Archive for January, 2010

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Q: Is it possible to choose an item from an infinite set of items such that each one has an equal chance of being selected?

Sunday, January 31st, 2010

The complete question was: The other day I was trying to explain the difference between “impossible” and “with probability zero” to a friend. I remarked “if you pick an integer, and I pick an integer, the mathematical probability that we picked the same number is zero, but it’s certainly not impossible.” A seemingly harmless statement, but only later did I think, what does it even mean to “pick ANY integer with EQUAL probability?” Is there any meaning to a random variable that can take on an infinite number of values with equal probability? It would be like a uniform distribution that has one or both bounds stretched to infinity. Does this kind of object have a name? What kind of mathematical properties would this even have? Oh dear, this is blowing my mind just thinking about it. Please help!

Mathematician: Developing a consistent theory of probability for sets with an infinite number of elements (like the set of all integers, the set of all real numbers, or the set of real numbers between 0 and 1) requires dealing with a handful of subtle and tricky problems. In fact, to determine whether it is possible to choose one object from an infinite number of objects such that each object has the same probability of being chosen, we must delve into what we really mean by “possible”. Various interpretations of this question are:


1. Can we work with probabilities (as they are defined in a formal, mathematical way) on infinite sets of items, and if so, can we assign equal probabilities to each item?

2. Can a (formal, mathematical) procedure be devised to sample uniformly at random from an infinite set?

3. Can an algorithm be designed that can carry out this sampling procedure on a real computer such that the algorithm will terminate in a finite amount of time?

Each of these questions, which I’ll address one by one, raises interesting considerations.

Can a formal theory of probability be developed for infinite sets of items? Absolutely. For example, the Gaussian distribution (often known as the bell curve or normal distribution) is defined on the set of real numbers (so when you draw a sample according to this distribution, you get some real number). Strictly speaking, it assigns a probability of zero to each of the individual real numbers, but assigns a non-zero probability to subsets of real numbers. For example, if we sample from a Gaussian distribution, there is a formula that can tell us how likely the number is to be less than any particular value X (so the set of all numbers less than X is assigned a positive probability). Why does the gaussian distribution not assign non-zero probabilities to the actual numbers themselves? The reason (loosely speaking) is because the probability of getting ANY number when we sample from a distribution must be 1 (which just means that some number must always occur). On the other hand, the set of real numbers contains so many numbers that, if each of them had a non-zero probability of occurring, it would not be possible for the total probability (which is, essentially, just the sum or integral of all of the numbers’ probabilities) to be 1. Another, more intuitive way to think about this is to consider a dart board. If we throw a dart at the board and are willing to assume that matter is infinitely divisible (sorry, Physicist) then we will always hit some point on the board (assuming our aim isn’t too terrible). But, at the same time, the chance of hitting any particular point is negligibly small (zero in fact) since there are so many possible points. So clearly, to describe the probability of hitting this board’s points, it is not sufficient to consider only the probability of each individual point being hit, but rather we have to consider how likely we are to hit various regions of the board, such as the region constituting the bullseye. Even though each particular point has a zero probability of being hit, some point is always hit in practice, and the set of points that make up the bullseye together have a positive probability of being hit with each throw.

Okay, so we can define probabilities on an infinite set, though as the Gaussian distribution case shows, we may have to actually let the probabilities be assigned to subsets of our original set, rather than to every object in the set itself. But can we do this in such a way that every object in our set is sampled with equal likelihood? The answer is again yes, though with some caveats. For instance, if we limit ourselves to the real numbers that are between 0 and 1, we can assign a uniform distribution to these numbers which will give them each an equal probability. The uniform distribution basically says that the probability that a given sampled number  will be between a and b (for 0<a<b<1) is equal to b-a. This fact implies that all numbers are equally likely to be sampled from this distribution.

