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On the planetary system ε Eridani
Abstract
Backman, Marengo et al (2008) report on new observations of the planetary system Epsilon Eridani, which provide hints of a structure similar to the Solar system’s. Using a simple method, we analyse the possible orbital parameters of the innermost planets of the system and comment on their possible habitability.
Introduction
In a recent article (Backman, Marengo et al. 2008; hereafter BM08), a team of astronomers report on new observations of the star ε Eridani with the sub-mm Spitzer and CSO telescopes. The star is very similar to the Sun, except for being much younger (online reference page, [4]). This data allows for a creation of a model of the system which would account for the observed spectral energy distribution profiles. The data shows several rings of dust orbiting the star, with the innermost one similar to the asteroid belt in our Solar system (see Figure). In the model presented in the article, gaps in the belts are mentioned as one of the possible indicators of several inner planets at resonant orbits. With this small amount of information, crude estimates of the inner planets‘ orbital parameters can be made and the temperature conditions in the planets discussed.
Method
We assume small eccentricities of all orbits, which is a reasonable claim for a fully-formed planetary system (see, e.g. Butler et al 2006 for a discussion pertaining to this particular system). We also assume the the particle density in the system to be small (this is likely to be true for an optically thin material, which is the case; cf. BM08, Fig. 8), so that all particles move in Keplerian orbits and fluid-like motion can be neglected. With these assumptions, one can calculate the rotation periods of small objects within the system, using Kepler‘s third law:
P = (4π2a3/GM)0.5, (1)
where all the symbols have their usual meanings. Gaps in this belt can appear due to a resonating cofocal motion of massive bodies (planets) in the system. Such planets will clear out a narrow annulus in the belt, provided that the period of a planet’s rotation is in ratio m:n with that of the asteroid belt, where m, n are integers (resonance condition). While theoretically any combination of m and n give a resonance, these effects become less pronounced with increasing m, n. In practice only several lowest combinations have any effect if the planet in question is of terrestrial size (i.e. up to 15MEarth). We assume that the inner planets are small (this is because we are interested in possible terrestrial planets in the inner parts of the system; in addition, a Jovian planet would most likely have been detected already), hence only ratios 1:2, 1:3 and 2:3 induce noticeable perturbations to asteroid motion.
The habitable zone of a star can be broadly defined as the region of space where the temperature of a planet’s surface would allow for liquid water. While it is far from a rigorous calculation, such an estimate allows for comparison of the ε Eridani system with that of the Sun. The estimate can be derived by equating the energy received by the planet from the blackbody radiation of the star to the energy radiated by the planet (which is also assumed to be a blackbody):
πR2L/(4πa2)= 4πR2σT4 -> T=(L/(16πσa2)0.25, (2)
where σ is the Stefan-Boltzmann constant, a is the semimajor axis of the planet’s orbit and the other symbols have their usual meanings.
Results and discussion
Using the data provided in BM08 and online, M = 0.85MSun. The luminosity of the star varies as M2.3 (Zeilik & Gregory), thus LEri ≈ 0.7LSun. We are interested in the inner belt (see Figure), which is located at a distance of around a = 3 AU. The main results are presented in Table 1.
|
Resonance |
Orbital period / yr |
Orbital distance / AU |
Surface temperature / K |
|
3:1 |
2 |
1.4 |
215 |
|
2:1 |
3 |
1.9 |
180 |
|
3:2 |
4 |
2.3 |
170 |
The calculated temperatures are lower than those required for liquid water (273 to 373 K), but if any of the planets are massive enough to support an atmosphere, the greenhouse effect may be enough to raise the temperatures to terrestrial levels. In comparison, the mean temperature of the Earth is increased by ~10K due to atmospheric effect, but the Venusian atmosphere provides heating of nearly 500K.
The accuracy of these results cannot be more than 25%, which is the uncertainty associated with the position and size of the inner particle belt in the system (BM08).
Conclusion
The existence of gaps in the innermost particle belt of the system ε Eridani allows to infer some information about the orbital characteristics of possible inner planets. With some reasonable assumptions, the distances of three such planets from their star have been calculated. None of these planets fall within the habitable zone of the star, however that is not a strong indicator against presence of life. In either case, subsequent detailed imaging of the system will reveal more precise information about its innermost parts.
References
1. D. Backman, M. Marengo et al. Epsilon Eridani’s Planetary Debris Disk: Structure and Dynamics based on Spitzer and CSO Observations. 2008 (to appear on ApJ) arXiv:0810.4564v1
2. A. Hatzes et al. “Evidence for a Long-Period Planet Orbiting ε Eridani” 2000, ApJ, 544, L145
3. R.P. Butler et al. “Catalog of Nearby Exoplanets” 2006, ApJ, 646, 505
4. Online data set at http://www.chara.gsu.edu/RECONS/TOP100.posted.htm
5. M. Zeilik, S.A. Gregory. Introductory Astronomy and Astrophysics, Fort Worth : Saunders College Pub., 1998.
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