SPIN AXIS VARIATIONS OF MARS: NUMERICAL LIMITATIONS AND MODEL DEPENDENCES

Transcription

1 The Astronomical Journal, 35:5 6, 8 April c 8. The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi:.88/4-656/35/4/5 SPIN AXIS VARIATIONS OF MARS: NUMERICAL LIMITATIONS AND MODEL DEPENDENCES S. Edvardsson andk.g.karlsson Department of Natural Sciences, Mid Sweden University, S-85 7 Sundsvall, Sweden Department of Technology, Physics and Mathematics, Mid Sweden University, S Harnosand, Sweden Received 7 February ; accepted 7 December 9; published 8 March 4 ABSTRACT Celestial mechanical integrations for the whole solar system are carried out over four billion years before and after the present. The orbital solution of the solar system is stable during this whole time period. An instantaneous spin axis model including triaxiality is developed. In particular, spin axis precession, the Martian obliquity, solar torques, and the orbital eccentricity/inclination of Mars are studied. Model dependences from general relativity, solar oblateness, the Martian moons, solar mass loss, etc. are tested. Although the obliquity of Mars sometimes shows dramatic variations, some structures are robust and seem insensitive to the initial spin precession rate. A full integration is carried out during a total of 8 Myr resulting in a Martian obliquity that is restricted between and 6. The problems with numerical noise due to limited machine precision, integration step sizes, and the chaotic sensitivity of the solutions are studied and discussed. The limited machine precision (5 6 figures) alone is found to limit the duration of exact obliquity solutions to about 8 Myr. Key words: celestial mechanics methods: N-body simulations methods: numerical planets and satellites: general solar system: general. INTRODUCTION During the last decade the number of spin axis investigations of Mars was rather sparse. Some examples are those by Touma & Wisdom (993), Bouquillon & Souchay (999), and Laskar et al. (, 4). Accurate determinations of, e.g., the spin precession rate of Mars were first achieved in the late 99s through the Mars Pathfinder mission (Folkner et al. 997) and later by the Mars global surveyor space mission (Yoder et al. 3). In a previous paper by Edvardsson et al. (), models and numerical methods were developed and a computer code was written in C. This work will be referred to as Paper I. The current paper is a continuation and presents possible evolutions of the Martian spin axis over periods of up to 8 Myr. Although substantial efforts are made to achieve maintained numerical stability, we will find that escalating numerical uncertainties limit exact obliquity calculations to 8 Myr. On the other hand, it is relatively straightforward to perform short calculations ( Myr). The long-term evolution presents an incredible challenge due to the huge CPU time, severe convergence difficulties due to numerical truncation errors, and limited machine precision. For example, the particular choice of the integration step size in the numerical model is closely connected to chaotic behavior. Numerical errors eventually grow so that the numerical solution curve diverges from the exact mathematical solution. Thus due to the uniqueness of the mathematical solution given its initial conditions, the long-term numerical solution curve will eventually correspond to a different set of initial parameters (e.g. positions and velocities). In fact, the chaotic sensitivity to the initial parameters such as the moments of inertia (or equivalently the initial mean spin precession rate) presents severe difficulties, as also noted by e.g. Laskar et al. (4), who made a substantial effort to approximate possible long-term evolutions (5 Myr) for the Martian spin axis. It was noted that the chaotic behavior puts a limit on the duration of trustworthy simulations. In the present work, we present independent calculations that are in excellent agreement with the short-term findings ( 5 Myr) of previous authors. We also study and emphasize several aspects not previously focused upon. Besides the influence of different initial parameters, we also study the effects of various model dependences, numerical truncation errors, and limited machine precision. It will be evident that all these effects are significant in the long term. The numerical analysis is based on the non-relativistic N- body library SWIFT by Levison & Duncan (). We have substantially modified this symplectic integrator: due to the very long simulations performed here, general relativity (GR) must be taken into account (see the appendix of Paper I). Solar oblateness is included as is the effect of solar mass loss. The influence on the Martian obliquity from the Earth Moon system is accounted for through the quadrupole-corrected acceleration described by Quinn et al. (99). We develop a new self-consistent method to deal with instantaneous precession rates of the spin axis, also including the triaxiality of Mars. We have also included a treatment to properly take the Martian moons into account in the spin axis calculation. More recent input