Beyond Point Masses. III. Detecting Haumea's Nonspherical Gravitational Field

Assuming a homogeneous density structure and using the equations found in Yoder (1995), the occultation-derived shape implies a J₂ of 0.24 (Ortiz et al. 2017). Allowing for differentiation decreases J₂ significantly. The model for a two-layer differentiated Haumea presented in Dunham et al. (2019) gives an overall J₂ of ∼0.16. Our model fitting to all available data (HST+Keck) is 3σ inconsistent with both of these models. However, the fit with only HST data is consistent with both, encouraging us to explore possible reasons Haumea's J₂ may be higher.

One possible reason could be the gravitational contributions from Haumea's ring. Assuming a circular ring, the following expression can be derived for the J₂ contribution of a ring:

$$\begin{eqnarray}&&{J}_{2}{R}^{2}\approx \displaystyle \frac{1}{2}\displaystyle \frac{{M}_{r}}{{M}_{P}}{r}^{2},\end{eqnarray} \tag{ 3 }$$

where M_r and M_P are the masses of the ring and Haumea, respectively, and r is the radius of the ring. For the ring to contribute ∼1% of the measured J₂ R² of our HST-only fit, the ring would need to have a mass of ∼10¹⁸ kg, about the mass of Namaka, given the known ring radius of 2287 km. For it to be the cause of Haumea's unexpectedly high J₂ in the HST+Keck fit, the ring would need to be 2 orders of magnitude more massive, equivalent to tens of Hi'iaka masses. While no mass constraints on the ring are found in the literature, this value is absurdly high. For comparison, Saturn's rings are ∼1 Hi'iaka mass (Iess et al. 2019). Hence, the ring is unlikely to contribute significantly to our measured J₂. There is a distinct possibility that more rings may be detected in future occultations (e.g., Pereira et al. 2023), but even when combined, a ring system is unlikely to contain enough mass to substantially contribute to Haumea's J₂.

Another potential source is an undetected satellite within Namaka's orbit. Averaged over an orbit, the putative inner satellite would act similar to a solid ring of material. Hence, using Equation (3) above to calculate the J₂ of a putative inner satelite orbiting at 10,000 km, we find that an inner satellite with a mass near Namaka's mass could significantly contribute to Haumea's measured J₂. Unfortunately, Burkhart et al. (2016) significantly ruled out satellites more than 60 km in diameter closer than 10,000 km by using a nonlinear shift-and-stack image-analysis technique. This diameter implies a mass approximately one-tenth of Namaka's mass, which would scarcely contribute to the overall J₂. Given this, we believe it is unlikely that our results could be caused by an unknown inner satellite. Likewise, undiscovered satellites external to Hi'iaka's orbit would not produce the observed signature.

An alternative reason could be an extreme mass anomaly on Haumea's surface, either positive or negative, which would cause Haumea's J₂ to be larger than expected. Unfortunately, the mass surplus (or deficit) would have to be substantial, of order ∼10% of Haumea's total mass. Maintaining a mass anomaly of that magnitude would require Haumea to have implausibly high material strength. Using the method developed by Johnson & McGetchin (1973), we optimistically estimate that Haumea may support a maximum topographic feature of ∼10 km, amounting to far less than the ∼100 km required to produce an unusually high J₂. Hence, it seems unlikely that a surface feature could plausibly explain the HST+Keck measurement.

Likewise, Haumea could have an unusual interior density distribution. If Haumea formed in the aftermath of a large collision, it may have an unusually shaped core left over as a remnant of this impact. Alternatively, its core could be offset relative to its external figure, potentially explaining Haumea's "Dark Red Spot" (Lacerda 2009). Assuming Haumea's core is triaxial and adopting the external figure of Haumea as measured by stellar occultation, we can calculate the J₂ of Haumea with an arbitrary triaxial core shape with an offset. We find that for any realistic core shape or offset, Haumea's J₂ cannot increase by more than 50%, still well below the constraint provided by the HST+Keck fit. In any case, the extreme versions of these hypotheses are unlikely to be geophysically viable as they ignore fluid-like relaxation of Haumea's core and mantle. We believe that an unusual interior is unlikely the cause of our results.

