Vol.0 (2008) No.2, 000–000

Jianghui Ji

11institutetext: Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210008, China
11email<EMAIL_ADDRESS>22institutetext: National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100012, China
33institutetext: National Astronomical Observatory, Mitaka, Tokyo 181-8588,
Japan
44institutetext: Department of Astronomy, Nanjing University, Nanjing 210093,
China

# The Dynamical Architecture and Habitable zones of the Quintuplet Planetary
System 55 Cancri

Jianghui JI 1122 Hiroshi Kinoshita 33 Lin LIU 44 Guangyu LI 1122aaaa

(Received 2008/09/08)

###### Abstract

We perform numerical simulations to study the secular orbital evolution and
dynamical structure in the quintuplet planetary system 55 Cancri with the
self-consistent orbital solutions by Fischer and coworkers (2008). In the
simulations, we show that this system can be stable at least for $10^{8}$ yr.
In addition, we extensively investigate the planetary configuration of four
outer companions with one terrestrial planet in the wide region of 0.790 AU
$\leq a\leq$ 5.900 AU to examine the existence of potential asteroid structure
and Habitable Zones (HZs). We show that there are unstable regions for the
orbits about 4:1, 3:1 and 5:2 mean motion resonances (MMRs) with the outermost
planet in the system, and several stable orbits can remain at 3:2 and 1:1
MMRs, which is resemblance to the asteroidal belt in solar system. In a
dynamical point, the proper candidate HZs for the existence of more potential
terrestrial planets reside in the wide area between 1.0 AU and 2.3 AU for
relatively low eccentricities.

###### keywords:

celestial mechanics-methods:n-body simulations-planetary systems-
stars:individual(55 Cancri)

##  1 Introduction

The nearby star 55 Cancri is of spectral type K0/G8V with a mass of $0.92\pm
0.05M_{\odot}$ (Valenti & Fischer 2005). Marcy et al. (2002) reported a second
giant planet with a long period of $\sim 14$ yr after the first planet
discovered in 1997. The 55 Cnc system can be very attractive, because first it
hosts a distant giant Jupiter-like planet about 5.5 AU resembling Jupiter in
our solar system. The second interesting thing is that this system may be the
only known planetary system in which two giant planets are close to the 3:1
orbital resonance, and the researchers have extensively studied the dynamics
and formation of the 3:1 MMR in this system (see Beaugé et al. 2003; Ji et al.
2003; Zhou et al. 2004; Kley, Peitz, & Bryden 2004; Voyatzis & Hadjidemetriou
2006; Voyatzis 2008). Still, the additional collection of follow-up
observations and the increasing of precision of measurements (at present $\sim
1$ ms-1 to $3$ ms-1) have indeed identified more planets. McArthur et al.
(2004) reported the fourth planet with a small minimum mass $\sim 14$
$M_{\oplus}$ that orbits the host star with a short period of 2.8 day, by
analyzing three sets of radial velocities. The improvement of the observations
will actually induce additional discovery. Hence, it is not difficult to
understand that more multiple planetary systems or additional planets in the
multiple systems are to be dug out supplemental data.

More recently, Fischer et al. (2008) (hereafter Paper I) reported the fifth
planet in the 55 Cnc system with the Doppler shift observations over 18 yrs,
and showed that all five planets are in nearly circular orbits and four have
eccentricities under 0.10. It is really one of the most extreme goals for the
astronomers devoted to searching for the extrasolar planets to discover a true
solar system analog, which may hold one or two gas giants orbiting beyond 4 AU
that can be compared to Jupiter and Saturn in our own solar system (Butler
2007, private communication; see also Gaudi et al. 2008). This indicates that
several terrestrial planets may move in the so-called Habitable Zones (HZs)
(Kasting et al. 1993; Jones et al. 2005), and the potential asteroidal
structure can exist. Considering the probability of the coplanarity and nearly
circular orbits for five planets (Paper I), the 55 Cnc system is suggested to
be a comparable twin of the solar system. Hence, firstly, in a dynamical
viewpoint, one may be concerned about the stability of the system over secular
timescale. On the other hand, the small bodies as terrestrial objects may
exist in this system and are to be detected with forthcoming space-based
missions (Kepler, SIM). In this paper, we focus on understanding the dynamical
structure and finding out suitable HZs for life-bearing terrestrial planets in
this system.

