The Elliptical Earth (And Satellites, Asteroids, Planets, Dwarf Planets) from 1672 to 2019

The elliptical Earth (and satellites, asteroids, planets, dwarf planets) From 1672 to 2019

Frédéric Chambat

École Normale Supérieure de Lyon Laboratoire de Géologie

Doctoral school, Barcelonette, 2019

ω y ds c g

ϕ r= a x

a − c 1 = = a − c = 21km a 300 1. How do we measure ? 2. How do we explain ? 3. How do we measure ? 4. How do we explain ? 5. England 1 - France 0 6. How do we measure in 2019? 7. Ellipticity and the interior of planets in 2019 8. How do we explain (Clairaut, My nice theory explained with modern maths, 2019)? Carlo Denis (1941-2019)

Born in Luxembourg, 1941. PhD in Sciences, in 1974, at the University of Liège, Belgium. He then pursued his entire career as a professor-researcher at the Institute of Astrophysics and Geophysics in the same university. Among the subjects he worked on, there were two on which he devoted more time and energy: the normal modes for his thesis and the hydrostatic equilibrium ﬁgures in the 1970s and 1980s. In both cases, it has developed its own numeric codes. He has supervised one doctoral student (Yves Rogister). He liked to travel (especially in his car), read and listen to classical music. He loved being independent and he was very independent. Carlo Denis wrote third order codes to integrate hydrostatic equation giving the shape of the Earth. He summarized his results in a long and useful paper published in a series:

Denis C. (1989). The Hydrostatic Figure of the Earth, Physics and Evolution of the Earth’s Interior 4, Gravity and Low-Frequency Geodynamics (edited by R. Teisseyre), Chap. 3, pp. 111–186. PWN-Polish Scientiﬁc Publishers & Elsevier, Amsterdam.

https://fr.wikipedia.org/wiki/Figure_de_la_Terre and articles on geodesy. Jean Richer (1672)

Map of America, maybe by Claude Bernou, 1681. Colbert introduces the members of the Royal Academy of Sciences to Louis XIV. Fictional scene. Henri Testelin (1616-1695), from Charles Le Brun (1619 – 1690), tapestry cardboard ordered by the king for the "Manufacture des Gobelins", around 1667. 1679 - 71 pp. Richer 1679

One of the most significant Observations I have made, is that of the length of the pendulum oscillating at a second of time, which was shorter in Caïenne than in Paris; because the same measure that had been marked in this place on an iron rod, according to the length that was necessary to make a pendulum oscillating at a second of time, having been brought to France, & compared with that of Paris, their difference was found to be one line & a quarter [2.81 mm], of which that of Caïenne is less than that of Paris, which is 3. feet 8. lines 3/5 [99.39 cm]. This Observation was repeated during ten full months, where there has not been a week during which it was not made several times with great care. The oscillations of the pendulum were very small, & remained very sensitive up to fifty-two minutes, & were compared to those of a very excellent clock, whose oscillations marked the second of time. Pendulum clock, with escapment (Huygens, Horologium, 1658)

Video

s ` 1 second = π γ s ` 1 second = π γ

In Paris: γ = π2` = π2 × 99.39 cm = 9.809 m/s2 dγ d` −2.81 mm = = = −2.8 × 10−3 γ ` 99.39 cm In Cayenne: γ = 9.781 m/s2 z

G x

 x  2 a = ω  y  0 In Paris: γ = 9.809 m/s2

In Cayenne: γ = 9.781 m/s2 dγ = −0.028 m/s2 Centrifugal acceleration diminishes the gravity as: ω2R cos2 (latitude) Between Paris and Cayenne the diﬀerence is −0.019 m/s2 6= −0.028 m/s2 About one third of the measured diﬀerence does not come from the centrifugal acceleration z

G x

 x  2 a = ω  y  0 gpole 6= gequator

ω y ds c g

ϕ r= a x

Equatorial radius a Polar radius c

Two mesurables quantities a−c Flattening: = a g −g Gravity relative diﬀerence: pole equator gequator 2 Order of magnitude: ≈ ω a γeq An important dimension-less number, the rotationnal parameter: ω2a 1 ω2R 1 = ≈ ≈ 3.10−3 ≈ γeq 289 g 300 Someone known

1643-1727

1687, 1713, 1723 1687, 1713, 1723 Emilie du Châtelet (1706-1749) par Maurice Quentin de La Tour

1756-59 1756-59

Emilie du Châtelet (1706-1749) par Maurice Quentin de La Tour Calculation of g ω y ds Calculation of the attraction g ? c g Book 1, propositions XC-XCI (p. 228). ϕ r= a x Calculation of g Calculation of g

g Z R 1 Y g = Gρ 2 2πr dr dY 0 d d α

d Y

R r Calculation of g

g Z R 1 Y g = Gρ 2 2πr dr dY 0 d d α

d Y Y g = 2πGρ 1 − √ dY Y 2 + R2

R r Calculation of g ω y

Y R Y c g = 2πGρ 1 − √ dY a Y 2 + R2 x ! Z 2c Y gpole = 2πGρ 1 − dY p 2 2 Y =0 Y + R (Y ) with R2(Y ) (c − Y )2 + = 1. a2 c2 Calculation of g ω y calculus calculus calculus... ⇒ Y R

c 4πGρa 1 a g ≈ 1 − x pole 3 5 Calculation of g ω y calculus calculus calculus... ⇒ Y R

c 4πGρa 1 a g ≈ 1 − x pole 3 5

... genius of Newton... the attraction at the equator is approximately the attraction at the pole of an ellipsoid whose equatorial radius is the mean of the two radius... ⇒ 4πGρa 2 ... ⇒ g ≈ 1 − eq 3 5 x dP = ρg dx pole pole c y dP = ρ(g − ω2y) dy eq eq a

∆P g c x x pole = pole dPpole = ρgpole d c ∆P g − ω2a a c c eq eq y y y dP = ρ(g − ω2a ) d a eq eq 1 − 5 a a a = (1 − ) 1 − 2 1 − ω2a 5 geq

∆Ppole = ∆Peq ⇒ 5 ω2a 1 = = . 4 geq 230

Gravity within a homogeneous ellipsoid is a linear function of coordinates. x g = g axe poles pole c y g = g axe eq eq a ∆Ppole gpole c = 2 ∆Peq geq − ω a a

1 − 5 = (1 − ) 1 − 2 1 − ω2a 5 geq

∆Ppole = ∆Peq ⇒ 5 ω2a 1 = = . 4 geq 230

ω y ds c g

ϕ r= a x

x dP = ρg dx pole pole c y dP = ρ(g − ω2y) dy eq eq a

x x dP = ρg d c pole pole c c y y y dP = ρ(g − ω2a ) d a eq eq a a a 1 − 5 = (1 − ) 1 − 2 1 − ω2a 5 geq

∆Ppole = ∆Peq ⇒ 5 ω2a 1 = = . 4 geq 230

∆Ppole gpole c = 2 ∆Peq geq − ω a a

ω y ds c g

ϕ r= a x

x dP = ρg dx pole pole c y dP = ρ(g − ω2y) dy eq eq a

x x dP = ρg d c pole pole c c y y y dP = ρ(g − ω2a ) d a eq eq a a a ∆Ppole = ∆Peq ⇒ 5 ω2a 1 = = . 4 geq 230

∆Ppole gpole c = 2 ∆Peq geq − ω a a

ω 1 − 5 y = (1 − ) ds 1 − 2 1 − ω2a c g 5 geq

ϕ r= a x

x dP = ρg dx pole pole c y dP = ρ(g − ω2y) dy eq eq a

x x dP = ρg d c pole pole c c y y y dP = ρ(g − ω2a ) d a eq eq a a a ∆Ppole gpole c = 2 ∆Peq geq − ω a a

ω 1 − 5 y = (1 − ) ds 1 − 2 1 − ω2a c g 5 geq

ϕ ∆Ppole = ∆Peq r= a x ⇒ 5 ω2a 1 = = . x 4 geq 230 dP = ρg dx pole pole c y dP = ρ(g − ω2y) dy eq eq a

x x dP = ρg d c pole pole c c y y y dP = ρ(g − ω2a ) d a eq eq a a a Shape of the Earth Comparison with the measures

7,35

measures

8,60

Newton: a − c = 27 km (instead of 21 km measured now) Mid-term conclusions

I Centrifugal force flattens the Earth at the poles a−c I We define flattening by = a ω2R 1 I Its order of magnitude is g ≈ 300 I The gravity on the polar axis is higher than that on the equatorial radius,

I To balance the pressure in the center, for a homogeneous, ellipsoidal, slightly ﬂattened Earth (Newton) it is necessary that:

5 ω2R 1 = = . 4 g 230 1718. Jacques Cassini. Traité de la grandeur et de la ﬁgure de la Terre. The Earth is elongated at the poles.

1737. Maupertuis. Report to the Royal Academy of Sciences. The Earth is ﬂattened. Newton was right.

Problem: the measures of ﬂattening (≈ 1/178) do not correspond well with Newton’s (1/230).

England 1 - France 0

1687. Newton. Philosophiae Naturalis Principia Mathematica. The Earth is ﬂattened at the poles. 1737. Maupertuis. Report to the Royal Academy of Sciences. The Earth is ﬂattened. Newton was right.

Problem: the measures of ﬂattening (≈ 1/178) do not correspond well with Newton’s (1/230).

England 1 - France 0

1687. Newton. Philosophiae Naturalis Principia Mathematica. The Earth is ﬂattened at the poles.

1718. Jacques Cassini. Traité de la grandeur et de la ﬁgure de la Terre. The Earth is elongated at the poles. England 1 - France 0

1687. Newton. Philosophiae Naturalis Principia Mathematica. The Earth is ﬂattened at the poles.

1718. Jacques Cassini. Traité de la grandeur et de la ﬁgure de la Terre. The Earth is elongated at the poles.

1737. Maupertuis. Report to the Royal Academy of Sciences. The Earth is ﬂattened. Newton was right.

Problem: the measures of ﬂattening (≈ 1/178) do not correspond well with Newton’s (1/230). The Earth is heterogeneous

Clairaut’s solution

At ﬁrst order, in the rotationnal parameter: 2 Clairaut equations ω R ρ(r) −−−−−−−−−−−→ ε(r) = C(r) g For an homogeneous model (Newton), C(r) = cst= 5/4.

For a 2000s’ model, as PREM, C(R) ≈ 0.96.

Still a problem between measures of flattening (1/178) and theory (Clairaut ≤ 1/230). But we get a theory for heterogenous planets. This will be Laplace’s work to reconcile measures and theory, which after several attempts, found in the 1820s an observed flattening between 1/304 and 1/310. Nowadays observed and hydrostatic flattenings are 1/298.25 and ≈1/299. Observe flattenings of planets in 2019

3 ways 1. Gravitationnal potential   2 ` GM R 0 0 X R m m U(r, θ, λ) = 1 + U Y (θ, λ) + U Y (θ, λ) r  r 2 2 r ` `  `,m

with

0 −3 U2 ' 10

m −6 U` . 10 .

0 Expression of U2 : C − A U0 = J := 2 2 MR2

C − A: moments of inertia diﬀerence Moments of inertia

Inertia is a degree 2 tensor that measures an object resistance to a change in rotation. 3 principal axis: x, y: equatorial axis z: polar axis Each one of the 3 moment of inertia is an indication of how far is the mass from the considered axis.

Mean equatorial moment of inertia: Z A = ρ(z2 + (x2 + y 2)/2) dV , V Polar moment of inertia: Z C = ρ(x2 + y 2) dV . V Moments of inertia difference: Z C − A = ρ((x2 + y 2)/2 − z2) dV . V The difference in moments of inertia (C − A) is an indication of how much excess mass is concentrated towards the equator. Thus their is a corresponding effect on the gravity field.