Fine, but can we define a probability distribution on the set of integers (rather than the real numbers between 0 and 1) such that they each occur with equal probability (i.e. does a uniform distribution on the integers exist)? The answer, sadly, is no. A probability mass function (which is the kind of probability distribution we need in this case) is defined to be a positive function that has a sum of values equal to 1. But any positive function that assigns an equal value to each integer must have probabilities that sum to either infinity or zero, so the desired distribution is impossible to construct. As a technical side note though, people sometimes try to get around this issue in Bayesian analysis by applying what are known as improper priors. Attempting to define a uniform distribution on the full set of real numbers also fails, for a very similar reason that it doesn’t work on the integers (it can not be the case that each real number (or equally sized interval of real numbers) has the same probability and the probability density function integrates to 1).

On to our second question of whether is it possible to come up with a formal mathematical procedure for sampling from infinite sets. The answer is yes, if we have an unlimited amount of time to spare. For real numbers between 0 and 1, we can use the following sampling procedure:

i.  Start with the number 0.0, and set n=1

ii. Set the nth digit of our number after the decimal point to a random number from 0 to 9.

iii. Increase n by 1 and return to step ii.

If this procedure were iterated forever, it would produce a single random number between 0 and 1, and all real numbers between 0 and 1 are equally likely to be generated. Essentially we are just constructing a number by choosing each of it’s decimal digits randomly.

But what if we wanted to carry out this procedure for the set of all integers? Here things get stranger. The natural choice is the following procedure:

i.  Start with the number 0.0, and set n=1

ii. Set the nth digit of our number BEFORE the decimal point to a random number from 0 to 9.

iii. Increase n by 1 and return to step ii.

If this procedure were carried out forever, it would seem as though it would produce an integer, and that this integer would have an equal probability of being any integer. This is true, in the sense that all integers that the algorithm produces have an equal likelihood of being produced (i.e. when it DOES produce integers those integers are each equally likely). But the algorithm does not actually do what we would like. We begin to see the problem when we pick any integer X, and ask the question, “what is the probability that this procedure would produce a number that is greater than X?”. The answer, is that there is a probability of 1 (i.e. a 100% chance), NO MATTER WHAT X IS. This makes sense, given that the integers stretch off to infinity, and that the number of integers close to 0 will always be dwarfed by the number of them far from it. But how can this procedure (when run forever) produce one, single integer, while at the same time having a probability of 1 of producing a number bigger than any particular number we choose? Well, each integer can be thought of as having an infinite sequence of zero digits to the left of its first non-zero digit. This algorithm will have a probability of 0 of producing a number with an infinite successive sequence of zeros, and therefore will have a zero probability of producing an integer! In other words, there is a probability of 1 that each number it produces will be (in a certain sense) “infinite”, so it does not serve the purpose that we hoped.

This brings us to our final question, regarding whether a terminating algorithm (that can be run on a real computer) can be created that will sample uniformly at random from an infinite set of numbers. The answer to this question is no, but with the footnote that this does not matter too much in practice (for reasons to be discussed). One way to understand why this is impossible is to consider how many bits it would take to transmit the number produced by such a sampling procedure. We would have to transmit some kind of code (agreed upon in advance) that represents the number we got from sampling, but since there are an infinite number of possible outcomes and since we have no knowledge (in advance) of what number will occur, we would need to have an infinite number of codes to represent all the outcomes. Hence, if  we think of the codes as numbers, and choose some number N, then some of the codes must contain more than N digits (since N digits is only enough to describe a finite number of different codes). But since this holds for all N no matter how large, this means that, on average, it would take literally forever to transmit one of these codes. But, if the number cannot be transmitted, no computer could ever make a copy of it, which implies that no computer could ever generate such a number. What this confusing, convoluted argument is getting at is the fact that a uniform distribution on an infinite set of items (if it existed) would have an infinite entropy, so the numbers sampled from such a procedure could never be transmitted or stored (as doing so, on average, would require an infinite number of bits) so there is no way that such an algorithm could be used in real life. One way to see that the entropy of such a distribution would be infinite is to note that if we define a uniform distribution on n items, that as we let n go to infinity the entropy of the distribution approaches infinity.

