In the search for extraterrestrial life, various methods are used to identify inhabitable planets in other star systems (so-called exoplanets or extrasolar planets). The most simple way to identify habitable exoplanets is based on the concept of the habitable zone1 (IZ) around stars. The IZ can be used as a yes-or-no criteria, where exoplanets are either habitable or not. However, it is clear that some exoplanets have a larger chance of being inhabited than others. Therefore, more advanced metrics are needed, such as distance to habitable zone (IZD), that can be used for example to order the over 600 exoplanets that are already confirmed to be habitable. Moreover, there are thousands of exoplanets that are waiting for confirmation of their status, and many more are likely to be discovered in the future. Procedures for ordering and classifying habitable exoplanets do not only help to prioritize observations, but also to compare results.

The IZ is usually defined by the area surrounding a star where fluid water can occur on the surface of a planet (Karsting et al., 1993), since this is an important requirement for the development of life as we know it. Within the IZ, smaller celestial bodies are not able to grip an atmosphere, while the atmosphere in which gigantic planets are captured in giant planets, causing them to compress every form of water in to ice. It is necessary that exoplanets have earthly measurements (between half and twice the intersection of the earth) to be declared potentially habitable. Exoplanets that are closer to their star, will lose all water to space and those that are further away will freeze. The IZ is the good area to search for habitable planets, with a width of about 0.3 AU2 for weak stars such as those with stellar classification3 M (red dwarfs4) and up to 2 AU for bright stars such as those with stellar classification F.

The IZ is defined in function of the stellar brightness and some other critical properties of stars (Karsting et al., 1993). A conservative approach for constructing the IZ uses a "recent Venus" model as boundary $$r_{is}$$ for the inner radius and an "early Mars" model as boundary $$r_{os}$$ for the outer radius (Underwood et al., 2003; Selsis et al. 2007). Other boundaries that are typically used are given in the below table.

model ID $$r_{is}$$ $$r_{os}$$
Mars-Venus (default model) 0 0,72 1,77
0% clouds 1 0,84 1,67
50% clouds 2 0,68 1,95
100% clouds 3 0,46 2,40

The inner $$r_i$$ and outer $$r_o$$ radius of the IZ (in AU) is given by \[ \begin{array}{rcl} r_i & = & \left( r_{is} - a_i (T_\text{eff} - T_\text{s}) - b_i (T_\text{eff} - T_\text{s})^2 \right) \sqrt{L} \\ r_o & = & \left( r_{os} - a_o (T_\text{eff} - T_\text{s}) - b_o (T_\text{eff} - T_\text{s})^2 \right) \sqrt{L} \end{array} \] where $$L$$ represents the stellar brightness5 (expressed relatively in relation to the luminosity of the sun), $$T_\text{eff}$$ is the actual temperature of the star (in Kelvin), $$T_\text{s} = 5700\,\text{K}$$, $$a_i = 2,7619 \times 10^{-5}$$, $$b_i = 3,8095 \times 10^{-9}$$, $$a_o = 1,3786 \times 10^{-4}$$ en $$b_o = 1,4286 \times 10^{-9}$$. Exoplanets are declared habitable if the mean distance to their stars is within these two boundaries. 

Wherever the IZ around a star gives a yes-or-no-answer to the question if the exoplanet is habitable, the IZ is converted to an analogue scale by the IZD. For this conversion, the distance of the exoplanet to the middle of the IZ is calculated. The distance is then normalized with half the width of the IZ. The IZA is thereby given by the formula \[ BZA = \frac{2r - r_o - r_i}{r_o - r_i} \] with $$r$$ representing the distance from the exoplanet to its star (expressed in AU). The advantage of the IZD is that the resulting values can easily be compared with the different star systems, because they always mean the same. IZD values that are between -1 and +1, always correspond to planets that are remote from the star (cold zone). An IZD that is equal to zero, corresponds with the center of the IZ. Do note that this does not necessarily imply that an exoplanet that is closer to the center of the IZ would we more habitable than an exoplanet that is further away from the center, but still within the IZ, since being habitable also depends on other properties of the planet.