Due to the implausibility of all of these solutions, we conclude that our result is due to factors dependent on our modeling techniques or data. One possible explanation is higher-order non-Keplerian dynamics taking place within the system. Since our model only includes Haumea's J₂, other gravitational harmonics may be needed to fully model the system. To investigate the effect of C₂₂, we ran a MultiMoon fit that included Haumea's C₂₂ potential. This fit gave nearly identical results and found no constraint on C₂₂, indicating that the orbital dynamics of the system are not strongly coupled to C₂₂. Indeed, previous work exploring the effects of C₂₂ found that it is only relevant when P_orb ∼ P_spin, or near a low-order SOR (Proudfoot & Ragozzine 2021). Beyond quadrupole dynamics, fourth-order, or hexadecapole, dynamics could contribute to orbital precession, but as previously discussed, their r⁻⁵ dependency ensures that their contributions are small. Taking the J₄ harmonic as an example, we find that the ratio of nodal precession of Namaka's orbit caused by the J₄ and J₂ harmonics is ${\dot{{\rm{\Omega }}}}_{{J}_{4}}/{\dot{{\rm{\Omega }}}}_{{J}_{2}}\approx 0.003\tfrac{{J}_{4}}{{J}_{2}}$. Apsidal precession has a similarly small strength. Typically, the J₄ harmonic is much smaller than J₂, implying the nodal precession induced by J₄ is a very small effect.

Odd harmonics (e.g., J₃, C₃₁, etc.) could, in theory, also play a role in the system's orbital dynamics. (Note that the "dipole" term is 0 due to using the center-of-mass as the coordinate system; a center-of-mass–center-of-figure offset can contribute to J₂, which was included in the calculations with the offset core above). Again, finding the ratio of nodal precession for Namaka, we find ${\dot{{\rm{\Omega }}}}_{{J}_{3}}/{\dot{{\rm{\Omega }}}}_{{J}_{2}}\approx 0.006\tfrac{{J}_{3}}{{J}_{2}}$ when ${\dot{{\rm{\Omega }}}}_{{J}_{3}}$ is at its maximum. Odd harmonics are typically extremely small in bodies at (or near) hydrostatic equilibrium, producing ∼no detectable precession. While it seems unlikely that J₃ or other odd harmonics cause significant changes in the dynamics on the timescale of our observational data, future investigations should explicitly test whether odd harmonics are necessary for accurate modeling of the Haumea system.

Another possible effect that we do not account for is the satellites' putatively nonspherical gravitational fields. Our model assumes that Hi'iaka and Namaka are point masses, although they are likely to be substantially nonspherical. In some cases, however, the gravitational harmonics of a system's secondary can play a major role in the overall orbital dynamics (e.g., Ragozzine & Wolf 2009). MultiMoon is well poised to explicitly test this assumption. Rather than adding six parameters to our overall model, which would be computationally expensive, we can simply add the satellites' harmonics as fixed values. We can then compare a model without their harmonics to one with them, allowing us to see the change in system dynamics. In this comparison, we use the characteristics of all our observations (HST+Keck) to explicitly connect the dynamical change to actual observability. In Figure 4, we show the change in orbit fit quality as a function of Hi'iaka and Namaka's J₂, when both satellites have a moderately high obliquity. We define change in fit quality as ${\rm{\Delta }}{\chi }^{2}\,={\sum }^{i}({x}_{i,{J}_{2}}-{x}_{i,{J}_{2}=0})/{\sigma }_{i,\mathrm{obs}}$, where ${x}_{i,{J}_{2}}$ and ${x}_{i,{J}_{2}=0}$ are synthetic astrometric measurements from models including satellite J₂ and those without and σ_i,obs are the measurement uncertainties for the system as tabulated in Table 1. Using those measurement uncertainties allows us to connect the system's dynamics to the actual observations. Overall, we find the fit quality is barely decreased when reasonable values for J₂ are tested. Hi'iaka would need J₂ > 2 for a detectable change, while Namaka would need J₂ > 10. For comparison, Arrokoth, a contact binary, has a J₂ ∼ 0.3. While not an exhaustive search of parameter space, this is strong evidence that Hi'iaka and Namaka's nonspherical shapes do not significantly contribute to the system's dynamics with the present data.