Despite these problems, some infinite sets have a nice property that the procedure of sampling (with equal probability) from them can be nicely approximated on a computer. For example, if we want to sample a real number between 0 and 1, we can approximate this procedure by limiting ourselves only to numbers with at most 40 digits after the decimal point, and then sampling uniformly at random from this restricted set. While this procedure is not perfect, it will produce numbers that (for most purposes) look like those we would get if we truly sampled from all real numbers between 0 and 1. On the other hand, sampling uniformly at random from the set of all integers cannot be approximated in any nice way (and hence, is in some sense an inaccessible procedure). The problem here is that, as noted, if you fix any number X that you like, 100% of integers are greater than that X, no matter what X is. Since real computers are limited in the size of the numbers they can store, any attempt to approximate the procedure of sampling from all integers will be limited to sampling from integers less than some number X, despite the fact that 100% of integers truly are above X. If we try to sample uniformly at random from the set of all integers (or the set of real numbers, for that matter) we are doomed to complete failure.


Posted in -- By the Mathematician, Math, Probability | 2 Comments »

Q: Do aliens exist?

Saturday, January 30th, 2010

Physicist: Yuppers.  In as much as the probability that they don’t is effectively zero.

The statistics on this are a little weak, since we only have one real data point.  If you define intelligent life as tool-using, then (based on the age of the oldest tools and the oldest fossils, and the progress of the Earth to date):  Intelligent life has existed 0.06% of Earth’s history, and animal life has existed for about 16% of Earth’s history.  Moreover, the vast majority of life on Earth (and the toughest) is microbial.  So by “yuppers”, I mean that space bacteria almost certainly exists.

As far as the fancy aliens (with their lasers and tentacles) that I assume the question is really about: probably.  The universe is crazy big.  However, stars are far apart (especially around here), and the likelihood of finding intelligent life is really low.

In the last decade there have been some surprising results from the panspermia people.  It seems to be entirely possible, even likely, for life to get kicked from planet to planet and even from star to star.  The three difficulties are getting off a planet, surviving in space, and landing somewhere else.  During a major impact the material immediately around the impact is vaporized.  A little farther out and things are pulverized.  Just beyond the “automatically dead zone” is a thin ring where material from the planet’s surface can be thrown into space smoothly (no more than a couple hundred G’s) and without excessive heating.  Although no animals could survive the shock, massive G forces have very little impact on single celled life (too small to slosh).

There’s a wide variety of life from Earth that does fine in space.  Things like Water Bears, and some bacteria can put up with the cold and radiation, and are more than happy to drop into a state of suspended animation for the trip (forever if they have to).  The classic example is a few cells of Streptococcus that survived on the moon (on Surveyor 3′s camera) between 1967 to 1969.

Something you may notice, if you collect large meteorites, is that although the surface tends to be pretty messed up, the interior is frequently quite intact.  Although the fall looks pretty impressive, the heat and fireball don’t have time to cook the meteors all the way.  In fact the hottest parts of the meteor vaporize during the fall, which serves to keep it cool (like sweat, but like… a rock version).  Although it’s unlikely for living things on any one rock to make it through all three stages intact, keep in mind that there are actually many rocks flying around that have been knocked off of planets in the past.  There are so many, that one of the cheapest ways to collect samples from Mars or Venus is to go to Antarctica.  (If you find a rock sitting on top of a 3 miles of ice, where do you think it came from?)  One of the biggest “life is out there” stories came from exactly this source.

Here’s the point: If there’s life anywhere it’s likely to spread everywhere, like… well, like life.  Panspermists think that life may have started on some other planet around some other star, and that this life then infected the Earth.  This would help explain why the Earth was covered with sophisticated (microbial) life almost immediately after it was capable of supporting life at all.  Or to spin it around, if there’s life here (check) it’s had over 3 billion years to get blasted out into the nearby universe.

 

Mathematician: There are compelling reasons to think that life exists on other planets (perhaps even on a huge number of other planets). If life spontaneously arose on earth from a soup of molecules through an evolutionary process, then all you need for life to be created is the right planetary conditions, the proper raw materials, and a sufficient amount of time. The right conditions may include things like being close enough to a sun that the planet is reasonably warm, but far enough from that sun so that it isn’t  burnt to a crisp. The right materials probably include carbon and water among other things. In any event, once you get these things right, you just add time (a billion years probably would suffice) and viola, life is born. That means that for earth to house the only living organisms in the universe, these requirements would have to have been met one time and one time only in all the billions of galaxies that have formed during the 14 billion year history of our universe. That sure sounds pretty unlikely.