parameters are applied. We will see that there are other important aspects than the chaotic sensitivity (due to uncertainties in the initial parameters) that are interesting to consider. Even in the future when the initial parameters (e.g. spin precession rate) are much more precisely known, the astrophysicist will still face problems with integration step sizes, limited machine precision, and model dependences, i.e., the question of which effect should be included and which can be neglected in a given model for the very long timescale. Solution curves within this work are available on request to sverker.edvardsson@miun.se.. NUMERICAL DATA AND TESTS OF MODELS The general approach to integrate the solar system and the Martian spin axis is given in Paper I. However, several modifications and improvements are made in the present work as indicated in the following... Initial Conditions and Physical Parameters The present work applies improved initial conditions for the numerical model compared with Paper I which applied slightly 5

2 5 EDVARDSSON & KARLSSON Vol. 35 different/older values. Although the number of figures is sometimes greater than the precision of the observational data, these are the initial values applied in our numerical integrations. The Martian moments of inertia applied are J z /MR =.366, J x /J z =.9949, J y /J z = (Bouquillon & Souchay, 999). The orientation of the spin axis was updated. Given right ascension α = and declination δ = of the Mars spin axis obtained from Horizon (JPL) 3 at the initial time February 6 :, and the Martian rotation rate ψ = ω = day (Bouquillon & Souchay, 999), we obtain the components in rad day : ω x () = () ω y () = () ω z () = (3) The angle between the instantaneous orbital plane and the spin axis (obliquity) is at the initial time. Further, the orientation of Mars at the initial time is given by the Euler angles in degrees (see Paper I): θ = φ = ψ = Knowledge of initial instantaneous spin rates θ and φ are not needed (see the updated treatment in Section 3). The remaining parameters such as planet masses, initial positions, and velocities are the same as in Paper I (DE46)... General Relativity The present work includes GR in the same way as in Paper I (see Appendix). The reason for this inclusion is that GR significantly influences the obliquity for integrations exceeding a few Myrs (Section 4). The relativistic force correction in a heliocentric system S for planet j is given by ( ) F = GM sm j rj 3 r j (β ppn + γ ppn ) GM s vj γ c ppn r j c +(+γ ppn ) GM sm j c rj 3 v j (r j v j ), (4) where pure GR corresponds to the PPN parameters β ppn = γ ppn =..3. Solar Mass Loss Including both electromagnetic radiation and particle radiation, the relative solar mass loss is (dm s /dt)/m s = yr (Noerdlinger ). This effect is expected to be small. Figure shows that this is indeed the case over Myr..4. Solar Oblateness For a non-spherical Sun and a planet j, the effective solar potential energy is expressed as [ W(r j,θ j ) = GM Sm j ( ) ] RS J n P n (cos θ j r j r ). (5) j n even 3 JPL database, in Obliquity difference (degrees) 3 x 3 3 Figure. The effect of solar mass loss. The difference between the obliquity with mass loss and without. This effect is insignificant for integrations running several tens of Myr. The force corrections acting on planet j due to the quadrupole term (n = ) become F xj = 3Gm j (J rj 5 z J x ) F yj = 3Gm j (J rj 5 z J x ) F zj = 3Gm j (J rj 5 z J x ) ( ( ( 5 z j rj 5 z j rj 5 z j rj ) ) x j y j ) 3 z j. It is assumed that J x = J y for the Sun and the usual normalized solar quadrupole moment is given by J = (J z J x )/M s R S. The primed coordinates are given in the solar principal axis system. The orientation of the z -axis (solar spin axis) is given by α = 86.3 and δ = on the celestial sphere. We have applied the value J =. 7 (DE46). The oblate Sun causes the orbit of Mercury to precess somewhat. For testing purposes, we consider the isolated two-particle problem and apply the current value for J. We then obtain the precession.59 arcsec cyr as compared with the purely relativistic effect of 6πGM s /[a( e )c ] = arcsec cyr.the combined effect becomes 43.7 arcsec cyr, in excellent agreement with the findings of Pireaux & Rozelot (3). We also made a calculation of the whole solar system, thus including the N-body effects, for 5 complete Mercury periods starting from MJD 56.5, both forwards and backwards in time. First, a purely classic N-body calculation and then an N-body calculation with GR and solar oblateness included were performed. The difference between the solutions in orbital precession then amounts to 4.9 arcsec cyr. This is the isolated GR/oblateness mean value during this period. Investigations by Laskar et al. (4) showed that the solar oblateness effect on the Martian orbital eccentricity becomes noticeable after about 3 Myr. In the case of the obliquity, it starts to be noticeable after Myr (see Figure ) and thus needs to be included for long integrations. (6)

