The search for exoplanets is made interesting by the spotting of Goldilocks, or habitable, zones, regions in space that could support life. A simple calculator to model these zones was proposed by a team from the University of Washington, and it throws up some curious results.

A bunch of scientists from the University of Washington have put together a habitable-zone (i.e., Goldilocks zone) calculator [link to HTML page] on FORTRAN, and I'm really kicked about it. You can find the FORTRAN code here*, and a pre-print copy of their paper here. In fact, the work's been accepted to the Astrophysical Journal.

Anyway, the calculator's a great thing because putting in a few numbers is all that's needed to throw up possibly useful and interesting insights into how habitable zones (HZ) are created around a star. At the moment, the input values required are effective temperature of the star and the stellar luminosity in solar units.

The effective temperature of a body is defined as the temperature of a blackbody that emits the same quantity of electromagnetic radiation as does the body. Many objects, like our Sun, emit different kinds of radiation - such as ultraviolet, X-ray, and gamma - because of different processes associated with them. In such cases, it becomes difficult to say what the temperature of the body is at some point in space and time.

The easier workaround is to think in terms of temperature, and figure out how hot or cold another body would have to be to be emitting the same /amount/ of radiation. The Sun's effective temperature is 5,780 K.

The stellar luminosity is a measure of the star's brightness. Basically, it's a product of the star's effective temperature and its surface area. To make matters simpler, our Sun is taken to be a benchmark of sorts, and its stellar luminosity is 1 solar unit.

The output values are the recent Venus limit, the runaway greenhouse limit, the water-loss limit, the max. greenhouse limit, and the early Mars limit (all measured in astronomical units - AUs).

1. The early Venus limit is the distance at which an Earth-like planet develops a Venus-like climate around a star like our Sun. The HZ closest to the star is assumed to lie beyong this point.

2. The runaway greenhouse limit is the distance at which the greenhouse effect is dominant on the planet with the oceans evaporating away entirely.

3. The water-loss limit is the distance at which the upper stratosphere of the planet becomes saturated with water vapour - this is a prerequisite for habitability.

4. The maximum greenhouse limit is the distance at which the planets has a greenhouse effect going that completely traps all infrared radiation in the atmosphere.

5. The early Mars limit is the distance at which a planet develops a Mars-like climate around a star like our Sun. The HZ closest to the star is assumed to lie before this point.

So, plug in different values into the calculator and see for yourself what different conditions persist around a star at different distances.

Model Moist greenhouse Runaway greenhouse Maximum greenhouse
Mars-sized planet (S.G. = 3.73 m/s2) 1.035 AU 1.033 AU 1.72 AU
Earth 0.99 AU 0.97 AU 1.70 AU
Super-Earth (S.G. = 25 m/s2) 0.94 AU 0.92 AU 1.67 AU
pCO2 = 5.2 x 10-3 bar 1.00 AU 0.97 AU -
pCO2 = 5.2 x 10-2 bar 1.02 AU 0.97 AU -
pCO2 = 5.2 x 10-1 bar 1.02 AU 0.97 AU -
pCO2 = 5.2 bar 0.99 AU 0.97 AU -

(Table from pre-print paper)

Here, S.G. is the surface gravity of the planet (Earth's S.G. is 9.81 m/s2); pCO2 is the amount of pressure that an atmosphere composed only of carbon dioxide would exert on the ground.

Here are some interesting plots from the paper.

OLR v. Wavenumber


A plot of the OLR (outgoing long-wave radiation) against the wavenumber (number of waves - one crest + one trough - per cm) for a water-loss limit modeled using the calculator - It shows a strong absorption (the downward peak) at around 675 per cm. (Image from pre-print paper)

Teff v. Flux


The distances of different recently discovered exoplanets are plotted on a "space-map" in which the habitable zone has been plotted using the calculator. Note that this map is said to accounted for stars of different sizes and effective temperatures. (Image from pre-print paper)

(HD 40307g, which sits smack in the middle of the HZ belt, is located 42 light-years away in the direction of the constellation Pictor.)

Planet mass v. Flux


Another interesting plot, this shows the "preferred" mass of a planet that lies in the HZ with some effective incident stellar flux (the amount of energy entering the planet's atmosphere from the star). The window is divided into three parts, with each indicating the kind of star that must be needed for a planet of some mass to be habitable. (Image from pre-print paper)

Interestingly, for an effective temperature between 4,000 K and 4,500 K, you'll notice that the values of the runaway greenhouse limit and the water-loss limit are almost identical! In other words, and the implication is that, there is a fine line between when stratospheric saturation occurs and when the oceans start to vaporise. This line is determined by the distance of an Earth-like planet from a Sun-sized star that's 69-78% as hot as our Sun is.

*Note: Per the request of the authors, please acknowledge the following papers if you're using the code or the calculator:

    Kopparapu et al. (2012), Habitable Zones around Main-sequence stars: New Estimates. Submitted to Astrophysical Journal.

    Ramirez et al. (2012), Greenhouse and Anti-greenhouse Effects in Dense CO2-CH4 and CO2-H2 Atmospheres AND Implications for the Habitable Zone. Submitted to Icarus.