Here’s another way to think about it: there is some probability p that a randomly selected planet will form life on it within a billion years. If p is sufficiently small, then there would be almost no chance of any life forming, including our own, and hence we should not exist. If p is sufficiently large, then life would exist almost everywhere in the universe. The only way that we should expect to be the one and only planet with life is if p is just right to produce about one planet with life over all the years and on all the planets that have ever existed. But we have no evidence whatsoever indicating that p should be perfectly balanced in this way, indicating that the chance of alien life is a good one.

But does technologically advanced alien life exist? Well, if life occurs on many other planets, then we should expect technologically advanced life to occur on at least some of them. Whatever caused natural or sexual selection to select for high levels of mammalian intelligence on earth could lead to intelligent aliens as well. On the other hand though, if technologically advanced civilizations tend to wipe themselves out fairly quickly (say, within a hundred thousand years) or if the process that creates highly intelligent life requires sufficiently rare conditions, then advanced aliens could certainly be the exception rather than the rule.

Posted in -- By the Mathematician, -- By the Physicist, Astronomy, Biology, Evolution, Physics | 1 Comment »

Q: Is it true that all matter is simply condensed energy?

Friday, January 29th, 2010

The complete question was: Is it true that all matter is simply condensed energy? Does that mean that the Big Bang was pure energy and coalesced into matter?

Physicist: Pretty much.  If you can get enough energy into one place (generally light or kinetic energy), then you’ll get a (mostly random) variety of particles popping out. The conversion between mass and energy is so ubiquitous in physics, that most physicists only know the mass of particles in the context of their equivalent energy.  If you ask a physicist “what is the mass of an electron?” they’ll say “0.5 MeV” (which is a unit of energy).  Frankly, it’s more important to know than the actual mass.  I mean, how hard is it to pick up an electron?  If you answered “I don’t care” or “zero”, you’re right.

The only thing that keeps particles from turning back into energy (again, usually light and kinetic) are “conserved quantities”.  If you’ve taken an intro physics course you should be familiar with conservation of energy and momentum.  In particle physics you also need things like: charge, Lepton flavor (which covers things like electrons and neutrinos), and Baryon number (which covers things like protons and neutrons).

The classic example is neutron decay:

Neutron decay: Tricky business.

A neutron is heavier than a proton, so you’d think it would decay into a proton and some extra energy (conserving energy and baryon number).  But that would violate conservation of charge (protons have 1, neutrons have zero).  So maybe it could decay into a proton and electron?  Now you’ve balanced charge, but violated lepton flavor (electrons have “electron flavor 1″).  To balance everything you need to add an anti-electron neutrino (electron flavor -1) to the mix.

The very early universe was a “particle soup”.  The mean energy of the photons flying about was more than enough to generate new particles.  These would pop into existence in balanced quantities and then cancel out again.  The big difference between now and then (why we don’t see particles being spawned off all the time) is that the average energy of photons today is closer to 660 meV (about 1 billionth of the energy needed to create electrons, the smallest particle).

Posted in -- By the Physicist, Physics, Quantum Theory | 1 Comment »

Q: Which is better: Math or Physics?

Thursday, January 28th, 2010

Physicist: Physics.

Mathematician: Math, of course. Can physics do this?

1 = \sum_{k=0}^{\infty} \frac{(2 \pi)^{2 k}}{(2 k)!} (-1)^{k}

Physicist: Lasers, dude.

Mathematician: Lasers, shmasers.

Posted in -- By the Mathematician, -- By the Physicist, Math, Philosophical, Physics | 1 Comment »

Q: Why is the number 1 not considered a prime number?

