyond a ≳ 0.05rH with either prograde or retrograde orbits that are not highly inclined (see figure 1 of ref. 26), whereas the widest known binary, 2001 QW322, with a ≈ 0.2rH, is expected to disrupt within a billion years16.

To test the feasibility of the semi-secular collision origin of MU69, we also need to account for the observed spin period of MU69. Angular momentum conservation enables us to find the resulting spin period depending on the impact angle and the primordial spins of each component. The final impact parameter at collision (which corresponds to an impact angle; see Methods) is uniformly distributed, and thus our model can robustly produce a wide range of possible final rotation periods, without any fine-tuned modelling of the composition and density of MU69, thus also alleviating the angular momentum problem of other models1.

Figure 3a shows the outcome of a collision at a 40° impact angle with high-material-strength composition, which reproduces the shape of MU69. Low- or medium-strength materials result in a deformed shape and are thus ruled out. If the density of MU69 is halved compared to the fiducial 1 g cm−3 value (as suggested by ref. 1), the escape velocity vesc—at which typical collisions occur—is lower, and thus using medium-strength-material parameters also produces an undeformed shape. Random collisions—even at relative velocities as low as 10vesc—destroy or heavily deform binaries with high-strength-material composition; they are likewise ruled out (see Methods). Figure 3b shows the expected spin-period dependence on the impact angle. An impact angle of about 40° reproduces the observed spin period (see Methods) for initially non-spinning objects. Taking a typical initial spin period of about 10 h with random orientations extends the range of plausible impact angles to about 20° and 70°, for the maximally aligned and anti-aligned configurations, respectively. Smoothed particle hydrodynamics (SPH) collision simulations agree with our simplified estimate and support our assumptions of undeformed, rigid bodies when modelled with high-strength material parameters, or else medium-high strength parameters if the density and impact velocity are slightly lower (see ref. 27 and Methods for details).

Fig. 3: Shape and spin period of MU69.
figure 3

a, Final collision outcome at an impact angle of 40° with high-material-strength composition (see Methods). b, Spin period as a function of the impact angle for MU69. The horizontal line is the observed period, 15.92 h. The solid blue line indicates initially non-rotating progenitors. The dashed red line indicates two initially aligned rotating progenitors with a period of 10 h, and the dotted green line indicates an anti-aligned configuration. The black crosses are the results obtained from SPH simulations (see Methods).

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Together, our dynamical and post-collisional modelling yields a coherent picture for the origin of MU69 from an ultrawide KBO binary. Such wide KBB progenitors could be a natural byproduct of KBO and KBB evolution in the early Solar System3,28,29. It is most probable that the characteristics of MU69 are not unique, and that secular or semi-secular evolution plays a major role in the evolution of many KBBs and in the production of low-velocity collisions between individual KBB components. In fact, modelling of the Pluto–Charon system also suggests a low-velocity impact origin30. Moreover, given the high obliquity of the Pluto–Charon system, it is possible that it also originated from an initially wide binary and followed a secular or semi-secular evolution, similar to MU69. Similar evolutionary scenarios might also apply to the evolution of other contact binaries such as (139775) 2001 QG298 (ref. 2), as well as moons and exo-moons, as all of these form hierarchical triple systems with their host star.

Methods

Lidov–Kozai secular evolution

Let us consider the evolution of a binary KBO due to LK secular evolution. Let the inner binary start with an initial separation of a0 = αrH, eccentricity e0 and mutual inclination i0. The Hill radius is rH = aout(min/3M☉)1/3. The most stable orbit is around αH ≈ 0.4 (ref. 8). The minimal eccentricity required for collision is

$${e}_{{\rm{coll}}}=1-\frac{{R}_{{\rm{tot}}}}{\alpha {r}_{{\rm{H}}}}$$
(2)

Using the standard LK formula, the maximal eccentricity is

$${e}_{max}^{2}=1-\frac{5}{3}(1-{e}_{0}^{2}){\cos }^{2}{i}_{0}$$
(3)

