Built on Facts

A couple days before I started teaching recitation sessions for Physics 208 (the E&M half of calc-based intro physics) this summer, I found out that in fact I was not teaching the second summer session, I’m teaching for the second half of the full summer session. Turns out there is a difference! For some bizarre reason, this means that the summer Physics 208 class is actually taught by one professor for the first half of the summer and another professor for the second half. Each 208 section switches TAs midstream as well - I’m therefore kind of a relief pitcher, I guess. Why is it set up this way? I have no idea, but if I had to guess it’s so professors have time off during the summer to go to more conferences in Hawaii and get more research done without interruption. Oh well, makes no difference to me, other than hoping the previous TA hasn’t been sloppy in his teaching. (It in a “he” in this case, though there are plenty of female TAs as well!)

This week we’re doing magnetic fields. They’re surprisingly weird. The fields encountered so far in introductory classes are gravitational and electric fields, in which force points along the field direction and a nice simple scalar potential exists. Neither of these are true with magnetic fields. The force on a particle with charge q in a magnetic field B is

We have a vector cross product involved (What’s a cross product? This.), and weirder still it involves the velocity v of the particle. A stationary particle feels no magnetic force, and a faster moving particle experiences a greater force. Since it’s a cross product the angle matters too. A particle moving parallel to the field experiences no force; a particle moving perpendicular to the field experiences the greatest force.

For a long time electricity and magnetism were treated as two completely different things before James Maxwell found a theory which could describe the relationship between the interactions. Einstein took this even farther and discovered that electricity and magnetism weren’t just related halves of the same theory, they are literally the same thing under a particular coordinate transformation. We don’t worry about the relativistic description in an intro class obviously, we’re just interested in communicating to students what magnetic fields do and where they come from. A little unusually, the question I always get asked is why is “B” used for the magnetic field instead of the more obvious “M”. The answer is that M is already taken by a quantity called magnetization.

The equation above is called the Lorentz force law, and theme and variations on that one equation make up the entire week’s worth of discussion. Let me quote the simple quiz I gave them (modified from one of the homework problems); working through it will help bring home the concepts.

A helium nucleus (charge +2e) moving horizontally from west to east with a speed of 1000.0 km/s experiences a magnetic force of 0.000500 nN vertically downward. What is the magnitude and direction of the weakest magnetic field required to produce this force? How could this same force originate from a stronger field?

Let’s take it from the top. The force from the cross product is going to have to be perpendicular to both the velocity and the field, so since the force is downward and the velocity is west-east, the field has to also be parallel to the ground. The cross product is a maximum where the velocity and the field are perpendicular, so the strongest force from that weak field will be when the field is pointing directly south. That’s the direction.

This reasoning also answers the second question: a stronger field parallel to the ground but not directly pointing south could produce the same force in the same direction as the weaker field pointing exactly in the right direction.

But what’s the magnitude? Well, we have (since we know v and B are perpendicular)

Solve for B.

Plugging in the values in the problem, I get around 1.56 Tesla. This is a pretty huge magnetic field, roughly comparable to that of an MRI.

That’s about as bare-bones of an introduction as you can get, but I hope it’s a good place to start for students who happen across this site. Next week is sources of magnetic field.

Physics is intimately bound up with probability and statistics for two main reasons. First, both thermodynamics and quantum mechanics are intrinsically probabilistic theories. So are some others, but those two in particular really embody the statistical concepts central to modern physics. Second, much of experimental physics is done at the bleeding edge of what our instruments can measure. It’s rare that a single measurement can adequately test a theory. Generally one has to conduct the same experiment numerous times to be sure whether an observed effect is real or just noise. This is especially true in experimental high-energy physics because by definition the effects you’re most interested in occur at the very top of the energy range available in your collider. As such, physicists spend a lot of time thinking about chance and statistics, and how those concepts affect the validity of their results. Scientists aren’t the only ones who think about chance, of course. How often have you heard the following?

