text:
11institutetext: Konkoly Observatory, Research Centre for Astronomy and Earth
Sciences, Konkoly Thege 15-17, H-1121 Budapest, Hungary
11email<EMAIL_ADDRESS>22institutetext: CSFK, MTA Centre of
Excellence, Budapest, Konkoly Thege Miklós út 15-17, H-1121, Hungary
33institutetext: ELTE Eötvös Loránd University, Institute of Physics,
Budapest, Hungary 44institutetext: Eötvös Loránd University, Pázmány Péter
sétány 1/A, H-1171 Budapest, Hungary 55institutetext: Max-Planck-Institut für
extraterrestrische Physik, D-85748 Garching, Germany 66institutetext:
Instituto de Astrofísica de Andalucía, IAA-CSIC, Glorieta de la Astronomía
s/n, E-18008 Granada, Spain 77institutetext: Department of Astronomy,
Institute of Geography and Earth Sciences, Eötvös Loránd University, H-1117
Budapest, Pázmány Péter sétány 1/A, Hungary 88institutetext: Centre for
Astrophysics and Space Science, Eötvös Loránd University, H-1117 Budapest,
Pázmány Péter sétány 1/A, Hungary 99institutetext: ELKH-SZTE Stellar
Astrophysics Research Group, H-6500 Baja, Szegedi út, Kt. 766, Hungary
1010institutetext: Wigner Research Centre for Physics, P.O. Box 49, Budapest
H-1525, Hungary

# Tidally locked rotation of the dwarf planet (136199) Eris discovered from
long-term ground based and space photometry

R. Szakáts 0000-0002-1698-605X 1122 Cs. Kiss 0000-0002-8722-6875 112233 J. L.
Ortiz 66 N. Morales 66 A. Pál 0000-0001-5449-2467 112244 T. G. Müller
0000-0002-0717-0462 55 J. Greiner 55 P. Santos-Sanz 0000-0002-1123-983X 66 G.
Marton 0000-0002-1326-1686 11 R. Duffard 66 P. Sági 0000-0001-7549-0348 112244
E. Forgács-Dajka0000-0002-5735-6273 7788991010

The rotational states of the members in the dwarf planet - satellite systems
in the transneptunian region are determined by the formation conditions and
the tidal interaction between the components, and these rotational
characteristics are the prime tracers of their evolution. Previously a number
of authors claimed highly diverse values for the rotation period for the dwarf
planet Eris, ranging from a few hours to a rotation (nearly) synchronous with
the orbital period (15.8 d) of its satellite, Dysnomia. In this letter we
present new light curve data of Eris, taken with $\sim$1-2m-class ground based
telescopes, and with the TESS and Gaia space telescopes. TESS data could not
provide a well-defined light curve period, but could constrain light curve
variations to a maximum possible light curve amplitude of $\Delta m$ $\leq$
0.03 mag (1-$\sigma$) for P $\leq$ 24 h periods. Both the combined ground-
based data and the Gaia measurements unambiguously point to a light curve
period equal to the orbital period of Dysnomia, P = 15.8 d, with a light curve
amplitude of $\Delta m$ $\approx$ 0.03 mag, i.e. the rotation of Eris is
tidally locked. Assuming that Dysnomia has a collisional origin, calculations
with a simple tidal evolution model show that Dysnomia has to be relatively
massive (mass ratio of q = 0.01–0.03) and large (radius of $R_{s}$ $\geq$ 300
km) to slow down Eris to synchronized rotation. These simulations also
indicate that – assuming tidal parameters usually considered for
transneptunian objects – the density of Dysnomia should be 1.8-2.4 g cm-3, an
exceptionally high value among similarly sized transneptunian objects, putting
important constraints on the formation conditions.

###### Key Words.:

Methods: observational, techniques:photometry, Kuiper belt objects: Eris-
Dysnomia

##  1 Introduction

The largest (D $\gtrsim$1000 km) solar system objects – the dwarf planets –
represent a separate class among transneptunian objects with distinct surface
characteristics and internal properties and also with a high incidence of
satellites (Brown et al. 2006; Kiss et al. 2017). The present rotational state
of these large bodies are expected to be a combined outcome of formation
conditions and tidal interactions in the case of a massive satellite, and
therefore their rotational light curves and the properties derived from them
are important clues to unravel their history. Among these objects, on one
side, the Pluto-Charon system is known to be tidally locked (Dobrovolskis et
al. 1997) while Haumea is an extremely fast rotator with a system of two
satellites in which the more massive satellite could not reach rotational
synchronization with its orbital period (Hastings et al. 2016). The rotation
periods of other transneptunian dwarf planets range from a few hours to a few
days (e.g. Quaoar, Gonggong and Makemake; Ortiz et al. 2003; Pál et al. 2016;
Hromakina et al. 2019), indicating a wide range of formation conditions and/or
tidal interactions. For these objects with relatively long rotation periods
the light curve is expected to be caused by albedo variegations on the surface
instead of being spin-shape driven. Eris is the most massive currently known
dwarf planet, with a satellite, Dysnomia (Brown & Schaller 2007). Recently
Holler et al. (2021) obtained an updated orbit of Eris’ satellite, Dysnomia,
with a corrected orbital period of $P_{orb}$ = 15.785899$\pm$0.000050 d. They
suggested various possible reasons for the observed non-Keplerian orbit of the
satellite, including the precession of Dysnomia’s orbit due to the oblateness
of Eris, an irregularly shaped Dysnomia, an unseen interior satellite, or
center-of-light versus center-of-body offsets.

Several light curve studies can be found in the literature providing very
different rotation periods. Lin et al. (2007) obtained a light curve period of
3$\aas@@fstack{h}$55 with an amplitude of $\Delta$m $\leq$ 0$.\\!\\!^{\rm
m}$05 using a 1 m-class ground based telescope. A low amplitude visual light
curve of Eris was tentatively detected by Roe et al. (2008), with a period of
P = 1$\aas@@fstack{d}$08$\pm$0$\aas@@fstack{d}$02, and with a peak-to-valley
amplitude upper limit of $\sim$0$.\\!\\!^{\rm m}$1 based on Swift satellite
data. They also reported that the shape of the light curve is likely not
sinusoidal, indicating the presence of a dark patch which is visible in part
of the rotation period only. Duffard et al. (2008) obtained a light curve
period of 13$\aas@@fstack{h}$7 at a high confidence level, also using a 1 m
class telescope. Carraro et al. (2006) did not obtain a definite rotation
period, just a lower limit of $\sim$5 d, and also Rabinowitz et al. (2007) and
Sheppard (2007) could not identify any period in their data. Rabinowitz &
Owainati (2014) reported on a possible synchronous rotation of the Eris-
Dysnomia system, with a dominant periodicity in the light curve matching the
orbital period of $\sim$15.8 days. Holler et al. (2020) suggested that the
rotation of Eris is near-synchronous, with a period of P = 14.56$\pm$0.01 d,
indicating that the system is not yet fully tidally evolved.

