stability of earthlike planets

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Celest Mech Dyn Astr (2008) 102:8395DOI 10.1007/s10569-008-9159-0

ORIGINAL ARTICLE

On the stability of Earth-like planets in multi-planet

systems

E. Pilat-Lohinger P. Robutel . Sli F. Freistetter

Received: 13 December 2007 / Revised: 26 July 2008 / Accepted: 30 July 2008 /

Published online: 19 September 2008 Springer Science+Business Media B.V. 2008

Abstract We present a continuation of our numerical study on planetary systems withsimilar characteristics to the Solar System. This time we examine the influence of threegiant planets on the motion of terrestrial-like planets in the habitable zone (HZ). Using theJupiterSaturnUranus configuration we create similar fictitious systems by varying Saturnssemi-major axis from 8 to 11 AU and increasing its mass by factors of 230. The analysisof the different systems shows the following interesting results: (i) Using the masses of the

Solar System for the three giant planets, our study indicates a maximum eccentricity (max-e)of nearly 0.3 for a test-planet placed at the position of Venus. Such a high eccentricity wasalready found in our previous study of JupiterSaturn systems. Perturbations associated withthe secular frequency g5 are again responsible for this high eccentricity. (ii) An increase ofthe Saturn-mass causes stronger perturbations around the position of the Earth and in theouter HZ. The latter is certainly due to gravitational interaction between Saturn and Uranus.(iii) The Saturn-mass increased by a factor 5 or higher indicates high eccentricities for atest-planet placed at the position of Mars. So that a crossing of the Earth orbit might occurin some cases. Furthermore, we present the maximum eccentricity of a test-planet placed inthe Earth orbit for all positions (from 8 to 11AU) and masses (increased up to a factor

E. Pilat-Lohinger (B)Institute for Astronomy, University of Vienna, Trkenschanzstrasse 17, 1180 Vienna, Austriae-mail: [email protected]

P. RobutelAstronomie et Systmes Dynamiques, IMCCE-CNRS UMR 2028, Observatoire de Paris,77 Av. Denfert-Rochereau, 75014 Paris, Francee-mail: [email protected]

. SliDepartment of Astronomy, Etvs University, XI. Pzmny Pter stny 1/A,1117 Budapest, Hungarye-mail: [email protected]

F. FreistetterAstrophysikalisches Institut, Friedrich-Schiller-Universitt Jena, Schillergsschen 2-3,07745 Jena, Germanye-mail: [email protected]

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84 E. Pilat-Lohinger et al.

of 30) of Saturn. It can be seen that already a double-mass Saturn moving in its actual orbitcauses an increase of the eccentricity up to 0.2 of a test-planet placed at Earths position. Amore massive Saturn orbiting the Sun outside the 5:2 mean motion resonance (aS 9.7AU)increases the eccentricity of a test-planet up to 0.4.

Keywords Planetary systems Planets: Jupiter, Saturn, Uranus Secular resonances Habitable zone Extra-solar planets

1 Introduction

From the about 300 Extra-solar planetary systems (EPS) near sun-like stars that have beendiscovered so far, we know about 29 multi-planet systems that can be divided from thedynamical point of view into the following classes (cf. Ferraz-Mello et al. 2005):

Ia: Planets in mean-motion resonance.Ib: Low-eccentricity near-resonant planet pairs.II: Non-resonant planets with significant secular dynamics.III: Hierarchical planet pairs.

In our study, we concentrated on class Ib systems. Even if they are not very numerousup to now,1 we are interested in such systems since they might have similar characteris-tics like our Solar System. Therefore, we studied the influence of JupiterSaturnUranusconfigurations on the motion in the so-called habitable zone (HZ). The boundaries 2 of this