“The lottery is a tax on those who don’t understand probability.”

I hear it all the time, and I used to say it. It’s partially true: most people don’t understand probability. Ignorance of the mathematical theory completely aside, most people have had their intuitive grasp of chance completely wrecked by the gambler’s fallacy and confirmation bias. The mathematical problem of the lottery is something like this.

Over the very long term, on average you expect that you’ll end up with per-ticket winnings of (lottery prize)*(odds of winning that prize). As an example, the multi-state Mega Millions lottery costs a dollar to play and the odds of winning are about 1 in 175 million. If you win, the prize varies depending on various obscure factors but the current value as of this writing is 43 million dollars. So for every dollar you spend, you can expect to win about 24 cents. Of course the vast majority of the time you fail to win the jackpot and you get nothing, and very rarely you’ll win millions of times the ticket cost. There’s smaller prizes too, so in reality the 24 cents is a low estimate. I haven’t done the math, but according to the lottery site the average payouts are roughly 50 cents on the dollar counting the smaller prizes. Still very much against you - much worse than most casino games.

But even if you win the lottery, if you play long enough you’ll lose your winnings. That’s why it’s called the tax on people who don’t understand probability.

On the other hand, insurance works the same way. On average, you’ll pay more into your homeowner’s insurance than you expect to get back from your house burning down - because (hopefully!) your house probably won’t burn down and thus you’ll never see any money back. Over the long term, you lose just as surely as with the lottery.

The reason people buy lottery tickets and fire insurance is that there’s more to the expected value than the money. The damage to one’s livlihood caused by losing a home without insurance is much more severe than just the dollar amount, and so people quite wisely purchase insurance. Lottery tickets aren’t exactly a necessity, but if people understand the odds and still pay the dollar for the fun of the wager that’s not irrational either.

Not that I’m encouraging you to gamble! I don’t gamble myself, and it’s a poor financial decision which can result in addiction with some people. Strictly on the math however, I no longer think it’s a tax on ignorance.

Note: The comments/permalink issue on this post has been fixed. I have no idea exactly what happened, but it’s not happening anymore which is good enough for me!

I saw Pixar’s Wall-E on opening night. Since even the most mediocre Pixar films are usually among the best in the business, I figured it would be worth the seven bucks. I was not at all disappointed. It is a beautiful, beautiful film - possibly the studio’s best and most touching. I cannot possibly recommend it highly enough.

There is a lovely scene where the titular robot uses a fire extinguisher to propel himself through the vacuum of space. There’s no sense in critiquing the physics of a gentle animated film, but it gives us an opportunity to talk about the principal challenge of moving about in space - there’s nothing to push against. On earth you push against the ground with your feet while walking, or with your tires when driving. If you’re in an airplane, the propellers or jet engines pull in still air in front of the plane and push it out the back at high speed. Boats do the same thing with water. It’s just Newton’s laws in action.

In space there’s just blank vacuum. You can spin your tires and turbines, flap your wings, and swing your feet but you’ll just be flailing in place. If you want to push against something, you’ll have to bring it with you. This something is rocket fuel. The faster you push it out the back of you spacecraft the faster you’ll go, which is why rocket fuel is ignited and blasted out the back as fast as its fiery chemistry can take it. But in theory you could fling rocks out the back by hand and it would accelerate you forward just as surely - if much more slowly. The problem is that eventually you run out of fuel or rocks to fling. How fast will you be going when you run out? Let’s assume (or pick an appropriate frame of reference) that you’re at rest when you start. The total momentum of the spacecraft/fuel system is zero, and since there are no outside forces it will remain zero during the process of flinging small rocks (or fuel) out the back. If we call the mass of the spacecraft M and its velocity V, and we call the mass of the rock m and its velocity v we can figure out what V is after you fling a rock.

