Interna

Nothing much new these days. I successfully moved an increasing amount of boxes, my furniture, and myself some floors upwards. I felt somewhat bad for letting the 70something year old landlord carry the couch, but he insisted. He did pretty well actually, though he broke the bathroom mirror so I am presently having a break from watching my own nose. It's somewhat difficult though to apply mascara without a mirror, and I came to realize an important drawback of these non-stick pans which are made of completely non-reflecting material.

The only reason why I'm telling you that is that I presently have nothing more intelligent so say. My most profound recent insight was yesterday, after 20 years, I finally realized why my grandmother always told me to turn the bedsheets inside-out before washing. But to be fair she also told me a lot of other things, like not to cut potatoes with the knife, and to always keep my knees together when sitting.

Interesting things to read today: the Smart makes it into the US, Your computer is a 'container' and it's perfectly legal for the US government to snoop around on it, and Nicholas Carr has big news:"Computers are technologies of liberation, but they're also technologies of control".

Topic of the day in Waterloo is the temperature, which reached 15°C in some places yesterday. All the snow has melted, and the park where Stefan took these lovely photos only a few days ago, has turned into a mudhole. Meaning, I ruined my new shoes, and the Canadians are running around in shorts and T-shirts, no kidding. A whole culture melts, if the Canadian kids can't iceskate in the backyard.

Have a nice day.

Ranking of Excellent European Graduate Programmes

The Centre for Higher Education Development (CHE) has published a Ranking of Excellent European Graduate Programmes in the natural sciences and mathematics. For their analysis they use four general indicators, and according to this ranking they then distribute 'gold', 'silver' and 'bronze' medals to the institutions. Achieving a gold (or silver) medal means that the respective institution belongs in the group of institutions which cover at least 25% (or 50%, if silver) of the total amount of publications. The share of medals therefore depends on the shape of the distribution: If there are few institutions with a large number of publications, there will be fewer medals than for a smoother distribution.

The general indicators for the classification are

• Number of publications
• Citations
• Outstanding researchers
• Number of projects in the Marie Curie programme

Those institutions who made it into the `excellence group' were then further examined in an in-depth analysis based on institutional questionnaires and online surveys. According to their summary:

"The results show that Europe in general provides a very high level of research and of graduate teaching in the academic fields that were analysed."

The figures below show the medal distribution in physics and mathematics (p. 17 and 18 of the full report)





I wasn't aware Germany is a maths paradise.

This project is intended to be transferred to other academic fields in the near future.

You can download the full report as a PDF here.

The Higgs Mass

In the Standard Model of particle physics, all particles are massless, unless they interact with a special field, called the Higgs field. We know of course that electrons and quarks and Z bosons have masses, so, taking the Standard Model at face value, the Higgs field should be real, and there should be particles associated with the Higgs field, the Higgs bosons. The Higgs boson in the Standard Model is expected to have a high mass, which, however, cannot be predicted within the current theoretical framework. Thus, the Higgs mass is a parameter that has to be determined by experiment.

As it comes out, no experiment so far has seen direct traces of the Higgs boson. This should be not too big a deception, since the Higgs mass is big - and, as it seems, too high for the Higgs to be produced in today's particle colliders. However, there are indirect means to estimate a probable range for the mass of the Higgs boson. The current expectations about the Higgs mass are encoded in the following figure:

A least-square fit to a number of precisely known data in electroweak physics using the Standard Model as theoretical framework and the Higgs mass as a free parameter yields an expectation value for the Higgs mass around the minimum of the parabola. [Source: Precision Electroweak Measurements and Constraints on the Standard Model by the LEP Collaborations and the LEP Electroweak Working Group, arXiv: 0712.0929v2, Figure 5.]

For while the Higgs mass could not be measured directly so far, it's a parameter which, within the framework of the Standard Model, influences all kinds of measurable quantities. An example of this reasoning is the analysis of the Z pole, the resonance peak of the Z boson in the cross section for the production of particles in electron-positron annihilation. There are many details of the Z pole that have been determined very precisely in experiment, for example the width, shape, asymmetry, or background slope - about a dozen or so parameters altogether. On the other hand, all these measured parameters can also be calculated within the Standard Model. In these calculations, there are five quantities that have to be assumed as input parameters: the coupling constants of QED and QCD at the Z pole, α(m²_Z) and α_s(m²_Z), the masses of the Z boson and the top quark, and the Higgs mass.

Now, one can search for those values of these five parameters that reproduce best the data measured in experiment. To do so, one minimises the squares of the differences between measured data and calculated values, weighted by the experimental error. This quantity to be minimised as a function of the free parameters is called χ² ("chi-square"). It is given in general by the formula


where X_i and &sigma_i are the experimental data and their respective errors, while the f_i are the theoretical predictions, depending on the free parameters p₁ ... p_N. In this case, there are five free parameters. Fixing the values for the coupling constants and the Z and top quark masses (parameters p₁ ... p₄) at their best fit values, one can then check how χ² changes if the Higgs mass (parameter p₅) is varied.

This yields the plot above: The parabolic curves show, as a function of the Higgs mass m_H, the corresponding χ² above its minimum at a mass of about 80 GeV/c². Note that the Higgs mass is shown on a logarithmic scale. The blue band represents theoretical uncertainties within the Standard Model calculations, and the dashed and dotted parabolas correspond to a different parametrisation for the running of the coupling constant α and the inclusion of more data into the fit, respectively. The area marked in yellow corresponds to Higgs masses which can be excluded by available experimental data: A Higgs boson with a mass below 114.4 GeV/c² would have been discovered at the Large Electron-Positron Collider LEP at CERN before it was shut down to give place for the construction of the Large Hadron Collider LHC.

Now, supposing that the Standard Model is the correct theory to describe electroweak data such as the Z pole, the fitting procedure predicts the Higgs mass to be around the minimum of the χ² curve. Together with the experimentally excluded mass range, this yields an upper limit on the Higgs mass of 182 GeV/c² at a 95% confidence level, meaning that if the Standard Model is right, there is a 95% probability that the mass of the Higgs boson is between 114 and 182 GeV/c².

This is a predictions that will be easy to check checked at the LHC.

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The Higgs mass plot is taken from the preprint Precision Electroweak Measurements and Constraints on the Standard Model by the The LEP Collaborations: ALEPH Collaboration, DELPHI Collaboration, L3 Collaboration, OPAL Collaboration, the LEP Electroweak Working Group, arXiv: 0712.0929v2. More plots and details can be found on the web site of The LEP Electroweak Working Group (LEP EWWG), which "combines the measurements of the four LEP experiments ALEPH, DELPHI, L3 and OPAL on electroweak observables, such as cross sections, masses and various couplings of the heavy electroweak gauge bosons, properly taking into account the common systematic uncertainties" to confront "theories such as the Standard Model of particle physics."

