About the various contributions in Venus rotation rate and LOD
aa r X i v : . [ a s t r o - ph . E P ] A p r Astronomy & Astrophysics manuscript no. LOD˙Venus c (cid:13)
ESO 2018October 10, 2018
About the various contributions in Venus rotation rate and LOD
L. Cottereau , N. Rambaux , , S. Lebonnois , J. Souchay Observatoire de Paris, Syst`emes de R´ef´erence Temps Espace (SYRTE), UMR 8630, CNRS, Paris, France(Laure.Cottereau, [email protected]) Universit´e Pierre et Marie Curie, Paris 6, IMCCE, UMR 8028, CNRS Observatoire de Paris, France([email protected]) Laboratoire de m´et´eorologie dynamique, UMR 8539, IPSL, UPMC, CNRS, France ([email protected])
ABSTRACT
Context.
Thanks to the Venus Express Mission, new data on the properties of Venus could be obtained in particular concerningits rotation.
Aims.
In view of these upcoming results, the purpose of this paper is to determine and compare the major physical processesinfluencing the rotation of Venus, and more particularly the angular rotation rate.
Methods.
Applying models already used for the Earth, the effect of the triaxiality of a rigid Venus on its period of rotationare computed. Then the variations of Venus rotation caused by the elasticity, the atmosphere and the core of the planet areevaluated.
Results.
Although the largest irregularities of the rotation rate of the Earth at short time scales are caused by its atmosphereand elastic deformations, we show that the Venus ones are dominated by the tidal torque exerted by the Sun on its solidbody. Indeed, as Venus has a slow rotation, these effects have a large amplitude of 2 minutes of time (mn). These variations ofthe rotation rate are larger than the one induced by atmospheric wind variations that can reach 25 −
50 seconds of time (s),depending on the simulation used. The variations due to the core effects which vary with its size between 3 and 20s are smaller.Compared to these effects, the influence of the elastic deformation cause by the zonal tidal potential is negligible.
Conclusions.
As the variations of the rotation of Venus reported here are of the order 3mn peak to peak, they should influencepast, present and future observations providing further constraints on the planet internal structure and atmosphere.
Key words.
Venus rotation
1. Introduction
The study of the irregularities of the rotation of a planetprovides astronomers and geophysicists with physical con-traints on the models describing this planet and allows abetter understanding of its global properties. AlthoughVenus is the planet sharing the most similarities with theEarth in terms of size and density, some of its characteris-tics are poorly understood like its atmospheric winds andits superrotation. With its thick atmosphere, the determi-nation of the period of rotation of Venus has been and stillis a very challenging task. Today thanks to Venus Express(the first spacecraft orbiting Venus since the MagellanMission in 1994) and Earth-based radar measurements,the period of rotation of Venus can be revised and itsvariations evaluated. So here for the first time a completetheoritical study of the variations of the rotation of Venuson a short time scale is presented as well as its implicationsconcerning the observations.Many studies on the rotation of Venus have alreadybeen made. Several authors studied this rotation on a longtime scale to understand why Venus spin is retrograde and why its spin axis has a small obliquity (Goldstein1964; Carpenter 1964; Goldreich and Peale 1970; Lago andCazenave 1979; Dobrovoskis 1980; Yoder 1995; Correiaand Laskar 2001, 2003), others studied the possible res-onance between the Earth and Venus (Gold and Soter,1979; Bills 2005; Bazs´o et al., 2010). But few studies havebeen made comparing the major physical processes influ-encing the rotation of Venus at short time scale. Karatekinet al. (2010) have shown that the variations of the rotationof Venus due to the atmosphere should be approximatively10 seconds for the 117 days time span characterizing theplanet’s solar day, but they neglected the main effect ofthe impact of gravitational torques on the solid body ofthe planet. Here, applying different models already usedfor the Earth, the irregularities of the rotation of Venus arecomputed. After clarifying the link between the length ofday (
LOD ) and the rotation rate of the planet, the effectsof the triaxiality of a rigid Venus on its period of rotationare evaluated. This model is based on Kinoshita’s theory(1977) and uses the Andoyer variables (Andoyer, 1923). AsVenus has a very slow rotation (-243.020 d), these effects
Cottereau et al.: About the contribution in Venus rotation rate variation have a large amplitude (2 mn peak to peak) and could beobservable as this is shown in Section 3.As we know, the variation of the period of rotation ofthe Earth of the order of 0 .
2. Rotation rate and LOD definition
A day is defined as the time between two consecutive cross-ings of a reference meridian by a reference point or body.While these periods vary with respect to time, we can usetheir average values for the purpose. For the Earth, thesolar day (24 h) and the sidereal day (23.56 h) are definedrespectively when the Sun and a star are taken as ref-erence. They are very close because the Earth’s period ofrotation is far smaller than its period of revolution aroundthe Sun. By contrast, as Venus has a slow rotation, its so-lar and sideral days are quite different. Indeed, the canon-ical value taken for its rotational period (sideral day) is243 .
02d (Konopliv et al, 1999), whereas its solar day variesaround 117d. In the following, to be consistent with thephysical inputs in the global circulation model (
GCM )simulations, we will fix the value of the solar venusian dayat 117d.There are mainly two methods to measure the rotation pe-riod, and its variations. The radar Doppler measurementsgive direct access to the instantaneous rotation rate ω ( t )that we can translate in an instantaneous period of rota-tion LOD ( t ) by LOD ( t ) = 2 πω ( t ) . (1)Because we will be interested in variation around a meanvalue, we further define ∆ ω ( t ) = ω ( t ) − ω and ∆ LOD ( t ) = LOD ( t ) − LOD where
LOD = 243 .