In this paper we present the analysis of long-term brightness monitoring data
collected from various instruments, including the TESS and Gaia space
telescopes, and several ground-based telescopes which covered different
possible period ranges from a few hours to a rotation synchronized with the
orbital motion of the satellite. Due to its sampling rate and duration TESS
data could be used to investigate periods from a few hours to a few days.
Ground based data were typically measured in blocks covering a few nights,
with (very) long gaps between the blocks which allowed us to search for light
curve periods in the few day – synchronized rotation (15.8 d) range. The
sparse sampling of the Gaia data allowed us to search for rotation periods in
this latter range, too. These observations and the data reduction are
described in detail in Sect. A.1. The summary of the analysis of these data
and the description of curve period identification are presented in Sect. 2.
Using the currently known characteristics of the system we applied a simple
tidal evolution model to try to match the rotation period which we obtained
from our light curve measurements (Sect. 3). Our conclusions are given in
Sect. 4.

##  2 Periods identified in the different data sets

###  2.1 TESS

TESS space telescope data could be used to search for possible light curve
periods in the range of a few hours to a few days (see Sect. A.1.1), and we
could identify a period with a residual minimum at a frequency of f =
0.411$\pm$0.018 cycle/day (denoted as c/d hereafter) which corresponds to a
period of P = 58.394$\pm$2.571 h, with a light curve amplitude of $\Delta$m =
0.132$\pm$0.037 mag, after correcting for instrumental effects. This
frequency, however, is considered to be tentative (1.8 $\sigma$) due to the
significantly increased noise at frequencies below 1 c/d. At frequencies above
1 c/d, however, a 1$\sigma$ upper limit of 0.03 mag amplitude (peak-to-peak)
could be obtained, indicating that no light curve period above this amplitude
level is present at these shorter periods (P $\leq$ 24 h).

###  2.2 Ground-based data

We used a large set of ground based data (Sect. A.3), which are in part new
measurements using several 1-2m-class telescopes (see Table 2), supplemented
by measurements taken from the literature, including ground-based data from
(Carraro et al. 2006), (Rabinowitz et al. 2007), (Sheppard 2007) and (Duffard
et al. 2008), and the Swift satellite data from Roe et al. (2008). We used a
residual minimalization method (see Sect. A.5) to find the light curve
amplitude and period best matched by this large data set. The efficiency of
the method was tested using synthetic light curves, using a sampling similar
to the real Eris data. We assumed that i) the light curve amplitude is the
same in any of the photometric bands used and ii) the light curve can be
characterised by a simple sinusoidal variation. With these assumptions each
model light curve can be described by four parameters: light curve amplitude,
period, phase-shift, and an offset from the photometric zero point. We allowed
a different zero-point offset for each measurement block (data consisting of
measurements of consecutive nights) even if the data were taken by the same
instrument and filter combination due to the sometimes year-long gaps between
the measurement blocks. The best-fitting light curve period and amplitude is
characterised by the minimum in the C(P,$\Delta m$) function, obtained by the
residual minimalization, where P and $\Delta m$ are the period and amplitude
of the light curve, respectively (see Sect. A.5).

Figure 1: C(P,$\Delta m$) contour map; The most prominent minima is identified
at a period of $\sim$16 d, very close to the orbital period of Dysnomia, 15.78
d (see the text for details).

The possible light curve periods were chosen in the range P $\in$ [1,17] d.
The upper limit was set to sufficiently cover the synchronized period (15.8
d), while the lower limit was set to 1 d as we used ’nightly’ average values
in many cases. We restricted our amplitude search range to $\Delta m$ $\in$
[0,0.2] mag (peak-to-peak, i.e. twice the sine amplitude), as the original
$\Delta m$ $\in$ [0,0.6] mag range was found to be too wide and did not
provide minima in the large amplitude domain. Our results are presented in
Fig. 1. While the C(P,$\Delta m$) contour map shows several shallower minima,
there is one main minimum, at P = 16.2$\pm$0.5 d, and $\Delta m$ =
0.027$\pm$0.005, very close to the orbital period of Dysnomia (15.8 d). To
check the robustness of this result, and obtain the period and amplitude
uncertainties, we repeated the process by modifying the photometric data
points by adding a random value with normal distribution using the specific
measurement errors as standard deviations, and repeating the fitting process
several times for the whole data set.

###  2.3 Gaia

Gaia G-band photometry data of Eris (see Sect. A.4) was analysed using a
residual minimalization algorithm to identify the possible frequencies in the
light curve (see Fig. 2, upper panel). There is one strong minimum identified
at the long period part of the residual spectrum at 15.87$\pm$0.22 d (5.5
$\sigma$), very close to the 15.78 d orbital period of Dysnomia. The Gaia
light curve folded with this period (Fig. 2, bottom panel) was fitted with a
sinusoidal curve using a Levenberg-Marquardt fitter which provided a peak-to-
peak amplitude of $\Delta$m = 0.031$\pm$0.001 mag.

Figure 2: Upper panel: Normalized residual spectrum of the Eris Gaia light
curve. The red dashed line is at P = 15.78 d, the orbital period of Dysnomia;
Lower panel: Gaia light curve folded with f = 0.063 c/d. The solid curve is
the best-fit sinusoidal, with a peak-to-peak amplitude of $\Delta m$ =
0.031$\pm$0.001.

Both the ground based data (Sect. A.3) and the Gaia data strongly suggest that
the light curve period of the Eris-Dysnomia system is equal to the orbital
period of Dysnomia. As Dysnomia is significantly fainter than Eris in the
visible range (1:0.0021, see Brown & Schaller 2007) a light curve variation of
$\sim$0.03 mag has to be associated with Eris, and not with Dysnomia. The P =
15.78 d period rules out all shorter periods in the few hours – few days range
(see a detailed list in Sect. 1), as well as the semi-synchronized rotation
period of P = 14.56 d obtained by Holler et al. (2020). We, however, cannot
exclude that the rotation period is not exactly the orbital period, but very
close to it. Considering the ground-based and Gaia data the latter one has a
smaller uncertainty in the period determination, providing an uncertainty of
$\sim$5 h. As we show below, the system must be extremely fine-tuned to have
an actual rotation period so close to but different from the synchronized
case, assuming a simple binary system. Therefore we argue that the rotation of
Eris and Dysnomia is double-synchronized, i.e. the Eris-Dysnomia system is
fully tidally locked.

As shown in Sect. A.2 Eris appears to have a considerable variability in the
near-infrared J-H colour, while all measurements show rather similar colours
in the visible. One explanation for this behavior could be that the surface
composition of Eris is not homogeneous and parts of the surface are covered
with ices which have characteristic bands in the near-infrared, but have a
reflectance similar to other materials in the visible (Fernández-Valenzuela et
al. 2021). E.g. Eris is known to have strong methane features in its
reflectance spectrum (Alvarez-Candal et al. 2020), especially between 1.5-1.8
$\mu$m, close to the H-band. A variegation in surface composition may lead to
a rotational variation in the near-infrared (J-H) colour, while leaving the
visible range colours unchanged.