Details of the fitting procedure (Which data have been fitted, actually? What are the fitting parameters?) are described in Section 8 (see specifically 8.5 and 8.6) of the paper Precision Electroweak Measurements on the Z Resonance by the ALEPH Collaboration, the DELPHI Collaboration, the L3 Collaboration, the OPAL Collaboration, the SLD Collaboration, the LEP Electroweak Working Group, the SLD electroweak, heavy flavour groups, arXiv:hep-ex/0509008v3, Physics Reports 427 (2006) 257.

χ² minimisation is done using the software package MINUIT. Different minimisation techniques and the χ² minimisation are described in a tutorial (PDF file).

Results of the direct Higgs search at CERN before the decommission of the LEP which have established the lower bound for the Higgs mass at 114.4 GeV/c² are described in Search for the Standard Model Higgs Boson at LEP by G. Abbiendi, et al., arXiv: hep-ex/0306033v1, Phys. Lett. B565 (2003) 61-75.

Physics World has a portrait of Peter Higgs, after whom the Higgs field has its name.

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This post is a latecomer to our A Plottl A Day series.

On The Edge

The Edge annual question 2008 is 'What have you changed your mind about?' and Sean over at CV summarized some of the answers for the average blogger's short attention span (try to recall the title of this post). I find it a tough question, given that I change my mind constantly about a lot of things. I also change my mind about whether I should consider this mental flexibility (better days), immaturity (worse days), or irrelevant (most days).

Either way, this reminded me that I promised previously to answer The Edge 2006 question 'What is your most dangerous idea?' (seems I missed out on 2007?). I said then I'd write this post when I'm in a particularly cynical mood, so therefore the parental advisory banner. There's no cursing here, and no discussion of my sexual preferences, just a pessimistic world view. Since I'm all for economic writing this is a read one get two, and I'll take care of '06 and '08 together: My most dangerous idea is that mankind is running into a downward spiral of reinforced constant unhappiness, and I've changed my mind about whether this is a realistic scenario or not. Though I've changed my mind back and forth about this a couple of times currently the status is back to disaster.

The point is simple. Intelligence is no longer an evolutionary advantage if the content of thought becomes increasingly abstract and theoretical. Our societies get more and more complex, and desperately need intellectuals, scientists, and thinkers to help them find their way in a world that's getting increasingly confusing every day. Yet, our societies don't listen to these voices, politics refutes any scientific method, leaders repeat mistakes, ignore warnings, and stick to believes that are scientifically wrong. It's a problem that has been around since thousands of years, but it is a problem that can be ignored for a long time - as long as trial an error works fast enough. Unfortunately though, the tolerance for mistakes gets smaller every year, and the consequences of mistakes larger.

The saddest example is maybe the present global warming discussion. All these political problems, the fact that capitalism alone fails to protect common goods, these have been discussed already decades ago. It is quite ironic to me, reading as news what we have been taught at school. May it be about the best way to provide incentives, saving energy, or reducing garbage. The climate change and energy shortening issue has been around since at least the Club of Rome report '72. It was the time of Greenpeace, remember that? Jute statt Plastik?

The energy problems I consider the much worse part because it will hit rather suddenly, yet despite all the hot air nobody actually does something about it. The obvious way out if oil gets short is power from nuclear fission. Face it - and think about that this will be a global problem. How many nations do you want to have in this world experimenting with their first nuclear fission reactors? We have, for better or worse, a global economy, but no global political system. It's a small wonder negotiations fail as long as the global marketplace has no balance in a political decision making process. And that's not a particularly new insight either. But hey, liberalism is still en vogue, lets wait some more decades.

I've been around in the blogosphere for long enough to realize that a significant fraction of our readers will now grind their teeth and say, girl stick to physics, you don't know nothing. Another part will think, gag, she shouldn't. And that brings me to the reason why I changed my mind about whether we will be able to resolve the present problems in a timely manner.

It seems I constantly hear people who don't know nothing about political systems, but are completely convinced their own is the best, and everything else is worse. I constantly hear people who don't know nothing about economical systems but are completely convinced their own is the best, and everything else is worse. And why so? Because they've been told so since the day they were born. And apparently that mode of thinking - which every serious scientist would immediately reject - is completely appropriate when it comes to political questions, never mind that a significant part of them can be addressed scientifically as well. And here we have it again, that gap between natural and social sciences, that gap between politics and our intellectual elite. A lack of communication, paired with increasing specialization, resulting in a society that doesn't listen to its thinkers. Hey, it's only a theory.

And that is what worries me. Not that other people's opinions might differ from mine, but that they are neither willing nor able to ask whether what they've been told is true, or still true. And they live very well with their ignorance that unfortunately reflects in the management of our societies. It's the way discussions are lead, the absence of scientific argumentation, the emphasis on advertisement over reason, rethorics over content, our inability to learn, that's what worries me.

A society that doesn't listen to its thinkers in times like this is a society that is destined to fail.

Our evolutionary developed mechanisms to improve 'fitness' of the human race don't work when we don't experience the consequences of our doing, or in an environment that changes too rapidly, too globally, too detached from our senses. And it's us who we change our world too fast to actually adapt to it, too fast so we can't accurately rate and act on the consequences. We just can't rely on our intuition in many regards, because our brains were never meant to deal with such situations.

Who cares if company X exploits underaged workers on the other side of the world, if their products are cheap and look nice. If you want consumers to change their mind, tell them the story of H. (name changed), with plenty photos of malnourished and sick children. That's the way the human brain works, and that's the game we play today.

So, we have an obvious tension between the immediate short-term goals that our neurons award with gratification (a personal masseur? fame? cigarette break?) and the problems that are either spatially or temporarily distant and get neglected.

What we have done to make our societies function efficiently in such a rapidly changing environment is we implemented fast working feedback mechanisms quite similarly to evolution, call it the survival of the fittest company, or idea. This works quite well to direct a fairly complex system towards optimization, allows capital to be invested into further development which is to the benefit of everybody. But then we forget that it was us who set up these mechanisms in the first place, for our own well-being, and that we might have to readjust them from time to time, asking whether they still do what we want them to do.

Now, we are living in a society where making money has become a self-purposeful action, where it should instead be a feedback mechanism to direct the economy. I am not against capitalism - it works well in many regards, and as long as improving the circumstances of living is correlated with economical growth of some kind, it is a good tool. But this correlation has its limitation, and the more advanced societies have reached them. Do I really have to tell you that money doesn't equal happiness? How many people are there today selling stuff they know is crap? How many of the pills advertised to make you younger, slimmer, more attractive are just wasting your time and potentially your money? How many advertisements are just blatant lies, but 100% guaranteed?

How many people spend their day trying to find a smart way to rip you off...

... and they will actually argue they make our lives better because it's good for the economy? Never mind that energy and resources are wasted into producing crap if one just adds enough psychologists for an irresistible advertisement. That's what I mean with reinforced unhappiness: The presently realized 'optimization' process blindly lead towards economical growth, on the possible expenses of the quality of humans living in that environment because not all factors are appropriately weighted by monetary value. Look at your leaders who talk about the economy, the economy, the economy all the time (unless they are busy talking about religion that is). What they should be talking about are their people first place. And then in the second place how the economy can be used to make our lives better (and they shouldn't be talking about religion at all). Capitalism, operation for profit, and market economy are tools. Useful tools, but as all tools they have a range where they can be applied, and others where they are inappropriate.

