Published on April 18, 2007

The Martin Theorem: A Continuation of the Drake Equation

Introduction

The Martin Theorem, also called the Martin Continuation, was hypothesized by Matthew Martin in 2007. It is an extension of the Drake Equation. The theorem conjectures that given a distribution of solar systems within the Milky Way galaxy, we can use Drake's Equation to limit those planets to a number that may contain intelligent life. Given that number, we can further estimate the probable time frame for when we may expect to receive return radio signals from those planets, assuming the civilizations have survived and are capable of sending and receiving electromagnetic transmissions.

Defining the Theorem

In defining this theorem, the following assumptions are made:

Radio waves leaving the surface of Earth travel outward at the speed of light.
Obstructions and deflections by extraterrestrial bodies (planets, moons, asteroids) and the curvature of space-time as represented by general relativity are not accounted for initially.
The distribution of planets is assumed to be roughly homogeneous throughout the Milky Way.

Refinement

To better reflect reality, a solid angle adjustment is introduced: electromagnetic emissions do not propagate uniformly in all directions. A simple approximation is to assume a hemispherical spread from Earth's surface rather than a full sphere, reducing the effective coverage volume by half.

Thus, for volume calculations:

V1 = \frac{2}{3}\pi t^3

V2 = \frac{4}{3}\pi D1^3

The effective ratio is adjusted accordingly.

Furthermore, a uniform distribution assumption across the galactic plane may be preferable to logarithmic scaling. This would better reflect real stellar distributions, which tend to be denser toward the galactic center and thinner at the edges.

If a uniform distribution is assumed, the time expectations can be recalculated linearly rather than logarithmically:

Tm = 2D1

Ta = 2 \left( \frac{D1 + D2}{2} \right)

Hypothesis

It is estimated that, using Drake's Equation, a probable calculation can be performed to determine average and maximum times it would take for an extraterrestrial civilization to respond if reached by Earth's radio transmissions.

Let

t = number of years since electromagnetic transmission from Earth began
D1 = distance in light-years to the most remote reachable solar system within the Milky Way
D2 = distance in light-years to the next reachable solar system beyond the current radius t
N = result of the Drake Equation (number of detectable civilizations)

Timeline

Question: When did our civilization first transmit electromagnetic waves?

Answer:

In 1878, David E. Hughes was the first to transmit and detect radio waves.
In 1925, John Logie Baird achieved the first television transmission.

We can reasonably use 1878 as our starting point. As of 2025, this is approximately 147 years.

Thus:

t = 147 \text{ years}

Typical assumed values:

D1 = 1000 \text{ light-years} D2 \geq t

Volume of Reached Space

Adjusted for hemisphere coverage:

Volume of space already reached by Earth's transmissions:

V1 = \frac{2}{3}\pi t^3

Total volume of space up to distance D1:

V2 = \frac{4}{3}\pi D1^3

The effective ratio of reached civilizations:

\text{Ratio} = \frac{V1}{V2} = \left( \frac{t}{D1} \right)^3 \times \frac{1}{2}

Distribution of Civilizations

Assuming N civilizations distributed uniformly across the galactic plane:

Number of civilizations already reached (W1):

W1 = N \times \left( \frac{t}{D1} \right)^3 \times \frac{1}{2}

Number of civilizations not yet reached (W2):

W2 = N - W1

Expected Contact Times

Maximum Expected Time (`Tm`)

The maximum expected return time (full out-and-back journey to D1):

Tm = 2D1

If adjusting for logarithmic scaling, optionally:

Tm = \frac{2D1}{\log_{10}(W2)}

Average Expected Time (`Ta`)

The average expected return time (halfway between t and D1):

Ta = 2 \left( \frac{D1 + D2}{2} \right)

Or, if using logarithmic scaling:

Ta = \frac{2\left( \frac{D1 + D2}{2} \right)}{\log_{10}(W2)}

Full Long Form (if logarithmic correction applied)

Ta = \frac{2\left( \frac{D1 + D2}{2} \right)}{\log_{10}\left(N - \left( \left( \frac{t}{D1} \right)^3 \times \frac{N}{2} \right)\right)}

Where:

log10 indicates base-10 logarithm.

Example Calculation

Let:

t = 147 years
D1 = 1000 light-years
N = 1000 civilizations

Then:

W1 = 1000 \times \left( \frac{147}{1000} \right)^3 \times \frac{1}{2} \approx 1.59 W2 = 1000 - 1.59 \approx 998.41

Thus:

If linear distribution assumed:

Tm = 2000 \text{ years}

Ta = 1147 \text{ years}

If logarithmic scaling used:

Tm = \frac{2000}{\log_{10}(998.41)} \approx 580.7 \text{ years}

Ta = \frac{1147}{\log_{10}(998.41)} \approx 333.1 \text{ years}

Conclusion

The Martin Theorem proposes a straightforward extension to the Drake Equation: by modeling the geometric propagation of Earth's electromagnetic emissions and assuming a distribution of civilizations, we can estimate both an average and maximum expected time frame to receive a reply.

Improvements to the original model, such as accounting for hemispherical emission and considering both uniform and logarithmic distributions, provide a more nuanced framework.

While heavily dependent on assumptions (such as the survival of civilizations and the clarity of signal paths), the model offers a rational and quantifiable framework to approach one of humanity's most profound questions: Are we alone?

References

Drake, Frank. (1961). "Project Ozma." Physics Today.
Hughes, David E. (1878). Early electromagnetic experiments.
Baird, John Logie. (1925). First television transmission experiments.
NASA ADS Database: Radio Wave Propagation and Interstellar Medium Studies.

The Martin Theorem

A Continuation of the Drake Equation