A pre-Caloris synchronous rotation for Mercury
Mark A. Wieczorek, Alexandre C. M. Correia, Mathieu Le Feuvre, Jacques Laskar, Nicolas Rambaux
AA pre-Caloris synchronous rotation for Mercury
Mark A. Wieczorek, ∗ Alexandre C. M. Correia, Mathieu Le Feuvre, Jacques Laskar, Nicolas Rambaux Institut de Physique du Globe de Paris, Univ Paris Diderot4 avenue de Neptune, 94100 Saint-Maur des Foss´es, France Departamento de F´ısica da Universidade de Aveiro, Campus Universit´ario de Santiago3810-193 Aveiro, Portugal Laboratoire de Plan´etologie et G´eodynamique, Universit´e de Nantes, France2 rue de la Houssini`ere, BP 92208, 44322 Nantes Cedex 3, France Astronomie et Syst`emes Dynamiques, IMCCE-CNRS UMR8028Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris, France Universit´e Pierre et Marie Curie–Paris 6; IMCCE-CNRS UMR8028Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris, France ∗ To whom correspondence should be addressed; E-mail: [email protected].
The planet Mercury is locked in a spin-orbit resonance where it rotates three times aboutits spin axis for every two orbits about the Sun [1, 2]. The current explanation for thisunique state assumes that the initial rotation of this planet was prograde and rapid, andthat tidal torques decelerated the planetary spin to this resonance [3–6]. When core-mantle boundary friction is accounted for, capture into the 3/2 resonance occurs with a26% probability, but the most probable outcome is capture into one of the higher-orderresonances [7]. Here we show that if the initial rotation of Mercury were retrograde, thisplanet would be captured into synchronous rotation with a 68% probability. Strong spa-tial variations of the impact cratering rate would have existed at this time, and these areshown to be consistent with the distribution of pre-Calorian impact basins observed byMariner 10 and M ESSENGER . Escape from this highly stable resonance is made possibleby the momentum imparted by large basin-forming impact events [8–10], and capture intothe 3/2 resonance occurs subsequently under favourable conditions. a r X i v : . [ a s t r o - ph . E P ] D ec ollowing the discovery that Mercury was in a 3/2 spin-orbit resonance [1, 2], a number ofstudies have attempted to explain how this planet might have come to occupy such a particularstate. It was shown that as the spin rate of this planet evolved from an initial rapid prograderotation, capture could occur not only in the 3/2 resonance, but other resonances as well [3–5].With a molten core [11], capture into many of the higher-order resonances, most notably the 2/1,would occur with high probability [5, 12]. Since non-synchronous resonant spin states can bedestabilized at times of low orbital eccentricity [6], and since Mercury’s eccentricity is knownto vary substantially over time [13], ultimate capture into the 3/2 resonance is made possible.The 26% probability of ending in the current configuration [6, 7] makes this scenario plausible,but at the same time, somewhat unsatisfactory.Planetary accretion models imply that the initial spin of the terrestrial planets could havebeen either prograde or retrograde with equal probability [14–16], leading us to investigate howan initial retrograde rotation of Mercury would have evolved. Given that the orbital evolutionof this planet is chaotic, such that it is not possible to predict precisely its evolution beyond afew tens of millions of years, we have performed a statistical study to characterize the likelyoutcomes of initial retrograde rotation. Using a numerical model that considers tidal dissipa-tion, planetary perturbations, and core-mantle boundary friction [7], the orbital and rotationalequations of motion of this planet were integrated for 1000 cases over a 100 million year timeperiod, starting with very close initial conditions and an initial −
10 day rotational period.The theoretical probability of capture in retrograde resonances is extremely low, and noevents were observed. The prograde 1/2 resonance is the first important resonance encountered,and capture in this state occurs about 29% of the time. Some of these captures are destabilizedat times of low eccentricity (Fig. 1), and the most likely outcome is to end in synchronousrotation, which occurs with a 68% probability (Table 1).Escape from the synchronous resonance cannot occur by variations in eccentricity alone [6],but evolution beyond this state is possible through the momentum imparted to a planet duringa basin-forming impact event [8–10]. Following such an