Sadly, nothing of what I've just said is new in any regard. If you want to measure the status of happiness in your society, look at the amount of people who need anti-depressants to get through their days because the quality of their living has so tremendously improved. It's a combination of feeling 'unfit' for the environment, amplified by the hopelessness that comes from believing they can't change anything about it - neither themselves, nor the external circumstances. We have changed our environment so much that we ourselves no longer 'fit', and are unable to resolve the mismatch.

You might notice the reasoning is similar to that of my earlier post on the Marketplace of Ideas, where I wrote about the gap between primary goals and secondary criteria. Secondary criteria are handy quantities to measure success, and optimizing them provides a sensible mechanism towards improvement possibly for a long time. The danger though is that fulfilling the secondary criteria becomes a self-purposeful action, and their optimization is pursued even though it is not the actual goal. Here it is making money (secondary criteria) that is confused with improving quality of living (primary goal). In the scientific community there's a bunch of criteria like number of publications, potential to obtain grants, being in fashion, connections in the community (secondary criteria) that are being confused with good research (primary goal).

Want more examples where people loose their primary goals out of sight for the immediate reward? Look at a couple of science blogs with advertisement banners. Meant to promote scientific knowledge, their websites also feature advertisements for crackpottery, intelligent design, the new quantum mechanics or other metaphysics. Sure I understand it's nice to make money. Know your priorities.

So to end with, why is this dangerous? Well, when I read the book 'What is your most dangerous idea?' I was wondering what actually is 'dangerous'. It crossed my mind that one could destroy the planet, or maybe the whole universe (yeah). But actually, I think suffering is worse than dying, so the most dangerous idea seemed to me that we can maneuver our society into a state of constant unhappiness because we're too stupid to correct our actions, and find the way out. Like the fly in your room that constantly bumps against the glass, yet is unable to learn from its mistakes. Stupid, stupid, stupid.

Actually, I think The Edge is one of the little efforts into the right direction, providing a forum for your societies' intellectuals to reach the public, as well as a possibility to bridge the communication gap that has developed between scientists of various disciplines. Let's hope enough people listen.

I just looked up the 2007 question. It was 'What are you optimistic about?' Ah well, I will come back to this when I have a better day.

This and That

If you're still in a pondering mood with the New Year, here are a few things to think about

• At his website The Edge, pop-science literature agent John Brockman compiles every year answers by scientists and writers to one "big question". This year's question is WHAT HAVE YOU CHANGED YOUR MIND ABOUT? WHY?, and it's very interesting to read through the answers by people with backgrounds ranging from physics over neuroscience to sociology and political science. See at Cosmic Variance for a few selected answers.



• Evaluating everything in numbers may be a controller's dream, but I am less sure about what to think about Bruce Knuteson's "Quantitative Measure of Experimental Scientific Merit" (arXiv:0712.3572v1). Building on Shannon's information entropy, Knuteson proposes a formula expressing the value of an experimental result by "how surprised you are that the particular result has been obtained", and applies his ansatz to experiments in particle physics over the last 35 years. The Collider Blog and Charm and Co are cautious about this measure of scientific merit - and indeed, it's not very well suited to evaluate the merit of precision experiments which, while maybe not very surprising, are nevertheless essential to solidify the trust we have in our theories.



• Chanda let us know of the 2008 joint annual meeting of the National Society for Black Physicists (NSBP) and the National Society of Hispanic Physicists (NSHP), in Washington (DC, USA) on February 20-24, 2008. Additional information on the program can be found at the NSBP web site.



• And, on a personal note, we were just informed that Sabine's post The Marketplace of Ideas made it into the second Science Blogging Anthology "Open Laboratory 2007". Yeah!

A Happy New Year to all of you!

Daemons

Yesterday, we went to see the movie 'The Golden Compass' based on the book by Philip Pullman. I had read the book quite some while ago, long enough to have forgotten most of it except for the fact that I didn't like it, and therefore never bothered to get the two other parts of the trilogy. Upon seeing the movie I recalled why I disliked the book, so if you were looking for praise of your new fav movie you're on the wrong blog. The movie adaption is as far as I can tell excellent, but I generally don't like a) movies with animals, b) movies with children, and c) movies with an overdose of moral and messages imposed on the innocent audience, so was destined to dislike The Golden Compass. If you translate movie's animals and children into underdeveloped characters, this disliking of mine applies to books as well. Combine that with the fact that Pullman's writing style didn't exactly strike me as outstanding, I keep wondering what people find so great about it.

It's not like I think the story is generally bad, it is without doubt a burst of creativity and imagination. I just think he mixed up too many things on the expenses of authentic persons and consistent storytelling. Take Lyra, the hero of the story, who is supposed to save not only her friend who has been kidnapped, but the whole world, the universe, and while we're at it, also all other parallel universes. Wouldn't one think she spends some time trying to cope with that news? But no, she just goes to the North pole, while other more or less flat characters drop in and out of the storyline. That's even worse than Frodo in the Lord of the Rings, which I disliked for essentially the same reason, but at least he wonders about his destiny - Rowling's Potter does so to a much more appropriate extend, and Harry develops some personality.

Either way, the idea that people's souls live outside their bodies in animal 'daemons' is interesting. Though here too I find the setting somehow insufficiently explored. I mean, are there other animals in that world (except ice bears!), and how do they get along with the daemons? Do the daemons get born with the humans? Can they have sex with other people's daemon's? Well, maybe some of that would be answered if I'd read the other two books.

I went home thinking that would make for a great online test, and sure enough the movie website allows you to find your own daemon, so have fun, you see mine above. I admit on taking the test twice, the first result happened to have one of my ex-boyfriend's names, and how likely is that? Gee, imagine your ex-boyfriend tied to your side for the rest of your life. It remained a tiger though, concluding "You are modest, a leader, assertive, solitary and inquisitive". As with other personality tests, it's not much of a match: I'm too modest to claim I'm modest, suspicious about people who follow me, and I am too solitary to be inquisitive. Or maybe too German, i.e. 'How are you' is about the maximum of inquisition I impose on strangers. Had I picked my daemon it would have been a dirty black cat I think.

So, in the hope to better get to know our commenters, what is your daemon?

I have to say though the movie's animations are really great. How difficult would it have been only a decade ago to get your armored bear running accross the ice with a girl on his back, and how smooth and realistic does this look nowadays. This really impresses me. I keep thinking the main reason why there seems to be an increasing amount of fantasy movies is simply that it has become possible.

Trend of arXiv submissions: Update

A couple of days ago I was wondering how the trend of average monthly submissions on the arXiv would continue this year. Here is the updated plot with the statistics from 2007:



As one sees the trends do more or less continue. The average number of submissions on the hep arXiv are shown in blue and include hep-th/hep-ph/hep-lat and hep-ex. Red is astro-ph, green is cond mat, violet is math with the pink addition being math-ph. The clear extensions are cross-links. The number of hep submission slightly increased again, so it seems the temporary drop was a fluctuation. It seems to stagnate around 730 papers/month.

