Rotation of Rigid Venus: a Complete Precession-Nutation Model

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c L S 2018 ESO S del which sthe is tion nt n s 2 L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model developments that allow us to obtain the precession and Venus is retrograde with a period of 243 days. Furthermore the nutation of the planet (sections 5, 6 and 7). Finally we the obliquity of Venus is very small (2.63) compared to give the precession, the tables of nutation and the polar that of the Earth (23.44). Last, as we will see in the fol- motion of Venus (section 8 and 9). We will discuss our lowing, the perturbating function for the rotational mo- results in section 10. In this study we make some approx- tion of the planets depends on the dynamical flattening 2C−A−B A−B imations : we suppose that the relative angular distance HV = 2C and on the triaxiality TV = 4C . The between the three poles (the poles of angular momentum, Earth has a fast rotation, therefore it has a relatively of figure and of rotation) is very small as is the case for the large dynamical flattening. Moreover its main moments Earth. Moreover we solve a simplified system where the of inertia A and B according to the X and Y axes can be second order of the potential does not appear. Last, we considered equal when comparing them to C. Thus the co- suppose that the small difference between the mean longi- efficient of triaxiality is very small with respect to the coef- tude and the mean anomaly of the Sun which corresponds ficient of dynamical flattening. On the other hand, Venus to the slow motion of the perihelion, can be considered as has slow rotation. Therefore its coefficient of dynamical constant. In sections 3 to 10, the orbital elements of Venus flattening is less important than that of the Earth but and their time dependence are required to carry out our its coefficient of triaxility is of the same order as the co- numerical calculations. In these sections we used the mean efficient of dynamical flattening (see Table.1). For these orbital elements given by Simon et al.(1994). These ele- reasons, it is important to account for the effect of triax- ments are valid over 3000 years with a relative accuracy iality on the rotation of Venus and it is interesting to see of 10−5. This prevents our theory being used in long term what difference will emerge with respect to the Earth’s calculations, our domain of validity being 3000 years. rotation. In Table 1, for the sake of clarity we have written the orbital parameters at J2000.0. However throughout this paper the orbital parameters are considered as a 2. Venus characteristics function of time (containing terms of first and third order From the point of view of its physical characteristics, in time). Venus can be considered as similar to sister of the Earth: the two planets have roughly the same size, mass and mean 3. Description of the motion of rotation and density (see Table 1.). torque-free motion of Venus In order to describe in the most complete way the rotation Earth Venus for a rigid model of Venus around its center of mass we

Semi-major axis 1.000001 U.A 0.723332 U.A need four axes : an inertial axis (arbitrairly chosen) , the ﬁgure axis, the angular momentum axis, and the axis of Eccentricity 0.0167 0.0068 rotation (see Fig.1.). Inclination 0 3.39

Ascending node 174.87 7667 IJ z Z Period of revolution 365.25 d 224.70 d If y G L H x 3 3 Density 5.51 g/cm 5.25 g/cm J g l P Equatorial radius 6378 km 6051 km h If γ I oV R Mass 5.97 ∗ 1024 kg 4.87 ∗ 1024 kg Q hf−h inertial plane of Venus at epoch J2000.0 Obliquity 23.43 2.63

Period of rotation 0.997 d -243.02 d plane normal to L A−B − ∗ −6 − ∗ −6 Triaxiality : 4C 5.34 10 1.66 10 equatorial plane to Venus

2C−A−B ∗ −3 ∗ −5 Dyn. ﬂattening : 2C 3.27 10 1.31 10 Fig. 1. Relation between the Euler angles and the Andoyer variables. B−A 2(2C−A−B) 0.0016 0.12 • As an inertial plane (0, X, Y) we choose the Venus Table 1. Comparison between the orbital (at J2000.0) osculating orbital plane at the epoch J 2000.0. We note and physical parameters of Venus necessary for our study 0V , the intersection between this orbital plane and the and the corresponding ones for the Earth. mean equator of Venus at J2000.0, which is the reference point on the (0, X, Y) plane. The choice of this plane is natural because we want to compute the variation of the Despite this fact, the rotation of the two planets has obliquity of Venus which is the angle between the orbit of with h distinctive characteristics. First, the rotation of Venus and the plane normal to L. L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model 3

• The angular momentum axis of Venus is the axis I = I + J cos g + O(J 2) (5) −→ f directed along G, with G the amplitude of the angular Φ= l + g − J cot I sin g + O(J 2). (6) momentum. We denote h = 0V Q and I as respectively the longitude of the node with respect to 0V and the hf and If correspond to the same definition as h and I, inclination with respect to (0,X,Y). g and J are the longi- but for the axis of figure instead of the axis of angular tude of the node and the obliquity of this same plane with momentum. The Hamiltonian for the torque-free motion respect to Q and to the equatorial plane . of Venus corresponding to the kinetic energy is : • The figure axis is the principal axis of Venus directed 2 2 2 1 sin l cos l 2 2 1 L along Oz and perpendicular to the equatorial plane. We F0 = ( + )(G − L )+ (7) choose it so that it coincides with the axis of the largest 2 A B 2 C moment of inertia C. We denote 0x the axis that is pointed where A, B, C are the principal moments of inertia of toward the origin meridian on Venus, which can be defined Venus. Then the Hamiltonian equations are: in a conventional way. The axis of the figure is determined d ∂F0 with respect to the inertial plane (O,X,Y) through the (L,G,H)= − Euler angles hf , If . The parameter Φ ≈ l + g + h gives the dt ∂(l,g,h) position of the prime meridian (O,x) with respect to 0V . d ∂F0 • The axis of rotation is determined through the vari- (l,g,h)= . dt ∂(L,G,H) ables hr, Ir, i.e. the longitude of the node and the inclination with respect to 0V and the orbit of Venus at J2000.0. The Hamitonian is the full energy of the system. We see For the sake of clarity, these parameters are not repre- the Hamiltonian (7) corresponds to the kinetic energy. sented in Fig.1. 4. Hamiltonian referred to a moving plane To describe the motion of the rotation of the rigid Earth, Kinoshita (1977) used Andoyer variables. This section is based on the pioneering work by Kinoshita Therefore we have introduced these variables to apply an (1977) for the rotation of the Earth. As a new reference analogous theory to Venus. The Andoyer variables (1923) plane, we choose the plane of the orbit of Venus at the date are (see Fig.1.) : t instead of the fixed orbital plane at J2000.0. The motion of this new plane is due to the disturbances of the planets and it is defined by two angles Π1 and π1 which depend – L the component of the angular momentum along the on time (cf Fig.2.). In subsection 4.3 we will determine 0z axis accuratly the numerical expressions of Π1 and π1. – H the component of the angular momentum along the 0Z axis plane normal to angular momentum vector – G the amplitude of the angular momentum of Venus – h the angle between 0V and the node Q between I’ orbit of Venus at t the orbital plane and the plane normal to the angular O momentum. h’− Π1 – g the angle between the node Q and the node P γ oV I orbit of Venus at Π1 π1 J2000.0 – l the angle between the origin meridian Ox and the N node P. M h− Π1