For a collision occur, we require that emax ≥ ecoll, which yields the critical inclination \({i}_{0}^{{\rm{c}}}\)

$$\cos \,{i}_{0}^{{\rm{c}}}\approx \sqrt{\frac{6}{5(1-{e}_{0}^{2})}\frac{{R}_{{\rm{t}}{\rm{o}}{\rm{t}}}}{\alpha {r}_{{\rm{H}}}}}=\frac{0.037}{\sqrt{1-{e}_{0}^{2}}}{\left(\frac{\alpha }{0.3}\right)}^{-1/2}$$
(4)

such that collisions occur for \(|\cos \,{i}_{0}|\le |\cos \,{i}_{0}^{{\rm{c}}}|\). Here, Rtot/rH ≪ 1 has been expanded to linear order. The probability for collision can be expressed in terms of integrating over the distribution function fa,θ(a, θ) and where θ ≡ cosi. For uniform independent distribution in cosi, as inferred from KBO observations9, the probability is

$$P({e}_{0})={\int }_{{a}_{min}}^{{a}_{max}}{f}_{a}(a){\theta }_{{\rm{c}}}(a){\rm{d}}a$$
(5)

where θc(a) is given by equation (4).

Inclination angle at impact

From the conservation of \(\sqrt{1-{e}^{2}}\cos i\,=\) \({\rm{c}}{\rm{o}}{\rm{n}}{\rm{s}}{\rm{t}}\equiv {j}_{z}\) the inclination at impact is

$$\cos {i}_{{\rm{c}}{\rm{o}}{\rm{l}}{\rm{l}}}=\frac{\sqrt{1-{e}_{0}^{2}}\,\cos {i}_{0}}{\sqrt{1-{e}_{{\rm{c}}{\rm{o}}{\rm{l}}{\rm{l}}}^{2}}}=\sqrt{1-{e}_{0}^{2}}\,\cos {i}_{0}\sqrt{\frac{\alpha {r}_{{\rm{H}}}}{2{R}_{{\rm{t}}{\rm{o}}{\rm{t}}}}}$$
(6)

To find the impact angle icoll, we invert equation (6):

$${i}_{0}=\,\arccos \left(\cos {i}_{{\rm{coll}}}\sqrt{\frac{2{R}_{{\rm{tot}}}}{\alpha {r}_{{\rm{H}}}}}\right)$$
(7)

To find the observed obliquity icoll = 98°, the critical inclination is at least |cosi0| ≤ 0.00585, (89.33° ≤ i0 ≤ 90.66°) for α = 0.1, which is unlikely. Moreover, for α ≱ 0.1, collisions are unlikely regardless of the initial inclination, owing to the effects of oblateness, as shown below.

Effects of oblateness

Small KBOs might not have spherical shapes, in which case their gravitational potential is not spherical. Such a configuration induces extra precession on the orbit which can considerably affect the secular evolution. The leading term is encapsulated in a dimensionless parameter J2, which is related to ratio of the axes, or the polar and equatorial radii of the bodies31. Planets are mostly spherical, and their deviation is small—J2 ≈ 10−3 for Earth and around J2 = 0.014 for Jupiter—and is related to the flattening of the planets induced by their rotations. In the case of the components of MU69, the objects are highly non-spherical and J2 could be large. Using the principal moments of inertia of an oblate spheroid, we have J2 = [1 − (c/a)2]/5, which is around J2 ≈ 0.18 for the primary component and J2 ≈ 0.1 for the secondary component.

The additional precession may quench the LK oscillations if it is too strong. To quantify the effects of the additional precession we can define a dimensionless quantity, εrot, that measures the ratio between the LK-induced and the oblateness-induced precessions32,33

$${\varepsilon }_{{\rm{r}}{\rm{o}}{\rm{t}}}=\frac{3}{2}\,{J}_{2}\frac{{m}_{{\rm{i}}{\rm{n}}}}{{m}_{{\rm{o}}{\rm{u}}{\rm{t}}}}\frac{{a}_{{\rm{o}}{\rm{u}}{\rm{t}}}^{3}{(1-{e}_{{\rm{o}}{\rm{u}}{\rm{t}}}^{2})}^{3/2}{R}_{1}^{2}}{{(\alpha {r}_{{\rm{H}}})}^{5}}$$
(8)

