Elsevier

Icarus

Volume 168, Issue 2, April 2004, Pages 223-236
Icarus

Obliquity variations of terrestrial planets in habitable zones

https://doi.org/10.1016/j.icarus.2003.11.017Get rights and content

Abstract

We have investigated obliquity variations of possible terrestrial planets in habitable zones (HZs) perturbed by a giant planet(s) in extrasolar planetary systems. All the extrasolar planets so far discovered are inferred to be jovian-type gas giants. However, terrestrial planets could also exist in extrasolar planetary systems. In order for life, in particular for land-based life, to evolve and survive on a possible terrestrial planet in an HZ, small obliquity variations of the planet may be required in addition to its orbital stability, because large obliquity variations would cause significant climate change. It is known that large obliquity variations are caused by spin–orbit resonances where the precession frequency of the planet's spin nearly coincides with one of the precession frequencies of the ascending node of the planet's orbit. Using analytical expressions, we evaluated the obliquity variations of terrestrial planets with prograde spins in HZs. We found that the obliquity of terrestrial planets suffers large variations when the giant planet's orbit is separated by several Hill radii from an edge of the HZ, in which the orbits of the terrestrial planets in the HZ are marginally stable. Applying these results to the known extrasolar planetary systems, we found that about half of these systems can have terrestrial planets with small obliquity variations (smaller than 10°) over their entire HZs. However, the systems with both small obliquity variations and stable orbits in their HZs are only 1/5 of known systems. Most such systems are comprised of short-period giant planets. If additional planets are found in the known planetary systems, they generally tend to enhance the obliquity variations. On the other hand, if a large/close satellite exists, it significantly enhances the precession rate of the spin axis of a terrestrial planet and is likely to reduce the obliquity variations of the planet. Moreover, if a terrestrial planet is in a retrograde spin state, the spin–orbit resonance does not occur. Retrograde spin, or a large/close satellite might be essential for land-based life to survive on a terrestrial planet in an HZ.

Introduction

More than 110 extrasolar planets around nearly 100 Sun-like stars are now known to exist (see, e.g., http://cfa-www.harvard.edu/planets/). The main technique used to detect these planets is to observe the reflex motion of host stars due to planetary motion through the Doppler shift of stellar spectral lines (e.g., Marcy et al., 2000). The planets so far found are as massive as Jupiter, so all the known extrasolar planets are considered to be Jupiter-like giant planets. However, it is reasonable to expect that terrestrial planets (relatively small rocky planets) could be present in some of the extrasolar planetary systems where a giant planet(s) has been detected. Are there terrestrial planets like the Earth that could habor life among extrasolar planetary systems?

In general, organisms which we are familiar with on Earth require liquid water during at least part of their life cycle. In order for life to emerge on a terrestrial planet, it is also necessary that its orbit remains confined to the habitable zone (HZ) of its host star over the length of time required for biological evolution. A stellar HZ is the range of distances from a star allowing the existence of liquid water at the surface of a terrestrial planet Abe, 1993, Kasting et al., 1993. The orbital stability of hypothetical Earth-like planets inside the HZs of extrasolar planetary systems has already been investigated through numerical simulations by many authors Gehman et al., 1996, Jones et al., 2001, Jones and Sleep, 2002, Noble et al., 2002, Menou and Tabachnik, 2003. Here, orbital stability means that the orbits of the hypothetical terrestrial planets remain confined to stellar HZs over the time required for the evolution of life. In general, if the orbits of giant planets are located in the vicinity of the HZs, terrestrial planets cannot exist long enough on stable orbits inside the HZs because of the strong perturbing influence of the giant planets. Menou and Tabachnik (2003) studied the orbital stability of 85 known extrasolar planetary systems. Their results indicate that more than half of the 85 systems are unlikely to harbor habitable terrestrial planets.

Not only long-term orbital stability of terrestrial planets in stellar HZs but also moderate climate on these planets may be necessary for the evolution of life, in particular for a land-based life. Planetary climate is greatly influenced by insolation distribution and intensity, which depend on the orbital parameters and spin motion of the planet, especially eccentricity, obliquity, and precession motion Milankovitch, 1941, Berger, 1984, Berger, 1989. Obliquity is the angle of a spin axis of a planet relative to its orbital plane. Because of their equatorial bulge, planets are subject to the torque arising from the gravitational force of a host star (and possibly of their satellites and of other planets). This torque causes precessional motion of the spin axis of the planet about its orbit normal. On the other hand, gravitational interactions with other planets cause the orbit normals to precess and nutate about the normal to the invariable plane of the system. So the obliquity of a planet generally changes periodically. We study the amplitude of obliquity variations of terrestrial planets in HZs perturbed by gas giants.

Obliquity changes the latitudinal distribution of yearly insolation on the planet, which could be an important driver of climate variations. Ward (1974) showed that the total fluctuation in the Earth's obliquity is about 1.5°–2.0° in a period of about 4×104 years. According to the more detailed calculation, the Earth's obliquity varies by ± 1.3° around its mean value of 23.3° Laskar and Robutel, 1993, Laskar, 1996. The glacial cycle of the Earth has been triggered by the periodic change of obliquity to some extent, through the variation of insolation at the edge of a glacier in the northern hemisphere (e.g., Crowley and North, 1991). Even such small variation of obliquity has produced significant variations in the Earth's climate, at least during the Quaternary.

