My bad: If fusion in the Sun suddenly stopped, what would happen?

Posted on October 24, 2012 by The Physicist

Physicist: A commenter from the original post was kind enough to point out a massively bone-headed mistake.  My bad.  In the calculation to figure out how long it would take the Sun to cool I had used Fourier’s law of thermal conduction.  What I should have used was the Stefan–Boltzmann law of thermal radiation.  So that calcu-estimation would have been perfectly valid if the Sun were floating in an expanse of very cold, and opaque, material and not valid if the Sun were floating in a whole lot of nothing.  Sadly, the universe has taken effectively none of my constructive suggestions, and the Sun persists with being in space.  After a revisit (included in the answer gravy below) it looks like we’d still have plenty of time to leave the planet, but longer time scale things would be done (continental drift, presidential campaigns, etc.).  Life has existed on Earth for about 4 billion years, but interesting, multicellular life only exploded onto the scene about half a billion years ago.  Assuming that this is because the Sun wasn’t doing it’s job at the time (the Sun gets brighter over time, and was dimmer in the past), that puts a floor of about 95% of its present output necessary for complex life.  If fusion stopped, we’d be there in a little over a million years.

Science is based on the little-acknowledged axiom that not a single one of us is particularly smart, but that there’s a good chance that somebody out there is busy not making mistakes today (and who tomorrow may rely on you).  To that end, what follows is a more direct calculation, for those of you interested in the details.

Answer Gravy:

The Stefan-Bolzman law says that the flow of heat out of the Sun should be \dot{q} = -S\sigma T^4, where \dot{q} is the rate of heat loss, T is the surface temperature, S = 6×1018m2 is the surface area, and σ = 5.67×10-8 is a physical constant.  If we assume that the surface temperature is proportional to the average temperature within the Sun (which is presently about 4 million °K), then we can find a relationship between the surface temperature, T, and the total heat energy, q.  This is a bad assumption over the long run, but should be decent enough for figuring out how long it would take the Sun to lose, say, 1% of its heat.

q = \underbrace{\left(2\times 10^{30} kg\right)}_{mass}\underbrace{\left(12500 \frac{J}{kg\cdot K}\right)}_{heat \, capacity}\underbrace{\left(680\right)}_{assumption}T = 1.7\times 10^{37}T

Plugging this into the Stefan–Boltzmann law:

\dot{T} = -\frac{(5.67\times 10^{-8})(6\times 10^{18})}{1.7\times 10^{37}}T^4 = -\left(2\times 10^{-26}\right)T^4

Fans of calculus can tell you that the solution of this is:

\begin{array}{ll}\dot{T} = -\left(2\times 10^{-26}\right)T^4\\\Rightarrow \frac{dT}{dt} = -\left(2\times 10^{-26}\right)T^4\\\Rightarrow T^{-4}dT = -\left(2\times 10^{-26}\right)dt\\\Rightarrow -\frac{1}{3}T^{-3} = -\left(2\times 10^{-26}\right)t +k\\\Rightarrow T^{-3} = \left(6\times 10^{-26}\right)t +k\\\Rightarrow T = \left[\left(6\times 10^{-26}\right)t +k\right]^{-\frac{1}{3}}\end{array}

Here k is an “integration constant”, and it’s set so that the temperature right now (at t=0) is what it should be: 5878.  Also, t is in seconds in σ.  To fix that, just multiply t by 31,500,000,000 (the number of second in a thousand years) to change it to millennia.

The surface temperature in °K vs. time in thousands of years.

So all together, if fusion were to stop today, the Sun’s surface temperature should be about T = \left[\left(6.3\times 10^{-19}\right)t +5\times 10^{-12}\right]^{-\frac{1}{3}}.  It doesn’t take much to end all life on Earth, so we would probably have, at most, a couple million years.  It would take a little over one hundred thousand years for the surface temperature of the Sun to drop by 1%.

Posted in -- By the Physicist, Mistake | 14 Comments

Q: If fusion in the Sun suddenly stopped, what would happen?

Posted on October 20, 2012 by The Physicist

Physicist: Almost nothing.  We’d have plenty time to pack our bags and leave.

Assume that the cause of the fusion-stopping doesn’t matter, maybe the Weak force suddenly changed, or maybe the Vorgons used the Tax Uthat on the Sun, or whatever.  The fusion just quietly stops (side note: if you got that reference, then it’s time to go outside).