Solve for V and you’re set - if you’re flinging just one rock. Flinging more rocks complicates things - M keeps changing and you have to iterate over and over. Worse, you’ll have to do so infinitely many times if you’re using a more-or-less continuous substance like rocket fuel as your propellant. Maybe we can improve matters if we figure out a way to describe this with an integral. Let’s see… we can leave little v alone because that’s the speed of whatever we’re flinging out - it stays constant. We can call the little bits of what we’re flinging out dM, since they were part of the original spaceship + fuel mass. We can call the change in speed of our spacecraft dV. Just like the above equation, after a bit of rearrangement this leaves us with

And to get our total change in velocity V, we just integrate this over the change in mass.

Where the i and the f mean the initial mass (the spaceship and the fuel) and the final mass (just the spaceship, emptied of fuel). Doing the integral gives

The ratio is called, appropriately, the mass ratio. If your fueled spacecraft is twice the mass of the unfueled spacecraft, you’ll end up moving at log(2) times whatever speed you were throwing fuel out the back. It’s pretty clear that even ludicrously huge mass ratios won’t do much good since the natural log is such a slowly growing function. You’re more or less confined to have a maximum speed on the order of the exhaust speed no matter how much fuel you pack.

Are there ways to get very high exhaust speeds? Sure, there’s lots of ways although most of them are pretty limited at the moment. Those will make a good subject for another time.

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Note from yesterday: Traffic to this site easily set a record yesterday when I posted about politics and religion on science blogs, with visits up from the average by about a factor of two. Either this means I’m right and people are happy to read someone disagreeing with the prevalence of those topics in science blogs, or I’m totally wrong and even mentioning those topics drives up interest! I’m sticking to my guns either way, and keeping those topics to a minimum here.

I propose an experiment. Head on over to Cosmic Variance or Bad Astronomy. Count the posts on the front page that have to do with their respective subject areas and how many have to do with politics or religion. Hold that ratio in your head for a moment.

Professor Orzel on Uncertain Principles is discussing the limitations of science blogging in terms of how much of a scientific education people can get from reading those blogs. He thinks there’s certain limitations intrinsic to the form, and he’s right. But he also makes a tangential point in passing:

Look, I don’t disagree with a thing he says about the incentive structure of science blogging, and blogging in general. He’s absolutely right that the desire for traffic pushes people to write about topics that will bring page views and comments, which all too often makes scienceblogs.com feel like ranting-about-religion-blogs.com. His analysis of the culture and processes of the science blogosphere is spot-on.

This I agree with, except the possible traffic motivation. Do those kinds of posts actually drive up traffic? After all, there’s a lot of blogs out there which are specifically about politics. Tons of them. Metric tons, even. There’s not exactly a shortage of religion blogs out there either, from the most strident atheism to mainline Christianity to obscure deranged cults. There is some supply and demand at work here. There’s a finite number of eyeballs interested in reading about those subjects, and a glut of supply is going to make it difficult to obtain traction with readership. If traffic honestly spikes noticeably after a post about one of those subjects I would be surprised.

There’s also the risk of alienating some readers. I personally tend to get a bit peevish when I head off on the blog circuit to read about science and end up reading a dozen angry screeds about the latest partisan outrage - even in those cases where I agree with the view of the writers. Blogs, however, are a free product of the writers and any person who doesn’t like the content is free to not read them. Many well-known science bloggers who like to write heavily about politics and religion have said just this. It’s a perfectly true and perfectly valid policy.

I wonder, however, why so many science writers make that particular decision at the risk of losing part of their audience who otherwise would love to read about science. I have a guess. In some sense, writing about science simply serves to make the readers a tiny bit more familiar with some interesting piece of the natural world. It’s good for public awareness of science and good to spread ideas and perspectives, but it’s an epiphenomenon of scientists, not science. Science would go on unimpeded if every science blog vanished tomorrow. After all, it went on just fine in the centuries before blogging was invented. Politics on the other hand involves the entire population, and in a democracy the direction of public policy is directly influenced by what the people are voting for. Writing to try to promote your politics views feels like directly Making A Difference in a way that writing about research does not. Every mind changed, changes the body politic. I think that, not traffic, is probably the motivation for political writing.  (Swans on Tea has some thoughts along these lines as well.)