Wednesday, January 27th, 2010

Mathematician: Note that when we say that a number is “prime”, all that we are doing is applying a definition that was devised by mathematicians. A prime number is generally defined to be any positive number that has exactly two distinct positive integer divisors (the divisors being 1 and the number itself). So 13 is prime, because it is divisible only by 1 and 13, whereas 14 is not prime because it is divisible by 1, 2, 7 and 14. Note that this excludes the number 1 from being prime. The biggest reason this definition of primality is used, as opposed to a slightly different one, is merely a matter of convenience. Mathematicians like to choose definitions in such a way that important theorems are simple and easy to state. Probably the most important theorem involving prime numbers is the Fundamental Theorem of Arithmetic, which says that all integers greater than 1 can be expressed as a unique product of prime numbers up to reordering of the factors. So, for example, 54 can be written as 54 = 3*3*3*2 which is a unique factorization assuming that we list the factors in decreasing order. Now, notice that if we counted 1 as a prime number, then this theorem would no longer hold as stated, since we would then be able to write

54 = 3*3*3*2*1 = 3*3*3*2*1*1 = 3*3*3*2*1*1*1

so there would not be a single, unique representation for 54 as the theorem requires. Hence, if we count 1 as a prime number, then the Fundamental Theorem ofArithmetic would have to be restated as something like, “all integers greater than 1 can be expressed as a unique product of prime numbers (not including 1) up to reordering of the factors.” This is a tiny bit more cumbersome, but not horrible. If you have to work with prime numbers day in and day out though, simplifying theorems just a little bit (by choosing your definitions carefully) may well be worth it. Nonetheless, if mathematicians chose a slightly different definition for primality that included the number one, while they would then be forced to modify many of their theorems involving primes, the world wouldn’t come crashing down on its head.

Posted in -- By the Mathematician, Math, Number Theory | No Comments »

Q: If the universe is expanding and all the galaxies are moving away from one another, how is it possible for galaxies to collide?

Tuesday, January 26th, 2010

Physicist: Because the universe isn’t expanding fast enough.  On average all the galaxies are moving apart, but often a given pair will be moving together.

Hubble observed that the farther things are away, the faster they’re receding.  Specifically, in the universe today, v = H_0 d where v is the relative velocity of two objects, d is the distance between them, and H_0 = 22 \pm 2 \, mm/s/Lyr (millimeters per second per light year).  Now this is an averaging thing, since galaxies are free to move however they like.

So for example, this equation says that the Andromeda galaxy which is 2.5 MLyr (million light years) away should be moving away at around 55km/s.  Instead it’s flying at the Milky Way at about 120 km/s.

As a side note: when Andromeda gets here (or we get there, or whatever) the collision of our gas clouds should set off a huge spike in star formations resulting in a liberal peppering of supernovas (bad for everyone).  But we’ve still got another 2.5 billion years, so don’t pack your bags just yet.

Posted in -- By the Physicist, Astronomy | 2 Comments »

Q: What happens when you fall into a blackhole?

Sunday, January 24th, 2010

Physicist: Terrible, terrible things.

The first thing you’ll be likely to notice as you approach the hole is the tidal forces.  Tidal forces are nothing more than the difference in gravitational force between the near and far side of an object, and they aren’t particular to blackholes.  For example, the tidal force of the moon on the Earth causes tides (hence the name).  For any reasonable sized blackhole (less than thousands of suns), the tidal force between different parts of your body will be greater than your body’s ability to stay intact, so you’ll be pulled apart in the up-down direction.  For much more obscure reasons, you’ll also be crushed from the sides.  These two effects combined are called “spagettification”.  Seriously.  Assuming that you somehow survive spagettification, or that you’re falling into an super-massive blackhole (which is ironically much more gentle than a smaller blackhole) then you can look forward to some bizarre time effects.

It’s been established for decades that “time moves slower the lower”.  For example, GPS satellites have to deal with an additional 45 microseconds every day due to their altitude (they move through time faster).  Also, one way to think about gravity is as a “bending” of the time direction downward.  In this way anything that moves forward in time will also naturally move downward.  At the event horizon of a blackhole (the outer boundary) time literally points straight down.  As a result, escaping from a blackhole is no more difficult than going back in time.  Once you’re inside all directions literally point toward the singularity in the center (since no matter what direction you move in will be toward the future).

