# Back-of-the-Envelope Calculations: The Distance Radio Waves Have Traveled

## Summary

*Question*

We have been broadcasting radio waves in all directions since the development of radio and television stations. How far could you be from the Earth and detect the faint signals of an early Star Trek broadcast? Have signals from Star Trek reached the nearest star yet?

## Assessment

*Answer*

Signals have gone out about 303 trillion kilometers, well past the nearest star. Here's how to figure it out. Remember that Star Trek broadcasts are sent out as electromagnetic radiation and therefore travel at the speed of light. You can figure how far they've gone by multiplying the speed of light (300,000 km/sec) by the number of seconds in 1 year (31,536,000) and multiplying that by 32 years to get about 303 trillion kilometers. But, you can save all that converting and calculating by using light years, the distance that light travels in a year. If we've been broadcasting for 32 years, then Star Trek broadcasts could be detected at a distance of over 32 light years, plenty of distance to be past the nearest star. Proxima Centauri is only 4.25 light years away (about 40 trillion kilometers). Our broadcasts haven't even come close to reaching the center of our galaxy, however—the center is 30,000 light years away! It will take another 29,968 years for Star Trek broadcasts to reach the center of the Galaxy...

## References and Resources

This SERC page describes the use of Back of the Envelope Calculations

A View from the Back of the Envelope (more info) : This site has a good number of easy simulations and visualizations of back of the envelope calculations.

The Back of the Envelope : This page outlines one of the essays in the book "Programming Pearls" (ISBN 0-201-65788-0). The book is written for computer science faculty and students, but this portion speaks very well to back of the envelope calculations in general.