Rare Earth equation
The following discussion is adapted from Cramer (2000). The Rare Earth equation is Ward and Brownlee's (W&B) riposte to the Drake equation. It calculates N, the number of Earth-like planets in the Milky Way having complex life forms, as:
N = N* x ne x fp x fpm x fg x fi x fc x fl x fm x fj x fme.
N* = number of stars in the Milky Way. This number is not well-estimated, because the Milky Way's mass is not well estimated. Moreover, there is little information about the number of very small stars. N* is at least 100 billion, and may be as high as 500 billion, if there are many low visibility stars.
ne = average number of planets in a star's habitable zone. For reasons given above, this habitable zone is fairly narrow, constrained by the requirement that the average planetary temperature be consistent with water remaining liquid over enough time to permit complex life to evolve. Thus ne =1 is likely to be a generous upper bound.
Hence we assume N* x ne = 5x1011. The rare Earth hypothesis, operationally speaking, asserts that the product of the other 9 Rare Earth equation factors listed below, all fractions, is no greater than 10-10 and could plausibly be as small as 10-12, in which case N could be as small as 0 or 1. Ward and Brownlee do not actually calculate the value of N, because the numerical values of quite a few of the factors below is conjectural at best.
fg = fraction of stars in the galactic habitable zone.
fp = fraction of stars in the Milky Way with planets. All known extrasolar planets orbit metal-rich stars, suggesting that planets may be peculiar to metal-rich stars.
fpm = fraction of planets that are rocky ("metallic") rather than gaseous.
fi = fraction of habitable planets where microbial life arises. W&B believe this fraction is unlikely to be small.
fc = fraction of planets where complex life evolve. For 80% of the time since life first microbial life appeared on the Earth, there was only bacterial life. Hence W&B argue that this fraction may be very small. Moreover, the Cambrian Explosion, when complex life really got off the ground, may have been triggered by extraordinary climactic and geological events.
fl = fraction of the total lifespan of a planet during which complex life is present. This fraction cannot be high because complex life takes so long to evolve.
fm = fraction of habitable planets with a large moon. The Earth's moon is doubly unusual. First, the other rocky planets in the Solar System either have no satellites (Mercury and Venus), or have tiny satellites that are captured asteroids (Mars). Second, the Moon as a fraction of its planet is much larger than any other satellite in the Solar System. If the giant impact theory of the Moon's origin is correct, fm will be small.
fj = fraction of planetary systems with large Jovian planets. This fraction could be large.
fme = fraction of planets with a sufficiently low number of extinction events. W&B argue that the low number of such events the Earth has experienced since the Cambrian explosion may be unusual. Such a low number again requires a very stable planetary system, with outer planets having nearly circular orbits, and no gravitational perturbations from passing stars.