Built on Facts

Leaked Climate Change Documents

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Around ScienceBlogs, people who don't accept global warming as a real phenomena tend to get called denialists. In the interests of full disclosure, I should admit that I'm not a denialist but rather a global warming defeatist. Doesn't matter how bad or not CO2 is, ain't nothin' gonna stop it. People will not give up electricity and transportation in the developed world (nor should they), and people in the developing world will not be give up the quest for developed-world living conditions (nor should they). As such it's either massive and immediate worldwide switches to nuclear power and electric transportation, or we'll just have to live with whatever happens, wherever it is on the spectrum from "nothing" to "Day After Tomorrow". CFLs and conservation and even the climate change bill currently stalled (and honestly pretty much admitted dead even by its supporters) in congress will only make the barest scratch in US emissions. The US is increasingly a side issue anyway - the 2,400,000,000 people in China and India will matter much more than the 304,000,000 in the US in terms of CO2 release.

And since such massive rollouts of nuclear power and electric transit don't seem to be forthcoming, I'm a defeatist.

But that's just disclosure. I know it's a minority opinion around here, so you should at least know that I hold it.

All this is just preliminary because ScienceBlogs needs an open thread on these leaked climate change documents. They're all over the web at this point, so you may have seen them mentioned already. (Here, for instance.) Allegedly there's all kinds of nasty misconduct possibly constituting scientific fraud.

I make ZERO claims about the credibility of any site discussing this. Since I generally don't follow the subject (on defeatist grounds), for all I know the sites discussing this might like to rail about the Illuminati and sell alien urine to pay their bandwidth bills. But it has been confirmed that the leaked emails are real by the Climate Research Unit, the organization that got hacked.

If the whole thing is overblown, I want to hear about it. If not, tell me about that too. You as a group of readers are pretty astute and I'm interested to hear your analysis of this mess. Thus: open thread.

Posted by Matt Springer at 1:36 PM • 30 Comments • 0 TrackBacks

Seeing Laser Beams

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Ok, see counselor Troi firing her phaser?

You see this kind of thing all the time on film in scifi. Whether it's Star Trek, Star Wars, or pretty much anything else, energy beams fired from future weapons are visible. Usually someone will point out that in fact laser beams are not visible in this manner. To see light, it has to reach your eyes. This is clearly not possible when all the light is actually traveling down the beam path. You can see this in action with laser pointers - only the spot where the light hits and diffusely reflects is visible. The path is not.

Writers of TV shows usually explain this by saying that the beam is not strictly light, but some stream of particles that slightly emits to the sides along its main path. While this has its own problems, at least it acknowledges the issue.

But what's even more interesting is that in fact there are already automatically particles present along the beam path in the atmosphere. Some of them are sizable particles like dust, others are individual atoms and molecules. Generally they don't scatter much light, but if the light is intense enough then the small amount they do scatter is enough to see. And so you have a visible laser beam. Here's one in my lab:

The beam scatters off the air and you can actually see it as a straight line. Apologies for the terrible camera phone picture, I really need to get a classy camera that can take nice pictures. This is not actually a laser I'm working on, so honestly I'm not sure which variant of frequency-doubled Nd:something laser this is. Probably Nd:YLF.

This is used to pump an infrared ultrashort-pulse laser with a repetition rate of 1 kHz. This can itself be focused to a point in the air, which becomes visible as a little stationary spark as the intense beam ionizes the air. This produces a 1 kHz buzz which can easily be heard by the unassisted ear.

I have to say it's a nice job perk that I can see old science fiction tropes come to life pretty much every day. :)

Posted by Matt Springer at 4:15 PM • 14 Comments • 0 TrackBacks

Al Gore and Geothermal

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There's a little bit of buzz burbling around over Al Gore's scientific goof during a Conan O'Brien interview. Discussing geothermal energy, he said the following:

It definitely is, and it's a relatively new one. People think about geothermal energy -- when they think about it at all -- in terms of the hot water bubbling up in some places, but two kilometers or so down in most places there are these incredibly hot rocks, 'cause the interior of the earth is extremely hot, several million degrees, and the crust of the earth is hot ...

Of course the interior of the earth is extremely hot, but not that hot. It's several thousand degrees rather than several million. If the earth were several million degrees it would be a rapidly diffusing cloud of metallic vapor. Even the center of the sun is only perhaps 13 million degrees C.

