## The Habitable Zone

One of the main ingredients for life as we know it is liquid water. Water exists as a liquid between 273K and 373K (unless the pressure is too low, in which case the water sublimates into gaseous water vapor). The region on the solar system (or any planetary system) where the temperature is in this range, is called the habitable zone.

Planets are in equilibrium with their surroundings: they are neither getting hotter nor colder. All planets absorb incident radiation from the Sun (this heats them up); to maintain equilibrium, they must radiate away the same amount of energy. The temperature of a planet can be approximated by assuming that it is a black body.

You determine the temperature by equating the planetary luminosity (proportional to its temperature raised to the fourth power, T4) to the solar irradiance (L/D2, where L is the solar luminosity and D is the distance to the Sun). The distance at which a planet is at temperature T is proportional to 1/T2. Merely plug in the values of the upper and lower temperature to get the radii of the inner and outer radii of the habitable zone.

To do this correctly, you need to take into account a number of effects:

• As stars evolve on the main sequence, they become brighter and hotter. This makes the habitable zone move out with time. The Sun is now some 30% brighter than it was 4 billion years ago, and will eventually double that brightness before evolving off the main sequence. In the figure at left, the habitable zone moves outward between times t0 and t1. The region labeled CHZ is the continuously habitable zone. About 4 billion years ago, Venus was located near the inner edge of the habitable zone; today it lies closer to the Sun than the inner edge of the habitable zone.

• Albedo. Planets do not absorb all incident light; much gets reflected. The albedo is the fraction of incident light reflected, not absorbed. The albedo of the Earth is 0.37; that of Venus is 0.65; that of the Moon is about 0.12 (cloud is highly reflective, basaltic rock is not). You must multiply the solar irradiance by the albedo. This extends the inner edge of the habitable zone inwards.

• Greenhouse effect. Planets are not ideal black bodies. Carbon dioxide, water vapor, and other atmospheric gases are opaque in the near-IR (where the peak of the black body emission would be). A less-than-ideal radiator must be hotter than a black body to radiate the same amount of luminosity. This extends the outer edge of the habitable zone outwards.

The likelihood of finding a planet in the habitable zone depends on the area in the habitable zone. This is proportional to Do2 - Di2, where Do and Di are the outer and inner boundaries of the zone, respectively. Since D2 is proportional to the stellar luminosity, the area of the habitable zone, and the likelihood of finding planets in it, is largest for the massive O, B, and A stars on the upper main sequence. In the figure at left, the habitable zone (yellow) is plotted as a function of spectral type for main sequence stars. The planets of our solar system are indicated. Planets inside the "tidal lock radius" are tidally locked to the star, i.e., they rotate once per year, or a fractional number of times per year. Mercury rotates three times every two Mercurian years. (The Moon is tidally-locked to the Earth, and rotates once per month).