## Habitable Value

Planets at Kapteyn’s Star (source)

Since the last post on this blog, there have been two additional habitable planet candidates announced. First, a two-planet system orbiting the very nearby, very old red dwarf Kapteyn’s Star was reported by Anglada-Escudé et al. The inner planet, the habitable zone world, at 4.8+0.9-1.0 ME is probably a mini-Neptune or micro-Jovian planet, based on its mass — the overwhelming majority of planets of this mass whose radii are known are clearly low-density worlds. The outer world is a cold super-Earth, and probably the same type of planet. Kapteyn’s Star is a member of the Galactic halo, and is quite ancient at ~11 Gyr old. The apparent fact that the Universe was assembling habitable planets when it was less than 3 Gyr old may have interesting implications for the Fermi Paradox, but I won’t go into that here.

Next is Gliese 832 c. In 2008, Bailey et al. reported the presence of a Jupiter-analogue orbiting the nearby red dwarf system GJ 832, and then last week, we learned of second planet, a super-Earth type planet straddling the inner edge of the habitable zone, reported by Wittenmyer, et al. It is almost certain that this planet is not habitable, certainly not to life “as we know it.” The planet’s mass comes in at 5.4±1.0 ME, and therefore likely a mini-Netune / micro-Jovian, much like Kapteyn’s Star b.

Then wandering through the news as I do on a daily basis, I found this

Note the description of the planet as “among the most habitable,” with artist images depicting oceans, lush green land, and so on, despite the description of the planet in the discovery paper as

However, given the large mass of the planet, it seems likely that it would possess a massive atmosphere, which may well render the planet inhospitable. Indeed, it is perhaps more likely that GJ 832c is a “super-Venus,” featuring significant greenhouse forcing.

And this was being generous! I personally thought the discovery of planets at Kapteyn’s Star was much more interesting than the discovery of GJ 832 c, but apparently news cycles have a different standard than I do as to what amounts to an interesting world. That standard, with respect to exoplanet discoveries, is the Earth Similarity Index (ESI) that the Planetary Habitability Laboratory uses to evaluate a planet’s habitability. A quick look at their site shows that, sure enough, GJ 832 c is the third most highly ranked exoplanet.

This is not the first time I have complained about the PHL. But this time I will instead work on providing an alternative method of evaluating a planet’s habitability. A child could look at the above diagram and tell you Kepler-186f was the most “Earth-like” of those planets based on their appearance, but to be rigorous and useful, we need a system to quantify a planet’s habitability. Let’s first look at how the ESI is determined.

$\displaystyle ESI=\prod_{i=1}^n\left(1-\left|\frac{x_i-x_{i_0}}{x_i+x_{i_0}}\right|\right)^\frac{w_i}{n}$

Where $x_i$ is the n-th property of the planet — in this case, either radius, density, escape velocity or surface temperature — $x_{i_0}$ is the value of this property for Earth, and $w_i$ is the weight exponent of a property. For the parameters usually used by the ESI, these values are

 Property Reference Weight Exponent Radius 1 $R_\oplus$ 0.57 Density 1 $\rho_\oplus$ 1.07 Escape Velocity 1 $V_{e_\oplus}$ 0.70 Surface Temperature 288 K 5.58

The formula I will use to evaluate the habitability of an exoplanet will be rather anthropocentric – for all I know, solid, hot super-Earth-type planets like Kepler-10 b may be the most frequently inhabited planets in the Galaxy, but all I know of is Earth-life, and so this formula will be centered around finding Earth-like life. It will effectively be based on Guassian distributions, and will take the form

$\displaystyle H = \prod_{i=1}^4 \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x_i-\mu)^2}{2\sigma^2}\right)$

Here, μ acts as a reference value much as in the ESI formula, σ describes the broadening of the distribution and will effectively be used to determine the tolerance of variation on a particular parameter, and $x_i$ is the parameter we look at. As the product sign suggests, we calculate this for each of four parameters and multiply the results. Here, the four parameters are the stellar temperature, planet mass, planet radius, and planetary insolation.

For the stellar temperature, I chose σ=0.001 and μ=5500, which is some 277 K cooler than our sun. It seems that early K dwarfs are probably a sort of “sweet spot” for planet habitability. As such, if you found an Earth-analogue around an early K dwarf, it would rank higher on this scale than Earth itself. For the planetary mass and radii, I chose μ=1.0 for obvious reasons, and chose σ=5 and σ=0.75, respectively — punishing radius pretty heavily. Lastly, I chose insolation values of μ=1 and σ=1. All values of σ are in terms of that of Earth. Lastly, the values were normalised to make 1 the highest achievable value.

Unsurprisingly, the Solar System is the clear winner, followed by Kepler-186 f, which I made a big deal about earlier this year. The GJ 581 system, which was celebrated as hosting the first habitable planet candidates in the latter years of the last decade, doesn’t even make it up to 10-5, nor does GJ 832 c.

 Planet H Earth 0.96635 Venus 0.61220 Kepler-186 f 0.18525 Kepler-62 f 0.09104 Mars 0.04304 Kepler-62 e 0.00530 Kepler-283 c 0.00005 Kepler-296 Af 0.00003

I would say this set-up makes a lot more sense than the one the PHL is using. Anything below 0.1 is probably not worth a raised eyebrow these days.