There’s a lot of loose talk about the Sun being a huge atom-smashing machine, but in reality the hydrogen that fuels the Sun can wander about in the core (where the fusion takes place) for several hundred million years on average without being used in fusion.  The rate of fusion that takes place in a fusion bomb, or that we hope to accomplish in fusion reactors, is more akin to an exploding star than a stable one.  The core of the Sun (which is huge) produces heat through fusion at about the same speed that the human body produces heat.  The difference is, while we can lose heat easily, the Sun can’t.

The thing about the Sun is not that it’s big, it’s that it’s so big.  And in general the bigger something is, the longer it takes to cool off.  A slightly warm cup of coffee (or “spot of tea”, for our UK readers) will drop to the surrounding temperature within minutes.  But a slightly warm lake will take weeks or months to assume the temperature of the surrounding area.  This is a big part of why the climate near big bodies of water tend to be more temperate; the water acts like a “thermal shock absorber”.

The amount of heat energy that something can hold is proportional to its mass.  That is; two cups of coffee have twice the heat energy of one cup of coffee.  But when something loses heat it loses that heat through its surface, whether that’s through radiating it away or through being physically in contact with cooler stuff.

The bigger something is the more mass is far away from the surface, and also the smaller the “surface area over mass ratio” becomes.

The bigger something is (assuming it’s not a really big and really flat) the more of the matter in it will be far from the surface.  More than that, as size increases mass increases faster than surface area.  So there’s more heat to get rid of, but proportionately less opportunity for it to get out.  Even though the Sun is exposed to the icy vacuum of space, and is seriously hot, it barely cools off at all.

A hot cup of coffee (spot of tea) is about 60°C hotter that the surrounding environment (room temperature), and the Sun is about 5,800°C hotter than its surrounding environment (infinite nothing), so each cup-sized-piece-of-surface on the Sun is losing heat about 100 times faster than the cup of coffee.  However, the Sun is approximately 30 billion times bigger across than a cup, and as a result it cools off about 30 billion times slower than a cup in similar circumstances (like a hot Russel’s teapot).  The average heat energy in the Sun is around 12,000 times greater than the average heat energy of water (coffee or tea), so it’ll take about that much longer to cool off.

Update: The estimates in what follows are wildly incorrect.  A correction with details can be found here.  The wrong kind of heat transfer was assumed, and as a result the time scales became a lot larger.  It looks like humanity would have no more than a hundred-thousand years (give or take), and the tougher parts of the biosphere would have no more than about a million.

Putting it all together, however long it takes for your hot-beverage-of-choice to cool off, it’ll take the Sun about 3.6 trillion times longer.  If it takes your cup, say, one minute to become noticeably cooler, then it’ll take the Sun about 6.8 million years to cool off noticeably.  The process is sped up a bit by the contraction of the Sun over time keeping its surface hot, but either way, if the Sun’s fusion stopped the only people who would notice right away would be the scientists studying solar neutrinos.  This may seem a bit surprising, but in general, stars take a long time to cool off on their own.

We’d have millions of years to figure out how to build ships and leave for greener pastures (full of working stars), or figure out how to live with a cooler star when it became incapable of supporting life, tens of million years down the road.

The last two paragraphs were terribly wrong, over-estimations.

A good ‘n scary sci-fi weapon wouldn’t be something that halts fusion, but something that directly cools stars suddenly, or goes the other way and speeds fusion up (which is the important step in a nova).

Posted in -- By the Physicist, Astronomy, Physics | 11 Comments

Q: Does opening a refrigerator cool down the room?

Posted on October 15, 2012 by The Physicist

Physicist: Briefly yes, or no, not at all.  If you think of the room as including the inside of the refrigerator, then opening the door does nothing.  Otherwise, it does almost nothing.  But ultimately, if you leave the door open the only end result will be spoiled food.

Refrigerators, and the opening of same.

It seems like it makes sense to say that things like refrigerators and air conditioners “make cold”, but like every other machine ever created (or ever to be created) they ultimately just make heat.  A refrigerator “creates cold” in very much the same way that a drain “creates lack-of-water”; it moves heat from its inside to its outside.  Specifically, it pulls heat out of the freezer, and drops it into the coils on the back side. (It’s worth noting that about the worst possible place to put cooling coils is in a tiny gap next to a wall).

It’s a general thermodynamic fact (a law even!) that generating cold is impossible.  You can generate heat, and you can move it around by taking advantage of the fact that heat always tries to “even out” (this is the idea behind all “cooling devices”), but that’s pretty much it.  So just like any other machine, refrigerators generate heat.  When you first open the door you’ll get a burst of cold air, but that’s about it.  An open fridge is like a water pump in the middle of the ocean, pointlessly moving stuff around.  It’ll cool the room a little, but also heat it up a lot more.