But I'll let him slide; pretty much everyone blanks out from time to time. And it gives us a chance to do a little thinking about just how much thermal energy is in the earth.

First of all, just because something is hot doesn't mean you can squeeze energy out of it. You can only squeeze energy out of temperature gradients - you need something hot and something cold. This is why we can't just set up a temperature-to-energy machine in the desert and have free energy. In your car, for instance, you need both the heat of the burning gasoline and the much cooler ambient temperature from the outside air via your radiator to turn the hot gasoline vapors into forward progress. Power plants frequently have large cooling towers for that very reason. It's not the energy of the hot substance, it's the process of moving that heat to a cooler place that creates useful work. Think of it in the same way as water flowing downhill can turn a paddlewheel - it won't work unless the water starts off high and ends up low.

But that's not a problem here. The interior of the earth is hot and the exterior is much colder. The difference in temperature is such that the efficiency of heat-to-work conversion could be near 100% in theory, though in practice it would be much lower. And we're not likely to run out of geothermal heat any time soon. As a slightly wild Fermi calculation, assume that the earth is uniformly iron at 3000 C. The specific heat of liquid iron is about 611 J/kg K, so cooling the earth to room temperature this yields about 1.8 million joules of energy per kilogram. Multiply by the mass of the earth, and the total energy content might be in the neighborhood of 10^31 joules. The total energy consumption of the world's human population is in the vicinity of 5e20 joules per year.

Divide out, the earth's geothermal energy could support that consumption rate for about 21 billion years. We're not likely to use it up.

So why isn't it in widespread use? After all, every nation has domestic access to it - all you have to do is drill straight down. The main problem is that the temperature really doesn't start getting ramped up until dozens of miles down. Drilling a hole that deep and pumping water (or whatever) down and up is technically unfeasible. Geothermal is at its best at those places which are close to geological activity that brings the heat closer to the surface. Volcanic and other geologically active locations often do very well with geothermal power. Iceland in particular produces vast quantities of usable energy from the internal heat of the earth. Most other places are much farther from the hot regions of the earth's interior and geothermal is correspondingly much more difficult to get.

Sadly Al Gore's hopes for geothermal as a major clean energy technology are probably futile until deep drilling develops into a much more mature form. It would be nice if that happened; the energy to be tapped is pretty close to inexhaustible.

Posted by Matt Springer at 4:33 PM • 21 Comments • 0 TrackBacks

Sunday Function

Category: Sunday Function

We're doing two functions today. If I'm not mistaken we've done each of them separately, but there's a famous and interesting relationship between the two that's always interesting to look at. Like very many interesting mathematical facts, it has to do with the prime numbers.

As such the first function is the log integral Li(x), usually defined in the following way:

We'll plot it in a minute, but if you're interested in a rough idea of it's behavior it so happens that Li(x) ~ ln(x)/x. That is, those two functions have a smaller and smaller percentage difference as x becomes larger.

Now the second function is π(x), which is the prime counting function. The notation is traditional; the pi has nothing to do with the number 3.14159..., rather the p in pi is supposed to suggest the word "prime". π(x) is defined as the number of prime numbers less than or equal to x. For instance, π(10) = 4, because there's 4 primes less than 10 (1 is not a prime, if you're curious.).

Now if we plot these two functions on the same graph we may get the accurate impression that the two are related.

The log integral and the prime counting function get closer and closer to each other in percentage terms as x gets large. This is the prime number theorem, and it is tremendously important in understanding the properties of the prime numbers.

You may also notice that while the log integral is a very good approximation, it is an approximation and over the interval of the graph it's always a little higher than the actual number of primes. This is true if you plot the first million primes or the first billion primes or the first trillion primes. For a while it was thought that this property of Li(x) > π(x) held for all x. There was some theoretical support for the idea that this was universal, but in math theoretical support isn't worth a whole lot. You want a rigorous mathematical proof. And it turns out that you wouldn't be able to prove that property because it's not true. In 1914 the great mathematician John Littlewood proved that in fact the property was not universal. At some large x, the prime counting function would pass up the log integral function (though of course their percentage difference would continue to decrease). Then it would in turn be passed up, and they would go on trading off an infinite number of times. But his proof of this fact was not a so-called "constructive" proof, which means that though he proved that some x existed where one passed the other, his proof did not in fact tell you what that x might be. Could be relatively small, could be unimaginably huge.