##  3 Possible tidal evolution scenarios

We used a simple tidal evolution model (see Sect. A.6) to find the possible
initial conditions and physical characteristics of Eris and Dysnomia that
could have led to the currently observed tidally locked rotation of Eris. Our
main assumption is that the Eris-Dysnomia system is formed in a giant
collision and started tidal evolution from a much more compact configuration,
Eris spinning significantly faster than today (Ragozzine & Brown 2009; Barr &
Schwamb 2016; Arakawa et al. 2019). A specific model is characterised mainly,
on one hand, by the properties of Eris relevant for tidal interactions,
including the tidal dissipation factor $Q_{p}$, the rigidity $\mu_{p}$ and/or
the second order tidal Love number $k_{2p}$; and by the mass $M_{s}$ and
effective radius $R_{s}$ (or, equivalently the density $\rho_{s}$) of the
satellite, Dysnomia. Previously Greenberg & Barnes (2008) studied the tidal
evolution of the Eris-Dysnomia system, but they restricted their calculations
to a specific Dysnomia radius of $R_{s}$ = 75 km, and mass of $M_{s}$ =
2.3$\cdot$1018 kg. In our model we considered a wide range of $R_{s}$ values
which are compatible with the brightness constraints and allows radius and
mass values as large as $R_{s}$ $\approx$ 370 km and $M_{s}$ $\approx$
5$\cdot$1020 kg (see Sect. A.6)

We first run our code for a large set of models which covered a wide range of
possible parameter values as shown in Fig. 3. For most of these cases the
final rotation periods of Eris, $P_{pf}$, remained below $P_{pf}$ $\leq$ 1 d,
but there is a well defined area in the $Q_{p}/k_{2p}$–$M_{s}$ plane where
$P_{pf}$ $>$ 1 d, or we reached synchronisation. This area is defined
approximately defined by the two dashed lines in Fig. 3 which roughly satisfy
a $Q_{p}M_{s}^{2}/k_{2p}$ = constant relationship, as expected from the
calculation of the spin rate change (Eq. 2). As we are interested especially
in those cases when the rotation of Eris slowed down considerably, we selected
starting parameters from the area defined above on the $Q_{p}/k_{2p}$ vs.
$M_{s}$ plot (Fig. 3) to further map the parameter space in a second set of
runs. The results of these runs are presented in Fig. 11.

Figure 3: $Q_{p}/k_{2p}$ versus the mass of Dysnomia $M_{s}$ in the first
trial runs. Orange symbols mark those cases which ended up in synchronized
Eris rotation; blue symbols represent the cases with P $<$ 1 d final Eris
rotation periods. Dashed lines mark the region of $M_{s}$–$Q_{p}/k_{2p}$
values that lead to slowed down rotation or tidal synchronization of Eris (see
text for details.)

Even in our simple model we have a rather wide range of parameters which lead
to synchronized or nearly-synchronized rotation rates for Eris. However, there
are some general conclusions which can be drawn from our simulations. First,
it is feasible that a massive Dysnomia can considerably slow down the rotation
of Eris, even forcing it to a synchronized rotation state. E.g. Eris rotation
periods of 10 d $\leq$ $P_{pf}$ $\leq$15.78 d can be reached for Dysnomia-to-
Eris mass ratios of 10-2 $\leq$ q $\leq$ 3$\cdot$10-2. To reach such a large
mass, Dysnomia has to be dark, with pVs $\lesssim$ 0.06, in contrast to the
very bright surface of Eris ($p_{Vp}$ = 0.96). While it is possible to
generate (near-)synchronous rotation for pVs $\lesssim$ 0.06, most of these
runs required pVs $\lesssim$ 0.04. Such a dark surface, and a corresponding
large size is also suggested by the submm detection of Dysnomia with ALMA
(Brown & Butler 2018), implying $R_{s}$ = 350$\pm$58 km and $p_{V}$ =
0.04${}^{+0.02}_{-0.01}$.

Although our simulations are run for a wide range of material/tidal parameters
assumed for Eris, mainly $Q_{p}$ and $\mu_{p}$, these parameters are
canonically chosen in a much narrower range. In Fig. 4 we selected those
simulation runs for which the tidal quality parameters of Eris was Q = 50, 100
or 200 ($\pm$10%) – $Q_{p}$ = 100 is the canonical value usually assumed in
the TNO tidal evolution calculations – and Eris’ rotation became tidally
locked. For $Q_{p}$ = 100 (red symbols in Fig. 4) synchronization is reached
for Dysnomia-to-Eris mass ratios of $q$ = 0.02-0.03, depending on the Eris
rigidity parameter $\mu_{p}$ which, for this $Q_{p}$ value, can be in the
range 4-20$\cdot$109 N m-2. These $\mu_{p}$ values correspond to the rigidity
of ice ($\sim$4$\cdot$109 N -2), or a mixture of ’ice and rock’, in the case
of the higher $\mu_{p}$ values (c.f. Grundy et al. 2011). To obtain mass
ratios of $q$ = 0.02-0.03 Dysnomia has to be large (D $\gtrsim$ 600 km) and
its density has to be in the range $\rho_{s}$ = 1.8-2.4g cm-3. These cases are
also associated with very low, $p_{V}$ = 0.02-0.03 geometric albedos. In the
case of a higher tidal quality parameter value, e.g. $Q_{p}$ = 200, even
higher mass ratios are required, and correspondingly the Dysnomia densities
are also higher, $\rho_{s}$ $\geq$ 2.0g cm-3. In these cases the allowed
rigidity of Eris is 3$\cdot$109 $\lesssim$ $\mu_{p}$ $\lesssim$ 8$\cdot$109 N
m-2. A lower $Q_{p}$, however, would allow smaller Dysnomia masses from q
$\approx$ 0.01, with a significant dependence on the rigidity as higher q is
required for higher rigidity values, up to $\mu_{p}$ = 3$\cdot$1010 N m-2. Due
to the lower required mass the density range allowed for Dysnomia is also
wider, $\rho_{s}$ = 1.2-2.4g cm-3. As indicated by the full range of
simulations (see Fig. 11) progressively smaller values of $Q_{p}$ will allow
smaller Dysnomia mass values to be compatible with a synchronized Eris
rotation. At $Q_{p}$ = 10, our smallest value chosen, a Dysnomia with a mass
ratio of q $\approx$ 0.006 and density of $\rho_{s}$ $\approx$ 1.0 g cm-3
would be massive enough to lock the rotation of Eris. Precise absolute
astrometry of the primary and secondary which could be performed with ALMA
(Brown & Butler 2019) may detect the barycentric wobble and obtain the mass
ratio, also putting constraints on the tidal quality factor $Q_{p}$.

Figure 4: Dysnomia-to-Eris mass ratio (q) versus the rigidity of Eris
($\mu_{p}$) in those simulation runs when Eris ended up in a synchronized
rotation. The orange, red and blue colours mark Eris tidal parameters of Qp =
50, 100 and 200 ($\pm$10%).