Setting εrot = 3/2 leads to the definition of the Laplace radius33,34 in terms of the Hill radius, rL ≡ αLrH where

$${\alpha }_{{\rm{L}}}={\left({J}_{2}\frac{{m}_{{\rm{i}}{\rm{n}}}}{{m}_{{\rm{o}}{\rm{u}}{\rm{t}}}}\right)}^{1/5}\frac{{({a}_{{\rm{o}}{\rm{u}}{\rm{t}}}^{3}{R}_{1}^{2})}^{1/5}}{{r}_{{\rm{H}}}}{(1-{e}_{{\rm{o}}{\rm{u}}{\rm{t}}}^{2})}^{3/10}=0.03{\left(\frac{{J}_{2}}{0.2}\right)}^{1/5}{\left(\frac{{R}_{1}}{11{\rm{k}}{\rm{m}}}\right)}^{2/5}$$
(9)

leading to32 εrot = 1.5(α/αL)−5. It has been previously shown that the maximal eccentricity attained is given by the implicit expression for cosi0 = 0 (their equation 50):

$$\frac{{\varepsilon }_{{\rm{r}}{\rm{o}}{\rm{t}}}}{3}\,\left[\frac{1}{{(1-{e}_{max}^{2})}^{3/2}}-1\right]=\frac{9}{8}{e}_{max}^{2}$$
(10)

Expanding in εrot ≪ 1 and \({e}_{{\rm{\max }}}^{2}\approx 1\)

$${e}_{{\rm{\max }}}\approx 1-\frac{2}{9}{{\varepsilon }}_{{\rm{rot}}}^{2/3}$$
(11)

A collision can occur only if ecoll < emax or if

$${\alpha }_{{\rm{coll}}} > {\left(\frac{3}{2}\right)}^{2/7}{\alpha }_{{\rm{L}}}^{10/7}\,{\left(\frac{2}{9}\frac{{r}_{{\rm{H}}}}{{R}_{{\rm{tot}}}}\right)}^{3/7}\approx 0.12$$
(12)

Note that a similar analysis can be performed for tidal distortions or relativistic corrections32. In this case, they are much weaker than the rotational effects.

Non-secular Lidov–Kozai evolution

In the previous section we considered the evolution due to secular LK evolution. In the semi-secular (semi-LK) regime6,7, short-term fluctuations can substantially change the evolution. In the following we discuss the overall effects of such short-term perturbations. The strength of the perturbations is encapsulated in the single averaging parameter7,17:

$${\varepsilon }_{{\rm{S}}{\rm{A}}}\equiv {\left(\frac{{a}_{1}}{{b}_{{\rm{o}}{\rm{u}}{\rm{t}}}}\right)}^{3/2}\sqrt{\frac{{m}_{{\rm{o}}{\rm{u}}{\rm{t}}}}{{m}_{{\rm{i}}{\rm{n}}}}}=\frac{{\alpha }^{3/2}}{\sqrt{3}{(1-{e}_{{\rm{o}}{\rm{u}}{\rm{t}}}^{2})}^{3/2}}\approx 0.1{\left(\frac{\alpha }{0.3}\right)}^{3/2}$$
(13)

One important quantity is the (averaged) z angular momentum \({\bar{j}}_{z}=\sqrt{1-{e}_{0}^{2}}\,\cos {i}_{0}\), assuming that i0 and e0 have their mean value.

In this case, jz is no longer conserved, but its value averaged over the outer orbit, \({\bar{j}}_{z}\), is conserved. The eccentricity of the orbit becomes unbound once the fluctuation in \({\bar{j}}_{z}\) (Δjz) is larger than its initial value, namely \(\Delta {j}_{z} > {\bar{j}}_{z}\). The fluctuation has been estimated analytically7,17, and can be used to show that the eccentricity is unbound if

$$\cos {i}_{0}\,\sqrt{1-{e}_{0}^{2}}\lesssim \frac{9}{8}{\mathop{\varepsilon }\limits^{ \sim }}_{{\rm{S}}{\rm{A}}}\approx 0.118{\left(\frac{\alpha }{0.3}\right)}^{3/2}$$
(14)

where \({\tilde{{\varepsilon }}}_{{\rm{SA}}}={{\varepsilon }}_{{\rm{SA}}}(1+2\sqrt{2}{e}_{{\rm{out}}}/3)\approx 1.039{{\varepsilon }}_{{\rm{SA}}}\) has been defined for convenience.

The width of the non-secular semi-LK regime increases with α, whereas the width of the secular LK regime decreases with increasing α. Comparing equation (4) and equation (14) yields the transitional separation αt found in equation (1).

Spin period

There is little evidence for structural changes of MU69 since its formation, and the spin period is believed to be primordial1. In our model, the collision is gentle and occurs at relatively low velocities (vesc = 442.4 cm s−1 for our nominal density and vesc = 3.128 cm s−1 for the lower density of 0.5 g cm−3 assumed in ref. 1), such that almost any impact parameter (or impact angle) is allowed. Therefore, to obtain the observed spin period, we can use the standard arguments of angular momentum conservation and derive the impact parameter (or impact angle) that yields the desired spin rate.