Mars' obliquity is believed to have suffered from a large-scale oscillation with an amplitude of more than ∼10° around its average of ∼25° on a time scale of ∼105–106 years Ward, 1973, Ward, 1974, Ward, 1979. Such a large variation is the result of a coupling between the precession rate of Mars' spin axis and one of the frequencies of its orbital precession. This is called the spin–orbit resonance. Laskar and Robutel (1993) showed that the obliquity of terrestrial planets other than the Earth could have experienced large and chaotic variations throughout their histories. As a result of chaotic variations on a longer timescale, the range of Mars' obliquity change can be as large as from 0° to 60°. Furthermore, obliquity of an Earth without the Moon would vary radically in the range from 0° to more than 85° Laskar and Robutel, 1993, Laskar et al., 1993, Laskar, 1996. If the Earth's obliquity had such a large variation, its climate would drastically change, providing a significant obstacle to the development of complex life on Earth.

The previous analyses of the behavior of planetary obliquity, however, have been concerned mostly with the present orbital configuration of the planets in our Solar System. We are interested in the evaluation of obliquity of possible terrestrial planets in extrasolar planetary systems with various orbital configurations of planets. For this purpose, it is useful to introduce generalized analytical formulae which approximately express the amplitude of the obliquity variations as functions of planetary dynamical parameters.

The oscillation amplitude of obliquity in nonresonant regions has been derived through a linear analysis of the equation of spin motion (e.g., Sharaf and Budnikova, 1969a, Sharaf and Budnikova, 1969b, Vernekar, 1972, Ward, 1974, Berger, 1978). With the analytical formulae for non-resonant regions, Ward et al. (2002) showed the amplitude for a mass less body in the place of Mars as a function of the rotational period and the semimajor axis of the particle. However, the nonresonant formulae cannot describe the oscillation amplitude of obliquity in the regions near spin–orbit resonances where the oscillation amplitude is not small. The variation of planetary obliquity is also written in Hamiltonian form (e.g., Kinoshita, 1977, Laskar and Robutel, 1993, Laskar, 1996), and analyses of the spin–orbit resonance in the Solar System are available (e.g., Colombo, 1966, Henrard and Murigande, 1987, Laskar and Robutel, 1993, Touma and Wisdom, 1993, Laskar, 1996). Analytical expression of obliquity in resonance is given by Ward, 1982, Correia and Laskar, 2003. In this paper, we reanalyze the spin–orbit resonance, assuming a simple planetary system to derive the generalized formulae for extrasolar planetary systems.

Calculating the obliquity variations using the analytical formulae, we investigate how strongly possible terrestrial planets in the known extrasolar planetary systems are affected by the spin–orbit resonances. We find that possible terrestrial planets in only 1/5 of the known extrasolar planetary systems exhibit both small obliquity variations (Δϵ⩽10°, where Δϵ is the variation amplitude of obliquity) and orbital stability over their entire HZs and most of them are systems with close-in giant planets. Our results show that terrestrial planets on stable orbits usually suffer large obliquity variations unless they have retrograde spin or a large/close satellite.

In Section 2, we briefly summarize the spin–orbit resonance and introduce the analytical expressions of obliquity variations. Using the analytical formulae, we discuss dynamical properties of giant planets causing large obliquity variations of terrestrial planets in HZs, in Section 3. In Section 4, we apply our results to the known extrasolar planetary systems. 5 Discussion, 6 Conclusion are devoted to discussion and conclusion.

Section snippets

Obliquity variation

In this section, we briefly summarize basic equations and analytical expressions of the amplitude of obliquity variations which we use in this paper. We investigate obliquity variations of a hypothetical massless terrestrial planet in a planetary system with a given giant planet(s).

Spin axis ŝ of a terrestrial planet precesses around its orbit normal (the vector normal to the orbital plane) n̂ because of the gravitational torque raised by other objects, following (e.g., Ward, 1974) dŝdt=α(ŝ·n

Obliquity variation in a habitable zone in extrasolar planetary systems

In Section 2, we have introduced analytical expressions for the amplitude of obliquity variation (|Δϵ|max). Hereafter, we denote |Δϵ|max by Δϵ. Using analytical formulae, we study dynamical properties of giant planets causing large obliquity variations of terrestrial planets in a habitable zone (HZ). As mentioned in Section 1, for land-based life to evolve and survive on a terrestrial planet in an HZ, small or limited obliquity variation of the planet may be required, in addition to the

Applications to known extrasolar planetary systems

We apply the results in the previous sections to known extrasolar planetary systems to evaluate obliquity variations of possible terrestrial planets in their HZs. The samples of extrasolar planetary systems used here are listed in Table 2, Table 3, Table 4 with relevant data on the stellar masses, and the planets' masses, semimajor axes and eccentricities. These data were taken from the Extrasolar Planets Encyclopedia (http://cfa-www.harvard.edu/planets/). We categorized extrasolar planetary

Discussion

In Section 4, we evaluated the probability of terrestrial planets in HZs having large obliquity variations in 87 known extrasolar planetary systems. The results suggest that the terrestrial planets with small obliquity variations in HZs may be uncommon. In this section, we discuss whether the resonances are avoidable.

In future surveys, giant planets at ≳5–10 AU could be detected as additional planets in known planetary systems. As suggested by the results in Fig. 7, such distant giant planets

Conclusion

We have investigated obliquity variations of possible terrestrial planets in HZs in extrasolar planetary systems. The amplitude of obliquity variations may be one of the most important factors for survival of land-based life on terrestrial planets.

Using the analytical expressions for the obliquity variations of terrestrial planets, we statistically studied general properties of obliquity variations of terrestrial planets in an HZ perturbed by a giant planet(s). We derived the existing

Acknowledgements

We thank William Ward for useful comments concerning the discussion about the known extrasolar planetary systems. We also thank Hidekazu Tanaka for valuable advice on the analytical formulation. The referees also suggested directions which bettered the quality of this paper a great deal.

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