bio engineering

Assignment 2: Big Numbers and Fermi Paradox (270 pts)

Goals: (1) understand orders of magnitude and (2) use them to address Fermi’s question

Preface

Big fleas have little fleas upon their backs to bite 'em,

And little fleas have lesser fleas, and so, ad infinitum .

And the great fleas, themselves, in turn, have greater fleas to go on;

While these again have greater still, and greater still, and so on.

— British mathematician and logician Augustus De Morgan

1 Quantities, Measurement, and Orders of Magnitude

Quantity is a property specifying amount or measure compared in terms of "greater", "less", "equal", or by assigning a numerical value in terms of a unit of measurement.

We use the International System of Units (SI), today’s metric system, for global communication, integration, and scientific endeavor.

The extent of a quantity can be described in terms of its scale.

An order of magnitude is a scale of 10. Thus, a number’s "order of magnitude" is the smallest power of ten needed to represent its quantity. Two quantities within about a factor of 10 of each other are said to be "of the same order of magnitude" [ source ]. That is, if either number can be multiplied or divided by a number less than 10 to equal the other, they are within the same order of magnitude.

Starting from the base unit (eg. 1 gram or 1 meter), we could go x orders smaller () or larger (), as shown along the columns in Table 1. Mathematically, there is no limit to how large or small something could be (x → ∞), but we are yet to confirm our physical limits.

Orders of magnitude are useful for estimating when details are unnecessary for our purposes, or when details are literally impossible to obtain (such as the size of the universe or quantum foam) [ source ].

Numeric Examples: The order of magnitude of 2500 is 3. We can see 2500 = 2.5 × 103 (in scientific notation ). So, we roughly compare 2500 to somewhere in the 103 = 1000’s, as opposed to 25 which is near the 101 = 10’s -- two orders smaller.

40 is of the same order of magnitude as 25.

These are somewhat trivial examples; orders become more useful when we start comparing extremely large or small quantities.

Table 1: Comparing some masses and lengths in orders of magnitude [ source ]

2 Measurement Errors and Bias [ source ]

As we observe and measure quantities, we must consider measurement flaws to keep errors to a minimum and refine our conclusions. Measurement errors in our data may show up as random error, measurement error, and measurement bias.

Measurement error results from flaws in the measurement instrument and human mistakes (e.g., misreading dial, incorrectly writing a number, not observing an important event, misjudging a particular behavior), thus contributing to the variability of the measurement and results of a study.

Random error is a nonsystematic measurement error beyond our control. Its effects average out over a set of measurements. In the presence of random noise, the average over several measurements should neither systematically overestimate nor systematically underestimate the pure (noise-free) value. For example, a scientist bumping a table, or small temperature fluctuations from an open door, or another machine vibrating or heating the area, can all change data slightly. This is importantly unbiased- it has nothing to do with the sample itself and is about equally likely to affect all samples, so it does not cause large problems or favor a result.

In contrast, measurement bias, or systematic error, favors a particular result. A measurement process is biased if it systematically overstates or understates the true value of the measurement. Bias can be produced by the observer (Anthropic Principle, Selection bias) or a measurement device, leading to unreliable results. The fishnet in the lake is an example of measurement bias- your measurement method impacts the accuracy of the result in a way of making the result seem larger. Anthropic Principle states that observations of an event must be compatible with the observer. In other words, you can’t observe phenomena if you’re not there to see it. For example, if our home planet or the universe were lifeless, we would not be there to observe anything. Being alive, this gives our observations an inherent bias about the prevalence of life in the universe.

Speaking of bias, often we want to extend our interpretation of data beyond the particular sample studied. Such generalizations are only valid if the sample is representative of the population. A representative sample has sample members with relevant characteristics generally the same as the characteristics of the population. If not, our sample may bias our generalization. [ source ]

2 The Drake Equation

Originally put forth by astronomer Frank Drake, the Drake Equation is one method of estimating the number of detectable civilizations which may exist in our galaxy. [ great source ]

Essentially: you need stars, with planets, suitable for life, where life actually occurs, and becomes intelligent, capable of communications, and doesn’t destroy itself too soon.

The equation looks as follows:

N = R* × fp × ne × fl × fi × fc × L

Where:

N -- the number of civilizations in the galaxy whose electromagnetic emissions are detectable

R* -- the rate of star formation per year in the galaxy

fp -- the percentage of stars that form planets

ne -- the average number of planets for each star with conditions suitable for life

fl -- the percentage of those planets on which life occurs

fi -- the percentage of those planets where intelligence develops

fc -- the percentage of intelligent species that produce interstellar communications

L -- the average lifetime of a communicating civilization

Over the years, by filling in numbers for the seven parameters R*, fp , …, L, scientists put forth many estimates for N: some very optimistic and others conservative. This lead to very large answer variations -- from 100 million to less than one.

These variations can be attributed to high uncertainties about parameters and large windows for error and bias.

