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# Drake Equation

Our sun is one of 100 billion stars in our galaxy. Our galaxy is one of billions of galaxies populating the universe. It would be the height of presumption to think that we are the only living things in that enormous immensity.

-- Wernher Von Braun

I think the surest sign that there is intelligent life out there in the universe is that none of it has tried to contact us.

-- Calvin (Bill Watterson in Calvin and Hobbes)

## Overview

• Develop estimation techniques
• Perform extrapolations from real data
• Examine the range and definition of each variable from the Drake Equation
• Evaluate how changes in the variables of the Drake Equation influence the outcome

## Introduction

### Estimating the Number of Civilizations in the galaxy

There are many instances in science where estimation is much more useful and efficient than counting. In particular, estimation techniques are important when analyzing a system for which counting is not actually possible. For example, it usually isn't practical to actually count the number of people who attend an event on the Diag. There are multiple entrances, and many people are just passing through. However, you can count the number of people in a small area, and multiply by the area where people are standing to get an estimate of how many people are there.

Sometimes, estimation also helps define what it is that we need to know. Estimating the number of civilizations in the galaxy is very complex, and involves several unknown variables. We can't really arrive at a good estimate, but the effort to do the estimate forces us to identify the things we want to learn.

The Drake Equation sets out, in equation form, many of the most important things we need to know in order to determine how many civilizations there might be in the galaxy with whom we could communicate. It is important to note that there are a number of other variables that could be considered when making this estimate that we will not include in our calculations.

The Drake equation:

Nc = ns * fp * nh * fl * fi * fc * L

Nc is the number of existing extraterrestrial civilizations in the Milky Way Galaxy that possess the technology to communicate beyond their home planet.

ns - Number of Stars: This number represents how many stars in the galaxy meet the following two criteria:

1. The star must be a second or third generation star formed from an interstellar cloud that included the necessary heavy elements for life (e.g., carbon, oxygen, etc.). The elements are created during the evolution of first generation, super-massive stars and supernova events that occurred early in the history of our galaxy. A reasonable estimate for this number is 200 billion stars.
2. The star must release enough energy to have a sizeable habitable zone. A habitable zone is the region around a star where liquid water could exist on an orbiting planet. 90% of the stars in our galaxy are too cool to have a sizable habitable zone. This eliminates stars with spectral type K5 and cooler. Of the remaining 10%, nearly a quarter of those have lifetimes too short for complex life to develop. This eliminates stars warmer with spectral type F8 and wanner as they have lifetimes shorter than 4 billion years.

Our Sun, a G2 star, fits both of these categories and thus is one of the target stars. Such target stars are often referred to as Sun-like stars. A reasonable estimate for the number of target stars is: 2 x1011 * 10% * 75% = 15 billion stars.

fp - Fraction with Planets: This number represents the fraction of those stars meeting the above criteria that also have planets or planet systems around them. Recent discoveries of numerous extra-solar planets suggest that most stars like our Sun probably have planets, so many astronomers argue that this number should be close to 1.

nh- Number of Habitable bodies: This number represents how many planets and/or moons there are at the right temperature for liquid water to exist (i.e. in the habitable zone). A reasonable estimate for this number is difficult to imagine. Based on our solar system, some reasonable numbers would one (Earth), two (Earth and Moon are both in the habitable zone, and if you were looking at it from another system it wouldn't be easy to tell the Moon had no water), four (adding Venus and Mars, which are on the edge of the habitable zone) or about half a dozen (adding a couple moons of Jupiter and Saturn which are tidally heated so they could potentially have liquid water). Note if either Jupiter or Saturn were close to the habitable zone, like many of the extrasolar planets, many of their moons would have to be included, pushing the number up to a couple dozen.

fl - Fraction where Life develops: This number represents the fraction of these planets where life actually develops. Some scientists believe that the evolution of life is inevitable when the conditions are right. Alternatively, we only know of one instance where life has successfully developed (Earth), therefore it is difficult to estimate this fraction.

fi - Fraction where the life is Intelligent: This number represents the fraction of these planets where at least one species of intelligent life evolves. Intelligent life could develop early on some planets and later on others and therefore again it is difficult to estimate this fraction.

fc - Fraction with long distance Communication: This number represents the fraction these planets where the technology to communicate beyond the planet exists. In our own civilization, we have been using television and radio signals for nearly a century. These signals have leaked into outer space and might be detectable by extraterrestrial civilizations. As before, it is extremely difficult to estimate this number.

L - Lifetime of the civilization: This number represents the fraction of the number of years that communicating civilizations have existed out of the total lifetime that the galaxy has existed. We call this fraction of years "Lifetime." This number depends both on social issues and technological issues. It is possible that intelligent civilizations elsewhere in the galaxy have existed for millions of years and mayor may not choose to communicate beyond their own planet. Alternatively, when civilizations develop the technology to communicate they might simultaneously develop technology capable o r making their environment uninhabitable (e.g., weapons of mass destruction). These factors make this number extremely difficult to estimate. L could range from 1 x 10-8 (100 years/10,000,000,000 years) to 1 x 10-4 (millions of years / 10,000,000,000 years) or more.

# Drake Equation Worksheet

1. Estimate the value for each of the terms in the Drake equation (explained in the introduction.) Enter the value, and an explanation for how you arrived at that estimate in table 1. In the last column, enter a number 1 - 5 for how reliable you think the source of information is. Use a scale where 1 is unreliable (you're not sure you can trust the source or your not sure the source has done all the required research), 3 is neutral, and 5 is very reliable (the source has done all the research and given you all the information, though the answer may not be very certain.)
Table 1: Estimates for the Drake Equation
Term Estimate Explanation Reli- ability
ns

fp

nh

fl

fi

fc

L

2. Calculate Nc. Show your work.

3. Compare your answer with two other groups. (If no other groups are finished, you can start on the conculding questions)
1. Record the values of fp, nh, fl, fi, fc L and Nc for the other groups in table 3.
Table 3, max and min values
Term group 1 group 2
fp

nh

fl

fi

fc

L

Nc

2. How did the other groups' numbers compare to yours - are the results similar, very different, pessimistic, optimistic, etc.? Which terms caused the biggest differences? Why? Which terms caused the least differences? Why? (note you may have to ask the other groups how they arrived at their estimates to answer these questions.)

### Concluding Questions:

1. Give at least one example where the ability to do an estimate like this would be useful. Also, state whether it is useful because it gives you a valuable number, or because it helps identify what you need to know.

2. The Drake equation is an estimate of the number of civilizations in the galaxy that have the potential for inter-planetary communication. What is the minimum value for this number? How do we know it?

3. How would you change your estimate if we discovered that early life developed on both Mars and Europa?

4. The stars that go into ns are almost exclusively in the disk of the galaxy, which has a radius R of about 5x104 light years. If the civilizations are distributed randomly in the disk we can set up a relation related to the density of civilizations in the galaxy: Nc/R2 = 1/r2 where r is the average between adjacent civilizations. Assuming our closest neighbor really is the average distance away, how long would it take a radio signal from us to reach them? Show your work and explain your result.
(Side note on how we arrived at the relation Nc/R2 = 1/r2 : we start with the ratio of the total number of civilizations in the entire area of the galactic disk, Nc / Agalaxy. The result tells us what area will only contain 1 civilization on average, or 1/ Aaverage. The disk is a circle, so Agalaxy= πR2 and Aaverage= πr2. If you set the two ratios equal and replace the areas with the equations for the areas, π gets canceled, leaving you with the relation above.)

Last updated: 6/13/13 by SAM