But does it actually make a difference, or does it just feel that way? I find it hard to imagine anyone is going to read this recent Cosmic Variance post and change their minds about war crimes charges for the Bush administration. A part of the audience will nod in agreement, the other part will roll its eyes in disagreement and possibly not return, and no one will leave knowing anything more about physics.

You may be able to tell I’m not a huge fan of the “I’ll write about politics anyway and if you don’t like it you can leave” school of science blogging (Though I have nothing personal against those with that policy!). So for this site, I’m going to try a different tactic at least for the time being.

I myself have strong opinions about politics and religion. It’s probable I may even write about them here from time to time. But I will do my best to keep my writings on those topics to a minimum. When I do write about them, I’ll try to see that they’re at least partially related to the putative subject of this site. And just for good measure I think I’ll put them under a link with a disclaimer urging you not to click it. I think you are smart enough to make up your own minds without me haranguing you.

Grab a hydrogen atom, hit it with a photon of the appropriate energy, and bump it up to its first excited state. After a while it will spontaneously release a photon and resume its quiescence in the ground state. Simple enough, and the general idea of atoms absorbing and spontaneously releasing photons is the basis of more physics than I can count.

It’s a surprisingly complicated process, and the standard quantum mechanics we start off learning can’t explain it. An energy eigenstate that’s not being disturbed by outside forces will stay in that eigenstate without bothering to decay. But atoms do spontaneously emit, and the reason our introductory treatments can’t explain this is that they only treat the quantization of the atomic electrons. They don’t treat the quantization of the electromagnetic field itself. This is the realm of quantum field theory - in this particular case the theory in question is quantum electrodynamics. Instead of just considering the atom by itself, we consider its interaction with the surrounding vacuum modes. Sure enough, it turns out that the atom sitting in an excited state has some finite expected lifetime before the system finds itself as an atom in the ground state and a photon propagating outward in the surrounding vacuum.

I think maybe the same kind of decay happens with TV networks. They all end up in a featureless ground state. Famously MTV and VH1 haven’t has much in the way of music programming in years, but that’s cliche at this point. Consider some others: TLC was The Learning Channel, and is now the home of several shows about babies, several shows about fashion, and several shows about wedding dresses. American Movie Classics is now AMC, which as of the day I write this is broadcasting Missing in Action 2 and Deathwish 3, which feature Chuck Norris and Charles Bronson respectively. The History Channel is now just calling itself History and while it does try gamely to more-or-less stay related to its putative subject, some shows like Ice Road Truckers are a stretch at best. Even TV Land has taken to broadcasting reality shows.

What does that have to do with physics? Not much, but it does give me the opportunity to point out the Discovery Channel as a notable exception. Originally a pretty standard nature show network, it developed into to something more like the unfortunate examples above before amazingly transitioning back into something which managed to combine modern entertainment tastes with its former goal of interesting programming about the real world in which we live. Mythbusters, Dirty Jobs, Survivorman and others aren’t exactly traditional educational material but if they don’t show science in all its messy real-world glory while making it incredibly fun to watch I don’t know what does. Good job guys. You ain’t perfect, but considering what else is out there on the airwaves I give it a solid A.

As you read this, I’m probably on the road back to my university to start teaching the second summer session for Physics 208 - Electricity, Magnetism, and Light.  It starts this Wednesday, and it’s the second half of the calculus-based intro physics class. Mostly engineers take it, though I’ve seen a few from scattered other majors. I haven’t done much preparing for it, but I taught the same class last semester and thus there’s not really anything new to put together. I have all my old lecture notes and materials so the only real thing I have to do is make up new quizzes. Students are good at nothing if not gaming the system and they’d notice repeated quizzes pretty quickly.

