An important part of science---of life, I would argue---is making inferences from data. (Contrary to some popular perception, it is not all that we do, but it is a large part.) This process is a lot more interesting than many people think, and this is best conveyed by stories where it went horribly wrong. I just published a paper rebutting a paper in which it went wrong; not horribly so, but to make this post understandable I will weave a few of these horrible stories into my tale.
The paper I rebutted claimed that one method (called weak gravitational lensing) of measuring the mass of a certain galaxy cluster gave an answer too low compared to the answers obtained through two other methods, and therefore the lensing method itself was suspect. The context is that astronomers find it very difficult to measure the mass of anything, since we are so far away. If the cluster is not changing over time, we can relate the velocities of the galaxies in the cluster to its mass (called the dynamical method) and we can also relate the cluster's X-ray emission to its mass. But that's a big if, and we would like a method which does not depend on this assumption. Lensing is such a method; it has weaknesses too, but I don't want to get too deeply into that here. The central question in this paper is really simple and applies to many situations: when numbers seemingly disagree, how do we characterize the strength of disagreement given that there is some uncertainty associated with each number?
The original paper made a model of the cluster using the X-ray method, and simulated weak lensing measurements of this model to see how often the simulated measurements gave answers as low as the actual weak lensing measurements. This is a great technique; it gives us what's called a p-value. By tentatively assuming that weak lensing is as effective as the X-ray method---the "null hypothesis"---we will see how often the inherent uncertainties in weak lensing would just randomly give us an answer as low as we got in real life. If the answer is "never" then we can state that our null hypothesis is wrong and weak lensing is not as effective as the X-ray method. More quantitatively, if the answer is "in 1 out of every 100 experiments" we would say p=0.01, which has the naive interpretation of "99% confidence that the null hypothesis is rejected." (One of the reasons it's naive is that if you tested, say, 100 different true hypotheses, you would still expect one to randomly come out with p=0.01. So the true interpretation is more nuanced. I will develop this further below.)
Now, what if this method gives you p=0.1 or so? You can't really reject the null hypothesis unless you have stronger proof than that, so you may go out and take more data, do more experiments, etc, to get the stronger proof. If you do so, make sure that the new experiments are independent of the original one. For example, if you want to prove that tall people are better basketball players than short people, the null hypothesis would be that they are the same and you might record the score from a scrimmage in which a tall person plays against a short one. If the tall person comes out slightly ahead, you will not have strong proof that the tall person is better, so you might replay the scrimmage. But if you play the same two people against each other, you can never prove that tall is better; the most you might prove is that player A is better than player B. To make the trials independent, you have to play a different tall person against a different short person. In more general terms, if you're trying to get an idea of the natural variation or "noise" in your measurement, you have to repeat the measurement in a way that actually incorporates those variations. What this paper did was equivalent to failing to recognize the nonindependence of identical triplet weak lensing players. They ran three scrimmages between an X-ray player and each of these three weak lensing players, mistakenly yielding a strong conclusion about X-ray vs weak lensing.
This idea of independence---and recognizing nonindependence even when it's subtle---is really important. Ben Goldacre in his book Bad Science relates the story of a woman suspected of murder because two of her kids died of sudden infant death syndrome. The chance of one baby dying of SIDS was stated as 1 in 8543. Prosecutors assumed that the chance of a second child dying of SIDS (over the course of years, not in the same incident) was independent of the chance of the first child dying of SIDS, so we can multiply probabilities and come up with a 1 in 73,000,000 chance of two babies dying of SIDS; so unlikely that we might suspect murder. But they're not independent. If SIDS has anything to do with genes or environment then they can't be independent, because the babies have the same parents and the same house. Given the shared genes and environment, the second baby's chance of SIDS may actually be quite high. In that case, we have no reason to suspect murder. The prosecutors vastly overstated the statistical case for murder by failing to recognize the non-independence. (That's not the only mistake the prosecutors made. I highly recommend Goldacre's book.)
A second mistake the authors of the weak lensing paper made was multiplying the p-values from the three experiments to obtain an overall p-value. Many people, even scientists, fall into the following trap: Say Experiment A gives p=0.10 and you interpret that as only a 10% chance that the null hypothesis is correct. Now independent Experiment B gives p=0.08, which you interpret as only an 8% chance that the null hypothesis is correct. It is natural to think that the experiments together imply only 8% of a 10% chance of the null hypothesis being correct, or p=0.008. But it's wrong! You have vastly underestimated the chance of the null hypothesis being correct, just as the paper we rebutted vastly underestimated the chance that the weak lensing measurements were actually consistent with the dynamical and X-ray measurements. Even if the experiments are independent, you should not multiply the p-values.
Here's an easy way to confirm that the above procedure is wrong: following an equivalent procedure you could also interpret p=0.10 as a 90% chance that the null hypothesis is incorrect and p=0.08 as a 92% chance that the null hypothesis is incorrect. Multiplying them, we would get an 82.8% chance that the null hypothesis is incorrect. But the same process in the previous paragraph yielded an 0.8% chance that it's correct, which doesn't match the 82.8% chance that it's incorrect. So something's wrong with the process! And it gets more wrong as you do more experiments and more multiplications. If we do 100 independent experiments and follow this line of reasoning, we will come up with a vanishingly small chance of the null hypothesis being correct, and a vanishingly small chance of the null hypothesis being incorrect, regardless of the specific p-values, because they are always less than one. You will rule out things which do not deserve to be ruled out. Goldacre gives the horrifying example of a nurse who was suspected of murdering patients and convicted largely on the basis of faulty statistics but was eventually freed. I'm going to make up the following numbers for simplicity. Let's say that some number of patients died while she was working, such that there was only a 10% chance that that would have happened randomly. So you start poking around, and find that at the previous hospital where she worked, there was only a 50% chance of that large a number of patients dying, and at the hospital before that only a 70% chance, and at the hospital before that only a 30% chance, etc. Multiplying all these together gives a really small chance that all these things occurred randomly. But you know by now that multiplying these is wrong.