##  4 Conclusions

In this paper we analysed long-term ground based photometric observations of
Eris, complemented by measurements with the TESS and Gaia space telescopes.
While the TESS data did not provide a conclusive rotation period, both the
combined ground-based measurements and the Gaia data unambiguously point to a
light curve period that equals to the orbital period of Dysnomia, i.e. a
tidally locked rotation of Eris. The synchronized rotation of Eris – which is
considered to be the consequence of a tidal interaction with Eris’ moon
Dysnomia – puts constraints on the key physical properties of the satellite,
as well as on those of Eris, as discussed in Sect. A.6. While the light curve
or tidal evolution results does not directly constrain the shape, due to its
very low spin rate the shape of Eris is expected to be very close to
spherical, similar to that observed for Pluto and Charon (Nimmo et al. 2017).
For Eris, both a homogeneous density interior model with a Maclaurin shape, or
the Darwin-Radau model (see e.g. Murray & Dermott 2000) with a two-component,
rocky core and an ice mantle interior provides flattening values of $\epsilon$
$\leq$ 0.0001. This also means that the occultation shape and size solution
(Sicardy et al. 2011) has to be very close to the spherical one with R =
1163$\pm$6 km, an important constraint e.g. for thermal emission models. Also,
(Holler et al. 2021) suggested an oblate Eris as a possible cause of the non-
Keplerian orbit of Dysnomia which now seems to be a less likely option.

Another explanation of the non-Keplerian Dysnomia orbit could be a centre of
light – centre of body (CoL-CoB) offset due to a large albedo pattern on the
surface or Eris. Considering our best fitting $\Delta m$ = 0.031 mag amplitude
we investigated this scenario by assuming a single spot on the equator,
coincident with the orbital plane of Dysnomia, and a viewing geometry as
defined by the ’combined’ solution in (Holler et al. 2021). In our simple
model the spot is visible in a fraction of the rotational phases and
completely invisible in others; in these latter cases Eris has a homogeneous,
high albedo surface. We varied the size of the spot with the spot albedo in a
way that it always produces the required light curve amplitude, and we
considered both Lommel-Seeliger and Lambert scattering laws. With this light
curve amplitude the maximum CoL-CoB offset that could be obtained is $\sim$40
km both for the Lommel-Seeliger and the Lambert scattering and roughly similar
values are obtained for the whole range of albedos considered. This is much
smaller than the 462 km offset obtained by Holler et al. (2021). Considering
the maximum possible size of $\sim$600 km for Dysnomia, it is not feasible
that the dominant part of the CoL-CoB offset could be due to features on the
surface of the satellite, i.e. CoL-CoB offset is not a likely reason for the
non-Keplerian orbit. The non-Keplerian orbit of Dysnomia could also be caused
by a non-spherical shape of the satellite. A recent study of Kepler/K2 light
curves of transneptunian objects (Kecskeméthy et al. 2022) shows that light
curve amplitudes of TNOs remain larger at large (D $\gtrsim$ 400 km) sizes
where the asphericity of main belt asteroids drops significantly (Vernazza et
al. 2021). While this could be due to an irregular-to-spherical transition at
larger sizes for TNOs, their general low densities and high porosities point
against this scenario. At the expected sizes of Dysnomia (D $\gtrsim$ 600 km)
objects should be fairly round, even with higher densities and a considerable
internal strength.

Assuming that the Eris-Dysnomia system formed in a giant impact the rotation
period of the post-impact Eris was probably much shorter, on the order of a
few hours. This fast rotation had to be slowed down by the tidal interaction
with Dysnomia. As we have shown above, to reach synchronized rotation periods
Dysnomia has to be relatively massive (mass ratio of q = 0.02-0.03), assuming
canonical values for the $Q_{p}$ tidal dissipation factor and $\mu_{p}$
rigidity of Eris. This mass ratio is the second largest value in the
transneptunian region after the $\sim$8:1 ratio in the Pluto-Charon system
(see Barr & Schwamb 2016; Arakawa et al. 2019; Kiss et al. 2019, for earlier
evaluations). (We note that currently the mass ratio in the Orcus-Vanth system
is rather uncertain). The relatively high mass ratio is also associated with
high Dysnomia densities of $\rho_{s}$ = 1.8-2.4 g cm-3 which are much larger
than the typical densities of transneptunian objects in this size range,
$\rho$ = 0.5-1.0 g cm-3 (see e.g. Bierson & Nimmo 2019). In our tidal
evolution model unconventionally low $Q_{p}$ tidal dissipation factors would
allow lower Dysnomia densities (down to $\rho_{s}$ $\approx$ 1.2g cm-3) and
Eris reaching synchronized rotation at the same time, however, these values
are still above the typical low densities of R $\approx$ 300 km objects, and
would also require a low level of porosity. Collisional simulations have shown
that in general intact moons with 10-3 $\leq$ q $\leq$ 10-1 could form in the
transneptunian region assuming a wide range of impact parameters (Arakawa et
al. 2019). More detailed impact and tidal evolution simulations should be able
to identify the conditions which could lead to the present high density Eris –
high density Dysnomia system.

###### Acknowledgements.

The research leading to these results has received funding from the K-138962
grant of the National Research, Development and Innovation Office (NKFIH,
Hungary). The data presented in this paper were obtained from the Mikulski
Archive for Space Telescopes (MAST). STScI is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
Support for MAST for non-HST data is provided by the NASA Office of Space
Science via grant NNX09AF08G and by other grants and contracts. This research
has made use of data and services provided by the International Astronomical
Union’s Minor Planet Center. Part of the funding for GROND (both hardware as
well as personnel) was generously granted from the Leibniz-Prize to Prof. G.
Hasinger (DFG grant HA 1850/28-1). We are grateful to the CAHA and OSN staff.
This research is partially based on observations collected at the Centro
Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by Junta de
Andalucía and Consejo Superior de Investigaciones Científicas (IAA-CSIC). This
research was also partially based on observation carried out at the
Observatorio de Sierra Nevada (OSN) operated by Instituto de Astrofísica de
Andalucía (CSIC). P.S-S. acknowledges financial support by the Spanish grant
AYA-RTI2018-098657-J-I00 “LEO-SBNAF”(MCIU/AEI/FEDER, UE). P.S-S., J.L.O.,
N.M., and R.D. acknowledge financial support from the State Agency for
Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa”
award for the Instituto de Astrofísica de Andalucía (SEV-2017-0709), they also
acknowledge the financial support by the Spanish grants AYA-2017-84637-R and
PID2020-112789GB-I00, and the Proyectos de Excelencia de la Junta de Andalucía
2012-FQM1776 and PY20-01309. We are also thankful to our reviewer for the
useful comments. This work made use of Astropy:111http://www.astropy.org a
community-developed core Python package and an ecosystem of tools and
resources for astronomy (Astropy Collaboration et al. 2013, 2018, 2022).

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##  Appendix A Supporting Material

###  A.1 Observations and data reduction

####  A.1.1 TESS data

The Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2015) observed
Eris in Sector 30 with its Camera 1 and CCD 3 (Fig. 5). The reduction of the
TESS data was performed in the same way as described in Pál et al. (2020)
which contains a detailed description of the reduction steps, photometry, and
the derivation of the residual spectrum for frequency analysis. We only
mention those steps here that are different from those in the Pál et al.
(2020) pipeline. The TESS photometry data of Eris is provided in Table 1.