02d and ω = πLOD . Atfirst order ∆ ωω ( t ) = − ∆ LODLOD ( t ). Thanks to infrared im-ages of the surface of the planet, we can also measure thelongitude of a reference point φ with respect to a givenreference system at several epochs where ˙ φ = ω ( t ). Aswe will discuss in Section 7, as ω ( t ) has periodic varia-tions, fitting φ by a linear function could lead to a wrongestimation of ¯ ω .
3. Effect of the solid potential on the rotationrate
A first and simplified model to describe the variation ofthe rotation of Venus is to consider that the atmosphere,mantle and core of Venus are rotating as a unique solidbody. This model enables us to use Kinoshita’s theory(1977) with the Andoyer Variables (1923) as those de-scribing the rotation of the Earth. The rotation of Venusis then described by three action variables (
G, L, H ) andtheir conjugate variables ( g, l, h ). G represents the ampli-tude of the angular momentum and L, H respectively itsprojections on the figure axis and on the inertial axis (axisof the reference plane) such as L = G cos J and H = G cos I (2)where I and J correspond respectively to the angles be-tween the angular momentum axis and the inertial axisand between the angular momentum axis and the figureaxis. Here the figure axis and the inertial axis coincide re-spectively with the axis of the largest moment of inertiaand the axis of the orbit of Venus at J2000.0. The angle J is yet unknown. As shown in appendix A, for plausi-ble value of this angle, its impact on the rotation is weakwith respect to the other effects taken into account in thisarticle. For sake of clarity, J is set to 0 in the follow-ing. In this coordinate system, the Hamiltonian related tothe rotational motion of Venus is (Cottereau and Souchay,2009): K = F + E + E ′ + U. (3) F is the Hamiltonian for the free rotational motion de-fined by F = 12 ( sin lA + cos lB )( G − L ) + 12 L C , (4)where
A, B, C are the principal moments of inertia ofVenus.
E, E ′ are respectively the components related tothe motion of the orbit of Venus which is caused by plane-tary perturbations (Kinoshita, 1977) and to the choice ofthe ”departure point” as reference point (Cottereau andSouchay, 2009). U is the disturbing potential of the Sun,considered as a point mass, and is given at first order by : U = G M ′ r h [ 2 C − A − B P (sin δ )+[ A − B P (sin δ ) cos 2 α i , (5)where G is the gravitational constant, M ′ is the mass ofde Sun, r is the distance between its barycenter and thebarycenter of Venus. α and δ are the planetocentric lon-gitude and latitude of the Sun with respect to the meanequator of Venus and a meridian origin (not to be con-fused with the usual equatorial coordinates defined withrespect to the Earth). The P mn are the classical Legendrefunctions given by: P mn ( x ) = ( − m (1 − x ) m n n ! d n + m ( x − n dx n + m . (6) ottereau et al.: About the contribution in Venus rotation rate variation 3 The hamiltonian equations are : dd t ( L, G, H ) = − ∂K∂ ( l, g, h ) (7) dd t ( l, g, h ) = ∂K∂ ( L, G, H ) . (8)As the components ω and ω of the rotation of Venusare supposed at first approximation to be negligible withrespect to the component ω along the figure axis, thisyields : G = p ( Aω ) + ( Bω ) + ( Cω ) ≈ Cω (9)and dd t ( Cω ) ≈ dd t ( Cω ) ≈ − ∂K∂g . (10)Splitting ω into its mean value ω and a variation ∆ ω , ω is given by : dd t ( C ∆ ω ) = − ∂K∂g . (11)Notice here that F and E + E ′ do not contain g , sothat the variations of the rotation of solid Venus are onlycaused by the tidal torque of the Sun included in the dis-turbing potential U . To make explicit the dependence onthe variable g , we express U as a function of the longitude λ and the latitude β of the Sun with respect to the orbitof Venus at the date t with the transformations describedby Kinoshita (1977) and based on the Jacobi polynomialssuch as : d ( C ∆ ω ) dt = − ∂∂g G M ′ r " C − A − B (cid:16) −
14 (3 cos I − −
34 sin I cos 2( λ − h ) (cid:17) + A − B (cid:20)
32 sin I cos(2 l + 2 g ) + X ǫ = ±
34 (1 + ǫ cos I ) cos 2( λ − h − ǫl − ǫg ) (cid:21) , (12)where β is, by definition, equal to 0 in this case. The firstcomponent at the right hand-side of Eq.(12), does not de-pend on the variable g , so that this yields:∆ ω = − C Z ∂∂g n (cid:16) ar (cid:17) A − B (cid:20)
32 sin I cos(2 l + 2 g )+ X ǫ = ±