In 1933 another mathematician named Stanley Skewes was able to shed some light on the issue. He wasn't able to find the specific number for the first crossover, but he was able to show that whatever it was, it was mathematically certain to be less than a number that's usually now called Skewes' number. That number was really unimaginably huge. If you packed the universe shoulder to shoulder with tiny print zeros you wouldn't even be able to write the number, much less the quantity that number represents. You have to use a tower of exponents to write the number. The number is approximately equal to 10^10^10^34.

Isaac Asimov tacked the subject of explaining such a huge number in simple terms in his classic essay "Skewered!" I don't think it's online but it's in his excellent essay collection "Of Matters Great and Small".

For a while it was the largest number to naturally appear in a mathematical proof. It's since been passed up by others which are much larger. On the other hand, since Skewes' number is an upper bound subsequent work has been able to show that the first crossing actually occurs at a vastly smaller number. The number is still huge, but not nearly so huge as Skewes' number - around 10^316, to be specific.

By physics standards this is a gargantuan number. But by mathematics standards it's not so much. After all, there's an infinite number that are larger.

Posted by Matt Springer at 5:49 PM • 3 Comments • 0 TrackBacks

The Physics of Gerrymandering

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Whew! Interesting day around here yesterday, no? There's more controversial topics out there: abortion, health care, gay marriage, Iraq, and a few others. But not many. It's good for sparking discussion, but I also know that some large (probably majority!) fraction of you would prefer to hear about physics. Which is good because generally I prefer writing physics, and I know that I can get irritated when my otherwise favorite nonpolitical blogs go off on political crusades for causes I dislike. So as always I'll try to continue to keep the partisan politics to relatively rare and easily skipped posts.

How about some of the mathematics of politics instead? I promise it's not partisan!

As you know, the various states are divided into congressional districts which each have their own representative in congress. These districts are drawn in very arcane and shady ways, with the majority trying to carve out districts in a way to give them the most votes, the minority trying to do the opposite, and politicians of any party pushing those considerations to the side to give their particular district the greatest possible incumbency advantage for themselves. Gerrymandering is nothing new, but it's especially bad these days.

There have been a number of mathematical proposals to fix this and take out any possibility of district manipulation. One of my favorite is the splitline algorithm, which works in the following way:

1. Start with the boundary outline of the state.

2. Let N=A+B where A and B are as nearly equal whole numbers as possible.
(For example, 7=4+3. More precisely, A = ⌈N/2⌉, B=⌊N/2⌋.)

3. Among all possible dividing lines that split the state into two parts with population ratio A:B, choose the shortest. (Notes: since the Earth is round, when we say "line" we more precisely mean "great circle." If there is an exact length-tie for "shortest" then break that tie by using the line closest to North-South orientation, and if it's still a tie, then use the Westernmost of the tied dividing lines. "Length" means distance between the two furthest-apart points on the line, that both lie within the district being split.)

4.We now have two hemi-states, each to contain a specified number (namely A and B) of districts. Handle them recursively via the same splitting procedure.

My own state is a stellar example of gerrymandering:

Using the splitline algorithm, the districts would be much more fairly and sensibly placed:

The algorithm only takes into account the geometry of the state and the density of the population, and produces a unique result. It does so in a transparent way, and prevents any tweaking by elected officials. Certainly it would immediately result in true contests for many formerly entrenched officeholders. I doubt we'll ever see anything like this implemented, but it's an interesting application of mathematics to the process of fair governance even if it remains purely theoretical.

Posted by Matt Springer at 12:25 PM • 25 Comments • 0 TrackBacks

Re: Nidal Hasan's Weapons

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God help me, I resisted mightily. If my fellow SB friend Greg wants to spin the Ft. Hoot shooting as a cause for gun control then frankly there's pretty much nothing further to say. You'd think a @#$% major in the @#$% army on a @#$% army base just might not have been terribly inconvenienced in procuring weaponry even if every civilian gun in the hemisphere vanished in a puff of sunshine and wishful thinking. But I was going to leave it alone, assuming that that particular point makes itself. To each his own.