Mean inertia: C + 2A 2 Z I = = ρ(x2 + y 2 + z2) dV . 3 3 V For a spherical model: 8π Z R I = ρ(r)r 4 dr. 3 0 Z R M = 4π ρ(r)r 2 dr. 0 The inertia factor I /MR2 is an indication of how much mass is concentrated towards the center of the Earth.

For the Earth, above values give I = 0.3308 MR2 Moon: I = 0.39 MR2 Homogeneous case: I = 0.4 MR2 An amazing but very useful relation: Radau approximation (1885)

With a density stratification not too strange, the first-order hydrostatic deformation of a body depends on the density stratification only via the inertia factor: ! 5/2 ω2R3 J2 ≈ − 1 25 3 I 2 3GM 1 + 4 1 − 2 MR2

Proof. In the hydrostatic Clairauts’ equation, Radau make the following function appear:

2 1 + η − η f (η) = √2 10 1 + η

and he notices that this function is very often close to one (for e.g. PREM the diﬀerence is < 10−4). Then Clairauts’ equations can be solved. Summary about moments of inertia

I Moment of inertia depends on distribution of mass

I For planets, C > A because mass is concentrated at the equator as a result of the rotational bulge

I The gravity ﬁeld is aﬀected by the rotational bulge, and thus depends on C − A (or, equivalently, J2)

I So we can measure C − A (e.g. by observing a satellite’s orbit)

I If the body has no elastic strength, we can also predict the shape of the body given I (or we can infer I by measuring the shape)

I If we know M and I , we gain some constrain on density. Example - Ganymede

Anderson et al., Nature 1996

rock MoI constraint ice

Two-layer• Two- modelslayer models (3 parameters) satisfying satisfyingmass and massMoI constraints and inertia constraints. • Again, ifFor we Ganymede, specify one Jupiter unknown satellite, (e.g. rockR = density),2634 km. then C/MR = . ± . J C the other2 two0 are3105 determined0 0028 determined from 2 and 22. • Here C/MR2=0.31 – mass v. concentrated towards the centre F.Nimmo EART162 Spring 10 2. Precession of the equinox c z

y !

Terre b + − " " Lune x a +

3 GM C − A 2π ϕ¯˙ = − m cos(ε) ≈ 2 ω d 3 C 26000 year

C − A ≈ 3.279 × 10−3. C LETTER RESEARCH

Munich b N 5 6 Mount Agliale Wendelstein 0.4 m 5 E 4 Wendelstein 200 km 2 m Lajatico Lajatico 4 ux f 3 Ondrejov

San Marcello San Marcel Pistoiese 3 c Normalized 2 Konkoly 0.6 m As Asiago lo i 2 a ago

Wendel Konkoly 1 m Konkoly 1 Skalnat 1 Munich 3. Topography: Radar (orbiter) or Occultation (far) stei Skalnate n e Pleso Ondrej (Ortiz, 2017) Crni Vr 0 0 0100 200 300400 0100 200 300400

ux this resonance, and if the latter, for what reason. However, answering f 3 Ondrejov these questions remains out of reach of the present work. drawn assuming that it is coplanar with the ring, with planetocentric San Marcello San Marcel = ± Another important property of Haumea is its geometric albedo (pV), elevation Bring 13.7° 0.5°; see Fig. 3. The pink ellipse indicates the Pistoiese 3 which can be determinedc using its projected area, as derived from the 1σ-level uncertainty domain for the ring centre, and the blue ellipse inside

Normalized 15 it is the corresponding domain for Haumea’s centre. To within error bars, 2 Konkoly occultation, and its absolute magnitude . We find a geometric albedo 0.6 m As the ring and Haumea’s centres (separated by 33 km in the sky plane) cannot Asiago lo p = 0.51 ± 0.02, which isi considerably smaller than the values of 2 a V ago be distinguished, indicating that our data are consistent with a circular . +.0 062 0.7–0.75 and 0804−0.095 derivedWendel from the latest combination of Herschel 8,16 ring concentric with the dwarf planet. The points labelled ‘a’, ‘ b’ and ‘c’ Konkoly 1 m and SpitzerKonkoly thermal measurements . The geometric albedo should be 1 Skalnat indicate the intersections of the a, b and c semi-axes of the modelled 1 even smaller if the contributions of the satellites and the ring to the Munich stei ellipsoid with Haumea’s surface. Skalnate absolute magnitude are larger than the 13.5% used here (see Methods). n e Pleso Ondrej 1,17,18 Because Haumea is thoughtCrni Vr to have a triaxial ellipsoid shape the figures of hydrostatic equilibrium, or on mass and previous volume 0 0 LETTER RESEARCH 0100 200 300400 0100 200 300400 with semi-axes a > b > c, the occultation alone cannot provide its determinations1. A value in the range 1,885–1,757 kg m−3 is far more Time since 03:06:00 UT (s) Time since 03:06:00 UT (s) three-dimensional shapeov unlessh we use additional information from in line with the density of other large TNOs and in agreement with the LETTER RESEARCH Figure 1 | Light curves of the occultation. Light curves in the form of the rotational light curve. From measurements performed in the days trend of increasing density versus size (see, for example, supplementary