Why is it wrong? Probabilities can be multiplied (1/2 chance of heads in each coin toss means 1/4 chance of two heads in two coin tosses), but despite its name, the p-value is not a simple probability that the null hypothesis is true. It's a measure of consistency which is constructed so that for any one experiment we interpret all but the lowest p-values as being consistent with the null hypothesis. Therefore p=0.5, say, is perfectly consistent with the null hypothesis; it does not mean a 50% chance of it being true or false. In the hypothetical example above, p=0.50 actually means that an average number of patients died on shift (deaths on random shifts rose to [at least] that level 50% of the time) and p=0.70 actually means that fewer than average patients died on shift (deaths on random shifts rose to [at least] that level 70% of the time). A correct way to combine p-values for independent experiments is Fisher's method. Had the paper we rebutted used that method, they would have seen that the dynamical and weak lensing measurements were entirely consistent, even without correcting the error regarding nonindependent trials. Correcting both errors makes it even more clear.
This stuff is complicated and it's easy to go wrong. Happily, many incorrect inferences in science are caught relatively quickly because so many scientists have so much practice in this kind of analysis. But Goldacre's book is an eye-opener. Things don't always work out so well so quickly.
Friday, December 20, 2013
Monday, September 2, 2013
"Just" a Theory?
A recently published letter to the New York Times reminds us that relativity is "just a theory" and so is the Big Bang. Scientists and science educators need to set the record straight on this "just a theory" meme any time we get a chance to discuss science with kids and grown-up nonscientists. So here's my shot at it.
A good analogy is to think of facts as being like bricks: solid and dependable, but one or a few bricks are not very useful by themselves ("an electron passed through my detector at 11:58:32.01" or "the high temperature in Davis, CA on September 1, 2013 was 96 F"). Only when we assemble lots (lots) of bricks into a coherent structure do we get the benefits of having a building (the theory of relativity, or a climate model). Not only is an isolated brick rather useless, but the building can easily survive the removal of a few bricks here and there. A good theory integrates millions or billions of observations into a coherent whole. Calling relativity "just a theory" is like calling the Great Wall of China "just a fence," the Panama Canal "just a ditch," or the Golden Gate Bridge "just a road."
There's a reason that calling the Great Wall of China "just a fence" sounds more outrageous than calling relativity "just a theory"---I used the word fence which connotes something less important than a wall. There's a rich vocabulary to describe to describe barriers: from weak to strong we might use tape, rope, cordon, railing, fence, and wall. But most people don't use a similarly rich vocabulary to describe levels of sophistication of mental models. From weak to strong I might suggest educated guess, working hypothesis, model, and theory, but most people in practice indiscriminately use the word theory for any of these. So it's our duty as scientists to make clear that well-accepted scientific theories integrate an incredible range of observations into a structure which is so coherent that it is difficult to imagine all those pieces fitting into any other structure. Maybe a better analogy to calling relativity "just a theory" is calling an assembled jigsaw puzzle "just one way to fit the pieces together."
Gotcha, the just-a-theory crowd says, by making that analogy you are showing that you are rigid in your thinking and unwilling to accept alternative explanations. Nonsense. Scientists are constantly trying to prove accepted theories wrong. Anyone who succeeds in disproving relativity, the Big Bang, or evolution will win a Nobel Prize and eternal fame, so we'd be happy to do so. But we know from experience that the most likely explanation for an isolated fact that seems to contradict relativity, the Big Bang, or evolution is that the fact itself was taken out of context or is not being properly interpreted, rather than that an extremely well-tested theory is wrong.
This doesn't mean that we will twist any fact to make it fit into our well-accepted theories. It does mean that surprising facts may end up extending the theory rather than replacing it. For example, Newton's theory of gravity explains a ton of observations about the motions of the planets and stars, but in a few extreme circumstances (such as very close to the Sun) it doesn't predict exactly what is observed. Einstein developed a theory of gravity (general relativity) which does correctly predict these situations. Einstein's theory is more complicated than Newton's, but in most situations the complicated parts of Einstein's theory have very little quantitative effect so we can simplify it a great deal and in those cases it turns out to be identical to....Newton's theory! This almost had to be the case, because Newton's theory accounted so well for so many observations that it would be hard to imagine that it was wrong rather than incomplete.
This example shows that a small number of facts can be critically important and that scientists do pay attention to facts which don't fit the theory. But we don't modify or overturn theories willy-nilly. When the planet Uranus didn't move exactly as Newton's theory predicted, modifications of the theory were considered but so was the possibility that some mass other than the Sun and the known planets was pulling on Uranus, and that led to the discovery of Neptune. If we rejected well-established theories at the first hint of any discrepancy with new observations, we would be giving undue weight to the new observations and too little weight to the vast range of previous observations explained by the theory. If you want to overthrow a theory because some new observation seems to contradict it, then give us a better theory which explains the new observation while still fitting the previous observations just as well as the old theory. That latter part seems to be conveniently forgotten by people who want to reject well-established theories.
