# Calculating the Number of ETIs

**Introduction **

The possibility of estimating the number of communicative and non-communicative extraterrestrial intelligent civilizations exists. Beginning with the Drake equation and modifying it to predict non-communicative ETIs, the number of ETIs in the universe will be predicted.

**Problem statement **

Given the existence of Earth civilization and the size of the universe it is perplexing that there has been no recognized ETI contact. The current model for predicting ETI behavior not only assumes an Earth-like civilization, it also assumes Earth's current level of technology represents a near epitome of development. This system model will make these same assumptions.

**System model **

ETI, Extra-Terrestrial Intelligence: Any anthropomorphic civilization capable of communicating by radio signals, whether intentional or unintentional.

Communication: Any artificial radio signal that is theoretically detectable and therefore considered an observable communication.

All ETI models draw heavily on Earth civilization as typical and so for the sake of argument, Earth civilization will sometimes be referred to as an ETI rather than as a special case of intelligent life.

**Related work**

Note: Here the term "ETI" is used in place of the original term "civilizations"

N = R* f_{p} n_{e} f_{l} f_{i} f_{c} L

where

N of observable ETIs in the galaxy

R* average rate of star formation

f_{p} fraction of stars with planets

n_{e} number of life capable planets per star with planets

f_{l} fraction of life capable planets that developed life

f_{i} fraction of living planets that developed into an ETI

f_{c} fraction of ETIs that are emitting detectable signals

L length of time detectable signals are emitted

This model predicts the number of observable ETIs in the Milky Way and can produce almost any answer depending on the assumptions for each term of the equation. The answer must be at least 1 since Earth civilization certainly exists and is communicative, albeit unintentionally. But the equation does suggest that with so many stars, even given low probabilities for the rest of the terms, an answer greater than 1 is almost certain.

**Solution **

The Drake equation can be divided into four parts. The rate of star formation, the likelihood of planets that can support life, the likelihood of life developing into observable ETIs, and the length of communication.

Extended to include all observable stars in the universe it becomes

N = R_{u} f_{lc} f_{i} L

where

N number of observable ETIs in the universe

R_{u} average rate of star creation in the universe

f_{lc} fraction of stars with life capable planets

f_{i} fraction of life capable planets that yield ETIs that become communicative

L average time during which ETIs are communicative

f_{lc} encompasses Drake's original terms f_{p} and n_{e}

f_{i} encompasses Drake's original terms f_{l}, f_{i} and f_{c}

L remains the same.

The average rate of star creation in the universe is

R_{u} = N_{s} / T_{u}

where

N_{s} number of stars in the universe

T_{u} age of the universe

Resolving L into two terms, length of a civilization and fraction of time spent communicating, yields

L = L_{ti} f_{tc}

where

L average time during which ETIs are communicative

L_{ti} average length of time an ETI exists

f_{tc} fraction of time an ETI is communicative during its existence

This equation yields an estimate of L that is exactly the same as in the original Drake equation but in two factors.

**The Number of Observable ETIs in The Universe **

Combining the above equations gives

N = N_{s} f_{lc} f_{i} L_{ti} f_{tc} / T_{u}

where

N number of observable ETIs in the universe

N_{s} number of stars in the universe

f_{lc} fraction of stars with life capable planets

f_{i} fraction of life capable planets that yield ETIs that become communicative

L_{ti} average length of time an ETI exists

f_{tc} fraction of time an ETI is communicative during its existence

T_{u} age of the universe

**The Total Number of ETIs in The Universe **

Assuming ETIs eventually become unobservable because their technology advances to a point where there are no emissions, the equation for total number of ETIs that exist is formulated by removing the term f_{tc}.

N = N_{s} f_{lc} f_{i} L_{ti} / T_{u}

where all terms are as defined in the previous section.

**Analysis **

This ETI model can not be experimentally verified because of the terms after f_{lc}. The f_{lc} term can be astronomically observed but even if life generated elements are detected in a planet's atmosphere it does not directly yield f_{i}. The f_{i} term is related to the odds of intelligent, communicative-capable ETIs developing, not microbial life. However, for this analysis it will be assumed that microbial life will always develop into an ETI eventually.

The most significant information that can be obtained from this analysis is a yes/no answer to the question of ETI existence. The large numbers, such as the number of stars, will be kept to the low end of the observations and estimated data will be kept to whichever extreme precludes ETIs.

**Estimate of Observable ETIs in the Universe**

N_{s} = at least 7 x 10^{20} stars

f_{lc} = at least 1 / 100 billion stars

f_{i} = assumed to be 1.0

L_{ti} f_{tc} = at least 100 years

T_{u} = no more than 14 billion years

Therefore

N = at least 50 observable ETIs in the universe

**Estimate of Total ETIs in The Universe **

N_{s} = 7 x 10^{20}

f_{lc} = 1 / 100 billion

f_{i} = 1.0

L_{ti} = at least 100,000

T_{u} = 14 billion

Therefore

N = at least 50,000 ETIs in the universe

**Conclusions **

Notice the low probability of life capable planets used. This was done to weigh against the huge number of stars. Even when the number of stars in the universe is set to the low end of observations, it is extremely large. However, the low probability of life capable planets used does mean there would likely be only one ETI in the entire Milky Way, specifically Earth.

The analysis section assumed microbial life always evolves into an ETI since this is true of Earth. It sets the length of communication period to 100 years since Earth civilization has been communicating roughly that long. The length of ETI existence is estimated at Homo Sapien's current 100,000 years. Considering the estimates used, these results for N are large enough to be virtual certainties, but small enough to be unobserved. The fact that the answer is always greater than zero, despite the extreme estimates used in the modeling factors, means Earth civilization can be presumed to be one of many ETIs.

This model fails to use current trends to predict Earth civilization's future development and apply this prediction to the ETI model. It is unlikely that current Earth civilization is near the epitome of development.

For a theory of how ETIs may develop, read The UETI Model.