Constraints on Ceres’ internal structure and evolution from its shape and gravity measured by the Dawn spacecraft

Constraints on Ceres’ internal structure and evolution
from its shape and gravity measured by the Dawn
spacecraft
A. I. Ermakov
1,2
, R. R. Fu
1,3
, J. C. Castillo-Rogez
2
, C. A. Raymond
2
, R. S.
Park
2
, F. Preusker
4
, C. T. Russell
5
, D. E. Smith
1,6
, M. T. Zuber
1
A. I. Ermakov, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA,
91109, USA. (eai@caltech.edu)
1
Department of Earth, Atmospheric and
Planetary Science, Massachusetts Institute
of Technology, Cambridge, MA, 02139, USA
2
NASA/Jet Propulsion Laboratory,
California Institute of Technology,
Pasadena, CA, 91109, USA
3
Lamont-Doherty Earth Observatory,
Earth Institute, Columbia University,
Palisades, NY, 10964, USA
This article has been accepted for publication and undergone full peer review but has not been through
the copyediting, typesetting, pagination and proofreading process, which may lead to diﬀerences
between this version and the Version of Record. Please cite this article as doi: 10.1002/2013GL057474
©2017 American Geophysical Union. All Rights Reserved.

Abstract. Ceres is the largest body in the asteroid belt with a radius of
approximately 470 km. In part due to its large mass, Ceres more closely ap-
proaches hydrostatic equilibrium than major asteroids. Pre-Dawn mission
shape observations of Ceres revealed a shape consistent with a hydrostatic
ellipsoid of revolution. The Dawn spacecraft Framing Camera has been imag-
ing Ceres since March 2015, which has led to high-resolution shape models
of the dwarf planet, while the gravity field has been globally determined to
a spherical harmonic degree 14 (equivalent to a spatial wavelength of 211 km)
and locally to 18 (a wavelength of 164 km). We use these shape and grav-
ity models to constrain Ceres’ internal structure. We find a negative corre-
lation and admittance between topography and gravity at degree 2 and or-
der 2. Low admittances between spherical harmonic degrees 3 and 16 are well
explained by Airy isostatic compensation mechanism. Different models of isostasy
give crustal densities between 1200 and 1400 kg/m3
with our preferred model
4
German Aerospace Center, Institute of
Planetary Research, D-12489 Berlin,
Germany
5
University of California Los Angeles,
IGPP/EPSS, Los Angeles, CA, 90095, USA
6
NASA/Goddard Space Flight Center,
Greenbelt, MD, 20771, USA

giving a crustal density of 1287−87
+70 kg/m3
. The mantle density is constrained
to be 2434−8
+5 kg/m3
. We compute isostatic gravity anomaly and ﬁnd evi-
dence for mascon-like structures in the two biggest basins. The topographic
power spectrum of Ceres and its latitude dependence suggest that viscous
relaxation occurred at the long wavelengths (>246 km). Our density con-
straints combined with ﬁnite element modeling of viscous relaxation suggests
that the rheology and density of the shallow surface are most consistent with
a rock, ice, salt and clathrate mixture.

1. Introduction
Ceres possesses both asteroid- and planet-like properties. It is suﬃciently large to
attain a shape close to hydrostatic equilibrium similar to planets. However, its surface
is uniformly heavily cratered more resembling asteroidal surfaces. Ceres’ mean density
of 2161 kg/m3
[Park et al., 2016] and its location in the Solar System (≈2.8 AU from
the Sun) indicate that it consists of a combination of silicates and volatiles. McCord
and Sotin [2005] estimate that Ceres has from 17% to 27% of non-mineralogically bound
water by mass. In this respect, Ceres is similar to icy satellites or even Trans-Neptunian
Objects (TNOs, see McKinnon 2008). Important factors in the evolution of Ceres include
[Castillo-Rogez and McCord, 2010; McCord et al., 2011; Neveu and Desch, 2015]:
1. abundance of water in its interior, which controls internal dynamics and energy
transport;
2. time of accretion, which determines the abundance of short-lived radioactive nuclides
(26
Al, 60
Fe) constituting the dominant heat source during the early epoch of evolution;
3. amount of long-lived radioactive elements (40
K,232
Th,235
U,238
U) that provide a con-
tinuous heat source throughout all of Ceres evolution;
4. surface temperature is an important boundary condition;
5. material thermophysical properties, speciﬁcally, whether the water is in the form of
ice or hydrates makes an impact on thermal conductivity and, therefore, on thermal state
evolution.
In this paper, we address constraints on the internal structure and composition of Ceres
from shape and gravity measurements from the Dawn mission. Two end-member models

of Ceres’ internal structure have been proposed. In the model proposed by McCord and
Sotin [2005] and elaborated on in Castillo-Rogez and McCord [2010], Ceres accreted as a
mixture of ice and rock just a few My after the condensation of Calcium Aluminum-rich
Inclusions (CAIs), and later differentiated into a water mantle and a mostly hydrated
silicate core. The possibility of an inner iron core was excluded based on the pre-Dawn
shape measurements [Thomas et al., 2005]. In the second model presented in Zolotov
[2009], Ceres formed relatively late from planetesimals consisting of chondritic materials.
This model suggests that Ceres could have undergone little internal evolution and is today
made up of porous hydrated silicates and has no free volatiles. However, Castillo-Rogez
[2011] and Neumann et al. [2015] argue that Zolotov [2009] is not correct to assume no
volatile is freed during thermal evolution. These two end-member models predict different
degrees of physical differentiation: the former being more differentiated and the latter
more homogeneous.
The two models also predict different surface morphologies. In the first model, viscous
relaxation is expected to be important in the outer icy crust whereas little viscous re-
laxation is expected for the second model. Bland [2013] predicts that craters as small
as 4 km in diameter should be substantially relaxed in the first model and, in addition,
suggests that there should be a strong latitude-dependent variation of crater morphology
due to the change in surface temperature.
A number of techniques had been used prior to Dawn to determine Ceres’ radius and
shape (see Section 3). Ground and space-based telescopic observations revealed that
Ceres has a shape consistent with hydrostatic equilibrium to the measurement accuracy
(typically order of 10 km) [Thomas et al., 2005; Carry et al., 2008; Drummond et al.,

2014]. Images from the Dawn spacecraft Framing Camera (FC) suggest that there is a
substantial non-hydrostatic component in the shape of Ceres. Equatorial axes of the body
differ by ≈2km [Park et al., 2016].
Gravity and shape data provide constraints on the internal structure and the degree of
physical differentiation. Internal structure models can be parameterized in a number of
ways. For this paper, we chose a two-layer model with uniform density layers, which is
the most parsimonious model of a differentiated Ceres than can explain the data. Such
a model has only five geophysical parameters: two densities, two radii and the rotation
rate of the body. While some previous work (e.g., Castillo-Rogez and McCord [2010])
has invoked more complicated internal structure models, we stress the inherent problem
of solution non-uniqueness when solving for free parameters given constraints only from
gravity and shape data, particularly when only low degree data is available.
This paper is organized as follows: we describe the methods of working with spherical
harmonic expansions of gravity and topography in Section 2; the available data is pre-
sented in Section 3; results are presented in Section 4; we discuss the implications of our
results in Section 5 and make conclusions in Section 6.
2. Methods
2.1. Spherical harmonic analysis
We expand Ceres’ shape model in a spherical harmonic series [Wieczorek, 2015]:
r(φ, λ) = R 1 +
∞
n=1
n
m=0
¯Anm cos(mλ) + ¯Bnm sin(mλ) Pnm(sin φ) , (1)

where R is the mean radius, ¯Anm, ¯Bnm are normalized coefficients of the spherical
harmonic expansion, Pnm are normalized associated Legendre functions, λ is longitude, φ
is latitude; and n and m are spherical harmonic degree and order, respectively.
The gravitational potential is similarly expanded in a spherical harmonic series [Wiec-
zorek, 2015]:
U(r, φ, λ) =
GM
r
1 +
∞
n=2
n
m=0
R0
r
n
Cnm cos(mλ) + Snmsin(mλ) Pnm(sin φ) , (2)
where R0 km is the reference radius of the gravity model (R0 = 470 km for Ceres),
{ ¯Cnm, ¯Snm} are normalized spherical harmonic coefficients of the gravity model. The zonal
coefficients ¯Jn are given as follows: ¯Jn = − ¯Cn0. The variance spectrum of topography V tt
n
is given by Bills et al. [2014]:
V tt
n = R2
n
m=0
( ¯A2
nm + ¯B2
nm).
Correspondingly, we can find the variance spectrum of gravity:
V gg
n =
n
m=0
( ¯C2
nm + ¯S2
nm),
To compare the gravity and shape datasets, we can find cross-variance spectra ,
V gt
n = R
n
m=0
( ¯Anm
¯Cnm + ¯Bnm
¯Snm),
The power spectral density (PSD) is given by dividing variance (or cross-variance) by the
number of coefficients for a given degree n:

Stt
n =
V tt
n
2n + 1
,
Sgg
n =
V gg
n
2n + 1
,
Sgt
n =
V gt
n
2n + 1
.
We will later use ¯αinm and ¯σinm, where i is either 1 or 2, to refer to the normalized shape
and gravity coefficients, respectively:
¯σ1nm = ¯Cnm, ¯σ2nm = ¯Snm;
¯α1nm = ¯Anm, ¯α2nm = ¯Bnm.
Admittance (Z) and correlation (R) are spectral characteristics that allow cross-analysis
gravity and shape data. The admittance estimate Zn is a ratio of gravity-shape cross-
power to the power of shape. This definition of the admittance estimate is appropri-
ate for the case when noise in the gravity data dominates noise in the topography data
[McKenzie, 1994]. Additionally, admittance analysis implicitly assumes that the gravity-
topography relationship is linear and requires that the deviations from linearity be uncor-
related with topography. In order to give admittance units of mGal/km, we multiply the
ratio byGM
R3 (n + 1):
Zn =
Sgt
n
Stt
n
·
GM
R3
(n + 1) (3)
The correlation coefficient is given by:
Rn =
Sgt
n
Stt
n Sgg
n
(4)

It can be seen that Zn can be expressed as GM
R3 (n + 1)Rn Sgg
n /Stt
n .
Computations related to spherical harmonics were performed using SHTOOLS software
package [Wieczorek et al., 2016] and Frederik Simons’ Matlab library [Simons et al., 2016].
2.2. Computing hydrostatic equilibrium
We use a numerical approach proposed by Tricarico [2014], which gives a solution for
a multilayer body in hydrostatic equilibrium by approximating the equipotential surfaces
with ellipsoids. In this approach, the equilibrium shape is found by minimizing the squared
differences of the total gravitational potential Utot(x, y, z) = Uell(x, y, z)+Urot(x, y) at the
outer surface and at the inner interfaces. Here Uell is the gravitational potential of an
ellipsoid and Urot(x, y) = 1
2
(x2
+y2
)ω2
is the rotational potential. The sum of the squared
potential differences ∆2
is given by:
∆2
=
N
k=1
{[Utot(ak, 0, 0) − Utot(0, bk, 0)]2
+
[U(ak, 0, 0) − Utot(0, 0, ck)]2
},
where N is the number of layers and ak, bk, ck are the longest, intermediate and shortest
ellipsoidal axes, respectively. For the case of Ceres, the equilibrium shape is rotationally
symmetric: bk = ck. In the case of a two-layer model, we fix rk, ρk and rotation period
T and treat fp,k = (ak − ck)/ak, where k is either 1 for the outer layer or 2 for inner
layer, as a free variable. Therefore, we can minimize ∆2
is a function of fp,1 and fp,2 for
a given set of r1, r2, ρ1, ρ2 and T. Given rk and fp,k , semiaxes can found in the following
way: ak = [rk/ (1 − fp,k)]1/3
and ck = r
1/3
k (1 − fp,k)2/3
. Normalized gravity coefficients of
individual layers can be subsequently found:

¯σ1,2,0,k = − ¯J2,k =
c2
k − a2
k
5
√
5R2
0
¯σ1,4,0,k = − ¯J4,k =
25
7
¯J2
2,k
Higher even degrees (n =6, 8, 10, ...) can be also found but non-hydrostatic signals
become dominant at those higher degrees. To find the hydrostatic normalized gravity
coefficients of the entire body, the coefficients of individual layers need to be weighted by
the fractional masses of the layers:
¯σhydrostatic equilibrium
i,n,m =
1
M
4
3
πρ1r3
1 ¯σi,n,m,1 +
4
3
π(ρ2 − ρ1)r3
2 ¯σi,n,m,2 (5)
Alternatively, classical methods of the theory of figures [Zharkov and Trubitsyn, 1978;
Moritz, 1990a] could be used. Rambaux et al. [2015] shows that integrating Clairaut’s
equation expanded up to third order in a small parameter is needed to accurately compute
hydrostatic shape and gravity of Ceres.
Fig. 1 shows the polar flattening factor fp,1 = (a1 − c1)/a1 of Ceres assuming homoge-
neous interior as a function of rotation period. Due to Ceres’ fast rotation, it is necessary
to take into account high-order effects when computing the body’s equilibrium shape.
The accuracy of the first order theory of figure methods such as Dermott [1979] is not
sufficient for rapid rotators such as Ceres or Vesta.
Even so, Ceres rotates sufficiently slowly such that it is in the Maclaurin regime, i.e.,
its equilibrium shape is a spheroid of revolution. In this case, we should note that the
shape of an equipotential surface can be an ellipsoid only for a homogenous body. This
fact is known as the Hamy-Pizzetty theorem: Hamy [1889]; Moritz [1990b]; Pohánka

[2011]; Rambaux et al. [2015]. However, we find that the accuracy of the Tricarico [2014]
approximation using ellipsoids of revolution to represent equipotential surfaces is on the
order of 10 m for a Ceres-like multilayer body. This estimate is found by evaluating
the potential at ellipsoidal interfaces and converting it to equipotential surface heights.
In other words, the exact equipotential surface is within ≈10 meters of the ellipsoidal
interface.
2.3. Gravity from shape
We compute gravity-from-shape spherical harmonic coefficients ¯σshape
inm as detailed by
Wieczorek and Phillips [1998], where gravitational coefficients are expanded in a series of
powers of shape:
¯σshape
inm =
3
2n + 1
·
R
Rvol
3
·
R
R0
n
·
n+3
h=1
h
¯αinm
Rhh!
·
h
j=1(n + 4 − j)
(n + 3)
, (6)
where h
¯αinm are the spherical harmonic coefficients of shape raised to the power h, R is
the mean radius of the shape, R0 = 470 km is the radius that the gravity coefficients are
referenced to and Rvol is the radius of a volume-equivalent sphere. The factor (R/R0)n
is
used to normalize the gravity field to the desired reference radius. The factor (R/Rvol)3
is typically small and can be ignored for nearly spherical bodies but could be significant
for irregularly shaped objects. For Ceres, Rvol = 469.7 km and R = 469.5 km. The first
term in this expansion corresponds to the mass sheet approximation. We find that due to
Ceres’ significant non-sphericity four terms (i.e. powers of topography h, max(h) = 4 in
Eq. 6) need to be retained in Equation 6 to compute gravity from shape with an accuracy
matching the accuracy of the observed gravity (Fig. 2).

3. Data
3.1. Shape model
Size and shape determinations of Ceres have been documented previously in the lit-
erature beginning with filar micrometer measurements by Barnard [1895]. Later, lunar
occultations [Dunham et al., 1974], polarimetry [Morrison and Zellner, 1979], radio [John-
ston et al., 1982], infrared imaging [Brown et al., 1982; Lebofsky et al., 1984], stellar oc-
cultations [Millis et al., 1987], adaptive optics systems [Saint-Pe et al., 1993; Drummond
et al., 1998; Carry et al., 2008; Drummond et al., 2014], radar [Mitchell et al., 1996], and
Hubble Space Telescope observations [Parker et al., 2002; Thomas et al., 2005] were used
to determine Ceres’ size and shape. Some of the methods were sensitive only to Ceres’
radius, others were able to constrain the body’s shape, typically approximating it with an
ellipsoid of revolution, with the exception of Drummond et al. [1998], who attempted to
solve for a triaxial ellipsoid. Table 1 summarizes pre-Dawn Ceres’ shape determinations
in chronological order.
Observations prior to the Dawn mission indicated that Ceres’ shape is oblate and con-
sistent with a hydrostatically-relaxed body. However, the accuracy of the pre-Dawn shape
determinations did not allow a robust estimation of non-hydrostatic effects such as tri-
axiality. Nevertheless, the determined rotationally-symmetric shapes in combination with
the hydrostatic equilibrium assumption have been cited as evidence for differentiation of
Ceres’ interior [Thomas et al., 2005; Carry et al., 2008]. In all pre-Dawn shape determi-
nations the formal error of the ellipsoidal fits was likely dominated by the noise in the
measurements as opposed to non-hydrostatic effects in the shape.