3 No. 4, 8 SPIN AXIS VARIATIONS OF MARS Phobos. JPL (shifted. upwards) and present work JPL Obliquity difference (degrees). Inclination (degrees) Deimos. JPL (shifted. upwards) and present work JPL.5 Figure. Effect of solar oblateness: the difference between the obliquity with oblateness and without..5. Torques from the Earth Moon System In Paper I, the Moon was treated as a separate body. However, for very long simulations (tens or hundreds of Myr), it is necessary for computational speed to avoid an explicit integration of the Moon (which would require a very small integration step size). Instead, we treat the Earth Moon system as a point mass in the barycenter with a correction due to the quadrupole acceleration in the same way as Quinn et al. (99); see their Equation (). We have tested this approximation by comparing it with a solution where the Earth and the Moon were integrated separately. After a Myr integration, the error is still less than. in the obliquity of Mars. Without the quadrupole correction, however, the obliquity error after Myr would be considerable (several degrees)..6. Torques from Phobos and Deimos The oblateness interaction between a planet and its moon was omitted in Paper I since rather short simulations were performed there. Although such an approximation is sufficiently good in the case of the Earth and the Moon (only about. difference in the obliquity after Myr), it is not so for the Martian moons and Mars. Indeed, for the oblate Mars the very close moons remain near the equatorial plane thus giving only a minor influence to the Martian obliquity. The force correction due to planet oblateness acting on moon j is here given in the Martian body system S (Paper I): F xj = 3Gm j (5I 3J rj 5 x J y J z )x j F yj = 3Gm j (5I 3J rj 5 y J x J z )y j F zj = 3Gm j (5I 3J rj 5 z J y J x )z j, where I = ( J x x j + J yy j + J )/ zz j r j is the moment of inertia about the direction of r j and m j is the mass of a moon. These forces are then transformed to the external reference system S (J.). An equal but opposite correction also acts on the planet. In Figure 3 we plot the orbital inclinations of Phobos and Deimos w.r.t. the Martian equator. We have also plotted (7) Time from present (yr) Figure 3. Inclinations of the Martian moons. Comparison between the present work and Horizon ephemerides from JPL; see TheJPL moon data (MAR63.DE45) are based on the work of Jacobson et al. (989). Only tiny model differences can be seen for Phobos. To clearly display the fine features, the Phobos JPL inclination is moved upwards by. and for Deimos,.. In the case of Deimos, the solution curves are essentially identical. the ephemeris inclinations from Horizon (JPL). The JPL moon data (MAR63.DE45) are based on the work of Jacobson et al. (989). We have shifted the curves slightly to facilitate comparison of the fine features (see Figure 3). In the case of Phobos, some tiny deviations are present possibly due to small differences in the initial parameters such as orientation of the spin axis and the moments of inertia. In the case of Deimos, no differences can be seen. We thus conclude that the calculations are sound. We have also calculated the evolution of the moon inclinations for, yr and both moons stay near the Martian equator during the whole time period. Although the effect on the obliquity from a moon near the equatorial plane is expected to be small, Bouquillon & Souchay (999) included Phobos and Deimos. Also Hilton (99)took the Martian moons into account. In order to see if the moons could be significant for the obliquity evolution at a long timescale, we decided to include the mean torques from the moons acting on Mars in the same way as described in Section. of Paper I. The mean torques are taken over full periods (Phobos) and full period (Deimos), corresponding to about 5 and 54 yr, respectively. In Figure 4, we display the torque from Phobos. Laskar (4) used a perturbative argument to show that the torques from the Martian moons are null to first order. The present work finds that the all order mean torque M = M x + M y + M z is.78 6 Nm for Phobos (Figure 4) and.54 6 Nm for Deimos. Compared with the solar torque of Nm, this is small but perhaps not negligible. In Figure 5 we display the long-term effect of including the moons. We see that the influence on the obliquity over a Myr period is as expected quite small, but for even longer integrations, the effect becomes increasingly noticeable. For example, at about 5 Myr before present the influence becomes of the order of degrees. It is therefore appropriate to include the Martian moons for long simulations..7. The Mean Torque from the Sun The time-averaged solar torque acting on Mars during a Martian year T can be expressed as (note the typographical error