34 (1 + ǫ cos I ) cos 2( λ − h − ǫl − ǫg ) (cid:21)! d t. (13)where G M ′ r has been replaced by n · a r , a and n beingrespectively the semi-major axis and the mean motion ofVenus defined by the third Kepler law n a = G M ′ . (14) Differentiating with respect to g in the Eq.(13), ∆ ω isgiven by :∆ ω = − Z A − B C n (cid:16) ar (cid:17) (cid:20) − sin I sin(2 l + 2 g )+ X ǫ = ± ǫ (1 + ǫ cos I ) sin 2( λ − h − ǫl − ǫg ) (cid:21) d t. (15)Notice that we can write l + g ≈ φ (Kinoshita, 1977) where φ is the angle between the axis which coincides with thesmallest moment of inertia and a reference point arbitrarlychosen as the ”departure point” on the orbit of Venus atthe date t (Cottereau and Souchay, 2009). The positionof this axis is given with respect to the origin meridianitself defined as the central peak in the crater Adriadne(Konopliv et al, 1999; Davies et al., 1992). In fact φ isthe angle of proper rotation of the planet ( ˙ φ ≈ ω ). Inthe following, φ will be used instead of l + g . To solveanalytically Eq.(15), the developments of ( ar ) sin 2 φ and( ar ) sin 2( λ − h ± φ ) are needed as a function of timethrough the variables M and L S (respectively the meananomaly and the mean longitude of Venus) taking the ec-centricity as a small parameter (Kinoshita, 1977). In a firstapproach, the orbit of Venus can be considered as circular(i.e : e =0, instead of : e =0.0068). This yields :∆ ω = 3 B − A C n (cid:20) sin I cos(2 φ ) 12 ˙ φ −
12 (1 + cos I ) cos 2( L s − φ ) 12 ˙ L s − φ + 12 (1 − cos I ) cos 2( L s + φ ) 12 ˙ L s + 2 ˙ φ (cid:21) . (16)Notice that this equation is similar to the equation givenby Woolard (1953) in his theory of the Earth rotation,using a different formalism based on classical Euler angles. VenusPeriod of revolution 224.70d (Simon et al., 1994)Obliquity 2 ◦ . A − B C -1 . − (Konopliv et al., 1999; Yoder, 1995) CM V R V . Table 1.
Numerical values. M V and R V are respectivelythe mass and the radius of Venus.Taking the numerical values of the Table 1, the largestterms of the variation of ∆ ωω are∆ ωω = 2 .
77 10 − cos(2 L S − φ )+6 .
12 10 − cos 2 φ − .
99 10 − cos(2 L S + 2 φ ) (17) Cottereau et al.: About the contribution in Venus rotation rate variation where 2 L S − φ, φ and 2 L S +2 φ correspond to the leadingterms with respective periods 58 d, 121 .
80 d and 1490d.As the two last terms scale as the square of the obliquity,they are significantly smaller than the term with argument2 L S − φ (2 .
77 10 − ). As Venus has a very slow rotationwhich appears in the scaling factor 3 B − A C n ω , the variationsof the rotation rate due to the solid torque are larger thanthe Earth ones which correspond to amplitudes of ∆ ωω ≈ − . Fig.1 shows the relative variations of the speed of - ´ - - ´ - - ´ - ´ - ´ - ´ - Venusian days D Ω (cid:144) Ω Fig. 1.
Variation of ∆ ωω during a 4 venusian solar daystime span (468d). One venusian solar day corresponds to117d.rotation of Venus due to the solid torque exerted by theSun during a four venusian days time span (468d). Theorientation of the bulge of Venus at t = 0 is given withrespect to the origin meridian. The mean orbital elementsare given by Simon et al. (1994). The choice of the timespan will allow us to compare in the following the differenteffects which act on the rotation of Venus.The physical meaning of the leading term with ar-gument 2 L S − φ can be understood using a simpletoy model. Consider a coplanar and circular orbit. Atquadrupolar order, the gravitational potential created bythe planet in its equatorial plane is equal to the one cre-ated by three point masses, one located at the center of theplanet and of mass M v − µ and two other ones symetri-cally located on the surface of the planet along the axis ofsmallest moment of inertia and of mass µ (see Fig.2) with µ = B − A R V , R V being the planet mean radius. The torqueexerted on the planet by the Sun can thus be computedusing the forces exerted on these three points only. Asshown in Fig.2, the sign of the resulting torque dependson the quadrant of the x, y plane the Sun is located in,and thus changes four times during a solar day, whose thecorresponding argument is 2 L S − φ .To evaluate the influence of the eccentricity of Venuson its rotation rate, we show in Fig.3 the residuals aftersubstraction of ∆ ωω when the eccentricity is taken into ac-count in the development of ( ar ) sin 2 φ and ( ar ) sin 2( λ − h ± φ ) with respect to the simplified expression given bythe Eq.(17). (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) Rv µ µ xyM− equatorial plane ++− − Fig. 2.
Diagram of the gravitational force exerted by theSun in the frame corotating with Venus. The − and +signs give the sign of the net torque in each quadrant. - ´ - - ´ - - ´ - ´ - ´ - ´ - Venusian days D Ω (cid:144) Ω - D Ω (cid:144) Ω @ e D Fig. 3.