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In our view, the only remaining option is the presence of unknown systematic errors plaguing our data set. Despite our use of novel statistical techniques, our model cannot account for all systematic errors arising from combining our data set. For example, time-varying distortions in the NIRC2 field cannot be appropriately accounted for by our model. Likewise, wavelength-dependent offsets between Haumea's center of light and center of mass, potentially caused by the known wavelength-dependent rotational variability known as the "Dark Red Spot" (Lacerda 2009) may introduce unwanted systematics when combining the data sets. Indeed, when combining the data sets, we find that the Keck data from the 2020s is incompatible with the HST visit from 2014. Our combined data set produces large residuals for the 2014 HST visit, while the HST-only fit shows no such effect. While disconcerting, this conclusion is not extremely surprising given a similar result in RB09, from which we draw much of our data. Those authors similarly found that the Keck data was inconsistent with the HST-only fits. We thus argue that unknown systematic errors are the source of our unusually high measurement of J₂. The HST instruments are extremely well studied and have been rigorously cross-validated, so we view the HST-only fit as more trustworthy. To remain as conservative as possible, we adopt the HST-only fit as our nominal model for the rest of the analysis in this work.

While our model does not fully constrain the masses of the satellites, especially Hi'iaka, it is still scientifically valuable given some assumptions. If we assume Haumea has the J₂ of a differentiated body (J₂ = 0.16 using the Dunham et al. 2019 model), we are able to estimate the mass of Hi'iaka to be roughly 1.6 × 10¹⁹, with uncertainties of ∼10%. This is similar to the mass determination found in RB09, albeit 10% smaller. Given an estimated radius of ∼160 km (RB09), this gives a density of ∼900 kg m⁻³, similar to other TNOs in the same size range (Grundy et al. 2019). Even given the undifferentiated model, Hi'iaka is not substantially less massive, altering the density marginally.

While the mass measurement for Hi'iaka must be improved to place better constraints on Hi'iaka's density, this needs to be matched with improvement in the radius determination. As density is proportional to r⁻³ (as opposed to linear in mass), uncertainties in radius have a stronger influence on the overall density uncertainty. Thermal observations and modeling of the satellites indicate high albedos and small sizes (Müller et al. 2019), but uncertainties are large. At this time, given the massive uncertainties in both radius and mass, robust interpretation of the density of Hi'iaka is impossible.

Fortuitously, however, a recent occultation campaign has captured a multi-chord stellar occultation of Hi'iaka (E. Fernandez-Valenzeula et al. 2024, in preparation). The results of this campaign are forthcoming, but early indications point toward Hi'iaka being larger than expected (E. Fernandez-Valenzeula 2024, personal communication). When using the results from those observations, future work should be able to tightly constrain the density of Hi'iaka if the Hi'iaka mass–J₂ degeneracy is broken.

While our uncertainty on the mass of Namaka is much smaller than for Hi'iaka, Namaka still suffers from a poorly constrained size. However, given size estimates from thermal measurements (Müller et al. 2019) and past photometric analysis (RB09), Namaka could have a radius of ∼75 km, giving a density of ∼650 kg m⁻³. While quite low, it is typical of small TNOs (Grundy et al. 2019). Such a low density is indicative of a porous interior, consistent with relatively gentle accretion in a satellite-forming disk around Haumea in the aftermath of an impact. To improve confidence in this measurement, Namaka should be a high-priority target for an occultation campaign.

Among TNOs, very few spin poles have been constrained. When disregarding non-Keplerian fitting, the only techniques currently able to characterize the spin poles of TNOs are long-term light-curve monitoring and occultations. Light-curve inversion techniques require observations of a TNO over a significant portion of their orbit, which is difficult due to TNOs' long heliocentric orbital periods. Occultations are extremely powerful for inferring spin poles, but observations of multiple multi-chord events are required, which are only available for a few TNOs. Non-Keplerian orbit fitting now adds an additional tool which can be used to understand the spin poles of TNOs. ⁸ Normally, non-Keplerian fitting cannot find a unique pole solution, but is able to determine the angle between the primary's spin pole and the secondary's orbit normal. However, since Haumea has two satellites, the spin pole can be found unambiguously.