This yields: Fig. 2. Motion of the orbit of Venus at the date t with L = G · cos J etH = G · cos I (1) respect to the same orbit at J2000.0. where I,J are respectively the angle between the angular momentum axis and the inertial axis (O,Z), between the angular momentum axis and the ﬁgure axis. By using the spherical trigonometry in the triangle (P, 4.1. Canonical transformations Q, R) (see Fig.1.), we determine the relations between the We denote (G′,H′,L′,g′,h′,l′) as the new set of variables : Andoyer canonical variables, wich play the same role as cos If = cos I cos J − sin I sin J cos g (2) (G,H,L,g,h,l) but with respect to the new moving reference plane instead of the inertial one. We denote I′ as sin(h − h) sin(Φ − l) sin g f = = . (3) the angle between the plane normal to angular momentum sin J sin I sin I f and the orbit of Venus at t. To prove that the transforma- The angle J is supposed to be very small, so by neglecting tion is canonical, we have to determine a complementary the second order we obtain (Kinoshita, 1977): function E depending on the new variables : J h = h + sin g + O(J 2) (4) ′ ′ ′ ′ f sin I Gdg + Hdh − F0dt = G dg + H dh − (F0 + E)dt. (8) 4 L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model

Because the tranformation does not depend on L,l these plane normal to angular momentum vector two variables do not appear in (8). In a given spherical tri- I’ angle (A,B,C) one can show that the following diﬀerential orbit of Venus at t O relation stands (see appendix) : hd γ oV I orbit of Venus at da c db B dc b C dA Π1 π1 J2000.0 = cos · + cos · + sin sin · . (9) N M Applied to the triangle (M,N,O) (see Fig.3.) this yields: h− Π1

γ oVt plane normal to angular momentum vector Fig. 4. Motion of the orbit of Venus at the date t with O regard to the orbit at J2000.0.

I’ orbit of Venus at the date t. This small transformation is particular to the present study and was not done in other g’−g h’−Π1 works (Kinoshita 1977, Souchay et al. 1999) We have hd = 0Vt M+MO where 0Vt is the so-called orbit of Venus at t Π −I1 ”departure point” for Venus, which constitutes our new π1 orbit of Venus at reference point. The choice of this point is justiﬁed by the M N J2000.0 condition of non-rotation wich characterizes it, described Fig. 3. Details of the spherical triangle M,N,O of Fig.2 in detail by Guinot (1979) and Capitaine et al (1986). which represents the motion of the orbit of Venus at the 0Vt is the natural point to measure any motion along date t with respect to the same orbit at J2000.0. the moving plane. hd is given by the equation :

hd = h + s (15) ′ ′ ′ d(g − g) = cos I · d(h − Π1) − cos I · d(h − Π1) with ′ ′ + sin(h − Π1) sin I · dπ1. s = (1 − cos π1)dΠ1 (16) (10) Z To prove that this transformation is canonical, we can Using the classical equation on the same triangle : determine a function E′ depending on the new variables ′ ′ ′ such that : cos I = cos I cos π1 − sin I sin π1 cos(h − Π1). (11) ′ ′ ′ ′ H dh − Kdt = H dhd − (K + E )dt. (17) Then, multiplying both sides by G and adding - Fo dt, we get : Because the tranformation does not depend on G and g, ′ ′ ′ ′ these two variables do not appear in (17). Using equation G · dg + H · dh − F · dt = G · dg + H · dh − K · dt (12) o (15) we have: ′ ′ ′ ′ ′ ′ where G = G , H = G cos I et K = F0 + E with H ds = −E dt. (18)

′ dΠ1 Thus : E = H (1 − cos π1) · ′ ds ′ dt − H = E . (19) dt ′ ′ dΠ1 ′ +G sin I · [ · sin π1 cos(h − Π1) (13) Notice that s is a very small quantity, so that to a first dt ′ dπ1 ′ approximation E is negligible. Finally, we have a canon- − · sin(h − Π1)]. ′ dt ical transformation with K as a new Hamiltonian of the (14) system : ′ ′ K = F0 + E + E We see that the time does not appear explicitly in Fo but appears in the new Hamiltonian K through the variables ′ ′ dΠ1 = F0 + G sin I · [ · sin π1 cos(hd − Π1) − π1 and Π1 expressing the moving orbital plane. Thanks dt to (12), we have shown that the transformation from dπ1 − · sin(hd − Π1)]. (G,H,L,g,h,l) to (G′,H′,L′,g′,h′,l′) is canonical.This dt canonical transformation is given in detail in Kinoshita (20) (1977). To calculate the new Hamiltonian, we have to de- ′ ′ termine the values of h − Π1, where h is an angle con- 4.2. Ecliptic coordinates of the orbital pole and structed on two different planes, the orbit of Venus at reference point 0V J2000.0 and the orbit of Venus at the date t. It is difficult to calculate this angle, so we can do a new transformation, To calculate the angles that characterize the moving plane ′ changing h by hd where hd is the sum of two angles on the of the orbit of Venus at the date t, a reference point on L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model 5 the orbit of Venus at J2000.0 is needed. For the Earth the orbital pole of Venus reference point is the vernal equinox 0 which coincides pole of Venus with the ascending node of the ecliptic of J2000.0 with the celestial equator of the same date. In a similar way we determine the coordinates of the ascending node 0V of the orbit of Venus at J2000.0 with respect to the equator Io Pov equator of Venus Pv of Venus at this same date. To calculate the coordinates −→ O of 0V we need Pov, the unit vector directed towards the orbital pole at t = 0 defined by : u orbit of Venus sin i0 sin Ω0 −→ γ Pov = − sin i0 cosΩ0 (21) oV   cos i0 Fig. 5. Relation between the venusian orbital pole and the   venusian pole of angular momentum at the date t = 0. where i0 is the angle between the orbit of Venus and the ecliptic at J2000.0. Ω0 is the longitude of the ascending node at the same date. Simon et al.(1994) give : 4.3. Determination of Π1 and π1

i0 =3.39466189 Ω0 = 76.67992019. (22) Using the previous section we can now determine the angles Π1 and π1, which are respectively the longitude of The relative angular distance between the three poles (the the ascending node and the angle between the two orbital poles of angular momentum, of ﬁgure and of rotation) is planes. The angle Π1 (see Fig.4.) is equal to the sum of supposed to be very small, as is the case for the Earth. the angle v and the angle v1 (see Fig.6.) : Throughout this study we will consider that the three −→ Π1 = v1 + v (28) poles coincide. So we denote Pv as the unit vector directed towards the pole of Venus at t = 0 deﬁned by :

0 0 0 0 orbit of Venus at t cos lp cos bp cos δp cos αp

−→  0 0   0 0  Pv = sin lp cos bp = P · sin αp cos δp (23)     p1 orbit of Venus at epoch  0   0  Π1 J2000.0  sin bp   sin δp  γ o     M     Ωt v ecliptic of J2000.0 0 0 Ωo it where lp and bp are the ecliptic coordinates, respectively io P L the longitude and the latitude of the pole of Venus. P is γ oV v1 the matrix that converts equatorial coordinates to eclip- 0 0 tic coordinates. αp,δp are the equatorial coordinates of the Venusian north pole with respect to the equator at 0 0 J2000.0. We take (αp,δp) according to the IAU report on Fig. 6. Motion of the orbit of Venus at the date t with cartographic coordinates(1991)(Habibullin, 1995). respect to the same orbit at J2000.0.