Since the arrival of the Dawn spacecraft at Ceres in May 2015, images from the
Framing Camera were used to construct Ceres shape model using two techniques. The
first technique is stereophotogrammetry (SPG) used by the German Aerospace Center
(DLR) [Preusker et al., 2015, 2016]. The second technique is stereophotoclinometry
(SPC) used by NASA’s Jet Propulsion Laboratory (JPL) [Park et al., 2016]. Ellip-
soidal fits to the shape SPC shape model reveal that Ceres shape can be best fit by
a 483.1 ± 0.2 × 481.0 ± 0.2 × 445.9 ± 0.2 km ellipsoid [Park et al., 2016]. The SPG shape
model can be best fit by a 483.2 ± 0.2 × 481.0 ± 0.2 × 446.0 ± 0.2 km ellipsoid.
The images acquired by the Dawn spacecraft enable Ceres’ shape determination to
much higher accuracy and precision. The residuals of the ellipsoidal fits are dominated
by actual non-hydrostatic topography rather than measurement noise. Dawn data reveal
that Ceres’ shape is significantly triaxial. The difference between the two equatorial axes
a and b is ≈2 km, which corresponds to an equatorial flattening factor fq = (a − b)/a of
0.0045. The equatorial flattening factor fq is a proxy for the body’s non-hydrostaticity as
a body in hydrostatic equilibrium would have fq = 0 at Ceres’ rotation rate. It can be seen
from Fig 1 that Ceres is less oblate than expected for a homogeneous body. However,
Ceres deviates from homogeneity by the same order as it deviates from hydrostaticity.
Therefore, non-hydrostatic effects need to be taken into account when interpreting Ceres’
shape and gravity field.
Ceres also possesses a moderate center-of-mass–center-of-figure (COM–COF) offset. We
define the COM–COF offset as the vector between the body’s center of mass and the
center of mass of the body assuming it has uniform density. The COM–COF offset can
be loosely used a a metric for planet’s general deviation from spherical symmetry with

regard to interior structure. The magnitude of the offset is ≈1.0 km or 0.2% of Ceres’
radius [Park et al., 2016]. This value is substantially larger than the COM–COF offsets
for the terrestrial planets but smaller than for Vesta and Eros (Table 2). This observation
again demonstrates the intermediate nature of dwarf planet Ceres. A global map of Ceres
topographic heights computed with respect to the best-fit ellipsoid with two degrees of
freedom (polar and equatorial axes) confirms its tri-axiality, which is evident as a residual
degree 2 signature in this map with two equatorial long-wavelength topographic highs at
longitudes of 40◦
and -140◦
(Hanami Planum), and two long-wavelength topographic lows
at longitudes of 130◦
and -50◦
.(Fig. 3).
The coefficients of the spherical harmonic expansion of the SPG HAMO shape are
available in the supporting information.
3.2. Gravity field model
We use a gravity field model derived from Dawn’s radio-tracking data from the approach
phase to Low Altitude Mapping Phase (LAMO) phase of the mission. At the time of this
writing, a degree 18 gravity model is available [Konopliv et al., 2017]. The model is
accurate up to degree and order 14-18 and includes data from Feb 2, 2015 to Sep 02, 2016
that is from the Approach phase to the Low Altitude Mapping Orbit (LAMO) phase. The
gravity model has a higher accuracy (≈10 mGal if expanded up to n = 16) and resolution
(n = 18) near the equator and worse accuracy (≈40 mGal if expanded up to n = 16) and
resolution near the poles (n = 14). The degree 2 gravity coefficients are shown in Table
3.
The major non-hydrostatic term in Ceres’ gravity is the sectorial term at degree 2.
Comparing the magnitude of the sectorial term ( ¯C2
22 + ¯S2
22) to that of the zonal degree

2 term ( ¯C20) reveals that Ceres’ gravity is non-hydrostatic at a 3% level at degree 2. On the
other hand, if we compute the same ratio for the shape, we get | ¯A2
22 + ¯B2
22/ ¯A20 |= 5%.
Therefore, Ceres’ shape is more non-hydrostatic than its gravity field, which indicates
that degree 2 topography could be at least partially compensated.
4. Results
4.1. Shape harmonic analysis
4.1.1. Comparison with terrestrial planets and Vesta
Fig. 4 shows the power spectrum of Ceres’ topography compared to those of the terres-
trial planets and asteroid 4 Vesta. Vesta is perhaps the most useful body to be compared
against Ceres due to its location in the asteroid belt, which suggests a similar impact his-
tory [O’Brien and Sykes, 2011; Marchi et al., 2016; Fu et al., 2015, 2017]. The topographic
power spectra of Vesta and Ceres have very similar slope but the spectrum of Ceres is
approximately one order of magnitude lower than that of Vesta. The surface gravity is
almost identical (0.2 - 0.3 m/s2
) and therefore, all other conditions being equal, Vesta and
Ceres are expected to have similar topographic powers. The difference in the topography
power is likely due to different target properties. Ceres near-surface material appears to
be a weaker target than Vesta. We also note that the Vesta spectrum at the lowest degrees
displays substantial variability, which is likely due to the effect of the two giant impact
basins Rheasilvia and Veneneia. The spectra shown in Fig. 4 are integral characteristics;
they contain information from all acting physical processes that can affect bodies’ shape.
For example, the spikes at degree 2 (second data point in all spectra indicated by yellow
circles) mostly represent the hydrostatic response to rotation. In subsequent figures we
remove this hydrostatic signal from the spectra. The power spectra for Ceres and Vesta

curve down at short wavelength (λ < 5 km) as they reach the resolution limit of the shape
reconstructing process.
4.1.2. Viscous relaxation in the spectral domain
Bland [2013] argues that if Ceres contains an outer water ice layer, craters as small as 4
km should be significantly relaxed at the equatorial regions. At mid-latitudes, all craters
older than 10 Ma and larger than 16 km should be completely relaxed. On the other hand,
if Ceres is a rocky body, Bland [2013] predicts that crater relaxation should be negligible.
The topographic power of Ceres behaves as a power law (Fig 4). However, we observe
a decrease of power at low degrees with respect to the power law. In order to assess the
statistical significance of the deviation of the power spectrum from a power law at low
degrees, we performed a least-squares linear fit between wavelengths of 10 and 400 km
and computed confidence intervals, which are shown in Fig. 5. Prior to making the fits,
the spectra were resampled in equal intervals in a logarithmic scale. The shape model
was assumed to be errorless since we have observed that at the relevant scales different
iterations of the shape model (both SPC and SPG) have converged. As shown in Fig. 5,
the deviation of the observed spectrum from a power law at low degrees is statistically
significant, i.e., the spectrum lies outside of the 95% confidence interval.
We interpret the decrease of power at low degrees to be due to viscous relaxation.
Viscous relaxation is a scale-dependent process: large scale features relax away faster
than small scale features. We observe that viscous relaxation is important only at large
scales. For n < 12 (spatial wavelength > 246 km) the observed topography PSD spectrum
is consistently outside of the 95% confidence region. The observed topographic spectrum
then implies that viscous relaxation does occur on Ceres but, unlike modeled in Bland

[2013], it is important only at the lowest degrees that correspond to scales of several
hundreds of km. There are only a few features on Ceres of that size and at least two of
them (a hexagonal-shaped basin named Kerwan with a diameter of 275 km and Coniraya
crater with a diameter of 131 km) are morphologically relaxed [Hiesinger et al., 2016;
Bland et al., 2016]. Therefore, due to a low sample size and the lack of confirmed craters
sampling longer wavelengths, crater morphology cannot be effectively used as a statistical
means to study viscous relaxation on Ceres at the low degrees (n < 12) where most
deviation from the power law is observed. In reality, the topographic power spectrum
reflects all ongoing processes (e.g., impact cratering, internal activity such as convection,
tectonic processes), which could balance viscous relaxation contribution to the spectrum
[Fu et al., 2017].
4.1.3. Regional variations of viscous relaxation
To confirm the occurrence of viscous relaxation on Ceres, we studied regional variations
of the topographic power spectrum. Since Ceres has relatively low obliquities (obliquity
varies between 2◦
and 20◦
Ermakov et al. 2017; Bills and Scott 2017), a strong system-
atic difference exists in the insolation between the polar and equatorial regions. As such,
equatorial regions are expected to have more relaxed topography due to their higher
temperatures [Bland, 2013]. Latitude-dependent crater relaxation that was predicted in
Bland [2013] has not been observed on Ceres by Dawn from crater morphology [Marchi
et al., 2016; Bland et al., 2016]. However, an insufficient number of confirmed impact
basins may exist in the wavelength range affected by viscous relaxation. There are only
two unambiguous impact features larger than 200 km in diameter and 12 craters with
diameters between 100 and 200 km. Unfortunately, this is not enough to distinguish lati-

tudinal trends in depth to diameter ratio from “geological noise” (i.e. from superimposed
effects of various geologic processes). Furthermore, crater morphology might be affected
by processes other than viscous relaxation, such as infilling with ejecta from subsequent
impacts. Therefore, we focus on a more fundamental surface property, the localized topog-
raphy power spectrum using a spectral-spatial localization algorithm described in Simons
et al. [2006]. We adopted a spherical cap localization window with a radius of 20 degrees
and a bandwidth of 20 spherical harmonic degrees and computed the topographic power
within each window. We tesselate a unit sphere using the 3rd order icosahedron tessel-
lation to find centers of the localization windows. This gives 642 points quasi-uniformly
distributed on a sphere.
Fig. 6 shows a strong dependence of topographic power at low spherical harmonic de-
grees on latitude. Specifically, topographic power is lower near the equator and increases
poleward for n< 40. This observation is consistent with the geometry of viscous relax-
ation as calculated by Bland [2013]. However, it is not consistent with the magnitude
of relaxation in the case of a water ice-dominated crust, which would predict nearly no
topography at equatorial latitudes.
At very high latitudes (| φ |> 80◦
), the topographic power is constant to slightly de-
creasing. This can be explained by the fact that Ceres’ shape models have lower resolution
near the poles due to poorer illumination conditions and therefore lack topographic power
in these regions.
Fig. 7 shows topographic spectral density localized in latitudinal bands. In this case, we
localized topography with irregularly shaped windows that represented latitudinal bands.
For the equatorial band, the window covers latitudes from -22.5◦
to +22.5◦
. The six non-