Our measurement in this work represents the first dynamically determined spin pole among TNOs. We are able to place tight constraints on the spin-pole direction of Haumea, finding $({\alpha }_{p},{\delta }_{p})=({282.9}_{-0.7}^{+0.6\circ} ,-{9.7}_{-1.0}^{+0.6\circ} )$. Interestingly, Haumea's pole is almost perpendicular to its orbit. Based on Haumea's heliocentric orbit, we find Haumea's obliquity to be 87.1°. Our dynamically determined spin-pole measurement lies 2.3° from the ring pole determined by stellar occultation (α_p , δ_p ) = (285.1° ± 0.5°, − 10.6° ± 1.2°; Ortiz et al. 2017). Given that our methods are completely different from those used in analyzing the stellar occultation and we use no constraints from that work, the close match is encouraging. Formally, however, these two spin-pole measurements are ∼2σ apart. The difference could be a real effect (i.e., Haumea's ring is inclined with respect to Haumea's nonspherical gravitational field), but the nodal precession of ring particles induced in this case is likely to erode the rings excessively (Marzari 2020). It seems much more likely that the ring is coplanar with Haumea's gravitational field, and the disagreement is due to model-dependent factors or random chance. For example, Ortiz et al. (2017) modeled Haumea's ring as a flat, circular annulus, but the true ring may have substantial eccentricity and/or thickness. Alternatively, since the spin pole in our fit is highly correlated with J₂, our large uncertainties may cause/contribute to the disagreement. When examining our HST+Keck fit, we find the spin-pole measurements are even further apart, lending further credibility to the HST-only model.

The rotational dynamics of the Haumea system are extremely interesting to study. For Haumea itself, the torque from both satellites on Haumea's nonspherical body cause a small amount of axial precession. We show a 1000 yr integration of Haumea's spin dynamics in Figure 5. The integration is performed by SPINNY, the spin–orbit integrator at the heart of MultiMoon (Ragozzine et al. 2024). We find that Haumea's axis precesses by <1° on a timescale of hundreds of years. Visible is a complex precession cycle in Haumea's spin-pole direction with one "fast" frequency and one "slow" frequency. These two frequencies are caused by the torque from each satellite, with periods corresponding to each satellite's nodal precession period. Namaka's nodal precession period is strongly coupled with both Hi'iaka–Namaka interactions and Namaka– J₂ interactions. Since Namaka is coupled to Hi'iaka's nodal precession, Namaka's nodal precession then weakly couples to Hi'iaka– J₂'s nodal precession, although this is a much smaller effect. Hi'iaka's nodal precession has a fast, low-amplitude component caused by Hi'iaka–Namaka interactions as well as a slow, high-amplitude component from the Hi'iaka– J₂ interaction. The low-amplitude, high-frequency precession of Haumea's pole caused by Namaka would produce little detectable change in Haumea's light curve or occultation shadow. The Hi'iaka coupled precession has a much higher amplitude, but has a period of hundreds of years, severely hampering detectability. Given Haumea's prolate shape, the satellites' torques can also alter Haumea's rotation period, however this effect is tiny due to Haumea's large angular momentum. Using SPINNY simulations, we estimate the period variations are ∼10⁻⁸ hr, approximately 2 orders of magnitude smaller than the uncertainty in the measured rotation period (Rabinowitz et al. 2006).