0 0 αp = 272.76 δp = 67.16. (24) The relationship between π1, v, i0, it, Ω0 and Ωt are derived from the spherical triangle (P,L,M) : To determine 0V we use the following equations (see Fig.5.) : −→ −→ cos I0 = Pov · Pv (25) cos π1 = cos i0 cos it + sin i0 sin it cos(Ωt − Ω0) (29)

cos v sin π1 = − sin i0 cos i +cos i0 sin i cos(Ω −Ω0) (30) −−→ −−→ −→ −→ t t t 0 V −→ −→ −→ 0 V Pv ∧ Pov sin I0 = Pv ∧ Pov ⇒ u = = (26) |0V | |0V | sin I0 sin v sin π1 = sin it sin(Ωt − Ω0) (31)

where i0, it are respectively the inclinations of the orbit where Pv and Pov are the unit vectors described previ- of Venus at J2000.0 and at the date t with respect to ously, I0 is the obliquity (angle between the axis of angular momentum and the 0Z inertial axis) and −→u is the unit the ecliptic at J2000.0. Ω0 et Ωt are the longitude of the −−−→ ascending node respectively at J2000.0 and at the date t. vector along 00 . V Simon et al. (1994) gave : We ﬁnally obtain : 2 3 it =3.39466189 − 30”.84437t − 11”.6783t + 0”.03338t 0 0 I0 =2.634, b = −0.103, l = 57.75. (27) 2 3 p p Ωt = 76.67992019 − 10008”.4815t − 51”3261t − 0”.58910t 6 L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model where t is counted in thousands of Julian years start- which is caused by planetary perturbations. The expres- ′ ing from J2000.0. In this section we choose to carry our sions of Fo and E + E have been set in the previous sec- calculations to third order in time. The relationships be- tion. U is the disturbing potential due to external disturb- 0 tween v1,bp and i0 are derived from the spherical triangle ing bodies. Here the external disturbing body is the Sun (OV GP) (see Fig.7.): (The perturbation due to the planets can be neglected), and its disturbing potential is given by :

π1 orbit of Venus at U = U1 + U2 epoch J2000.0 M

ecliptic of Ωt ′ J2000.0 γο it GM 2C − A − B A − B Ωo 2 G io L U1 = 3 [ P2(sin δ)+ P2 (sin δ)cos2α] lp° P r 2 4 bp° (38) v1 γο V orbit of Venus at t ∞ ′ n GM MV aV U2 = [J P (sin δ) rn+1 n n Fig. 7. Motion of the orbit of Venus at the date t with n=3 regard to the same orbit at J2000.0. Xn m − Pn (sin δ) · (Cnm cos mα + Snm sin mα)] (39) m=1 0 X sin v1 sin i0 = sin bp. (32) where G is the gravitationnal constant, M ′ is the mass of Using the equations (29), (30), (31) we obtain : the Sun, r is the distance between its barycenter and the barycenter of Venus. α and δ are respectively the plan- v1 = 18.92 etocentric longitude and latitude of the Sun (not to be confused with the usual equatorial coordinates), with re- 2 Π1(t) = v1 + v = 381348”. − 901.”346 t − 0.0910822 t spect to the mean equator of Venus and with respect to a m 3 4 meridian origin. The P are the classical Legendre func- −0”.000759571 t + O(t ) (33) n tions given by : 2 m + 2 (−1)m(1 − x ) 2 dn m(x − 1)n 2 3 4 P m(x)= . (40) π1(t) = 59”.2626 t+0.0285354 t +0.00028575 t +O(t ) n 2nn! dn+mx (34) where t is counted in thousands of Julian years starting The Hamiltonian equations are : from J2000.0. For the Earth we have : d ′ ′ ′ ∂K” (L , G ,H )= − ′ ′ ′ (41) 2 dt ∂(l ,g ,h ) Π1E(t) = 629543.” − 867.927 t + 0”.1534 t (35) 3 4 +0”.0000053 t + O(t ) d ∂K” (l′,g′,h′)= . (42) dt ∂(L′, G′,H′)

2 In the following the prime notations used above are omit- π1E(t) =46”.9973 t − 0”.0335053 t ted for the sake of clarity. Using the following equations 3 4 −0”.00012374 t + O(t ). (36) seen in section 3 : H L The relative motion of the orbital plane at t with respect cos I = and cos J = . (43) to the orbital plane at t=0 is signiﬁcantly greater for Venus G G than for the Earth. We used here the mean orbital ele- The canonical equations (41) and (42) become (Kinoshita, ments given by Simon et al.(1994). These elements are 1977): valid over 3000 years with a relative accuracy of 10−5 and dl 1 ∂K thus so too are our results = − sin J (44) dt G ∂J

5. The Hamiltonian of the system and the dg 1 ∂K ∂K = [cot J + cot I ] (45) equation of the motion of rotation dt G ∂J ∂I

5.1. The equation of the motion of rotation dh 1 ∂K = − (46) As explained in section 4, the Hamiltonian related to the dt G sin I ∂I rotational motion of Venus is : and ′ dI 1 dG 1 dH K”= Fo + E + E + U (37) = [cot I − ] dt G dt sin I dt ′ where Fo is the Hamiltonian for the free motion, E +E is 1 1 ∂K ∂K = [ − cot I ]. (47) a component related to the motion of the orbit of Venus, G sin I ∂h ∂g L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model 7

Replacing I by J in the precedent equation yields the this transformation we obtain: variation of J. I is the obliquity and J is the small angle between the angular momentum axis and the figure axis. To solve the six differential equations (41) and (42), 1 2 1 2 Kinoshita (1977) used the Hori’s method (1966). Here we P2(sin δ) = (3 cos J − 1) (3 cos I − 1)P2(sin β) solve a simplified system, i.e the potential U2 does not 2 "2 appear in our study. To know the motion of precession 1 and nutation of Venus, we have to solve only the following − sin 2IP 1(sin β)cos2(λ − h) 2 2 equations : # 3 + sin 2J − sin 2IP2(sin β)cos g dI 1 dG 1 dH " 4 = [cot I − ] dt G dt sin I dt 1 − (1 + ǫ cos I)(−1+2ǫ cos I) 1 1 ∂K ∂K 4 = [ − cot I ] (48) ǫ=±1 G sin I ∂h ∂g X P 1(sin β) sin(λ − h − ǫg) dh 1 ∂K 2 = − (49) 1 dt G sin I ∂I − ǫ sin I(1 + ǫ cos I) 8 ǫ=±1 X where I characterizes the obliquity and h the motion of 2 2 P2 (sin β) cos(2λ − 2h − ǫg) + sin J precession-nutation in longitude. These equations can also # be written as : 3 2 1 sin IP2(sin β)cos2g + ǫ sin I 4 4 dI 1 dG 1 dH " ǫ=±1 = [cot I − ] X dt G dt sin I dt 1 1 (1 + ǫ cos I)P2 (sin β) sin(λ − h − 2ǫg) − 1 1 ∂U1 ∂U1 16 = [ − cot I ] (50) G sin I ∂h ∂g 2 2 (1 + ǫ cos I) P2 (sin β)cos2(λ − h − ǫg) ǫ=±1 # X (54)