Figure 5: Eris’ path through the field-of-view of Sector 3 / Camera 1 / CCD 3
of the TESS space telescope between Julian dates 2459115.89 and 2459142.52.
The red and green parts mark those points that were excluded / considered for
light curve analysis. Table 1: TESS photometry data of $(136199)$ Eris
(sample).

Julian Date | $m$ | $\delta m$
---|---|---
| (mag) | (mag)
2459119.81621 | 18.99754 | 0.46354
2459119.85788 | 18.65064 | 0.33787
2459119.89954 | 18.69310 | 0.31456
… | … | …

Note. We are listing here the Julian date, brightness (m) and its uncertainty
($\delta$m) in the TESS photometric band. The table is available in its
entirety in electronically readable format. All data points have 60 min
integration time.

Figure 6: Top: Normalized residual spectrum of the TESS light curve of Eris.
The insert shows the residual spectrum of the pixel-wise x- (blue) and
y-direction (green) subpixel centroid positions. The most prominent
characteristic frequency of f = 0.411 c/d is marked by a red vertical dashed
line both in the main figure and the insert. Bottom: TESS light curve of Eris
folded with f = 0.411 c/d. The red dots mark the binned light curve. These
light curve data are presented in Table 1) in electronically readable format.
Figure 7: Fourier spectrum of the Eris TESS light curve. The blue curve is the
r.m.s. amplitude, calculated using a running box and sigma clipping.

A significant portion of the light curve data had to be excluded due to Eris’
encounter with nearby background sources which left a dominant feature in the
background-subtracted image. As shown in Fig. 5 the two ’green’ zones, where
the vicinity of Eris was relatively clean, covers two blocks with lengths of
2.3 d and 6.6 d. In these blocks readout-to-readout variation of the
background was estimated to be $\sim$3 mag lower then the typical ($\sim$18.5
mag in the TESS bandpass) brightness of Eris, in the same measuring aperture.
The residual spectrum obtained from the data of these two blocks merged is
shown in Fig. 6.

As TESS has large, 21″-sized pixels, the photometry of the source is affected
by the relative position of the source inside the actual pixel, and the
projection of the total source flux into the neighbouring pixels. This is
expected to introduce a periodic signal as the target moves through the field-
of-view, and the characteristic frequency depends on the actual apparent speed
of the target in the X and Y (pixel-wise) directions. To look for this effect
we checked the spectrum of the X and Y pixel fractions of centroid positions
of Eris’ TESS photometry. The results are presented in the insert in Fig. 6.
These residual spectra show well defined minima at fx = 1.73 c/d (X-direction,
blue curve in Fig. 6) and fx = 0.14 c/d (Y-direction, green curve). The
residual spectrum of the TESS photometry data (main figure, black curve) shows
a well-defined minimum at f = 0.411$\pm$0.018 c/d (P = 58.394$\pm$2.571 h).
The uncertainty of the frequency is obtained as the FWHM of Gaussian fitted to
the main frequency minimum in the residual spectrum. This frequency is
different from the fx and fy frequencies identified above, and could not be
associated with any currently known instrumental effects. The light curve
folded with this frequency is shown in Fig. 6. The peak-to-peak amplitude of
this folded light curve is $\Delta m$ = 0.132$\pm$0.037 mag, obtained the
maximum-minus-minimum of the binned light curve as amplitude, and using
standard error propagation in the calculation of the uncertainty.

We also derived the Fourier spectrum of the Eris TESS light curve (Fig. 7,
note the factor of two conversion between the Fourier and peak-to-peak
amplitudes). The r.m.s. Fourier amplitude (blue) curve shows that the 1
$\sigma$ noise is $\sigma_{f}$ $\approx$ 0.015 mag in the frequency range 1–12
c/d, and it increases considerably for lower frequencies, reaching $\sim$0.065
mag at the lowest ones. These r.m.s. amplitudes can be used to estimate light
curve detection upper limits ($\Delta m_{lim}$) for specific frequencies.
Considering the detection limit as 3 $\sigma_{f}$ we obtain $\Delta m_{lim}$ =
0.045 mag in the f = 1–12 c/d (P = 2–24 h) range, and $\Delta m_{lim}$ = 0.155
mag at f $\approx$ 0.4 c/d, i.e. at the frequency where a prominent peak was
identified in the residual spectrum, as discussed above. While a peak in the
Fourier spectrum can be identified at the same frequency (f = 0.411 c/d) as in
the residual spectrum, the signal-to-noise value is significantly lower here,
$\sim$1.8 $\sigma$. Due to this limitation we consider the f = 0.411 c/d (P =
58.394 h) peak as tentative. Due to the limited length of the TESS light curve
blocks considered in the analysis (2.3 d and 6.6 d) it was not possible to
detect light curve periods longer than $\sim$3 d, also excluding the
possibility to detect periods close to the orbital period (15.78 d). However,
the Fourier spectrum shows that we can _exclude_ rotation periods in the range
P = 2–24 h which would be associated with peak-to-peak amplitudes $\Delta m$
$\geq$ 0.09 mag, with a 3 $\sigma$ confidence, i.e. if Eris had a rotation
period in this range, it would have a very small amplitude, likely in the
order of 0.03 mag, or below.

###  A.2 GROND observations

Observations of Eris with the GROND instrument (Greiner et al. 2008) on the
MPG 2.2m telescope at La Silla were made in 3 nights, with details given in
Tab. 3. Observations consisted of 8m4td observation blocks, i.e. individual
120 s (115 s) exposures in JHK (g’r’i’z’) at each of the four telescope dither
(td) positions, except for the first epoch with 2 s (20 s) exposures each at 2
td-positions. Since the Eris motion on the sky is below
0$.\\!\\!^{\prime\prime}$2/10 min, the 4 different dither pointings were co-
added. GROND data were reduced in the standard manner (Krühler et al. 2008)
using pyraf/IRAF (Tody 1993; Krühler et al. 2008). The optical imaging data
($g^{\prime}r^{\prime}i^{\prime}z^{\prime}$) was calibrated against the Sloan
Digital Sky Survey (SDSS)222http://www.sdss.org catalogue (Eisenstein et al.
2011), and the NIR data ($JHK_{\rm s}$) against the 2MASS catalogue (Skrutskie
et al. 2006). This results in typical absolute accuracies of $\pm$0.03 mag in
$g^{\prime}r^{\prime}i^{\prime}z^{\prime}$ and $\pm$0.05 mag in $JHK_{\rm s}$.
Since the GROND dichroics were built after the Sloan filter system (Greiner et
al. 2008), the colour terms are very small, below 0.01 mag, except for the
$i^{\prime}$ band which is substantially narrower than the SDSS i’ band:
$i^{\prime}_{SDSS}-i^{\prime}_{GROND}=(-0.023\pm 0.010)+(0.216\pm
0.054)\cdot(i^{\prime}_{SDSS}-z^{\prime}_{SDSS})$
333https://www.mpe.mpg.de/$~$jcg/GROND/calibration.html. In order to minimize
the impact of variability in the comparison stars, relative photometry was
done against the same observation (2010-08-31T08:51-09:03). Since Eris moved
by about 1′ over the 3-day observing period, care was taken to select
comparison stars such that they were covered in all observations. GROND
photometry data are listed in Table 2. The GROND J and H colours in Table 2
and the J-H colour index in Table 5 are given in the AB photometric system. To
convert them to the Vega system we used the conversions $J_{Vega}=J_{AB}-0.91$
and $H_{Vega}=H_{AB}-1.38$ from (Blanton & Roweis 2007).