before and after the occultation, and given that we knowMunich Haumea’s information in ref. 5, and refs 21 and 22). We also note that the axial normalized flux versus time (at mid-exposure) were obtained from the Figure 2 | Haumea’s projected shape. The blue lines are the occultation 16 b N different observatories that recorded the occultation (Table 1). The black chords of the main5 body projected rotation onto period6 the withsky plane, high asprecision seen from, ninewe determined the rotational ratios derived from the occultation are not consistent with those points and lines represent the light curves extracted from the observations. Mount Agliale Wendelstein observing sites (Table 1). Thephase red at segments the occultation are the uncertainties time.0.4 m It turns (1 outσ level) that Haumea was at its abso- expected from the hydrostatic equilibrium figures of a homogeneous 5 E 23 The blue lines show the best square-well-model fits to the main body and 200 km on the extremites4 of each chord,lute brightness as derived minimum, from the timingWendelstein which uncertainties means that the projected area of the body for the known rotation rate and the derived density. It has 2 m the ring at Konkoly, with square-well models derived from the assumed ring Lajatico Lajatico 24 in Table 1. We show the chordbody from was Crni at its4 Vrh minimum in dashed during line because the occultation. it is previously been hypothesized that the density of Haumea could ux width and opacity (W = 70 km and p′ = 0.5) at other sites. The red points and f −3 considerably uncertain.3 For theThe observatories magnitude change for which fromOndrejov two minimum telescopes to maximum absolute bright- be much smaller than the minimum value of 2,600 kg m reported San Marcello San Marcel lines correspond to the optimal synthetic profile deduced from the square- Pistoiese 3 were used we show only theness best determined chord. Celestial from north the Hubble (‘N’) and Space east Telescope(‘E’) is 0.32 magc (using previously, if granular physics is used to model the shape of the body well model fitted at each data point (see Methods). The rectangular profile in Normalized 2 Konkoly 7 are indicated in the upper rightimages corner, that togetherseparated with Haumea the0.6 m scale. and The Hi’iaka blue ). Using equation (5)As in instead of the simplifying assumption of fluid behaviour. From figure Asiago lo i green corresponds to the ring egress event at Skalnate Pleso, which fell in a arrow shows the motion of the star relative2 to the body. Haumea’s limb a ago ref. 19 together with the aspect angle in 2009 (when the observationsWendel 4 of ref. 25, we determine an approximate angle of friction of between readout time of the camera (see Fig. 3). The light curves have been shifted in 7 Konkoly 1 m Konkoly (assumed to be elliptical)1 has been fitted to the chords, accounting for the Skalnat were taken )1 and the occultation ellipse parameters, we derive the 10° and 15° for Haumea given the c/a ratio of about 0.4 that we deter- steps of 1 vertically for better viewing. ‘Munich’ corresponds to the Bavarian Munich stei uncertainties on the extremity of each chord (red segments).Skalnate The limb has = ± n following values for the semi-axes of the ellipsoid: a 1,161e 30 km, mined here. For reference, the maximum angle of friction of solid rocks σ Pleso Ondrej Public Observatory. Error bars are 1 . Crni Vr semi-major axis a0 ′ = 852 ± 2 km and semi-minor0 axis b′ = 569 ± 13 km, Extended Data Figure 2 | Map of Earth showing the locations of the observatories can be found in Table 1. The red marks at Trebur and Valle 0100 b200 = 8523004 ± 00 4 km01 and00 c200 = 5133004 ± 0016 km (see Methods). The resulting observatorieson Earth that recorded isthe occultation 45° (greenand dots). Thethat solid ofD’Aosta a fluid observatories indicateis 0°. the two Also,closest sites to the differentiation shadow path that and other lines mark the limits of the shadow path. Mount Agliale is indicated in recorded a negative occultation. The coordinates of Trebur observatory Time since 03:06:00 UT (s) Time since 03:06:00 UT (s) ov h the latter having a position angle Plimb = − 76.3° ± 1.2° counted positively20 −3 blue because the occultation by the main body was not positive there, are 49° 55′ 31.5″ N, 8° 24′ 40.6″ E and the coordinates of Valle D’Aosta 26 density of Haumea, using its known mass , is 1,885 ± 80 kg m , and buteffects the occultation by may the ring was detected.have The dashed an line important denotes observatory arerole 45° 47′ 22 ″ inN and 7°determining 28′ 42″ E. The shadow motion is from the final shape . Figure 1 | Light curves of the occultation. Light curves in the form of the centre of the shadow path. Note that Munich corresponds to the the bottom to the top of the figure. this resonance, and if the latter, for what reason. However, answering from the celestial north to the celestial east. Haumea’s equator has been location of the Bavarian Public Observatory. The complete names of the normalized flux versusits time volume-equivalent (at mid-exposure) were obtained diameter from the is 1,595Figure ± 2 | Haumea’s11 km. projected This diameter shape. The blue is lines deter are the- occultationChariklo, a body of around 250 km in diameter with a Centaur orbit these questions remains out of reach of the present work. drawn assumingdifferent that observatories it is coplanar that recorded with the occultation the ring, (Table with 1). The planetocentric black mined under the assumption that chordsthe ringof the main does body not projected contribute onto the sky plane, to asthe seen from(between nine the orbits of Jupiter and Neptune), was the first Solar System =points and lines± represent the light curves extracted from the observations. observing sites (Table 1). The red segments are the uncertainties (1σ level) Another important property of Haumea is its geometric albedo (p ), elevation Bring The 13.7° blue lines show 0.5°; the best see square-well-model Fig. 3. The fits pink to the mainellipse body andindicates the 7 V total brightness. For an upper limiton of the 5% extremites contribution of each chord, as (see derived Methods), from the timing uncertainties object other than the giant planets found to have a ring system . Shortly 1σ-level uncertaintythe ring at Konkoly,domain with for square-well the ring models centre, derived from and the assumed the blue ring ellipsein Table 1inside. We show the chord from Crni Vrh in dashed line because it is which can be determined using its projected area, as derived from the width and opacity (W = 70 km and p′ = 0.5) at other sites. The red points and 15 it is the corresponding domainthe real for Haumea’samplitude centre. of the To rotational within errorconsiderably light bars, curve uncertain. increases,For the observatories and for whichhence two telescopesafter that discovery, similar occultation features that resembled those occultation, and its absolute magnitude . We find a geometric albedo lines correspond to the optimal synthetic profile deduced from the square- were used we show only the best chord. Celestial north (‘N’) and east (‘E’) 8,9 the ring and Haumea’swell model fitted centres at eachthe data a(separated semi-axispoint (see Methods). by increases 33 The km rectangular in too. the profile skyThe in plane) volume-equivalentare indicated cannot in the upper right corner, diameter together within thethis scale. Thefrom blue Chariklo’s rings were found on Chiron , another Centaur. These pV = 0.51 ± 0.02, which is considerably smaller than the values of green corresponds to the ring egress event at Skalnate Pleso, which fell in a arrow shows the motion−3 of the star relative to the body. Haumea’s limb +. be distinguished,readout indicating time of the cameracase that (seeis our Fig.1,632 3 ).data The km light are curvesand consistent havethe been density shifted with in isa circular1,757 kg m . These two densities discoveries directed our attention to Centaurs and phenomenology 0 062 (assumed to be elliptical) has been fitted− to the chords, accounting for the 0.7–0.75 and 08. 04− . derived from the latest combination of Herschel steps of 1 vertically for better viewing. ‘Munich’ corresponds to the Bavarian uncertainties on the extremity of each chord3 (red segments). The limb has 0 095 ring concentric with the dwarfare considerably planet.σ The points smaller labelled than the ‘a’, ‘lowerb’ and limit ‘c’ of 2,600 kg m based on related to them to explain our unexpected findings. The discovery of and Spitzer thermal measurements8,16. The geometric albedo should be Public Observatory. Error bars are 1 . semi-major axis a′ = 852 ± 2 km and semi-minor axis b′ = 569 ± 13 km, indicate the intersections of the a, b and c semi-axes of the modelledthe latter having a position angle Plimb = − 76.3° ± 1.2° counted positively even smaller if the contributions of the satellites and the ring to the ellipsoid with Haumea’sthis resonance, surface. and if the latter, for what reason. However, answering fromOrtiz the celestial et north al., to the celestial 2017 east. Haumea’s equator has been 12 OCTOBER 2017 | VO L 550 | NAT U RE | 221 these questions remains out of reach of the present work. drawn assuming that it is coplanar with the ring, with planetocentric absolute magnitude are larger than the 13.5% used here (see Methods). elevation© B 2017ring = 13.7° Macmillan ± 0.5°; see Fig.Publishers 3. The pink Limited, ellipse indicates part theof Springer Nature. All rights reserved. Another important property of Haumea is its geometric albedo (pV), © 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. 1,17,18 which can be determined using its projected area, as derived from the 1σ-level uncertainty domain for the ring centre, and the blue ellipse inside Because Haumea is thought to have a triaxial ellipsoid shape the figures of hydrostaticoccultation, and its equilibrium, absolute magnitude 15or. We on find mass a geometric and albedo previous it is the volumecorresponding domain for Haumea’s centre. To within error bars, 1 −3the ring and Haumea’s centres (separated by 33 km in the sky plane) cannot > > pV = 0.51 ± 0.02, which is considerably smaller than the values of with semi-axes a b c, the occultation alone cannot provide its determinations . A value+. 0in062 the range 1,885–1,757 kg m be is distinguished, far more indicating that our data are consistent with a circular 0.7–0.75 and 08. 04−0.095 derived from the latest combination of Herschel 8,16 ring concentric with the dwarf planet. The points labelled ‘a’, ‘ b’ and ‘c’ three-dimensional shape unless we use additional information from in line with theand density Spitzer thermal of measurementsother large. TheTNOs geometric and albedo in shouldagreement be indicate with the intersections the of the a, b and c semi-axes of the modelled the rotational light curve. From measurements performed in the days trend of increasingeven smaller density if the contributions versus size of the (see, satellites for and example, the ring to the supplementary ellipsoid with Haumea’s surface. absolute magnitude are larger than the 13.5% used here (see Methods). before and after the occultation, and given that we know Haumea’s information inBecause ref. 5 Haumea, and isrefs thought 21 to and have a 22 triaxial). We ellipsoid also shape note1,17,18 thatthe figures the ofaxial hydrostatic equilibrium, or on mass and previous volume 16 with semi-axes a > b > c, the occultation alone cannot provide its determinations1. A value in the range 1,885–1,757 kg m−3 is far more rotation period with high precision , we determined the rotational ratios derivedthree-dimensional from the shapeoccultation unless we use additionalare not information consistent from inwith line with those the density of other large TNOs and in agreement with the phase at the occultation time. It turns out that Haumea was at its abso- expected fromthe the rotational hydrostatic light curve. From equilibrium measurements performed figures in theof daysa homogeneous trend of increasing density versus size (see, for example, supplementary 23 before and after the occultation, and given that we know Haumea’s information in ref. 5, and refs 21 and 22). We also note that the axial lute brightness minimum, which means that the projected area of the body for the rotation known period withrotation high precision rate16, weand determined the derived the rotational density. ratios derived It has from the occultation are not consistent with those phase at the occultation time. It24 turns out that Haumea was at its abso- expected from the hydrostatic equilibrium figures of a homogeneous body was at its minimum during the occultation. previously beenlute brightness hypothesized minimum, which meansthat thatthe the density projected area of of Haumeathe body23 for could the known rotation rate and the derived density. It has −3 24 The magnitude change from minimum to maximum absolute bright- be much smallerbody was than at its minimumthe minimum during the occultation. value of 2,600 kg m previously reported been hypothesized that the density of Haumea could The magnitude change from minimum to maximum absolute bright- be much smaller than the minimum value of 2,600 kg m−3 reported ness determined from the Hubble Space Telescope is 0.32 mag (using previously, if nessgranular determined physics from the Hubble is used Space toTelescope model is 0.32 the mag shape (using previously,of the body if granular physics is used to model the shape of the body 7 instead of theimages simplifying that separated assumption Haumea and Hi’iaka of 7fluid). Using behaviour.equation (5) in Frominstead of figure the simplifying assumption of fluid behaviour. From figure images that separated Haumea and Hi’iaka ). Using equation (5) in ref. 19 together with the aspect angle in 2009 (when the observations 4 of ref. 25, we determine an approximate angle of friction of between ref. 19 together with the aspect angle in 2009 (when the observations 4 of ref. 25, wewere determine taken7) and the an occultation approximate ellipse parameters, angle we of derive friction the 10° of and between 15° for Haumea given the c/a ratio of about 0.4 that we deter- 7 following values for the semi-axes of the ellipsoid: a = 1,161 ± 30 km, mined here. For reference, the maximum angle of friction of solid rocks were taken ) and the occultation ellipse parameters, we derive the 10° and 15° forb = Haumea 852 ± 4 km and given c = 513 the ± 16 ckm/a (see ratio Methods). of about The resulting 0.4 that on Earthwe deteris 45° and- that of a fluid is 0°. Also, differentiation and other following values for the semi-axes of the ellipsoid: a = 1,161 ± 30 km, mined here. For density reference, of Haumea, usingthe itsmaximum known mass20, angleis 1,885 ± of 80 kgfriction m−3, and ofeffects solid may rocks have an important role in determining the final shape26. its volume-equivalent diameter is 1,595 ± 11 km. This diameter is deter- Chariklo, a body of around 250 km in diameter with a Centaur orbit b = 852 ± 4 km and c = 513 ± 16 km (see Methods). The resulting on Earth is 45°mined and under that the assumptionof a fluid that is the 0°. ring Also, does not differentiation contribute to the (between and theother orbits of Jupiter and Neptune), was the first Solar System 20 ± −3 total brightness. For an upper limit of 5% contribution (see Methods), object other 26than the giant planets found to have a ring system7. Shortly density of Haumea, using its known mass , is 1,885 80 kg m , and effects may havethe real an amplitude important of the rotational role in light determining curve increases, and the hence final after shape that discovery,. similar occultation features that resembled those its volume-equivalent diameter is 1,595 ± 11 km. This diameter is deter- Chariklo, athe body a semi-axis of around increases too. 250 The km volume-equivalent in diameter diameter with in this a Centaur from Chariklo’s orbit rings were found on Chiron8,9, another Centaur. These case is 1,632 km and the density is 1,757 kg m−3. These two densities discoveries directed our attention to Centaurs and phenomenology mined under the assumption that the ring does not contribute to the (between the areorbits considerably of Jupiter smaller than and the Neptune),lower limit of 2,600 was kg m the−3 based first on Solarrelated Systemto them to explain our unexpected findings. The discovery of 7 total brightness. For an upper limit of 5% contribution (see Methods), object other than the giant planets found to have a ring system . Shortly12 OCTOBER 2017 | VO L 550 | NAT U RE | 221 the real amplitude of the rotational light curve increases, and hence after that discovery, similar occultation© 2017 features Macmillan Publishers that Limited, resembled part of Springer Nature.those All rights reserved. the a semi-axis increases too. The volume-equivalent diameter in this from Chariklo’s rings were found on Chiron8,9, another Centaur. These case is 1,632 km and the density is 1,757 kg m−3. These two densities discoveries directed our attention to Centaurs and phenomenology are considerably smaller than the lower limit of 2,600 kg m−3 based on related to them to explain our unexpected findings. The discovery of

M: observed by satellite orbit or other

C−A C−A 1. J2 := MR2 satellite, C spin (precession or libration) −→ A, C, I : by exact relations. C−A 2. J2 := MR2 satellite, ω axial rotation, and hydrostatic hypothesis = Radau Approximation: −→ A, C, I by hydrostatic hypothesis, approximative, and for small ω. a−c 3. := a geodesy or occultation, and 2 3 3J2 ω R = + . 2 2GM then same as 2. −→ A, C, I by hydrostatic hypothesis, and for small ω.

Amazing : it’s only because planets are not spherical that we can observe the mean (spherical) inertia I . Moment of inertia factors in the Solar System

Inertia factors for planets and satellites (C/MR2) https://en.wikipedia.org/wiki/Moment_of_inertia_factor The tides on satellites of giant planets 25

Ganymede includes components of degree and order 4 due to mass concentration that could be identified by disk-cap mass anomaly modeling (Palguta et al. 2009). The situation is even worse in the case of Europa and Callisto because the gravity passes at these objects where all in near equatorial orbit, so that only C22 could be determined (Anderson et al. 1998; Anderson et al. 2001b). However, the gravity field of Callisto presents a non-zero S22 coefficient suggesting that another source of potential may affect Callisto (Anderson et al. 2001b). The shape data available for the Galilean satellites are all consistent with ellipsoids in hydrostatic equilibrium (Anderson et al. 2001b; Schubert et al. 2004). As a consequence, constraints on the density profile may obtained from inferring the secular Love number k f from C22 through Eq. (54) and the Radau-Darwin equation Eq. (58). Results displayed in Table 3 indicate that the satellites are not homogeneous because their C/MR2 values are smaller than 0.4, i.e., the upper limit corresponding to a homogeneous spherical body. Models of these satellites match- ing both their axial moments of inertia C/MR2 and mean densities indicate that Io, Europa, and Ganymede present a core enriched in rock, while Callisto is mostly Momentundifferentiated of inertia (Schubert factorset al. 2004 of some and references Jupiter therein). and Saturn satellites

Table 3 Gravity data provided by the Galileo (a) and Cassini (b) missions. (a) Schubert et al. 2004, (b) MacKenzie et al. 2008. Rhea is not in hydrostatic equilibrium and the Radau-Darwin approximation can not be applied, (c) Iess et al. 2010 . The data for the gravity ﬁeld of Enceladus are not available yet.

6 6 2 Satellites C20(10 ) C22(10 ) C/MR k f

Io(a) -1859.5 2.7 558 0.8 0.37824 0.00022 1.3043 0.0019 ± ± ± ± Europa(a) -435.5 8.2 131.5 2.5 0.346 0.005 1.048 0.0020 ± ± ± ± Ganymede(a) -12753 2.9 38.26 0.87 0.3115 0.0028 0.804 0.018 ± ± ± ± Callisto(a) -32.7 0.8 10.2 0.3 0.3549 0.0042 1.103 0.035 ± ± ± ± Rhea(b) - 931 12 237.0 4.5 - - ± ± Titan(c) -31.808 0.404 9.983 0.039 0.3414 0.0005 1.0097 0.0039 ± ± ± ± Rambaux, Castillo-Rogez, 2013, Tides on Satellites of Giant Planets.