As an example, we will compare the exoplanets of the star Gliese 5816 with those from our own solar system7 (Image 1). Gliese 581 is a red dwarf with four confirmed exoplanets that orbit the star in less than 1 AU. In this example, however, we use the six-planet-solution that was proposed by Vogt et al. (2010). The IZD was calculated on the basis of the above formulas. In the upper pannel of Image 2, we use the absolute value of the IZD to sort the exoplanets on the basis of how habitable they are. The closer the value is to zero, the better the exoplanet can be inhabited. In this comparison it is also important to look at the sign of the IZD (lower frame of Image 2). Candidates with an IZD just below -1 (hot zone) can be ignored as habitable exoplanets, since these probably resemble Venus a lot. Values that are just above +1, however, might be interesting if there would be severe greenhouse effects.

afstand tot de ster (AE)

Image 1. Comparison of the distance from the star Gliese 581 to its exoplanets (red dots) with the distance between our sun and the planets of our solar system (blue dots). Because of the clearness, only the earthly planets are given on the linear axis of the upper panel, and the gas giants are used as a reference on the logarithmic axis of the lower pannel. the green areas indicate the habitable zone in both cases. Red areas indicate the hot zone and blue areas indicate the cold zone. On this distance, it is harder to compare how to order the planets in comparison with their habitable zones.

afstand tot het centrum van de bewoonbare zone

Image 2. Distance to the center of the habitable zone (IZD) of the exoplanets of Gliese 581 (red dots) in comparison with the planets of our solar system (blue dots) The upper pannel uses the absolute value of the IZD, while the lower panel uses the IZD value itself. The green areas indicate the habitable zones in both cases. The red areas indicate the hot zone and the blue areas indicate the cold zone. The yellow area combines the hot and cold zone. Note that Gliese 581 c (IZD = -1.2) can be catalogued as less habitable than Venus (IZD = -1.0) and that Gliese 581 g (IZD = -0.2) seems more habitable than the Earth (BZA ~ -0.5). Only 2 IZD units sufficed to represent all exoplanets, except for Gliese 581 f, and all earthly planets, the other cases are outside the scale. 

Sources

Assignment

The HEC Full Database of Exoplanets contains observed and modeled parameters for the current list of confirmed exoplanets and NASA Kepler8 candidates. This databank can be downloaded9 in comma separated value (CSV) format. This format can be read by Microsoft Excel and most scientific analyses software. An overview of the meaning of the different columns in this CSV can be found here.

Example

In the interactive Python session below, we assume that the CSV file exoplanets.csv10 is in the current directory.

>>> IZ(1.0)
(0.72, 1.77)
>>> IZ(0.012, 3498)
(0.08325382874078525, 0.2234714169113722)
>>> IZ(0.012, 3498, 3)
(0.05477225575051661, 0.29248445915702315)

>>> IZD(1.0, 1.0)                     # Earth
-0.4666666666666666
>>> IZD(1.524, 1.0)                   # Mars
0.5314285714285715
>>> IZD(0.210, 0.012, 3498)           # Gliese 581 d
0.8078498270133836

>>> habitablePlanets('exoplanets.csv', star='Gl')
Gl 581           0.012  3498   0.084   0.224  Gl 581 d           0.210   0.797
Gliese 876       0.015  3350   0.092   0.247  Gliese 876 b       0.210   0.524
Gliese 876       0.015  3350   0.092   0.247  Gliese 876 c       0.130  -0.509
>>> habitablePlantes('exoplanets.csv', star='Gliese')
Gliese 876       0.015  3350   0.092   0.247  Gliese 876 b       0.210   0.524
Gliese 876       0.015  3350   0.092   0.247  Gliese 876 c       0.130  -0.509