region in the Solar System are at 0.93AU and 1.37AU according to the pioneering work byKasting et al. (1993). A planet moving in this zone is habitable if it has a terrestrial oceanof superficial water, where the carbonate-silicate cycle controls the CO2 level in equilibriumwith a surface temperature above the freezing in the HZ. Numerous numerical investiga-tions about planetary motion in the HZ have been carried out during the last decade, whereeither specific extra-solar planetary systems were examined (see e.g. Rivera and Lissauer2000, 2001; Jones and Sleep 2002, Laughlin et al. 2002; Menou and Tabachnik 2003; Dvo-rak et al. 2003; Barnes and Raymond 2004; Asghari et al. 2004; rdi et al. 2004; Jones et al.2005, 2007; Raymond et al. 2006; Ji et al. 2005; Rivera and Haghighipour 2007; Schwarzet al. 2007; and many others) or general stability studies were performed (see e.g. Sndor

et al. 2007; Sli et al. 2007).This investigation is a continuation of our previous study (Pilat-Lohinger et al. 2008),

where we used the orbital parameters of Jupiter and Saturn and created similar fictitiousconfigurations by varying Saturns semi-major axis (aS from 8 to 11 AU) and increasing itsmass by factors 2 to 30. Now we add a third giant planet (i.e. Uranus) to our dynamicalsystem and study the influence of three gas giants on the motion of test-planets in the regionfrom 0.6 to 1.6AU. This enlargement of the Suns HZ is interesting from the dynamicalpoint of view, as we obtain additional information for the positions of Venus and Mars.The resulting max-e maps representing the (at p, aS)-plane3 display clearly the perturbations

1 The OGLE-06-109L system belongs to this class. Taking in mind, that this system discovered by micro-lensing (Gaudi et al. 2008) is one of only 6 far away planetary systems detected by this method, one canconclude that there are many similar systems that might be detected in the future.2 The inner boundary (at 0.93AU) defines the distance when H2O will no more become a major atmosphericcompound; and the outer boundary (at 1.37AU) depends on the CO2 that condensates in the atmosphere andproduces CO2 clouds which can affect the temperature-CO2 coupling significantly.3 at p and aS are the semi-major axes of the test-planets and Saturn, respectively.

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On the stability of Earth-like planets in multi-planet systems 85

of the giant planets on the motion in the HZ resulting from mean motion (MMR) and secularresonancesespecially the secular frequency g5 plays an important role in the dynamicalmaps. A comparison with the results of Pilat-Lohinger et al. (2008) shows the influence ofUranus on the motion in the HZ, that appears especially for Saturn-masses increased by

factors 2 to 9. From the numerous computations that have been carried out for this studywe will discuss in detail: (1) The JupiterSaturnUranus configuration, where an increaseof the eccentricity for a test-planet placed in the position of Venus was found; (2) Systems,where the masses of Saturn and Uranus were increased by a factor 3. Especially the higherUranus-mass changed the max-e map significantly, which indicated additional perturbationsdue to the secular resonances g = g7 and s = s7; (3) Systems with a larger Saturn-massthan that of Jupiter show perturbations in the whole HZ defined by Kasting et al. (1993),especially if Jupiter and Saturn are close to the 5:2 MMR.

In the following, we describe our dynamical model and the computations in Sect. 2, thenwe present the results for the Solar System configuration in Sect. 3. The results for systems

with a higher Saturn-mass as well as a comparison with Pilat-Lohinger et al. (2008) arediscussed in Sect. 4.

2 Dynamical model and computations

Based on Pilat-Lohinger et al. (2008), we took the orbital parameters of Jupiter, Saturn andUranus (see Table 1) and varied the initial semi-major axis of Saturn (aS) from 8 to 11 AUin steps of 0.1 AU, while the orbits of Jupiter and Uranus were fixed to the parameters given

in Table 1. We computed for Saturn-masses multiplied by a factor S of 1, 3, 4 and 5 theorbits of test-planets in the region from 0.6 to 1.6 AU (in steps of 0.02 AU),4 while forthe other mass-factors (S = 2, 6, 7, 8, 9, . . . , 30) a smaller region for the terrestrial-likeplanetsfrom 0.8 to 1.4AU in steps of 0.02AUwas studied. The other orbital elementsfor the test-planets (with negligible mass) were those of the Earth:

the eccentricity et p = 0.0167the inclination it p = 0.0008the argument ofperihelion t p = 103.946the node t p = 358.859 and the mean anomalyMt p = 206.900.