equatorial windows were spanning between 22.5◦
, 45◦
, 67.5◦
and 90◦
. It is evident that the
difference between the topographic spectra is more pronounced at the low degrees. The
error bars of the topographic spectrum for the equatorial band starts overlapping the error
bars of the other bands at n ≈ 30. The fact that Ceres’ equatorial regions have a reduced
topographic power also implies that the dwarf planet has not experienced a significant
true polar wander (TPW) or obliquity change over geologic time scales. By significant
change we mean a change can make the equator cooler than the poles. Quantifying TPW
is beyond the scope of this paper and should be addressed in a separate study.
We also note that at n > 45, the southern hemisphere has lower topographic power
than the northern hemisphere. We interpret this to be due to vast regions covered by
smooth ejecta blankets in the southern hemisphere (around Kerwan, Urvara and Yalode
craters) compared to ubiquitous heavily cratered terrains in the northern hemisphere.
4.2. Internal structure constraints
As indicated above, a two-layer model is the most parsimonious structure possible to
explain the topography and gravity data. We therefore adopt this basic structure and
explore the parameter space of two-layer models constraining the permitted range of
plausible internal density structures. We will refer to the outer layer as “crust” and inner
layer as “mantle”. However, we note that the reader should not take these terms with their
terrestrial connotations. We reserve the term “core” for potential future use to describe a
deeper compositionally and/or rheologically distinct layer.
Fig. 8 shows the parameter space for the two-layer model and a family of hydrostatic
solutions given the observed shape and gravity field of Ceres (see subsection 2.2). The
difference between the equatorial axes and the difference between shape and gravity so-

lutions demonstrate the degree of Ceres’ non-hydrostaticity. The family of solutions lies
above the homogeneous line indicating an increase of density towards the center of Ceres.
The two-layer model solution based on the gravity field appears to be more differentiated
than the solution based on the shape. This is likely due to some topography being, at
least partially, compensated at degree 2 (see below).
There are multiple ways to normalize the moment of inertia of Ceres. For the two-layer
model, we use the following notation: λ is the normalized moment of inertia:
λ =
2
5
ρ1V1r2
1 + 2
5
(ρ2 − ρ1) V2r2
2
Mr2
1
, (7)
where V1 and V2 are the total volume and the volume of the mantle, respectively; M is
the total mass of the body; r1 is the volume equivalent body radius, r2 is the volume-
equivalent mantle radius. If the body’s rotation is stopped and the body relaxes to
hydrostatic equilibrium and attains a spherical shape, then the polar moment C and
equatorial moment A will approach λMr2
1. The normalized moment of inertia λ is ≈ 0.373
(which corresponds to C/Mr2
1 ≈ 0.392) based on the gravity solution and ≈ 0.390 based
on the shape solution. The nomalized polar moment of inertia of Ceres is greater than
that of the Earth (0.33, Yoder [2013]), Mercury (0.35, Margot et al. [2012]; Smith et al.
[2012]) and Mars (0.366, Folkner et al. [1997]), but more similar to that of the Moon
(0.39, Williams et al. 2014). We note that for these bodies the difference between polar
and mean moments of inertia or different normalizations is in the third digit. As observed
by Dawn, Ceres appears to be less differentiated than based on some of the pre-Dawn
shape models [Thomas et al., 2005].

Also, we note that the contour lines for constant λ on Fig. 8 do not exactly correspond
to the contour lines of constant ¯J2 or ¯A20 . The crustal densities between 900 kg/m3
and 1800 kg/m3
correspond to λ between 0.3734 and 0.3738. This range is rather small.
Effectively, this means that for Ceres’ mean density and rotation rate there is a one to one
correspondence between λ and ¯J2 or ¯A20. The Radau-Darwin (RD) relation is typically
used to find the relationship between the mean moment of inertia I = (A + B + C)/3 and
¯J2 (e.g. Rambaux et al. 2015):
I
Mr2
1
=
2
3
1 −
2
5
4 − kf
1 + kf
1/2
(8)
where kf is the fluid Love number, J2 = 1
3
qkf and q is the ratio of the centrifugal to
gravity forces at equator: q = ω2
r3
1/GM. For the case of Ceres, RD gives I/Mr2
1 = 0.370,
or equivalently C/Mr2
1 = 0.387, quite close to the value based on Tricarico [2014] and our
two-layer model.
The magnitude of the sectorial degree 2 term indicates that there could be a non-
negligible non-hydrostatic contribution in the observed ¯J2. It is possible to separate hy-
drostatic from non-hydrostatic contributions in ¯J2 by employing two assumptions. First,
hydrostatic and non-hydrostatic contributions are not coupled and the observed quan-
tity can be represented as their sum. Second, non-hydrostatic contributions to gravity
and topography behave isotropically. These assumptions have been used several times
in the literature for constraining the internal structures of Enceladus and Dione [Iess
et al., 2014a; McKinnon, 2015a; Hemingway et al., 2016; Beuthe et al., 2016]. Prior to
computing admittance, hydrostatic contributions should be subtracted from the degree
2 spherical harmonic coefficients since rotationally induced flattening is not isotropic.

If non-hydrostatic gravity and topography are isotropic, gravity-topography admittance
should not depend on spherical harmonic order m. Therefore, Znh
20 should be the same as
Znh
22 . Unlike in Eq. 3, here we compute admittance component by component as can be
seen in Eq. 9.
Znh
22 =
¯C22
¯A22 + ¯S22
¯B22
¯C2
22 + ¯S2
22
¯A2
22 + ¯B2
22
3GM
R3
(9)
We have computed a solution family for the two-layer model using isotropic admittance
assumption:
¯Cobs
20 − ¯CHE
20
¯Aobs
20 − ¯AHE
20
3GM
R3
= Znh
22 , (10)
where ¯CHE
20 and ¯AHE
20 are the hydrostatic values computed using Tricarico [2014] for the
two-layer model; the factor 3GM/R3
was used in order to have notation consistent with
Eq. 3.
Fig. 9 shows the mean crustal thickness as a function of crustal density. Ceres is more
oblate in its topography than in its gravity. Therefore, Ceres’ internal structure solution
is more diﬀerentiated based on gravity compared to topography. If Ceres was perfectly
in hydrostatic equilibrium, the two solutions would be identical. Some of this degree
2 topography can be compensated. However, the mechanism for creating such degree
2 pattern in crustal thickness is unclear. Also, we refer the reader to the Supporting
Information (Section 1) for the two-layer solution in a table form.
4.2.1. Possible despinning
The present day rotation period of Ceres is ≈9.07 hours [Chamberlain et al., 2007;
Konopliv et al., 2017]. As seen in Fig. 8, the hydrostatic solutions from the gravity and

shape are different for the present day rotation period. Mao and McKinnon [2018] suggest
that the gravity and topography can be both made hydrostatic at degree 2 and order 0 by
changing the rotation period. In fact, the solution for the internal structure from gravity
( ¯J2) and shape ( ¯A20) are identical for a rotation period of 8.46 hours. The moment of
inertia of such hydrostatic equilibrium model is smaller (λ ≈ 0.353) than for the shape
or gravity solutions given the present rotation rate implying stronger differentiation. The
required despinning is significant (≈7.2%). One possible mechanism capable of imparting
the necessary angular momentum is a giant impact. Mao and McKinnon [2018] suggested
that an impactor with a mean density of Ceres and a diameter of ≈80 km colliding
with Ceres at a 45◦
angle and 5 km/sec impact velocity can provide enough despinning.
However, Ceres lacks big impact features on its surface. Marchi et al. [2016] suggest that
the low topography region centered at 135◦
E and 20◦
N is an ancient, heavily eroded basin
some 700 km in diameter. However, the observational evidence is so far inconclusive as
to whether this topographic depression is an impact feature. The impactor that formed
this hypothetical basin could have potentially been large enough to change the rotation
rate of Ceres by 7%. Alternatively, Mao and McKinnon [2018] suggest that Ceres could
have been despun by the cumulative effect of numerous impacts through the angular
momentum drain effect of Dobrovolskis and Burns [1984]. AlternativelyFinally, solar tides
could also affect the rotation rate of Ceres. However, tidal response of Ceres is unknown
and, therefore, quantitative study of solar tides is beyond the scope of this paper.
Mao and McKinnon [2018] state that the faster rotation hydrostatic solution is appli-
cable for modeling Ceres’ internal structure if the fossil bulge is frozen in. In other words,
Ceres’ shape coefficient ¯A20 and gravity coefficient ¯J2 should not have appreciably changed