event, the spin rate could have beenincreased beyond that of the 3/2 resonance, and capture into the present configuration couldhave occurred as tides decelerated the planet. Alternatively, the impact could have been justlarge enough to have unlocked the planet from the synchronous resonance, and the spin ratecould then have been tidally accelerated to the 3/2 resonance at times of high orbital eccentricity.For a given impact velocity and an average impact geometry, we determine the minimumbolide size for these two unlocking scenarios to occur, and then use standard impact craterscaling laws to estimate the corresponding crater size [10]. An event forming a basin with adiameter between about 650 and 1100 km is required for a direct transfer from synchronousrotation to the 3/2 resonance (Fig. 2). About 14 such basins are known to exist (SupplementaryTable 1), but only half of these would have formed with an impact direction that would have2ncreased the rotation rate of the planet, and some of these could have formed before the planetwas captured into synchronous rotation. The formation of the 3.73 billion year old Calorisimpact basin [17], which is the largest basin on the planet with a diameter of about 1450 km,would have been the last event capable of performing such a direct transfer. This basin wasformed by an oblique impact [18] that likely increased the rotation rate [19], and given its greatsize, it is possible that this event could have spun the planet up to even higher-order resonances,such as the 4/1. The crater sizes necessary to have just increased the rotation rate of Mercurybeyond the synchronous resonance are smaller, between about 250 and 450 km, and about 40suitably sized basins are known to exist.Additional integrations of the rotational and orbital evolution of Mercury were used to quan-tify the likely outcomes following an impact that destabilizes the synchronous resonance. Forthree different post-impact rotation rates, 1000 simulations were run starting 3.9 billion yearsago and ending at the present day (Supplementary Fig. 1). For the case where the impact justunlocks the planet from synchronous rotation, the planet is recaptured into this resonance 20%of the time, and capture into the 3/2 resonance occurs 56% of the time (Table 1). When the post-impact rotation rate lies between the 3/2 and 2/1 resonances, capture into the 3/2 resonance isnearly assured with a 96% probability. When the post-impact rotation rate is greater than thatof the 4/1 resonance, the probability of capture into the 3/2 resonance is 26%, analogous toprevious studies that assume initial rapid prograde rotation.In contrast to previous work, our scenario for capture into the 3/2 resonance makes specificpredictions about the density of impact craters on the surface of this planet. Though the cra-tering rate should be spatially uniform for a planet in non-synchronous rotation, the crateringrates would have been highly asymmetric during periods of synchronous rotation. The trajecto-ries of most asteroids and comets intersect the orbit of Mercury at high angles, and geometricconsiderations imply that the highest impact rates should occur in the centre of the daylit andunilluminated hemispheres. The Earth also exhibits this behaviour, as is indicated by the radi-ants of sporadic meteors [20]. During synchronous rotation, the axis of the planet’s minimumprincipal moment of inertia would have been directed towards the Sun, and this axis passesthrough ◦ and ◦ E on the equator [21].The expected cratering rates for synchronous rotation are calculated numerically [17, 22].Impact probabilities with a model population of planet-crossing objects [23] were first deter-mined, the relative approach velocities and inclinations were used to determine the impact co-ordinates on the planet, and scaling laws were used to convert the projectile diameters into craterdiameters. Synchronous rotation predicts that the average cratering rate should have varied sys-tematically across the surface by more than a factor of ten (Fig. 3), with the lowest crateringrates being located on the equator at ± ◦ longitude. The impact velocities vary by a factorof two between the western and eastern hemispheres, and similar results are obtained using the3rbital elements of the known near-Earth asteroids (Supplementary Fig. 2).We test whether spatial variations in the ancient impact-cratering rate ever existed on Mer-cury by analyzing the distribution of large impact basins, the vast majority of which are olderthan the