In the dynamical model of the restricted 5 body problem (R5BP)where the massless test-planets move in the gravitational field of the Sun, Jupiter, Saturn and Uranus without per-turbing their orbitswe computed for each map representing the (at p, aS)-plane 1581 orbitsover 107 years, using the hybrid-symplectic integrator5 MERCURY 6 ofChambers (1999).For all orbits we determined the maximum eccentricity (max-e), which is a crucial orbitalparameter for studies in the HZ. By calculating max-e over the whole computation time, wewere able to distinguish between (i) orbits being in the HZ during the whole computationtime, (ii) orbits being in the HZ for most of the time during its revolution and (iii) highlyeccentric orbits, that probably cause problems for the habitability. In this context, we have tomention that Williams and Pollard (2002) have shown that the Earth could also be habitable,according to the standard definition of habitability, if its orbital eccentricity would be 0.7.But this leads certainly to strong variations in the surface temperature.

4 We made this choice to be able to compare the results with those of Pilat-Lohinger et al. (2008).5 For the study presented in Pilat-Lohinger et al. (2008), we tested carefully this integration method regardingthe accuracy of the results.

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Table 1 Orbital elements of the giant planets

Planet a (AU) e inc. (deg) (deg) (deg) M (deg) Mass (m Su n )

Jupiter 5.203 0.0483 1.305 275.201 100.471 183.898 0.95479e3

Saturn 9.530 0.0533 2.486 339.520 113.669 238.293 0.28588e3Uranus 19.235 0.0473 0.773 99.866 74.033 111.688 0.43554e4

The max-e maps show regions of higher eccentricities resulting either from MMRsbetween the gas giants or from secular perturbations. Planetary frequencies were deter-mined with the aid of the frequency analysis ofLaskar (1990)see also Robutel and Gabern(2007)where the secular frequencies g and s are deduced by the following secular linearapproximation (see e.g. Murray and Dermott 1999):

g = n

4

mJ

MSu n2Jb

(1)3/2(J) +

m S

MSu n2Sb

(1)3/2(S) +

mU

MSu n2Ub

(1)3/2(U)

(1)

g = s (2)

whereJ = at p/aJ,S = at p/aS,andU = at p/aU; (aJ, aS and aU arethesemi-majoraxes

of Jupiter, Saturn and Uranus, respectively) and b(1)3/2 is a Laplace coefficient. The test-planetsmust have nearly zero initial eccentricities and inclinations.

Thesolutionsoftheequations6 in a,m S for g(at p, aS,m S) = g5(aS,m S)and g(at p, aS,m S) = g6(aS,m S) can be found as black bold lines in all figures showing the (at p, aS)-

plane. One can see the good agreement of the numerical frequency analysis with the resultof the max-e study. Secular perturbations due to Uranus ( g(at p, aS,m S) = g7(aS,m S) ands(at p, aS,m S) = s7(aS,m S))areonlyvisibleintheHZofthesystem,wheretheUranus-masswas increased, otherwise g7 < g so that the solution does not appear.

3 The habitable zone in the JupiterSaturnUranus system

The study of the influence of Jupiter, Saturn (with S = 1) and Uranus on test-planets (withnegligible mass) moving in the region from at p = 0.6 to 1.6 AU for different semi-major axes

of Saturn (aS = 8 to 11 AU) is summarized in Fig. 1. The max-e result shows different greyscales defining various maximum eccentricities, where white labels the lowest and black thehighest max-e. Like in the JupiterSaturn system (see Pilat-Lohinger et al. 2008), the plot isdominated by a striking arched band of higher eccentricities, which is of secular origin. Theapplication of the numerical frequency analysis ofLaskar (1990) assigned this feature to thesecular frequency g5. In addition to this significant arched structure, some smaller areas ofhigher max-e appeared mostly in the outer HZ (near the position of Mars which is markedby the vertical line labeled with M). They are either close to the 2:1 MMR between Jupiterand Saturn (the dashed horizontal lines show the most important MMRs between Jupiterand Saturn) or close to the 3:1 MMR. In the latter case, the perturbation results from the g

6frequency.A comparison of the results derived for the JupiterSaturnUranus and JupiterSaturn

(Pilat-Lohinger et al. 2008) systems shows more or less the same dynamical behavior. So

6 Indices 5 and 6 denote secular frequencies associated with the precession of perihelion of Jupiter and Saturn,respectively.