I am still wondering how it comes though. Several people have argued this is to be expected and just suggests 100% participation from that community (this is also the explanation you find on the arXiv website). I don't think though this can be the full story. I would expect the average number of submissions to be roughly a product of

- the total number of people in the community
- their average productivity
- the fraction of them using the arxiv

First, it is plausible to me the fraction of people using the arxiv in the hep community is saturated in North America and Europe, but given that there are still new countries coming into the game and the arXiv is as global as can be I am not sure whether this actually holds. That is to say, globally seen I'd still expect that fraction to be increasing.

Second, since the world population generally is increasing, one should think this general trend underlies the statistics, unless the fraction of people in the hep-community relative to the total population decreases.

Think China, Japan, India. Then have a look at the submitter's affiliations and tell me one should not expect further growth. I mean, yeah, there are a lot of Germans, but we don't outweigh the rest of the world.

Taken together this means, if people stick to the arxiv either the number of people in the community doesn't grow as I'd expect, or they publish less. Another possibility, as Dylan mentioned, would be that people just submit papers to other arXivs. E.g. it might be that some stuff that used to end up on hep, now goes into physics general, or history of physics. Or, maybe now that there are more online journals that publish more or less immediately, people don't submit all their papers to the pre-print server? Anybody noticed something like this?

The other thing that I found surprising is the sheer number of math publications. Not the trend - I understand that probably that community is still not completely arxiv-ed - but gee, look at all those math papers! I always had the impression mathematicians publish only sparingly, seems I was somewhat mistaken there.

For more statistics see here.

Did you know... (VI)

... that the Eskimos have 98 words for snow?

Yeah, me too, but it's actually bullshit. One of the more useful side effects of the internet is the busting of urban legends. Though it's useful only if one actually looks for it: Googling 'Eskimo Words for Snow' gives you easily several references that explain not only where the myth comes from, but also what's wrong about it.

The brief explanation is that besides there being several 'Eskimo languages' these are polysynthetic, meaning one can put several nouns with describing adjectives together into one word -- which gives a new word. I.e. there is snow, there is frozen-snow, frozen-and-dirty-snow, frozen-and-dirty-snow-with-a-crust-that-breaks-if-one-steps-on-it, and then there is snow-on-my-outside-chair-waiting-for-springtime.

Reference: Laura Martin, American Anthropologist, Vol. 88, No. 2 (Jun., 1986), pp. 418-423

"Eskimo words are the product of extremely synthetic morphology in which all word building is accomplished by multiple suffixation [...] Furthermore, precisely identical "whole" words are unlikely to recur because the particular combination of suffixes used with a "snow" root, or any other, varies by speaker and situation as well as by syntactic role."

The paper is actually quite entertaining in the way she clarifies earlier claims ("A minimal knowledge of Eskimo grammar would have confirmed the relevance of these facts to the central hypotheses [...]" Ouch.)

Either way, I was shocked to see that the above publication is from '86, since I must have read about it repeatedly, and definitely after '86.

The interesting question is much longer will that story to survive? So, take the poll below and answer the question whether you had heard of the story that the Eskimo's have so-and-so-many words for the one English word 'snow' (the precise number of words doesn't matter)

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See also: Did you know...

• ... that the can opener was invented 48 years after invention of the can?
• ... the opposite of eloquent?
• ... why the toast is called toast?
• ... what the Baconian method is?
• ... why Google is called Google?

Trend of ArXiv Submissions

The plot above shows the average monthly submission rates on the arXiv from '91 until December last year. The colors indicate the different areas blue: hep, green- cond-mat, red: astro-ph, purple: math(+pink: math-ph). The clear extensions of the bars indicate cross-links. Note how hep stagnates and drops, while math is taking the lead. Below is the statistics for hep only



I'd be interested to see the statistics from 2007, do you think this trend continued?

You find more of the arxiv statistics here.

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This post is a latecomer to our A Plottl A Day series.

Winter Wonderloo

Photos taken in Waterloo, Ontario. (Hey, Google maps has updated the satellite photos!).

More photos.

Merry Christmas!

So, here is the promised quiz to conclude our seasonal program. It turned out to be not so easy to come up with questions that aren't immediately googleable, but still seemed answerable with an appropriate amount of effort.

1. The light deflection from the sun computed within Newtonian gravity differs from the result obtained in General Relativity by a factor ...
   
   (If your answer is smaller than one, take the inverse).



2. The ratio of these both scientist's name is the commonly used symbol for ....
   
   
   
   

   /

   
   
   [My husband complains this might be too tough, so here is a hint. The first guy is a Dutchman and you're supposed to know his name if you've made it through the physics undergrad courses. The second guy had something to do with geese. Don't bother checking the source code for the filenames.]
   



3. "I didn't attend the funeral, but I sent a nice letter saying I approved of it." ~ Mark Twain
   
   Is an example for .......
   
   Take the first four letters.



4. Two divided by zero. Always on my mind. I wouldn't normally do this kind of thing. It's a ...



5. Electromagnetism is the gauge theory of the group .... Drop the first letter.

Write down your answers in one line, add two spaces in the right places and translate into Greek.

The first correct answer wins a PI mug, see photo. And no, I didn't steal it in the kitchen, I actually bought it. (If you are not willing to provide a mailing address, or aren't interested anyhow, please do not post your answer in the comments). A Merry Christmas and nice holidays to all of you!

Update: The mug is gone, congratulations to Samuel! If you want to continue guessing, spoiler warning: the correct answer is in the comment section.

iPostDoc Playlist - Christmas Edition

I'm temporarily indulging in a certain amount of self-pity while my family is opening the sparkling wine, and I am sitting in my apartment packing my household back into boxes. For those of you who haven't followed my moving fun, due to an unfortunate time-ordering-problem of the previous tenants paired with my own stupidity, I will have to move some floors upwards during the next days. So I won't be home for Christmas this year. Otoh, I don't have to repeat my annual summary a dozen times to update all the neighbors, friends, and relatives. And no brothers will have picked away all the nougat pralines while I am blogging.