A closely parallel situation is that of criminal investigators and prosecutors who present their "theory of the crime" to a jury. ("Model of the crime" would better fit my vocabulary hierarchy, but this is the word actually used.) A lot of facts may be introduced into evidence ("a car with the suspect's license plate was recorded crossing the Tappan Zee Bridge at 2:20am on August 31"), but by themselves they don't mean anything important. A good theory of the crime provides a coherent explanation of so many different facts that the jury is forced to conclude that it is true beyond a reasonable doubt. If you want to call it "just a theory" then offer us a different theory which fits the facts just as well. The defense is given sufficient time and strong motivation to offer a good alternative theory, so failure to present one is damning.
A good analogy is to think of facts as being like bricks: solid and dependable, but one or a few bricks are not very useful by themselves ("an electron passed through my detector at 11:58:32.01" or "the high temperature in Davis, CA on September 1, 2013 was 96 F"). Only when we assemble lots (lots) of bricks into a coherent structure do we get the benefits of having a building (the theory of relativity, or a climate model). Not only is an isolated brick rather useless, but the building can easily survive the removal of a few bricks here and there. A good theory integrates millions or billions of observations into a coherent whole. Calling relativity "just a theory" is like calling the Great Wall of China "just a fence," the Panama Canal "just a ditch," or the Golden Gate Bridge "just a road."
There's a reason that calling the Great Wall of China "just a fence" sounds more outrageous than calling relativity "just a theory"---I used the word fence which connotes something less important than a wall. There's a rich vocabulary to describe to describe barriers: from weak to strong we might use tape, rope, cordon, railing, fence, and wall. But most people don't use a similarly rich vocabulary to describe levels of sophistication of mental models. From weak to strong I might suggest educated guess, working hypothesis, model, and theory, but most people in practice indiscriminately use the word theory for any of these. So it's our duty as scientists to make clear that well-accepted scientific theories integrate an incredible range of observations into a structure which is so coherent that it is difficult to imagine all those pieces fitting into any other structure. Maybe a better analogy to calling relativity "just a theory" is calling an assembled jigsaw puzzle "just one way to fit the pieces together."
Gotcha, the just-a-theory crowd says, by making that analogy you are showing that you are rigid in your thinking and unwilling to accept alternative explanations. Nonsense. Scientists are constantly trying to prove accepted theories wrong. Anyone who succeeds in disproving relativity, the Big Bang, or evolution will win a Nobel Prize and eternal fame, so we'd be happy to do so. But we know from experience that the most likely explanation for an isolated fact that seems to contradict relativity, the Big Bang, or evolution is that the fact itself was taken out of context or is not being properly interpreted, rather than that an extremely well-tested theory is wrong.
This doesn't mean that we will twist any fact to make it fit into our well-accepted theories. It does mean that surprising facts may end up extending the theory rather than replacing it. For example, Newton's theory of gravity explains a ton of observations about the motions of the planets and stars, but in a few extreme circumstances (such as very close to the Sun) it doesn't predict exactly what is observed. Einstein developed a theory of gravity (general relativity) which does correctly predict these situations. Einstein's theory is more complicated than Newton's, but in most situations the complicated parts of Einstein's theory have very little quantitative effect so we can simplify it a great deal and in those cases it turns out to be identical to....Newton's theory! This almost had to be the case, because Newton's theory accounted so well for so many observations that it would be hard to imagine that it was wrong rather than incomplete.
This example shows that a small number of facts can be critically important and that scientists do pay attention to facts which don't fit the theory. But we don't modify or overturn theories willy-nilly. When the planet Uranus didn't move exactly as Newton's theory predicted, modifications of the theory were considered but so was the possibility that some mass other than the Sun and the known planets was pulling on Uranus, and that led to the discovery of Neptune. If we rejected well-established theories at the first hint of any discrepancy with new observations, we would be giving undue weight to the new observations and too little weight to the vast range of previous observations explained by the theory. If you want to overthrow a theory because some new observation seems to contradict it, then give us a better theory which explains the new observation while still fitting the previous observations just as well as the old theory. That latter part seems to be conveniently forgotten by people who want to reject well-established theories.
A closely parallel situation is that of criminal investigators and prosecutors who present their "theory of the crime" to a jury. ("Model of the crime" would better fit my vocabulary hierarchy, but this is the word actually used.) A lot of facts may be introduced into evidence ("a car with the suspect's license plate was recorded crossing the Tappan Zee Bridge at 2:20am on August 31"), but by themselves they don't mean anything important. A good theory of the crime provides a coherent explanation of so many different facts that the jury is forced to conclude that it is true beyond a reasonable doubt. If you want to call it "just a theory" then offer us a different theory which fits the facts just as well. The defense is given sufficient time and strong motivation to offer a good alternative theory, so failure to present one is damning.
Tuesday, June 11, 2013
Great Balls of Fire
After learning about gravity and taking the midmorning break, the Peregrine 3-4 graders and I worked on understanding nuclear fusion in the core of the Sun and where elements come from.
I started by setting the context. The students had studied atoms and molecules the previous year so I started by drawing a molecule of water (two hydrogen atoms and one oxygen atom) and reminding them of the evidence for atoms and molecules. Then we zoomed in to one hydrogen atom and discussed the Rutherford experiment showing that atoms are very fluffy; most of their volume is nearly empty while nearly all their mass is concentrated in a tiny volume in the center (nucleus). Then we zoomed in further by a factor of 10,000 to the nucleus. For a hydrogen atom, the nucleus is a single positively charged particle called a proton. I held up a ping-pong ball as a proton and said that if protons really were that size, the atom would have to be the size of South Davis.