A clever thing to do would be to put the heating coils outside of the room.  The room would get cooler, the outside would get warmer (slightly), and you’d have re-invented air-conditioning.

Posted in -- By the Physicist, Engineering, Entropy/Information, Physics | 32 Comments

Q: What is the probability of an outcome after it’s already happened?

Posted on October 10, 2012 by The Physicist

Physicist: There are a lot of subtleties to this.  Reading the question, your gut reaction should be “Duh, it’s 100%!  Wait, is this really a question?”.

And yet, there are many times in which you may find yourself estimating probabilities on things that have already happened.  If you flip a coin and cover it or go looking for a lost dog, the “true” probability is always 100%: the coin is definitely either heads or tails, and Fluffins (the wonder dog) has a 100% chance of being exactly where it is.

In liar’s dice all the bets and estimations are based on events that have already occurred, but are unknown.

Probabilities are usually defined in terms of the uncertainty in what’s known.  Liar’s dice is a beautiful example (so are most card games for that matter); all of the dice are what they are, and yet in the picture above, if you’re the player on the left, then there’s a chance of 1 that all of your dice are 5’s, but there’s an even chance that your opponent’s dice could be any combination.  From the left player’s perspective, there’s some chance that the dice on the right are, say, “1,2,3,4,5”, even though from the right player’s perspective, that chance is zero (the right player knows their dice are not “1,2,3,4,5”).

Long story short: probability is extremely subjective.  Whether an event happened in the past or will happen in the future doesn’t make too much difference, it’s the knowledge you have about an event that defines its probability (for you).  That said, in terms of gaining knowledge about an event, it helps a lot for it to have been in the past.  If you were some kind of time traveler it would be a lot easier to determine the result of a coin toss by just looking at it after it’s happened, rather than going to the trouble of predicting it before it’s happened.  That’s why there are Futures Markets, but not Pasts Markets.

But the spirit of this question is really about some kind of “objective probability”.  Maybe you don’t know how something in the past turned out, but surely if you somehow had access to all of the information in the universe you’d be able to determine that the probability is 100% or 0%.  Surely everything in the past either happened or didn’t, it’s just a matter of finding it out.

Very, very weirdly; no.  You have to root around in quantum mechanics to see why, but it turns out that even things in the past, in the most objective possible sense, are also uncertain.  This doesn’t mean that, for example, the Nazi’s may have won the war (since it’s pretty well-known that they didn’t), but it does mean that if an event is so small and fleeting that it leaves no real trace, then it may have happened in multiple ways (quantum mechanically speaking).

Answer gravy: This is high on the list of the weirdest damn things ever.

Way back in the day, the double slit experiment demonstrated that a particle (and later much larger things) can literally be in two places at once.  This means that the question “where did I leave my quantum keys?” doesn’t have a definite answer.  The probability that the particle will be found going through one slit or the other is non-zero, not just because the position isn’t known, but because it can’t be known (essentially, there’s nothing definite to know).  The first reaction that any half-way reasonable person should have is “dude, you missed something, and that particle totally has a definite position, you just don’t have a way to figure out what it is”.  But physicists, being clever and charming, found a way to prove that that isn’t the case.  It can be shown that, regardless of what you do or how you measure, quantumy things don’t have a definite position.  This is basically what Bell’s theorem is all about.

Not comfortable with reality being merely a little weird and uncomfortable, a dude named Franson proposed an experiment to demonstrate that the past is in a similar superposition of states.  Not only can things be in multiple places now, but they can do it at multiple times.

In the Franson experiment a photon is emitted at a random time and shot toward a beam splitter, which allows it to take one of two paths; a long path and a short path.  You’d “expect” that the photon would be emitted at a particular time, then take one (or both) paths, and then randomly exit (diagram below).  The thinking is that, since the two versions of the photon arrive at the second beam splitter at different times, there’s no way for them to interfere.

The Franson experiment: You’d think that a photon would be emitted at a definite time (left) and then move through the paths and exit randomly, either sooner or later.  But (right) we find evidence of interference at the exit, so the photon must exit at one time and enter at two.

However!  When this experiment is done (with a random photon source) interference is seen.  Therefore the photon must be arriving at the second beam splitter from both paths (similar to how the double slit experiment creates interference).  But that means that the photon must have been released at two different times.

There’s some subtlety that I’m not including, such as the fact that paths above are only half of the device (the other half is identical), and that the experiment requires entangled photons, but if you’re interested in the details you can read the original paper.

What’s really horrifying is that this experiment is done pretty regularly!  Nothing special.  The past genuinely is in multiple-states, and as a result the probabilities of events in the past can be damn near anything.