We don’t experience time moving at different rates or being position dependent, so when we start talking about messed up spacetime it’s useful to look at things from more than one point of view.

From an outsider’s perspective (far from the blackhole): As someone falls in they will move slower and slower through time.  They will appear redder, colder, and dimmer.  As they approach the event horizon their movement through time will halt, as they fade completely from view.  Technically, you’ll never actually see someone fall into a blackhole, you’ll just see them get really close.

From an insider’s perspective (falling into the blackhole): First, torn apart and crushed.  Things farther from the blackhole move through time faster, so the rest of the universe will speed up from your point of view.  As a result the rest of the universe becomes bluer, hotter, and brighter.  The blue shift of the incoming light turns it into gamma rays.  So, right before you pass through the event horizon, you’ll get nuked with a universe’s lifetime worth of starlight and microwave background radiation turned into nuking nastiness.  The event horizon itself is only special from an outside perspective.  If you fall in you should pass right through it.  However, what you see in the moment that you pass through the horizon is dependent on things we don’t know yet.

-If the blackhole lasts until the universe ends (assuming that the universe ends), then you’ll see the entire history of the universe whip by (bluely).  You’ll then find yourself face to face with the singularity.  At that point you go away, according to the math.  However, the universe is slippery like a greased up eel fresh from the bar exam.  It always finds a way to not have singularities where the math predicts it.  So, to be safe, I’ll say “know one knows what happens then”.

-If the blackhole evaporates, then all the matter that (almost) gets to the horizon will be torn apart and reappropriated as Hawking radiation.  If you were to survive, then you would find yourself as close to the horizon as (for uncertainty reasons) it is possible to be, and you would ride it in as it shrinks.  In a blink you’d suddenly find yourself floating around right next to an amazing explosion, as the last of the blackhole evaporates.

Posted in -- By the Physicist, Astronomy, Physics | No Comments »

Q: Is the total complexity of the universe growing, shrinking or staying the same?

Friday, January 22nd, 2010

The complete question was:

If you were to look at the universe as an organism, was the early universe a simpler organism than the present-day organism?  Is the total complexity of the universe growing, shrinking or staying the same?  And how do you measure that?

Physicist: Absolutely.  The total complexity of the universe is increasing, due to the inevitable march of entropy (or information), which is exactly the measure of complexity.  A more intuitive way to talk about complexity and entropy is: can you predict what you’ll see next?  If you look at part of a checker board, you can probably guess what the whole thing looks like, so the board is predictable and has low entropy.  In the early universe matter was distributed pretty uniformly, almost all of it was hydrogen, almost everything was the same temperature, and there were no complex chemicals of any kind (going back far enough everything was ionized).  So if you’d seen one part of the universe, you’ve pretty much seen all of it.

This is actually a chess board.

No surprises.

Nowadays the universe is full of a wide variety of different elements with very complicated ways to combine together, matter shows up hot, cold, as plasma, as proteins, in stars, and clouds, and not at all.  The amount of data it would take to accurately describe the universe as it is now utterly dwarfs the amount that it would take to describe the early universe.  On an atom-by-atom basis, in the early universe you could grab an atom at random and feel fairly confident that: it’s hydrogen, it’s ionized, it’s about “yay” far away from the other nearby hydrogen, etc.  Today you’d probably be right if you guessed “hydrogen” (about 3/4 of the universe’s mass is still hydrogen), but you’d have a really hard time predicting anything beyond that.

Oddly enough, life is surprisingly uncomplex compared to say, dirt or sea water.  If you look at a single cell in your body, you’ve already got a pretty good idea of what you’ll see everywhere else in your body.  Admittedly, we are more complex than single celled life, but most of that is a symptom of being physically bigger.

Posted in -- By the Physicist, Biology, Entropy/Information | 2 Comments »

Q: If two trains move towards each other at certain velocities, and a fly flies between them at a certain constant speed, how much distance will the fly cover before they crash?