dh 1 ∂U1 = − (51) dt G sin I ∂I and

Integrating these equations, we obtain the variations of the angles h and I : 2 2 1 2 P (sin δ) cos2α = 3 sin J − (3 cos I − 1)P2(sin β) 2 2 1 1 ∂ ∂ " ∆I = [ U1dt − cot I U1dt] (52) 1 G sin I ∂h ∂g cos2l + sin 2IP 1(sin β) sin(λ − h − 2ǫl) Z Z 4 2 ǫ=±1 X 1 2 2 + sin IP2 (sin β)cos2(λ − h − ǫl) 8 # 1 ∂ ∆h = − U1dt. (53) G sin I ∂I + ρ sin J(1 + ρ cos J) Z ρ=±1 X 3 5.2. Development of the disturbing function − sin 2IP2(sin β) cos(2ρl + g) " 2 The disturbing potential (38) is a functions of the modiﬁed 1 2 − (1 + ǫ cos I)(−1+2ǫ cos I) Legendre polynomials P2(sin δ) and P2 (sin δ) and of the ± 2 planetocentric longitude α and latitude δ of the Sun, with ǫ= 1 1X respect to the mean equator of Venus and with respect to P2 (sin β) sin(λ − h − 2ρǫl − ǫg) a meridian origin. From the ephemerides we only know the 1 − ǫ sin I(1 + ǫ cos I) longitude λ and the latitude β of the Sun with respect to 4 ǫ=±1 the orbit of Venus. So we use the transformation described X by Kinoshita (1977) and based on the Jacobi polynomials, 2 P2 (sin β) cos(2λ − 2h − 2ρǫl − ǫg) which expresses α and δ as a function of λ and β. Applying # 8 L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model

1 2 2 6. Determination of the precession and the + (1 + ρ cos J) − 3 sin IP2(sin β) cos(2l +2ρg) 4 nutation of Venus ρ=±1 " X − ǫ sin I(1 + ǫ cos I) Replacing U1 in (52) and (53) we obtain : ǫ=±1 ′ X 1 ∂ GM 2C − A − B 1 ∆h = − [ P2 (sin β) sin(λ − h − 2ρǫl − 2ǫg) G sin I ∂I r3 2 1 2 Z + (1 + ǫ cos I) 1 2 3 2 4 − (3 cos I − 1) − sin I cos2(λ − h) ǫ=±1 4 4 X h A − B 3 2 i 2 + sin l cos(2l +2g) P2 (sin β)cos2(λ − h − ρǫl − ǫg) . 4 2 # 3h + (1 + ǫ cos I)2 (55) 4 ǫ=±1 X Using (40) we obtain : cos2(λ − h − ǫl − ǫg) dt ! 1 2 i P2(sin β)= (3 sin β − 1) 2 (59)

1 2 1 P (sin β)= −3 sin β(1 − sin β) 2 ′ 2 1 1 ∂ GM 2C − A − B ∆I = 2 2 G sin I ∂h r3 2 P2 (sin β) = 3(1 − sin β) " Z 1 3 where β is the latitude of the disturbing body with respect − (3 cos2 I − 1) − sin2 I cos2(λ − h) to the orbit of Venus at the date t. In our problem the 4 4 h A − B 3 i disturbing body is the Sun, so that β = 0. This yields : + sin2 I cos(2l +2g) 4 2 3h 1 + (1 + ǫ cos I)2 P2(sin β)= − 4 2 ǫ=±1 X 1 P2 (sin β)=0 cos2(λ − h − ǫl − ǫg) dt ! 2 i P2 (sin β)=3. ∂ GM ′ 2C − A − B − cot I 3 Assuming that the angle J is small (J ≈ 0),as it the ∂g r " 2 case for the Earth, (54) and (55) can be written : Z 1 3 − (3 cos2 I − 1) − sin2 I cos2(λ − h) 1 2 3 2 4 4 P2(sin δ) = [− (3 cos I − 1) − sin I cos2(λ − h)] (56) h A − B 3 i 4 4 + sin2 I cos(2l +2g) 4 2 and 3h + (1 + ǫ cos I)2 3 4 P 2(sin δ)cos2α = [ sin2 I cos(2l +2g) ǫ=±1 2 2 X 3 + (1 + ǫ cos I)2 · cos2(λ − h − ǫl − ǫg)]. (57) cos2(λ − h − ǫl − ǫg) dt . 4 #! # ǫ=±1 i X (60) Thus the disturbing potential is simplified in the following form : To simplify the calculations, we study separately the components depending on the dynamical flattening and those ′ GM 2C − A − B 1 2 depending on the triaxiality of Venus. As the dynamical U1 = 3 − (3 cos I − 1) r " 2 4 flattening corresponds to the symmetric part at the right hand side of (59) and (60) although the triaxiality cor- 3 − sin2 I cos2(λ − h) responds to the antisymmetric one, in the following the 4 coefficients depending on dynamical flattening will be de- A − B 3 + sin2 I cos(2l +2g) noted with an index ”s” and the coefficients depending on 4 2 the triaxility with an index ”a”. Moreover the expression ′ ′ GM GM a3 3 3 in (59) and (60) can be replaced by 3 · 3 where + (1 + ǫ cos I)2 · cos2(λ − h − ǫl − ǫg) . r a r 4 a is the semi-major axis of Venus given by : ǫ=±1 # X (58) n2a3 = GM ′. (61) L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model 9

Finally we have the following equations : where u is the eccentric anomaly, e the eccentricity, n the mean motion and M the mean anomaly of Venus, we have 1 ∂ ∆hs = − W1dt (62) the following classical development: G sin I ∂I Z 1 a 1 3 1 27 ( )3 = (1 + e2)+ (3e + e3)cos M 2 r 2 2 2 8 1 1 ∂ 9 2 53 3 ∆Is = W1dt (63) + e cos2M + e cos3M. (72) G sin I ∂h 4 8 Z For comparison we also give the corresponding coeﬃcients and : of the development of the Earth (see Table 2). We have 1 ∂ ∆ha = − W2dt (64) G sin I ∂I M LS period − − Z days cos ×10 7 t cos ×10 7

0 0 1 (1 + 3 e2) = 5000344 [5002093] -96 [-105] 1 1 ∂ ∂ 2 2 ∆Ia = W2dt − cot I W2dt (65) 1 0 224,70 [365,26] 1 (3e + 27 e3) = 101584 [250708] -7164 [-6305] G"sin I ∂h ∂g # 2 8 Z Z 9 2 2 0 112,350 [182,63] 4 e = 1032 [6282] -146 [-316] where 53 3 3 0 74.900 [121.753] 8 e = 200 [308] -40 [-23] GM ′ a 2C − A − B 1 3 2 3 W1 = 3 ( ) − (3 cos I − 1) 1 a a r 2 4 Table 2. Development of 2 r of Venus (the corre- 3 2 sponding values for the Earth are in square brackets). t − sin I cos2(λ − h) (66) 4 is counted in Julian centuries. ′ GM a 3 A − B 3 2 W2 = ( ) sin I cos(2l +2g) the following definition : a3 r 4 2 3 h 2λ = 2(ω+v)=2(ω+M+(v−M))=2LS+2(v−M) (73) + (1 + ǫ cos I)2 4 ǫ=±1 where λ is the true longitude of the Sun, ω the longitude X of the periapse of Venus, M the mean anomaly and LS cos2(λ − h − ǫl − ǫg) . (67) the mean longitude of the Sun. From Kepler’s equation i we have the following classical development : We adopt the following notations : e3 5 11 13 v−M = (2e− ) sin M+( e2− e4) sin 2M+ e3 sin 3M. 4 4 24 12 ∆ψ = ∆h = (∆hs + ∆ha) (68) (74) From the trigonometric equation : ∆ǫ = ∆I = (∆Is + ∆Ia). (69) cos2λ = cos2LS cos2(v−M)−sin 2LS sin 2(v−M) (75) As the rotation of Venus is retrograd the convention used we obtain : is not the same as IAU to the Earth. a a ( )3 cos2(λ − h) = ( )3(cos2L cos2(v − M) r r S 7. Development of the disturbing function − sin 2LS sin 2(v − M)). (76) Using the developments (74) and (76) and the classical Considering the level of accuracy and the time interval trigonometric relationships we obtain a development of involved (3000 y), truncating the function e to first order ( a )3 cos2(λ − h): in time is a sufficient approximation. Therefore we take r a 5 the eccentricity of Venus from Simon et al.(1994) as : ( )3 cos2(λ − h) =(1 − e2)cos2L r 2 S e =0.0067719164 − 0.0004776521t. (70) 7 123 +( e − e3) cos(2L + M) 2 16 S 1 1 3 7.1. Terms depending on the dynamical flattening +(− e + e ) cos(2LS − M) 2 16 To solve the equations (62) and (63) which give the nuta- 17 2 +( e ) cos(2LS +2M) tion coming from the dynamical flattening, it is necessary 2 1 a 3 a 3 to develop ( ) and ( ) cos2(λ − h) with respect to 845 3 2 r r + e cos(2LS +3M) time (i.e with respect to the mean anomaly and the mean 48 1 longitude of Sun). Using Kepler’s equation : + e3 cos(2L − 3M). 48 S u − e sin u = n(t − t0)= M (71) (77) 10 L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model