The thing I most want to improve on from last semester is the quality of the lab reports.  We have pre-printed lab manuals which describe the experiment and procedure and provide places to record data.  Generally the analysis and calculations involving that data have been pretty decent, but where students really fall apart is writing the conclusions.  Three sentences where one of them is “I learned a lot in this lab” will not cut it (though a conclusion that bad is of course rare).  To fix this, I think I’m going to have to start counting the conclusions as a larger fraction of the lab grade.  Previously they were only about 20% of the grade and thus even a low-credit conclusion wouldn’t hurt too badly.

Why is a conclusion important in the first place?  That’s easy: because conclusions represent your ability to both understand what you’ve done and what the results are, and communicate those things clearly to someone who hasn’t done the experiment.  This is what they’ll be doing in pretty much any career where they report results to their supervisors.

I’m going to get some conclusions from good physics papers in the published literature that should be good guidance as examples.  I will additionally make it very clear that I’m looking for a conclusion that describes the method, results, and interpretation sufficiently so that a person unfamiliar with the experiment would understand what happened.  Then I’m going to nuke any bad conclusions especially hard during the first couple of weeks.  I hope it works.

If you have any suggestions for your favorite especially lucid (and not too long) examples of concluding paragraphs in the professional literature (not necessarily just physics), I’d love to hear them!

A few days ago I wrote a pretty basic post about the trajectories of a football.  Dr. Pion suggested the real physics in football was in the inertia tensor of the ball itself.  True enough.  As such, this post is going to be a little tough in places for people who haven’t seen this kind of thing before but I’ll try to keep it at least conceptually tractable for everyone.

Angular momentum and angular velocity are intimately bound up with each other, and in introductory physics they’re essentially treated as the same thing.  Multiply angular velocity by the moment of inertia and you get angular momentum.  This works great for objects spinning about their axis of symmetry but it simply is not correct in general.  Nothing in the world prevents there from being angular momentum components about axes other than the axis of rotation.  We don’t even need to look at particularly weird scenarios to see examples of this.  A tetherball might have an axis of rotation corresponding to the pole, but its angular momentum about its point of connection to the pole is not parallel to the pole.

This has a lot of implications in the study of dynamics.  The most serious of these is that we can no longer describe the relationship between angular velocity and angular momentum by a single number called the moment of inertia.  We have to make the leap to a more complicated mathematical object called the moment of inertia tensor.

Strictly speaking, a tensor is not a matrix.  However, tensors can often be meaningfully written as a matrix and the moment of inertia tensor is no exception.  The moment of inertia tensor in all its glory is usually written as a 3×3 matrix:

where

This is pretty intimidating if you’ve never seen it before, so here’s two things to keep in mind.  The tensor is symmetric about the diagonal, and for a continuous mass the sum over all particles is replaced by an integral of the mass density over the volume.  Since the matrix is Hermitian, it is diagonalizable - this just means there’s some change of coordinate axes we can do that makes the off-diagonal elements zero.  The axes we find upon doing this procedure are called the principle axes.  These are the axes of symmetry if the object is symmetric, and in general the principle axes are the ones you can spin the object smoothly around without it trying to “wobble”.

How does this apply to a football?  Some apologies first: Dr. Pion is probably going to be unhappy about the lack of rigor here, and Carl Brannen is going to want me to focus more on forces. Both are good criticisms!  But I think this is probably the clearest way to think about it without worrying about the mathematical details of the problem.  Besides, I have no idea what equation describes the shape of a football, and I really think the resulting integrals would be too hideously bad to do without approximation anyway.  But let’s look qualitatively.

Lay the football down on the ground, long axis parallel to the floor.  If you spin it gently, it will rotate about its center.  The principle axes of the football will be along the symmetry axes, so that means the long axis is a principle axis.  It’s basically symmetric about that line, and so we can take the other principle axis to be the one going through the center pointing straight upward.  The third principle axis is perpendicular to the other two but it’s not important here.