4.4.3 Saturnian satellites

The gravity and topography of the Saturnian satellites have been inferred from observations obtained by the Cassini-Huygens mission that arrived in the system on July 1st , 2004. The Cassini orbiter performed several flybys of all major satellites but only a few of these flybys have been dedicated to radio science tracking that en- ables gravity field measurement. So far, only the gravity fields of Titan, Enceladus, and Rhea have been obtained to degree two. For the other medium-sized satellites, only the mass has been determined from radio tracking of Cassini, so far (see a review in Krupp et al. 2010). Trans-Neptunian Objects • Composed of ice-rich bodies and are left over from the formation of the Solar system • Large TNO (>400 km) are supposed to be in hydrostatic equilibrium (e.g. Tancredi and Favre 2008)

• Observations by Earth-based ground telescope, space telescopes, Spitzer, Herschel, HST and the New Horizon extended mission • The accuracy of stellar occultation observations has significantly improved over the past decades and now reaches about a 20 km level for distant TNOs (e.g., Braga-Ribas et al. 2013).

• MacLaurin or Jacobi ellipsoids are used as proxy to interpret the measurements.

2 (Quaoar prediction 07-18-2018; LuckyStar) Most likely dwarf planets

https://en.wikipedia.org/wiki/Dwarf_planet The Astrophysical Journal Letters, 850:L9 (5pp), 2017 November 20 Rambaux et al.

Table 1 Physical Properties of Five Large TNOs

Bodies—Parameters Equivalent Mean Density Rotation Period Rotation Ratio References W2 −3 Radius (km)(kg m )(hours) prG (136472) Makemake 715±3.5 K 7.771±0.003 K (a), (b) +430 (50000) Quaoar 535±19 2180-360 8.84±0.01 0.085 (c), (d) 17.6788±0.0004 (c), (e) 555±2.5 1990±460 8.84±0.01 0.093 (f), (d) (90377) Sedna 497.5±40 K 10.273±0.002 K (g) +150 a (90482) Orcus 458.5±12.5 1530-130 10.47 0.087 (c), (d) +290 b (120347) Salacia 427±22.5 1290-230 6.5 0.267 (c), (h)

Notes. a Preferred period of Thirouin et al. (2010). b Preferred period of Thirouin et al. (2014). References. (a) Brown (2013); (b) Heinze & de Lahunta (2009); (c) Fornasier et al. (2013); (d) Thirouin et al. (2010); (e) Ortiz et al. (2003); (f) Braga-Ribas et al. (2013); (g) Gaudi et al. (2005); (h) Thirouin et al. (2014).

Figure 1. (a) Comparison between MacLaurin (violet) and the numerically computed solution at first (red), second (green), and third (blue) order. The Jacobi ellipsoid 2 case is expected for W = 0.374 but is not represented here. (b) Absolute difference between the equatorial radius δa=a −a (polar radius δc) computed from prG ML 3 the MacLaurin equation and Clairaut’s equations numerically solved at first (red), secod (green), and third (blue) order. The results in δa are plotted with lines, whereas the results of δc are represented with the red cross line at first order, the green square line at second order, and the blue circle line at third order. The horizontal line represents 2.5km whereas the vertical line represents e=0.67. The radius of the body is assumed to be 500 km.

Figure 2. Difference δν=νML−ν3=(a−c)ML/R−(a−c)3/R computed for the homogeneous case (MacLaurin ML) and a two-layer model, developed to Figure 3. Solution of ν3=(a−c)3/R computed for the two-layer model, the order of 3. The horizontal lines represent the uncertainties in the mean developed to the order of 3, for the parametric space covered in this study and densities. The error bars on the rotation periods are too small to be legible on this for speciﬁc bodies. σ=0 corresponds to a homogeneous body of density equal −3 plot. σ=0 corresponds to a homogeneous body of density equal to 920 kg m−3, to 920 kg m , whereas σ=1 corresponds to a homogeneous body of density −3 whereas σ=1 corresponds to a homogeneous body of density 3000 kg m−3. 3000 kg m . 2 Selection of large TNO 30 Eris 25

20 (h) 15

Period Orcus Sedna 10 Chariklo Makemake Ceres Quaoar 5 Salacia Haumea 0 0 200 400 600 800 1000 1200 Radius (km)

Rotation period determined from light curve, radius from light curve or stellar occultation method 3 1544 SelectionA. C. Barr andof M. large E. Schwamb TNO

3.0 Sedna Makemake PC 10-1 OV Haumea 2.5 Eris

) OV 3

Triton ED

q -2 2.0 Quaoar 10 Pluto Orcus Charon Hau Orcus Density (g/cm 1.5 -3 10 QW Salacia 1.0 0500100015002000 1.0 1.5 2.0 2.5 3.0 3 Radius (km) Primary Density (g/cm ) 5 Figure 2. Satellites-to-total mass ratio, q as a function of primary density Figure 1. Updated values for the densitiesAdapted and sizesfrom ofBarr the dwarfand Schwamb planets,(2016) for the dwarf planet systems Pluto/Charon, Orcus/Vanth, Eris/Dysnomia, including Neptune’s satellite Triton. Quaoar/Weywot, and Haumea. Table 2. Satellite-to-total system mass ratio, q for each of the dwarf planet systems. the differentiation state of the precursor objects (see e.g. Leinhardt & Stewart 2012 for discussion). Primary Satellite q We hypothesize that the two categories represent two classes of Eris Dysnomia 0.0253 0.006 collision, each of which has a different effect on the densities of the ± Haumea Hi’iaka 0.0049 0.007 final objects. Systems with large moons originate in low-velocity, ± Orcusa Vanth 0.0292 grazing collisions between undifferentiated precursors. These colli- ∼ Orcusb Vanth 0.0794 sions involve little-to-no vaporization or melting, and so both bodies ∼ Pluto Charon 0.1086 0.001 retain their primordial compositions. Thus, the primordial com- ± Quaoar Weywot 0.000 53 0.0002 position of Kuiper belt material can be inferred from the present ± Notes. aAssuming Orcus and Vanth have equal albedo. bAssuming compositions of members of these systems. Systems with rock-rich the albedo of Vanth is half that of Orcus (Brown et al. 2010). primaries and small moons originate in a different type of collision (see Brown 2008 and references therein), perhaps a ‘graze and between size and density; the updated masses and radii determined merge’ (Leinhardt, Marcus & Stewart 2010), or another type of in the last few years seem to support this conclusion. We are also collision yet to be identified by numerical simulations. interested in the relative sizes of the satellites versus the primary, which we describe as q Msatellites/(Msatellites Mp ), the ratio between the combined masses= of the satellites and+ the total system 3.1 Lower density, large moons mass, where M is the mass of the primary. p 3.1.1 Pluto/Charon Table 2 summarizes the q valuesforeachofthesystems.Toderive q , we assume equal albedos and densities for all of the bodies in the Successful hydrocode simulations of impact scenarios for the for- Eris and Quaoar systems. Thus, our values of q are upper limits; if mation of the Pluto system involve an impactor-to-total mass ratio, the small satellites are more ice-rich than their primaries, the true q γ 0.3–0.5, implying that precursor bodies range from 0.3 MT = values for Eris and Quaoar could be smaller. For Haumea, we use to 0.7MT (Canup 2011), where the total mass of the Pluto/Charon 25 estimates of the masses of each satellite from the multibody orbital system, MT 1.463 10 g (Stern et al. 2015). The success- fits of Ragozzine & Brown (2009). For Orcus/Vanth, the assumption ful collisions= are gentle,× with impact velocities v v , where imp ≈ esc of equal albedo is known to be inaccurate. Following (Brown et al. vesc √(2GMT )/(Ri Rt ), where Ri and Rt are the radii of the im- 2010), we report two values of q ,oneassumingthebodieshavethe pactor= and target, respectively+ (Canup 2011); for the Pluto/Charon -1 same albedo, and the other where we assume the albedo of Vanth is system, vesc 1kms . The peak shock pressure at the point of ∼ 2 a factor of 2 lower than Orcus, which may be more consistent given impact, P ρvimp 1 GPa (Melosh 1989), is barely high enough the colour difference observed between the two bodies. to melt ice∼ (Stewart∼ & Ahrens 2005;Barr&Citron2011). Impact simulations show that the temperature rise in the interiors of both bodies #T tens of K (Canup 2005). The successful collision is so 3TWOCLASSESOFDWARFPLANET oblique that∼ the bodies do not undergo significant mixing or modify DENSITIES their original densities. We find that the systems fall into two categories when classified by Canup (2011) shows that the mixed ice/rock composition of 3 q (see Fig. 2). Excluding Triton, bodies with densities >2gcm− , Charon is best reproduced by impacts in which one or both objects Haumea, Quaoar, and Eris, have extremely small satellites, indi- are undifferentiated (see also Desch 2015). The most successful cated by small q values. More ice-rich bodies, including Pluto, cases are those in which 90 per cent of both precursor bodies are Charon, and Orcus, are part of systems with large moons. In a col- composed of an intimate mixture of ice and rock, with the remain- lision, the mass of the final moon and the system’s final q value ing 10 per cent of their masses composed of pure water ice, likely depend on the speed, angle, and impactor-to-total mass ratio, and in an outer ice shell (Canup 2011). If both bodies have undergone

MNRAS 460, 1542–1548 (2016) Solutions for large TNO

220 200 Makemake The Astrophysical 180 Journal Letters, 850:L9 (5pp), 2017 November 20 Rambaux et al. 160 140 120