To reinforce the sense of scale, I showed the movie Powers of Ten. This classic ten-minute movie should be seen by anyone wanting to understand the universe. I also took the time to answer questions about it.
The basic rules of nuclear physics are actually understandable by anyone. Last year we investigated the effects of electrical charge, and concluded that like charges repel while opposite charges attract. Atoms beyond hydrogen in the periodic table have more protons. But why do the protons stick together if they repel each other? There must be some form of glue. I demonstrated two magnets which repelled each other. They were "donut" magnets threaded onto a rod so they didn't flop around and the repulsion was clear. But when I turned the rod vertically and one magnet fell with enough speed onto the other one, they touched briefly. That was enough for the velcro on their surfaces to attach and keep them together. The velcro is a short-range force, like the strong nuclear force which keeps a nucleus together.
But protons alone can't generate sufficient strong nuclear force to keep nuclei together. Another type of particle, with similar mass but no charge and called a neutron, provides the glue. Nuclei need roughly equal amounts of protons and neutrons to be stable. I modeled this with a bunch of ping-pong balls I had wrapped with velcro. The "protons" had velcro hooks and the "neutrons" had velcro loops, so that you needed roughly equal numbers of each to build up a large nucleus. (The different types were also different colors to make the idea plainly visible.) Adding a neutron to a nucleus adds mass, but doesn't otherwise change the properties of the atom. For example, a proton plus a neutron is still hydrogen, but we call it a different isotope of hydrogen. Similarly, carbon-12 (usually written with a superscript 12 on the left) and carbon-14 are different isotopes of carbon which differ by two neutrons.
With that in mind, we can start building up more complicated elements from hydrogen. Element number 2 (two protons) is helium, and we need two neutrons to provide the glue so the most common isotope of helium is helium-4. The protons have to be smashed together at very high speed if they are to ever get close enough for the "velcro" of the strong nuclear force to make them stick, so we need very high temperatures to make this fusion process happen. (High temperature means the individual microscopic particles are wiggling or bounding around at high speed.) We find it difficult to make these high temperatures on Earth, but the core of the Sun is 15 million degrees (Celsius; tens of millions of degrees if you think in Fahrenheit) and this happens quite routinely. In fact, most stars turn hydrogen into helium in their cores.
Fusing helium into even heavier elements is harder, but most stars will do that as well by the ends of their lives. It turns out that crashing two heliums together results in an unstable isotope of element 4 (beryllium), which quickly decays back into two helium-4 nuclei. But if you manage to crash a third helium into the two heliums before the two-helium complex has a chance to decay, you make carbon-12 (the most common isotope of element 6, carbon; again, equal amounts of protons and neutrons). Then, if you crash another helium into that, you get element number 8: oxygen. Another helium into that produces element 10, argon. These helium capture reactions are common in massive stars (substantially more massive than the Sun), and they create more of the even-numbered elements than the odd-numbered elements (nitrogen, fluorine, etc). They go all the way up to iron (element 26). I modeled all this with the velcro-covered ping-pong balls.
Have you noticed what we've done here? We've explained the origin of the elements using basic, well-understood physical processes. That's pretty cool! Here's a graph of the observed abundances:
You can see that hydrogen is the most abundant, followed by helium, then the even-numbered elements carbon, oxygen....through iron. But why are there elements beyond iron if stars only make up to iron? Well, stars make up to iron when they are in equilibrium. But when they explode (a supernova), so much energy is released that even more complicated nuclei can be made. I won't explain the details here, but the abundances of all the elements beyond iron are well understood as consequences of supernovae. That we can understand all the features of the above plot is, to me, one of the most amazing things in all of science.
The supernova explosions are also what throw the newly-manufactured elements back into space, where they can mix into gas clouds that eventually collapse to form new stars. That means that the atoms in your body were once inside another star. (Not from the Sun, because new atoms made in the Sun won't escape until the end of its life.)
Supernovae make some unstable elements, like uranium. The most common type of decay for a heavy element is to violently eject a "bullet" made of two protons and two neutrons, in other words a helium nucleus (again I modeled this with the ping-pong balls). This is why there is helium on Earth; our gravity is too weak to hold on to helium gas, but helium produced by radioactive decays is trapped in rocks underground. When we drill for natural gas, we can capture some of this helium and eventually use it to fill balloons. When it escapes from the balloon, it eventually escapes into space.
I started by setting the context. The students had studied atoms and molecules the previous year so I started by drawing a molecule of water (two hydrogen atoms and one oxygen atom) and reminding them of the evidence for atoms and molecules. Then we zoomed in to one hydrogen atom and discussed the Rutherford experiment showing that atoms are very fluffy; most of their volume is nearly empty while nearly all their mass is concentrated in a tiny volume in the center (nucleus). Then we zoomed in further by a factor of 10,000 to the nucleus. For a hydrogen atom, the nucleus is a single positively charged particle called a proton. I held up a ping-pong ball as a proton and said that if protons really were that size, the atom would have to be the size of South Davis.
To reinforce the sense of scale, I showed the movie Powers of Ten. This classic ten-minute movie should be seen by anyone wanting to understand the universe. I also took the time to answer questions about it.
The basic rules of nuclear physics are actually understandable by anyone. Last year we investigated the effects of electrical charge, and concluded that like charges repel while opposite charges attract. Atoms beyond hydrogen in the periodic table have more protons. But why do the protons stick together if they repel each other? There must be some form of glue. I demonstrated two magnets which repelled each other. They were "donut" magnets threaded onto a rod so they didn't flop around and the repulsion was clear. But when I turned the rod vertically and one magnet fell with enough speed onto the other one, they touched briefly. That was enough for the velcro on their surfaces to attach and keep them together. The velcro is a short-range force, like the strong nuclear force which keeps a nucleus together.