Table 2: Ground-based photometry data of $(136199)$ Eris (sample).

Telescope | Julian Date | m |  $\delta$m | Filt. |  texp |  ttot
---|---|---|---|---|---|---
|  | (mag) | (mag) |  | (min) | (min)
GROND | 2455436.91528 | 19.031 | 0.031 | $g^{\prime}$ | 8 | 8
GROND | 2455438.69348 | 19.097 | 0.010 | $g^{\prime}$ | 8 | 8
GROND | 2455438.75578 | 19.099 | 0.010 | $g^{\prime}$ | 8 | 8
…
1.5m | 2453647.53342 | 18.450 | 0.035 | R | 293.3 | 450.1
1.5m | 2453648.55623 | 18.459 | 0.025 | R | 353.3 | 393.0
1.5m | 2453649.56174 | 18.430 | 0.025 | R | 383.3 | 428.8
…

Note. The table is available in its entirety in electronically readable
format. The columns of the table are: Telescope name; mean Julian day of the
measurement; target brightness (m) and its uncertainty ($\delta$m) in the
respective filter band; filter; texp: sum of the individual exposure times
used for this photometric point (as explained in the text); ttot: difference
between the end of the last, and the start of the first measurement used to
obtain this specific photometric point.

We compared the colours from our GROND photometry with values from the
literature, as shown in Table 5. We converted the GROND g,r,i,z colours to
Johnson/Cousins BVRI, as described in Smith et al. (2002). Near-infrared
colours were converted from the respective systems used in the specific papers
to a common 2MASS system, as described in the table caption. The colours from
the new GROND measurements are in a relatively good agreement with the values
from other studies in the visible bands. However, in the near-infrared, the
J-H colours show large variations. J-H values range from -0.290$\pm$0.045 to
0.287$\pm$0.114, i.e. a $\sim$0.58 mag difference between the lowest and
highest colour values.

###  A.3 Other ground based observations

Table 3: Summary table of ground-based observations.

Telescope | Instrument | Date range | Number of nights | Filter | Reference
---|---|---|---|---|---
TESS | Camera 1 | 2020.09.27.-2020.10.16. | 19 | TESS Bandpass | (Ricker et al. 2015)
MPG 2.2m | GROND | 2010.08.28.-2010.08.31. | 3 |  $g^{\prime}r^{\prime}i^{\prime}z^{\prime}$JH* | (Greiner et al. 2008)
1.5m | Roper, Andor | 2005.10.03.-2020.10.16. | 18 | Johnson R, Clear | 444https://www.osn.iaa.csic.es/en/page/15-m-telescope
CA2.2 | CAFOS2.2 | 2007.01.11.-2007.01.16. | 6 | Johnson R | 555http://www.caha.es/CAHA/Instruments/CAFOS/
La Hita | SBIG STX-16803-3 | 2014.10.24.-2014.10.28. | 5 | Clear | 666https://fundacionastrohita.org/instrumental/

Note. We are listing here the telescope, instrument (camera), date range,
number of nights, and the filers used. *:GROND JH magnitudes are in the AB
system. These observations, taken between JD = 2453647.53342 and 2459139.5323,
covered heliocentric and observer distances of $r_{h}$ = 95.9-96.9 au,
$\Delta$ = 95.0-96.6 au, and phase angles $\alpha$ = 0.12-0.58 deg.

We obtained ground based photometry data of Eris from four telescopes (see
Table 3). All the data were reduced using standard calibration steps with the
FITSH (Pál 2012) software package, i.e. bias, dark and flat corrections were
applied. Then we performed aperture photometry on Eris and on selected
comparison stars. Using the comparison stars as standard stars we obtained
magnitudes from the Pan-STARRS DR2 catalogue and used these magnitudes to do
an approximate standard calibration of Eris via simple linear fitting the
observed and the catalogue magnitudes. In most cases only one filter was used,
so more sophisticated standard calibration was not possible. When it was
needed the Pan-STARRS magnitudes were converted to Johnsons R magnitude using
the method described in Smith et al. (2002). Typical observations consisted of
a few individual integrations per night, covering a few hours interval. As our
aim with these measurements was to look for long-term variations we produced
an average ’per-night’ photometric point from the individual integrations
which have considerably improved the signal-to-noise ratio, too. These ’per-
night’ photometry data are presented in Table 2. Data obtained with a specific
telescope/instrument have been divided into measurement blocks which typically
contain data of a few consecutive nights and are separated from the other
blocks by a longer period (up to a year). To avoid problems with absolute
calibration we allowed a different absolute brightness zero point offset for
each of these blocks in the subsequent light curve analysis. Due to the slow
apparent motion of Eris heliocentric and observer distance, and phase angle
corrections were applied only in those cases when the measurements covered a
longer period – this was the case for some of the literature data (e.g.
Rabinowitz et al. 2007), but not for our own measurements which have a typical
measurement block length of a few days. In the case of the GROND and the
Rabinowitz et al. (2007) data observations were performed by alternating
between the $g^{\prime}$–$r^{\prime}$ and the $V$–$B$ filters. For our light
curve period search we converted the $g^{\prime}$ data to the $r^{\prime}$
band using a mean $g^{\prime}-r^{\prime}$ colour, and the $B$ data to $V$ data
using a mean $B-V$ colour to increase the number of data points for these
measurement sequences. Assuming that the spin pole of Eris is coincident with
the orbital pole of Dysnomia, and using the pole solution by (Holler et al.
2021) we estimated that the aspect angle of Eris’ pole changes between
$\vartheta$ $\approx$ 129-133 deg, i.e. $\Delta\vartheta$ $\approx$ 4 deg
between the first and last date of the ground-based measurement sequence (see
Table 3). As the aspect angle is not at its extremes, this does not affect the
light curve amplitudes and the detectability of the light curves through
different data sets notably.

###  A.4 Gaia data

The Eris Gaia data is available in the third Gaia Data Release (Gaia
Collaboration et al. 2022), accessible in the Gaia Science
Archive777https://gea.esac.esa.int/archive/ through the
gaiadr3.sso_observation table. The table contains data obtained during the
transit of the source on a single CCD, during a single transit. More details
about the SSOs in the Gaia DR3 are discussed in Tanga et al. (2022).

Table 4: Gaia photometry data of $(136199)$ Eris

Julian Date | $m$ | $\delta m$
---|---|---
| (mag) | (mag)
2456900.12065 | -1.21999 | 0.01012
2456900.12071 | -1.21999 | 0.01012
2456900.12076 | -1.23254 | 0.01313
… | … | …

Note. The table lists the Julian date, brightness (m) and its uncertainty
($\delta$m) in the TESS photometric band. Brightness has been corrected for
heliocentric and observer distance, and phase angle.