since the despinning event, which likely happened early in cerean history. However, ana-
lytic modeling of Ceres’ relaxation based on the propagator matrix approach [Hager and
Clayton, 1989] suggests that the fossil bulge will relax to isostatic equilibrium in approx-
imately 0.5 Gy using a viscosity of 1025
Pa s for the crust (see Supporting Information,
Section 2). As such, the fossil bulge is not frozen in and the faster rotation solution (shown
in magenta in Fig. 8) is likely not representative of the internal structure of Ceres.
4.3. Bouguer anomaly and crustal thickness variations
We introduce the free-air gravity anomaly (¯σFA
inm) as the difference between the spherical
harmonic coefficients of the observed gravity field and the spherical harmonic coefficients
of the gravity field of a body in the state of hydrostatic equilibrium (see Eq. 5) for the
present-day rotation rate:
¯σFA
inm = ¯σobs
inm − ¯σhydrostatic equilibrium
inm (11)
Naturally, the magnitude of the free-air anomaly is a measure of the body’s non-
hydrostaticity. The maps of the free-air anomalies are provided in the Supporting In-
formation (Section 4). In order to judge about subsurface mass distribution, we compute
the Bouguer anomaly. The Bouguer anomaly (BA) is computed as the difference between
the observed gravity coefficients and the gravity coefficients of the two-layer model, where
the internal interface corresponds to a hydrostatic equilibrium shape (its gravity corre-
sponds to the second term in Eq. 5) for the present rotation rate and the outer surface is
the observed outer surface of the body:
¯σBA
inm = ¯σobs
inm − ¯σshape model
inm + ¯σinner equilibrium surface
inm (12)

Since the model is assumed to be hydrostatic, the degree 2 zonal term in the Bouguer
anomaly is zero. The Bouguer anomaly is presented in Fig. 10A, which shows that
regions of negative Bouguer anomaly correspond to regions of positive topography. Fig.
10B shows a map of Ceres’ crustal thickness based on the solution for the crust-mantle
interface that minimizes the Bouguer anomaly (using Eq. 18 in Wieczorek and Phillips
1998). Finite-amplitude correction was taken into account both for the crust-mantle
interface and outer surface. Gravity and shape data do not allow an absolute estimation
of the crustal thickness since either layer densities or the shape of the interface between
different density layers must be assumed. A mantle density of 2434 kg/m3
and crustal
density of 1287 kg/m3
were assumed to compute the crustal thickness shown in Fig.
10B. This choice of densities comes from admittance analysis and is justified further
below (see Sec. 4.4). We would also like to note that, while the choice of densities affects
the amplitude of the Bouguer anomaly and the amplitude of the crust-mantle interface
variations, it does not affect the spatial distribution of the anomalies.
4.4. A self-consistent two-layer isostatic model
It is clear from the Bouguer anomaly map (Fig 10A) that there is a negative correlation
between the Bouguer anomaly and topography, which implies compensation of topogra-
phy. Finite element modeling of Fu et al. [2017] constrains the rheological structure of
Ceres, which, given the existence of plastic failure, is consistent with no significant intact
elastic lithosphere at 4 Gy time scale. Therefore, isostatic compensation is expected for
Ceres. Low values of admittance are also suggestive of isostatic compensation. Due to
lack of evidence for significant global scale compositional heterogeneity [Williams et al.,
2017a], we considered only Airy isostatic compensation. However, Pratt isostasy cannot

be fully eliminated based on the Dawn data. In the case of Airy isostatic compensation in
cartesian geometry (inﬁnite half-space), the amplitude of topography at the bottom of the
compensated layer is proportional to the amplitude of topography on its top by a factor of
ρ2−ρ1
ρ1
, where ρ1 is the crustal density, ρ2 is the mantle density and ∆ρ = ρ2−ρ1. In the case
of spherical geometry, if we keep the same proportionality constant, gravity-topography
admittance can be computed as follows:
ZAiry
n =
GM
R3
3(n + 1)
2n + 1
ρ1
¯ρ
1 −
R − Dcomp
R
n+2
, (13)
where ¯ρ is the mean density of the body, Dcomp is the depth of compensation, which
is taken to be the mean crustal thickness in the two-layer model and n is the spherical
harmonic degree. We deﬁne the ratio of crust-mantle relief amplitude to the outer surface
amplitude as w. In this model, w is independent of wavelength and is equal to ∆ρ
ρ1
. We
note that Eq. 13 is accurate if the departure from sphericity is small. In a more general
case, if w is wavelength-dependent we can compute admittance as:
ZAiry, analytic
n =
GM
R3
3(n + 1)
2n + 1
ρ1
¯ρ
−
∆ρ
¯ρ
R − Dcomp
R
n+2
wn , (14)
where we use a subscript n to indicate the wavelength-dependence of w. In the limit of thin
crust, wn approaches ρ1
∆ρ
and Eq. 13 is recovered from Eq. 14. It is commonly assumed
in the literature (e.g. Phillips and Lambeck [1980]; Rummel et al. [1988]; Wieczorek
and Phillips [1997]; McKinnon [2015b]; Wieczorek [2015]) that for a spherical body w
= ρ1
∆ρ
R
R−Dcomp
2
. Under this condition crustal columns have equal masses in spherical
geometry. The factor R
R−Dcomp
2
is introduced to take into account the convergence of
the verticals. Konopliv et al. [2017] explored a range of two and three-layer models and

found that the gravity and topography of Ceres are consistent with the Airy isostatic
compensation under an assumption of equal mass spherical columns for crustal densities
between 1200 and 1600 kg/m3
with a crustal thickness between 27 and 43 km. Unlike in
the present paper, Konopliv et al. [2017] used admittance between gravity and gravity-
from-topography, which gave a more stable solution for the depth of compensation as a
function of spherical harmonic degree.
Beuthe et al. [2016] argue that the equal mass spherical columns condition should be
replaced by the minimum deviatoric stress condition. Additionally, Hemingway and Mat-
suyama [2017] point out that the equal masses condition is not physically justified and
argue to replace it with an equal pressure condition under which w = ρ1
∆ρ
g1
g2
, where g1 is
the gravitational acceleration at the outer surface and g2 is the gravitational acceleration
at the interface between crust and mantle. Hemingway and Matsuyama [2017] argue that
the isostatic equilibrium condition requires no lateral flow which would occur due to pres-
sure gradients. The absolute difference between the two definitions of isostasy is larger
if the thickness of the compensated layer constitutes a large fraction of the body’s ra-
dius. Both Beuthe et al. [2016] and Hemingway and Matsuyama [2017] conclude that the
crustal thickness of Enceladus was overestimated by a factor of two if equal mass columns
condition is used. Additionally, although Ceres is close to hydrostatic equilibrium, its fast
rotation distorts the shape to the point that non-linear effects in computing gravity from
the shape become important. Non-linearity also affects admittance and correlation.
Since viscous relaxation has been a dominant large-scale topography modification pro-
cess on Ceres [Fu et al., 2017] , we employ a viscous relaxation based approach to isostasy.
We compute isostatic compensation using an analytical theory of Stokes flow in spherical

geometry [Hager and Clayton, 1989]. A similar approach for mantle dynamics was also
used in an ealier work [Hager and O’Connell, 1979]. We impose a single degree surface
topography over a hydrostatic two-layer structure and evolve the system using a prop-
agator matrix approach. As the system viscously relaxes, the ratio of the crust-mantle
interface topography to the outer surface topography approaches an asymptotic value wn
(for more details see Section 2 in Supporting Information). For convenience, we will refer
to this model of isostasy based on Hager and Clayton [1989] as “dynamic isostasy model”.
We note that this term was not used in Hager and Clayton [1989]. A similar approach
was presented for a viscoelastic medium in Zhong and Zuber [2000].
We note that the concept of isostatic equilibrium is not truly an equilibrium, since,
given a finite viscosity throughout the body, it will eventually relax to a hydrostatic
equilibrium state (a sphere for a non rotating body). For a simplified two-layer model,
where two layers have uniform densities and viscosities, viscous evolution of the crust’s
top and bottom interfaces can be modeled as a sum of two exponential modes (see Section
2 in Supporting Information). The mode for which the two interfaces move in the same
direction (symmetric mode) has a shorter time scale compared to the mode for which
the interfaces move in the opposite directions (antisymmetric mode). The time scale
of the symmetric mode can be thought of as the time scale of isostatic compensation.
The time scale of the antisymmetric mode is the time scale associated with relaxation
toward hydrostatic equilibrium. However, the time scale of relaxation to hydrostatic
equilibrium could be longer the age of the Solar System, which would effectively allow
treating isostatically compensated structures as static ones.