##  2 Dynamical Analysis

In this paper, we adopt the orbital parameters of the 55 Cancri system
provided by Paper I (see their Table 4). For the convenience of narration, we
re-label the planets according to the ascendant semi-major axes in the order
from the innermost to the outermost planet (e.g., B, C, D, E, F), while the
original names discovered in the chronological order are also accompanied but
in braces (see Table 1). Furthermore, McArthur et al. (2004) derived the
orbital inclination $i=53^{\circ}\pm 6^{\circ}.8$ with respect to the sight
line for the outermost planet from HST astrometric data by measuring the
apparent astrometric motion of the host star. In the simulations, we adopt
this estimated orbital inclination of $53^{\circ}$ and further assume all the
orbits to be coplanar. With the planetary masses $M\sin i$ reported in Table
1, then we obtain their true masses. Specifically, the masses of five planets
are respectively, 0.03 $M_{Jup}$, 1.05 $M_{Jup}$, 0.21 $M_{Jup}$, 0.18
$M_{Jup}$ and 4.91 $M_{Jup}$, where $\sin i=\sin 53^{\circ}=0.7986$. Thus, we
take the stellar mass $M_{c}$ of 0.94 $M_{\odot}$ (Paper I), and the planetary
masses above-mentioned in the numerical study, except where noted. We utilize
N-body codes (Ji, Li & Liu 2002) to perform numerical simulations by using
RKF7(8) and symplectic integrators (Wisdom & Holman 1991) for this system. In
the numerical runs, the adopted time stepsize is usually $\sim$ 2% \- 5% of
the orbital period of the innermost planet. In addition, the numerical errors
were effectively controlled over the integration timescale, and the total
energy is generally conserved to $10^{-6}$ for the integrations. The typical
timescale of simulations of the 55 Cnc system is from 100 Myr to 1 Gyr.

###  2.1 The Stability of the 55 Cancri Planetary System

####  2.1.1 case 1: 5-p for $10^{8}$ yr

To explore the secular stability of this system, firstly, we numerically
integrated the five-planet system on a timescale of $10^{8}$ yr, using the
initials listed in Table 1. In Figure 1, a snapshot of the secular orbital
evolution of all planets is illustrated, where $Q_{i}=a_{i}(1+e_{i})$,
$q_{i}=a_{i}(1-e_{i})$ (the subscript $i=1-5$, individually, denoting Planet
B, C, D, E, and F) are, respectively, the apoapsis and periapsis distances. In
the secular dynamics, the semi-major axis $a_{1}$ and $a_{2}$ remain unchanged
to be 0.0386 and 0.115 AU, respectively, for $10^{8}$ yr, while $a_{3}$,
$a_{4}$ and $a_{5}$ slightly librate about 0.241, 0.786, and 6.0 AU with quite
small amplitudes over the same timescale. The variations of eccentricities
during long-term evolution are followed, where $0.23<e_{1}<0.28$,
$0.0<e_{2}<0.03$, $0.034<e_{3}<0.069$, $0.0<e_{4}<0.013$, and
$0.056<e_{5}<0.095$, implying that all the eccentricities undergo quasi-
periodic modulations. In Fig.1, we note that the time behaviors of $Q_{i}$ and
$q_{i}$ show regular motions of bounded orbits for all five planets and
indicate their orbits are well separated during the secular evolution due to
small mutual interactions, which again reflect the regular dynamics of the
eccentricities over secular timescale. In the numerical study, we find the
system can be dynamically stable and last at least for $10^{8}$ yr. Thus our
numerical outcomes strengthen and verify those of Paper I for the integration
of $10^{6}$ yr, which also showed the system can remain stable over 1 Myr and
the variations of all planetary eccentricities are modest.

Secondly, we further performed an extended integration for the planetary
configuration simply consisting of four outer planets over timescale up to 1
Gyr (see Figure 2). The longer integration again reveals that the orbital
evolution of four planets are quite similar to those exhibited in Fig.1, and
then strongly supports the secular stability of this system. In a recent
study, Gayon et al. (2008) show that the 55 Cnc system may remain a stable
chaos state as the planetary eccentricities do not grow over longer timescale.
Therefore, it is safely to conclude that the 55 Cnc system remain dynamically
stable in the lifetime of the star.