Gaia G-band data of Eris was corrected for heliocentric and observer distance
and phase angle, using spacecraft-centric data obtained from the NASA Horizons
system (Giorgini et al. 1996). We applied a linear phase angle correction
using the heliocentric and observer distance corrected brightness values. We
used these reduced magnitudes, as provided in Table 4 for period search.

###  A.5 Period finding method

We used a residual minimalization method to find the best-fitting light curve
period and amplitude in our long-term ground based photometry data (Sect.
A.3). We chose an amplitude $\Delta m$ and period P, and determined the best
fitting light curve phase using a Levenberg–Marquardt minimization algorithm.
With these best-fit phase models we calculated the following C(P,$\Delta m$)
value for each P-$\Delta m$ pair:

$C(P,\Delta
m)=\sum_{i}{\frac{w_{i}}{N_{i}}}\sum_{j}\Bigg{(}\frac{m^{mod}_{ij}-m^{obs}_{ij}}{\delta
m_{ij}}\Bigg{)}^{2}$ (1)

where the index $j$ denotes the individual, night-averaged values, and $i$
denotes the measurement blocks; $m^{obs}$ and $m^{mod}$ are the measured and
model photometry values, respectively, $\delta m$ is the photometric
uncertainty, and $N_{i}$ is the number of individual data points in the
measurement block. $w_{i}$ are the weights of the individual measurements
blocks which have been chosen to be $\sqrt{N_{i}}$. We expect that the best-
fitting period-amplitude values provide the lowest C(P,$\Delta m$) values. We
searched the period range P $\in$ [1d, 17 d], where the upper limit is set to
cover the 15.8 d is the orbital period of Dysnomia (and it would correspond to
a synchronised rotation). The 1 d lower limit has to be set due to the ’per
night’ photometry points used in the case of most ground-based measurements.

To check the efficiency of our period/amplitude finding method, we generated a
synthetic sinusoidal light curve with a peak-to-peak amplitude of A = 0.040
mag. We sampled this light curve exactly at the same dates as our real data,
and divided these photometry points into the same blocks as the original ones,
as described in Sect. A.3. We used the mean photometry error in each
measurement block, as assigned a photometric uncertainty to each photometry
point in this specific block by assuming a random value with a normal
distribution with a standard deviation equal to the mean error. We generated a
large sample of synthetic light curves using random light curve phases and
different random error assignments, and run our period/amplitude finding
residual minimalization method. The results show that the period/amplitude can
be well recovered with our method, and the expected uncertainty is $\delta$P
$\approx$ 0.5 d in the period and $\delta$m $\approx$ 0.005 mag in the light
curve amplitude. An example of these $\chi^{2}$ results is shown in Fig. 8.

Figure 8: C(P,$\Delta m$) map obtained with the period-amplitude finding
algorithm using a synthetic signal with P = 15.78 d (the orbital period of
Dysnomia), as described in detail in the main text.

While the true period and amplitude of the original signal is correctly
identified within the uncertainties (dark blue region in Fig. 8) there are
other, shallower minima popping up at various frequencies, caused by aliasing.
E.g. one of these periods is at P = 1 d due to the single, combined photometry
points per night in the case of a number of measurement blocks.

The match of measurements and the best-fit light curve obtained by the
residual minimalization method is shown in Fig. 9 for several measurement
blocks.

Figure 9: Best matching sine functions (red curve, 15.78 d, the orbital period
of Dysnomia) fitted to the long term photometry data of Eris for some of
telescopes considered in our analysis. Time is defined with respect to the
start of the measurement block, and brightness is in the actual apparent or
reduced system, depending on the telescope and measurement block, as marked in
the bottom left corners and described in the text.  Table 5: Colours derived
from GROND photometry.

SDSS | g-r | g-i | r-i | g-z | J-H
---|---|---|---|---|---
photometry | (mag) | (mag) | (mag) | (mag) | (mag)
this work |  0.499$\pm$0.004 |  0.623$\pm$0.006 |  0.124$\pm$0.004 |  0.594 $\pm$ 0.013 |  -0.479$\pm$0.089
Johnson/Cousins | B-V | V-R | R-I | R-J | J-H
photometry | (mag) | (mag) | (mag) | (mag) | (mag)
this work |  0.709$\pm$0.040 |  0.358$\pm$0.030 |  0.359$\pm$0.02 |  0.64$\pm$0.04 |  0.024 $\pm$0.028
Carraro et al. (2006) |  0.823$\pm$0.023 |  0.391$\pm$0.023 |  0.386$\pm$0.012 | - | -
Fulchignoni et al. (2008) |  0.71$\pm$0.02 |  0.45$\pm$0.02 |  0.33$\pm$0.03*** | - |  -0.29$\pm$0.045*
DeMeo et al. (2009) | - | - | - | - |  0.054$\pm$0.070****
Perna et al. (2010) | - | - | - | - |  0.022$\pm$.070****
Snodgrass et al. (2010) |  0.78$\pm$0.01 |  0.45$\pm$0.03 |  0.33$\pm$0.02 |  0.52$\pm$0.02 |  0.287$\pm$0.114**
Tegler et al. (2016) |  0.75$\pm$0.02 |  0.43$\pm$0.02 | - | - | -
Alvarez-Candal et al. (2020) |  0.782$\pm$0.003 |  0.393$\pm$0.003 | - | - | -
Fernández-Valenzuela et al. (2021) |  0.74$\pm$0.06 |  0.39$\pm$0.05 |  0.38$\pm$0.08*** | - | -
Verbiscer et al. (2022) |  0.805$\pm$0.015 |  0.389$\pm$0.049 | - | - | -

Note. The first row shows the colours directly derived from GROND photometry
using the g, r, i, z, J and H bands in this work. Note that the J, H
magnitudes are int the AB system. The second raw lists the colours transformed
to the B, V, R, I, J, H bands, as described in Smith et al. (2002) and Blanton
& Roweis (2007). The additional rows represent the values obtained earlier in
other studies. *: From (Brown et al. 2005) **: Snodgrass et al. (2010)
converted to J-H from $J-H_{s}$ ***: Calculated from V-R and V-I., ****:
converted from UKIRT JH to 2MASS JH, using (Cutri et al. 2003)

###  A.6 Tidal evolution model

The satellites of the largest Kuiper belt objects are thought to be formed by
large collisions (Barr & Schwamb 2016) and remarkably all large Kuiper belt
objects with diameters D $\gtrsim$ 1000 km have confirmed satellites (Kiss et
al. 2017). The tidal evolution of these systems depends on the size, mass,
formation distance, and material properties of the bodies. Tidal evolution has
certainly led into a double-synchronous state in the case of the Pluto-Charon
system (Dobrovolskis et al. 1997), but in the Haumea system even the larger
satellite Hi’iaka could not reach synchronous rotation (Hastings et al. 2016).
Although the spin period of Eris seems to be rather well defined now by this
present work, it is an interesting question whether the possible rotation
periods of Eris – ranging from a few hours to the orbit-synchronous state –
are feasible in terms of tidal evolution, using the current knowledge on the
system components. This is especially interesting after the likely detection
of Dysnomia in the ALMA 870 $\mu$m data which suggests that Dysnomia could be
a massive satellite with a diameter of $\sim$700 km (Brown & Butler 2018).