Thursday, January 21st, 2010

The brain teaser comes in a many variations. For example:

Trains A and B, 700 miles apart, are heading toward each other on a straight piece of track. Train A is going 85 mph while train B is going 55 mph. At the same moment, a bee that flies 110 mph is sitting on the nose of train A and begins flying toward train B. When it reaches train B it makes an instantaneous reversal of direction and flies back toward train A. It continues to change direction every time it runs into a train until both trains and the bee meet in a spectacular crash. What total distance did the bee fly before the big collision?

Mathematician: The difficult way to solve this problem is to figure out how much distance the bee (or fly) traveled before turning around each time it approached a train, and then sum these distances together. The easy way to solve it is simply to figure out how long it took the trains to crash, and then calculate how far the bee, which travels at a constant speed, must have gone during this amount of time.

More specifically: The bee always travels at the same speed V. If we can figure out how much time, T, the bee flew before the trains collided with each other, then the total distance D it flew will just be V T, the product of the velocity and time. We know V, so all that remains is to figure out T. To do this, we just need to calculate how long it takes for the trains to crash. If the first train has velocity v1 and the second v2, and the distance between them initially is d, then the time T before the crash will just be d/(v1+v2), which is equivalent to the amount of time that it takes a train going velocity v1+v2 to travel the distance d. The total distance traveled by the bee is given by:

D = V d / (v1+v2)

= (700 miles)  * (110 mph)/((55 mph)+(85 mph))

= 550 miles

Posted in -- By the Mathematician, Brain Teaser, Math | 1 Comment »

Q: Why does oxygen necessarily indicate the presence of life?

Wednesday, January 20th, 2010

Physicist: Short answer: Life is the only thing that makes lots of oxygen.

This question comes in the context of a conversation about the Kepler mission.  So far (as of January 11, 2010) 424 “exoplanets” have been discovered and confirmed in orbit around other stars.  It’s worth pausing to take a minute and say, “holy shit!”.  Most civilizations throughout the ages have been aware of Mercury, Venus, Mars, Jupiter, and Saturn, and so was it for tens of thousands of years.  Between 1781 and 1930 we found 4 more: Uranus, Ceres, Neptune, and Pluto.  (It’s been slow.)

Since 1992 we’ve found over 400 new planets around other stars and, depending on where you draw the line, between 7 and several new dozen dwarf planets around our star.  It may have nothing to do with the question, but I think it’s worth knowing.

Pluto has friends.

New Dwarf Planets

Unpause.  Due to the difficulties in measurement, the vast majority of the exoplanets discovered so far are bigger than Jupiter, and orbit their parent star closer than Earth orbits the Sun.  Kepler holds the promise of detecting Earth sized planets, and brings us a step closer to detecting life around other planets (even if there were life on gas giants, it would be so alien we wouldn’t know what to look for).  Kepler works by waiting for eclipses.  When a planet passes in front of its star, the star appears to dim a little (if someone around another star were to see Earth do this, the Sun would appear to dim by about one part in 10,000).  Even better, by staring really (really) hard you can actually see light that has passed through the atmosphere of those planets, and now you’re talking chemical analysis!

If you look around at the other planets in our solar system you’ll notice that they all have something in common: their atmospheres are all chemically stable.  The other atmospheres, CO2, Hydrogen, Helium, Methane, etc., don’t do much more than just blow around.  You can’t start a fire anywhere other than Earth.

This is why we can't have nice things

Oxygen: The jerk of the elements. It's corrosive, burns like crazy, and is generally reactive and unstable.

Oxygen, on the other hand, is about as stable as a drunk unicyclist.  When you find oxygen in nature (and by “nature” I mean “other than Earth”) it’s always already tied up in the molecules of something else (such as in water or granite).  As soon as oxygen is released it tends to immediately combine with things around it.  It has been estimated that, left on it’s own, atmospheric oxygen will be completely absorbed by chemical processes within a few hundred years, and that’s not including big fires and whatnot.

The only known process that actually releases O2 into the air in any real quantity is photosynthesis.  So, observing oxygen in the atmosphere of other planets implies photosynthetic life.

Posted in -- By the Physicist, Astronomy, Biology, Physics | No Comments »

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