We note that the rate of viscous relaxation depends on the wavelength. It takes more
time for small-scale features (high spherical harmonic degrees n) to reach a state at which
wn does not appreciably change. Once wn is computed for a range of internal structure
parameters (i.e. density of the crust and the mantle as well as crustal thickness), we can
compute admittance using Eq. 14 and choose which point on the of the hydrostatic family
(Fig. 8) fits best with our isostatic model. We provide more details on computing wn in
the Supporting Information (Section 2). We find the best-fit isostatic solution by varying
the crustal density between 1100 kg/m3
and 1500 kg/m3
while staying on the two-layer
model solution derived based on hydrostatic equilibrium (i.e. on the blue curve in Fig. 8)
in order to minimize the admittance misfit:
χ2
=
nmax
n=3
Zobs
n − ZAiry, analytic
n
2
σ2
Zobs
n
, (15)
where σ2
Zobs
n
is the variance of the observed admittance computed from the covariance
matrix of the gravity field model and assuming that shape model is errorless. nmax should
be chosen considering the degree-dependent isostatic compensation time scale. Fu et al.
[2017] shows that the relaxation time scale becomes comparable to the Solar System age
at n ≈ 11. We therefore chose nmax = 11. The gravity coefficient ¯C40 and shape coefficient
¯A40 were zeroed out prior to computing admittances in Eq. 15 since these coefficients have
strong contributions from rotation. Isostatic anomaly coefficients ¯σIA
nm are then computed
similarly to the Bouguer anomaly, i.e., by subtracting the gravitational attraction of an
isostatic model from the observed gravitational attraction:
¯σIA
inm = ¯σobs
inm − ¯σisostatic model
inm (16)

The isostatic anomaly is the difference between the observed gravity field and a super-
position of the gravity from the outer shape and from compensating mantle topography
at depth. Due to non-hydrostaticity, the exact hydrostatic value of ¯J2 is uncertain. An
additional constraint is needed to find the hydrostatic value of ¯J2. This could be an
assumption of isotropic admittance, which has been employed for Enceladus [Iess et al.,
2014b; McKinnon, 2015b]. However, in the case of Ceres, negative admittance for the
sectorial component of degree 2 cannot be explained just by isostasy. The nature of the
mechanism that creates the negative admittance is unknown. In order to estimate the un-
certainty in the isostatic solution, we vary ¯J2 by 3% (which is the ratio of non-hydrostatic
terms to the hydrostatic one) and find confidence intervals for the crustal and mantle den-
sity as well as the mantle radius. Naturally, different models of isostasy lead to different
estimates of the two-layer model parameters. The results are summarized in Table 4. It
can be seen that the mantle density is essentially independent from the isostasy model.
On the other hard, the crustal density and crustal thickness vary by as much as 245 kg/m3
and 14.7 km, respectively. In our preferred analytical model of dynamic isostasy [Hager
and Clayton, 1989], the admittance misfit is minimized for the crustal density of 1287−87
+70
kg/m3
. Here and elsewhere the superscript and subscript denote the solutions for upper
and lower bounds on hydrostatic ¯J2. This corresponds to a crustal thickness of 41.0−4.7
+3.2
km and a mantle radius of 428.7+4.7
−3.2 km. These are the parameters that are used for
computing isostatic anomaly maps (Fig. 12-14) as well as in the Bouguer anomaly map
(Fig. 10). The modeled and observed admittances as well as observed correlation spectra
are shown in Fig. 11. Additionally, in Table 4, we provide a solution based on the isotropic

admittance assumption (Eq. 10), which yields a slightly less differentiated interior (with
a higher moment of inertia λ ≈ 0.377, as can be seen in Fig. 8).
When the isostatic anomaly is positive in a region of negative topography (e.g. in an
impact basin), this region can be thought of as supercompensated. If we assume the only
source of the anomaly is the topography of the crust-mantle interface, a supercompensated
state would mean a stronger uplift of the crust-mantle interface than required for isostatic
balance. Similarly, a region of positive topography associated with a negative isostatic
anomaly also implies a supercompensated state since the inferred crustal root is deeper
than required for isostatic balance. In summary, the opposite signs of topography and
isostatic anomaly are indicative of a supercompensated state. Alternatively, matching
signs of topography and isostatic anomaly are indicative of a subcompensated state.
We show a map of the isostatic anomalies for degrees from 1 to 16 in Fig. 12A. The
range of the isostatic anomaly (from -129 mGal to +132 mGal) is smaller than that of the
Bouguer anomaly (from -287 mGal to +222 mGal), indicating that isostasy can indeed
explain a significant fraction of the Bouguer anomaly. The sectorial degree 2 component
dominates the isostatic anomaly. Possible sources of this signal include fossil topography
of a compositionally different core or a bottom buoyancy-driven load within the mantle
(see Subsection 5.2). If we compute the isostatic anomaly for degrees from 3 to 16 (Fig.
12B), the range is smaller (from -96 mGal to +93 mGal). This makes it easier to analyze
small scale features since the anomaly map is not overwhelmed by a long-wavelength
signal. At high latitudes, the resolution of the gravity field model is slightly worse due
to higher spacecraft altitudes. Therefore, we compute the isostatic anomaly only up to
n = 14 (Fig. 12C-D).

5. Discussion
5.1. Elastic support
Based on the observed latitudinal patterns of topography power and comparison to rocky
bodies such as Vesta, we attribute the reduction of topographic power at low degrees as due
to viscous relaxation. We have also explored other processes that may contribute to the
observed topography. For instance, Nimmo et al. [2011] discussed how a flexural response
can reduce low-degree power. We repeated the analysis of Nimmo et al. [2011] for Ceres.
To model the effect of elastic support on the topographic power spectrum, we multiplied
the power law fit of Ceres’ topography (Fig. 5) by the factor F2
n = 1/ 1 + ρ1
∆ρ
Cn
2
, where
Cn is the degree of compensation (Eq. 27 in Turcotte et al. [1981]) for a spherical harmonic
degree n. The parameter Cn depends on such lithospheric properties as Young’s modulus,
effective elastic thickness, Poisson’s ratio, as well as crustal and mantle densities. We
varied the effective elastic thickness (d) between 0.5 km and 5 km, Young’s modulus (E)
between 1 GPa and 100 GPa. We note that E=9 GPa is typical for water ice and E=40
GPa is typical for rocks. Crustal and mantle densities were taken according to the gravity
solution in Fig. 9. We fixed Poisson’s ratio at ν = 0.25 .
Given these reasonable ranges, we were not able to achieve a satisfactory fit to the
observed Ceres topography spectrum with a purely elastic model. As it can be seen in Fig.
13, it is not possible to simultaneously match the magnitude of the drop of topographic
power at low degrees and the location of the drop. Therefore, a flexural response is likely
not the source of the low-degree topography power reduction.

5.2. Degree 2 anomaly
The correlation between observed gravity and gravity-from-shape for the degree 2 sec-
torial term is ≈ −0.7. The admittance is therefore also negative ≈ −36 mGal/km (Fig.
11). This term dominates non-hydrostatic signal at degree 2. Negative admittances are
rare (see discussion in Hemingway et al. 2013). The negative correlation between grav-
ity and topography at {n = 2, m = 2} is a puzzle (Table 3). Here we enumerate three
possible explanations for this observation as well as discuss their implications and con-
sistency with other data. The first proposed explanation is for a static interior and the
other two involve a dynamic interior. The proposed explanations differ in terms of the
depth at which the density anomaly is placed: the first one being the deepest, the last
one being the shallowest. These three mechanisms are not mutually exclusive. Therefore,
a combination of these mechanisms is also allowable.
First, it is possible that viscosity increases at some depth due to a compositional bound-
ary forming a rigid core, therefore making this deep interior resistant to relaxation and able
to support a significant deep interface topography over geologic time-scales. Such sharp
viscosity change could be due to rock dehydration from high temperatures achieved in
the course of Ceres history at some depth in the mantle. This hypothetical compositional
boundary, if it exists, is constrained to be deeper than 100 km [Fu et al., 2017]. Non-
hydrostatic topography at this boundary could potentially explain the observed negative
correlation if this rigid core’s longer equatorial axis is aligned approximately perpendicu-
larly to the longer equatorial axis of the outer shape.
Second, a combination of bottom buoyant loading from a mantle plume with a
rigid/thick lithosphere could explain the observed negative correlation of the sectorial

degree 2 component. The load’s contribution to the gravity is negative and the surface
uplift contribution is positive. However, if the crust is rigid, it would not bend enough
to compensate for the negative contribution of the load and the total effect is negative.
This could imply a present-day global-scale convection or a frozen-in anomaly for such
convection in the past. Travis et al. [2015] suggests that liquid water could still be present
and active in Ceres’ interior and that hydrothermal convection in a mud ocean and wet
rocky mantle is currently ongoing. The Travis et al. [2015] model can potentially explain
the observed negative correlation, although further modeling is necessary to establish that
the magnitude of the gravity and topography anomaly due to this source is quantitatively
consistent with the data. However, there is no evidence of global lithosphere based on
the finite-element modeling (Fu et al. [2017] and Subsection 5.1) on long timescales. This
could indicate that the source of the anomaly is geologically recent.
Finally, another mechanism that can produce a negative anomaly in the Hanami planum
region and contribute to negative correlation is salt tectonics. Buczkowski et al. 2016
identifies extensional features that are consistent with a salt dome. This mechanism
would imply a local salt diapir rising due to its lower density and bending the surface.
Compositional data derived from the VIR instrument [De Sanctis et al., 2016] shows that
Occator bright spots are a mixture of sodium carbonates and ammonium salts, possibly
ammonium chlorides, which provides a connection between salt tectonics mechanism and
the bright spots.
The negative correlation of the sectorial degree 2 term also introduces crustal thick-
ness variation as seen in the crustal inversions (Fig. 10B). This allows estimating the
magnitude of the regional heterogeneity of Ceres’ outer layer. Two regions with larger