Influence of the eccentricity on the variation of thespeed of rotation of Venus.We can observe that the eccentricity of Venus acts on ∆ ωω with an amplitude of ≈ − for the terms 2 L s + M − φ (46.34 d) and 2 L s − M − φ (78.86 d). These variationsare smaller than the leading coefficient seen in Eq.(17) bytwo orders and more important than the other coefficientswith arguments 2 φ and 2 φ + 2 L s .From the relation ∆ ωω ( t ) = − ∆ LODLOD ( t ), the variationsof the LOD(t) due to the torque of the Sun are 120 s (i.e2 mn or 0 . . ± . . ± . ottereau et al.: About the contribution in Venus rotation rate variation 5
4. Effects of the elasticity on the rotation rate :deformation due to the zonal tidal potential
For the Earth, an important variation of the speed of ro-tation is due to the zonal potential which causes temporalvariations of the moment of inertia C . In this section, thesezonal effects are evaluated for Venus, considered as a de-formable body. The zonal part of the potential exerted bythe Sun on a point M at the surface of the planet is givenby the classical formula V , S ( δ M , δ ) = 94 G M ′ R V r (sin δ −
13 )(sin δ M −
13 ) , (18)where δ and δ M represent respectively the planetocentriclatitude of the Sun (the disturbing body) and of the point M and r the distance between the barycenter of the Sunand the Venus one. The corresponding bulge produced hasa potential (Melchior, 1978):∆ V , V ( δ M , δ, r ′ ) = k V , S R V r ′ , (19)where k and R V are respectively the Love number andthe radius of the planet. Differentiating the potential pro-duced by Venus at its surface (MacCullagh’s formula),∆ V , V can be expressed as a function of the principal mo-ments of inertia∆ V , V ( δ, r ′ ) = 32 G r ′ ( d C − d A )(sin δ −
13 ) , (20)where we take d A = d B as the deformation is purely zonal.By identification we obtain: k M ′ M V (sin δ −
13 )( R V r ) = d C − d A M V R V . (21)Because this deformation does not induce a change in thevolume of the isodensity surfaces in the planet to first or-der, d I = d C + d A = ∆ C +∆ A = 0 ( see Melchior, 1978,for a rigorous demonstration). The Euler’s third equationis Cω = constant , to first order this yields ∆ CC = − ∆ ωω .Thus the variations of the angular rate of rotation of Venusare given by :∆ ωω = − ∆ CC = − k (cid:16) R V a (cid:17) (cid:16) ar (cid:17) M ′ M V (sin δ −
13 ) M V R V C (22)where M V is the mass of Venus. Expressing sin δ as afunction of I and λ where I , λ are given in Section 3 andrepresent the obliquity of Venus and the true longitude ofthe Sun, Eq.(22) becomes∆ ωω = − k M V R V C M ′ M V (cid:16) R V a (cid:17) (cid:16) ar (cid:17) (cid:16) sin I − cos 2 λ ) − (cid:17) . (23) Using the developments of ( ar ) and ( ar ) cos 2 λ with re-spect to time from Cottereau and Souchay (2009), Fig.4shows the relative variation of the speed of rotation ofVenus due to the zonal potential. The numerical valuesof k = 0 . ± .
066 and the ratio CM V R V = 0 . LOD ( t ) of 0 . − s) with respect to the 1 d rotation (Yoder et al., 1981;Souchay and Folgueira, 1998). Note that the semi-annualcomponents are particularly small for Venus because itseccentricity and obliquity are much smaller than their re-spective value for the Earth. - ´ - ´ - ´ - Venusian days D Ω (cid:144) Ω Fig. 4. ∆ ωω due to the zonal potential during a 4 venusiandays time span (468d).
5. Atmospheric effects on the rotation of Venus
As Venus has a denser atmosphere than the Earth, andas we know presently that the Earth atmosphere acts onits rotation in a significant manner (Lambeck, 1980), itwill be interesting to study the corresponding effects onVenus. In this section, the core and the mantle of Venus aresupposed to be rigidly coupled and its atmosphere rotatesat a different rate. The variations of the speed of rotationof Venus due to its atmosphere are given by :∆ GG = ∆ ωω , (24)where G, ∆ G represent respectively the angular momen-tum of the rigid Venus and its variation due to the atmo-sphere ∆ G = − ∆ G atm . The angular momentum ∆ G atm ofthe atmosphere can be split into two components : – The matter term G M which is the product of ω withthe inertia momentum of the atmosphere – The current term G w which is due to the wind mo-tions with respect to the frame solidly rotating withthe planet Cottereau et al.: About the contribution in Venus rotation rate variation
From Lebonnois et al. (2010a) we have : G atm = (1 + k ′ ) G M + G w = (1 + k ′ ) ωR V g Z Z s P s cos θ d θ d φ + R V g Z Z Z v cos θv θ d h d θ d φ. (25)where θ, φ, R V stand respectively for the latitude, longi-tude and radius of Venus, v θ is the zonal wind and k ′ is the load Love number of degree 2 (Karatekin et al.,2010). Here to determine the variations of the angularmomentum, two simulations made with the global circu-lation model ( GCM ) of the Laboratoire de MeteorologieDynamique (LMD) (Lebonnois et al., 2010b) are used andcompared. These simulations have been obtained with theLMD Venus General Circulation Model, using conditionssimilar to those presented in Lebonnois et al. (2010a), ex-cept for the boundary layer scheme. The first simulation(
GCM
1) was integrated from a zero wind state and isvery close to the simulation published in Lebonnois et al.(2010a), though the winds in the deep atmosphere (0 to40 km altitude) are slightly higher, due to the updatedboundary layer scheme. The second one (
GCM
2) was in-tegrated with initial winds in superrotation where the re-sulting winds in the deep atmosphere are close to observedvalues. The results presented in Fig.5 were obtained after a200 days integration for
GCM
GCM - ´ - - ´ - - ´ - ´ - Venusian days D Ω (cid:144) Ω Fig. 5. ∆ ωω due to the atmosphere during a 4 venusiandays time span for the GCM1 (red curve) and the GCM2(blue curve).Figure 5 shows the relative variation of the speed ofrotation of Venus due to the atmosphere for a 4 venu-sian days time span (470 d) with the GCM
GCM .