But he wrote a follow-up post asserting a few points of fact, pretty much all of which rather wildly miss the mark. As a physicist, semi-pro educator (this blog!), and enthusiastic firearms owner and advocate, I simply can't help setting the record straight as to the points of fact. The political gun-control points I will grit my teeth and let slide. Let's begin:

He apparently carried two pistols, and both are designed to be effective killing weapons. The more newly designed Five-sevN that he had purchased under the noses of the FBI who was busy investigating him is specifically designed to be very effective at killing large numbers of people in close quarters, to have more controlled "follow-up shots" and to pierce body armor.

The Five-Seven (weird trademark capitalization is goofy even when Apple does it) is not designed to kill large numbers of people in close quarters, except insofar as any pistol is most effective relatively close. It's a pistol like any other, and does the same thing. With the exception of the last three words, you could replace Five-Seven with pretty much any centerfire pistol except the niche wilderness big-bore revolvers and have a statement that works just as well.

But the Five-Seven was designed to fire a rather unusual 5.7x28mm round which is itself designed to pierce body armor. That much is entirely true. But what's not true is that the armor-piercing handgun ammunition is available in the US. You cannot buy it, it is a violation of federal law. To be clear: if Hasan bought ammo for this pistol in the civilian market he bought ordinary, standard 5.7x28mm ammo. And the 5.7x28mm round sucks at pistol velocities for the purposes of incapacitating or killing. Though Greg sarcastically speculates it might be good for hunting moose, in fact in many states it would be banned for hunting even small critters on the grounds that such a small round would be cruel to the animal by virtue of the injury being too small to reliably kill quickly. For that matter the armor-piercing ammo wouldn't have been much better - the ability to penetrate armor is more or less precisely the inverse of what's needed to damage tissue. This is why police and self-defense ammo is almost exclusively hollow-point, which is good for quick incapacitation but terrible at armor penetration.

In short, the very fact that the pistol was designed for military use against armor-wearing opponents makes the pistol poor for anything else. Even if he had the armor-piercing bullets, which he didn't.

The other gun was a magnum, a.k.a., miniature cannon.

Magnum doesn't mean that. I suppose you might think so if you didn't know anything about guns, but that wouldn't make you correct. Magnum means all kinds of things depending on context. The .357 magnum Hasan possessed but apparently didn't use is a solid but not particularly unusual round. The .22 magnum is a very small bullet. The .44 magnum is ginormous, but no one uses it for actual combat because it's unwieldy. Magnum shotgun shells have more pellets but they're slower-moving. Some of the largest pistol bullets (.50 AE, .454 Casull, etc) aren't labeled magnum at all.

None of this is to say the weapons Hasan chose weren't dangerous and lethal. Certainly they were, and for his actions he deserves nothing but a short drop and a sudden stop. It is to say that a matter of empirical fact the weapons he carried were relatively ordinary, and that the weapon he actually used was in fact among the least effective weapons he could possibly have picked - certainly orders of magnitude less dangerous than a standard combat rifle. (The M-16, by the way, actually shoots a slightly narrower 5.56mm projectile, but the bullet is about twice as heavy and moving around 50% faster. This makes it much more effective, and yet it's still the target of continual controversy among military circles for its not-always-impressive terminal performance. In my state it's illegal for hunting deer for that very reason - too high risk of an escaped, injured deer rather than a quick kill. UPDATE: The previous sentence is not quite right - the Texas requirement is just that the round be centerfire, so effectively the 5.56 mm is the smallest legal hunting round. Other states vary, some do enforce a larger cutoff.)

You can make up your own mind about the politics of gun control (on an army base!) - as I said I'm not going to argue it here now - but at least now we can be clear as to what actually happened.

Posted by Matt Springer at 11:56 PM • 50 Comments • 0 TrackBacks

Veterans Day (and the scientists who help them win)

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On Veterans Day we commemorate the living veterans of the American armed forces. On Memorial day we commemorate those who lost their lives. We should also spend a moment to remember those who helped make sure more soldiers fit into the first category. Though I've made the point before on this blog, I'd like to commemorate two men in particular who between them likely saved the lives of tens or hundreds of thousands of Allied soldiers during the Second World War, simply by doing brilliant science. Their names are Alan Turing and Robert Watson-Watt.