crustal thicknesses are centered near the equator at 45◦
E and 135◦
W. The magnitude
of the crustal thickening in the two regions is different (Fig. 10B) due to the non-zero
COM–COF offset.
Mao and McKinnon [2018] propose that the {n = 2, m = 0} anomaly (i.e. the difference
between the gravity and shape hydrostatic solutions, see Fig. 8) could be coupled to
the {n = 2, m = 2} anomaly. Mao and McKinnon [2018] argue that a combination of
despinning and a deep-seated density anomaly due to low order convection or upwelling
can explain the observed degree 2 in gravity and topography.
5.3. Regional anomalies
Fig. 14 shows an isostatic anomaly from n = 3 to n = 16 for four regions of interest.
These regions have strong gravity anomalies associated with geomorphologic features.
These features are: Occator and Hanami Planum, Kerwan crater, Urvara and Yalode
crater and Ahuna Mons. Below, we describe the structure of these gravity anomalies as
well as discuss geophysical implications.
5.3.1. Hanami planum and Occator
Hanami planum is the region of high standing terrain around the Occator crater (Fig.
14 upper right). Occator has a unique set of bright spots [Scully et al., 2017] in its
topographic center named Cerealia Facula (central spot) and Vinalia Faculae (group of
off-center spots). The Occator crater corresponds approximately to the location of highest
elevation within the Hanami planum. This region is associated with the strongest negative
Bouguer anomaly (-250 mGal, Fig. 10) on Ceres. Hanami planum is large enough (≈600
km) to be resolved in the current gravity model. The regions of high topography and
negative Bouguer anomaly surrounding the Occator crater are associated with so called

“bluish material”. This material is presumed to be Occator’s ejecta. The ejecta material
itself is probably too thin to create the observed gravity anomaly but it could indicate
that either the crater is very young [Nathues et al., 2017] or that the subsurface material
that was excavated has a unique composition.
Since this region has a positive topography and negative gravity anomaly, it is expected
that, at least partially, the negative Bouguer anomaly is a consequence of isostatic compen-
sation. However, even after computing the isostatic correction, the gravitational anomaly
is still negative, indicating supercompensation. A possible explanation for this observa-
tion could be a combination of a buoyancy-driven anomaly with a high-rigidity/thick shell
at the time scales relevant to the buoyant process. Alternatively, a relatively low regional
density can create such a negative isostatic anomaly.
Additionally, the Occator crater itself is located near a local maximum in the isostatic
gravity anomaly. However, the extent of this local maximum is at the limit of the model’s
resolution.
5.3.2. Ahuna Mons
Ahuna Mons is a pyramid-shaped mountain of possible cryovolcanic origin (Fig. 14 lower
left) located near the equator in the western hemisphere of Ceres (11.5◦
S and 44.2◦
W).
This feature has a unique morphology that has not been found anywhere else on Ceres.
It was proposed that salts with low eutectic temperatures and low thermal conductivities
enabled the presence of brines within Ceres, which facilitated the formation of Ahuna
Mons via cryovolcanism [Ruesch et al., 2016]. Ahuna Mons is associated with a strong
positive Bouguer and isostatic anomaly. However, since the feature is only 17 km wide, it
is not possible to associate the anomaly with the feature itself, but only with the general

surrounding area. The positive anomaly in the Ahuna Mons region might be explained by
an extrusion of high-density brines to the surface. Alternatively, we note the possibility of
a thin crust being the source of, or at least contributing to, the positive gravity anomaly at
Ahuna Mons, analogous to the correlation between thin crust and maria on the Moon. But
why is there only one Ahuna Mons observed on Ceres? It was proposed that cryovolcanic
domes might have local ice enrichment such that their topography quickly relaxes [Sori
et al., 2017]. However, this ice enrichment is at odds with the observed strong positive
gravity anomaly.
5.3.3. Mascon-like signatures in impact basins
The isostatic anomaly map shows a structure similar to the lunar mascons for the two
biggest impact basins Kerwan (Fig. 14 upper right) and Yalode (Fig. 14 lower right).
With a diameter of 281 km, Kerwan is the biggest unambiguous impact basin on Ceres
[Williams et al., 2017b]. It is located near the equator in Ceres’ eastern hemisphere. Its
subdued topography could indicate that this feature has experienced viscous relaxation
[Bland et al., 2017]. The isostatic anomaly map shows a positive anomaly in the basin’s
center and a negative annulus near the rims. Yalode is the second biggest impact basin
with a diameter of 271 km. The structure observed in the isostatic anomaly map is
similar. The center of the basins has a local maximum in isostatic anomaly surrounded by
a negative annulus closer to the rims of the basins. The observed gravity signatures could
potentially mean a superisostatic mantle uplift beneath Kerwan and Yalode, which we
propose as a hypothesis for future study. The third biggest basin is Urvara with a diameter
of 163 km. The isostatic anomaly is negative throughout the basin. However, it is possible
that the central positive peak in Urvara is not resolved in the present gravity ﬁeld model.

Unlike Kerwan, Urvara and Yalode have well-defined rims and unrelaxed morphology.
Additionally, Yalode possesses an inner ring structure. This pristine morphology could
be explained by either their younger age – 0.48 and 1.8 Gy, respectively, for Urvara and
Yalode against 2.8 Gy for Kerwan (Wagner et al. [2016], using lunar based chronology; for
asteroid based chronology all ages are much younger Hiesinger et al. [2016]) – or their lower
rate of viscous relaxation due to lower temperatures since the two features are located at
higher latitudes than Kerwan. Curiously, despite morphological differences, the gravity
anomaly structures are very similar in Kerwan and Yalode.
It is possible that other big craters might also possess mascon-like structures but unre-
solved in the gravity-field model. The only other craters that show a significant isostatic
anomaly are Ezinu and Dantu (diameters of 116 and 126 km, respectively). They have
negative isostatic anomalies indicating a subcompensated state. Since these features are
smaller, it takes more time for them to relax. Therefore, the isostatic compensation state
has likely not been achieved.
This rather remarkable observation of mascon-like structures will be very useful for
improving our understanding of the thermal history and rheology of the cerean crust.
Melosh et al. [2013] showed that a similar pattern observed for lunar mascons resulted from
impact basin excavation and collapse followed by isostatic adjustment with subsequent
cooling and contraction of a voluminous melt pool. The Dawn observations show that
mascons can be found on a dwarf planet where crustal structure and composition is
markedly different from that of terrestrial planets or the Moon, which implies that mascon
formation might be an ubiquitous outcome of the impact process in the Solar System.

5.4. Isostatic constraints
Isostasy can provide the upper bound on crustal density. Isostatic admittances are low
compared to the uncompensated case. However, if there is a lithosphere, it will be able
to, at least partially, support topographic loads. Therefore, admittance will be higher in
presence of an elastic lithosphere. Consequently, as can be seen in Fig. 11, a lower crustal
density will be required to match the observed admittance.
Using a model of dynamic isostasy [Hager and Clayton, 1989], we find that a density
of 1287−87
+70 kg/m3
is most consistent with full isostatic compensation at degrees 3 to 11
(see subsection 4.4). This yields a crustal thickness of 41.0−4.7
+3.2 km and mantle density of
2434−8
+5 kg/m3
, which is close to the density of CI chondrite [Consolmagno et al., 2008]
assumed in Park et al. [2016]. The inferred low mantle density for Ceres implies strong
mantle hydration and/or porosity, possibly in the form of cracks opened upon cooling
[Neveu and Desch, 2015], which indicates a relatively cool thermal history and points to
either a late accretion without short-lived radioisotopes or efficient early heat transfer due
to hydrothermal circulation. This low density does not support a scenario where Ceres
accreted as a mixture of chondritic material [Zolotov, 2009]. Castillo-Rogez and McCord
[2010] show that this starting composition would lead to the differentiation of a several
hundred km thick dehydrated silicate core with a density in excess of 3200 kg/m3
. Instead,
models that assume Ceres went through a phase of extended hydrothermal circulation at
the global scale [Travis et al., 2015] or multiple phases of hydrothermal circulation in a
fractured mantle [Neveu and Desch, 2015] are better suited to explain the mantle density
inferred from this study.