26 10 − peak to peak for GCM .
44 10 − respectively for GCM
2. For the two models, the large am-plitude of ∆ ω ¯ ω of the order of 10 − comes from the current term and depends on winds of Venus whereas the matterterm acts on the rotation with an amplitude of the orderof 10 − .Using the definition of the LOD ( t ) given in Section 2,the atmospheric contributions to Venus rotational speedcorrespond to peak to peak variations of the LOD ( t ) of27s with the GCM1 and 51s with the GCM2. These valuesare consistent with the value of LOD ( t ) of the planet of7 .
9s given by Karatekin et al. (2009, 2010), who used thesimulation presented in Lebonnois et al. (2010a). The dif-ferences are related to the amplitudes of the zonal windsin the region of maximum angular momentum (10-30 kmof altitude), which vary between the different simulations.The most realistic values for these winds are obtained withthe
GCM
GCM
GCM
6. Core effects on the rotation of Venus
The interior of Venus is probably liquid as inferred fromthe orbiting spacecraft data (Konopliv & Yoder 1996).The internal properties of Venus are expected to be likethe Earth with a core radius around 3120 km (Yoder1995), but with a noteworthy difference because there isno dynamo effect on Venus (Nimmo 2002). In this Section,we investigate the impact of such a core on the rotationalmotion of Venus and especially on the variation of its
LOD ( t ). For that purpose, we numerically integrate therotational motion of Venus by taking into account the in-ertial pressure torque. The equations governing the rota-tional motion of two-layer Venus are the angular momen-tum balance for the whole body d H d t + ω ∧ H = Γ (26)and for the core d H c d t − ω c ∧ H c = 0 (27)(Moritz and Mueller 1987) where H , H c are respectivelythe angular momentum of Venus and of the core (see alsoe.g. Rambaux et al 2007). The vector Γ is the gravita-tional torque acting on Venus. Here, we assume that thecore has a simple motion has suggested by Poincar´e in1910 and we neglect the core-mantle friction arising atthe core-mantle boundary. Then, the rotational motion ofeach layers of Venus is integrated simultaneously with thegravitational torque due to the Sun acting on the triaxialfigure of Venus. The orbital ephemerides DE421 (Folkneret al., 2008) are use for the purpose. ottereau et al.: About the contribution in Venus rotation rate variation 7 First we double-checked the good agreement betweenthe numerical and the analytical solutions given byEq.(17) for a simplified model of one layer solid rigidVenus case. The very small differences obtained (at thelevel of a relative 10 − ) may result from the use of differ-ent ephemerides (DE421 for the numerical approach andVSOP87 for the analytical one). Then we applied the pro-cedure for the two layers case. As the moments of inertia I c of the core are not yet constrained by measurements, weused internal models (Yoder, 1995) for which I c is takenfrom 0 .
01 to 0 .
05, according to the size of the core. Noticethat the flattening of the core is scaled to the flatten-ing of the mantle by the assumption that the distribu-tion of mass anomalies is the same. Table 2 shows thenew amplitudes of ∆ ωω for the 4 main oscillations witharguments 2 L s − φ, L s − φ + M, L s − φ − M, φ and with corresponding periods 58 . . . . I c = 0 .
05, we obtain an increase in the oscillation of2 L S − φ of 17% corresponding to a variation of +20 . Period / 58.37d 46.34d 78.86d 121.51d I c L s − φ L s − φ + M L s − φ − M φ Table 2.
Resulting amplitude for Venus ∆ ωω with a fluidcore (expressed in unit of 10 − ). The last line is computedfor a model without a fluid core. - ´ - - ´ - ´ - ´ - Venusian days D Ω (cid:144) Ω - D Ω (cid:144) Ω @ c o r e D Fig. 6. ∆ ωω due to the core during a 4 venusian days timespan with a I c = 0 .
05 (red curve) and a I c = 0 .
01 (bluecurve). Figure 6 shows the relative variation of the speed of ro-tation of Venus due to the core during a 4 venusian daystime span with I c = 0 .
05 (red curve) and I c = 0 .
01 (bluecurve). These variations have peak to peak amplitudes re-spectively of 1 . − and 9 . − . Using the definition ofthe LOD ( t ) given in Section 2, the core contributions toVenus rotational speed correspond to peak to peak vari-ations of the LOD ( t ) between 3 .
6s and 20 . I c = 0 .
05 the core effectsare respectively nearly two and six times smaller than theatmospheric and solid ones.
7. Comparison and implication for observations
The variations of the rotation of Venus presented in thispaper are quasi-periodic and mainly due to three kinds ofeffects: solid, atmospheric and core. Compared to them,the zonal potential has a negligible influence. Finally ω ( t )can be written in the form: ω ( t ) = ¯ ω + ∆ ω = ¯ ω + h X i ( a s,i + a c,i ) cos( ω i t + ρ i )+ X j a a,j cos( ω j t + ρ j ) i ¯ ω, (28)where ω i , ρ i and a s,i , a c,i (see Table 3) are the frequen-cies, the phases and the corresponding amplitudes of thevariation of rotation of Venus due to the solid and thecore, and ω j , a a,j (see Table 4) those due to the atmo-sphere. The atmospheric coefficients as well as their pe-riods have been obtained from a fast fourier transform(FFT) where ρ j are the phases. Note that the power spec-trum of the atmospheric variation is complex and can varysignificantly from one model to another. As a consequence,only the most important frequencies are shown for the GCM
Period P i Argument solide a s,i Core a c,i πω i cos ω i t cos ω i t
58d 2 L s − φ .