Consider triaxial ellipsoidal bodies i with masses mi and axes ai ≥ bi ≥ ci, with i = 1, 2. We assume that the major axes ai are parallel, similar to the observed object, and that the collision occurs in parallel with the major axes. The largest moment of inertia is \({I}_{3}^{(i)}={m}_{i}({a}_{i}^{2}+{b}_{i}^{2})/5\).

After the collision the distance between the centre of masses of the joint body and each centre of the ellipsoid is ri. Then the principal moment of inertia of the joint body is

$${I}_{3}^{{\rm{tot}}}={I}_{3}^{(1)}+{I}_{3}^{(2)}+{m}_{1}{r}_{1}^{2}+{m}_{2}{r}_{2}^{2}=\mathop{\sum }\limits_{i=1}^{2}\frac{{m}_{i}}{5}({a}_{i}^{2}+{b}_{i}^{2}+5{r}_{i}^{2})$$

Now, the ellipsoids collide with relative velocity vesc and impact parameter b. The orbital angular momentum is Lz = μbvesc, where μ = m1m2/(m1 + m2) is the reduced mass. If the two bodies are non-rotating, then the joint angular frequency is

$${\Omega }=\frac{{L}_{z}}{{I}_{3}^{{\rm{tot}}}}=\frac{5\mu b{v}_{{\rm{esc}}}}{{m}_{1}({a}_{1}^{2}+{b}_{1}^{2}+5{r}_{1}^{2})+{m}_{2}({a}_{2}^{2}+{b}_{2}^{2}+5{r}_{2}^{2})}$$
(15)

If the individual bodies are rotating around the z axis with frequencies Ωi, the additional angular momentum of each body is \({I}_{3}^{(i)}{{\Omega }}_{i}\) for i = 1, 2, and thus equation (15) becomes

$${\Omega }=\frac{5\mu b{v}_{{\rm{esc}}}+{m}_{1}({a}_{1}^{2}+{b}_{1}^{2}){{\Omega }}_{1}+{m}_{1}({a}_{1}^{2}+{b}_{1}^{2}){{\Omega }}_{2}}{{m}_{1}({a}_{1}^{2}+{b}_{1}^{2}+5{r}_{1}^{2})+{m}_{2}({a}_{2}^{2}+{b}_{2}^{2}+5{r}_{2}^{2})}$$
(16)

For an impact angle θ, the distance of the point of contact to the centre of each ellipsoid is \({\xi }_{i}=\sqrt{{a}_{i}^{2}{\cos }^{2}\theta +{b}_{i}^{2}{\sin }^{2}\theta }\). The impact parameter is related to the impact angle by sinθ = b/(ξ1 + ξ2) = b/d. The distances from the centre of mass are r1 = m2d/(m1 + m2) and r2 = m1d/(m1 + m2) and the spin rate is

$${\Omega }=\frac{5\mu d{v}_{{\rm{esc}}}\,\sin \,\theta +{m}_{1}({a}_{1}^{2}+{b}_{1}^{2}){{\Omega }}_{1}+{m}_{1}({a}_{1}^{2}+{b}_{1}^{2}){{\Omega }}_{2}}{5\mu {d}^{2}+{m}_{1}({a}_{1}^{2}+{b}_{1}^{2})+{m}_{2}({a}_{2}^{2}+{b}_{2}^{2})}$$
(17)

Figure 3 shows the spin-period dependence on the impact angle for the typical parameters of MU69. The spin period is P = 2π/Ω, where Ω is given by equation (17), and when there is no internal rotation, Ω1 = Ω2 = 0. We see that an impact angle of about 40° gives the observed spin period. We have performed hydrodynamical simulations based on the code of ref. 27 that qualitatively agree with our assumptions and produce similar results. Typical classical KBOs could have primordial spin periods35 with comparable contributions to the angular momentum budget. Recently, it was found21 that the mean cold classical KBO spin period is 9.48 ± 1.53 h. Generally, there is no reason for the spin vectors of each body to be correlated, so on average the contribution is zero. In extreme cases, the spin vector of both objects could be aligned or anti-aligned with the orbital angular momentum. In these cases, a large range of impact angles and spin configurations are possible, resulting in the observed spin period after the collision. For a typical period of 10 h, the impact angle is about 20° in the aligned case, and about 70° in the anti-aligned case.