-- Richard Marx: Right Here waiting -- Oceans apart day after day, And I slowly go insane -- Michael Buble: Home -- And I know just why you could not come along with me, 'Cause this was not your dream, but you always believed in me -- Chicago: Hard to say I'm sorry -- After all that we've been through, I will make it up to you. I promise to. -- Bangles: Hazy Shade of Winter -- Time, time, time, See what's become of me, While I looked around, For my possibilities -- Thornley: So far so good -- So far so good 'cause no one knows I'm faking, I wish I could show you the toll it's taking, Sometimes I live as if there's no tomorrow, So far so good -- NIN: The Wretched -- Just a little reminder, Of all the what abouts, And all the might have, Could have beens, Another day, Some other way -- Ron Sexsmith: Imaginary Friends -- For they've gone where the action is, And they've crossed you off their list -- Coldplay: Fix you -- When you try your best, but you don't succeed, When you get what you want but not what you need, When you feel so tired but you can't sleep, Stuck in reverse? -- Enya: Only Time -- Who can say where the road goes, Where the day flows, only time? -- ABBA: Waterloo (don't miss this video, Sweden '74!) -- Waterloo. Couldn’t escape if I wanted to. --

The Unitary Triangle

[The Unitary Triangle, by the CKMfitter Group]

My husband and I, we both agreed the Unitary Triangle above is the prettiest plot of contemporary physics. Unfortunately, it turns out neither of us knows very much about the actual experiments that constrain the parameter space, so we'll have to be a bit brief in this regard and just refer you elsewhere for the details.

The quantities constrained in this plot are parameters of the Cabibbo-Kobayashi-Maskawa matrix, V_CKM, which maps the weak eigenstates of the quarks to the mass eigenstates. This matrix it not diagonal, and mixes quark flavors in a similar way how the neutrino mixing works.


Here, the primes indicate the weak eigenstates. For example, it is the d' quark that transforms into an up quark in the weak decay mediated by the charged W boson - this is, on the quark level, the process which is going on in the radioactive decay of the neutron into a proton, an electron, and an antineutrino. The CKM matrix is unitary, which means the elements have to fulfil the relation

and other similar relations, but this is the one frequently used. This is a sum of three complex numbers that add up to zero. One can draw these numbers as vectors in the complex plane, and since they sum up to zero they will have to form a closed triangle. The relative angles between two sides are the arguments of the ratios of these complex numbers

Since it's a triangle one has α+β+γ = π. In addition, one can normalize one of the sides to length one and turn it such that it lies on the real axis. The resulting triangle is what you see in the plot above, the task is now to constrain it's shape, i.e. the angles or side lengths respectively. The entries of the CKM matrix affect a lot of particle physics processes, so one has to figure out smart ways how to extract them from the data. Most experiments look in detail at the weak decay of hadrons such as Kaons and B mesons. In these decays, strange and bottom quarks decay into up and charm quarks, respectively. Moreover, some of the experiments involve penguins.

Some experiments constrain an angle with a certain precision - this you see in the plot as the light blue shaded outgoing beams. Different shades indicate confidence levels. The angle β is studied through the interference between decays of neutral B⁰ mesons; γ is more challenging to measure and requires studies of rare decay processes; and also α is difficult to measure because quantum effects interfere with direct measurements. Other experiments constrain the length of a side which results in the green and orange circles. The triangle’s left side is measured through the decay rate of bottom quarks into up and charm quark; the triangle’s right side is constrained by the rate the B⁰ meson spontaneously turns into its antiparticle. And then there is the boomerang shaped green region which results from measurements of CP violation in Kaon decays.

Taken together this data constrains the top of the diagram to lie in the region marked with the red boundary.

The CKM matrix is one of the less beautiful parts of the standard model of particle physics, despite the aesthetic appeal of the unitary triangle. There are no explanations for the entries of the matrix, which just have to be taken as some parameters of the standard model. Nevertheless, the CKM matrix is relevant to quantify CP violation, which helps to understand the predominance of matter over antimatter in the universe.

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This post is part of our 2007 advent calendar A Plottl A Day.

The Quantum Hall Effect

One of the basic rules of electricity says that in order to maintain an electrical current I in a wire, a voltage U is necessary that is proportional to the current. This is Ohm's law, U = RI, where the constant of proportionality is the resistance, R. If the wire with the current is exposed to a magnetic field, the Lorentz force acts on the wire, normal to both the direction of the current and the magnetic field. If the current flows through a sheet, as a result of the Lorentz force on the electrons in the current, a voltage can be measured that is oriented along the direction of the Lorentz force. This is the Hall voltage, named after the American physicist Edwin Herbert Hall, who was the first to observe this phenomenon. The Hall effect is used today, for example, to measure magnetic fields.

Ohm's law and the Hall effect can be understood in terms of classical physics: electrons travelling through the wire bump into the atoms of the material of the wire, which causes the resistance. However, as we know for example from the study of the electronic band structure of materials, electric current is carried by particles subject to the rules of quantum mechanics. And if the temperature of the conducting material is low enough, thus reducing the strength of thermal effects, quantum effects can become apparent, leading to such striking phenomena as superconductivity. In a superconductor, charges set in motion do not scatter, since quantum mechanics doesn't allow it, and hence, there is no resistance.

Another quite strange quantum phenomenon can be observed if the current is constrained to a very thin sheet, so that it's essentially two-dimensional, and if a strong magnetic field is applied normal to the sheet. In this case, there is a special class of states for the electrons called Landau levels, and depending of the occupancy of the Landau levels, a new macroscopic phenomenon similar to superconductivity can set in: the Quantum Hall Effect.

[Source: K. v. Klitzing, G. Dorda, M. Pepper: New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45 (1980) 494-497, Figure 1.]

The experimental data shown in the plot stem from the experiment in which the Quantum Hall effect was discovered. In the experiment, shown schematically in the inset, a thin sheet structure of semiconducting material, carrying a constant current of 1 µA, was exposed to a strong magnetic field at the temperature of liquid helium, and the voltage drop along the current and the Hall voltage normal to the current and the magnetic field have been measured. These voltages are plotted on the vertical axis, and the curves are labelled as U_H for the Hall voltage, and U_pp. The horizontal axis shows another voltage, the so-called gate voltage V_g, which squeezes the electrons in a thin, two-dimensional sheet, and thus determines the occupancy of the Landau levels. Now, with varying gate voltage, a curious phenomenon can be observed in the Hall and standard voltage: the standard voltage drops to zero several times, meaning that the current flows without resistance, as in a superconductor, and at the same time, the Hall voltage takes on constant values. There is series of step-like plateaux in the Hall voltage, where it stays constant.

Now, what looks like quite an esoteric effect - take some contrived semiconductor structure, cool it down to a few Kelvin, put it in the strongest magnetic field your high-field lab can provide - has an extremely interesting twist to it:

The Hall resistance, the quotient of Hall voltage and the current, in the step-like structures comes in a regular series, R = R_K/n, where n is an integer, and R_K, the von Klitzing constant, is R_K = 25.812807557 kΩ. And the really fascinating thing is that as a consequence of the theory of Landau levels, the von Klitzing constant does not depend on the details of the material, but is universal: It is given by R = h/e² = μ₀ c/2α, where h is Planck's constant, e is the charge of the electron, and α = 1/137 is the fine structure constant, the coupling constant of electrodynamics.

Thus, the quantum Hall effect can be used, for example, to set a standard for electrical resistance, or as a means to measure the fine structure constant. Klaus von Klitzing, the German physicist who discovered the effect in February 1980 and immediately grasped these implications, was awarded the Nobel Prize in Physics 1985 for the discovery of the quantized Hall effect.