The coeﬃcients of this development are given in Table a 3 5 2 ( ) cos2(λ − h +Φ) =(1 − e ) cos(2LS + 2Φ) 3. The periods in Tables 2 and 3 have been computed r 2 7 123 +( e − e3) cos(2L + M + 2Φ) 2 16 S 1 1 +(− e + e3) cos(2L − M + 2Φ) 2 16 S M LS period days cos ×10−7 t cos ×10−7 17 2 +( e ) cos(2LS +2M + 2Φ) − 5 2 2 0 2 112,35 [182,62] (1 2 e ) = 9998853 [9993025] 161 [351] 845 3 -1 2 224,70 [365,22] (− 1 e + 1 e3)= −33859 [−83540] 2388 [2101] + e cos(2LS +3M + 2Φ) 2 16 48 7 − 123 3 1 1 2 74,90 [121,75] ( 2 e 16 e ) = 236993 [584444] -16718 [- 14713] 3 + e cos(2LS − 3M + 2Φ) 2 2 56,17 [91,31] ( 17 e2) = 3898 [23730] -550 [-1193] 48 2 (79) 845 3 3 2 44.94 [73.05] 48 e = 54 [821] -4 [-20]

-3 2 -224.70 [-365.25] 1 e3 = 0[0] 0[0] 48 1 a 1 3 ( )3 cos(2Φ) = ( + e2) cos(2Φ) 3 2 r 2 4 Table 3. Development of a cos(2(λ − h) of Venus (the r 3 27 3 corresponding values for the Earth are in square brackets). +( e + e ) cos(M − 2Φ) 4 32 t is counted in Julian centuries. 3 27 +( e + e3) cos(M + 2Φ) 4 32 9 + e2 cos(2M − 2Φ) 8 9 +( e2) cos(2M + 2Φ) through the mean elements of the planets given by the 8 ephemerides VSOP87 of Simon et al.(1994). Because the 53 + e3 cos(3M + 2Φ) eccentricity of Venus is smaller than the eccentricity of 16 the Earth, the coeﬃcients of the development for Venus 53 + e3 cos(3M − 2Φ). are smaller than the corresponding ones for the Earth, 16 except for the leading one in Table 3. (80) These are given in Tables 4,5 and 6 7.2. Terms depending on the triaxiality M LS Φ period − − days cos ×10 7 t cos ×10 7

0 0 2 -121.51 ( 1 + 3 e2) = 5000344 -48 To solve the equations (64) and (65) it is necessary 2 4 1 a 3 a 3 to develop 2 ( r ) cos2Φ, ( r ) cos(2(λ − h) − 2Φ) and 1 0 2 -264.6 ( 3 e + 27 e3) = 50792 -3582 a 3 4 32 ( r ) cos(2(λ − h)+2Φ) with respect to the mean anomaly 1 0 -2 78.86 ( 3 e + 27 e3) = 50792 -3582 M, the mean longitude LS of Sun and to the angle 4 32 9 2 l + g ≈ Φ according to (6). Using classical trigonomet- 2 0 2 1490.35 ( 8 e ) = 516 -72 ric equations and developments similar to those used in 2 0 - 2 58.37 ( 9 e2) = 516 -72 section 7.1, we obtain the following : 8 53 3 3 0 2 195.26 16 e = 10 0

53 3 3 0 -2 46.34 16 e = 10 0

3 Table 4. Development of a cos(2Φ) of Venus. t is a 5 r ( )3 cos2(λ − h − Φ) =(1 − e2) cos(2L − 2Φ) counted in Julian centuries. r 2 S 7 123 +( e − e3) cos(2L + M − 2Φ) 2 16 S 1 1 +(− e + e3) cos(2L − M − 2Φ) 2 16 S 17 +( e2) cos(2L +2M − 2Φ) 2 S 845 + e3 cos(2L +3M − 2Φ) 48 S 1 + e3 cos(2L − 3M − 2Φ) 48 S (78) L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model 11

M LS Φ period M LS Φ period − − − − days cos ×10 7 t cos ×10 7 days cos ×10 7 t cos ×10 7

− 5 2 − 5 2 0 2 - 2 58.37 (1 2 e ) = 9998853 161 0 2 2 1490.35 (1 2 eV ) = 9998853 161

− 1 1 3 − − 1 1 3 − -1 2 - 2 78.87 ( 2 e + 16 e )= 33859 2388 -1 2 2 -264.6 ( 2 e + 16 e )= 33859 2388

7 − 123 3 7 − 123 3 1 2 - 2 46.34 ( 2 e 16 e ) = 236993 -16718 1 2 2 195.26 ( 2 e 16 e ) = 236993 -16718

17 2 17 2 2 2 - 2 38.41 ( 2 e ) = 3898 -550 2 2 2 104.47 ( 2 e ) = 3898 550

1 3 1 3 -3 2 -2 264.59 48 e = 0 0 -3 2 2 224.70 48 e = 0 0

845 3 845 3 3 2 -2 32.80 48 e = 54 -4 3 2 2 71.31 48 e = 54 -4

a 3 a 3 Table 5. Development of r cos(2(λ−h)−2Φ) of Venus. Table 6. Development of r cos(2(λ−h)+2Φ) of Venus. t is counted in Julian centuries. t is counted in Julian centuries.

We note that the coefficients in Tables 5 and 6 are the same as the coefficients of Table 3, because the calculations are similar and they do not depend on the frequency associated with each argument, but only on the eccentricity of Venus. Nevertheless the corresponding periods are different because their calculation includes the argument Φ, the sideral rotation of Venus given by : 2Π Φ ≈ l + g = −( ) t (81) TV where TV is the sidereal period of the rotation of Venus. The numerical value for TV is TV = 243.02 days. Notice also that because the rotation is retrograde, Φ has a nega- tive sign, and that the period of LS (225 d) is close to the period of Φ (243 d). Therefore the period for the leading componant 2LS + 2Φ in Table 6 (1490.35 d) is much longer than the period of the leading component 2LS − 2Φ in Table 5 (58.37 d). We have LS = ωo t, M = ωoω t andΦ = ωr t where ωo = (2Π/224.70080)rd/d, ωoω = (2Π/224.70082)rd/d and ωr = −(2Π/243.02)rd/d. The very small difference between ωo and ωoω is due to the longitude of periapse ω which corresponds to a very low frequency. As a consequence we do not change our calculations significantly by considering that L˙S = M˙ (LS = M + ω).

