The Drake Equation, Fermi Paradox, and SETI


Drake Equation

N = R x fs x ne x fl x fi x fc x L

R: average rate of star formation
fs: fraction of stars with planetary systems
ne: numbers of planets in "habitable" zones
fl: fraction of planets where life develops
fi: fraction of planets where life becomes intelligent
fc: fraction of intelligent civilizations which want to communicate

L: lifetime of "communicative life"

People have long wondered whether we are unique or not. Frank Drake of Cornell University formulated what is now known as the Drake Equation in 1960 as a way to open the discussion on the possibility of intelligent life in the galaxy. At left is the Drake Equation and the meaning of the terms in the Drake Equation. We consider each term in the Drake Equation and come up with an answer as to the number of intelligent civilizations likely to exist in the Milky Way galaxy. The terms in blue are probabilites and facts which can be deduced from astrophysics. These terms can, in principle, be determined quantitatively. The sections in green are probabilities and numbers which come from biology, psychology, and sociology. These are much more difficult to estimate.


Rate of star formation x Lifetime = # of Stars in the Galaxy:

The number of stars in the Galaxy is determined by the rate at which stars are formed and their lifetimes. We argue why this is so below but first, we give the stellar birthrate for our Galaxy.

As an initial guess, the Milky Way galaxy contains 200 billion stars. If the Milky Way galaxy is ten billion years old, then the average rate of star formation has been

R = 20 stars per year

over the lifetime of the Galaxy. This is roughly correct as most measures for other galaxies suggest that around 10 Solar masses of material goes into star formation per year in typical galaxies. Estimates for our galaxy run from several stars per year to tens of stars. Although, in principle, the star formation rate is potentially the most reliable of the terms in the Drake Equation, it is currently uncertain.

Number of Stars = Rate of Star Formation x Lifetime?

At birth, the galaxy contains no stars. If the rate of star formation is 1 per year and the lifetime of a star is 10 years, let's consider how the stellar population evolves with time.

  • After 1 year, there will be 1 star in the galaxy.
  • After 2 years, a second star will be born and there will be 2 stars in the galaxy.
  • After 3 years, a third star will be born and there will be 3 stars in the galaxy.
  • This continues for 10 years, after which there will be 10 stars in the galaxy.
  • What Happens after the Eleventh Year? Well, an eleventh star was born, but because the lifetime of a star is 10 years, the first star born dies; there remains 10 stars in the galaxy.
  • From hereon, every time a star is born, one dies and the number of stars remains fixed at 10, which we note is

    Number of stars = Rate of Star Formation x Lifetime
The appropriate value for the Lifetime will be discussed later.


Fraction of stars with planetary systems:

Beta Pictoris is 62.9 light years from the Sun and harbors an A5 V star surrounded a dusty disk which contains the so-called planet b, a Jupiter-sized planet, mass ~ 8 times that of Jupiter, whose orbit is like that of Saturn, 7.6-8.7 Astronomical Units. The bottom images show Planet b as it moved from one side of the disk to the other.

We know that approximately 60 % of all stars are in multiple star systems. Out of double star systems a handful are known to contain planets. Consequently, we suggest that the fraction of stars with planetary systems may be as large as 40-50 %. This, however, assumes that all places where planets can form that they actually do form. Support for the idea that planetary formation is natural and common has gotten stronger over the last 20 years. For a discussion of the techniques used to discover planets go to the NASA Planet Quest link above or here. Around 5,000 confirmed extra-solar planets have been discovered in nearly 4,000 distinct planetary systems. This suggests that when planetary systems can form, they do form and we estimate that the fraction of stars that have planetary systems is (optimistically)

fp = 0.5


Number of habitable (Terrestrial) planets per planetary system:

In our Solar System,

The planets are divided into three classes, the Terrestrial planets, the Jovian planets and the dwarf planets. Life As We Know It (LAWKI) requires that a planet have conditions such that they have liquid water (oceans). For the Solar System when we consider oceans that form on the surfaces of the planets we get the traditional Habitable Zone which stretches from around 0.7 Astronomical Units to (optimistically) 2 Astronomical Units. This range includes Venus, Earth, and Mars.

The size of the habitable zone as a function of the central star. The habitable zones of low mass stars fall near the stars. For our Solar System, Venus, Earth, and Mars fall within the habitable zone. Today, only the Earth has liquid oceans and so falls within this narrowly defined habitable zone. Why is this so?

The notion of liquid oceans is key to the size of the Habitable Zone. Without liquid oceans, it is expected that our atmosphere would have experienced a Runaway Greenhouse Effect (as did Venus) driving our surface temperature up to many hundreds of degrees Farenheit (on Venus, the surface temperature is 800 to 900 F) or that we could not have maintained a thick atmosphere and we would be an airless planet (so, why is Mars at a distance of 1.5 Astronomical Units from the Sun have an atmosphere which is only 1 % the mass of the atmosphere of the Earth?).

Implicit in the above discussion is that there are Earth-like planets (which are necessary for LAWKI) in the extra-Solar planetary systems which have been discovered. Presently, the number of Earth-like planets in habitable zones is around 30 (exoplanets.org); most planets which have been discovered are referred to as super-Earths and Neptune-like. To explore this and other questions, go to the website Exoplanets.org and use the plotting tool to see the discovered exoplanets and exosystems compare to us.

Artist's rendition of the two Saturn-sized planets in orbit about their star, named Kepler-9. The planets were named Kepler-b and Kepler-9c.

Important work for discovering Earth-like planets is being performed by the NASA satellite Kepler. Kepler announced the discovery of two transiting planets in one planetary system. The two planets are around the mass of Saturn (~1/3 that of Jupiter) with tight orbits, periods 19 days and 38 days. The orbit of Saturn is ~9.6 Astronomical Units and Saturn's orbital period is around 29.5 years. Interestingly, there is evidence for a third planet with Earth-like properties, 1.5 times the mass of the Earth but in an orbit which is only 1.6 days long. The planet is not in the Habitable Zone, however, it orbits way too close to its parent star.

In our Solar System, we have 1-2 planets in our traditional Habitable Zone. The overall situation is not clear because our view of the habitable is changing; we now include the moons of Jupiter and Saturn Europa and Enceladus within the Habitable Zone of our Solar System. Let's set (optimistically)

ne = 4



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