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More amenable to detectability is possible precession of Hi'iaka's rotational axis. In Hastings et al. (2016), the light curve of Hi'iaka was studied using resolved photometry from HST images. They found that Hi'iaka is rapidly rotating (∼9.8 hr double-peaked period), with an unusual sawtooth-shaped light curve of amplitude Δm ≈ 0.23. Using a simplified model of axial precession, they found that Hi'iaka's axial precession would be detectable on decade timescales if there was significant obliquity (with respect to its orbit). The detectability is significantly enhanced by Hi'iaka's high-amplitude light curve. Detection of any precession would require long-term monitoring of Hi'iaka's light curve, which would slowly change amplitude and/or shape across the precession cycle. SPINNY provides an ideal framework for validating this possible method. Using the best fit from our nominal orbit model, we have explicitly modeled Hi'iaka's axial precession. In Figure 6, we show the evolution of Hi'iaka's rotation axis assuming differing starting obliquities. Then, in Figure 7, we illustrate how that precession translates to variation in Hi'iaka's light-curve amplitude, assuming triaxial shapes as in Hastings et al. (2016). Interestingly, even in the case where Hi'iaka's pole is initially aligned with its orbit, precession still occurs. While initially surprising, the precession is due to Hi'iaka's nodal precession in its orbit around Haumea, which will always misalign Hi'iaka's spin pole. Encouragingly, even for a relatively small obliquity of 10°, the precession is substantially different from the no-precession case. This allows us to confirm previous results (e.g., Hastings et al. 2016) and show that small perturbations (e.g., Hi'iaka's eccentric orbit, torques from Namaka, etc.) seem to make little difference to the overall evolution. In the future, SPINNY and/or MultiMoon could be modified to explicitly model changes in light-curve amplitudes. This method would provide a detailed model with which to understand the spin dynamics of Hi'iaka; we defer this to future work.

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Even though Namaka has been solidly detected in several epochs of HST observations, no periodic brightness variations have been found, although photometry from HST programs is suggestive of a long rotation period (>1 day). Purely based on theoretical dynamical arguments, Namaka is expected to be significantly despun, except if its initial rotation period was extremely short (Hastings et al. 2016). Given Namaka's eccentric orbit, overlap between SORs inevitably causes spin–orbit chaos, very similar to Saturn's satellite Hyperion (Wisdom et al. 1984). As an illustration of chaotic rotation, the resonance overlap criterion, which predicts chaotic spin–orbit evolution if satisfied, near the 1:1 and 3:2 SORs is

$$\begin{eqnarray}&&\sqrt{\displaystyle \frac{3({ \mathcal B }-{ \mathcal A })}{{ \mathcal C }}}\geqslant \displaystyle \frac{1}{2+\sqrt{14e}},\end{eqnarray} \tag{ 4 }$$

where ${ \mathcal A }$, ${ \mathcal B }$, and ${ \mathcal C }$ are Namaka's principal moments of inertia and e is Namaka's orbital eccentricity. In this case, to avoid spin–orbit chaos Namaka's shape would have to satisfy $\tfrac{({ \mathcal B }-{ \mathcal A })}{{ \mathcal C }}\lt 0.025$. Satellites that are similar in size to Namaka generally have $\tfrac{({ \mathcal B }-{ \mathcal A })}{{ \mathcal C }}\gtrsim 0.1$. For example, with a mass between Namaka and Hi'iaka (∼5 × 10¹⁸ kg), Hyperion has $\tfrac{({ \mathcal B }-{ \mathcal A })}{{ \mathcal C }}\approx 0.26$.

While the above calculation is a simplistic example comparing just two resonances, in general resonance overlap and spin–orbit chaos are expected for a slowly rotating Namaka. In Figure 8, we show the initiation of chaotic tumbling where Namaka's attitude (i.e., pole direction) and rotation period evolve chaotically. Given the difficulty of acquiring resolved photometric observations of Namaka and the long time span needed to detect a slow (and possibly chaotic) rotation, confirming it may be extremely difficult. Similarly, searching for Namaka's light curve in unresolved photometry of the entire system is extremely difficult as Namaka's brightness only contributes ∼1% of the total system flux.

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Haumea's ring, first discovered during a stellar occultation, lies close to Haumea's equatorial plane, as shown in the previous section. This matches theoretical expectations, which show that ring particles should collisionally damp to the equatorial plane in the presence of Haumea's J₂. When accounting for interactions between ring particles and Haumea's satellites, however, ring particles can preferentially settle in the satellite orbit plane (Marzari 2020). However, the satellites are too far away and/or not massive enough to cause this to occur for the ring. The satellites' main dynamical roles are to act as small perturbers that increase velocity dispersion. Marzari (2020) studied the increase in collision velocity of ring particles under the influence of Haumea's satellites and found that collision velocities still remained low, with typical velocities <1 m s⁻¹.