77 10 − .
51 10 − < a c < .
85 10 − .
34d 2 L s + M − φ .
24 10 − .
01 10 − < a c < . − .
86d 2 L s − M − φ .
27 10 − . − < a c < . − .
80d 2 φ .
12 10 − . − < a c < . − Table 3.
Variation of ∆ ω ¯ ω due to the solid and core effects.Comparing the amplitudes given in Tables 3 and 4, wesee that the most important effect on the rotation rateof Venus is due to the solid potential exerted by the Sunon its rigid body. If all effects are taken into account, thevariations of the LOD ( t ) can reach 3mn which could beobservable in the future.As most past studies used infrared imaging of the sur-face to measure the evolution in time of the longitude of Cottereau et al.: About the contribution in Venus rotation rate variation
Period P j a a,j ρ j ω j cos ω j t + ρ j .
17 10 − . .
22 10 − . .
02d 1 .
74 10 − . .
09d 1 .
41 10 − . .
6d 1 .
28 10 − . .
5d 1 .
21 10 − . Table 4.
Variations of ∆ ω ¯ ω due to the atmosphere effectsmodeled by GCM φ at Venus surface, let us consider howthis variable behave in our model. As ˙ φ ≈ ω ( t ) we have φ ( t ) = φ , + ωt + Z tt ∆ ω d t. (29)Fig.7 shows the variations ∆ φ during four venusian daystime span, caused by the combined effects of the solid, thecore and the atmosphere modeled by the GCM2. -
505 Venusian days D Φ H t L * R v H m L Fig. 7.
Variations of ∆ φ × R V in meter during a fourvenusian days time span.Like the variations of the rotation rate, these variationsare periodic with a peak to peak amplitude reaching 12 m(time differences are converted in distance at the surface).In precedent studies that measured the mean rotation rateof Venus, these variations where neglected. This impliedfitting the measurements of the phase angle φ with aline of constant slope, this slope giving the mean rotationrate. Because the variations of the rotation rate discussedabove are not negligible, we show in the following thatthis approach can yield large errors on the derived valueof ¯ ω , especially for a short interval of time. Similarly toLaskar and Simon (1988) let us consider the error madeon the mean rotation rate when fitting the signal given byEq.(29), keeping only the most important frequency forsimplicity, by a function of the type : φ ( t ) = φ , obs + ω obs t (30) over a time span [ t , t + T ]. For a least square fittingprocedure, the residual is given by : D = Z t + Tt ( φ , + ¯ ωt + A sin ω t − ( ω obs t + φ ,obs )) d t (31)where A and ω = π rd/d correspond respectively tothe larger amplitude of Eq.(29) and its corresponding fre-quency. Minimizing this residual yields ω obs = ¯ ω − AT ω " T ω h cos( ω t ) + cos ω ( T + t ) i +2 h sin( ω t ) − sin ω ( T + t ) i (32) φ ,obs = φ , + 2 AT ω " T (2 T + 3 t ) ω cos( ω t )+ T ( T + 3 t ) ω cos ω ( T + t )+3( T + 2 t ) h sin( ω t ) − sin ω ( T + t ) i (33)We can see that ω obs depends on both the time of thefirst observation ( t ) (i.e phase) and interval ( T ) betweenobservations. If the phase of the effect is unknown, themax error made on the mean rotation rate is given by :∆ LOD obs = max t | ω obs ( t , T ) − ¯ ω | π ¯ ω (34)Fig.8 shows this maximum error ∆ LOD obs in seconds asa function of the interval T between observations. H Days L D L OD ob s Fig. 8.
Maximum error of ∆
LOD obs ( T ) in seconds givenby Eq.(34). The dotted line is the uncertainty given byMagellan (Davies, 1992) on the rotation of Venus.As we can see, even if the mean angular velocity is re-trieved for long baseline observations, the error yielded bythe linear fit can be large for short duration observations.In addition, the use of such a simple model for φ pre-vents any measurement of the amplitude of the variationsdetailed in this paper. To measure the amplitude of thesevariations, modeling ω and φ with Eq.(28) and Eq.(29)respectively during the data reduction is necessary. ottereau et al.: About the contribution in Venus rotation rate variation 9 As the most important effect is due to the torque ofthe Sun on the Venus rigid body with a large amplitudeon a 58d interval, it could be interesting to substract themeasured signal by a fitted sinusoid of this frequency. Thedirect measurement of the amplitude of the sinusoid wouldgive information on the triaxiality of Venus at 3 to 17%of error because of the core contribution.To disentangle atmospheric effects, a multi frequencyanalysis of the data will be necessary. Indeed after hav-ing determined the larger amplitude on 58d as explainedpreviously, it could be substracted to the analysis. Thenas the atmospheric winds (described by Eq.(25)) are thesecond most important effects on the rotation of Venus,the residuals obtained could constrain their strength. Inparallel it should be interesting, also, to fit the signal ob-tained by a sinusoid on the period of 117d because onlythe atmosphere acts on the rotation with this periodicity.Note that this period is also present when using an al-ternative atmospheric model
GCM . ± . . . ± . GCM I c = 0 .