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Klaus von Klitzing's Nobel Lecture The Quantized Hall Effect gives a good introduction into theory and experiment of the Quantum Hall Effect (PDF file).

More details can be found in the proceedings of the Poincaré Seminar of November 13, 2004, dedicated to the Quantum Hall Effect. Among the talks:

Klaus von Klitzing: 25 Years of Quantum Hall Effect (QHE) A Personal View on the Discovery, Physics and Applications of this Quantum Effect (PDF file)

Benoît Douçot and Vincent Pasquier: Physics in a Strong Magnetic Field (PDF file), about Landau levels etc...

Beat Jeckelmann and Blaise Jeanneret: The Quantum Hall Effect as an Electrical Resistance Standard (PDF file), about the implications of the Quantum Hall Effect for metrology.

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This post is part of our 2007 advent calendar A Plottl A Day.

Treehugging

Done! Done with Christmas shopping!

My husband just told me he bought a flashlight for my younger brother. Totally ingenious, it has a mode with a red warning signal and can be tied around the head. I'm not entirely sure what my brother is supposed to do with that. But the batteries are allegedly durable for ten years, so maybe something springs into mind.

Either way, I read today this article in the Globe and Mail:

"The hush-hush regreening of Europe"

According to which "[Europe's] forest cover has expanded by almost 10 per cent since 1990, and a much larger greening seems to be under way, reversing centuries of deforestation. The greatest share of this growth is a result of deliberate policies designed to turn farmland into woodland."

Having grown up seeing the forests in my neighborhood shrink every year, it made me very happy reading this. I welcome the trend to support afforestation of farmland not only because I like trees (yeah sure, I talk to my plants), but because the financial support for farmers whose products are not consumed either way and just rot away is nothing but a waste of resources.

That further caused me to look up the websites of the European Environment Agency where one finds a lot of data and statistics, and websites of the Ministerial Conference on the Protection of Forests in Europe (MCPFE), more data and statistics, in my impression trends are overall good. Below you see Fig. 46 from their 2007 Report on the State of Europe's Forests (beware, it's more than 16 MB worth of plots and diagrams). It shows the share of protected area of the forest by country:

[Share of protected area of the total forest and other wooded land area for biodiversity (MCPFE Classes 1.1–1.3) and for landscape (MCPFE Class 2) (%), by country in the MCPFE region, 2005, classified according to the share of forest protected for biodiversity, Source: MCPFE. The * means only data for forest available. Click to enlarge]

I admit on deliberately chosing a figure where Germany looks good. And the pie below shows the share of marketed forest products other than wood... seasons greetings!


[Marketed non-wood forest products from forest and other wooded land in Europe. Share of total value in countries (based on available data). Click to enlarge.]

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This post is not part of our 2007 advent calendar A Plottl A Day.

Winter Solstice

Today is the Winter Solstice, the shortest day of the year in the Northern Hemisphere. At 06:08 UT (1:08 AM EST, 07:08 MEZ), the Sun will reach its southern turning point on the celestial sphere, culminating above some point in the Indian Ocean on the Tropic of the Capricorn. In Frankfurt am Main, Germany, sunrise will be at 08:22, and sunset at 16:26, yielding in principle 8 hours and 4 minutes of sunshine.



Today's plot shows the time of sunrise, Sun's culmination, and sunset for Frankfurt over the course of the year. Date is on the horizontal axis, and time of the day on the vertical axis. The shortest day is marked by the vertical line. Sunrise and sunset are shown in orange, culmination - that is, "the true noon", when the Sun is exactly at South - in red. The lines in black, purple, and blue mark the times of astronomical, nautical, and civil twilight, respectively, when the Sun is 18°, 12°, and 6° below the horizon.

Besides the changes in the length of the day, which is roughly between 8 and 16 hours over the course of the year, the plot shows one more unexpected phenomenon: The time of the culmination of the Sun, the true Noon, is not always at the same time - it varies by about half an hour over the year. As a result, the time span of daylight not only varies in length, but also shifts around slightly with respect to the hours of the day. The difference between true Noon and the time when one would expect the Sun at South from the longitude (the "mean Noon", 12:25 MEZ for Frankurt at 8°41 East, for example) is called the Equation of Time. It is caused by the combined effect of the elliptical shape of the Earth's orbit and the tilt of the Earth's axis.

As a side effect of the Equation of Time, the earliest sunset is not at the Winter Solstice, but a few days before. The figure on the left shows a detail of the plot above: Sunrise, culmination, and sunset between November and February. One can see clearly that there is an offset of about 15 days between the earliest sunset, around mid-December, and the latest sunrise, in the first days of January.

The rapid shift in the time of culmination - about half an hour between beginning of November and end of January - can be understood from the fact that the Earth is at the perihelion, the smallest distance to the Sun on its orbit, around January 3. As described by Kepler's law of areas, encoding the conservation of angular momentum, the Earth's angular velocity as seen from the Sun is biggest at the perihelion. But this means that the Earth has to spin a little bit longer from culmination to culmination than at other phases of the orbit. As a consequence, culmination is always a little late from day to day, and the "true Noon" shifts from a bit later than 12 o'clock to 12:40 between November and February. Summer days, instead, will be a little bit shorter.

But at the meantime, those of us in the Northern Hemisphere who are suffering from the winter's darkness can look forward to the daylight coming back.

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All data for the Sun shown in the plots have been obtained from the Data Service of the U.S. Naval Observatory.

Here is a link to a Daylight Applet that you can use to plot sunset, sunrise and twilight times for any location on the planet.

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This post is part of our 2007 advent calendar A Plottl A Day.

The Cosmological Parameters

[Figure: Supernovae Cosmology Project]

The most puzzling experimental result of the last decade that has inspired hundreds, if not thousands of papers, is the matter composition of our universe. The measurements indicate that the usual matter we are made of is only a small fraction, 4%, of all the matter content of the universe. According to present data, 23% is dark matter, whereas the remaining 73% is dark energy. The matter densities are usually normalized to the so-called critical density ρ_crit = 8 π G/3 H², upon which one obtains the more convenient dimensionless parameters Ω_M for the fraction of matter (us + dark), and Ω_Λ for the dark energy part.

The plot above shows best fit confidence regions in the Ω_Λ versus Ω_M plane. It combines data from supernovae redshift, galaxy clustering, and analysis of the Cosmic Microwave Background. The shaded upper left corner indicates a region where there would be no Big Bang (scale factor doesn't go to zero). The diagonal line is a flat universe, and divides the areas of closed and open models. The slightly upward bended line divides region with and without recollapse (derivative of the scale factor can have a zero). If Ω_Λ = 0, then a closed universe recollapses eventually. Roughly speaking, more matter requires more dark energy to have continuing expansion, so the line dividing continuing expansion from recollapse bends upwards. (Sean Carroll explains you how to compute these boundaries here.)