But protons alone can't generate sufficient strong nuclear force to keep nuclei together. Another type of particle, with similar mass but no charge and called a neutron, provides the glue. Nuclei need roughly equal amounts of protons and neutrons to be stable. I modeled this with a bunch of ping-pong balls I had wrapped with velcro. The "protons" had velcro hooks and the "neutrons" had velcro loops, so that you needed roughly equal numbers of each to build up a large nucleus. (The different types were also different colors to make the idea plainly visible.) Adding a neutron to a nucleus adds mass, but doesn't otherwise change the properties of the atom. For example, a proton plus a neutron is still hydrogen, but we call it a different isotope of hydrogen. Similarly, carbon-12 (usually written with a superscript 12 on the left) and carbon-14 are different isotopes of carbon which differ by two neutrons.
With that in mind, we can start building up more complicated elements from hydrogen. Element number 2 (two protons) is helium, and we need two neutrons to provide the glue so the most common isotope of helium is helium-4. The protons have to be smashed together at very high speed if they are to ever get close enough for the "velcro" of the strong nuclear force to make them stick, so we need very high temperatures to make this fusion process happen. (High temperature means the individual microscopic particles are wiggling or bounding around at high speed.) We find it difficult to make these high temperatures on Earth, but the core of the Sun is 15 million degrees (Celsius; tens of millions of degrees if you think in Fahrenheit) and this happens quite routinely. In fact, most stars turn hydrogen into helium in their cores.
Fusing helium into even heavier elements is harder, but most stars will do that as well by the ends of their lives. It turns out that crashing two heliums together results in an unstable isotope of element 4 (beryllium), which quickly decays back into two helium-4 nuclei. But if you manage to crash a third helium into the two heliums before the two-helium complex has a chance to decay, you make carbon-12 (the most common isotope of element 6, carbon; again, equal amounts of protons and neutrons). Then, if you crash another helium into that, you get element number 8: oxygen. Another helium into that produces element 10, argon. These helium capture reactions are common in massive stars (substantially more massive than the Sun), and they create more of the even-numbered elements than the odd-numbered elements (nitrogen, fluorine, etc). They go all the way up to iron (element 26). I modeled all this with the velcro-covered ping-pong balls.
Have you noticed what we've done here? We've explained the origin of the elements using basic, well-understood physical processes. That's pretty cool! Here's a graph of the observed abundances:
You can see that hydrogen is the most abundant, followed by helium, then the even-numbered elements carbon, oxygen....through iron. But why are there elements beyond iron if stars only make up to iron? Well, stars make up to iron when they are in equilibrium. But when they explode (a supernova), so much energy is released that even more complicated nuclei can be made. I won't explain the details here, but the abundances of all the elements beyond iron are well understood as consequences of supernovae. That we can understand all the features of the above plot is, to me, one of the most amazing things in all of science.
The supernova explosions are also what throw the newly-manufactured elements back into space, where they can mix into gas clouds that eventually collapse to form new stars. That means that the atoms in your body were once inside another star. (Not from the Sun, because new atoms made in the Sun won't escape until the end of its life.)
Supernovae make some unstable elements, like uranium. The most common type of decay for a heavy element is to violently eject a "bullet" made of two protons and two neutrons, in other words a helium nucleus (again I modeled this with the ping-pong balls). This is why there is helium on Earth; our gravity is too weak to hold on to helium gas, but helium produced by radioactive decays is trapped in rocks underground. When we drill for natural gas, we can capture some of this helium and eventually use it to fill balloons. When it escapes from the balloon, it eventually escapes into space.
Big Bang
I left out one detail in the story above: most of the helium in the plot was actually made in the Big Bang. Some of the kids had expressed interest in the Big Bang previously, so I used the remaining time to talk about that. I used the usual balloon-with-stickers demo, and I also showed this interactive tool made by an undergraduate student of mine. The point of the tool is to show that although we see all galaxies moving away from us, observers in all other galaxies also see all galaxies moving away from them. So we are not at the center of anything. If we think back in time, all galaxies were closer to each other, so the universe was denser (and hotter). Far enough back in time, the universe was so hot (everywhere) that a fair amount of hydrogen fused into helium. This is called Big Bang nucleosynthesis (BBN). We can look at the abundance of various BBN byproducts, like hydrogen-2 (aka deuterium) and confirm that this really happened.Wrapping up
Most of this trimester we worked on understanding the immense size of space. If this makes you feel insignificant, remember that you are made of atoms from another star. You are a part of the universe which can actually understand itself.Monday, June 10, 2013
The Gravity of the Situation
Friday was my last day doing astronomy with the 3-4 graders at Peregrine School. The one standard I hadn't yet covered was gravity, so we did gravity before the break (this post) and after the break we discussed nuclear fusion in the Sun's core (next post).
I reviewed some ideas about motion we had discussed last year. If you roll a marble, you expect it to go in a straight line unless something (another kid, perhaps, or a wall) interferes by pushing (exerting a force) on the marble. That's Newton's first law of motion. I then put a donut on a string and spun the donut in a circle over my head. What will happen if the string is cut? Will the donut continue in a circle, fly off in a straight line, or fly off in a curve? We took a vote. I always clarify that the question is about what happens immediately, not about what happens eventually, like the donut falling due to the gravity in the room. This means that when we do the experiment, they have to really pay attention!