The kinetic energy of that spinning football will be

Wait, how did that get so bizarre looking?  Why isn’t it just the usual (1/2)Iω²?  Well, now the moment of inertia and the angular velocity are explicitly matrices and vectors, and thus we have to treat them as such and pay attention to how they’re actually supposed to be multiplied.  But!  We can make it simpler.  For our football spinning about the short axis of symmetry, only one component of ω will not be 0, and only one component of the diagonal matrix will multiply that.  Thus if we give that short axis the label “1″, the kinetic energy will be

Now imagine the football balanced with the long axis vertical - the way it is before a field goal.  If it’s set up like that and spinning around the long axis (which we’ll label “2″), the kinetic energy will be

Here’s the important part.  For the football laid horizontally spinning about the short axis, its moment of inertia about that axis is large because most of the football mass is far from the axis.  Remember the inertia tensor equation - it involves an r squared term.  For the football balanced on its tip and spinning like a top, the mass of the football is closer to the axis and thus the inertia about that axis is much smaller.  The #2 moment of inertia is smaller than the #1 moment.  Thus for a given angular velocity, the rotating football lying horizontally will have a larger rotational kinetic energy.

But remember all that Lagrangian stuff we did a few posts back?  The potential energy of the ball is also important.  Lying flat on the ground the potential energy is small because the mass is mostly close to the ground.  Balanced on the tip the potential energy is larger since more of the ball’s mass is farther above the ground.

Let’s put all this together.  There’s nothing adding or subtracting energy from the spinning football, and so the total energy is conserved.  The Lagrangian is

Where K is the kinetic energy from rotation and U is potential energy due to gravity.  For a slowly rotating football, K is small and thus U dominates the motion.  Since the time integral of the Lagrangian is going to be a stationary point with respect to small variations in the orientation of the ball, U must therefore be also a stationary point.  Sure enough, any small tilting of the football away from its horizontal spinning position will raise the potential energy and so the forces will conspire to keep the system horizontal and close to the ground.

But what if the football is spinning very quickly?  K dominates the motion and takes U along for the ride.  Since K is proportional to I, therefore the football will pick out a motion that minimizes I.  That’s going to happen when the football is rotating about the long axis - when it’s spinning like a top balanced on its point.

What about for those speeds of rotation where K and U are comparable?  The situation becomes much more complicated because the football is no longer spinning about a principle axis.  It will start wobbling crazily as we’ve all seen.  So a football spun at a high speed will tend to rise and orient itself to spin like a top on its point.  As it slows, it will wobble crazily and then finally lie horizontally and rotate about the short axis through the center of mass.

Crazily heuristic and thoroughly unmathematically rigorous?  Yes, but mostly true nonetheless. Mostly.  Physicists familiar with this stuff will recognize some serious simplifications and smoothing over.  The biggest thing I’ve ignored is the possibility of gyroscope-type motion, where the football is spinning about the long axis but not oriented vertically.  This is certainly possible in theory, but in the particular case of a football it’s not something that you’re likely to see for long.  The leather surface of the ball is not a low-friction surface and thus a quickly spinning ball will tend to roll about the point of contact in sort of a precession-type motion. It will be close to vertical, but not quite there unless it’s spinning quite fast.

Nonetheless I hope it’s a helpful explanation on why footballs behave as they do.  There’s serious physics lurking in the old pigskin.

You may have heard of the actinides, lanthanides, oxides, and various other -ide names from chemistry. Here’s another: pnictide. Rolls right off the tongue, no?

Well, you may be hearing more about them in the future.  In the race - now more like a marathon - to understand superconductivity, research has focused on the various copper oxides.  The most famous high-temperature superconductor is probably yttrium barium copper oxide, which broke the liquid nitrogen barrier for superconductivity in 1987.  The current state of the superconducting art is critical temperatures at standard pressure of about 135K.  This is much too hot to be explained by BCS theory, but of course it’s nowhere near the holy grail of room-temperature superconductivity.  We won’t know if that’s possible or not until someone either gets lucky and finds a material with that elusive property or until someone comes up with a decent theory of high-temperature superconductivity that either confirms or denies that such a thing is possible.  So far we’re not very close.