In order to assess the stability of 55 Cnc with respect to the variations of
the planetary masses, we first fix $\sin i$ in increment of 0.1 from 0.3 to
0.9. In the additional numerical experiments, we simply vary the masses but
keep all orbital parameters (Table 1), again restart new runs of integration
for the five-planet system for 100 - 1000 Myr with the rescaled masses. As a
result, we find the system could remain definitely stable for the above
investigated timescale with slight vibrations in semi-major axes and
eccentricities for all planets, indicating the present configuration is not so
sensitive to the planetary masses. Subsequently, we again examine the
stabilities of different orbital configurations within the error range of the
Keplerian orbital fit given by Paper I. Herein 100 simulations are carried out
for 10 Myr, and the numerical results show that all the runs are stable over
the simulation timescale, indicating that this five-planet system is fairly
robust with respect to the variational planetary configurations.

####  2.1.2 case 2: 7-p for $10^{8}$ yr

However, Paper I argued that the 6th or more planets could exist and maintain
dynamical stability in the large gap between Planets E and F in this system.
Next, we also integrate the 55 Cnc system with additional planets (2 massive
terrestrial planets, Earth at 1 AU and Mars at 1.52 AU) to mimic the situation
of the inner solar system. In this runs, we examine the configuration
consisting of 5 planets and 2 terrestrial bodies to study the coexistence of
multiple objects. This means that we directly place Earth and Mars into the 55
Cnc system to simulate ”the inner solar system”, where the orbital elements
for above terrestrial planets are calculated from JPL planetary ephemerides
DE405 at Epoch JD 2446862.3081 corresponding to the outermost companion (see
Table 2), e.g., the semi-major axes are respectively, 1.00 and 1.524 AU. The
five planets are always assumed to be coplanar in the simulations, thus the
inclinations for 2 terrestrial planets refer to the fundamental plane of their
orbits. In this numerical experiment, we find that the 7-p system can remain
dynamically stable and last at least for $10^{8}$ yr. In Figure 3 are shown
the time behaviors of $Q$ (yellow line) and $q$ (black line) for Mars, Earth
and Planet E. The numerical results show the regular bounded motions that
their semi-major axes and eccentricities do not dramatically change in their
secular orbital evolution, and this is also true for the other four planets in
the 55 Cnc. It is not so surprised for one to realize that two additional
terrestrial planets could exist for long time because the gravitational
perturbations arise from other planets are much smaller. In the following
section, we will extensively explore this issue on the dynamical architecture
for the Earth-like planets in the system.

##  3 Dynamical Architecture and Potential HZs

To investigate the dynamical structure and potential HZs in this system, we
extensively performed additional simulations with the planetary configuration
of coplanar orbits of four outer companions with one terrestrial planet. In
this series of runs, the mass of the assumed terrestrial planet selected
randomly in the range 0.1 $M_{\oplus}$ to 1.0 $M_{\oplus}$. The initial
orbital parameters are as follows: the numerical investigations were carried
out in [$a,e$] parameter space by direct integrations, and for a uniform grid
of 0.01 AU in semi-major axis (0.790 AU $\leq a\leq$ 5.900 AU) and 0.01 in
eccentricity ($0.0\leq e\leq 0.2$), the inclinations are
$0^{\circ}<I<5^{\circ}$. The angles of the nodal longitude, the argument of
periastron, and the mean anomaly are randomly distributed between $0^{\circ}$
and $360^{\circ}$ for each orbit. Then each terrestrial mass body was
numerically integrated with four outer planets in the 55 Cnc system. In total,
about 10,750 simulations were exhaustively run for typical integration time
spans from $10^{5}$ to $10^{6}$ yr (about $10^{6}$ \- $10^{7}$ times the
orbital period of Planet C). Then, our main results now follow.