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Fig. 1 Stability map for test-planets in the HZ calculated in the SunJupiterSaturnUranus system. The

panel shows the max-e result of the computations over 107

years. According to the grey scale, the eccentricityis the lowest in the white area and the highest in the black area. The solution of the frequency analysis is givenby black bold lines. Horizontal dashed lines show the most important MMRs between Jupiter and Saturn. Thepositions of the planets Venus, Earth and Mars are shown by the vertical full lines and that of Saturn by thehorizontal full line

that no significant influence of Uranus on the HZ was detected within 107 years, when usingthe standard values of Uranus mass (mU) and semi-major axis (aU).

Pointing the attention to the position of Venus (vertical full line labeled with V), werecognize a maximum eccentricity of nearly 0.3 for a test-planet placed in this region whenSaturn moves in its actual orbit (see the horizontal full line marked with Sat).

Compared to the JupiterSaturn system (Pilat-Lohinger et al. 2008) we observe a furtherincrease of the test-planets eccentricity from 0.22 (dashed line in Fig. 2) to 0.265 due toUranus (see full line in Fig. 2). The signal is quite regular but we do not observe such a higheccentricity for Venus in our Solar System. Like in Pilat-Lohinger et al. (2008) we had to

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2e+06 4e+06 6e+06 8e+06 1e+07

eccentricity

time [yrs]

JSU-modelJS-model

JSUE-model

Fig. 2 The evolution of the eccentricity for a test-planet placed in the orbit of Venus and calculated in threedifferent dynamical models: (i) in the JupiterSaturnUranus system (full line), (ii) in the JupiterSaturnsystem (dashed line) and (iii) in the JupiterSaturnUranusEarth system (dotted line)

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Fig. 3 Stability map for Earth-like planets in the HZ calculated in the SunJupiterSaturnUranusEarthsystem. The panel shows the max-e result of the computations over 107 years. The max-e isaccording tothe grey scalethe lowest in the light grey area and the highest in the black area. The bold solid, dashed anddashed-dotted lines represent the secular resonances g5, g6 and g3, respectively, that are derived from thesecular frequency analysis. The hatched area shows the region influenced by the Earth and the thin solid lineat 0.72 AU marks the actual position of Venus. For details see the text

include the Earth in our dynamical model to decrease the eccentricity of the test-planet inthe region of Venus significantly (see dotted line of JSUE-model in Fig. 2). In this context,we have to point out that a high eccentricity of Venus due to the absence of the EarthMoonsystem was also found by Innanen et al. (1998) in a stability study of the inner Solar System.

Moreover, a recent numerical study of the Solar System showed an escape of Mercurywhen its orbit is influenced by the g5 frequency (see Laskar 2008).

Comparing Figs. 1 and 3 one recognizes a significant change of the dynamical structure inthe (at p, aS)-plane when adding the Earth-Moon system. This results from the well-knownfact that in the linear secular theory the frequency g has a singularity located at 1AU in theJupiterSaturnUranusEarth system, which does not appear in the JupiterSaturnUranus

system. The max-e result for all test-planets in the (at p, aS)-plane computed in the JupiterSaturnUranusEarth system is given in Fig. 3. The change in g shifts the position of Venusinto the low-eccentric area (where e < 0.1) for the Solar System parameters. In this context,we still have to check the influence of the high order MMR (13:8) between Venus and Earth.

In the new system we observe a stronger influence of the g6 frequency (which is restrictedto the outer HZ for aS > 10 AU in Fig. 1). Following the bold dashed line labeled by g6 werecognize an influence for nearly all aS even inside the Earths orbit. In contrast thereto, theperturbations due to the g5 frequency are restricted to a quite small areasee the bold solidline marked with g5 for at p < 0.85 and aS between 8 and 8.7AU. Near at p = 0.62AUperturbations resulting from the g3 frequency7 have been found and they are given by thevertical bold dashed-dotted line marked with g3. There is obviously no influence on theposition of Saturn between 8 and 11 AU. Moreover, the g3 frequency is quite close tothe 2:1 MMR between the test-planet and the Earth (around 0.63 AU). We also recognize aslight increase of the max-e in this region.