(a-c) (km) 100 Salacia 80 Sedna 60 Quaoar 40 20 Orcus 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 920 kg/m3 σ 3000 kg/m3 18 Figure 4. Solutions of the five selected TNOs (see Table 1) for (a − c) as a function of the mass silicate ratio σ. The line with circles represents the heterogeneous solutions computed from Clairaut’s equations and the line with squares are plotted for the MacLaurin solutions. The two sets of solutions are close at σ=0 and 1 as expected (see Figure 1(a) for an estimation of the differences). The figure shows that the difference between the homogenous and stratified solutions can be larger than the observation uncertainties, at least for Makemake and Salacia. is defined as e2=1−(c/a)Observational2, with a and c as the uncertainties equatorial between on radius 6 and 13≈ hr and25 the km. interval for the silicate mass ratio and polar radii, and it is determinedObservational via (e.g., Chandrasekhar uncertaintiesbetween on a 0− andc 1.≈ An50 icy homogeneouskm. body with a density of 1969) Observation of a − c →constraint920 kgon m− density3 corresponds stratification to σ=0, whereas the σ=1 case is for a purely silicate body with a density of 3000 kg m−3. Note W2221- e 1 - e2 = 32arcsin6--ee2 ,2 that the silicate mass ratio σ is not a linear function of the mean prG e3 ()e2 ()density. Figure 2 presents the differences δν=νML−ν3 between where Ω is the angular spin velocity, G is the gravitational the homogeneous cases (index ML for MacLaurin) and constant, and ρ is the density. stratified cases computed to the order of three (index 3). We fi The MacLaurin equation does not apply to strati ed bodies, have superimposed in this figure the data with uncertainties for whose shapes are not ellipsoids anymore (Hamy 1889; the five bodies listed in Table 1. The white area located at the Moritz 1990). On the other hand, by comparing the Maclaurin left corner of this figure is excluded, because it corresponds to solution with that of Clairaut’s equations for a homogeneous bodies with an eccentricity larger than 0.67 for the stratified body, we can explore the validity of the Clairaut solution as a case (see Section 3). The fast rotating and low density models function of the eccentricity. Figure 1(a) shows the solutions for present a large signature of their stratification in the computed the MacLaurin and numerical Clairaut solutions for a shapes and the maximum difference in δν reaches a value of homogeneous body. In order to determine the difference 0.108. This value, converted to δ (a−c)=(a−c) − between the two solutions directly in the shape, we estimate the ML (a−c)3, is equal to 54 km for a 500 km radius body. difference in the equatorial (and polar) radius for MacLaurin The limit δν=0.07, or δ (a−c)=35 km for a 500 km and numerical solution at orders of one, two, and three. These radius body, is on par with the current level of ground-based differences, quoted δa (and δc), are represented in Figure 1(b) observational uncertainties taking 24 km for the equatorial as a function of e. In this paper, we will require that δa be fi radius uncertainty (e.g., Braga-Ribas et al. 2013) and assuming smaller than 2.5 km to be sure to have a suf cient numerical the same value for the polar radius uncertainty. Consequently, precision with respect to the observationnal uncertainties. all bodies above that curve i.e., δν 0.07 (dark red and black Indeed, this value is 10 times smaller than the uncertainty in colors) would potentially present a detectable signature of equatorial radius from Braga-Ribas et al. (2013; of 24 km). It internal stratification with current state-of-the-art measurements. also corresponds to e0.67 (Figure 1(b)), and for that reason In addition, similar mapping in period P and silicate mass we truncate our results of the next section at that threshold. ratio σ, excluding the area where e>0.67, provide an ’ The Clairaut s equations can be scaled by the mean spherical estimation of the impact of the order on the solution. The radius R (i.e., Lanzano 1974; Chambat et al. 2010). Conse- difference between second and first order ν −ν is at quently, the quantity ν=(a−c)/R, corresponding to the ratio 2 1 maximum 0.03 or (a−c)2−(a−c)1=15 km, where the between the equatorial and polar radii difference divided by the index represents the order in the Clairaut’s equations. The mean radius, is independent of R. The advantage of using ν is difference between third and second order, ν −ν , reaches to be able to compare the solutions for TNOs selected in 3 2 0.01 or 5 km when converted into (a−c) −(a−c) . These Table 1. 3 2 values show the necessity to use the second- and third-order development for the solutions located at the boundary at the left 4. Results and Discussion corner maps of Figure 2 for accuracy about 2.5 km. Below we This section presents the equilibrium figures computed as a describe the results specific to each of the five bodies in Table 1 function of the rotation period P and the silicate mass ratio σ and Figures 2–4. The last figure represents the (a− c) for each with comments for the five TNOs. The rotation period ranges body as a function of the mass silicate ice ratio σ for

3 Hydrostatic equilibrium: general solution

The general question: ﬁnd the shape of a rotating body (= shape of equidensity surfaces) that is self-gravitating and in hydrostatic equilibrium.

gradp = ρ gradW ∇2W = −4πGρ + 2ω2. (1) p = 0 at surface Property: In an hydrostatic body, equipotential sufaces are equipressure and equidensity surfaces. Interfaces are such surfaces.

Proof gradp = ρ gradW (1)

gradρ ∧ gradW = 0. Taking the curl of (gradp)/ρ shows that they are also equidensity surfaces. Take the jump of (1), it yields JK gradp = ρ gradW J K J K Decompose ∂p gradp = n + grad p, ∂n T use the fact that the pressure is continuous:

gradT p = gradT p = 0, J K J K then ∂p n = ρ gradW J ∂n K J K that is: gradW // n i.e. an interface is an equipotential surface (and equidensity and equipressure). The surface of a lake is horizontal. The general question becomes: ﬁnd the shape of a body (= shape of equidensity surfaces) which equidensity surfaces are equipotential surfaces, with the density and potential related by

∇2W = −4πGρ + 2ω2.

Problem: the potential depends on the shape. No mathematical method to solve this implicit question.

Solutions • For ω = 0: the spherical stratification is the only solution. • For ω 6= 0: no general solution. More precisely: - For ρ = cst. For rotation rate not too large, we know 2 solutions: an ellipsoid of revolution (Maclaurin ellipsoid), a triaxial ellipsoid (Jacobi ellipsoid). Analytical exact solutions. But we don’t know if they are the only ones. - For ρ 6= cst. We don’t know if any exact solution exists. For small ω, we can determine an approximate solution: the solution of the Clairaut’s equations (differential or integro-differential). For greater ω, more numerical calculations exist. Non-perturbative methods, for any ω

Some brut-force numerical methods since the 1990s’.

The ’concentric MacLaurin spheroid’ method (Hubbard 2012, 2013, Debras 2018). Used to interpret Juno mission data.

Mapping of the potential onto a reference sphere (Maitra & Al-Attar 2019). Approximate solution for small ω (Clairaut et al.)

ω2R ε(r) = C(r) g

ω2R Deﬁne m = g C(r) depends on the density stratiﬁcation ρ(r) and on m. If m is small, we write

2 3 ε = C(r) m = C1 m + C2 m + C3 m ...

All Ci depends on ρ(r) and can be computed numerically.

The ﬁrst order C1 can is computed by a resolution of the classical Clairauts’ o.d.e., and can be approached by the Radau solution. Clairaut’s theory modernized and explained A good way to learn how to use perturbations

r s(r, θ, λ)

Actual hydrostatic conﬁguration (Earth) in black, spherical reference conﬁguration in red.

s(r, θ, λ) = r + h(r, θ, λ) with h r

W (r, θ, λ) = W0(r) + δW (r, θ, λ) with δW W0 (2) ρ(r, θ, λ) = ρ0(r) + δρ(r, θ, λ) with δρ ρ0, r s(r, θ, λ)

Density on a sphere of radius r is chosen equal to that on the corresponding deformed surface: ρ(s, θ, λ) = ρ0(r) i. e. ρ(r + h, θ, λ) = ρ0(r + h, θ, λ) + δρ(r + h, θ, λ) = ρ0(r). A ﬁrst-order development in h/r gives:

∂ρ0 ρ0(r + h) ≈ ρ0(r) + h. ∂r By noting with a dot the radial derivative, we then obtain

δρ(r, θ, λ) = −ρ˙0(r) h(r, θ, λ). (3) r s(r, θ, λ)

The fact that the surfaces considered are equipotential reads:

W (s, θ, λ) = W0(r) + C(r). By proceeding in the same way as before :

δW (r, θ, λ) = −W˙ 0 h(r, θ, λ) + C(r) = −g0 h(r, θ, λ) + C(r). (4) Poisson equation relate the potential to the density and rotation: ∇2W = −4πGρ + 2ω2 in the Earth 2 ∇ W0 = −4πGρ0 in the spherical reference Substraction: ∇2(δW ) = −4πGδρ + 2ω2. With the previous two relationships, and by choosing C so that its laplacian is 2ω2 (C is therefore the spherical part of the centrifugal potential), we obtain a partial diﬀerential equation veriﬁed by h : 2 −∇ g0(r)h(r, θ, λ) = 4πGρ˙0(r) h(r, θ, λ). (5)

ρ0(r) is known, which allows to calculate g0(r). We then want to determine h(r, θ, λ). Spherical harmonic functions

Spherical harmonics are the equivalent of Fourier series on the sphere: they apply to a function of the two variables θ and θ λ of the spherical reference frame. Their property is that any function h(θ, λ) can be broken down into the form:

∞ ` X X m m h(θ, λ) = h` Y` (θ, λ), `=0 m=−` λ Historique Mesure et observation Analyse spatiale Variabilité temporelle Harmoniques sphériques m Représentation where ` is the degree of the harmonic, m is the order, h are the m ` Comment retrouver l’allure d’une harmonique sphérique ? coefficients of development, and Y` are the spherical harmonic I Y s’annule sur 2m méridiens et l m parallèles lm I Exemples : functions that thus form a basis for the functions defined on the !# s’annule sur !" s’annule sur !" s’annule sur Y20" s’annule sur Y22" s’annule sur Y$42 s’annule sur - 0 cercle méridien - 2 cercles méridiens sphere. 2 0 = 0 méridiens, 2 - 22= cercles4 méridiens, méridiens 2 2 = 4 méridiens, ⇥- 2 parallèles ⇥- 0 parallèles ⇥- 2 parallèles m et 2 0 = 2 parallèles : et 2 2 = 0 parallèle : et 4 2 = 2 parallèle : The Y oscillate in θ and λ. They cancel each other out on m ` meridian circles, and ` − m parallel (fig.). For example, the first harmonic is constant : Y 0 = 1, and to 0 represent a flattened Earth at the poles, we use Y 0, which only 2 cancels out on two parallels: Examples of spherical harmonics. The color indicates the 3 cos2 θ − 1 value of the function : red for Y 0(θ) = . positive values, blue for nega- 2 2 tive values. An essential property is that they are eigen-functions of the Laplacian operator: `(` + 1) ∇2Y m = − Y m. (6) ` r 2 ` Let’s decompose h thanks to spherical harmonics :

∞ ` X X m m h(r, θ, λ) = h` (r)Y` (θ, λ). `=0 m=−` Inject in: 2 −∇ g0(r)h(r, θ, λ) = 4πGρ˙0(r) h(r, θ, λ). (7) Use the eigen-function property, and properties :

2g0 div(g0) =g ˙0 + = −4πGρ0, r 2f˙ ∇2f (r)Y m(θ, λ) = f¨, Y m + , Y m + f ∇2Y m, ∀f , ` ` r ` ` as well as by introducing ρ¯, the average density in the radius sphere r, which can be deﬁned by : 4πGρ¯r g0(r) = ... 3 ...equation (7) becomes : ∞ l X X 2 ρ0 (` − 1)(` + 2) h¨m(r) + 3 − 1 h˙ m(r) + hm(r) Y m(θ, λ) = 0. ` r ρ¯ ` r 2 ` ` l=0 m=−l m Since the Y` forms a base, it comes : 2 ρ0 (` − 1)(` + 2) h¨(r)m + 3 − 1 h˙ m(r) + hm(r) = 0 ∀ `, m, r. (8) ` r ρ¯ ` r 2 ` Boundary conditions 1. In the center, there must not be a hole; therefore, the surfaces s(r, θ, λ) =cst must converge to the center, which is expressed as s(0, θ, λ) = 0 i.e h(0, θ, λ) = 0, ∀(θ, λ). i.e. m h` (0) = 0. (9)

2. At the surface W and gradW are continuous. 2 2 Outside the Earth: ∇ (δWext) = 2ω . This gives an additional condition on δW , which depends on ω: rh˙(R) − (` − 1)h(R) = 0 if (`, m) 6= (2, 0) 5 ω2R2 (10) rh˙(R) − (` − 1)h(R) = − if (`, m) = (2, 0). 3 g0(R) Case ` = 0. Only deﬁnes the sphere in relation to which the height is measured h. By convention choosing the sphere that has the same average radius as the deformed 0 Earth, we have the same zero solution h0 = 0 (this also comes from the choice of C).

Case ` 6= 0. For (`, m) 6= (2, 0) the system 8-9-10 (equation and boundary conditions) is completely homogeneous : the zero function, h = 0, is the obvious solution. Poincaré (1902) showed that it was the only solution, whatever the density ρ0(r), as long as it decreases with the radius. The general solution therefore contains only one term, in (`, m) = (2, 0) : 0 0 h(r, θ, λ) = h2(r) Y2 (θ, λ). Physically, this is due to the fact that the rotation potential is a sum of only two spherical harmonics, the harmonic ` = 0 and the harmonic (`, m) = (2, 0).