N-body stopping conditions and tests

We impose a stopping condition that the distance between the two bodies is less than their mutual radius. During the non-secular highly eccentric passage, the change in the pericentre q is much faster than the inner orbital period (this is the definition of the non-secular regime), and hence the orbital elements are not reliable at this stage. Once the simulation stops it records the orbital elements at impact, which we use for our statistics, but these are not involved in the stopping condition. From the output we know the closest approach at impact. We have tested the stopping condition by varying it to be slightly smaller or larger than the q found in the first run. Indeed, when the stopping condition was below q the objects did not collide and the code continued running. We thereby concluded that the collision is physical and reliable.

Extended Data Table 1 shows the merger fractions from the simulations. The total merger fraction fi is the total number of mergers divided by the initial number of runs, multiplied by the relative fractions of the inclination distribution, assuming that no mergers occur outside the sampled inclination distribution. The fraction f80−100 is calculated in the same way, except that the only mergers considered are those where the mutual inclination during the merger is within the designated boundaries of 80°−100°. For example, for α = 0.2, the merger fraction is 78/200 = 0.39. Multiplied by the range of the inclination distribution, fi = 0.39 × 0.3 ≈ 0.12 and the high-obliquity merger fraction is f80–100 = 9/200 × 0.3 ≈ 0.014. For α = 0.3, the merger fraction is 99/250 = 0.396. Multiplied by the range of the inclination distribution, fi = 0.396 × 0.4 = 0.158 and the high-obliquity merger fraction is f80–100 = 12/250 × 0.4 ≈ 0.019.

Impact modelling

We perform hydrodynamical collision simulations using our SPH code27, which treats self-gravity, gas, fluid, elastic and plastic solid bodies that have a material strength, including a porosity and fracture model that can be applied for small-body collisions36,37. In order to treat numerical rotational instabilities, a tensorial correction scheme38 is implemented. The miluphCUDA code is implemented with CUDA, and runs on graphics processing units (GPUs), with an improvement of approximately one to two orders of magnitude for a single GPU compared to a single central processing unit (CPU). The code has previously been successfully applied to several studies involving impact processes36,37,39,40,41,42,43,44,45,46,47.

For the porosity treatment, we implement the P−α model48,49, in which the pores are much smaller than the spatial resolution and cannot be modelled explicitly. Here, the total change in the volume depends both on the compaction or collapse of the pore space and on the compression of the solid material that constitutes the matrix. The dependence is expressed in terms of a porous material pressure P and density \({\varrho }\) as \(P/{\varrho }={P}_{{\rm{s}}}/{{\varrho }}_{{\rm{s}}}\), where Ps and \({{\varrho }}_{{\rm{s}}}\) are the pressure and density of the solid matrix material, respectively. The distention parameter \(\bar{\alpha }={{\varrho }}_{{\rm{s}}}/{\varrho }\) is the ratio between the solid matrix material and the porous material densities, and relates to the porosity ψ via \({\psi }=1-1/\bar{\alpha }\). For the solid matrix material we use the Tillotson equation of state (EOS) parameters50 with a reduced bulk modulus of A = 2.67 × 108 Pa (the leading term in the EOS) to take into account the smaller elastic wave speeds in porous materials compared to solid materials, consistent with previous work51. Our matrix density is chosen to be 2 g cm−3, about the same as that used previously52, which leads to a 50% porosity for our fiducial bulk density for MU69, 1 g cm−3. The matrix density and the initial porosity are both in rough agreement with what might be expected from an object of this origin and size range. In particular, the former constrains the rock–ice mass ratio to be about 3–4 (depending on the exact choice of silicate grain density), which could be compatible with this type of KBO53,54,55,56. However, we note that given the uncertainties involved, we seek only to obtain a rough estimate of the density that will permit us to test our working hypothesis. We then also run simulations with 75% porosity and half the previous bulk density to establish the qualitative differences between these two setups.