Caloris basin. Impact basins identified by Mariner-10 based geologic mapping [24–26]and M ESSENGER flyby images [27] (Supplementary Table 1) are utilized. Since the spatialdensity of craters smaller than 100 km in diameter have been affected by more recent geologicprocesses [27], and since the numerous craters that are somewhat larger than this might be insaturation [27], we use only those basins that are larger than 400 km (Fig. 3). These are dividedinto four classes based on the reliability of their identification and stratigraphic age. The firstthree contain only those basins that are known to be equal in age or older than Caloris, whereasthe last corresponds to basins that have an unknown or uncertain relative stratigraphic age withrespect to this event.The distribution of large impact basins is striking. First, there are very few large basins onthe hemisphere from 0 to 180 ◦ E longitude. This is consistent with the variations predicted bysynchronous rotation, but as the solar illumination conditions over much of the flyby images ofthis hemisphere are sub-optimal for crater identification [27], this agreement may be equivocal.Second, there is a clear deficit of basins centred near the equator and 90 ◦ W, which is alsoconsistent with synchronous rotation. No basin centres are located within a circle with a ◦ angular radius close to this point, even though the low sun elevations of the M ESSENGER imagesin this region are favourable to the detection of impact basins.The significance of these observations is quantified using Monte Carlo simulations of theaverage angular distance of basins to the point (90 ◦ W, 0 ◦ N). Only those basins that lie onthe Mariner-10 imaged hemisphere are analyzed since this hemisphere has been subjected todetailed geologic mapping, and since this hemisphere has been imaged under favourable illu-mination conditions [27]. The probability that the observed value would occur by chance isonly 3% when considering all possible pre-Calorian basins, and increases to 6% when usingthe smaller number of most reliably identified basins. The inclusion of basins with uncertainstratigraphic ages increases the probability to 11%, but this is to be expected as some portion ofthese likely formed after the Caloris impact when this planet was certainly in non-synchronousrotation. The observed value of the average angular distance to (90 ◦ W, 0 ◦ N) is nearly identicalto that predicted for synchronous rotation, and similar results are obtained using basins withdiameters greater than 300 km (Supplementary Tables 2-3 and Supplementary Fig. 3). Craterswith diameters less than 300 km show no such asymmetry, consistent with crater saturation [27].If Mercury was ever in a state of synchronous rotation, one hemisphere would have beenextremely cold, and the other extremely hot. Substantial quantities of volatile deposits wouldhave accumulated on the unilluminated hemisphere, just as is believed to have occurred withinthe permanently shadowed craters near the poles [28]. If these deposits were thick enough, it4ight be possible to find relict geomorphological differences between the two ancient hemi-spheres, as well as hydrous minerals that might have formed when these deposits were melted.The large differences in temperature between the two hemispheres would have given rise tospatial variations in the thickness of the lithosphere, and viscous relaxation of surface featureswould have occurred more quickly on the daylit hemisphere. Since the average impact veloc-ity would have differed between the two hemispheres by a factor of about two, the volumeof impact melt generated during an impact would have varied by a factor of about five [29].The consequences of synchronous rotation will be illuminated by data obtained from NASA’smission M
ESSENGER [30].
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Earth Planet. Sci. Lett. ,441–455 (2003). 7 e cc en t r i c i t y time [Gyr] ω / n Figure 1: Capture into synchronous rotation from initial retrograde rotation. The rotation rate ω normalized by the orbital mean motion n (top, red), and orbital eccentricity (bottom) are plottedover a 100 million year time period showing capture into the / resonance, escape from thisresonance at times of low eccentricity, and ultimate capture into synchronous rotation. Theequilibrium limit value of the rotation rate (green) is given by a delayed average that dampsfast eccentricity-induced variations and is always greater than 1. The statistical results areinsensitive to the starting time of the simulations.8 C r a t e r d i a m e t e r ( k m )
