Math and the Final Frontier

"Albert Einstein (Walter Matthau): Tell me Ed, do you think we will find intelligent life on other planets?
Ed Walters (Tim Robbins): I am still not sure there is intelligent life on this planet." - From the movie
IQ

The beginning of August brought some interesting news to us amateur star gazers. A new planet was discovered in another solar system that could actually be reached by space ship in our life time. The Star Epsilon Eridani, a star about as big as the Sun, was found to have a Jupiter sized planet in orbit. The full story can be found here.

While dozens of extra solar planets have been discovered, what makes this one so interesting is that the star is a mere 10 light years away. It is also the first planet discovered around a star easily visible to the naked eye. Epsilon Eridani, as its name implies, is the fifth brightest star in the constellation Eridanus. To find it yourself, just locate the easy to find constellation Orion and use the following map:


Map is from a screen shot of Redshift 3 software. This time of year (september) requires getting up early before sunrise to see it. In winter, it will be visible in the evening.

This news, along with three other astronomy related news articles this month, got me thinking optimistically about space exploration in general, and those cold pessimistic mathematical formulas that say what little chance we have at being successful at it. I thought a Q&A approach might be best, so here goes:


Q: Could there be a planet like Earth around Epsilon Eridani?

A: Possibly. Stars with planets usually have more than one. This planet (a gas giant like Jupiter) is far enough away to allow the possibility of closer planets similar to Earth. That's the good news. The bad news is that the planet they did discover has an awkward orbit. That's how they discovered it. A large planet with such an awkward orbit could likely cause other nearby planets into awkward orbits. A stable near circle orbit is needed to support life.

Q: Could there be an advanced civilization in the Epsilon Eridani (EE) system?

A: No. The system is close enough to us that if an advanced civilization did exist there, we should be able to point a radio telescope there and pick up the latest news, sports, and weather (in Eridani language of course, and the news would be 10 years old). EE was targeted by Project Ozma (a precursor to Project SETI) at the National Radio Observatory in Green Bank, West Virginia during the summer of 1960. Results were negative, of course. Since there are no signals, there is no advanced civilization.

Q: Could there be any life around EE?

A: Maybe. The solar system of EE is a young one by the amount of gas we detect. Probably a couple of billion years younger than ours. If our evolutionary track is typical, then a life supporting planet around EE would be in the pre-cambrian period (single cell organisms). The important thing is that if such a planet does exist, it could be a possible colony world we could explore or settle in as little as a hundred years or so (it would take 50 years just to get there).

Q: When will we know if such a planet exists around EE?

A: Could be by the end of the decade. Another astronomy news article from August is this one, where scientists are proposing an upgrade to the Hubble Telescope which will increase its range ten times what it has now. If the proposal goes forward, visually spotting planets around close stars (and consequently finding out what kind of atmosphere they support via spectrum analysis), would be easy.

Q: If there is such a planet around EE, how do we get there and how long would it take?

A: Yet another recent article (click here) points out the increasing feasibility of plasma rockets that could provide constant acceleration for space travel as opposed to current technology that gets us going and we have to coast the rest of the way. Such rocket technology is being developed for future missions to Mars, cutting travel time more than half.

Q: Could such technology get us close to the speed of light?

A: Here is where the mathematical pessimism comes in to play. Popular wisdom aside, the speed of light is not the ultimate speed limit when it comes to space travel. In truth, we could never go faster than a third of the speed of light. The problem is that space is not a vacuum. It has trace amounts of Hydrogen everywhere. Even this little bit of hydrogen can cause friction on a space ship, friction generates heat, and heat melts everything eventually. Even the most heat resistant substances (ceramics) would melt if we travelled at even a third the speed of light. Plus you have the added burden of G forces on humans putting a limit on how fast ships can accelerate. Barring the discovery of exotic physics like those in Sci-Fi stories (warp speed, worm holes, etc.), interstellar travel will take centuries to accomplish.

Q: How many civilizations are there in the galaxy?

A: We can use the famous "Green Bank" formula (named after the location of Project Ozma mentioned above)to estimate. The formula is:

N = R* fp ne fl fi fc L where:

N = number of civilizations in the galaxy capable of interstellar communication.
R* = average rate at which stars develop during lifetime of galaxy.
fp = fraction of stars with planets.
ne = average number of planets per star capable of supporting life.
fl = fraction of potential life sustaining planets that actually have life.
fi = fraction of planets with life that have intelligent life.
fc = fraction of planets with intelligent life capable of interstellar communication.
L = average lifetime of society capable of interstellar communication.

Only the R* is well documented. Now that we can start detecting planets around other stars we may come to a consensus on fp. Everything else is educated guessing.

The measure we cannot even guess at is L. Even here on Earth we have only gone about 80 years since we could first send signals to outer space. If L is 10,000 years, then there may be about 1,000 other advanced civilizations in the galaxy. If L is more like 100,000 years, then there may be 10,000 other advanced civilizations.

Even assuming the most liberal estimates of 10,000 other advanced civilizations in the galaxy, we need to remember that the galaxy is 100,000,000,000,000 cubic light years big. Assuming an even distribution of these civilizations, there is one advanced civilization for each 10,000,000,000 cubic light years of space. Thus the average distance in light years between advanced civilizations is the cube root of this value, or 2200 light years. That grows to 4500 light years if there are only 1000 advanced civilizations.

On the other hand, according to the movie Contact the broadcast of the Berlin games in 1936 was the first transmission strong enough to reach outer space, and the A-bomb capable of destroying society was built only 9 years later in 1945. We can imagine that the value of L may be much smaller than 10,000 years.

Q: Speaking of sci-fi vs. mathematical pessimism, how is project SETI coming along?

A: Let me point out another recent article (click here) from Scientific American magazine that makes the case that the population of advanced intelligent life in the galaxy may not be much higher than the population of Earth. The view that there are about 1000 civilizations in our galaxy, goes against what current trends are telling us. This article is so good about explaining the mathematics of life in the universe, I wanted pull all sorts of interesting tidbits for this Q&A, but I though you should just read it for yourself.

Q: Does this mean we are alone in the universe?

A: There are approximately 100 billion galaxies in the known universe, each with approximately 100 billion stars each, chances are extremely good we are not alone. To quote Carl Sagan, "If we are alone, it would be an awful waste of space."

On the other hand there is this quote from the Hitchhikers Guide to the Galaxy by Douglas Adams:

The Universe... Population: None.

It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not everyone is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the universe is said to be zero. From this it follows that the population of the whole Universe is also zero, and any people you meet from time to time are merely the products of a deranged imagination. ("The Restaurant at the End of the Universe" pg. 148)

I'll let you find the mathematical mistake in that quote for yourself.

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