Turing was a mathematician and computer scientist who lead the British code-breaking efforts which broke the Nazi Enigma code and various other forms of wartime message encryption. Though it's impossible to accurately judge counterfactual history, I have read more than one historian speculate that the Allied codebreaking successes may have shortened the war in Europe by a year or more.

Watson-Watt was one of the early pioneers of radar, and the first person to develop it into a practical means of finding range and direction of enemy aircraft. The Battle of Britain might have been a very different story if his invention had not allowed the vastly outmanned and outgunned Royal Air Force to hold its ground against the Luftwaffe. (In a cute coda, many years later he was cited for speeding in Canada by a policeman with a radar gun)

We owe these two men a lot more credit than they get. May their names never be forgotten.

Posted by Matt Springer at 1:07 PM • 12 Comments • 0 TrackBacks

Sunday Function

Category: Sunday Function

Again I have to apologize for the sparseness of posting lately, but I've got two research projects going full blast and time has not been something I have a lot of. I'll still be writing at least a few times a week, and you can't beat the price. ;) In any case once things cool down just a little I should be back to a more regular schedule.

Today's function isn't interesting because of the function itself, the interest comes from what we'll do with it. Let's say we have a function like this:

If we want to see where the function is equal to zero, it's clear that 0 = x^2 - 2 is solved by x equal to the square root of two, either positive or negative. That's the snappy analytic solution, but let's say we want to use this function to compute a decimal approximation to whatever accuracy we feel like. The brute force method is just to try various decimal numbers and see what gets us closest, refining our guess each time. But this is ugly, slow, and requires a lot of continual work. We'd prefer a faster method where we can just turn the crank easily and get a good answer. To do this, let's plot the function and (for reasons I'll explain in the next paragraph) its tangent line at the point x = 4.

Let's say we picked x = 4 for our initial guess as the value of the square root of two. It's an awful guess, obviously, but that's ok since we're looking for a procedure that can turn any terrible guess gradually into better and better approximations of the actual answer. So we pick our initial point and draw the tangent line. We notice that it cuts the x-axis pretty close to the square root of two (which is exactly where the parabola cuts the x-axis, since the square root of two is by definition the number that makes the function equal to zero). So why don't we take the location where the straight line cuts the x-axis, and use that as our second guess, and draw the tangent line there:

This line cuts the axis even closer to the point x = the square root of 2. If we take that point as our next guess and repeat the process, we should be closer still. So first we need to actually work this procedure into mathematical language so we can actually get some numbers out of it. The equation of a line is:

Where m is the slope and b is the y-intercept. We know the slope of a tangent line is just the derivative of the function at that point, and y is just the value of the function at that point. That means we can solve for b:

Where the prime denotes differentiation, and we're subscripting the x so we know it's the particular value of the guess we're using. Our new x for the next iteration of the guess will be the value that makes our line equation y = mx + b equal zero, i.e., 0 = m x + b. Substituting all the previous stuff into this equation and solve for x. After a little simplification, we get:

Now this is a very general expression that works to find the zeros of an arbitrary function f, whatever it happens to be. Our particular function can be plugged in (for us, f'(x) = 2x by a little bit of calculus), which gives us the complete procedure:

Let's give this a try. Plug in our initial guess of 4 and the procedure tells us our next guess is 2.25. Plug that in and the procedure gives us 1.56944. Plug that in and we get 1.42189. Repeat again and get 1.41423. So on and so forth closing in on the rounded-off real value of 1.41421, and if we didn't round off at 5 decimal places as I'm doing eventually we'd get as many digits of the square root as we wanted to arbitrary accuracy.

This procedure is called Newton's Method, and it's a fine way of calculating the zeros of a function if you know its derivative. The method is not quite perfect - if a function has more than one zero the one you get will depend on your initial guess in a not-always-predictable way. And while the method is quite general, there are certain functions that don't fulfill the relatively generous convergence conditions. Still, it's a great method with a long and continuing history of use. It's pretty likely that your pocket calculator uses a very similar method when you hit the square root button. And now if you ever find yourself without such a button, you can do it yourself if you're patient.