Figure 4 shows the contours of the surviving time for Earth-like planets
(Upper) and the status of their final eccentricities (Lower) for the
integration over $10^{5}$ yr, and the horizontal and vertical axes represent
initial $a$ and $e$ of the orbits. Fig. 4 (Upper) displays that there are
stable zones for the Earth-like planets in the region between 1.0 and 2.3 AU
with final low eccentricities of $e<0.10$. The extended simulations ($10^{6}$
yr) for the objects of the above region also exhibit the same results. This
zone may be strongly recommended to be one of the potential candidate HZs in
the 55 Cnc, and our results coincide with those by Jones et al. (2005), who
showed the possible HZs of 1.04 AU $<a<$ 2.07 AU. Still, the outcomes
presented here have confirmed those in §2.1.2, where we show that the stable
configuration of Earth at 1.00 AU and Mars at 1.523 AU in this five-planet
system. The sixth planet or additional habitable bodies may be expected to
revealed in this region by future observations111The semi-amplitude of wobble
velocity $K\propto{M_{p}\sin i}/{\sqrt{a(1-e^{2})}}$ (with $M_{p}\ll M_{c}$),
herein $M_{c}$, $M_{p}$, $a$, $e$ and $i$ are, respectively, the stellar mass,
the planetary mass, the orbital semi-major axis, the eccentricity and the
inclination of the orbit relative to the sky plane. This means that planets
with larger masses and (or) smaller orbits could have larger $K$. For example,
a planet of 1.0 $M_{\oplus}$ at 1 AU in a nearly circular orbit may cause
stellar wobble about 0.10 m/s. In this sense, much higher Doppler precision is
required to discover such Earth-like planets in future..

In general, the planetary embryos or planetesimals may be possibly captured
into the mean motion resonance regions or thrown into HZs by a giant planet
under migration due to the planet-disk interaction and could survive during
the final planetary evolution over the secular timescale after complex
scenarios of secular resonance sweeping, gravitationally scattering, and late
heavy bombardments (Nagasawa et al. 2005; Thommes et al. 2008). We note that
there are strongly unstable orbits 222We define an unstable orbit as an Earth-
like planet is ejected far away or moves too close to the parent star or the
giant planets, meeting the following criteria: (1) the eccentricity approaches
unity, (2) the semi-major axis exceeds a maximum value, e.g., 1000 AU, (3) the
assuming planet collides with the star or enters the mutual Hill sphere of the
known giant planets. for the low-mass planets initially distributed in the
region $3.9$ AU $<a<5.9$ AU, where the planetary embryos have very short
dynamical surviving time. In the meantime, the eccentricities can be quickly
pumped up to a high value $\sim 0.9$ (see Fig. 4, Lower). We note that the
orbital evolution is not so sensitive to the initial masses. In fact, these
planetary embryos are involved in many of MMRs with the outermost giant in the
55 Cnc system, e.g., 7:4 (4.063 AU) and 3:2 MMRs (4.503 AU). The overlapping
resonance mechanism (Murray & Dermott 1999) can reveal their chaotic behaviors
of being ejected from the system in short dynamical lifetime $\sim
10^{2}-10^{3}$ yr, furthermore the majority of orbits are within the sphere of
3 times Hill radius ($R_{H}={\left({M_{5}}/(3M_{c})\right)}^{1/3}a_{5}$,
$3R_{H}\doteq 2.10$ AU) of the 14-yr planet. Using resonance overlapping
criterion (MD99; Duncan et al. 1989), the separation in semi-major axis
$\Delta{a}\approx 1.5{\left({M_{5}}/M_{c}\right)}^{2/7}a_{5}\doteq 1.95$ AU,
then the inner boundary $R_{O}=a_{5}-\Delta{a}$ for Planet F is at $\sim 3.95$
AU. And the orbits in this zone become chaotic during the evolution because
the planets are both within 3 $R_{H}$ and in the vicinity of $R_{O}$.
Similarly, there exist unstable zones for the nearby orbits around Planet E
(0.78 AU $<a<$ 0.90 AU), which may not be habitable in dynamical point.

