The search for life in the universe requires methods to identify and characterize habitable planets in other stars. The concept of habitable zones (HZ) around stars provides the easiest procedure to identify habitable exoplanets. The HZ is used as a yes-no criteria where exoplanets are either habitable or not. Clearly, some will be more habitable than others and there is a need for procedures to rank them due to almost 600 that had been confirmed, and thousands that are waiting for confirmation and will be detected in the future. Ranking or classification procedures not only help to prioritize observations but also to compare results. Therefore, here we present a simple method to convert the traditional HZ definition to an analog scale that is useful to sort exoplanets within and outside the HZ as a basic metric of planetary habitability.
The HZ is generally defined as the region around a star were a planet could maintain liquid water at their surface, an essential ingredient for the development of life as we know it (Kasting et al. 1993). Small bodies within the HZ will not be able to hold an atmosphere while giant planets will trap dense atmospheres that will "crush" any water to ice. So it is also essential that they are near terrestrial size (between half to two times Earth's diameter) to be considered potentially habitable. Exoplanets closer to their star will lost all water to space and those farther away will freeze out. The HZ is thus the right region with a width of about 0.3 AU for dim stars like M stars (red dwarfs) and up to 2 AU for bright stars like F stars.
The HZ is a function of the stellar luminosity and some critical stellar fluxes values (Kasting et al. 1993). In this analysis we considered a conservative HZ bounded by a "recent Venus" model for the inner edge and an "early Mars" model for the outer edge (Underwood et al., 2003; Selsis et al. 2007). The inner ri and outer ro boundaries of the HZ in AU units are given by
where L is the stellar luminosity in solar units, Teff is the stellar effective temperature in K units, Ts = 5700 K, ai = 2.7619e-5, bi = 3.8095e-9, ao = 1.3786e-4, bo = 1.4286e-9, ris = 0.72, and ros = 1.77. Habitable exoplanets are those that their mean distance from the star fall between this boundary. [Attached below is a IDL function (habitable_zone.pro) that can be used to calculate the HZ including other boundaries (explanation available in the code)].
Our approach to convert the HZ into an analog scale, and not just a yes-no condition, consist in calculating the distance of the exoplanet from the middle point of the HZ and normalized to half-width of the HZ. This in effect creates a new metric or distance function in habitable space and habitable zone units (HZU). The habitable zones distance (HZD) is therefore given as
where r is the distance of the exoplanet from the star in AU units and HZd is the HZD in HZU units. These units are very practical because they mean the same thing independently of the stellar system under consideration. HZD values between -1 and +1 HZU always correspond to planets within the HZ. [Attached below is a IDL function to calculate the HZD (hz_distance.pro, requires habitable_zone.pro)].
The HZD value tells how relatively far the exoplanet is from the center of their stellar HZ, or even outside the HZ for values outside the -1 to +1 range. Negative signs are for exoplanets in the side closer to the star (Hot Zone) and positve values for those farther (Cold Zone). A HZD of zero corresponds to the exact center of the HZ. Note that this does not necessarily means that an exoplanet at the center of the HZ is more habitable than one a bit farther from this center, still in the HZ, as this also depends on other planetary properties. Other metrics, like the Earth Similarity Index (ESI), can be used when more information about the exoplanet is known (Schulze-Makuch et al., 2011). In the absence of more information, the HZD is the easiest proxy for planetary habitability. It only requires stellar luminosity and effective temperature together with distance of the exoplanet for calculations.
As an example, we compared the exoplanets of Gliese 581 with the Solar System (Figure 1). Gliese 581 is a red dwarf star with four confirmed exoplanets that orbit at no more than 1 AU, but the six planets solution was used for this example (Vogt et al., 2010). The HZD was calculated using the equations described above. The absolute value of the HZD was used to easily sort out the their habitability, values closer to zero are better (Figure 2, top frame). The use of the sign is important too for comparisons (Figure 2, bottom frame). Candidates with HZD values just below -1 (Hot Zone) might be discarded as habitable planets, as they are probably Venus-like, however, HZD values just above +1 (Cold Zone) are still interesting if strong greenhouse effects are involved.
In conclusion, the proposed Habitable Zone Distance (HZD) provides a simple way to rank and compare the habitability of exoplanets, not only those within the HZ, but also how those outside the HZ rank among them. The HZD converts the spatial scale of stellar systems to a convenient habitable space with corresponding Habitable Zones Units (HZU). This analysis used a conservative HZ boundary but it can be extended to other HZ boundaries. The HZD for exoplanets with highly elliptical orbits can also be averaged, which might be useful to compare cases with long excursions outside the HZ. Future analysis will compare the HZD and other planetary quantities of confirmed exoplanets and Kepler candidates.
As a commentary, we also suggest to alternatively name the Habitable Zone Distance (HZD) simply as the "Kasting Distance" because this analysis was a simple adaptation of the decisive contribution of Jim Kasting from Penn State to our current understanding of habitable zones and planetary habitability.