Posted in -- By the Physicist, Philosophical, Probability, Quantum Theory | 11 Comments

Q: How do you answer a question scientifically?

Posted on October 5, 2012 by The Mathematician

Mathematician:

Suppose you’re interested in answering a simple question: how effective is aspirin at relieving headaches? If you want to have conviction in the answer, you’ll need to think surprisingly carefully about how you approach this question. A first idea might simply be to take aspirin the next time you get a headache, and see if it goes away. But as we’ll see, that won’t be nearly enough.

First of all, since all headaches go away eventually, whether it disappears isn’t the relevant question. It would be better to ask how quickly the headache goes away. But even this phrasing doesn’t capture what you care about, because even if aspirin doesn’t relieve your headache completely, a significant reduction in pain is still worthwhile.

To deal with this issue, you settle on the following plan. When you next get a headache, you’ll make a record of how you feel when it first starts. Then, you’ll take an aspirin, wait 30 minutes, and record how you feel again. You’ll write things like “dull, throbbing pain of low intensity” or “sharp, searing pain over one eye”.

Unfortunately, just examining how you feel after taking aspirin a single time won’t be adequate, since the aspirin may be more helpful some times and less helpful others. For example, it could be the case that it works on moderate headaches but not on severe ones, so if your next headache happened to be really severe, it would look like aspirin was useless. To solve this problem, and give yourself more data, you might resolve to make these records of how you’re feeling for each of the next 20 headaches you get.

There is still a problem though. These subjective descriptions of headaches are difficult to compare to each other. It you take aspirin and your headache goes from a sharp pain over one eye to an intense ache over the entire head, have you made things better or worse? It would be difficult to aggregate the information from these varied descriptions over 20 different headaches to make a final assessment of how well aspirin is working.

Your analyses would be a lot simpler if you scored how unpleasant each headache was on a simple scale from 1 to 5 (1 meaning slight unpleasantness, 3, moderate unpleasantness, and 5, extreme unpleasantness). That way, you can simply look at all the scores you got just before taking aspirin and average them together. You can then compare this to the average of the scores 30 minutes after taking the aspirin. That way, you can see if the amount of unpleasantness you feel really does drop substantially.

Recall that our goal here is to determine how effective aspirin is at relieving headaches, but all you’ve investigated so far is measure how good it is at relieving your own headaches. Perhaps you are more or less sensitive to aspirin than other people, or perhaps your headaches are more severe and harder to treat than most other people’s. To solve this problem, you enlist 40 people who are frequent headache sufferers. You get them to agree that, over the next 6 months, any time they begin to notice that they have a headache they will record how they feel on your 1 to 5 scale. They will then take aspirin and record how they feel 30 minutes later.

But what if people take different doses? You might think that the aspirin isn’t working for some of them, but it’s only because they haven’t taken enough. To fix this problem, you hand them each an identical bottle of pills and tell them to take two whenever they get a headache (the maximum recommended dose, so that if aspirin does work you’ll be using a high enough dose to detect the effects, but not putting people at significant risk of side effects). Handing out aspirin bottles also has the added benefit that everyone will be taking the same brand. That way if you find out that the aspirin really does work, other people can try to replicate your results by using the same brand you did. To make your experiment even better, you also provide everyone with a timer, to help them be more accurate about recording their pain 30 minutes after a headache starts.

There is still a problem though. You know that headaches sometimes become less severe within 30 minutes or so even when left untreated. That means that even if someone’s pain score tends to have fallen 30 minutes after taking the aspirin, you don’t really know whether it is the medicine that caused the reduction in pain or if it would have occurred regardless. To remedy this, you decide that only half of the 40 participants will take aspirin when they have a headache (the rest taking nothing), though everyone will still keep a record of their pain. Then, to see how well the aspirin worked, you can compare the average pain scores of the 20 people who took the aspirin with the scores of the 20 people who didn’t. If the aspirin group’s pain fell a lot more than the non-aspirin group, it would seem then that the aspirin probably was the cause.

But is it possible that the pain levels people record could be influenced by the act of taking a pill, independent of the chemical effect of the active ingredients? Perhaps since people expect aspirin to work, the people in the group taking the pills are more aware of signs of improvement. Or perhaps people’s expectations of improving can even influence how much pain they experience. If either of these things are true, the aspirin might seem to work better than it really does, because some of the reported improvement would be coming from expectations people have about the chemical, rather than just the chemical itself.

Fortunately, this problem is easily fixed. Instead of giving half the people nothing, you instead give that half pills in an aspirin bottle that look just like aspirin, but which are known to have no effect on headaches. Sugar pills are a good choice because they are cheap and free of side effects, and because a small amount of sugar is unlikely to alter a person’s headaches.