8. Precession and nutation which depend on the longitude and obliquity In the following, we calculate the precession and the nutation of Venus depending on both the dynamical ﬂattening and the dynamical triaxiality.

8.1. Precession and the nutation of Venus depending on the dynamical ﬂattening According to (62) and (63) we compute the precession- nutation in longitude (∆hs) and obliquity (∆Is) due to the dynamical ﬂattening :

1 ∂ GM ′ 1 2C − A − B ∆hs = − 3 W1dt (82) sin I ∂I a3 G 2 Z Ks ∂ = − W1dt sin I ∂I Z 12 L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model ′ 1 ∂ GM 1 2C − A − B Table 7. ∆Ψs = ∆hs: nutation coefficients in longitude ∆Is = 3 W1dt (83) sin I ∂h a3 G 2 of Venus depending on its dynamical flattening. Z Ks ∂ = W1dt sin I ∂h Z Argument Period sin t sin cos where days arc second arc second/Julian cy arc second − − − a 1 1 (10 7) (10 7) (10 7) 3 2 2 2Ls 112.35 21900468 352 0 W1 = ( ) − (3 cos I − 1) − sin I cos2(λ − h) (84). r 12 4 M 224.70 -889997 62765 61 ∆hs and ∆Is in (82) and (83) depend directly on the 2Ls + M 74.90 346057 - 24412 -7 scaling factor Ks : 2Ls − M 224.70 -148323 10461 10 GM ′ 2C − A − B K =3 . (85) 2Ls + M 56.17 4269 - 602 0 s Ga3 2 h i Using the third Kepler ’s law we obtain : 2M 112.35 -4521 640 0 n2 2C − A − B 3n2 Ks =3 = HV (86) ω 2C ω Table 8. ∆ǫs = ∆Is : nutation coefficients in obliquity of Venus depending on its dynamical flattening. where n is the mean motion of Venus. The dynamical flat- 2C−A−B tening, defined as HV = 2C , is a dimensionless parameter. We supposed here to the first order of approx- Argument Period cos t cos sin imation that the components ω1 and ω2 of the rotation days arc second arc second/Julian century arc second − − − of Venus are negligible with respect to the component ω3 (10 7) (10 7) (10 7) along the figure axis (0,z), as is the case for the Earth 2Ls 112.35 -100741 -16 0 (Kinoshita, 1977). So we have 2Ls + M 74.90 -15919 1123 0

G = Aω1 + Bω2+ Cω3 ≈ Cω (87) 2Ls − M 224.70 6822 -481 0 with the values of A,B,C adopted in section 10 and that of 2Ls + 2M 56.17 -196 27 0 the dynamical ﬂattening given which is deduced directly from them (given in Table 1), we ﬁnd :

Ks = −8957”.55 ± 133.29/Julian. cy. (88) These coefficients are of the same order of magnitude as those for the Earth due to the action of the Sun found In the case of the Earth, the scaling factor due to the by Kinoshita (1977) and Souchay et al. (1999). The largest gravitational influence of the Sun is (Souchay et al.1999) nutation coefficient has roughly a 2” amplitude whereas : for the Earth it has a 1” amplitude. As in the case of the E Ks = 3475”.36/Julian. cy. (89) precession, this difference comes from the larger value of Then we have the following result : the constant Ks and is slightly compensated for after inte- gration by the fact that the period of revolution of Venus K n2 ω H around the Sun (225 d) is shorter than that of the Earth. s = ( ) · ( E ) · ( V ) ≈−2.577. (90) E 2 The largest nutation coefficient in obliquity is compara- Ks nE ω HE tively very small with a 0.1” amplitude whereas for the Thus for Venus, the value of the constant of the precession- Earth it has a 0.5” amplitude. The difference is due to the nutation due to the Sun is roughly 2.6 times bigger that small obliquity of Venus (I ≈ 3), our results depending on the corresponding one for the Earth. Through the devel- sin I. opment done previously (cf section 7) and according to the conventional notations (68) and (69), we get the precession and the nutation of Venus depending on its dynami- 8.2. Precession and nutation of Venus depending on cal flattening. We will see in the following sub-section that dynamical triaxiality the triaxiality does not contribute to the precession. So we Starting from (64) and (65), we can compute the nutations obtain the following result for the precession : ∆ha and ∆Ia respectively in longitude and in obliquity, 3 due to the triaxiality of Venus : ψ˙ = K cos I (1 + e2)dt = 4474”.35t − 0.021t2 (91) s 2 Z where t is counted in Julian centuries . The nutation coef- 1 ∂ GM ′ 1 A − B ∆ha = − 3 W2dt (92) ficients in longitude and in obliquity are given respectively sin I ∂I a3 G 4 in Tables 7 and 8. All our results are given with respect Z Ka ∂ = − W2dt to the moving orbit at the date t. sin I ∂I L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model 13

′ 1 ∂ GM 1 A − B Table 9. ∆Ψa = ∆ha: nutation coeﬃcient in longitude of ∆Ia = 3 3 W2dt Venus depending on its triaxiality. "sin I ∂h a G 4 Z ∂ GM ′ 1 A − B − cot I 3 3 W2dt (93) ∂g a G 4 # Argument Period sin t sin cos Z days arc second arc second/Julian cy arc second − − − Ka ∂ ∂ 10 7 10 7 10 7 = W2dt − Ka cot I W2dt (94) sin I ∂h ∂g 2Φ -121.51 -5994459 58 0 Z Z where 2LS − 2Φ 58.37 -2880826 -46 0 M + 2Φ -264.6 -132590 9351 - 11 a 3 1 2 W2 = ( ) sin I · cos(2Φ) r 2 M − 2Φ + 2LS 46.34 -54201 3823 0 1 2 M − 2Φ 78.86 39519 - 2787 0 + (1 + ǫ cos I) 4 ǫ=±1 2LS + 2Φ 1490.35 38866 0 0 X · cos2(λ − h − ǫΦ) . (95) −M − 2Φ + 2LS 78.86 13179 -929 0

∆ha and ∆Ia in (92) and (93) depend directly on the 2M + 2Φ 1490.35 7587 - 1058 -7 − scaling factor Ka : 2M 2Φ + 2LS 38.41 -739 104 0 2M − 2Φ 58.37 297 -41 0 GM ′ A − B K =3 (96) a Ga3 4 M + 2Φ + 2LS 195.26 121 - 8 0 h i −M + 2Φ + 2LS -264.66 23 -2 0 where M ′, G, G and a have been deﬁned previously. Still using Kepler’s third law we get : 2M + 2Φ + 2LS 104.47 1 0 0

n2 A − B Ka =3 . (97) ω 4C Table 10. ∆ǫa = ∆Ia :nutation coeﬃcients in obliquity of Venus depending on its triaxiality. A−B The triaxiality, deﬁned as TV = 4C , is also, as HV , a dimensionless parameter. here we assume again that :