For collisions between small porous bodies, compressibility is limited by the crush curve for \(\bar{\alpha }\) for typical pressures, instead of by the Tillotson EOS parameters. We thereby choose three sets of crush-curve parameters52, using a simple quadratic crush curve57:

$$\bar{\alpha }=1+({\bar{\alpha }}_{0}-1)\frac{{({P}_{{\rm{s}}}-P)}^{2}}{{({P}_{{\rm{s}}}-{P}_{{\rm{e}}})}^{2}}$$
(18)

where \({\bar{\alpha }}_{0}=2\), Pe is the transition pressure between the elastic and plastic regimes and Ps is the pressure of full compaction. Both Pe and Ps are listed in Extended Data Table 2. As ref. 52 treats comet 67P/Churyumov–Gerasimenko, which belongs to a class of much smaller and active objects, we assume MU69 is probably fluffier and more porous. Hence, our low-strength crush-curve values correspond to the previous high-strength values52, and taking the same modelling approach we then increment the parameters in each subsequent model by one order of magnitude.

Fracture and brittle failure are treated using the Grady and Kipp fragmentation prescription58,59,60, which is based on randomly distributed flaws in the material following a Weibull distribution with material-dependent parameters. The lowest activation threshold strain, κ, derived from the Weibull distribution, is given by κ = kV−1/m, where V is the volume of the brittle material and k and m are the material-dependent Weibull parameters. We adopt m = 9.5 for pressure-dependent failure61. The volume is calculated given the dimensions of the MU69 binary. From the material strength parameters K and G, Young’s modulus E may be calculated as E = (9KG)/(3K + G). Here K = 2.67 × 108 Pa, which is the leading term in the Tillotson EOS, and G = 1.6 × 108 Pa. Finally, for undamaged material, κ = YT/E, where YT is the tensile strength given in table 30 of ref. 52. k may thus be extracted and is k = 1047, 2 × 1039 and 2 × 1028 m−3 for the low-, medium- and high-strength-material setups, respectively. Damage accumulates when the local tensile strain reaches the activation threshold of a flaw.

For the plasticity model we use a pressure-dependent yield strength62 following the implementation of ref. 61. The yield stress Yi is different for damaged and intact material. For intact material, the yield stress is Yi = Y0 + μiP/(1 + μiP/(YM − Y0)), where Y0 is the cohesion (again see table 30 of ref. 52), μi is the coefficient of friction and YM is the shear strength at P = ∞. We adopt μi = 1.5 (ref. 61) and a typical YM = 1.5 × 109 Pa (ref. 60), which is appropriate for an object composed of ice, rock and organics. For P = 0, we recover the pressure-independent form Yi = Y0. For damaged material the yield stress is Yd = μdP, where μd is the coefficient of friction of the damaged material. Here we take μd = 0.6, following ref. 61, and thus fully damaged particles still undergo some shear stress.

Extended Data Fig. 1 shows additional results of our simulated impacts. We obtain the rotation period of MU69 using the nominal density of 1 g cm−3 only when using the high-strength-material parameters. Medium-strength (Extended Data Fig. 1a) or low-strength (Extended Data Fig. 1b) materials deform MU69 and do not produce the observed shape of a gently merged contact binary. If the nominal density is halved (0.5 g cm−3), the impact velocity vesc is lower, which produces less deformation in our simulations. Even medium-strength material parameters generate a gently merged contact binary for virtually all impact angles (Extended Data Fig. 1c, d). Here, we used the 55° impact angle, for which the observed spin period of MU69 is approximately obtained. In Extended Data Fig. 2 the shape is considerably deformed after the collision if the impact velocity is larger than v = 10vesc, using high-strength material parameters. The same velocity with weaker material parameters leads to a complete disruption of MU69.

Our simulations were performed for a grid of impact angles, assuming pre-alignment of the two lobes. Simulating higher impact angles and low- and medium-strength-material compositions causes the two lobes to interact in other ways: in some cases they hit and create contact craters and then roll on top of each other; in other cases they collide, bouncing off each other instead of rolling, and then return to re-collide following a (now) shorter orbital period. These formation channels may also generate compatible shapes but, so far, not the exact rotation period of MU69. A full investigation of the collision phase space must also include the initial self-rotation of each lobe in addition to inclined hits. This requires a huge collision phase space, exceeding the scope of this work, and necessitates a dedicated hydrodynamical study. Preliminary results (in preparation) indicate that such an approach may yield more channels through which the unique orientation of MU69 might be generated, besides the successful cases for pre-aligned binary components that are shown here.

Our standard resolution is 5 × 105 SPH particles. We have additionally preformed simulations with 105 and 2.5 × 105 particles. Test simulations were performed on the TAMNUN GPU cluster at the Technion Institute in Israel and production runs on the bwForCluster BinAC at the University of Tübingen, Germany.