Posted by Matt Springer at 12:13 PM • 8 Comments • 0 TrackBacks

Being an Absentee President

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Of late president Obama has taken a little bit of heat for his frequent (and mostly male) golf outings. Before him, president Bush took the same sort of heat for his golf and vacations. If you were willing to dig a bit through the news archives, I'd bet you could find similar tut-tutting about previous presidents taking time off. It's a common theme for criticism of just about any important federal officeholder - it's no coincidence that so many congressional "fact-finding" missions are to tropical paradises or European vacation spots. In that case it's a criticism I vigorously share, as the taxpayer dime ought not be spent ferrying already rich congressweasels around the globe. What they do on their own dime I don't worry too much about.

But the president comes in for special criticism no matter who pays for his downtime, in view of the fact that his job is so critical and demanding. Critical I'll grant, but contra the received opinion I'd like to argue that in fact it's easily possible to be an perfectly effective president while spending shockingly little time behind the Resolute desk. In fact for much of early American history presidents did just that - ie, very little. The presidency and the country have both changed, but in my opinion even today the president simply doesn't have to do much to do perform his job with great competence. Now I don't expect that any modern president will actually take as little time as I'm going to suggest; the very type of person who is attracted to the job and can campaign effectively is necessarily the kind of person who's willing, able, and eager to manage as much as possible. But he doesn't have to be. Let's go down the list of his constitutional responsibilities:

Sign or Veto Laws
How many of these does congress generate each week? I don't know, but given the glacial pace at which anything of consequence gets done in congress it can't be many. I believe it's a few hundred per year. The president has ten days to sign or veto each bill, so there's nothing wrong with just knocking out the previous week's bills over a Monday morning. No need to read them in their entirety - God knows congress doesn't bother. Working with official summaries, your various adviser's opinions, and the opinion of your constituents ought to be enough to decide the fate of a piece of legislation.

Take Oath of Office
Takes about a minute.

Be Military Commander-in-Chief
A huge responsibility, to be sure. But the president's job here is to set overall policy and strategic objectives at the highest levels. Getting involved in the details is both inefficient and actively a bad idea (Most famously in history, Hitler's terrible mis/micromanagement of his armed forces was a huge boon to the Allies). Given the fact that the president has a full-time civilian SecDef (with a huge staff) whose only job is to manage the military, the National Security Council, the Joint Chiefs, and a huge stack of generals and other military officials to translate strategic objectives into concrete plans, being CinC should be something the president could do each Monday after bill signings and before lunch.

Appoint Government Officials
The president has to appoint the cabinet and various other officials. This is also an important responsibility, but it's something that's pretty much done after the first week or two in office at least for the cabinet. The scores of lower officials that continually have to be appointed might be a pain, but that can easily (and in practice is) delegated to the cabinet and other advisers and then signed off on.

Make Treaties
Obviously the president doesn't actually write these. If he needs a treaty he tells the Secretary of State what he wants, and then subject to senate approval it happens. Treaties don't happen much anyway, so if there happens to be one in the pipeline give it an hour after lunch for progress reports and making any needed changes.

State of the Union
We think of it as a speech, but it is not thus mandated and in fact plenty of presidents just wrote up a SotU letter and mailed it to congress. So if you happen to have one coming up and want to actually do the speech, pencil in another after-lunch hour for practice and revisions with the speechwriters.

Ancillary Executive Duties
These amount to a few procedural matters with respect to congress, executive appointments as already dealt with above, and most importantly "receive Ambassadors and other public Ministers; he shall take Care that the Laws be faithfully executed". The ambassador stuff is easy: the State Department does all that. Faithfully executing the laws is obviously highly critical, but as chief executive this boils down to making sure your subordinates are doing their jobs with competence and fairness. So we'll say from 2-5 pm Monday grill your cabinet officials and lower agency officials (and heck, even random lower employees and the general public they affect) about the way their agencies are doing their jobs. Bring down the hammer when poor governance is spotted.

And that's that. Assuming your staff is even marginally committed to their jobs you can do your job perfectly well working one day a week, maybe less. Now sure you're neglecting the statecraft presidents like to do: visiting dignitaries, traveling the country giving speeches, campaigning for your party, wheeling and dealing with recalcitrant congresscritters, and all that. But will the wheels actually come off the country if those things don't happen? I seriously doubt it.

So go ahead Mr. Obama, golf to your heart's content.

Posted by Matt Springer at 10:30 AM • 10 Comments • 0 TrackBacks