It may be possible that if other rings exist external to the currently known ring, satellite–ring dynamics may be more important. As the strength of satellite–ring interactions increases, the collision velocities between ring particles may become large enough to completely disperse the ring. Rings external to the Roche limit may be possible at the temperatures of the Kuiper Belt (Morgado et al. 2023; Pereira et al. 2023), but in Haumea's case they are likely to be in a regime where perturbations make them long-term unstable.

To enable future observations of the Haumea system, we have created an ephemeris of the system containing the predicted ${\rm{\Delta }}\alpha \cos \delta$ and Δδ of each satellite and the uncertainties on the positions. We compute the ephemeris using 500 random draws from our posterior distribution. The ephemeris contains the predicted positions every 8 hr between 2005 and 2035. In Table 3, we show the first 10 lines of the ephemeris. The ephemeris is published in its entirety in machine-readable format. The uncertainties on our predictions for both satellites are quite small (≲50 mas) through the 2020s until 2030, at which time the uncertainties begin to grow rapidly, especially for Namaka. By 2035, the uncertainties on Namaka's position are of order ∼01. Rapid growth in uncertainty is attributable to the large degeneracy in our model (see Table 2 and Figure 3).

Table 3. System Ephemeris

Julian Date	Date	ΔxN	ΔyN	${\sigma }_{{\rm{\Delta }}{x}_{{\rm{N}}}}$	${\sigma }_{{\rm{\Delta }}{y}_{{\rm{N}}}}$	rN	${\sigma }_{{r}_{{\rm{N}}}}$	ΔxH	ΔyH	${\sigma }_{{\rm{\Delta }}{x}_{{\rm{H}}}}$	${\sigma }_{{\rm{\Delta }}{y}_{{\rm{H}}}}$	rH	${\sigma }_{{r}_{{\rm{H}}}}$
		('')	('')	('')	('')	('')	('')	('')	('')	('')	('')	('')	('')
2453371.500	2005-01-01 00:00:00	0.068	−0.190	0.002	0.005	0.202	0.004	0.057	0.649	0.001	0.003	0.651	0.003
2453371.833	2005-01-01 08:00:00	0.068	−0.121	0.002	0.005	0.139	0.004	0.043	0.600	0.001	0.003	0.601	0.003
2453372.167	2005-01-01 16:00:00	0.068	−0.051	0.002	0.005	0.085	0.003	0.029	0.549	0.001	0.003	0.550	0.003
2453372.500	2005-01-02 00:00:00	0.067	0.019	0.002	0.005	0.069	0.002	0.015	0.497	0.001	0.003	0.497	0.003
2453372.833	2005-01-02 08:00:00	0.065	0.090	0.002	0.006	0.111	0.004	0.000	0.444	0.001	0.003	0.444	0.003
2453373.167	2005-01-02 16:00:00	0.062	0.159	0.002	0.006	0.171	0.005	−0.014	0.391	0.001	0.003	0.391	0.003
2453373.500	2005-01-03 00:00:00	0.059	0.227	0.002	0.006	0.234	0.005	−0.028	0.336	0.001	0.003	0.337	0.003
2453373.833	2005-01-03 08:00:00	0.055	0.291	0.002	0.006	0.296	0.005	−0.042	0.281	0.001	0.003	0.284	0.003
2453374.167	2005-01-03 16:00:00	0.051	0.352	0.002	0.005	0.355	0.005	−0.056	0.225	0.001	0.003	0.232	0.003
2453374.500	2005-01-04 00:00:00	0.045	0.407	0.002	0.005	0.410	0.005	−0.070	0.168	0.001	0.004	0.182	0.003