05) effectspresented in this paper. Note that the value of Davies etal. (1992) set to 243 . ± . – the variations of the period of rotation (0 . – if the true mean value of the period of rotation of Venusis close to the IAU value, the variations cannot explainthe most recent value obtained by VIRTIS (Muelleret al., submitted paper 2010). Indeed, the differencebetween them of 7mn implies larger variations.The different values of the rotation of Venus since 1975could be explained by the variations of 0 . ± . r o t a ti onp e r i od (cid:144) d a y s Fig. 9.
Values of the period of rotation of Venus and theirerror bars given since 1975. The dotted lines representvariations of 0 . .
8. Conclusion
The purpose of this paper was to detemine and to comparethe major physical processes influencing the angular speedof rotation of Venus. Applying different theories alreadyused for the Earth, the variation of the rotation rate aswell as of the
LOD was evaluated.Applying the theory derived from Kinoshita (1977),the effect of the solid potential exerted by the Sun on arigid Venus was computed. Considering in the first stepthat the orbit of Venus is circular, we found that thevariations of the rotation rate have a large amplitude of2 .
77 10 − with argument 2 L s − φ (58d). Taking into ac-count the eccentricity of the orbit adds periodic variationswith a 10 − amplitude. On the opposite of the Earth, asVenus has a very slow rotation, the solid potential has aleading influence on the rotation rate which correspondsto peak to peak variations of the LOD of 120sConsidering Venus as an elastic body, we then evalu-ated the impact of the zonal tidal potential of the Sun.These variations correspond to peak to peak variations onthe
LOD of 0 . LOD of 25 −
50s and 3 . − .
4s respectively at differentperiods. Despite its thickness, the impact of the venusianatmosphere modeled by our most realistic simulation on the rotation is 2 . LOD due to the core, which increase with its size, are stillsmaller.At last we have shown that the variations of ω and R V φ which reach 3mn and 12m respectively, need to betaken into account in the reduction of the observations.Ignoring these variations could lead to an incorrect esti-mation of ¯ ω . With the steadily increasing precision of themeasurements, carrying a frequency analysis of the datamodeled by either Eqs.(28) or (29) will hopefully enableto put physical constraints on the physical properties ofVenus (triaxiality, atmosphere, core).To conclude, the variations shown in this paper, wouldexplain different values of the mean rotation of Venusgiven since 1975. The difference of 7 mn between the IAUvalue (243 . . ± . Acknowledgements.
L.C wishes to thanks Pierre Drossart,Thomas Widemann, Jean-Luc Margot and Jeremy Leconte fordiscussion on Venus.
Appendix A: Influence of the angle J on therotation rate In Section 3, we assumed that the angle J between the an-gular momentum axis and the figure axis of Venus can beneglected. Here we reject this hypothesis of coincidence ofthe poles and we evaluate the impact of J on the rotation.According to Eq.5 the variations of ∆ ω is given by : dd t ( C ∆ ω ) = − ∂U∂g . (A.1)where U = G M ′ r h [ 2 C − A − B P (sin δ )+[ A − B P (sin δ ) cos 2 α i , (A.2)As it was done in Section 3, we express P (sin δ ) and P (sin δ ) cos 2 α as functions of the longitude λ and thelatitude β of the Sun with the transformation described by Kinoshita (1977) without supposing J = 0 such as: P (sin δ ) = 12 (3 cos J − "
12 (3 cos I − P (sin β ) −
12 sin 2 I sin( λ − h ) P (sin β ) −
14 sin 2 IP (sin β ) cos 2( λ − h ) + sin 2 J " −
34 sin 2 IP (sin β ) cos g − X ǫ = ±
14 (1 + ǫ cos I )( − ǫ cos I ) P (sin β ) sin( λ − h − ǫg ) − X ǫ = ± ǫ sin I (1 + ǫ cos I ) P (sin β ) cos(2 λ − h − ǫg ) + sin J "
34 sin IP (sin β ) cos 2 g + 14 X ǫ = ± ǫ sin I (1 + ǫ cos I ) P (sin β ) sin( λ − h − ǫg ) − X ǫ = ± (1 + ǫ cos I ) P (sin β ) cos 2( λ − h − ǫg ) (A.3)and P (sin δ ) cos 2 α = 3 sin J " −
12 (3 cos I − P (sin β )cos 2 l + 14 X ǫ = ± sin 2 IP (sin β ) sin( λ − h − ǫl )+ 18 sin IP (sin β ) cos 2( λ − h − ǫl ) + X ρ = ± ρ sin J (1 + ρ cos J ) " −
32 sin 2 IP (sin β ) cos(2 ρl + g ) − X ǫ = ±
12 (1 + ǫ cos I )( − ǫ cos I ) P (sin β ) sin( λ − h − ρǫl − ǫg ) − X ǫ = ± ǫ sin I (1 + ǫ cos I ) P (sin β ) cos(2 λ − h − ρǫl − ǫg ) ottereau et al.: About the contribution in Venus rotation rate variation 11 + X ρ = ±