It is suggested that MMRs can play an important role in determining the
orbital dynamics of the terrestrial bodies, which are either stabilized or
destabilized in the vicinities of the MMRs. The outermost giant, like Jupiter,
may shape and create the characteristic of dynamical structure of the small
bodies. Most of the initial orbits for planetary embryos located about 3:1
(2.837 AU), 5:2 (3.204 AU), and 4:1 MMRs (2.342 AU), are quickly cleared off
by the perturbing from Planet F. In the region of $2.4$ AU $<a<3.8$ AU, stable
zones are separated by the mean motion resonance barriers, e.g., 3:1 and 5:2
MMRs. Note that the initial orbits for the relatively low eccentricity (under
0.06) for 4:1 MMR can remain stable over the simulation timescale. However,
the terrestrial bodies about 7:3 MMR (3.354 AU) and 2:1 MMR (3.717 AU) are
both on the edge of the stability, and the former are close to 5:2 MMR, while
the latter just travel around the inner border of $3R_{H}$ at $\sim 3.80$ AU.
The extended longer integrations show that their eccentricities can be further
excited to a high value and a large fraction of them lose stabilities in the
final evolution. The above gaps are apparently resembling those of the
asteroidal belt in solar system. In the simulations, several stable orbits can
be found about 3:2 MMR at 4.503 AU, which is analogous to the Hilda group for
the asteroids in the solar system, surviving at least for $10^{6}$ yr. In
addition, the other several stable cases are the so-called Trojan planets (1:1
MMR), residing at $\sim$ 5.9 AU. The studies show that the stable Trojan
configurations may be possibly common in the extrasolar planetary systems
(Dvorak et al. 2004; Ji et al. 2005; Gozdziewski & Konacki 2006). Indeed,
terrestrial Trojan planets with circular orbits $\sim$ 1 AU could potentially
be habitable, and are worthy of further investigation in future.

##  4 Summary and Discussions

In this work, we have studied the secular stability and dynamical structure
and HZs of the 55 Cnc planetary system. We now summarize the main results as
follows:

(1) In the simulations, we show that the quintuplet planetary system can
remain dynamically stable at least for $10^{8}$ yr and that the stability
would not be greatly influenced by shifting the planetary masses. Account for
the nature of near-circular well-spaced orbits, the 55 Cnc system may be a
close analog of the solar system. In addition, we extensively investigated the
planetary configuration of four outer companions with one terrestrial planet
in the region 0.790 AU $\leq a\leq$ 5.900 AU to examine the existence of
potential Earth-like planets and further study the asteroid structure and HZs
in this system. We show that unstable zones are about 4:1, 3:1 and 5:2 MMRs in
the system, and several stable orbits can remain at 3:2 and 1:1 MMRs. The
simulations not only present a clear picture of a resembling of the asteroidal
belt in solar system, but also may possibly provide helpful information to
identify the objects when modeling multi-planet orbital solutions (Paper I) by
analyzing RV data. The dynamical examinations are helpful to search for best-
fit stable orbital solutions to consider the actual role of the resonances,
where some of best-fit solutions close to unstable islands of MMRs can be
dynamically ruled out in the fitting process. As well-known, the extensive
investigations in the planetary systems (Menou & Tabachnik 2003; Érdi et al.
2004; Ji et al. 2005, 2007; Pilat-Lohinger et al. 2008; Raymond et al. 2008)
show that the dynamical structure is correlated with mean motion and secular
resonances. The eccentricities of the planetesimals (or terrestrial planets)
can be excited by sweeping secular resonance (Nagasawa & Ida 2000) as well as
mean motion resonances, thus the orbits of the small bodies can undergo mutual
crossings and then they are directly cleared up in the post-formation stage.
In conclusion, the mentioned dynamical factors perturbing from the giant
planets will influence and determine the characteristic distribution of the
terrestrial planets in the late stage formation of the planetary systems, to
settle down the remaining residents in the final system.

(2) As the stellar luminosity of 55 Cnc is lower than that of the Sun, the HZ
should shift inwards compared to our solar system. It seems that the newly-
discovered planet at $\sim 0.783$ AU can reside in the HZs (Rivera &
Haghighipour 2007), and this planet may be habitable provided that it bears
surface atmosphere to sustain the necessary liquid water and other suitable
life-developing conditions (Kasting et al. 1993). In a dynamical
consideration, the proper candidate HZs for the existence of more potential
terrestrial planets reside in the wide area between 1.0 AU and 2.3 AU for
relatively low eccentricities, and the maintenance of low eccentricity can
play a vital role in avoiding large seasonal climate variations (Menou &
Tabachnik 2003) for the dynamical habitability of the terrestrial planets.
Moreover, our numerical simulations also suggest that additional Earth-like
planets (§2.1.2) can also coexist with other five known planets in this system
over secular timescale. This should be carefully examined by abundant
measurements and space missions (e.g. Kepler and TPF) for this system in
future.