4 54 EDVARDSSON & KARLSSON Vol. 35 M x..8.6 Torques from Phobos ( 8 Nm) M y M z 4 Obliquity difference (degrees) Time before present (yr) Figure 4. Instantaneous torques acting on Mars from Phobos in the x-, y- and z-directions (J.). in Equation (3) of Paper I where the denominator 4 should have been ) M G T = 3GM ( s J r 3 z J ) x + J y sin θ. (8) T Here, θ is the obliquity and r is given by the equation of the orbit (e.g. Danby, 964): r = a e +ecos v. a With the period T = π 3 µ, µ = G(M s + m) andr dv = dt a( e )µ, wefind = T r 3 T T r dt = 3 a 3 ( e ). 3/ This result and Equation (8) will be applied in Figure SELF-CONSISTENT INSTANTANEOUS SPIN PRECESSION In contrast to the usual mean approach, the spin axis calculation performed here is instantaneous. The present spin axis calculation thus deals with the spin evolution at each time step and works on all timescales. Also, the triaxiality of Mars is fully taken into account. The spin axis rates may be derived as follows. We start with Equation (5) of Paper I: dω dt = A [M G B( φ, θ)ω] = C( φ, θ), (9) where φ is the spin precession rate, θ is the change of the angle between the Martian equator and the J. ecliptic plane and the planet s rotational rate is given by ψ = ω =ω. The spin vector is a function of the Euler angles, ω = ω(φ, θ, ψ), so dω dt = dω dφ φ + dω dθ θ + dω dψ ω. () Figure 5. Effect of the inclusion of Phobos and Deimos: the difference between the obliquity with the moons and without. Explicitly, since the spin vector ω = ωe ω = ω(sin θ sin φ, sin θ cos φ,cos θ), we obtain dω/dφ = ω sin θ(cos φ,sin φ,) dω/dθ = ω(cos θ sin φ, cos θ cos φ, sin θ) dω/dψ = (M Gz /J zω)e ω. The last expression is valid since dω dψ = dω dψ e ω = dω dt dt dψ e ω = (M Gz /J zω)e ω. The torque M Gz w.r.t. the spin axis is related to the moments of inertia J x J y (see Equation () of Paper I) so if J x = J y this term vanishes. However, in the case of Mars J x J y.in order to start an integration, we need access to several initial conditions of which almost all are readily accessible from JPL. The exceptions are the instantaneous rates φ and θ.however,in the following we see that these are self-contained in the above equations. By equating Equations (9) and(), the following system of equations is obtained: θ = (M Gz /J z)cosθ C z ( φ, θ) () ω sin θ and [ ] ω sin θ cos φ ωcos θ sin φ ω sin θ sin φ ω cos θ cos φ][ φ θ [ Cx ( φ, θ) (M = Gz /J ] z)sinθ sin φ C y ( φ, θ)+(m Gz /J () z)sinθ cos φ This system is solved iteratively at each time step in the simulation. We must find the pair ( φ, θ) so that Equations ()and() are simultaneously fulfilled. At the start of the iterations, we set ( φ, θ) = (, ), then solve Equation (), then insert (, θ)into Equation () thus obtaining a new pair ( φ, θ) which then is resubmitted into Equation () and then Equation (). The iteration process continues until self-consistency is reached (convergence within machine precision). The initial guess ( φ, θ) = (, ) is not needed later in the simulation. Instead the pair from the previous time step, ( φ(t t), θ(t t)), is used to start the iterations resulting in improved convergence speed.

5 No. 4, 8 SPIN AXIS VARIATIONS OF MARS 55 Spin precession rate (arcsec/yr) Time (yr) Figure 6. Instantaneous Mars spin precession rate φ. A period of Martian year is clearly seen. Variations in amplitude are due to the substantial eccentricity of the Mars orbit. The necessary number of cycles is usually less than and the iterative process takes negligible time. We thus conclude that this approach yields precise instantaneous values of the pair ( φ(t), θ(t)) at each time step of the simulation. This is a nice improvement to Paper I. As an example, we plot the instantaneous spin precession rate over 6 Martian years in Figure 6. This result also included an explicit integration of both the Martian moons which was carried out with the extreme time step t = days. It is evident that the curve is numerically stable including a very nice fine structure (compare with Figure 8 in Paper I). For this period, we find that the mean φ =7.59 arcsec yr which is near the recent value ±.5 arcsec yr of Yoder et al. (3). The mean precession rate could also have been calculated from (see e.g. Folkner et al. 997) φ =α cos ε = 3 J cos ε( e ) 3/ MR n /ωj z. (3) With our data at the initial time, we again obtain 7.59 arcsec yr. We conclude that our instantaneous spin axis model is sound. 4. RESULTS 4.. Obliquity Evolution and Mean Precession Rates Figure 7 shows the obliquity for both the relativistic and classic cases. Also the nominal solution of Laskar et al. (4) is plotted. A comparison with the solution of Laskar et al. shows good agreement during most of the time period. The exception occurs from about 8 Myr into the future and onwards. It is plausible that this difference is related to the chosen mean spin precession rate at the initial epoch. In our instantaneous model, it corresponds to studying the uncertainties in the moment of inertia (J ). The mean spin precession rate in Equation (3) can then be expressed as φ = 3 cos ε( e ) 3 (n /ω) J z k(j x + J y )/, J z where k = corresponds to our chosen parameters, leading to φ =7.59 arcsec yr. The most recent value of Yoder LA4 5 GR 5 Classic 5 Figure 7. Obliquity solutions. Top: Laskar (4); see Equipes/ASD/insola/mars/La3-4/index.html. Middle: The present work with GR effects included. Bottom: Same as middle, but without relativistic effects. et al. (3) is ±.5 arcsec yr. The maximum limits 7.6 and 7.57 then correspond to k = and k =.34, respectively. We also give the result for the nominal value arcsec yr and arcsec yr of Folkner et al. (997). The obliquity evolutions for the various cases are displayed in Figure 8. The lowest subplot is very similar to the LA4 4 result in Figure 7. It is thus plausible that the LA4 solution is based on the Folkner et al. value φ =7.576 arcsec yr.itisalsointeresting to note in Figure 8 that some structures are robust ( to +7 Myr), while others (e.g. +7 to + Myr) are strongly dependent on these initial parameters. Further, our GR solution in Figure 7 shows quite large changes in the mean level of the obliquity. We confirm the findings of Touma & Wisdom (993) that there was a decrease in obliquity that started 5 Myr ago. Contrary to the findings of Laskar et al. (4), we find that, with the most recent physical parameters, the Martian obliquity will continue to increase from +7 to + Myr into the future. On the other hand, the classic solution (Figure 7) shows a simple quasi-periodic behavior at an essentially constant mean level. Although not visualized, this behavior was already noted by Touma & Wisdom (993). It is also interesting to note for how long a classic calculation can be considered valid (± Myr). The sudden level changes of the GR curve seen in Figure 7 are not at all unique. As will be evident in Section 4.6, the very long-term evolution (Gyr) shows repeated changes of the obliquity mean levels occurring at all timescales. 4.. The Obliquity and Solar Torque The relativistic and classic obliquity solutions are very different (Figure 7). We feel that it is worthwhile here to briefly give some insights into the reasons for this peculiar behavior. By applying Equation (8), we start by plotting the torque from the Sun for both the relativistic and classic cases; see Figure 9. It is seen that the solar torque is intimately related to the corresponding obliquities (see Figure 7). The torque from the Sun appears to be the driving force for the evolution of the 4 LA4, 4, in

6 56 EDVARDSSON & KARLSSON Vol present work:7.59 arcsec/yr Yoder:7.597 arcsec/yr upper limit:7.6 arcsec/yr lower limit:7.57 arcsec/yr Folkner:7.576 arcsec/yr Figure 8. The effect of various initial mean spin precession rates and obliquity. Torque from Sun ( Nm) GR Classic 5 5 Figure 9. Comparison of the relativistic and non-relativistic solutions of the acting torque on Mars due to the Sun alone. Martian obliquity. However, the Sun alone cannot explain the obliquity evolution. We feel that it would be beneficial to some readers to recount the basic science here. We therefore briefly study the effect of including various planets and their influence on the Martian obliquity. Figure (a) shows the relative importance of the most important planets for the obliquity. First, the simple two-body solution of Mars and the Sun is given by the horizontal curve at the top of Figure (a). In the two-body problem, the Martian orbit is essentially conserved (the GRinduced orbital precession is very tiny) and so is the mean torque and therefore the mean obliquity, see Equation (8). Thus, the apparent importance of the torque from the Sun and its influence on the Martian obliquity needs further consideration. The three curves following below give the results when Saturn, the Earth Moon system and Jupiter, respectively, are subsequently added (together with the Sun). Finally, the bottom curve is the solution including the whole solar system. Obviously, the evolution of the obliquity is entirely governed by the N-body effects in the solar system. This is explained by the following. (a) The planets affect the Martian orbit which in turn changes the solar torque acting on Mars and (b) the torques from the other planets change Fine structure Time from present (yr) Figure. (a) The yr obliquity evolution for five different cases. Top curve: The two-body solution of Mars and the Sun; next curve: inclusion of Saturn; next curve: the Earth Moon system is added; next curve: Jupiter is added; lowest curve: The full N-body solution. (b) The yr fine structure of the obliquity for the full N-body solution. The period is one-half Martian year. the orientation of Mars which in turn gives a change in the torque from the Sun. Thus, although the solar torque gives by far the most important contribution, it is the N-body effects that are decisive for the obliquity evolution. The above considerations explain the differences between the relativistic and classic cases (Figures 7 and 9), since they result in quite different orbital solutions. Finally, we also display the fine structure of the full N-body solution in Figure (b) for yr. The period seen is one-half Martian year Solar Oblateness and the Martian Moons In Figure, we show the obliquity calculated excluding and including solar oblateness. The solutions are similar up to about Myr before the present. However, near 5 Myr the solar oblateness curve shows a large dip. It is reasonable to wonder what might have been causing this peculiar behavior. One could expect that the Martian orbit, and thus the solar torque, was significantly altered 