7 Index 3 denotes the secular frequency associated with the precession of perihelion of the Earth.

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Besides the secular perturbations we observe also higher eccentricities near the 2:1 MMRbetweenJupiterandSaturn(around aS = 8.2AU)andverystrongperturbationsforthebiggerpart of the HZ, when Saturn is placed at 8.6 and 8.7AU (see the horizontal dark stripe from0.6 to 1.2AU). The origin of this perturbation can be seen in Fig. 1, where the max-e plot

shows for the Earth position at aS = 8.7 AU an increase of the eccentricity (up to 0.4) dueto the g5 frequency. This high eccentricity for a massive Earth at 1AU avoids the existenceof other planets between 0.6 and 1.2AU for this aS.

The hatched area between 0.92 and 1.04AU is strongly perturbed by the Earth, wheremost of the test-planets escaped from the system. We did not study this area in detail in thecourse of this investigation, since we were mainly interested in the region near the positionof Venus (see the vertical solid line close to 0.7AU).

4 Results for higher Saturn-masses

The same study for higher Saturn-masses (the multiple of the mass is given by the factor S)is presented in this section, where we compare the results of the computations of differentJupiterSaturnUranus systems with the according ones of the JupiterSaturn configurationpresented in Pilat-Lohinger et al. (2008). This comparison indicates the strongest perturba-tions in the HZ due to the third giant planet in systems where the Saturn-mass is increasedby S = 2 to 9. From the numerous systems that have been computed, we selected two caseswhich will be discussed in detail:

4.1 Results for Saturn-mass Jupiter-mass

In the max-e plot for S = 3 we observe a shift of the dominant secular structure towards theouter border in the HZ (see Fig. 4). This moves the position of Venus in the low-eccentricregion, while test-planets around 1 AU enter in the area influenced by the g5 frequency,which induces higher eccentricities. This behavior near the Earths orbit (see the verticalline labeled by E in Fig. 4) can be observed for aS from 9.15 to 10.3AU. It follows thata more massive Saturn placed at its actual semi-major axis (labeled by the horizontal solidline at 9.53 AU) would cause a higher eccentricity for the Earth. This was already found inthe JupiterSaturn configuration when S = 3 (see Fig. 5). A comparison of Figs. 4 and 5

illustrates the influence of Uranus that is visible due to the additional perturbations in Fig. 4.

Fig.4 Samemax-emapasFig. 1 but with a higher Saturn-massincreased by a factor 3 (labeled by [S = 3])

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Fig. 5 Same max-e map like Fig. 4 but for test-planets calculated in the Sun JupiterSaturn system

It seems that they result mainly from MMRs between Saturn and Uranus: i.e. the 3:1 MMRnear 9.2AU, the 8:3 MMR near 10 AU and the 5:2 MMR near 10.4AU. Following the boldblack line, they appear also in the solution of the numerical frequency analysis. The gaps andfluctuations indicate positions of MMRs, where the method cannot be applied.

To obtain more information about the influence of the third giant planet on the motion inthe HZ, we increased the Uranus-mass by factors 3 and 5. Since the results of both systems arequite similar we show in Fig. 6 only that for 3 Uranus-masses. Of course the perturbationsare stronger now. Comparing Figs. 4 and 6 we see higher eccentricities along the MMRs

especially at the 5:2 and 8:3 MMRs between Saturn and Uranus at 10.4AU and 9.2AU,respectively. For aS between 9.2AU and 10.8AU stronger secular perturbations are visible.Applying the numerical frequency analysis (Laskar 1990), we detected the dominant arched

Fig. 6 Max-e map of the HZ in the Sun-JupiterSaturnUranus system, where the masses of Saturn andUranus are increased by a factor 3. Both results (max-e and numerical frequency analysis) are plotted in thispanel and coincide quite well

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Fig. 7 The max-i in the (at p, aS)-plane computed for test-planets in the Sun-JupiterSaturnUranus systemwhen the masses of Saturn and Uranus are increased by a factor 3. The slanted lines of higher inclinationindicate the perturbation of the secular frequency s7