###### Acknowledgements.

We would like to thank the anonymous referee for valuable comments and
suggestions that help to improve the contents. We are grateful to G.W. Marcy
and D. A. Fischer for sending us their manuscript and insightful discussions.
This work is financially supported by the National Natural Science Foundations
of China (Grants 10573040, 10673006, 10833001, 10203005) and the Foundation of
Minor Planets of Purple Mountain Observatory. We are also thankful to Q.L.
Zhou for the assistance of computer utilization. Part of the computations were
carried out on high performance workstations at Laboratory of Astronomical
Data Analysis and Computational Physics of Nanjing University.

## References

  * [Beaugé et al.(2003)] Beaugé, C., et al. 2003, ApJ, 593, 1124
  * [Dvorak et al. (2004)] Dvorak, R., et al. 2004, A&A, 426, L37
  * [Duncan et al.(1989)] Duncan, M., Quinn, T., & Tremaine, S. 1989, Icarus, 82, 402
  * [Érdi et al.(2004)] Érdi, B., et al. 2004, MNRAS, 351, 1043
  * [Fischer et al.(2008)] Fischer, D. A., et al. 2008, ApJ, 675, 790 (Paper I)
  * [Gaudi et al.(2008)] Gaudi, B. S., et al. 2008, Science, 319, 927
  * [Gayon et al.(2008)] Gayon, J., Marzari, F., & Scholl, H. 2008, MNRAS, 389, L1
  * [Gozdziewski(2006)] Gozdziewski, K., & Konacki, M. 2006, ApJ, 647, 573
  * [Ji et al.(2002)] Ji, J. H., Li, G.Y., & Liu, L. 2002, ApJ, 572, 1041
  * [Ji et al.(2003)] Ji, J. H., Kinoshita, H., Liu, L., & Li, G.Y. 2003, ApJ, 585, L139
  * [Ji et al.(2005)] Ji, J. H., Liu, L., Kinoshita, H., & Li, G.Y. 2005, ApJ, 631, 1191
  * [Ji et al.(2007)] Ji, J. H., Kinoshita, H., Liu, L., & Li, G.Y. 2007, ApJ, 657, 1092
  * [Jones et al.(2005)] Jones, B. W., et al. 2005, ApJ, 622, 1091
  * [Kasting et al.(1993)] Kasting, J. F., et al. 1993, Icarus, 101, 108
  * [Kley et al.(2004)] Kley, W., Peitz, J., & Bryden, G. 2004, A&A, 414, 735
  * [Marcy et al.(2002)] Marcy,G. W., et al. 2002, ApJ, 581, 1375
  * [McArthur et al. (2004)] McArthur, B.E., et al. 2004, ApJ, 614, L81
  * [Menou & Tabachnik(2003)] Menou, K., & Tabachnik, S. 2003, ApJ, 583, 473
  * [Murray Dermott(1999)] Murray, C. D., & Dermott, S. F. 1999, Solar System Dynamics (New York: Cambridge Univ. Press) (MD99)
  * [Nagasawa & Ida(2000)] Nagasawa, M., & Ida, S. 2000, AJ, 120, 3311
  * [Nagasawa (2005)] Nagasawa, M., Lin, D.N.C., & Thommes, E. 2005, ApJ, 635, 578
  * [Pilat-Lohinger(2008)] Pilat-Lohinger, E., et al. 2008, ApJ, 681, 1639
  * [Raymond et al.(2008)] Raymond, S. N., Barnes, R., & Gorelick, N. 2008, ApJ, 689, 478
  * [Rivera & Haghighipour(2007)] Rivera, E., & Haghighipour, N. 2007, MNRAS, 374, 599
  * [Thommes et al.(2008)] Thommes, E. W., et al. 2008, ApJ, 675, 1538
  * [Valenti et al. (2005)] Valenti, J.A., & Fischer, D. A. 2005, ApJS, 159, 141
  * [Voyatzis & Hadjidemetriou(2006)] Voyatzis, G., & Hadjidemetriou, J. D. 2006, CeMDA, 95, 259
  * [Voyatzis(2008)] Voyatzis, G. 2008, ApJ, 675, 802
  * [Wisdom Holman(1991)] Wisdom, J., & Holman, M. 1991, AJ, 102, 1528
  * [Zhou et al.(2004)] Zhou, L.-Y., et al. 2004, MNRAS, 350, 1495