Argument Period cos t cos sin G ≈ Cω. (98) days arc second arc second/Julian cy arc second

Then, with the value of the triaxiality (given in Table 1), 2Φ -121.51 275453 -3 0 we get : 2lS − 2Φ 58.37 -132365 -2 0 M + 2Φ -264.6 6093 -430 0 Ka = 1131”.23 ± 17.09 Julian/cy. (99) M − 2Φ + 2LS 46.34 -2491 176 0

In the case of the Earth, the scaling factor due to the 2lS + 2Φ 1490.35 -1786 0 0 gravitational inﬂuence of the Sun is (Souchay et al.1999) M − 2Φ 78.86 -1816 128 0 : E −M − 2Φ + 2LS 78.86 606 -43 0 Ka = −5”68 Julian/cy. (100) 2M + 2Φ 1490.35 -348 47 0 Thus we have the following ratios : 2M − 2Φ + 2LS 38.41 -35 5 0 2 Ka nv ωE TV 2M − 2Φ 58.37 -14 2 0 E = ( 2 ) · ( ) · ( ) ≈−199.16. (101) Ka nT ω TE M + 2Φ + 2LS 195.26 -6 0 0

This result shows that the coefficients of nutation due to −M + 2Φ + 2LS -264.6 -1 0 0 the triaxility are considerably larger for Venus than for 2M + 2Φ + 2LS 104.47 0 0 0 the Earth. Thanks to the development done previously (cf section 4.1) and according to the equations (92) and (93), we get the nutation of Venus depending on its triaxiality. The nutation coefficients in longitude and in obliquity are From the Tables 7 to 10, we choose to calculate nu- given respectively in Tables 9 and 10. All our results are merically from their analytical expression the precession- given with respect to the moving orbit at the date t. nutation with respect to t, over a period of 4000 d. Fig.8. For Venus, the largest nutation coefficient in longitude represents the combined precession and nutation of Venus for the terms depending on its triaxiality has a 0”.6 ampli- in longitude. The periodic part stands for the nutation and tude and the largest nutation coefficient in obliquity has the linear part for the precession (see (91)). In Fig.9. and a 0”.03 amplitude. Their period corresponds to half the Fig.10., we represent the nutation of Venus respectively period of rotation of the planets, 121.5 d. in longitude and in obliquity depending on its dynami- 14 L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model

L 3

L 2 arcsecond 400 H

1 arcsecond 300 H 0 200 -1

100 - nutation in longitude 2

0 -3 precession and nutation in longitude 0 1000 2000 3000 4000 0 1000 2000 3000 4000 tempsHdaysL tempsHdaysL

Fig. 8. The precession and the nutation of Venus in lon- Fig. 11. The nutation of Venus in longitude for a 4000 gitude for a 4000 day time span. day time span, from J2000.0

0.10 2 L L 0.05 1 arcsecond H arcsecond H 0.00 0 -0.05

-1 -0.10 nutation in obliquity nutation in longitude -2 0 1000 2000 3000 4000 0 1000 2000 3000 4000 tempsHdaysL tempsHdaysL Fig. 12. The nutation of Venus in obliquity for a 4000 Fig. 9. The nutation in longitude of Venus depending day time span, from J2000.0. on its dynamical ﬂattening (cyan) and depending on its triaxiality (blue) for a 4000 day time span, from J2000.0. of Venus depending on its triaxiality are of the same or-

0.10 der of magnitude as the nutation coeﬃcients depending on

L its dynamical ﬂattening, whereas for the Earth they are

0.05 small and negligible in comparison ( Kinoshita,1997 and

arcsecond Souchay et al. 1999). Two reasons explain this : H Ka 1131 1 0.00 – the ratio of scaling factors is Ks = 8953 ≈ 8 for Venus, whereas for the Earth it is two orders of magnitude E -0.05 Ka 5 1 smaller : E = ≈ Ks 3475 695 nutation in obliquity – the frequency of the sidereal angle Φ˙ ≈ l˙ +g ˙ which -0.10 0 1000 2000 3000 4000 enters in the denominator during the integrations in tempsHdaysL (92) and (93) is roughly 243 times larger for the Earth than for Venus Fig. 10. The nutation in obliquity of Venus depending on its dynamical flattening (cyan) and depending on its . triaxiality (blue) for a 4000 day time span, from J2000.0. Thus we have pointed out a significant difference between the nutations of the two planets. cal flattening (black curve) and its triaxiality (red curve). 9. Motion of the pole of Venus in space In Fig.11 and Fig.12., we represent the nutation respectively in longitude and in obliquity when combining the From the previous section, it is possible to describe the two components above. In Figs.8 to 12. we choose the 4000 motion of the pole of Venus with respect to its or- d time span to clearly see the leading oscillations. Recall bital plane at the date t. The rectangular coordinates that our study is valid over 3000 years. The nutation of (Xp, Yp,Zp) of the pole are given by :

Venus depending on its triaxiality is dominated by a si- Xp − sin(I0 + ∆I) sin(h + ∆h) nusoid with a period of 121.51 d and another one with Yp = − sin(I0 + ∆I) cos(h + ∆h) (102)     a period of 58.37 d. On the other hand the nutation de- Zp cos(I0 + ∆I) pending on its dynamical ﬂattening is largely dominated     by a single sinusoid with a period of 112.35 d. We can also The precession in longitude Ψ is given by remark (Fig.9. and Fig.10.) that the nutation coeﬃcients Ψ= Ψ˙ t = −h = 4474”.35t ± 66.5 (t in Julian/cy) (103) L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model 15

C I0 is the nominal value of the obliquity : with these values and adopting MR2 =0.3360, we get:

C − A −5 C − B −6 I0 = −ǫ0 = −2.63. (104) =1.643 × 10 , =9.79 × 10 , C C B − A − =6.631 × 10 6. (106) Replacing ∆I and ∆h by their numerical values from C Tables 7 to 10, we show in Fig.13. the motion of the Venus figure axis in space over a one century time interval, for All our calculations in this paper have been done with these values. When studying the rotation of Venus, which the combined motion of precession and nutation C appears clearly. Habbibulin (1995) has taken MR2 = 0.340 and the values of the Stokes parameters obtained by Williams (1983) From (103) we directly determine the period of preces- : sion of Venus, i.e 28965.10±437 years. It is slightly longer C − A − C − B − than the period of precession of the Earth, i.e. 25712.4 =2.3324 × 10 5, =1.2618 × 10 5, y (Bretagnon et al. 1997). Notice that in the case of the C C Earth, the Moon contributes roughly 2/3 and the Sun 1/3 B − A −5 =1.0706 × 10 . (107) to the precession rate. In the case of Venus, only the Sun C contributes significantly to the precession rate. However, In Table 11 we compare our results with those ob- as we have shown in section 8, this solar contribution is tained by directly using the values of Habbibulin (1995) roughly 2.5 times greater than the corresponding one for for the physical parameters of Venus, thus showing large the Earth. This explains why the periods of precession are differences. rather equivalent.