Fig. 8 Same max-e map as Fig. 4 but for a Saturn-mass increased by a factor of 5 (labeled by [S = 5])

structure resulting from the influence of the g5 frequency between at p = 1 and 1.1AU.Additionally, perturbations due to the g = g7 resonance at at p = 0.8 AU resulting from theincreased Uranus-mass were found. As well as several small areas of higher eccentricity fol-lowing a slanted line that corresponds to perturbations from the s7 frequency. The calculationof the maximum inclination (max-i) for the test-planets in the JupiterSaturn(3m)Uranus(3m)

system8 shows also this slanted stripe of higher inclination (see Fig. 7) which coincides quitewellwiththe s = s7 solutionofthenumericalfrequencyanalysis.Finally,wehavetostatethatan increase of the Uranus-mass up to a factor of 5 does not invoke significant perturbationsdue to three-body resonances in the region of the HZ.

8 (3m) indicates that the masses of Saturn and Uranus are increased by a factor 3.

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Fig. 9 Max-e map for a test-planet moving at 1AU in the different SunJupiterSaturnUranus systems:the x-axis shows the different starting positions of Saturn and the y-axis the different Saturn-masses. Theperturbation of the test-planet is given by the max-e (according to a grey scale), where white indicates lowestvalues of max-e and black the highest ones

4.2 Saturn-mass> Jupiter-mass

Similar perturbations in the HZ due to a third giant planet appeared for Saturn-massesincreased by factors of S > 3. In Fig. 8 we show the max-e result for S = 5, wherethe higher Saturn-mass causes stronger perturbations in the outer part of the HZ that can

be ascribed to stronger interactions between Saturn and Uranus. We observe eccentricitiesbetween 0.1 and 0.3 for test-planets placed at the position of Mars when aS is between 9and 10AU. Even higher eccentricities (between 0.4 and 0.6) were found if Saturn is eitherbetween 8 and 8.7 AU or around 11 AU. There are only few positions of Saturn, where theorbits of Venus, Earth and Mars are all in the region of nearly circular motion. Moreover, sucha high Saturn-mass moving in its actual orbit increases the eccentricities of the test-planetsplaced at the positions of Earth and Mars up to 0.15, which could lead to a minimum distanceof the two planets of only 0.14 AU.

As soon as S = 10 (or larger) the strong interactions between Saturn and Uranus are nolonger visible in the max-e maps of the HZ. For such high masses of Saturn we got plots

showing an arched band of higher eccentricities associated with the g5 frequencywhichmainly influences the test-planets at Earth position for aS between 9.6 and 10.8AU. Inaddition, three horizontal bands of higher eccentricity indicating the 3:1, 5:2 and 2:1 MMRsbetween Jupiter and Saturn. They would cause quite high eccentricities for test-planets atMars position. In some panels for S > 25 we have found another horizontal band of highereccentricity indicating the 7:3 MMR of Jupiter and Saturn.

Interesting is as well that the perturbations close to the 2:1 MMR of Jupiter and Saturnare stronger in the 2-planet systems (compare e.g. Figs. 4 and 5) when S < 10.

A summary of the max-e results for a test-planet moving at 1AU in the different

JupiterSaturnUranus systems (for

S=

1 to 30) is given in Fig. 9. We see the influ-ence of the giant planets depending on their mutual distances (due to the variation of aS)and on the mass of Saturn. The max-e in the different 3-planet systems is defined accordingto the grey scale, where white areas label the regions of lowest max-e and black labels thatof highest max-e. The map shows: (i) escapes of test-planets for very high Saturn-masses,when Jupiter and Saturn are in 2:1 MMR (i.e. aS near 8AU); (ii) a quite large region (i.e.white area) with nearly circular motion of the test-planets during the whole computation time

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of 107 years and (iii) a large band of higher max-e between 9.6 and 10.8AU indicating thatthe test-planets might leave the HZ periodically in certain configurations. Already for thedouble-mass of Saturn and the standard aS a max-e of about 0.14 for the test-planet placed atEarth position can be observed. In this case the perihelion of the orbit would be at 0.86AU

and, therefore, no more in the HZ according to the definition of Kasting et al. (1993).