5 Myr ago but this is not the case. An investigation of the orbital elements does not show any substantial difference between the two orbital solutions. The plausible explanation is instead that the orientation of Mars was tilted so that its equator came closer to its orbital plane (probably due to Jupiter). This then resulted in a reduced solar torque which evidently has such a large impact on the obliquity. In Figure, we display the solutions considering the account of the Martian moons. Their impact on the obliquity is seen to be small. However, before 4 Myr, differences in the solution curves become evident The Long-Term Evolution and Numerical Problems It is truly a numerical challenge to perform simulations beyond several tens or even hundreds of Myr. Checking numerical convergence with respect to the integration step is extremely time consuming, let alone that it might even be impossible to achieve convergence due to the event of severe chaotic behavior. The Lyapunov exponent is a frequently used measure in chaos theory to estimate the divergence of solution curves. However, the real Lyapunov exponents are difficult to determine. For example, there are widely different values reported in the literature, see e.g. Hayes (7). Apart from variations in meth- (a) (b)

7 No. 4, 8 SPIN AXIS VARIATIONS OF MARS 57 5 without Solar oblateness Convergence tests with Solar oblateness Various step sizes Figure. The effect of Solar oblateness: the obliquity evolution with oblateness and without without Martian moons with Martian moons Figure. The effect of Phobos and Deimos: the obliquity with the moons and without. ods, convergence problems (integration step sizes, see Newman et al. ), and numerical noise, further serious problems were recently pointed out by Hayes (7). He demonstrated that the Lyapunov times are very sensitive to the initial outer planetary positions. A whole range of Lyapunov times were possible within the observational error. The conclusion is that much more precise positional observations of the outer planets are required in order to determine trustworthy Lyapunov exponents. Therefore, we choose to not provide any new Lyapunov exponents in the present work. We wish to emphasize that it is too simple to only consider chaotic behavior and the sensitivity to the initial conditions. In practice, we are often forced to use numerical analysis for physical problems where the analytical solutions are not available. For such cases, the numerical approach is limited by integration step size and/or machine precision. Eventually, the numerical solution curve will deviate significantly from the exact mathematical solution curve. This means that, from then on, the simulated solution corresponds to a different. Figure 3. Convergence tests for several integration steps (see left; units in yr). The black curves indicate how far the solution is converged. The gray curves are unphysical. Further convergence beyond about 8 Myr is not possible due to the limited double precision of 5 6 figures. set of initial conditions. Due to the chaotic sensitivity, the numerical solution might subsequently diverge severely. Even when the integration step size is decreased considerably, we still face the problem with limited machine precision. These errors develop in various ways so that convergence in the numerical solution curve may not be achieved at all for a long run. The discrete numerical method in itself always introduces truncation errors that become spread throughout the solution curve. It is therefore of interest to estimate for what length of period convergence may be achieved given the limitations of current computer technology (CPU speed and machine precision). A different aspect was studied by Yao (5) who discussed the application of using discrete numerical integration for a chaotic system and its extreme sensitivity to the integration step. He found that in contrast to systems having oscillatory error behavior, the exponential growth of errors present in a chaotic system leads to the fact that there exists no solution which is independent of the integration time step. The question is only for how long it is practical to decrease the integration step and possibly increase the machine precision (e.g. into quadruple or even octuple) before serious divergence will occur. Our goal is thus to simply test (with our choice of initial parameters) for how long convergence in the obliquity can be achieved. Since in our approach the solar system and the spin axis are integrated simultaneously for the same integration time step, it is straightforward to test a series of declining integration steps. Typical CPU time on a Sun Ultra SPARC 9 Mhz is day for a Myr simulation (at t =.5 yr). In Figure 3 we plot the results of this numerical experiment. The black curves indicate how far converged patterns are obtained given a certain integration step (using the simple criterion that no differences can be detected by the eye), while the gray curves are divergent (numerical noise). The application of even smaller time steps (e.g..5) is meaningless. It was found that the truncation errors then became of the same order as the machine precision errors, leading to results that can no longer be improved. The convergence cannot be taken further due to