In reality I don't cut the string, but the string pulls through the soft donut, and it flies off in a straight line---Newton's first law again. This is a pretty vivid demonstration that the Moon wouldn't keep going around the Earth, nor the planets around the Sun, unless there was a force keeping them from flying off in a straight line. Kids this age already know that we call that force gravity, but gravity is also the force that makes things fall when I drop them. Why do we call these two forces by the same name?
I also have a tennis ball on a string so I can demonstrate circular motion as much as needed. I do this and ask the kids what direction the force must be in. It must be towards the center of the circle, where my fist is holding the string. That's clear because the only direction a string can exert a force is pulling along the string! So whatever force is pulling on the Moon, it must be pointed toward the center of the Earth. And that's exactly what we observe about gravity on Earth! (It helps to draw an Earth and how the arrow of gravity points in your location vs in, say, Australia.) So it's quite plausible that these two forces are really the same force.
To bolster the argument that these are the same force, we should look not just at the direction, but also the strength. I had the kids whirl the tennis ball on a string at various speeds, and feel whether the higher speed requires more force, less force, or the same force (the answer is more). So let's look at the planets' speeds around the Sun and see if we can relate that to the force of gravity. I asked the kids for suggestions as to what would affect the planet speed. The two main suggestions were planet size, and planet distance from the Sun. It would have been great to investigate both of these possibilities, but we were running short on time so we just did planet distance from the Sun. I had the kids make graphs of planet speed vs planet distance from the Sun. We took our time doing this right, figuring out how to draw the axes with reasonable scales, and adding planets one by one, starting with the most familiar ones.
A pattern did emerge: more distant planets are slower, as the graph below shows.
By our tennis ball experiment, slower circular motion implies a weaker pull (less acceleration). Therefore this graph implies that more distant planets feel a weaker pull, and planets closer to the Sun feel a stronger pull. Does this make sense if the Sun's gravity is what keeps the planets from flying off in straight-line paths? The kids agreed that it did.
[If we had also made the graph of speed vs planet size, we would not have seen such a clear pattern. It happens that the outer planets tend to be bigger, so that there would be a tendency for bigger planets to be slower, but it would only be a tendency, not a law, because the biggest planet happens to be the nearest (fastest) of the outer four. And the pattern would really be broken if we also included Pluto, which is a very distant (hence very slow), small object, providing a counterexample to the fast inner planets which happen to be small and which therefore might give someone the false impression that small means fast.]
I liked this 40-minute activity and I think it worked well. I did simplify some details to avoid getting bogged down (eg the distinction between force and acceleration), but I think it was appropriate for 3-4 graders who wanted to focus on astronomy rather than physics. We also got in some more practice with graphs, which is important. And we learned something which in Newton's time was revolutionary: the same laws of physics which we can deduce here on Earth also apply to objects in the sky. This was one of the most wonderful discoveries in the history of science, and it's what allows us to understand the universe.
I reviewed some ideas about motion we had discussed last year. If you roll a marble, you expect it to go in a straight line unless something (another kid, perhaps, or a wall) interferes by pushing (exerting a force) on the marble. That's Newton's first law of motion. I then put a donut on a string and spun the donut in a circle over my head. What will happen if the string is cut? Will the donut continue in a circle, fly off in a straight line, or fly off in a curve? We took a vote. I always clarify that the question is about what happens immediately, not about what happens eventually, like the donut falling due to the gravity in the room. This means that when we do the experiment, they have to really pay attention!
In reality I don't cut the string, but the string pulls through the soft donut, and it flies off in a straight line---Newton's first law again. This is a pretty vivid demonstration that the Moon wouldn't keep going around the Earth, nor the planets around the Sun, unless there was a force keeping them from flying off in a straight line. Kids this age already know that we call that force gravity, but gravity is also the force that makes things fall when I drop them. Why do we call these two forces by the same name?
I also have a tennis ball on a string so I can demonstrate circular motion as much as needed. I do this and ask the kids what direction the force must be in. It must be towards the center of the circle, where my fist is holding the string. That's clear because the only direction a string can exert a force is pulling along the string! So whatever force is pulling on the Moon, it must be pointed toward the center of the Earth. And that's exactly what we observe about gravity on Earth! (It helps to draw an Earth and how the arrow of gravity points in your location vs in, say, Australia.) So it's quite plausible that these two forces are really the same force.
To bolster the argument that these are the same force, we should look not just at the direction, but also the strength. I had the kids whirl the tennis ball on a string at various speeds, and feel whether the higher speed requires more force, less force, or the same force (the answer is more). So let's look at the planets' speeds around the Sun and see if we can relate that to the force of gravity. I asked the kids for suggestions as to what would affect the planet speed. The two main suggestions were planet size, and planet distance from the Sun. It would have been great to investigate both of these possibilities, but we were running short on time so we just did planet distance from the Sun. I had the kids make graphs of planet speed vs planet distance from the Sun. We took our time doing this right, figuring out how to draw the axes with reasonable scales, and adding planets one by one, starting with the most familiar ones.
A pattern did emerge: more distant planets are slower, as the graph below shows.
By our tennis ball experiment, slower circular motion implies a weaker pull (less acceleration). Therefore this graph implies that more distant planets feel a weaker pull, and planets closer to the Sun feel a stronger pull. Does this make sense if the Sun's gravity is what keeps the planets from flying off in straight-line paths? The kids agreed that it did.