5 Conclusion

This numerical investigation examined the influence of three gas giants on Earth-like planetsmoving in the HZ of a sun-like star. It is a continuation of a previous paper (Pilat-Lohingeret al. 2008), where the same study was done for two giant planets. The numerical work wasbased on the JupiterSaturnUranus configuration, for which a cloud of test-planets in theregion from 0.6 to 1.6AU (i.e. an extended HZ) was studied for different semi-major axes of

Saturn (from 8 to 11AU) over a time-interval of 107 years. The resulting max-e maps repre-senting the (at p, aS)-plane show perturbations due to mean motion and secular resonances.We analyzed the stability in the HZ (i) for the JupiterSaturnUranus system, (ii) for systems,where the Saturn-mass the Jupiter-mass and (iii) for systems, where the Saturn-mass >the Jupiter-mass.

Using the masses of the Solar System we have found an increase of the eccentricity(up to nearly 0.27) for a test-planet moving in the orbit of Venus. This high eccentricity wascaused by the secular resonance associated with the precession of perihelion of Jupiter. Wehave found in both studies (Pilat-Lohinger et al. 2008 and the present study), that the only

way to decrease the eccentricity in this area is to include the Earth in our dynamical model.Since the adding of the Earth changed the g frequency significantlyand, therefore, as wellthe solution for g5(aS)the position of Venus was no more in the area influenced by the g5frequency in the Solar System configuration.

An increase of Saturns mass by a factor of 2 shifted the secular resonance towards theouter border of the HZ, while further increases of its mass caused very small displacementsof this secular resonance. Consequently, Venus was no longer perturbed by the secular reso-nance but then test-planets placed in the area near the Earth were influenced. The calculatedeccentricities of the test-planets led to a crossing of the inner border of the HZ in some cases,so that they left this region periodically. This would certainly produce significant changes

in the surface temperature but maybe it does not exclude the habitability (see Williams andPollard 2002).

A comparison of the dynamical maps of the JupiterSaturn and the JupiterSaturnUranus systems showed the influence of the third gas giant. We analyzed in detail the addi-tional perturbations in the HZ due to the presence of Uranus for the case where Jupiterand Saturn have nearly equal masses. The increase of the Saturn-mass showed perturbationsresulting from MMRs between Saturn and Uranus. An additional increase of the Uranus-massby a factor 3 or 5 displayed also secular perturbations assigned to the perihelion and node ofUranus. While significant perturbations due to three-body resonances were not detected.

Systems, where the Saturn-mass is larger than the Jupiter-mass indicated strongerperturbations in the outer HZ due to Uranus (especially up to a mass-factor 9 for Saturn).Quite high eccentricities for test-planets at Mars position occurred for certain positions ofSaturn, which could lead to close approaches or even crossings with the Earth. For Saturn-masses increased by a factor of 10 or larger, we have found quite similar results showingperturbations either due to the g5 frequency or due to the 2:1, 5:2 and 3:1 MMRs betweenJupiter and Saturn. The max-e maps showed that the region near the Earth was influenced

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by the g5 frequency while that near Mars was mainly affected by MMRs. It seems that ahigher Saturn-mass (>Jupiter-mass) would cause problems for the stability of the inner SolarSystem. In this context, we have to point out, that the existence of a JupiterSaturn config-uration for a Saturn-mass Jupiter-mass is questionable since the study by Morbidelli and

Crida (2007), showed a migration into the 3:2 or the 2:1 MMR for such systems.Another interesting result of this study are the perturbations around the position of Marswhen Jupiter and Saturn are in 2:1 MMR, especially in connection with the so-called Nice-model (see Tsiganis et al. 2005). A first investigation about the influence on the inner SolarSystem has been done by Agnor and Lin (2007).

Acknowledgements The work of this paper was supported by the Austrian FWF project no. P19569-N16,especially EP-L and S benefited from this project. Most of the numerical integrations were carried out onthe NIIDP (National Information Infrastructure Development Program) supercomputer in Hungary.

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