Table 1: The orbital parameters of 55 Cancri planetary system.

Planet |  $M$sin$i$($M_{Jup}$) |  $P$(days) |  $a$(AU) | $e$ |  $\varpi$(deg) | $T_{p}$
---|---|---|---|---|---|---
Planet B (e) | 0.0241 | 2.796744 | 0.038 | 0.2637 | 156.500 | 2447578.2159
Planet C (b) | 0.8358 | 14.651262 | 0.115 | 0.0159 | 164.001 | 2447572.0307
Planet D (c) | 0.1691 | 44.378710 | 0.241 | 0.0530 | 57.405 | 2447547.5250
Planet E (f) | 0.1444 | 260.6694 | 0.785 | 0.0002 | 205.566 | 2447488.0149
Planet F (d) | 3.9231 | 5371.8207 | 5.901 | 0.0633 | 162.658 | 2446862.3081

a. The parameters are taken from Table 4 of Fischer et al. (2008). The mass of
the star is 0.94 $M_{\odot}$.

Table 2: The orbital elements for 2 terrestrial planets at JD 2446862.3081 (From DE405). Planet |  $a$ (AU) | $e$ |  $I$(deg) |  $\Omega$(deg) |  $\omega$(deg) | M(deg)
---|---|---|---|---|---|---
Earth | 1.000 | 0.0164 | 0.002 | 348.33 | 115.231 | 61.647
Mars | 1.524 | 0.0935 | 1.850 | 49.60 | 286.352 | 85.614

Figure 1: Snapshot of the secular orbital evolution of all planets is
illustrated, where $Q_{i}=a_{i}(1+e_{i})$, $q_{i}=a_{i}(1-e_{i})$ (the
subscript $i=1-5$, each for Planet B, C, D, E, and F) are, respectively, the
apoapsis and periapsis distances. $a_{1}$ and $a_{2}$ remain unchanged to be
0.0386 and 0.115 AU, respectively, while $a_{3}$, $a_{4}$ and $a_{5}$ slightly
librate about 0.241, 0.786, and 6.0 AU with smaller amplitudes for $10^{8}$
yr. The simulations indicate the secular stability of the 55 Cnc.

Figure 2: Numerical simulations for four outer planets for $10^{9}$ yr. The
apoapsis and periapsis distances: $Q_{i}=a_{i}(1+e_{i})$,
$q_{i}=a_{i}(1-e_{i})$ (the subscript $i=2-5$, each for Planet C, D, E, and
F). The long-term simulation shows that the system can remain stable over 1
Gyr.

Figure 3: Simulation for 7-p case. The system can be dynamically stable and
last at least for $10^{8}$ yr, the time behaviors of $Q$ (yellow line) and $q$
(black line) each for Mars, Earth and Planet E. The results show that their
semi-major axes and eccentricities do not dramatically change in the secular
orbital evolution, and it is also true for the other four planets in the 55
Cnc.

Figure 4: Upper: Contour of the surviving time for Earth-like planets for the
integration of $10^{5}$ yr. Lower: Status of their final eccentricities.
Horizontal and vertical axes are the initial $a$ and $e$. Stable zones for the
Earth-like planets in the region between 1.0 and 2.3 AU with final low
eccentricities of $e<0.10$. Unstable islands, e.g., 3:1 and 5:2 MMRs, have
separated the region of $2.4$ AU $<a<3.8$ AU. Strongly chaos happen for the
low-mass bodies initially distributed in $3.9$ AU $<a<5.9$ AU, and their
eccentricities can be quickly pumped up to a high value $\sim 0.9$.