The shape of the Earth is therefore known, it is given by : 0 0 s(r, θ, λ) = r + h2(r) Y2 (θ, λ). It is an (nearly) ellipsoid of revolution around the axis of rotation.

We can express the relation with the ﬂattening: s(θ = π/2) − s(θ = 0) 3h0(r) = = − 2 . r 2r The new condition at the center is obtained with a series development of r in the vicinity of the center. All eqs. become: Clairaut’s 1st order equations

ρ ˙ ρ ¨+ 6 0 + 6 0 − 1 = 0 ρ¯ r ρ¯ r 2 ˙(0) = 0 5 ω2R R˙(R) + 2(R) = 2 g0(R) At discontinuities in density, , ˙ are continuous (because the potential and its derivative are continuous).

Knowing ρ0(r) from an average model of the Earth such as the PREM model, the angular rate of rotation of the Earth ω, we are therefore able to solve this differential equation numerically, and to deduce from it the hydrostatic shape of the Earth. The calculated flattening of the Earth is about 1/300 at the surface and 1/415 at the centre. This calculation can be improved by doing it up to the second order in (and up to the third order for some dwarf planets). We then find about 1/299 for the surface flattening, very close to the observed one, which is 1/298.25. Hydrostatic equilibrium is a very good model for the shape of the Earth. Second order code (soon third order): http://frederic.chambat.free.fr/geophy/ 1/e (k) & 1/e2 (r) 420

400

380 %

& 360

340

320

300

280 0 1 2 3 4 5 6 7 6 x 10 Rayon de la Terre (m)

The density (g/cm3)andgravity(m/s2) functions use the left scale. The mass (1024 kg) pressure (1011 Pa) and 3 2 24 binding energy (1026 MJ),Figure: use the right.Left: The PREMdensity model of (pink, the Earth inprovides g/cm the following), gravity numbers : (green, Binding in m/s ), mass (blue, in 10 11 Energy = 24.84608 x 1025kg),MJ, Pressure pressure at center (cyan, of Earth =in 363.65 10 GP,Pa) Surface and Gravity gravitational = 9.8129 m/s2,Radius energy (red) derived from the PREM model.=6371km. The density and gravity functions use the left scale. The mass, pressure and gravitational energy (1026 MJ), use the right. The PREM model of the Earth provides the following numbers: Binding Energy = 24.84608 x 1025 MJ, Pressure at center of Earth = 363.65 GP, Surface Gravity = 9.8129 m/s2, Radius = 6371 km. Ref: http://www.preearth.net/worlds-collide.html Right: Evolution of the inverse of flattening as a function of radius, determined by injecting the PREM model into the differential equation in . The flattening is minimal at the center of the Earth. Thanks to Clairaut’s calculation, we know how to calculate it in such a way as to correct gravimetric, seismological and other observations for the ellipticity effect:

1/e (k) & 1/e2 (r) 420

400

380 %

& 360

340

320

300

280 0 1 2 3 4 5 6 7 6 x 10 Rayon de la Terre (m) −→

The density (g/cm3)andgravity(m/s2) functions use the left scale. The mass (1024 kg) pressure (1011 Pa) and binding energy (1026 MJ), use the right. The PREM model of the Earth provides the following numbers : Binding

Energy = 24.84608 x 1025 MJ, Pressure at center of Earth = 363.65 GP, Surface Gravity = 9.8129 m/s2,Radius

=6371km. Near-end conclusions

I Celestial bodies are ﬂattened by rotation

I For large bodies (>several 100 km radius), hydrostatic theory is most often a good approximation to determine the shape

I For a small rotation rate, the shape is close to an ellipsoid of revolution

I Clairaut equations are very nice to calculate this shape (up to several order in the rotation rate)

I For a larger rotation rate, more complex numerical codes are required

I The ﬂattening of a body can be observed by several ways:

I a. geometrical (the shape), I b. gravitational (potential or gravity), I c. rotational (precession or libration)

I They help to constrain the inner density stratiﬁcation

I b+c is the best

I If we have a. or b., hydrostatic hypothesis has to be used to constrain the density stratiﬁcation. Post-conclusion: Haumea fascinating case

Ellipsoid with 3 inequalLETTER axis RESEARCH May be a Jacobi ellipsoid

Munich b N 5 6 Mount Agliale Wendelstein 0.4 m 5 E 4 Wendelstein 200 km 2 m Lajatico Lajatico 4 ux f 3 Ondrejov

San Marcello San Marcel Pistoiese 3 c Normalized 2 Konkoly 0.6 m As Asiago lo i 2 a ago

Wendel Konkoly 1 m Konkoly 1 Skalnat 1 Munich stei Skalnate n e Pleso Ondrej Crni Vr 0 0 0100 200 300400 0100 200 300400

Time since 03:06:00 UT (s) Time since 03:06:00 UT (s) ov h Figure 1 | Light curves of the occultation. Light curves in the form of normalized flux versus time (at mid-exposure) were obtained from the Figure 2 | Haumea’s projected shape. The blue lines are the occultation different observatories that recorded the occultation (Table 1). The black chords of the main body projected onto the sky plane, as seen from nine points and lines represent the light curves extracted from the observations. observing sites (Table 1). The red segments are the uncertainties (1σ level) The blue lines show the best square-well-model fits to the main body and on the extremites of each chord, as derived from the timing uncertainties the ring at Konkoly, with square-well models derived from the assumed ring in Table 1. We show the chord from Crni Vrh in dashed line because it is = ′= width and opacity (W 70 km and p 0.5) at other sites. The red points and considerably uncertain. For the observatories for which two telescopes lines correspond to the optimal synthetic profile deduced from the square- were used we show only the best chord. Celestial north (‘N’) and east (‘E’) well model fitted at each data point (see Methods). The rectangular profile in are indicated in the upper right corner, together with the scale. The blue green corresponds to the ring egress event at Skalnate Pleso, which fell in a arrow shows the motion of the star relative to the body. Haumea’s limb readout time of the camera (see Fig. 3). The light curves have been shifted in (assumed to be elliptical) has been fitted to the chords, accounting for the steps of 1 vertically for better viewing. ‘Munich’ corresponds to the Bavarian uncertainties on the extremity of each chord (red segments). The limb has σ Public Observatory. Error bars are 1 . semi-major axis a′ = 852 ± 2 km and semi-minor axis b′ = 569 ± 13 km, the latter having a position angle Plimb = − 76.3° ± 1.2° counted positively this resonance, and if the latter, for what reason. However, answering from the celestial north to the celestial east. Haumea’s equator has been these questions remains out of reach of the present work. drawn assuming that it is coplanar with the ring, with planetocentric = ± Another important property of Haumea is its geometric albedo (pV), elevation Bring 13.7° 0.5°; see Fig. 3. The pink ellipse indicates the which can be determined using its projected area, as derived from the 1σ-level uncertainty domain for the ring centre, and the blue ellipse inside occultation, and its absolute magnitude15. We find a geometric albedo it is the corresponding domain for Haumea’s centre. To within error bars, the ring and Haumea’s centres (separated by 33 km in the sky plane) cannot pV = 0.51 ± 0.02, which is considerably smaller than the values of +.0 062 be distinguished, indicating that our data are consistent with a circular 0.7–0.75 and 08. 04−0.095 derived from the latest combination of Herschel 8,16 ring concentric with the dwarf planet. The points labelled ‘a’, ‘ b’ and ‘c’ and Spitzer thermal measurements . The geometric albedo should be indicate the intersections of the a, b and c semi-axes of the modelled even smaller if the contributions of the satellites and the ring to the ellipsoid with Haumea’s surface. absolute magnitude are larger than the 13.5% used here (see Methods). Because Haumea is thought to have a triaxial ellipsoid shape1,17,18 the figures of hydrostatic equilibrium, or on mass and previous volume with semi-axes a > b > c, the occultation alone cannot provide its determinations1. A value in the range 1,885–1,757 kg m−3 is far more three-dimensional shape unless we use additional information from in line with the density of other large TNOs and in agreement with the the rotational light curve. From measurements performed in the days trend of increasing density versus size (see, for example, supplementary before and after the occultation, and given that we know Haumea’s information in ref. 5, and refs 21 and 22). We also note that the axial rotation period with high precision16, we determined the rotational ratios derived from the occultation are not consistent with those phase at the occultation time. It turns out that Haumea was at its abso- expected from the hydrostatic equilibrium figures of a homogeneous lute brightness minimum, which means that the projected area of the body23 for the known rotation rate and the derived density. It has body was at its minimum during the occultation. previously been hypothesized24 that the density of Haumea could The magnitude change from minimum to maximum absolute bright- be much smaller than the minimum value of 2,600 kg m−3 reported ness determined from the Hubble Space Telescope is 0.32 mag (using previously, if granular physics is used to model the shape of the body images that separated Haumea and Hi’iaka7). Using equation (5) in instead of the simplifying assumption of fluid behaviour. From figure ref. 19 together with the aspect angle in 2009 (when the observations 4 of ref. 25, we determine an approximate angle of friction of between were taken7) and the occultation ellipse parameters, we derive the 10° and 15° for Haumea given the c/a ratio of about 0.4 that we deter- following values for the semi-axes of the ellipsoid: a = 1,161 ± 30 km, mined here. For reference, the maximum angle of friction of solid rocks b = 852 ± 4 km and c = 513 ± 16 km (see Methods). The resulting on Earth is 45° and that of a fluid is 0°. Also, differentiation and other density of Haumea, using its known mass20, is 1,885 ± 80 kg m−3, and effects may have an important role in determining the final shape26. its volume-equivalent diameter is 1,595 ± 11 km. This diameter is deter- Chariklo, a body of around 250 km in diameter with a Centaur orbit mined under the assumption that the ring does not contribute to the (between the orbits of Jupiter and Neptune), was the first Solar System total brightness. For an upper limit of 5% contribution (see Methods), object other than the giant planets found to have a ring system7. Shortly the real amplitude of the rotational light curve increases, and hence after that discovery, similar occultation features that resembled those the a semi-axis increases too. The volume-equivalent diameter in this from Chariklo’s rings were found on Chiron8,9, another Centaur. These case is 1,632 km and the density is 1,757 kg m−3. These two densities discoveries directed our attention to Centaurs and phenomenology are considerably smaller than the lower limit of 2,600 kg m−3 based on related to them to explain our unexpected findings. The discovery of

Extended Data Figure 6 | Rotational light curve of Haumea. The relative (absolute brightness minimum) is reached at the time of the occultation magnitude versus rotational phase obtained two days after the occultation (arbitrarily located at a phase of 0 here), which means that the projected with the Valle D’Aosta 0.81-m telescope with no filters is shown. The area of Haumea was also at its minimum. The continuous line is a fit to rotational zero phase was established at the time of the occultation and the data. The peak-to-peak amplitude of the light curve is 0.25 ± 0.02 mag. the rotation period used was 3.915341 h. Superimposed is a fit to the Error bars are 1σ. observational data. As can be seen, the absolute maximum in magnitude

Video

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. The recent occultation of Haumea indicate that its axes are ≈ 1161 km x 852 km x 523 km and its bulk density is ≈ 1885 kg m−3.

The ﬁrst triaxial (non axisymmetric) hydrostatic celestial object, predicted theoretically by Jacobi (1834).

This is not consistent with an hydrostatic equilibrium Jacobi ellipsoid (Ortiz, 2017), but with a diﬀerentiated triaxial ellipsoid ﬂuid in hydrostatic equilibrium (Dunham 2019).

Problem: this is not consistent with Hamys’ theorem (1889): unless homogeneous, a self-gravitating ﬂuid in hydrostatic equilibrium cannot be composed of (exact) ellipsoidal layers (even if for small rotation rate, the ﬁgure is close to an ellipsoid).