This raises ethical considerations. People might agree to take aspirin, but that doesn’t mean they are willing to take sugar pills. So before you start your study, you’ll need to tell everyone that there is a 50% chance they won’t be getting medication. You can’t let them know which kind of pill they got until after the analyses is over though, because that could skew your results. You’ll also want to get them to agree to not take any other headache medication during the experiment.

So half of your group (20 people) will be taking aspirin, and the other half (another 20 people) will get sugar pills. But who should get which? If, for example, the 20 people getting aspirin have headaches that typically last much longer than those of the 20 people that aren’t getting anything, it could make aspirin seem less effective than it really is.  So, if possible, you don’t want there to be any substantial differences between the two groups of people. A simple way to make sure this sort of thing is unlikely is to use a computer to randomly assign individuals to the two groups.

It is even better if someone else is in charge of this te randomization (secretly recording which person is assigned to which type of pill). That way, when you talk to the subjects about the experiment, there is no chance that you accidentally tip them off  (with body language, or subtle cues) to which type of pill they are getting. Furthermore, when you analyze the final results, you won’t have the temptation to make the data come out a particular way (since you won’t know until you are done which subject was taking the aspirin and which was taking the sugar pill).

Unfortunately, even with all the precautions, if you carried out this experiment on multiple occasions, you’d get somewhat different results every time. Even if you used the same study participants, the intensity of their headaches might vary from one 6 month period to the next, which would influence how the results turn out.

But, if the results fluctuate randomly, that implies that sometimes, just by luck alone, the aspirin might seem to be effective even if it isn’t. Likewise, the aspirin might, through bad luck, seem to be ineffective, even though it does work. So whatever your experiment shows, how can you be sure you’re getting the right answer?

Unfortunately, since chance is involved, certainty is not possible. But a statistician can easily calculate for you the probability that you would get results (in favor of aspirin reducing headache pain) that are at least as strong as the ones that you got, if in fact aspirin is no more effective than the sugar pill. If this probability is large, then based on your experiment you do not have sufficient evidence to conclude that aspirin helps headaches. If this probability is small (say, less than 5%), then aspirin likely is effective. In order to increase the likelihood that the results of your test are conclusive, you would need only to increase the number of participants involved.

We see that to answer questions with a high degree of certainty, a well thought out approach is necessary. Most elements of good study design become obvious when we reflect logically on the ways that data may mislead us. When possible, experiments should be triple-blind (the study participants, researchers, and statisticians all should not know who is receiving each treatment). A control group should be used, consisting either of a placebo control, or another treatment whose effectiveness is already know. Studies should have large sample sizes (20 participants in each treatment group, at the bare minimum). They should use standardized dosages, and should apply a standardized (and predetermined) method for measuring results. They should employ careful statistical analyses.

Unfortunately, many studies don’t follow these important rules. But without such rules, studies are hard to trust, even when they’re just trying to answer a simple question about aspirin.

Posted in -- By the Mathematician, Math, Philosophical, Probability | 4 Comments

Q: Why are the days still longer than nights, until a few days after the fall equinox?

Posted on September 29, 2012 by The Physicist

Physicist: The issue here is the equinoxes are the two days of the year when the length of the day should be exactly as long as the night.  And yet you’ll find that on the equinox the day is always slightly longer than 12 hours.  As it happens, there’s nothing particularly special about the exquini.  There’s just more daytime than nighttime overall.

Kind of inspiring!

Because the Sun is bigger than the Earth its light wraps just a little more than halfway around our kick-ass planet. This picture is wildly out of proportion; the Earth is actually a few dozen times smaller and a couple hundred times farther away.

But there’s one more slick trick that Earth has for keeping the lights on.  Before the light from the Sun can get to us (here on the ground) it has to pass through the atmosphere.  While the atmosphere itself doesn’t do too much (aside from scattering blue light and helping things breathe) the transition from the airlessness of space to the airfullness of here does change the direction that light travels.  This is just refraction, which is the same effect responsible for things not being where they appear to be at the bottom of swimming pools, lenses working, and blurry vision under water.

The shallower the angle at which the sunlight hits the top of the atmosphere, the more it bends. This effect ramps up close to the horizon so much that it warps the shape of the Sun. That is, the image of the Sun is raised but the bottom of the Sun is raised more than the top.

Basically, when light has to pass between a medium where it can travel fast (space counts) to one where it travels slower, the light will bend toward the slower medium.

So, every day a handful of physical laws conspire to give us a couple more minutes of sunlight than you might otherwise expect (even on the equinoctes!).

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