Several models with inverted density structure have been proposed (e.g. Neveu and
Desch 2015). In such models, the primordial crust overlays either a pure water ice or
ice-rock mixture layer. For such structure, the density of the top layer is higher than
the density of the lower layer, therefore it is gravitationally unstable. Formisano et al.
[2016] studied whether or not Rayleigh-Taylor instability could reverse all or part of the
crust. A range of structures was identiﬁed for which the crust is stable over the age of
the Solar System. We note that the concept of isostasy is inconsistent with such inverted
structure as it requires a lighter material on top of denser material. Therefore, it is possible
that either the instability occurred and the primordial crust foundered leaving a lighter
material on the surface or such inverted structure never existed.
5.5. Rheological constraints
Rheology can give a lower bound on the crustal density. Ceres’ topographic power spec-
trum provides an important constraint on the rheologic structure of the interior. Finite-
element modeling of viscous relaxation [Bland et al., 2016; Fu et al., 2017] shows that a
pure ice crust is not consistent with the observed spectrum as relaxation in an icy crust
would occur too quickly. The simplest explanation is that Ceres’ crust is neither purely
ice (the case studied in Bland, 2013) nor purely rock but an ice-rock-salt-hydrates mixture
that allows some relaxation at the longest wavelengths but is able to support the topog-
raphy on the shorter wavelengths over geologic time scales. This suggested composition
is supported by Visible and Infrared Spectrometer (VIR) observations, which detected
salts in the form of sodium carbonates and ammonium salts at Occator [De Sanctis et al.,
2016], as well as sodium carbonates at Ahuna Mons [Zambon et al., 2016]. Adding sil-

icates or salts into water ice increases crustal density, implying that the crustal density
must exceed a minimum value in order to meet the required mechanical strength.
Without invoking any geochemical evolution models, in order to constrain the crustal
composition, we follow Fu et al. [2017] and adopt 35 vol% or less for the weak phase (water
ice + void space) to limit viscous relaxation and the mean crustal density of 1287 kg/m3
to satisfy the admittance analysis. This bulk density is rather low and we need to identify
a material which is not too dense and rheologically strong (rock-like) to simultaneously
satisfy the rheology and density constraints.
We have considered three-component mixtures of water ice with the following materi-
als: 1) meridianiite (hydrated magnesium sulfate), which is essentially the lightest salt
identiﬁed in planetary setting [Peterson and Wang, 2006] with a density of 1504 kg/m3
[Fortes et al., 2008]; 2) phyllosilicates, taking the density of chrysotile (2500 kg/m3
) as a
reference, and 3) methane hydrate clathrate, whose density is ≈950 kg/m3
–similar to that
of water ice – and yet it is much stronger than water ice [Durham et al., 2003]. In order
to illustrate the rheological and density constraints, we plot three-component mixtures
on a ternary density diagram (Fig. 15). We note that water ice is always the least dense
component. Fu et al. [2017] estimate the bulk crustal porosity to be 10%. We have con-
servatively considered 15% to be the upper limit on bulk void porosity within the crust.
In summary, we have to satisfy the following system:



ρmin
1 < (xaρa + xbρb + xcρc) (1 − χ) < ρmax
1
xa < 0.35 − χ
0 < χ < 0.15
(17)
where χ is porosity, xi are volume fractions and ρi are densities of materials (the subscript
a refers to the water ice; subscripts b and c refer to the other two materials), ρmin
1 = 1221

kg/m3
and ρmax
1 = 1343 kg/m3
are the bounds on crustal density from admittance analysis
using dynamic isostasy model.
In Fig. 15A, we use a mixture of phyllosilicates, methane hydrate clathrate and water ice
for crustal composition. Since the densities of water ice and methane hydrate clathrate are
similar, the bulk density is controlled mostly by the phyllosilicates content. The system of
equations 17 provides upper and lower bounds on volume fractions of the components as
outlined by cyan curves in Fig. 15. Methane hydrate clathrate is constrained to constitute
from 37 to 82 vol%, whereas phyllosilicates are constrained to constitute from 17 to 43
vol%.
In Fig. 15B, we use a mixture of meridianiite, methane hydrate clathrate and water
ice. Similarly to Fig. 15A, the density is mostly controlled by the densest material -
meridianiite. In this case, methane hydrate clathrate can constitute between 0 and 51
vol%, whereas meridianiite can be between 48 and 100 vol%.
In Fig. 15C, for a mixture of water ice, meridianiite and phyllosilicates, we can see
that the rheological constraint (<35 vol% of water ice) is inconsistent with the density
constraint from the admittance analysis for any combination of materials’ volume fractions
assuming zero porosity. Again, allowing up to 15% porosity, we ﬁnd that meridianiite can
constitute between 57 and 100 vol%, whereas phyllosilicates can constitute between 0 and
23 vol%.
An alternative approach is to assume a mixture of water ice with a CI chondrite material
for the crust – the most aqueously-altered meteorite type known. CIs were estimated to
consist of 14.6% salts by mass based on Kargel [1991]. The density of these meteorites
is 2260 kg/m3
after removing the eﬀect of porosity [Britt and Consolmagno S.J., 2003].

35% water ice and 65% of CI yields a crustal thickness of 104 km and a crustal density of
1791 kg/m3
– much more dense than inferred from admittance analysis.
While insight from the topographic spectrum (see subsection 4.1.3) shows that Ceres’
crust is not currently consistent with an icy composition, it could have had more ice in
its early history. Castillo-Rogez et al. [2016] predicts removal of >20 km of ice due to
impact-driven sublimation during the first 200 My of Ceres’ history. This scenario would
imply a bulk density change on the order of 10%. Alternatively, Ceres’ low density was
hypothesized to be due to its porous interior in the absence of a substantial ice content
[Zolotov, 2009]. However, Castillo-Rogez [2011] and Neumann et al. [2015] argue that
high interior porosity is unlikely, although the porosity of the upper layers can be greater
due to impact gardening.
Surprisingly, the range of crustal densities/thicknesses allowed both by rheological (see
subsection 5.5) and isostatic constraints is quite small. However, we want to emphasize
that given all simplifications in our model (two uniform density layers, compensation
happens at the only internal interface), the small range allowed by rheological and isostatic
constraints is likely not representative of the true uncertainty in the internal structure, as
perhaps best manifested by the negative admittance for the sectorial degree 2 component,
which cannot be explained within the framework of an isostatic model.
6. Conclusions
Gravity and shape data indicate that Ceres is a physically-differentiated object. How-
ever, the extent of differentiation is less than based on some of the pre-Dawn shape data
[Thomas et al., 2005] and is more consistent with more recent ground telescope-based
adaptive optics studies [Carry et al., 2008; Drummond et al., 2014]. The tri-axiality of

Ceres indicates the magnitude of the non-hydrostatic influence is on the order of its devi-
ation from a uniform interior. Ceres’ topography at degree 2 appears to be overcompen-
sated as indicated by the negative correlation between sectorial gravity and topography
at degree 2. This negative correlation cannot be explained by an isostatic model and its
source is likely located at a great depth.
We observe that the topographic power of Ceres deviates from a power law at low
degrees and that equatorial regions have lower topographic power. This indicates that
viscous relaxation played a role at Ceres. However, we find that viscous relaxation is
important only at low degrees (n<12) that correspond to spatial scales of more than 246
km. At smaller scales, or, equivalently, higher spherical harmonic degrees, there is no
a deviation of the topography power from a power law and the latitudinal variations of
topography power are significantly smaller.
An isostatic model reproduces the observed gravity reasonably well as demonstrated by
the lower magnitude of isostatic gravity anomalies compared to the Bouguer anomalies.
The best fit two-layer dynamic isostasy model has the crustal density of 1287−87
+70 kg/m3
and mean crustal thickness of 41.0−4.7
+3.2 km based on dynamic isostatic model [Hager and
Clayton, 1989]. We compute the isostatic gravity anomaly for the best-fit two-layer model
parameter and find evidence for mascon-like structures in the two biggest basins: Kerwan
and Yalode.
Finite element modeling of Ceres’ topography [Fu et al., 2017] shows that the topo-
graphic power cannot be supported by a solely ice rheology over billion year timescales.
Using a lower bound for crustal density based on rheology, we derive constraints on the
crustal thickness using the assumption of hydrostatic equilibrium. A low-density, high-

strength mixture is required to explain the inferred crustal density and rheology. The
latter does now allow more than 43 vol% silicates assuming 15% void porosity in the
crust. Therefore, lower density materials, such as salt or gas (clathrate) hydrates, are
required.
Acknowledgments. We thank Francis Nimmo and Peter James for discussions re-
garding the effect of flexure; Doug Hemingway and Isamu Matsuyama for discussions of
different definitions of isostasy; James Keane and an anonymous reviewer for very con-
structive comments that have helped to improve the clarity of the manuscript. Part of
this research was carried out at the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with the National Aeronautics and Space Administration.
The gravity and shape model used in this work are available on the NASA’s Planetary
Data System.
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