[If we had also made the graph of speed vs planet size, we would not have seen such a clear pattern. It happens that the outer planets tend to be bigger, so that there would be a tendency for bigger planets to be slower, but it would only be a tendency, not a law, because the biggest planet happens to be the nearest (fastest) of the outer four. And the pattern would really be broken if we also included Pluto, which is a very distant (hence very slow), small object, providing a counterexample to the fast inner planets which happen to be small and which therefore might give someone the false impression that small means fast.]
I liked this 40-minute activity and I think it worked well. I did simplify some details to avoid getting bogged down (eg the distinction between force and acceleration), but I think it was appropriate for 3-4 graders who wanted to focus on astronomy rather than physics. We also got in some more practice with graphs, which is important. And we learned something which in Newton's time was revolutionary: the same laws of physics which we can deduce here on Earth also apply to objects in the sky. This was one of the most wonderful discoveries in the history of science, and it's what allows us to understand the universe.
Friday, May 31, 2013
Light and Telescopes
In the second half of this morning's activities with the 3-4 graders, we discovered some things about light and telescopes. I handed out diffraction gratings and we looked at the spectrum of the Sun and of the fluorescent lights in the room, discovering that white light is actually composed of many colors. We also looked at discharge tubes filled with different elements, with mercury and helium being the stars. We found that each element emits a unique "fingerprint" of spectral lines. To see a great 2-minute video of everything the kids saw, check this out. This is how we know what stars and other planets are made of.
We then discussed how the colors always appear in a certain order in a rainbow or a diffraction grating: red, orange, yellow, green, blue, violet. Could there be any light which appears before red? Yes, it's called infrared, and we can build cameras to see it even though our eyes can't. I showed this nice video demonstrating the properties of infrared light. Could there be any light which appears after violet? Yes, ultraviolet, and after that would be X-rays and finally gamma rays. We talked about X-rays for a while because some kids were worried about it being dangerous. (Like many other things, they are safe if used properly, but dangerous if not. A yearly dental X-ray is ok, but how do we protect the parts of our bodies which don't need to be X-rayed? And how do we protect the workers who administer dozens of X-rays each day?) I extended that discussion to the ultraviolet and sunlight.
All this was a springboard for discussing telescopes, which is one of the last astronomy standards I hadn't covered yet. Specialized telescopes are built to look at all kinds of light, from gamma rays to the infrared and radio. I showed pictures of some of the big telescopes I have used in my research, and that led to all kinds of interesting questions. We ran out of time, so I may start next Friday by answering more telescope questions.
We then discussed how the colors always appear in a certain order in a rainbow or a diffraction grating: red, orange, yellow, green, blue, violet. Could there be any light which appears before red? Yes, it's called infrared, and we can build cameras to see it even though our eyes can't. I showed this nice video demonstrating the properties of infrared light. Could there be any light which appears after violet? Yes, ultraviolet, and after that would be X-rays and finally gamma rays. We talked about X-rays for a while because some kids were worried about it being dangerous. (Like many other things, they are safe if used properly, but dangerous if not. A yearly dental X-ray is ok, but how do we protect the parts of our bodies which don't need to be X-rayed? And how do we protect the workers who administer dozens of X-rays each day?) I extended that discussion to the ultraviolet and sunlight.
All this was a springboard for discussing telescopes, which is one of the last astronomy standards I hadn't covered yet. Specialized telescopes are built to look at all kinds of light, from gamma rays to the infrared and radio. I showed pictures of some of the big telescopes I have used in my research, and that led to all kinds of interesting questions. We ran out of time, so I may start next Friday by answering more telescope questions.
Scale Model Solar System Complete!
This morning I guided the 3-4 graders through assembling our scale model solar system. I wanted them to really think about how to make a scale model, so I returned to each student the graph they had made last time and I asked them to use the graph to figure out where they would put their planet, given that I had put Teacher Moné's beautiful Earth poster 2.5 meters from the Sun poster. Of course, I found that I needed to break this task into smaller chunks for them to process. We began by revisiting some of the steps we had done last week. Each child identified his/her planet on the graph, read its distance off the graph, and then we thought about what that distance means. For example, Jupiter is at a distance of 5 on the graph. Five what? The graph doesn't say. But the graph itself is a scale model of the solar system. We don't really care what the actual distance is because we are simply stretching this scale model to become a larger scale model which will fill the school. All we need to do is choose a reference point and stretch everything else accordingly. The graph made this easy because it shows Earth as being at a distance of 1. So if Jupiter is at 5, we simply need to put Jupiter 5 times farther from the Sun than Earth is from the Sun; in other words 5x2.5 meters or 12.5 meters.
To help the kids visualize this, I took a rubber band and marked three dots on it, representing Sun, Earth, and Jupiter. This is a scale model much like the graph (if we ignore the vertical dimension of the graph). If I stretch the rubber band, will Jupiter still be 5 times more distant from the Sun than the Earth is from the Sun? Some kids said no and some said yes, so we took a vote. Having to commit to a vote made the kids think harder and they voted overwhelmingly yes. After the vote I did stretch the rubber band and I did get a bigger scale model. In principle, if we got a really long rubber band, I could mark all the planets' distances at the scale of the graph and then stretch it out to get a giant scale model as big as the school, and that would tell us where to put each planet poster. But since that's impractical, we do the math instead.
This seems to have been more or less the right level of conceptual challenge and the right level of math for the kids. They found it a bit of a challenge, but a doable one that became satisfying rather than frustrating. After looking over each child's computation, we practiced some metacognition. Alex was concerned that his number didn't make sense given what he knew about the relative positions of the Sun, Earth and Venus. It turned out that he was misinterpreting his number as the Earth-Venus distance, but the point was a really important one: always check that your numerical results make sense! I have had so many students make a mistake punching numbers into a calculator, and get a number that obviously doesn't make sense given a moment's thought, but blithely write down the number as if any number displayed by a calculator must be correct. In this case we wrote out the multiplication rather than use a calculator, but the same principle applies: check that the results actually make sense! This goes not only for numbers that you compute, but also for numbers that other people compute for you.