10 20 30 40 50 60 70 80 90
Impact velocity (km s ) -1 synchronous to 3/2unlock synchronous ρ = 8000 kg m -3 ρ = 500 kg m -3 Figure 2: Impact crater size and impact velocity required to unlock Mercury from synchronousrotation. Curves in red correspond to the conditions to increased the spin to the 3/2 resonance,and curves in blue represent the conditions necessary to increase the rotation beyond the syn-chronous resonance. Calculations assume an average impact geometry, and projectile densitiesof 500 and 8000 kg m − . The impact velocity on Mercury today ranges from about 10 to90 km s − , with an average of 42 km s − [17]. The planet is assumed to be completely solid,but if the mantle were uncoupled from the core with no core-mantle friction, the crater diameterswould be smaller by about 10%. 9 (cid:176) (cid:176) -90 (cid:176) -180 (cid:176) -270 (cid:176) B A B AB
Relative cratering rate (D > 100 km) (cid:176) (cid:176) -90 (cid:176) -180 (cid:176) -270 (cid:176) B A B AB
Figure 3: Predicted cratering rates and observed impact basins. (top) Synchronous rotation im-pact cratering rate normalized to the average value for craters larger than 100 km. ◦ longitudecorresponds to the subsolar point. (bottom) Impact basins on Mercury with diameters largerthan 400 km. Solid red circles (class 1) correspond to the centres of definite basins equal in ageor older than the Caloris basin, solid orange circles (class 2) correspond to probable basins [26],solid blue circles (class 3) correspond to possible basins [26], and gray circles (class 4) cor-respond to basins with uncertain ages with respect to Caloris. None of the centres of theseimpact basins lie within the dashed white circle. The solid white great circle encompasses theMariner-10 imaged hemisphere of Mercury. Solid black and white circles mark the axes of theminimum A and intermediate B moments of inertia of Mercury, respectively. Data are plottedon a USGS mosaic of M ESSENGER and Mariner 10 images, and is displayed in a Mollweideprojection with a central meridian of ◦ W longitude.10able 1: Probability of capture in spin-orbit resonances. Probabilities are given in percent forinitial retrograde rotation and three different post-impact spin rates. ω is the initial angularrotation rate and n is the orbital mean motion.spin-orbit pre impact post impactresonance ω /n = − . ω /n = 1 . ω /n = 1 . ω /n = 4 . − − − − − − − − − − − − − − − − − none − upplementary Information Name Latitude a Longitude a Diameter a Age b Class c Source ◦ N ◦ E kmCaloris 30.9 159.7 1456.7 C 1 [24–27]Andal-Coleridge -43 -49 1300 pT 1 [26]Tir 6 -168 1250 pT 1 [26]Eitoku-Milton -23 -171 1180 pT 1 [26]Bartok-Ives -33 -115 1175 pT 3 [26]Donne-Moliere 4 -10. 1060 pT 3 [26]Sadi-Scopas -83 -44 930 pT 3 [26]Matisse-Repin -24 -75 850 pT 1 [26]Budh 17 -151 850 pT 2 [25, 26]Sobkou 33.4 -133.5 785.3 pT 1 [24–27]Borealis 72.1 -80.9 785.2 pT 1 [26, 27]Mena-Theophanes -1 -129 770 pT 1 [24, 26]Rembrandt -33.1 87.7 696.7 ∼ C 4 [27, 32]Vincente-Yakovlev -52.6 -162.1 692.5 pT 1 [26, 27]Ibsen-Petrarch -31 -30 640 pT 2 [26]Beethoven -20.8 -123.9 632.5 ∼ C 4 [24–27]Brahams-Zola 59 -172 620 pT 1 [26]Tolstoj -17.1 -164.6 500.6 T 1 [24–27]Hawthorne-Riemenschneider -56 -105 500 pT 2 [26]Gluck-Holbein 35 -19 500 pT 3 [26]Dostoevskij -45.0 -176.2 430.4 pT 1 [24–26, 33](unnamed) 0.6 93.4 428.4 4 [27](unnamed) -39.0 -101.4 420.3 4 [27](unnamed) -44.5 -93.2 411.4 4 [27]Derzhavin-Sor Juan 50.8 -26.9 406.3 pT 1 [24, 26, 27](unnamed) -2.6 -56.1 392.6 4 [27](unnamed) 27.9 -158.6 389.0 pC 1 [24, 25, 27, 34]Vyasa 50.7 -85.1 379.9 pC 1 [27, 35]Shakespeare 48.9 -152.3 357.2 pT 1 [24–27]Hiroshige-Mahler -17.0 -23.0 340.3 pT 1 [25–27]Chong-Gauguin 57.1 -107.9 325.6 pT 1 [26, 27]Raphael -20.3 -76.1 320.4 pC 1 [24, 25, 27, 36]Goethe 81.5 -54.3 319.0 pC 1 [24, 25, 27, 37](unnamed) -2.5 -44.6 311.4 4 [24, 25, 27](unnamed) 28.9 -113.8 307.9 4 [24, 27](unnamed) -25.0 -98.8 307.6 4 [27](unnamed) -17.3 -96.8 303.4 4 [27] a Locations and diameters are from the most recent source. One impact basin tabulated by [24] (27.3 ◦ N, 146.1 ◦ E, D=379 km), but not identified in other studies, is not included. Two basins, Homer (-36.5 ◦ E, -1.4 ◦ N) and anunnamed crater (101.78 ◦ E, 70.23 ◦ N), that have measured diameters less than 300 km by [27], but larger than thisby [38], are not included. b C, Calorian; T, Tolstojan; pT, pre-Tolstojan. pC refers to craters that are presumed to be older than the Calorisbasin based on mapped c1 or c2 degradation states [39]. c Class 1 corresponds to definite basins of [26] and craters from [27], class 2 corresponds to probable basinsfrom [26] and class 3 corresponds to possible basis from [26]. All class 1–3 basins are younger than, or equalin age to, the Caloris basin, whereas class 4 basins have uncertain or undefined stratigraphic ages with respect toCaloris. ¯ θ of basins on the Mariner-10imaged hemisphere from the point ( ◦ W, ◦ N), and the probability (in percent) of obtainingthis value by chance from a uniform cratering rate. Craters with diameters less than kmtaken from [27]. Diameter range, km Class Number ¯ θ P ( > ¯ θ ) >