14 (1 + ρ cos J ) " − IP (sin β )cos(2 l + 2 ρg ) X ǫ = ± ǫ sin I (1 + ǫ cos I ) P (sin β ) sin( λ − h − ρǫl − ǫg )+ X ǫ = ±
14 (1 + ǫ cos I ) P (sin β ) cos 2( λ − h − ρǫl − ǫg ) . (A.4)Assuming that β = 0 and removing the components whichdo not depend on the variable g this yields: d ( C ∆ ω ) d t = − ∂∂g G M ′ a (cid:16) ar (cid:17) " C − A − B sin 2 J (cid:20)
34 sin 2 I
12 cos g − X ǫ = ± ǫ sin I (1 + ǫ cos I )3 cos(2 λ − h − ǫg ) (cid:21) + sin J (cid:20) −
34 sin I
12 cos 2 g − X ǫ = ± (1 + ǫ cos I ) λ − h − ǫg ) (cid:21)! + A − B X ρ = ± ρ sin J (1 + ρ cos J ) h
32 sin 2 I
12 cos(2 ρl + g ) − X ǫ = ± ǫ sin I (1 + ǫ cos I )3 cos(2 λ − h − ρǫl − ǫg ) i + X ρ = ±
14 (1 + ρ cos J ) h I l + 2 ρg ) + X ǫ = ±
14 (1 + ǫ cos I ) λ − h − ρǫl − ǫg ) i! , (A.5)where G M ′ r has been replaced by n ( ar ) .The variations of the speed of rotation of Venus due tothe solid torque exerted by the Sun are obtained by devel-oping Eq.(A.5) as functions of time through the variables M and L s taking the eccentricity as a small parameter(Kinoshita, 1977). Fig.A.1 shows the residuals after sub-straction of ∆ ωω obtained by the Eq.(A.5) when the angle J = is taken into account and e = 0 with respect to thevariations given by Eq.(17) in Section 3. We arbitrarilytake the angle J = 0 . ◦ . - ´ - ´ - Venusian days D Ω (cid:144) Ω - D Ω (cid:144) Ω @ J D Fig. A.1.
Influence of the angle J = 0 . ◦ on the variationof the speed of rotation of Venus.The influence of the angle J = 0 . ◦ on the rotationwith amplitudes of ≈ − is smaller than the leadingcoefficients seen in Eq.(17) by three orders of magnitudeand than the influence of the eccentricity of Venus by oneorder of magnitude. As J is probably much smaller than0 . ◦ as it is the case for the Earth (1”), its influence on therotation can be neglected with respect to the other effectstaken into account in this paper. References
Andoyer H. 1923, Paris, Gauthier-Villars et cie, 1923-26.Barnes, R. T. H., Hide, R., White, A. A., & Wilson, C. A.1983, Royal Society of London Proceedings Series A, 387,31Bazs´o, ´A., Dvorak, R., Pilat-Lohinger, E., Eybl, V., & Lhotka,C. 2010, Celestial Mechanics and Dynamical Astronomy,107, 63Bills, B. G. 2005, Journal of Geophysical Research (Planets),110, 11007Carpenter R. L. 1964, AJ, 69, 2Correia A. C. M., & Laskar J. 2001, Nature, 411, 767Correia A. C. M., & Laskar J. 2003, Icarus, 163, 24Cottereau, L., & Souchay, J. 2009, A&A, 507, 1635Davies, M. E., et al. 1987, Celestial Mechanics, 39, 103Davies, M. E., Colvin, T. R., Rogers, P. G., Chodas, P. W., &Sjogren, W. L. 1992, LPI Contributions, 789, 27Dobrovolskis A. R. 1980, Icarus, 41, 18Folkner W.M, Williams, J.G, Boggs, D.H : the planetary andlunar ephemeris DE421 JPL IOM 343R-08-003, March 31,2008Gold, T., & Soter, S. 1979, Nature, 277, 280Goldreich P., & Peale S. J. 1970, AJ, 75, 273Goldstein R. M. 1964, AJ, 69, 12Karatekin, O., Dehant, V., & Lebonnois, S. 2009, Bulletin ofthe American Astronomical Society, 41, 561Karatekin, ¨O., deViron.O, Lambert.S, Dehant.V,Rosenblatt.P, Van Hoolst.T and LeMaistre.S 2010,Planetary and Space Science 10.1016Kinoshita H. 1977, Celestial Mechanics, 15, 2772 Cottereau et al.: About the contribution in Venus rotation rate variationKonopliv, A. S., & Yoder, C. F. 1996, Geophys. Res. Lett., 23,1857Konopliv, A. S., Banerdt, W. B., & Sjogren, W. L. 1999, Icarus,139, 3Lago B., & Cazenave A. 1979, Moon and Planets, 21, 127Lambeck, K. 1980, Research supported by the Universit´ede Paris VI, Universite de Paris VII, Institut Nationald’Astronomie et de Geophysique, DGRST, CNES, andAustralian National University. Cambridge and New York,Cambridge University Press, 1980. 458 p.,Laskar, J., & Simon, J. L. 1988, Celestial Mechanics, 43, 37Lebonnois, S., Hourdin, F., Eymet, V., Crespin, A., Fournier,R., & Forget, F. 2010a, Journal of Geophysical Research(Planets), 115, 6006Lebonnois S., Hourdin F., Forget F., Eymet V., Fournier, R.,2010b. Venus Express Science Workshop, Aussois, France,june 20-26.Melchior, P. 1978, Oxford: Pergamon PressMoritz, H., and Mueller, I, 1987, Ungarn Publ. Comp., NewYorkMunk, W. H., & MacDonald, G. J. F. 1960, CambridgeUniversity Press, 1960.,Nimmo, F. 2002, Geology, 30, 987Poincar´e, H., 1910,
Bull. Astron.27