8 58 EDVARDSSON & KARLSSON Vol. 35 Eccentricity Eccentricity LA4 5 GR 5 Classic 5 Figure 4. The eccentricity: LA4 and the present work. Figure 5. Long-term eccentricity; compare with Figure 5 of Laskar et al. (4). this limitation and thus the maximum trustworthy time period of about 8 Myr is established. Unfortunately, switching to quadruple precision (33 34 significant figures) would result in simulations that would be far too time consuming and thus impractical Long-Term Orbital Properties In Figure 4 we show the eccentricity of Mars. The top curve is the LA4 solution and the middle curve is from the present work. Clearly, there are no significant differences during this period. Also, our GR inclination (not plotted) is very similar to the findings of Laskar et al. (4). The lowest curve is the classic result and in this case there are, as expected, substantial differences already a few Myr from the present. Figure 5 displays the eccentricity for a longer period. This curve should be compared with Figure 5 of Laskar et al. (4). The agreement is again very nice until about 5 Myr where the bottom of their curve is higher. The same is seen at 3 Myr. Earlier than 3 Myr, the solutions become quite different. Although differences develop in the long run, our orbital solution is quite similar to the LA4 solution during a fewtensofmyr. The deviation d 3.5 x Venus Mars Earth.5.5 Figure 6. The deviations for Venus, Mars, and Earth; t =.,.5 yr. (Mercury is included in all the calculations but not plotted.) The deviation d Saturn Uranus Neptune Pluto Jupiter.5.5 Figure 7. The deviations for the outer planets. The lowest curve is Jupiter and the curve directly above is Pluto and so on as indicated in the figure; t =.,.5 yr. In the very long term (hundreds of Myr), it is not easy to calculate trustworthy orbital elements. In fact, due to the chaotic behavior we expect that precise planet positions can be very tricky to obtain (see, e.g., Lecar et al. ; Hayes, 7). There are of course several reasons for this complication. Aspects that are involved include (a) completeness of the model, (b) accuracy of the initial conditions, (c) numerical truncation errors, and (d) limited machine precision. For long periods, we will see that the limited machine precision/numerical truncation errors alone can be major obstacles. We certainly do not have access to the exact dynamical solution of the solar system, but we can give some indication of the problem by performing two integrations using very tiny integration steps (. and.5 yr). As mentioned above, these tiny integration steps will result in truncation errors of the same order as the machine precision ( 5 ). Even so, we see in Figures 6 and 7 that the machine/numerical errors grow rapidly, sometimes even exponentially although exactly the same initial parameters have been used. These positional

9 No. 4, 8 SPIN AXIS VARIATIONS OF MARS Figure 9. A possible obliquity evolution for Mars during ±4 Myr; t =.5 yr.. eccentricity. 5 5 inclination in degrees 5 Figure 8. The orbits of the four inner planets during ±4 Myr plotted once every 3, yr; t =.5 yr. errors (deviations) are calculated according to the formula d(t) = ( ( r(t; t) r t; t )), a r=x,y,z at each integration step where a is the semi-major axis of the considered planet. Obviously when d, the planet position is completely lost, although substantial discrepancies develop much earlier. It is particularly interesting to note that Saturn is difficult to keep in place (Figure 7). By assuming an exponential continuation, we find that this planet position is already lost after 4.5 Myr (d ). However, the orbit of Saturn is stable as we shall see below (Section 4.6) sothe deviation is mainly angular in character. On the other hand, the inner planet positions may be kept for a few tens of Myr (Figure 6). Although a somewhat different approach, it is interesting to note that the Lyapunov times w.r.t. changing an initial planet position with 5 4 AU are of the same order (Sussman & Wisdom 99). The uncertainty in Saturn s position is not crucially important for the Martian obliquity (Figure (a)). We also see in Figure 3 that reliable results are possible for much longer periods of time. Nevertheless, we conclude that the planetary angular positions will eventually become very uncertain and consequently the same is true for the evolution of orbital elements and predictions of obliquites. It would seem that changing to quadruple precision could be helpful, but the computational speed is currently insufficient for an integration of all the planets during several hundreds of Myr A Numerical Experiment: 8 Myr Simulation Although it is clear that precise solutions are not feasible with the current computer technology, it is still of interest to study a possible long-term behavior of the solar system. For example, in the literature it has sometimes been speculated that Mercury might be ejected