3 Fermi Paradox: "Where is everybody?"

Optimistic estimates for the Drake Equation lead us to Italian astrophysicist Enrico Fermi’s question:

If intelligent life is not unique in the galaxy, why have we not had contact with alien species?

Hypothetical explanations for the paradox include Filters, doubts about Von Neumann Probes, Simulated Reality, the Zoo Hypothesis, that human beings have not existed long enough, that we are not listening properly, that advanced civilizations tend to isolate themselves, etc.. A longer list can be found here .

View Parts 1 and 2 of “The Fermi Paradox” by Kurzgesagt – In a Nutshell

Part 1: https://youtu.be/sNhhvQGsMEc

Part 2: https://youtu.be/1fQkVqno-uI

You might also want to read the “Wait but Why?” blog on the Fermi Paradox:

https://waitbutwhy.com/2014/05/fermi-paradox.html

Activity 1

To get a sense of magnitude, we will observe and compare the relative sizes of objects using an interactive viewer.

Start by opening “Scale of the Universe 2”: http://htwins.net/scale2/

(Note: This requires the latest version of Adobe Flash Player. If you are still unable to view, make sure to enable flash player in your browser)

Tip: To zoom, use left and right arrow keys or move the bar.

Click on objects for more information.

For recording lengths/sizes answers, use scientific notation.

1) How many Planck lengths can fit in a human length? What is the order of magnitude of your answer? (You can find the Planck length inside the viewer by zooming in to the very end) (10 pts)

Size of human: __× 10n m Size of Planck length: We can fit ___ Planck lengths in a human length. The order of magnitude is __.

2) Currently, how does a Transistor compare to DNA width in orders of magnitude? How does a Transistor compare to a cortical neuron? (20 pts)

Cortical neuron: 2 × 10-7 m (not pictured in the game)

DNA width: Transistor gate: Cortical neuron: Comparison:

3) How many orders of magnitude larger is the Earth compared to a human? (10 pts)

Size of human: Size of Earth: Earth is ____ orders of magnitude larger than a human.

4) List 4 objects in the viewer on the same order of magnitude as Pluto (2.5 × 106 m). (20 pts)

5) List 2 objects in the viewer on the same order of magnitude as a Light-Day (2.6 × 1013 m). (10 pts)

6) How many meters are there in a light-year? How many meters across is the milky way? Combining these, how many light-years does it take to cross the Milky Way Galaxy? You may write the light-years without scientific notation. (10 pts)

Light-year: Milky Way Galaxy: Crossing the Milky Way would take approximately ____ light-years.

7) In the viewer, find the bar representing the distance in light years from the Milky Way to the Andromeda Galaxy. How many light-years does it take to travel from the Milky Way to Andromeda? How does this compare in orders of magnitude to the light years needed to cross just the Milky Way (your answer from question 4)? (20 pts)

Distance between Milky Way and Andromeda: ____ light-years Distance across the Milky Way: ____ light years Comparison:

8) How many orders of magnitude smaller is Milky Way Galaxy compared to the Observable Universe? (10 pts)

Milky Way Galaxy: Observable Universe: The Milky Way Galaxy is ____ orders of magnitude smaller than the Observable Universe.

Activity 2

Now, let’s use our intuition of big numbers to calculate the possibility of intelligent communicating life existing in our universe.

In 1961, Drake and his colleagues used the following 'educated guesses' ( source ):

· R∗ = 1/yr (1 star formed per year, on the average over the life of the galaxy; this was regarded as conservative)

· fp = 0.2 to 0.5 (one fifth to one-half of all stars formed will have planets)

· ne = 1 to 5 (stars with planets will have between 1 and 5 planets capable of developing life)

· fl = 1 (100% of these planets will develop life)

· fi = 1 (100% of which will develop intelligent life)

· fc = 0.1 to 0.2 (10–20% of which will be able to communicate)

· L = 1000 to 100,000,000 years (which will last somewhere between 1000 and 100,000,000 years)

For Questions 1 and 2, refer to the estimates from 1961 shown above. Write the equation with numbers you used and the final answer.

1) For the given parameters, what is the most optimistic value of N? (40 pts)

2) For the given parameters, what is the pessimistic value of N? (40 pts)

3) How many orders of magnitude does the optimistic differ from the pessimistic estimate? (10 pts)

4) Based on your response to question 3, what does this say about our certainty regarding N? (10 pts)

5) Explain how Anthropic bias may contribute to flaws in our estimates for extraterrestrial intelligent communicating lifeforms. (60 pts)

Just for fun: https://xkcd.com/718/

Related to Class Discussion: Bonus video on Boltzmann Brains and Entropy

Extra Credit:

Research the Internet and Google Scholar for other estimates of the parameters for the Drake equation. Quote your sources and state why you believe they are better estimates than those used by Drake in 1961. (30 pts)

Blog questions: (30 pts)

1. Are we alone in the universe? Why is there life now? (30 pts)

Full 30 points are earned by participating in answering this Blog or commenting 5 times on other student’s responses.