The shaded lower right corner would imply the the universe was younger than the oldest observed stars. The supernovae results and the CMB data constrain combinations of both parameters, such that the best fit regions lie on diagonal ellipses, whereas the cluster data is dominantly sensitive to the amount of matter, though the ratio of X-ray gas mass to total mass depends on depends on both parameters, so it yields some weak constraints also on Ω_Λ (Allen, Schmidt and Fabian, Mon. Not. Roy. Astron. Soc. 334 (2002) L11 arxiv:astro-ph/0205007).

The present data is compatible with a flat universe. Though one has to be somewhat careful what this analysis actually shows. It shows that the ΛCDM model - a flat cosmology with the above mentioned fractions of dark matter and dark energy - is a parametrization of observed effects that is the best fit to the presently available data. But so far we have no experimental knowledge about the microscopic nature of the unknown constituents of the universe.

Actually, writing this series of posts about today's data and how it is compatible with our theories I am impressed how much the homo sapiens sapiens has learnd about the world around him, ;-).

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This post is part of our 2007 advent calendar A Plottl A Day.

The J/Psi and the Charmonium Spectrum

Atoms emit light at very specific wavelengths - their so-called spectrum is their characteristic fingerprint. The origin of the discrete lines in the spectrum can be understood best for the most simple atom: In the hydrogen atom, which consists of an electron bound to a proton, the electron can exist only with certain distinct energies. Such are the rules of quantum mechanics, and as a consequence, the bound system of the electron and the proton can absorb or set free energy only in specific amounts, which correspond to light with the frequencies, or wavelengths, that show up as lines in the spectrum. The analysis of the spectrum of the hydrogen atom has revealed a lot of details about the electromagnetic interaction between electron and proton, and between electrical charges in general.

[Source: Kay Königsmann: Radiative decays in the Ψ family, Physics Reports 139, Issue 5, June 1986, Pages 243-291, Fig. 5.]

One may wonder, is there a similar phenomenon for bound systems between other particles, say, between quarks, which carry a so-called colour charge and interact via the strong force described by quantum chromodynamics, QCD? The answer is an emphatic yes - and one can learn a lot from it. The figure shows the spectrum of excited states of the J/Ψ meson, a bound charm quark-antiquark pair. Like the electron-proton pair in the hydrogen atom, the quark-antiquark pair can have only specific energies, it can be excited to a series of states with higher energies, and it can emit and absorb light when transiting between these states. The corresponding photon spectrum - the numbers of photons within a small range of energy counted in a detector - is shown in the plot, with a specific pattern of lines superimposed on a smooth background. The numbers help to identify the lines with the transitions between different states, which are shown schematically in the lower part of the figure. Data have been measured at the Crystal Ball, a spherical detector that completely encloses the quark-antiquark pair.

The photons counted in the spectrum do not correspond to visible light, however, but are short-wavelength gamma rays with an energy in the range between 100 and 500 MeV - that's much more than the 1.9 eV corresponding to the red Hα line in the hydrogen spectrum. This should not come as a surprise: The energy scale of the hydrogen atom is set by the Rydberg constant, R = 13.6 eV, which is proportional to α²m, where α ≈ 1/137 is the fine structure constant - the coupling constant of QED - and m is the mass of the electron. Now, a charm quark has about 3000 times more mass than an electron, and the coupling constant of QCD, the strong interaction, is about 100 times bigger than that of QED. Hence, one could expect spectral lines with energies by a factor 30 million times bigger for the charm-anticharm pair than for the hydrogen atom - as you can see, that's not that bad an estimate.

However, there is an essential difference between the hydrogen atom and charmonium, as the bound charm-anticharm pair is usually called, and it can be spotted in the known spectrum of charmonium states, which is shown in this plot.

[Source: Ted Barnes: The XYZs of charmonium at BES, Int. J. Mod. Phys. A21 (2006) 5583-5591 (arXiv: hep-ph/0608103v1), Fig. 1.]

Here, the known charmonium states are shown as black lines according to their energy, or mass, on the vertical axis, and grouped as per orbital angular momentum of the quark-antiquark pair, labelled by S, P, D, F, along the horizontal axis. In contrast to the hydrogen spectrum, there is no series limit at high energies, which in the hydrogen atom corresponds to ionisation, the separation of the electron and the proton. Instead, energy levels increase rather uniformly, but cross a line, labelled "DD". Above that energy, the charmonium system can decay in a D meson, made up of charm quark and a light antiquark, and the respective antiparticle, the anti-D. This is the manifestation of very characteristic feature of QCD, called colour confinement: No isolated quarks, or other colour charges, can be observed. Instead, if one tries to separate, say, the charm quark and the anticharm quark of the J/Psi; by adding energy, a new quark-antiquark pair will be created, and two mesons will be formed, the D/anti-D pair.

Because of confinement, it is clear that the interaction energy between quarks can not be described by a simple analogy to the Coulomb law for electrical charges. To model confinement, it is stipulated that a Coulomb-type interaction has to be amended by some energy which increases linear with charge separation. In fact, one has tired to reverse-engineer an interaction potential between quarks starting from the charmonium spectrum: Making an ansatz for the interaction energy, one can calculate the corresponding spectrum, and fit the parameters to match the observed spectrum. The most popular ansatz is often called Cornell potential and looks like this:

Here, the first term is the Coulomb energy, with the strong coupling constant α_s instead of that of electrodynamics, the second term is the energy linear in distance, which enforces confinement, and the third term contains spin-dependent terms to model the fine structure of the spectrum. The constant κ is the the so-called string tension. In the spectrum shown in the figure, the best fit to the experimental data with this spectrum is shown in red. It corresponds to a mass of the charm quark of m_c = 1.46 GeV/c², a coupling constant α_s = 0.55, and a string tension (called "b" in the plot) of κ = 0.72 GeV/fm - meaning that an energy of 0.72 GeV is needed to separate the quark-antiquark pair by 10^-15 meters.

Understanding confinement of colour charge is an open problem in physics, and the details of the interaction between quarks forming a hadron still contain many riddles. The analysis of the spectrum of quark-antiquark pairs as in charmonium can help to a better understanding of these issues - that's why the spectrum of charmonium is still an active area of research.

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The best electrodynamcial analogue to charmonium is not the hydrogen atom, but the bound state of an electron and a positron, called positronium. The nomenclature of charmonia, and the name charmonium itself, derives from positronium physics.

The Cornell potential got his name after the group of physicists at Cornell who had used it within weeks of the discovery of the J/Ψ to analyse the spectrum of its excited states - see E. Eichten et al.: Spectrum of Charmed Quark-Antiquark Bound States, Phys. Rev. Lett. 34 (1975) 369.

Charmonium physics will be one main topic of the PANDA (Proton ANtiproton DArmstadt) experiment at the new "Facility of Antiproton and Ion Research" (FAIR) at GSI, Darmstadt, Germany (see, e.g., Bertram Kopf: Physics with Antiprotons at PANDA, J. Phys.: Conf. Ser. 69 012026).

The Crystal Ball Detector is now in use at the Mainzer Mikrotron (MAMI), Germany.

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This post is part of our 2007 advent calendar A Plottl A Day.