from the solar system in a few Gyr from the present (see e.g. discussions in Lecar et al. ). In Figure 8, we therefore plot the orbits of the inner planets during ±4 Myr. Even for such a long period, there are no signs of orbital instabilities or ejections, although the exact planet positions are Figure. A possible evolution of the eccentricity/inclination for Mars during ± Myr; t =.5 yr. (due to a disk crash the ±4 Myr ( t =.5) orbital solution was lost). certainly lost as previously described. The use of the rather large integration step (.5 yr) is necessary for computational speed. It should be noted that even a much smaller step size would still only give indicative results. The apparent circular shapes are mainly due to the precession of the orbits. Although not plotted, the same stable behavior is also seen for the outer planets. In fact, this result is in agreement with Lecar et al. (, p. 48). They used the result of Ito & Tanikawa () to estimate that the Mercury orbit might be stable for as long as 3 4 Gyr. For curiosity, we also display in Figure 9 the corresponding obliquity evolution ( t =.5). This is the longest simulation so far in the literature (±4 Myr). The general behavior of the mean level is semi-stable with a typical obliquity of around 39. However, we also see very sudden transitions, e.g. the transition at 8 Myr. It is interesting that over this entire time interval the obliquity is restricted between and 6. Touma & Wisdom (993) found 49 during their 8 Myr interval. In Figure, we also display the eccentricity and inclination ( t =.5) during ± Myr. Again, this is the longest full integration we know of. Our solution indicates that in the past Myr, the e-value varied between and.4. For the future, e ranges from to.45. The last zero appeared.3 Myr ago (Figure 5) and in the future it will happen again in about 3.6 Myr. The mean value for the whole period is.6 as compared with today s value of.934. As for the inclination, it varied between and 9.3 in the past and and 8.9 in the future (mean i is 5.4 compared with the present value of.85 ). 5. CONCLUDING REMARKS We have investigated both spin axis and orbital properties, primarily for Mars. We have confirmed the findings of Touma & Wisdom (993) that there was a decrease in obliquity that started 5 Myr ago. Contrary to the findings of Laskar et al. (4), we find, with recent physical parameters, that the Martian obliquity will increase from +7 to + Myr into the future (see Figures 7 and 8 and Section 4.). We have performed the longest simulation of the Martian obliquity (±4 Myr), although it

10 6 EDVARDSSON & KARLSSON Vol. 35 only represents an indicative solution. In this case, the obliquity is confined to 6. We also presented the eccentricity and inclination for ± Myr. Although formally chaotic, the solar system during this whole period (8 Myr) is practically stable. However, in about Myr the planetary positions become very uncertain (Figures 6 and 7). The effect of general relativity, solar oblateness, the Martian moons, solar mass loss, etc., and their quantitative importance have been studied. We have also discussed several uncertainties both in the model applied and in the choice of initial parameters, e.g. the spin precession rate (or moments of inertia). Numerical difficulties limit trustworthy long-term integrations and these problems are related to convergence properties, machine precision, truncation errors, and limited CPU speed. We also point out that as numerical errors evolve in the integration, the numerical solution eventually deviates from the exact mathematical solution and the numerical curve then corresponds to a different initial condition and can evolve away with chaotic sensitivity according to the new initial condition in effect. REFERENCES Bouquillon, S., & Souchay, J. 999, A&A, 345, 8 Danby, J. M. A. 964, Fundamentals of Celestial Mechanics, (New York: Macmillan), 5 Edvardsson, S., Karlsson, K. G., & Engholm, M., A&A, 384, 689 Folkner, W. M., Yoder, C. F., Yuan, D. N., Standish, E. M., & Preston, R. A. 997, Science, 78, 749 Hayes, W. B. 7, arxiv:astro-ph/779 v Hilton, J. L. 99, AJ,, 5 Ito, T., & Tanikawa, K., in Proc. 3nd Symp. Celest. Mech., ed. H. Arakida, Y. Masaki, & H. Umehara (Japan: Graduate Univ. Advanced Studies), Jacobson, R. A., Synnott, S. P., & Campbell, J. K. 989, A&A, 5, 548 Laskar, J. 4, A&A, 46, 799 Laskar, J., Correia, A. C. M., Gastineau, M., Joutel, F., Levrard, B., & Robutel, P. 4, Icarus, 7, 343 Laskar, J., Levrard, B., & Mustard, J. F., Nature, 49, 375 Lecar, M., Franklin, F. A., Holman, M. J., & Murray, N. W., ARA&A, 39, 58 Levison, H., & Duncan, M. The SWIFT library., in Newman, W. I., Varadi, F., Lee, A. Y., Kaula, W. M., Grazier, K. R., & Hyman, J. M., BAAS, 3, 859 Noerdlinger, P. D., pdnoerd/smassloss.html Pireaux, S., & Rozelot, J-P. 3, Ap&SS, 84, 59 Quinn, T. R., Tremaine, S., & Duncan, M. 99, AJ,, 87 Sussman, G. J., & Wisdom, J. 99, Science, 57, 56 Touma, J., & Wisdom, J. 993, Science, 59, 94 Yao, L.-S. 5, arxiv:nlin/5645 Yoder, C. F., Konopliv, A. S., Yuan, D. N., Standish, E. M., & Folkner, W. M. 3, Science, 3, 99