Notes. The predicted R.A. and decl. positions of Namaka (N) and Hi'iaka (H) from 2005 to 2035. Δx and Δy are the predicted R.A. and decl., r is the total separation, and σ are the uncertainties on each value. All values are given in arcseconds. Predicted positions, separations, and uncertainties are taken from a sample of 500 random posterior draws. We display the first 10 rows of the table, with the rest of the table available as a machine-readable table.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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To ascertain whether future observations may be able to break the mass– J₂ degeneracy in our fits, we have analyzed an ephemeris (similar to that presented above) where the predicted positions are also a function of Haumea's J₂ (or equivalently Hi'iaka's mass). We find that the future positions of Hi'iaka and Namaka are strong functions of Haumea's J₂ (or Hi'iaka's mass), indicating that the degeneracy will be broken with additional HST observations. Based on our fits, we can roughly estimate that ∼2–4 new epochs of observations in the mid-2020s (or beyond) are necessary to constrain Haumea's J₂ with ∼10% precision. Past work has shown that timing observations at certain, well-selected times can substantially improve the quality of the resulting orbit fits (Proudfoot et al. 2024). These times occur when the uncertainties in ${\rm{\Delta }}\alpha \cos \delta$ and Δδ in Table 3 are at (local) maximum. We recommend that continued observations be taken by HST to prevent any systematic errors from arising, as we found in our HST+Keck fits. As these observations will probe Haumea's interior and place strong constraints on its differentiation, we regard them as high priority.

As well as its use for future observations, our ephemeris can evaluate past predictions of mutual events in light of our new orbit solution. We find that the mutual-event predictions made in RB09 are generally accurate, even given their sign error in the system astrometry. We predict events that are similar in depth and duration, but are somewhat offset in time. The timing differences are only a few hours in 2009–2011, but steadily grow to tens of hours by the end of the mutual-event season. Since the mutual-event timings are quite sensitive to Namaka's eccentricity, and we find an eccentricity ∼15% lower than previously found, it is unsurprising that we find timing differences. The next mutual-event season will occur in approximately half a heliocentric Haumea orbit, about midway through the 2200s. The exact time frame will be dependent on Namaka and Hi'iaka's precise precession rates, which future observations will be able to precisely measure.

Our new information on the properties of the Haumea system give more accurate insight into the formation and evolution of Haumea and its satellites.

Presumably, Haumea's satellites (and possibly its ring) formed in the same event that created the Haumea family. For a full review of family formation proposals, see Proudfoot & Ragozzine (2019). The most compelling proposals involve mass shedding due to an excess of angular momentum, explaining Haumea's rapid rotation and the family members' low ejection velocity. There is still much debate in the literature on the original reason that proto-Haumea had too much angular momentum: Proudfoot & Ragozzine (2022) propose the merger of a near-equal-mass binary similar to Pluto-Charon, perhaps triggered by Kozai cycles initiated due to Haumea's placement on a higher inclination orbit. Noviello et al. (2022) point out that internal evolution could change Haumea's moment of inertia, spinning it up past the point of breakup. Ortiz et al. (2012) and others suggest that it may be the cumulative effect of many smaller impacts, though starting with a rapid rotation would significant increase the probability that a random walk would lead to such a rapid rotation.

Whatever the origin, the general agreement is that Haumea goes through a phase of excess angular momentum where water-ice chunks are ejected at low velocities (relative to catastrophic collisions) from the tips of Haumea. Thus, a plausible starting point for the formation of the satellites is a disk of satellitesimals ejected from the fast-spinning Haumea, mostly composed of pure water ice with sizes similar to known satellites and family members. Given that the distribution of ejection (or sub-ejection) velocities should be somewhat smooth, it is likely that satellitesimals are both ejected and remain in orbit.

This initial disk of satellitesimals should rapidly evolve through ejections and collisions. In this scenario, the unejected material experiences a collisional cascade that leads to a final configuration of a ring of near-circular, near-coplanar disk of collisional debris. This debris would eventually reaccrete into Hi'iaka and Namaka. Haumea's ring could also derive from the parts of this initial ring that did not form into satellites. Combining smoothed particle hydrodynamic modeling with long-term dynamical evolution is necessary to understand this process. We strongly encourage work on this problem, which may provide understanding applicable to the formation of other TNO satellites.