Table 11. Comparison between our results computed with the recent values of the moment of inertia and those 2,6328 obtained with the values of Habbibulin et al.(1995)

2,6326

2,6324 Principal parameter Habbibulin(1995) This study Coordinate Y of pole Venus (degree) − − 2,6322 A−B − ∗ 6 − ∗ 6 Triaxiality : 4C 2.67 10 1.66 10

-0,06 -0,05 -0,04 -0,03 -0,02 -0,01 0 2C−A−B −5 −5 Coordinate X of pole of Venus (degree) ∗ ∗ Dyn. ﬂattening : 2C 1.70 10 1.31 10

V Fig. 13. (X,Y) motion of Venus polar axis in space for a Ks −12275”.32/cy −8957”.55/cy ± 133.29 one century time span V Ka 1829”.34/cy 1133”.28/cy ± 17.09

Precession 6133”.77t 4474”.35t ± 66.5

Period of precession 21156 years 28965.10 ± 436.99 years

2C−A−B A−B Largest nutation coeﬃcient 3”.00 2”.19 10. Numerical values of 2C and 4C and comparison of results

From the various calculations done in the previous sec- Table 11 shows that the values of the moments of iner- tions, we know that the accuracy in the determination of tia are crucial for our computations.The comparison with the precession rate Ψ˙ and of the coefficients of nutation the final results computed by Habbibulin (1995) is not pos- depends directly on the quality of determination of the − − sible because in his paper different angles, parametrisation dynamical ellipticity H = 2C A B and of the triaxiality V 2C and more complex developments were used. In another T = A−B . Therefore for this purpose it is crucial to ob- V 4C study, Zhang (1988) found a precession rate of 7828”.6/cy tain accurate values of the moments of inertia. The value C corresponding to a period of 16555 years. Altough no pre- of ratio 2 is inperfectly known. Kozlovskaya (1966) MR cise information could be extracted from this paper, the obtained a value in the [0.321;0.360] interval, Shen and difference is probably due to different values of the mo- Zhang (1988) in the [0.321;0.350] one and Yoder(1995) in ments of inertia. What we highlight here is that different the [0.331;0.341] one. Williams (private communication) values of moment of inertia yield fairly different results for gives the following values relative to the differences of the the precession and the nutation as shown in Table 11. moments of inertia:

C − A C − B =5.519 × 10−6, =3.290 × 10−6, 11. Conclusion and Prospects MR2 MR2 In this paper we investigated by the rotation of Venus. B − A − =2.228 × 10 6 (105) MR2 We adopted a well suited theory of the rotation of a rigid 16 L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model body set up by Kinoshita (1977) and developed in full de- we obtain: tail by Souchay et al.(1999) for the Earth. In a first step a da b c c b A db we defined the reference frames to the various polar axes − sin · = (− sin cos + sin cos cos ) · of the planet (axis of angular momentum, rotation axis, +(− cos b sin c + sin b cos c cos A) · dc figure axis) and on its moving orbital plane. We have also +(− sin b sin c sin A) · dA. given in full detail the parametrization of Venus rotation (108) starting from the set of Andoyer canonical variables with respect to the moving orbital plane. In particular we have Dividing by - sin a we have the following equation: determined precisely the motion of the orbit of Venus at sin b cos c − sin c cos b cos A t with respect to the orbit at J2000.0, through the pa- da = · db sin a rameters Π1 and π1, for which we have given polynomial expressions. Moreover we have checked the value of the cos b sin c − sin b cos c cos A + · dc Venus obliquity (263). sin a Then we calculated the disturbing function due to sin b sin c sin A + · dA. the gravitational interaction with the Sun. Applying sin a Kinoshita’s Hamiltonian analytical developments, we cal- (109) culated the precession constant of Venus, with a precision and an accuracy better than in previous works (Habillulin, In the spherical triangle A,B,C the following equations 1995, Zhang, 1988). Our value for the precession in longi- holds : ˙ tude is ψ = 4474”.35t/cy ± 66.5. which is more than two c a b c A sin sin sin sin sin b C times larger than the corresponding term for the Earth C = A ⇒ a = sin sin . due to the Sun (ψ˙ = 1583”.99/cy) and slightly smaller sin sin sin than the combined effect of the Moon and of the Sun for Thus : ˙ the Earth (ψ = 5000.3”/cy). We have shown that the ef- a C c b b c A fect of the very small value of the dynamical ellipticity of sin cos = cos sin − sin cos cos − sin b cos c − sin c cos b cos A Venus (H = 1.31 × 10 5), which should directly lower C V ⇒ a = cos the amplitude of its precession and nutation, is more than sin fully compensated for by its very slow retrograde rotation. Moreover, one of the specificities of Venus is that − sin a cos B = cos b sin c − sin c cos c cos A its triaxiality (T = −1.66 × 10 6.) is of the same or- V sin c cos b − sin b cos c cos A der as its dynamical ellipticity (see value above), unlike B ⇒ a = cos what happens for the Earth for which it is considerably sin smaller. Consequently, we have performed a full calcula- Replacing in (109), we obtain (9). tion of the coefficients of nutation of Venus due to the Sun and presented the complete tables of nutation in lon- References gitude (∆Ψ) and obliquity (∆ǫ) due both to the dynamical ellipticity and the triaxiality.These tables have never been Andoyer H. 1923, Paris, Gauthier-Villars et cie, 1923-26. presented in previous works. We think that this work can Bretagnon P., Rocher P., & Simon J. L. 1997, A&A, 319, 305 , be a starting point for further studies dealing with Venus Capitaine N., Souchay J., & Guinot B. 1986, Celestial rotation, for which it has set up the theoretical founda- Mechanics, 39, 283 tion (parametrization, equations of motion etc...), such as Carpenter R. L. 1964, AJ, 69, 2 a study of precession-nutation over a long time scale, the Correia A. C. M., & Laskar J. 2001, Nature, 411, 767 Correia A. C. M., Laskar J., & de Surgy O. N. 2003, Icarus, calculation of Oppolzer terms, the effects of the atmo- 163, 1 sphere. Correia A. C. M., & Laskar J. 2003, Icarus, 163, 24 Dobrovolskis A. R. 1980, Icarus, 41, 18 Goldreich P., & Peale S. J. 1970, AJ, 75, 273 Goldstein R. M. 1964, AJ, 69, 12 12. Appendix Guinot B. 1979, Time and the Earth’s Rotation, 82, 7 Habibullin S. T. 1995, Earth Moon and Planets, 71, 43 Demonstration of equation 9 : Hori G. 1966, PASJ, 18, 287 Kinoshita H. 1977, Celestial Mechanics, 15, 277 Kozlovskaya, S. V. 1966, AZh, 43, 1081 da = cos c · db + cos B · dc + sin b sin C · dA. Lago B., & Cazenave A. 1979, Moon and Planets, 21, 127 McCue J., & Dormand J. R. 1993, Earth Moon and Planets, 63, 209 Derivating the following classical relation : Simon J. L., Bretagnon P., Chapront J., Chapront-Touze M., Francou G., & Laskar J. 1994, A&A, 282, 663 Souchay J., Loysel B., Kinoshita H., & Folgueira M. 1999, cos a = cos b cos c + sin b sin c cos A A&AS, 135, 111 L. Cottereau and J. Souchay: Rotation of rigid Venus : a complete precession-nutation model 17

Yoder C. F. 1995, Icarus, 117, 250 Shen, M., & Zhang, C. Z. 1988, Earth Moon and Planets, 41, 289 Williams, B. G., Mottinger, N. A., 1983, Icarus, 56, 578 Zhang C.-Z. 1988, Chinese Astronomy and Astrophysics, 12, 38