Needed: a method to calculate the equilibrium ﬁgure of an heterogeneous triaxial body. References

I Chandrasekhar, Subrahmanyan, 1987, Ellipsoidal figures of equilibrium, New Haven, Yale Univ. Press, 1969 ; 2 e éd., New-York, Dover. [Maclaurin and Jacobi ellipsoids]. I Dahlen, F. & Tromp, J., 1998. Theoretical global seismology, Princeton university press. [First order Clairaut equations and Radau approximation]. I Chambat F. Ricard Y., Valette B., 2010, Flattening of the Earth: further from hydrostaticity than previously estimated, G.J.I., 183, 727-732. doi: 10.1111/j.1365-246X.2010.04771.x. The Nature highlight: 14 oct. 2010, v. 467, p. 755. [Second order Clairaut equations]. I Rambaux N., Chambat F., Castillo-Rogez J. C., 2015, Third-order development of shape, gravity, and moment of inertia for highly flattened celestial bodies. Application to Ceres. Astronomy & Astrophysics, 584, A127. doi: 10.1051/0004-6361/201527005. [Third order Clairaut equations]. I Rambaux N., Baguet, D., Chambat F., Castillo-Rogez J. C., 2017, Equilibrium shapes of large trans-neptunian objects. Ap. J. Letters, 850:L9. doi: 10.3847/2041-8213/aa95bd. [Application of third order equations]. I Tassoul, Jean-Louis, 1978, Theory of rotating stars, Princeton, Princeton Univ. Press. [General stuff on equilibrium figures; only in this first edition]. Historical references

I Clairaut, Alexis Claude, 1743, Théorie de la Figure de la Terre, David Fils ou Durand, Paris, seconde édition chez Courcier, Paris, 1808. I Deparis, Vincent, Legros, Hilaire, 2000, Voyage à l’intérieur de la Terre, CNRS- Éditions, Paris. I Lacombe, Henri, Costabel, Pierre (dir.), 1988, La figure de la Terre du xviiie siècle à l’ère spatiale, Acad. Sci., Paris, Gauthier-Villars. I Tisserand, Isaac, 1891, Traité de mécanique céleste, Tome II, Théorie de la figure des corps célestes et de leur mouvement de rotation, Gauthier-Villars, Paris. I Todhunter, Isaac, 1873, A History of the mathematical theories of attraction and the figure of the Earth from the time of Newton to that of Laplace, Londres, Macmillan, 1873 ; réimpression New York, Dover, 1962. I Trystram, Florence, 1979, Le procès des étoiles , Seghers, Paris.

Newton I Newton, Isaac: Philosophiae naturalis principia mathematica, Londres, Jussu Socie- tatis Regiae, 1687; 2e éd., Cambridge, Auctior et Emendiator, 1713; 3e éd., Londres, G. & J. Innys, 1726. I Newton, Isaac: The Mathematical Principles of Natural Philosophy, Translation Andrew Motte, London, at the Middle-Temple-Gate. 1729. Second english translation I. Bernard Cohen, Anne Whitman, University of California Press, Berkely, 2016. I Newton, Isaac: Philosophiae naturalis principia mathematica, éd. et commentaires PP. Thomas Le Seur et François Jacquier, Genève, Barrilot et fils, 1739 (vol. I), 1740 (vol. II), 1742 (vol. III et IV). I Newton, Isaac: Principes mathématiques de la philosophie naturelle, trad. fr. et commentaires Mme du Châtelet, 2 vol., Paris, Dessaint & Saillant et Lambert, 1756 ou 1759. Reprint Blanchard, 1966, Gabay, 1990. I Newton, Isaac, Du Châtelet, 2015, Principes mathématiques de la philosophie naturelle. La traduction française des Philosophiae naturalis principia mathematica. Édition critique du manuscrit par Michel Toulmonde, 2 vol, Ferney-Voltaire, Centre Int. d’étude du xviiie s I Chandrasekhar, Subrahmanyan, Newton’s Principia for the Common Reader, Oxford University Press, 1995. Clairaut’s Theory of the figure of the Earth (1743) is a major work of hydrostatics and in my opinion the most important on the shape of the Earth. You will find quality information on Clairaut, and in great quantity, on the extraordinary site directed by Olivier Courcelle: www.clairaut.com

You can watch some beautiful presentations ﬁlmed at the Academy of Sciences for the tricentenary of Clairaut (2013). Have a nice second week in Barcelonette.

End N. Rambaux et al.: Ceres Hydrostatic equilibrium

Table A.1. Preferred model of Ceres based on Thomas et al. (2005) and Drummond et al. (2014) observations with a 5 km frozen ocean layer. 3 3 The mean density used for the calculation is 2077.02 kg m and 2156.00 kg m , respectively, to facilitate comparison with these papers.

Composition Density Mean Radius Mean Radius Mean Radius Mean Radius 3 (kg m ) C1 (km) C2 (km) C3 (km) C4 (km) Fe-FeS 5500 - - 80 80 Serpentine Assemblage 2600 - - 413.981 418.870 Silicate 2700 - 408.989 - - Salts 1500 - - 418.981 423.870 Ice 1000 - 476.2 476.2 470.5 Mean density 2077.02 476.2 - - -

McCord, T. B. & Sotin, C. 2005, Journal of Geophysical Research (Planets), 110, 5009 McKinnon, W. B. 1997, Icarus, 130, 540 Milliken, R. E. & Rivkin, A. S. 2009, Nature Geoscience, 2, 258 Millis, R. L., Wasserman, L. H., Franz, O. G., et al. 1987, Icarus, 72, 507 Moritz, H. 1990, The ﬁgure of the Earth : theoretical geodesy and the Earth’s interior (Karlsruhe : Wichmann, c1990.) Murray, C. D. & Dermott, S. F. 1999, Solar system dynamics (Cambridge Univ. Press) Polanskey, C. A., Joy, S. P., Raymond, C. A., & Rayman, M. D. 2014, in 13th International Conference on Space Operations 2014, SpaceOps Conferences - AIAA 2014-1720, 2116 Rambaux, N., Castillo-Rogez, J., Dehant, V., & Kuchynka, P. 2011, A&A, 535, A43 Raymond, C. A., Jaumann, R., Nathues, A., et al. 2011, Space Sci. Rev., 163, 487 Rivkin, A. S. & Volquardsen, E. L. 2008, in Lunar and Planetary Science Conference, Vol. 39, Lunar and Planetary Science Conference, 1920 Schubert, G., Anderson, J., Zhang, K., Kong, D., & Helled, R. 2011, Physics of the Earth and Planetary Interiors, 187, 364 Scott, H. P., Williams, Q., & Ryerson, F. J. 2002, Earth and Planetary Science Letters, 203, 399 Tassoul, J.-L. 1978, Theory of rotating stars (Princeton Series in Astrophysics, Princeton: University Press, 1978) Thomas, P. C., Parker, J. W., McFadden, L. A., et al. 2005, Nature, 437, 224 Tricarico, P. 2014, ApJ, 782, 99 Equilibrium shape of large differentiatedZharkov, V. N., Leontjev, V. V., & Kozenko, V.TNO A. 1985, Icarus, 61, 92

N. Rambaux (1), F. Chambat (2), J.C. Castillo-Rogez (3), D. Baguet (4,1) (1) IMCCE, Obs. Paris – PSL Research University, Sorbonne Universités – UPMC univ. P06, univ LilleAppendix 1, CNRS, 77 Av. A: Denfert-Rochereau, Table for the poster 75014, Paris, France, [email protected] These models include an ice layer thickness of density 1000-1200 kg/m3 varying between 0 and 80 km (step 20 km); it overlays (2) LGLTPE, CNRS UMR5276, ENS de Lyon, Site Monod, 15 parvis René Descartes; Lyon, F-69007, France 3 (3) Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, United States a mixture of ice, salts, and silicates (clays) ranging from 0 to 220 km. We fixed the density of that layer at 1000-2000 kg/m , 50 (4) Institut UTINAM, CNRS-UMR 6213, Observatoire de Besançon, Université de Franche-Comté, BPkg 1615,/m3 25010 steps forBesançon computer Cedex, time France sake. The deepCredit: NASA/JPL-Caltech/UCLA/MPS/DLR/IDA interior is made up of a hydrated silicate layer with a density varying between 2610 and 2990 kg/m3, that may overlay a metallic core of density of 5500 kg/m3 of up to 80 km radius. We solve the equation of mass conservation and deduce R , the hydrated silicate radius. Figure 2 shows the results of (a c) to third order as a function of Si Hydrated Introduction Methods:R Siheterogeneous Hydrated with color points spheroids as function of the densities chosen for the salt-clay layer. The large trans-Neptunian objects (TNO) with radii larger than 400 km are Homogeneous Interior: MacLaurin’s ellipsoid is a shape solution of self- thought to be in hydrostatic equilibrium. Their shapes can provide clues regarding gravitating, rotating, and homogeneous bodies in hydrostatic equilibrium. For this 2 2 their internal structures that would reveal information on their formation and MacLaurin's ellipsoid,It has the been eccentricity long known e defined that as an e ellipsoid= 1 – (c/a) can, with be a anand hydrostatic c the self-gravitating equilibrium figure of a rotating homogeneous evolution. We model equilibrium shapes of moonless TNO assuming equatorial and polar radii, respectively, is given by (e.g. Chandrasekhar 1969) homogeneous and heterogeneous interior models by numerically integrating planet. For this Maclaurin’s ellipsoid, the eccentricity e (e.g. Chandrasekhar 1969) is given after Clairaut’s equations of rotational equilibrium expanded up to third order in the ⌦2 p1 e2 1 e2 geodetic parameter. Indeed, a level of accuracy better than a few kilometers is = (3 2e2) arcsin e 3 , (A.1) required for modeling those objects that are rapid rotators. We show that the 2⇡G⇢ e3 e2 difference between the equilibrium figures for homogeneous and heterogeneous assumptions can reach several kilometers for fast rotating and low density 2 where ⌦ is the angular spin velocity, G the gravitational constant, ⇢ the density and e the eccentricity defined by e2 = 1 c , with bodies. Such a difference could be measurable by ground-based techniques. where Ω is the angular spin velocity, G the gravitational constant, and ρ the mean a a and c the equatorial and polar radii, respectively. density. ⇣ ⌘ Differentiated Interior: In this case the MacLaurin equation does not apply and the 15 ⌦2 Inputs and interior models The equation Eq.(A.1) is either solved numerically or expanded in series of the small parameter " = 8 ⇡G⇢ . This yields the shape is not expansionan ellipsoid ofanymore (a c ):(Hamy 1889; Moritz 1990). As a consequence a Dwarf planets are observed by Earth-based telescopes, space telescopes, and numerical integration of third-order Clairaut equations developed by Kopal (1960) the Dawn mission in the case of Ceres. Their properties are determined via and Lanzano (1974) has been1 solved23 numerically655 (Chambat et al. 2010; Rambaux et infrared spectroscopy, photometry, and stellar occultations. The determination al. 2015, 2017).a Thec = numericalR " +estimate"2 is+ valid until"3 + a ocertain("3) . value of eccentricity. (A.2) e 2 168 7056 of the bulk densities is realized through the tracking the position of satellites or From FigureMacLaurin 1, which presents thesolutions comparison between (homogeneous MacLaurin! and numerical ) by fitting a MacLaurin or a Jacobi ellipsoid to the observed shape. solution for an homogeneous planet, we infer this limit as e = 0.6. 3 2 TheTable Astrophysical 1 lists Journal 10 Letters,dwarf850:L9 planets(5pp), 2017 with November their 20 physical properties (radii, bulk densities, Rambaux et al. 0.5 where Re = pa c is the equivolumetric radius of the ellipsoid. We will use this expansion to quantify the uncertainties in the and rotation periods determined by theTable measurements.) 1 3d order Physical Properties of Five Large TNOs computation of the shape parameters at each2d order order. 1st order Bodies—Parameters Equivalent Mean Density Rotation Period Rotation Ratio References W2 0.4 −3 Radius (km)(kg m )(hours) prG (136472) Makemake 715±3.5 K 7.771±0.003 K (a), (b) +430 (50000) Quaoar 535±19 2180-360 8.84±0.01 0.085 (c), (d) Figure 1: Solution for the MacLaurin ellipsoid. En rouge valeur pour Ceres. 17.6788±0.0004 (c), (e) ρ 0.3 555±2.5 1990±460 8.84±0.01 0.093 (f), (d)