After their initial formation, a variety of physical processes can influence the evolution of the satellites until they reach their current configuration. The most important effects are expected to be tidal evolution, Hi'iaka–Namaka interactions, excitation from passing TNOs and binaries, and possible interactions from previous satellites (Ćuk et al. 2013; Hastings et al. 2016). These are discussed in detail in Ćuk et al. (2013) and we focus here only things that are updated in our new fit.

Ćuk et al. (2013) found that long-term orbital stability would be significantly improved if the satellites were ∼50% of their nominal masses reported in RB09. Indeed, our new results are most consistent with satellites that are ∼60% of the initially estimated masses with the Namaka/Haumea mass ratio of 3.0 ± 0.6 × 10⁻⁴ and the Hi'iaka/Haumea mass ratio of 3.1 ± 0.8 × 10⁻³ in the HST-only fit (see Table 2), though this is strongly affected by the degeneracy discussed above.

Ćuk et al. (2013) also investigate in detail the effect of the 8:3 and 3:1 mean-motion resonances between the satellites, especially in the presence of tidal evolution. This was based on the observation by RB09 that the satellites were possibly in or near the 8:3 resonance, suggesting that tidal evolution in resonances could explain the source of the moderately eccentric and inclined orbits. This idea was further strengthened by the observation—still true with the new orbit fit—that the excitation is similar in eccentricity and inclination and is inversely proportional to the masses of the satellites. This means that the "angular momentum deficit" (Laskar 1997) is approximately evenly partitioned between the two satellites, suggesting they have been strongly dynamically coupled at some point in the past (or present). We find a period ratio of Hi'iaka and Namaka of 2.689 ± 0.004 (including corrections to Newton's version of Kepler's third law from J₂), slightly closer to the 8:3 mean-motion resonance that was reported in RB09. With the residual uncertainty in Hi'iaka's mass and Haumea's J₂, we leave to future investigation whether the 8:3 resonance is currently dynamically active in the system. Confirmation of an active resonance is key to understanding the recent history of the satellites. Overall, however, the general conclusion of Ćuk et al. (2013) that resonance passage could explain the excited orbits remains consistent with the updated fits.

The primary challenge in explaining the current orbital configuration of the satellites is their distant orbits from Haumea, at ∼36 and ∼70 primary radii. Tidal evolution to such distances is challenging in standard tidal theories, requiring Haumea tides to be extremely dissipative with an implausible combination of tidal parameters. This is exacerbated by a factor of ∼2 with the lower masses for the satellites, but Quillen et al. (2016) find that the triaxial shape of Haumea increases tidal evolution by a factor of a few (though not as large as hoped for by RB09 and Ćuk et al. 2013). It is possible that more realistic tidal and geophysical models would be able to resolve these issues. It is interesting to note that the satellites are close to the expected outcome of tidal evolution given their mass and semimajor axis ratios.

One potential resolution to the tidal dissipation problem is to form Hi'iaka and Namaka near their current locations. Since tidal expansion is very strongly dependent on separation, even starting at ∼90% of the present distance does not relieve pressure on tidal theories. Given that we observe objects ejected from this disk, it could have been extended out to the Hill sphere. A disk of such size may allow satellite formation further out than previously thought. Along these lines, we note that, although the satellites seem well-separated, dynamically speaking they are only separated about 5 mutual Hill radii, suggesting they are dynamically packed. Hence, intermediate satellites could not fit dynamically, so perhaps Namaka and Hi'iaka are the natural outcome of an extended disk near their present locations. Further modeling is needed to understand the plausibility of this scenario.

In conclusion, the formation and evolution of Namaka and Hi'iaka are plausibly connected to the same process that spun up Haumea and created the family. One possible formation hypothesis is that water-ice chunks which do not escape to form the family eventually collide and grind down to a disk near the present location of the satellites. Namaka and Hi'iaka perhaps form directly from this disk and recent dynamical interactions (e.g., from the nearby 8:3 resonance) lead to the orbits seen today, as proposed in Ćuk et al. (2013). Once Hi'iaka's mass is better known, a more detailed investigation into the secular, resonant, and tidal dynamics could confirm or refute this hypothesis. However, the most important next step is more detailed modeling of the post-spin-up and family-ejection process, extending to the longer timescales needed to understand the subsequent epoch of satellite formation.