An especially effective way to double-check your number is to perform some completely different procedure; if you just perform the original procedure again, you may easily make the same mistake again. So I thought of a way we could all check our numbers without recomputing anything. I made a list of the students' results, starting with the closest planet and proceeding outward. If the distance numbers didn't increase steadily, that would be a smoking gun indicating a mistake. And we did find a mistake this way, so it was instructive.
Once we had our final numbers, we split into groups to measure off the distances and attach the posters to the walls. We couldn't quite fit Neptune into the school grounds, and Orcus wasn't even close, but we put them up at the far end with a note saying where they should really be. Even after choosing a scale so large that the orbit of Neptune was just outside the fence, the sizes of the planets are really small, smaller than a grain of sand for most planets. Even Jupiter is only 2.4mm across. Space is really big!
Looking at the finished product, I am really happy we did it and spent enough time on it to do it right. We certainly appreciate the solar system much better now, but we also learned new ways of thinking.
To help the kids visualize this, I took a rubber band and marked three dots on it, representing Sun, Earth, and Jupiter. This is a scale model much like the graph (if we ignore the vertical dimension of the graph). If I stretch the rubber band, will Jupiter still be 5 times more distant from the Sun than the Earth is from the Sun? Some kids said no and some said yes, so we took a vote. Having to commit to a vote made the kids think harder and they voted overwhelmingly yes. After the vote I did stretch the rubber band and I did get a bigger scale model. In principle, if we got a really long rubber band, I could mark all the planets' distances at the scale of the graph and then stretch it out to get a giant scale model as big as the school, and that would tell us where to put each planet poster. But since that's impractical, we do the math instead.
This seems to have been more or less the right level of conceptual challenge and the right level of math for the kids. They found it a bit of a challenge, but a doable one that became satisfying rather than frustrating. After looking over each child's computation, we practiced some metacognition. Alex was concerned that his number didn't make sense given what he knew about the relative positions of the Sun, Earth and Venus. It turned out that he was misinterpreting his number as the Earth-Venus distance, but the point was a really important one: always check that your numerical results make sense! I have had so many students make a mistake punching numbers into a calculator, and get a number that obviously doesn't make sense given a moment's thought, but blithely write down the number as if any number displayed by a calculator must be correct. In this case we wrote out the multiplication rather than use a calculator, but the same principle applies: check that the results actually make sense! This goes not only for numbers that you compute, but also for numbers that other people compute for you.
An especially effective way to double-check your number is to perform some completely different procedure; if you just perform the original procedure again, you may easily make the same mistake again. So I thought of a way we could all check our numbers without recomputing anything. I made a list of the students' results, starting with the closest planet and proceeding outward. If the distance numbers didn't increase steadily, that would be a smoking gun indicating a mistake. And we did find a mistake this way, so it was instructive.
Once we had our final numbers, we split into groups to measure off the distances and attach the posters to the walls. We couldn't quite fit Neptune into the school grounds, and Orcus wasn't even close, but we put them up at the far end with a note saying where they should really be. Even after choosing a scale so large that the orbit of Neptune was just outside the fence, the sizes of the planets are really small, smaller than a grain of sand for most planets. Even Jupiter is only 2.4mm across. Space is really big!
Looking at the finished product, I am really happy we did it and spent enough time on it to do it right. We certainly appreciate the solar system much better now, but we also learned new ways of thinking.
Friday, May 24, 2013
Planet Posters
Two weeks ago each student chose a planet (or other solar system object) to research and make a poster about. Today they brought in their posters, and each student told the class what they learned in their research. The kids were very engaged and asked so many good questions that we spent all morning doing this. So next week we will put up the posters at the appropriate distances from the Sun poster (which I made and put up near the school entrance today) to make a scale model of the solar system. The discussions today were so full, frank, and wide-ranging that I can't hope to capture them in a blog post. I will simply leave you with a short video with amazing images of Jupiter's moon Europa.
I think the posters were quite successful as a learning experience. The kids learned by researching and making them, but they also learned by listening to other kids talk about their posters, and they all learned when I answered numerous questions in more depth as they arose. I think a key to real learning is that the posters should not be just a laundry list of facts, but should really be based on the students' questions. When I issued the assignment, I offered some questions they might be interested in answering:
I think the posters were quite successful as a learning experience. The kids learned by researching and making them, but they also learned by listening to other kids talk about their posters, and they all learned when I answered numerous questions in more depth as they arose. I think a key to real learning is that the posters should not be just a laundry list of facts, but should really be based on the students' questions. When I issued the assignment, I offered some questions they might be interested in answering:
- What would it be like to visit? What is the temperature? Is there a solid surface? Would the Sun look bright from that distance? If the temperature is extreme, think about ways to convey how extreme it is.
- Does the planet have moons or rings? If you chose a moon to begin with, briefly describe the host planet.
- What are seasons like on that planet? This depends on how tilted the planet is with respect to its orbit.
- How long is the year on that planet? How long is a day?
- Are there volcanoes? Rocks? Rivers/lakes/oceans? (If so, are they made of water or some other substance? Moons of Jupiter and Saturn are especially interesting in this respect.) Clouds? Earthquakes? Storms? Lightning?
- Could you possibly find life there?
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