400 1
13 67.5 6.2 >
400 1 −
16 66.3 6.7 >
400 1 −
20 67.1 3.1 >
400 1 −
25 63.6 10.9 >
300 1
20 64.7 7.0 >
300 1 −
23 65.2 7.0 >
300 1 −
27 65.5 3.5 >
300 1 −
37 58.5 42.7 −
400 1 −
12 49.1 90.7 − −
24 57.5 52.8 − −
215 55.2 95.3 − − ¯ θ of basins on the Mariner-10imaged hemisphere to the point ( ◦ W, ◦ N). Two cases are given for synchronous rotation asthe subsolar point could correspond to either 0 ◦ or 180 ◦ longitude.Case ¯ θ in degreesUniform cratering 57.7Synchronous rotation a (apex = 90 ◦ W) 62.6Synchronous rotation a (apex = 90 ◦ E) 68.9Synchronous rotation b (apex = 90 ◦ W) 67.0Synchronous rotation b (apex = 90 ◦ E) 64.2 a Using the near-Earth asteroid model of [23]. b Using the observed near-Earth asteroids with
H < [31]. ω / n Supplementary Figure 1. Typical cases of capture into the / spin-orbit resonance. The rota-tion rate (red) and limit value (green) are plotted as a function of time following an impact event.Scenario ( a ) corresponds to an increase in the rotation rate to ω /n = 4 . . This is the classicalscenario of capture, where the eccentricity is always less than . and the limit value of therotation rate is always lower than that of the / resonance. Scenario ( b ) corresponds to anincrease in the rotation rate to ω /n = 1 . , just above that of the 3/2 resonance. In this scenario,the rotation rate tracks the limit value until it crosses and is captured into the / resonance.Scenario ( c ) corresponds to the case where the post-impact rotation rate was just large enoughto have escaped synchronous rotation, with ω /n = 1 . . In this case, the rotation rate tracks thelimit value until it crosses and is captured into the / resonance.16 ottke et al. (2002) B A B AB 90 ° ° -90 ° -180 ° -270 ° Bottke et al. (2002)
B A B AB 90 ° ° -90 ° -180 ° -270 ° H < 18
B A B AB 90 ° ° -90 ° -180 ° -270 ° Relative cratering rate (D > 100 km) H < 18
B A B AB 90 ° ° -90 ° -180 ° -270 °
25 30 35 40 45 50 55
Average impact velocity (km s ) -1 Supplementary Figure 2. Predicted impact cratering rate normalized to the average value forcraters greater than 100 km in diameter (left), and predicted average impact velocity (right)for synchronous rotation. The upper two images were obtained using the Bottke et al. [23]model population of near-Earth objects, and the lower two images were obtained using theorbital elements of the known near-Earth objects with absolute magnitudes less than 18 [31]. ◦ longitude corresponds to the average subsolar point. Solid black and white circles mark the axesof the minimum A and intermediate B moments of inertia of Mercury, respectively. Crateringrates are only marginally sensitive to the employed crater diameter.17 (cid:176) (cid:176) -90 (cid:176) -180 (cid:176) -270 (cid:176) B A B AB
Supplementary Figure 3. Impact basins on Mercury with diameters larger than 300 km (datafrom Supplementary Table 1). Solid red circles correspond to the centers of definite basins equalin age or older than the Caloris basin (class 1), solid orange circles correspond to probablebasins (class 2), solid blue circles correspond to possible basins (class 3), and gray circlescorrespond to basins with uncertain ages with respect to Caloris (class 4). The solid whitegreat circle encompasses the Mariner-10 imaged hemisphere of Mercury. Solid black and whitecircles mark the axes of the minimum A and intermediate B moments of inertia of Mercury,respectively. Data are plotted on a USGS mosaic of M ESSENGER and Mariner 10 images, andis displayed in a Mollweide projection with a central meridian of ◦ W longitude.18 eferences [31] Lowell Observatory. astorb.dat. ftp://ftp.lowell.edu/pub/elgb/astorb.html (downloaded6/10/2011).[32] Watters, T., Head, J., Solomon, S., Robinson, M., Chapman, C., Denevi, B., Fassett, C.,Murchie, S., and Strom, R. Evolution of the Rembrandt impact basin on Mercury.
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