(90377) Sedna 497.5±40 K 10.273±0.002 K (g) G

+150 a π (90482) Orcus 458.5±12.5 1530-130 10.47 0.087 (c), (d) MacLaurin b +290 / (120347) Salacia 427±22.5 1290-230 6.5 0.267 (c), (h) 2 3.2 Calcul approché Ω 0.2 Equilibrium Figure of DwarfNotes.Table Planets 1 : Physical properties of five TNOs, (a) Brown (2013); (b) Heinze & October de Lahunta (2009); 2016 (c) Fornasier etDraft No 1. a al.Preferred (2013); period (d) of Thirouin et al.et (2010al. (2010);). (e) Ortiz et al. (2013); (g) Gaudi et al. (2005); (h) Thirouin et al. (2014). b Preferred period of Thirouin et al. (2014). On chercher àrésoudre l’équation Eq. (2) avec un développement limitéen posant comme

References. (a) Brown (2013); (b) Heinze & de Lahunta (2009); (c) Fornasier et al. (2013); (d) Thirouin et al. (2010); (e) Ortiz et al. (2003); (f) Braga-Ribas et al. 15 ⌦2 (Barr2013); (g )andGaudi etSchwamb al. (2005); (h) Thirouin (2016) et al. (2014 have). suggested that large Kuiper Belt Objects may 0.1 petit paramètre " = et l’excentricitése développe alors comme 8 ⇡G⇢ c 2 be classified in two groups. One group presents a primordial composition with a Figure 1:e = Solution1 for the MacLaurin ellipsoid.Article number, En rouge page 9 of valeur 9 pour Ceres. mean density of 1800 kg/m3 whereas the second group could lost icy material 1 2 III. Interior models e2 = " "2 + "3 +ro("3). a (6) during collisions and present a larger density. Here, we range possible interior 0 7 49 ⇣ ⌘ models for our pull of objects assuming a simple two-layer structure assuming 0 0.2 0.4 0.6 0.8 1 silicate core and icy shell with densities: ρ = 920 kg/m3 and ρ = 3000 kg/m3. Le coecient 15/8 dans la définition de " sert àavoir un développement plus simple. ice sil La di↵érencee (a c) est alors égale à The silicate mass fraction f isThen given we express by our results in terms of silicate mass fraction f as a function of 3.2 Calcul approché the bulk density and layer densities: Figure 1 : comparison between MacLaurin and the numerically 1computed23 1st, 2d655 • Whichand 3rd order. order The Jacobiis required? ellipsoid case is expect(a forc )=Ω2/πRgρ = "0.374+ but"2 is+ not "3 + o("3) . (7) On cherchere 2 168 àrésoudre7056 l’équation Eq. (2) avec un développement limitéen posant comme ⇢sil(⇢ ⇢ice) represented here. ⇣ 2 ⌘ f = – (1)Expansion of the MacLaurin 15 ⌦ ⇢(⇢ ⇢ ) Remarque 1 petit paramètre " = 8 ⇡G⇢ et l’excentricitése développe alors comme Figure 1. (a) Comparison between MacLaurin (violetsil) and the numericallyice computed solution at first (red), second (green), and third (blue) order. The Jacobi ellipsoid formulae 2 7 case is expected for W = 0.374 but is not represented here.(b) Absolute difference between the equatorial radius δa=a −a (polar radius δc) computed from prG ML 3 On pourrait vouloir introduire le paramètre géodétique(Rambaux et al.q 2015)définit par the MacLaurin equation and Clairaut’s equations numerically solved at first (red), secod (green), and third (blue) order. The results in δa are plotted with lines, whereas 1 2 the results of δc are represented with the red cross line at first order, the greenResults: square line at second order,Shape and the blue circle Estimates line at third order. The horizontal for line Layered Hydrostatic Bodies 2 2 3 3 where ⇢sil is the silicate density,represents 2.5⇢kmice whereasthe the vertical density line represents e=0.67. of The radius ice of the and body is assumed⇢ tois be 500 the km. mean density. ⌦2a3 e = " " + " + o(" ). (6) q = . 7 (8) 49 GM 3 3 13920 kg/m 3000 kg/m 0.09 Figure 2 shows the difference between Il représente le rapport entreLe l’accélération coecient centrifuge 15/8 et dans l’accélélération la définition gravitationnelleà de " sert àavoir un développement plus simple. (a-c) computed for a layered model and l’équateur. Maintenant, si R représente le volume équivalent et la relation suivante est IV. Hydrostatic equilibrium model 0.08 e La di↵érence (a c) est alors égale à 12 (a-c) for a homogeneous model obtenue en exprimant la masse totale en fonction de a etb : (MacLaurin). This difference increases 220 0.07 2 with increasing rotation periods and 200 4 b ⌦ 1 23 2 655 3 3 11 Makemake q = = m (a c)=Re " + (9) " + " + o(" ) . (7) Un corps en rotation prend la forme d’équilibre imposéeOrcus par son 0.06 potentieldecreasing centrifuge densities (small et son value of f). 180 3 a ⇡G⇢ ✓ ◆ 2 168 7056 The difference may reach 20~25 km that 160 10 Sedna 5 ⇣ ⌘ auto-gravité(voir les références). Pour cette étude, nous utilisons 0.05 les équationsis of the order of de possible Clairaut detection has in et qui conduit à " = 2 m et on retrouve le premier terme de développement dans (a c)=

ν 140

5 δ Makemake detection (Ortiz et al 2012, Re 4 m. Cependant la définitionRemarque du rayon équivolumétrique 1 R se complique lorsque la développées jusqu’au 3eme ordre 9 (Lanzano 1974) qui ont étéutiliséjusqu’au 0.04 2eme ordre 120 P (hours) Brown 2013). forme n’est plus ellipso¨ıdale et est di↵érente de Re. Voir par exemple l’expression (30)

Quaoar (a-c) (km) 100 Salacia pour l’étude de la Terre (Chambat)Figure 2. Difference δν et=νML au−ν3=(a troisième−c)ML/R−(a−c)3/R computed ordre pour 0.03 Cérès (Rambaux etal 2015; dans Chambat and Valette (2001). On peut aussi utiliser un rayon moyennésur la sphère for the homogeneous 8 case (MacLaurin ML) and a two-layerMakemake model, developed to Figure 3. Solution of ν3=(a−c)3/R computedFigure for the 3 two-layer shows model, solutions of the five 80 On pourrait vouloir introduire le paramètre géodétique q définit par the order of 3. The horizontal lines represent the uncertainties in the mean developed to the order of 3, for the parametric space covered in this study and R (voir les emailsSedna du 15/06/2015 avec F. Chambat). densities. The error bars on the rotation periods are too small to be legible on this for specific bodies. σ=0 corresponds to a homogeneous body of density equal s −3 −3 selected TNOs (see Table 1) for (a − c) as 60 Park etal 2016). plot. σ=0 corresponds to a homogeneous body of density equal to 920 kg m , to 920 kg m , whereas 0.02σ=1 corresponds to a homogeneous body of density Quaoar −3 −3 2 3 whereas σ7=1 corresponds to a homogeneous body of density 3000 kg m . 3000 kg m . a function of the mass silicate ratio σ. The 40 ⌦ a Salacia 2 0.01 line with circles represents the Remarque 2 Orcus q = . (8) e>0.6 20 Estimation de la précision de la méthode en comparant avec MacLaurinheterogeneous et 1er-3eme solutions ordre. computed from 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GM 6 Dans la formule Eq. (7) on utilise R et dans le calcul numérique R . La di↵érence entre Clairaut’s equations and the line with e s 0 0.2 0.4 0.6 0.8 1 σ Description du sphéroide/ellipsoide. squares are plotted for the MacLaurin les deux s’exprime en termesIl représente des aplatissements le rapport àl’ordre 2. entre Donc, l’accélération seuls les termes à centrifuge et l’accélélération gravitationnelleà The figurel’ordre shows 2 qui that se the combine difference avec between epsilon the pour homogenous être des termes d’ordre 3 interviennent Ecrire σ solutions. The two sets of solutions are l’équateur. Maintenant, si Re représente le volume équivalent et la relation suivante est close at σ=0 and 1 as expected. and stratifiedles expressions solutions. can be larger than the observation uncertainties, at least for Makemakeobtenue and Salacia en. exprimant la masse totale en fonction de a et b : 3 2 3 (a c)=Re ...O(" )... = Rs(1 1/4f2 +7/12f4 + ....)(...O(" )...) (10) Conclusion 3 2 2 = Rs(...O(" )... 1/8f " +7/24f4" + ....) 4 b (11) ⌦ 2 q = = m (9) Stellar occultation techniques have significantly improved over the past decade and now reach an accuracy of a few kilometers for distant dwarf planets. In 3 a ⇡G⇢ preparation for the interpretation of future observations in terms of interior structure, it is important to develop accurate shape models. As many of the dwarf ✓ ◆ planets are fast rotators the development of the Clairaut equations to high order is required to properly represent their shapes. 5 et qui conduit à " = 2 m et on retrouve le premier terme de développement dans (a c)= References: Chambat, F., Ricard, Y., & Valette, B. 2010, Geophysical Journal International, 183, 727, Chandrasekhar, S. 1969, ed. Yale University Press, Hamy, M. 1889, Annales5 de l’Observatoire de Paris, 19, F1, Kopal, S. 1960, ed. The University of Wisconsin Press, Lanzano, P. 1974, Astrophysics and Space Science, 29, 161, Moritz, H. 1990, ed. Karlsruhe : WichmannRe ,,m Rambaux,. Cependant N., Baguet D., la définition du rayon équivolumétrique R se complique lorsque la Chambat, F., and Castillo-Rogez, J., 2018, ApJ, 850:L9. Acknowledgments: NR is grateful to the Observatory of Paris grant. Part of this work was carried out at the Jet Propulsion4 Laboratory, California Institute of Technology, under contract to NASA. forme n’est plus ellipso¨ıdale et est di↵érente de Re. Voir par exemple l’expression (30) dans Chambat and Valette (2001). On peut aussi utiliser un rayon moyennésur la sphère Rs (voir les emails du 15/06/2015 avec F. Chambat).

Remarque 2

Dans la formule Eq. (7) on utilise Re et dans le calcul numérique Rs. La di↵érence entre les deux s’exprime en termes des aplatissements àl’ordre 2. Donc, seuls les termes à l’ordre 2 qui se combine avec epsilon pour être des termes d’ordre 3 interviennent Ecrire les expressions. (a c)=R ...O("3)... = R (1 1/4f 2 +7/12f + ....)(...O("3)...) (10) e s 2 4 = R (...O("3)... 1/8f 2" +7/24f " + ....) (11) s 2 4

4 Caveats measured (a-c) • Assumes hydrostaticity equilibrium • Presence of topography mountains or deep craters may also perturb the interpretation of the shape in terms of hydrostatic equilibrium shape.

Body Altitude (km) Makemake 6.81 Quaoar 8.53 Sedna 9.79 Ceres 9.87 (From Johnson & McGetchin 1973) Orcus 20.21 Salacia 30.52 19