Dr. Paul Grant, writing in Nature, has an interesting article about a new experimental effort with new materials - the pnictides.  These are group 15 compounds, especially those involving iron, which had not yet thus far been adequately studied in terms of superconducting properties.  Right now these compounds have been demonstrated to have superconducting properties up to the 55K range.  But this isn’t particularly high in an absolute sense despite the fact that it’s vastly higher than previous records with these compounds.  In fact, there’s some reason to believe these might actually be on the bleeding edge of the range that standard BCS theory may be able to explain.  This potentially carries a great deal of interest, as the transition between regimes might finally shed some light on how high-temperature superconductivity works.  Or it might not.  There’s never guarantees.  As Dr. Grant said

Will T_c in the pnictides continue to go up, and perhaps double or triple as happened in 1987–88? I doubt it. We’ve now been on standby for several months, and to my mind the best hope is that the discovery of pnictide high-temperature superconductivity will help us understand better the physics of the cuprates. The iron age has yet to dawn.

Let’s hope it does.  Or if it doesn’t let’s hope we can figure out why.  From a pure science standpoint, both are equally fascinating possibilities.

“I returned day before yesterday,” he answered, while he leaned his arm on the keys, bringing forth a crash of discordant sound.

“Day before yesterday!” she repeated, aloud; and went on thinking to herself, “day before yesterday,” in a sort of an uncomprehending way. She had pictured him seeking her at the very first hour, and he had lived under the same sky since day before yesterday; while only by accident had he stumbled upon her. Mademoiselle must have lied when she said, “Poor fool, he loves you.”

- The Awakening, by Kate Chopin

In 12th grade English we read Kate Chopin’s The Awakening. I wasn’t a fan. For a supposedly feminist piece of literature the protagonist Edna is really not quite a model of empowerment. She emancipates herself and promptly wrecks her life pretty comprehensively. Most of the self-sufficient women I know have managed their lives quite adroitly. I am not a literary critic however, and I could be missing the point. But my point here is quite different. The point is that Edna is disappointed that Robert has been under the same sky for two days without visiting her.

Under the same sky? That always struck me as a rather beautiful turn of phrase. Stand outside and look to the horizon. Consider the horizon to be the edge of the dome of the sky and whatever is within your horizon is under the same sky as yourself. You might like to know just exactly how far that is. Literature wonders, science answers.

Call the radius of the earth R. Call your own height h. Call the line from your eyes to the horizon x. Let’s let Apollo 17 provide the image, with h greatly exaggerated for clarity.

We’re interested in the distance x; or more accurately we’re interested in the arc length of the earth just beneath that line. If the angle between the lines R and R + h is denoted by θ, we know that arc length (the distance to the horizon) is simply Rθ. And θ is by simple trig just

Therefore our answer, the ground distance for your beloved to be under the same sky (call it S) with you is R times the above quantity.

That’s an exact answer, but we need not be so precise. The height h is very small compared to the other parameter of interest, R. Therefore nothing prevents us from very accurately approximating the above expression by the first term in its series expansion about h = 0. After a little bit of work, I get:

This approximation is extraordinarily accurate for human-sized values of h. We can get the same thing by just using the Pythagorean theorem to find x and dropping second-order terms in h. This is what we should expect, since for small angles the arc length approaches x. Nothing is stopping us from expressing the quantity (2R)^(1/2) in appropriate units to make our life even simpler. With S in miles and h in feet, the expression most convenient for calculation is

If Edna is 5 feet tall (really this should be the height of her eyes), her horizon is about 2.74 miles away. If you are closer to her than that, you and she are under the same sky.

