“The Drake Equation – Estimating the Number of Civilizations in the Milky Way Galaxy”

Students estimate the number of civilizations in the galaxy by first estimating the number of craters on the Moon and then by performing estimates of multiple-variable systems culminating in the use of the Drake Equation. In this three-part activity, students use estimation techniques to describe complex situations. First, students are given a close-up photograph of a small portion of the Moon’s surface. Using the scale provided on the image, students count the number of large craters in the image and extrapolate to find the number of such craters on our Moon. In the second part, students are given a list of variables that describe a particular population of students. Students estimate the portion of the population that match the given characteristics and answer questions about how their estimates change with alternatively defined variables. Finally, students utilize a form of the Drake Equation to estimate the number of communicating civilizations that exist in the Milky Way Galaxy. Students examine the range and definition of each variable comprising the Drake Equation and evaluate how changes in the variables influence their result.

STUDENT INSTRUCTION AND ANSWER SHEET

Part I – Exploration: How many craters are on the Moon?

Below is a 2.7 square kilometer (2.7 km²) image of the Apollo 14 landing site on the Moon. You can divide the Moon’s surface up into 14,000,000 such patches. Write a step-by-step plan for estimating the number of craters on the Moon that are larger than a football field. After you have shown your plan to your teacher, carry out your plan and compare your results to the class average.

File written by Adobe Photoshop® 5.2

Image Source: http://www.nasm.edu:80/APOLLO/AS14/a14landsite.htm

A. Record your step by step plan in the space below. What was your estimate for the number of football field sized craters?

B. Describe why your estimate might increase or decrease if a different picture of the Moon’s surface were used.

C. How would your estimate change if you were estimating the number of craters that are smaller than a football field or larger than your entire school including the parking lot and sports fields?

Part II – Concept Introduction: Making Complex Estimates

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. Complete the following estimation task.

PREDICTION: How many females in the 9^th grade with long hair are scheduled to be at lunch between 12:15 and 12:30 and are eating in the cafeteria and having french-fries with ketchup? __________________

TASK: To check your prediction, complete the following table by estimating the:

Variable	Estimated Value	Notes
n – total number of students in your school
f_f - fraction of females in your school
f_f,9 - fraction of those females in 9^th grade
f_f,9,L - fraction of those females in 9^th grade with long hair
f_f,9,L,t - fraction of those females in 9^th grade with long hair at lunch between 12:15 and 12:30
f_f,9,L,t,c - fraction of those females in 9^th grade with long hair at lunch between 12:15 and 12:30 in the cafeteria
f_f,9,L,t,c,FF - fraction of those females in 9^th grade with long hair at lunch between 12:15 and 12:30 in the cafeteria eating french-fries
f_{f,9,L,t,c,FF,k} - fraction of those females in 9^th grade with long hair at lunch between 12:15 and 12:30 in the cafeteria eating french-fries with ketchup
F - Fraction Of Total Population (the product of all the fractions)
T - Number of People at Your School Meeting The Criteria (product of fraction of total population F, and the total school population, n)
CLASS AVERAGE, T_avg

Reflection Questions about the Lunchroom Equation

T = n x ( f_f x f_f,9 x f_f,9,L x f_f,9,L,t x f_f,9,L,t,c x f_f,9,L,t,c,FFx f_{f,9,L,t,c,FF,k} ) = n x (F)

A. What value of T did you determine and how did it compare with the class average?

B. Give specific reasons why your value was different than the class average?

C. Predict how the calculation will change if you observe males instead of females. Check your prediction by performing the necessary calculation.

D. Explain how the value of T would be different if you changed your definition of long hair? Provide an example calculation with your written explanation.

E. Suppose that you were making this estimate for an all-female school where lunch is served from 12 to 1 pm. How would your estimate change? Why?

Part III – Concept Application: Using The Drake Equation

N = R_* x f_p x n_e x f₁ x f_i x f_c x L

In part II we estimated the number of students that had particular characteristics. In this activity we will use the same estimate techniques to discover the number of existing extraterrestrial civilizations that possess the technology to communicate beyond their home planet. Your task is to complete the table below and use those values to solve the Drake Equation in order to estimate the number of intelligent civilizations in the Milky Way. You might wish to review the Drake Equation Background Information Sheet before making your estimation. After you make the calculation, answer the reflection questions.