We used the simple tidal evolution model by (Murray & Dermott 2000), also used
by Hastings et al. (2016), to calculate the evolution of the satellite orbit
(only the semi-major axis in this approximation) and the spin evolution of
Eris and Dysnomia. In this model the satellite orbit and the equator of Eris
are assumed to be co-planar. Some of the main characteristics of the system
originate from the orbit of Dysnomia (Holler et al. 2021), as it defines the
current semi-major axis of the satellite orbit, $a_{f}$, the orbital period,
$P_{orb}$, and the system mass, $M_{sys}$. Eris is expected to be nearly
spherical and the radius and V-band geometric albedo of Eris, $R_{p}$ =
1163$\pm$6 km and $p_{Vp}$ = 0.96 are known quite precisely from a stellar
occultation (Sicardy et al. 2011). The rate of change of the spin frequency
depends on the ratio of the second-order tidal Love number and the tidal
quality factor, $k_{2p}/Q_{p}$, the mass and size of Eris, $M_{p}$ and
$R_{p}$, the actual semi major axis of the satellite orbit $a$, and the mass
of Dysnomia, $M_{s}$:

$\dot{\omega_{p}}=-sign(\omega_{p}-n)\frac{15}{4}\frac{k_{2p}}{Q_{p}}\frac{M_{s}^{2}}{M_{p}}\bigg{(}\frac{R_{p}}{a}\bigg{)}^{3}\frac{G}{a^{3}}$
(2)

The $Q_{p}$ tidal quality factor was chosen in the range of 10–1000 (Goldreich
& Soter 1966; Murray & Dermott 2000) allowing an order-of-magnitude variation
around the canonical Q = 100. The second order tidal Love number k2 is
calculated from the rigidity $\mu$ following Hastings et al. (2016). The
$\mu_{p}$ rigidity of Eris was also allowed to vary in a wide range from 109
to 1011 N m-2 which should be sufficiently wide to cover typical values from
icy to rocky interiors (Murray & Dermott 2000).

The evolution of the semimajor axis can be expressed by the equation below,
following (Hastings et al. 2016):

$a(t)=({a_{f}}-{a_{0}})(t/T)^{2/13}+{a_{0}}$ (3)

where T is the age of the Solar System, ${a_{0}}$ is the initial semimajor
axis and ${a_{f}}$ is the present semimajor axis.

The brightness ratio of Eris to Dysnomia in the F606W band of the HST is
0.0021 (Brown & Schaller 2007) which defines the size of Dysnomia, $R_{s}$,
for a specific Dysnomia geometric albedo chosen. Among trans-Neptunian objects
and satellites a very wide range of albedos are possible from extremely dark
surfaces to very bright ones. While Dysnomia is probably large and dark (Brown
& Butler 2018) we choose geometric albedo in the range $p_{Vs}$ = 0.02–0.8 for
Dysnomia for our model calculations. Trans-Neptunian objects with diameters
below $\sim$500 km are expected to have high porosity and low bulk density
(Grundy et al. 2019). As a reasonable range we first assumed $\rho_{s}$ =
0.5–2.4 g cm-3, which, together with $R_{s}$ obtained above, defines the mass
of Dysnomia, $M_{s}$. The lower limit is the typical density of objects in the
few hundred km size in the Kuiper belt (see e.g. Grundy et al. 2019). The
upper limit in density, 2.4g cm-3 is the bulk density of Eris, and in extreme
cases Dysnomia might have a similar density (as argued e.g. in Holler et al.
2021). The mass of Eris is obtained as $M_{p}$ = $M_{sys}-M_{s}$, and we use
the mass ratio $q$ = $M_{s}/M_{p}$ to characterise the system in this sense.

Figure 10: Top panel: Demonstration of the sensitivity of the model to the
$Q_{p}$ parameter. Each dot in this plot corresponds to the $Q_{p}$ and the
final Eris rotation period $P_{pf}$ of a specific run. All other model
parameters were kept the same. Bottom panel: Evolution of the rotation period
of Eris, using the models presented in the upper panel. The colours of the
curves correspond to the colour of the symbols on the upper panel, the red
curve is the evolution of the orbital period.

In addition to the system parameters described above, following (Canup 2005)
the initial semi-major axis of the satellite orbit is assumed to be $a_{s}$ =
1.2$\cdot a_{R}$, and $a_{R}$ = $2.456\cdot
R_{p}\cdot(\rho_{p}/\rho_{s})^{1/3}$ is the Roche limit. The initial spin
period of Eris is set by the breakup limit – $\omega_{c}$ =
$(GM_{p}/R_{p}^{3})^{1/2}$ = $(4\pi G\rho_{p}/3)^{1/2}$ – and we allowed spin
periods [1–2] $\cdot\omega_{c}$.

We run the tidal evolution model for a large number of cases assuming
independent, random values of $k2_{p}$, $Q_{p}$, $\rho_{s}$ and $p_{Vs}$
chosen in the intervals described above. We also assume that once Eris has
reached synchronous rotation in a simulation run it remains in this state and
the rotation period just changes with the changing semi-major axis and orbital
period.

In Fig. 10 we demonstrate the sensitivity of the model to the $Q_{p}$ tidal
parameter. The model runs presented in this figure led to very different final
Eris rotation periods from $P_{pf}$ $\approx$ 1 d to synchronized rotation
when all starting parameters were kept the same except $Q_{p}$ which was
varied in the range 13.5 $\leq$ $Q_{p}$ $\leq$ 17.5. A very similar
sensitivity is seen for the $k_{2p}$ parameter (see Eq. 2), however, this
parameters is calculated from the $\mu$ or $\mu_{eff}$ values.

We also present in Fig. 11 scatter plots of the key parameters of the tidal
evolution model (mass ratio $q$; $k_{2p}$ and $Q_{p}$ tidal parameters,
$\mu_{p}$ rigidity and $\omega_{0}$ starting angular speed of Eris; density
$\rho_{s}$, albedo $p_{Vs}$ and radius $R_{s}$ of Dysnomia) which resulted in
Ppf $>$ 1 d final Eris rotation periods. The different colours correspond to
ranges of different Ppf values or a final synchronized state.

Figure 11: Scatter plots of model parameters for the selected $Q_{p}$–$M_{s}$
range cases. The colours correspond to ranges of the final rotation period of
Eris ($P_{pf}$), as indicated in the legend boxes.

authors:
R. Szak\'ats, Cs. Kiss, J. L. Ortiz, N. Morales, A. P\'al, T. G.
  M\"uller, J. Greiner, P. Santos-Sanz, G. Marton, R. Duffard, P. S\'agi, E.
  Forg\'acs-Dajka

submitter: R\'obert Szak\'ats