Indirect Detection of Gravitational Radiation

[Source: J.M. Weisberg, J.H. Taylor: Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis, arXiv: astro-ph/0407149v1 Figure 1.]

One of the predictions of Einstein's General Theory of Relativity is the existence of gravitational waves: Two large masses in orbital motion will create small, wavelike distortions in spacetime that propagate like ripples on a pond, and carry away energy.

Despite big efforts and huge detectors, no gravitational waves could be measured so far. But direct discovery notwithstanding, physicists are confident that gravitational waves are real, since there is very compelling indirect evidence for their existence, and for their compliance to the rules of General Relativity.

A typical source of gravitational waves would be a couple of very massive, compact stars in close orbits. Binary neutron stars are good candidates for sources of gravitational radiation. The energy oss by the emission of gravitational waves would result in a slowly decaying orbit: the stars would et ever closer the more energy would be radiated away as gravitational waves. And if one of the stars is a pulsar, there are chances that one can measure the orbit, and establish the indirect consequences of the emission of gravitational waves.

That is exactly what has happend with the binary pulsar Binary Pulsar PSR 1913+16: In December 1973, astronomy student Russell A. Hulse was at the Arecibo Radio Observatory in Puerto Rico, collecting data on pulsars for his Ph.D. thesis. One of the pulsars he was observing showed a curious, periodic variation in the pulsation frequency. It soon became clear that this variation could be best understood as the periodic Doppler shift of the pulsar in an elliptic, Keplerian orbit around another star. This allowed for the very precise reconstruction of the orbit of the pulsar.


Schematic picture of the binary pulsar PSR1913+16: Two neutron stars circle each other closely in elliptical orbits. Their mean separation is only a few times the distance Earth-Moon, and one orbit takes only about eight hours. When the stars pass close to each other, they emit large amounts of gravitational radiation. [Source: Nobel Foundation, 1993.]

The orbital parameters of this binary system, however, are so extreme - masses on the order of the Sun orbiting within eight hours at 1 permille of the speed of light at a distance on the order of the distance Earth-Moon - that Kepler's laws for the motion of two masses under the influence of gravitation are not sufficent anymore: As Hulse's advisor Joseph Taylor noted, the full-scale apparatus of General Relativity was necessary to describe the orbit, instead of the simple Newton law of gravitation, For example, the binary pulsar shows a big motion of the periastron - the equivalent of the perhelion shift of Mercury, whose share not accounted for by the perturbations of the other planets in the Solar System was explained by Einstein as the first "postdiction" of his new theory of gravitation. What's more, the time evolution of the orbit can be measured precise enough to check for the consequences of the emission of gravitational waves!

The data points in figure show the observed change in time of periastron over the last 30 years, since the first data on pulsar PSR 1913+16 have been available. The parabola illustrates the theoretically expected change in periastron time for a system emitting gravitational radiation, according to general relativity.

Not only the Nobel committee in Stockholm considered this an extremely important result - but they awared to Russell A. Hulse and Joseph H. Taylor, Jr., the Nobel Prize in Physics 1993 for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation.

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Taylor's Nobel Lecture is on Binary Pulsars and Relativistic Gravity (PDF file), while Hulse in his Nobel Lecture talks about The Discovery of the Binary Pulsar (PDF file).

The Binary Pulsar PSR 1913+16 is a nice short introduction that explains in more detail how the pulsar data are analysed, and how the periastron shift can be extracted.

For the latest data about the "Taylor-Hulse" binary pulsar PSR1913+16, check out Wikipedia.

For two recent related rviews, see Duncan R. Lorimer: Binary and Millisecond Pulsars, Living Rev. Relativity 8, (2005), and Ingrid H. Stairs: Testing General Relativity with Pulsar Timing, Living Rev. Relativity 6, (2003).

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This post is part of our 2007 advent calendar A Plottl A Day.

Neutrino Masses and Angles

Neutrinos come in three known flavors. These flavors correspond to the three charged leptons, the electron, the muon and the tau. The neutrino flavors can change during the neutrino's travel, and one flavor can be converted into another. This happens periodically. The neutrino flavor oscillations have a certain wavelength, and an amplitude which sets the probability of the change to happen. The amplitude is usually quantified in a mixing angle θ. sin²(2 θ) = 1, or θ = π/4 corresponds to maximal mixing, which means one flavor changes completely into another, and then back. For a brief introduction, see also our earlier post Neutrinos for Beginners.

This neutrino mixing happens when the mass-eigenstates of the Hamiltonian are not the same as the flavor eigenstates. The wavelength λ of the oscillation turns out to depend (in the relativistic limit) on the difference in the squared masses Δm² (not the square of the difference!) and the neutrino's energy E as λ = 4E/Δm². The larger the energy of the neutrinos the larger the wavelength. For a source with a spectrum of different energies around some mean value, one has a superposition of various wavelengths. On distances larger than the typical oscillation length corresponding to the mean energy, this will average out the oscillation.

The plot below from the KamLAND Collaboration shows an example of an experiment to test neutrino flavor conversion. The KamLAND neutrino sources are several Japanese nuclear reactors that emit electron anti-neutrinos with a very well known energy and power spectrum, that has a mean value around some MeV. The average distance to the reactors is ~180 km. The plot shows the ratio of the observed electron anti-neutrinos to the expected number without oscillations. The KamLAND result is the red dot. The other data points were earlier experiments in other locations that did not find a drop. The dotted line is the best fit to this data.



[Figure: KamLAND Collaboration]

One sees however that there is some kind of redundancy in this fit, since one can shift around the wavelength and stay within the errorbars. These reactor data however are only one of the measurements of neutrino oscillations that have been made during the last decades. There are a lot of other experiments that have measured deficites in the expected solar and atmospheric neutrino flux. Especially important in this regard was the SNO data that confirmed that indeed not only there were less solar electron neutrinos than expected, but that they actually showed up in the detector with a different flavor, and the KamLAND analysis of the energy spectrum that clearly favors oscillation over decay.

The plot below depicts all the currently available data for electron neutrino oscillations, which places the mass-square around 8×10^-5 eV², and θ at about 33.9° (i.e. the mixing is with high confidence not maximal).




[Figure: Hitoshi Murayama, see here for references on the used data]

The lines on the top indicate excluded regions from earlier experiments, the filled regions are allowed values. You see the KamLAND 95%CL area in red, and SNO in brown. The remaining island in the overlap is pretty much constrained by now. Given that neutrinos are so elusive particles, and this mass scale is incredibly tiny, I am always impressed by the precision of these experiments!

To fit the oscillations between all the known three neutrino flavors, one needs three mixing angles, and two mass differences (the overall mass scale factors out and does not enter, neutrino oscillations thus are not sensitive to the total neutrino masses). All the presently available data has allowed us to tightly constrain the mixing angles and mass squares. The only outsider (that was thus excluded from the global fits) is famously LSND (see also the above plot), so MiniBooNE was designed to check on their results. For more info on MiniBooNE, see Heather Ray's excellent post at CV.

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This post is part of our 2007 advent calendar A Plottl A Day.