R - Number of target stars in the galaxy that: · are second generation stars with heavy elements · are hot enough to have a large habitable zone · have a long enough lifetimes for life to develop	R =
f_p - Fraction (percentage) of those stars with planets or planet systems.	F_p =
n_e -Number of "earth-like planets" in a planetary system that are at the right temperature for liquid water to exist (in the habitable zone).	N_e =
f_l - Fraction (percentage) of earth-like planets where life actually develops	F_l =
f_i - Fraction (percentage) of earth-like planets with at least one species of intelligent life	F_i =
f_c - Fraction (percentage) of earth-like planets where the technology to communicate beyond their planet develops	F_c =
L - "Lifetime" of communicating civilizations (years) - Note: This number must be divided by the age of the galaxy, 10 billion years when you make your final calculation.	L =

N - Number of communicative civilizations	N =

Reflection Questions about the Drake Equation

N = R_* x f_p x n_e x f₁ x f_i x f_c x L

A. What value did you get for the number of civilizations?

B. How does the value change if you double the lifetime of communicating civilizations?

C. How does the estimate change if we discover that only 1/3 of Sun-like target stars have planets?

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

E. Determine the most reasonable maximum and minimum values that your group believes the terms f_p, n_e, f₁, f_i, and f_ccould have. Record your values for each term below.

F. Calculate the range of values for “N” that result from using the maximum and minimum values that your group recorded in the previous question.

G. Do the maximum and minimum values that you calculated make sense to your group? Explain why you think they might be too large or too small or just right.

H. How many intelligent, communicating species in the galaxy do we actually know about? What then is the actual minimum value for “N.” Hint it is not zero. Explain your reasoning.

In this paragraph we will offer some values for several of the terms in the Drake equation that are often used by scientists when making these estimates. If we think that all stars that are like are sun have planets than we could estimate f_p = 1 to represent 100%. If we use our solar system as a model then there is only one planet in the habitable zone that we know has liquid water on its surface (Earth) so we could imagine setting n_e =1. Since Earth is the only planet in our solar system that we know to have developed life, it seems reasonable to set f_l = 0.1 to represent that about one out of every 10 planets has life. It is essentially impossible to know the fraction of species that develop on a planet that turn out to be intelligent and able to communicate so a conservative estimate for f_i and f_c that we might use is 0.1 for each term. As a rough guess we might imagine that across the galaxy intelligent communicating civilizations last for about 20,000 years out of the 10 billion year existence of the galaxy, which sets L = 2 x 10^-6_.

I. What value do you get if you use the estimates provided in the preceding paragraph? How does this value compare to your original estimate, your estimate for a maximum value, or your estimate for a minimum value?

CHALLENGE PROBLEM: Scientists recently discovered a massive gas giant planet orbiting the star 51 Peg. This planet orbits in the star’s habitable zone (where liquid water can exist). Describe how might this finding change your estimate.

Drake Equation Background Information Sheet

N = R_* x f_p x n_e x f₁ x f_i x f_c x L

R – This number represents how many billions of 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 400 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 life to develop. This eliminates stars warmer with spectral type F8 and warmer 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
400e9 * 10% * 75% = 30 billion stars.

f_p – 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.

n_e –This number represents how many "earth-like planets" there are at the right temperature for liquid water to exist (i.e. in the habitable zone). Recent discoveries suggest that we should also consider including moons around gas giant planets that are orbiting their central star in the habitable zone. A reasonable estimate for this number is difficult to imagine. In our solar system, the number ranges from one to three depending on if you include Venus or Mars. If Saturn were to migrate into the habitable zone, its 22 moons would make this number much larger.

f_l – This number represents the fraction of earth-like 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.

f_i – This number represents the fraction of earth-like 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.

f_c - This number represents the fraction of earth-like 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 – This number represents 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 may or may not choose to communicate beyond their own planet. Alternatively, when civilizations develop the technology to communicate they might simultaneously develop technology capable of making their environment uninhabitable (e.g., weapons of mass destruction). These factors make this number extremely difficult to estimate. L could range from only 100 years to many millions of years.