Complications? Yes, there are a few. The earth is not perfectly spherical. Both its global oblateness and its local roughness will affect the result. The atmosphere provides an additional distorting effect, and we have not considered the potential problem of how to characterise a person who is only partially visible above the horizon. But perhaps we have subjected literature to enough for one day.

The problem with particle physics is the difficulty of acquiring new data.  The major way particle physicists get data is to accelerate individual particles to tremendous energies and smash them into each other.  If you have enough E, the universe will obligingly generate some mc² in the form of new particles flying out from the collision point.  The more energy, the more kinds of particles and interactions you’ll be able to observe, and the more hard facts you’ll have to test old theories and generate new ones.

So these physicists build particle accelerators to generate these high-energy particles in order to learn more about the fundamentals of the universe.  The current most powerful particle accelerator is the Tevatron at Fermilab outside of Chicago, which can accelerate protons up to about 1 TeV.  It’s really hard to describe these energies on a human scale.  Though the collision is not really at anything approaching statistical equilibrium enough to have a well-defined thermodynamic temperature, 1 TeV per particle is roughly equivalent to a temperature of twenty million trillion degrees Fahrenheit (around half that in Celsius).

But after a years of running and tweaking an individual accelerator, eventually you’ve squeezed most of the fundamental physics you’re going to get from it.  You simply can’t get much more at that energy and you need something bigger to push our knowledge forward.  In order to do this and move beyond the aging Fermilab, the Large Hadron Collider is being brought online near Geneva.  Each accelerated proton will have around 7 TeV of energy, or in the thermodynamic analogy about 140 million trillion degrees F.

Not only will this give particle physicists a whole world of new data to explore, the discoveries made will almost certainly have dramatic effects in other areas of physics as well, particularly cosmology.  Cosmology has always been notoriously data starved - if astronomy is like studying a forest while tied to a tree, then cosmology is like studying plate tectonics while tied to a tree.  But since cosmology is intrinsically involved with the high-energy origin of the universe it will likely improve its own models with the LHC data.

That is, if the LHC doesn’t get shut down by the people who are suing because they’re convinced it will destroy the world.  I’ve never really understood this line of argument.  The physicists building the LHC would understand the potential risks better than anyone.  Both them and their families live on earth.  Therefore it’s unlikely that every single one of them would be perfectly willing to gamble their existence on an experiment.  Now I suppose the report of the CERN scientists finding no risk won’t convince any of the alarmists.  I suppose the expert outside review of the report coming to the same conclusions won’t convince them either.  Simply put, there’s nothing fundamentally different about the LHC that the universe doesn’t already do on a much larger and more powerful scale millions of times per second.

It’s not the first time physicists have put some thought into avoiding the end of the world. Decades ago, there was some concern that nuclear weapons might be able to ignite the entire atmosphere in a fusion chain reaction. It was analyzed and found to be impossible insofar as their understanding of the processes allowed them to judge. Here’s LA-602, the original Manhattan Project report.  And compared to the supposed danger from the LHC, at least that had a vague aura of plausibility about it.

However, I understand the point of the lawsuit-wielding crowd even if I think it’s mistaken.  In the midst of all the “science verses the luddites” type argumentation going on here, the blog Overcoming Bias makes a good point.

It is not obvious to my inexpert eyes that the assumptions in the RHIC Review are any more firm than those in LA-602 - they both seem very firm - but the two papers arise from rather different causes.

To put it bluntly, LA-602 was written by people curiously investigating whether a hydrogen bomb could ignite the atmosphere, and the RHIC Review is a work of public relations.

You should read the whole thing.  Good PR is not worth slouching into the trap of letting the conclusions write the report even if the conclusion is absolutely correct. It can breed a culture of a lack of self-examination in science and engineering, which has had fatal results more than once. While the earth is not going to be a casualty of the LHC by any stretch, let’s not let it lead us into overconfidence elsewhere.