As you probably surmised, BEER isn’t the beverage: it’s an observational technique standing for BEaming, Ellipsoidal, and Reflection/emission modulation. (That wins the award for the most awkward acronym I’ve seen in some time. As Mary Roach would say, it’s “an example of PLEASE—Pretty Lame Excuse for an Acronym, Scientists and Experimenters.”) The method looks for a fluctuation in the host star’s light due to motion caused by the gravitational pull of its planet, along with the tiny additional effects due to distortions in the star’s shape and the reflection off the exoplanet.

The BEER technique scored its first major victory: astronomers used it to determine a signal from Kepler observatory data was due to a planet and not some other source. (Kepler is probably dead, alas, but we’ll see results from its observations for many months or even years to come.) Previously BEER was used on previously detected exoplanets, but the sample is small — mainly because the signals BEER is designed to detect are themselves very small. The promise of BEER isn’t necessarily as a detection method, though. Instead, it opens up a bottle-full of possible physical properties of planets and stars unavailable to other techniques: direct mass measurement, rotation rates, and the distortion of the shape of the star.

The exoplanet — labeled Kepler-76b, since it was found in Kepler data — was first hinted as a fluctuation in the light of its host star. Using BEER, astronomers in Israel, the United States, and Denmark determined not only that this was a planet, but found evidence of its rotation and an estimate of its mass.

We can drive it home with one headlight

Simulation of relativistic beaming, for a star moving to the right (relative to us) at 55% of light-speed. [Credit: moi]

Beaming is in addition to the Doppler effect, where the star’s light will be pushed to shorter wavelengths — blueshifted — if it’s moving toward you, and stretched longer — redshifted — if it’s moving away. [Credit: moi]

An exoplanet or other companion will cause a star to move slightly as they mutually orbit. (All planets have this effect, though only Jupiter in our Solar System is massive enough to produce a measurable result on the Sun.) Thanks to beaming, whichever angle we look at a star system, we should see a small variation in the star’s brightness. It will appear slightly brighter when the star is moving toward us, and slightly fainter when moving away, with the pattern repeating itself every time the companion object completes an orbit. The effect will obviously be strongest if we observe the system in the plane of the companion’s orbit (since we’d be directly in line of the “beam”). However, as you can see from the movie, beaming has an effect even if we’re seeing the orbit from a steep angle, where the Kepler method of observing a transit isn’t possible.

In Heaven, there is no BEER; that is why we use it here

Periodic fluctuations of light alone won’t tell us if the star has a companion, so beaming by itself isn’t enough. This is the reason for the Ellipsoidal and Reflection/emission portion of BEER.

If the companion is close enough to produce orbital wobbling in the host star, then it’s also close enough to squeeze the star via tidal forces. The change in shape from spherical to very slightly ellipsoidal means the star’s light will no longer be emitted in the same way in all directions. Measuring the variations in light output is a means of measuring the shape of the star; the amount of ellipsoidal deformation provides a way of determining the mass of the companion object.

Finally, the star’s light will heat up its companion, producing a hot spot on the surface closest to the star. That hot spot itself emits light (in the infrared), which modifies the spectrum and variation in the light we see from the system. If the companion is gaseous — the most likely possibility — then the hot spot will probably migrate as atmospheric effects carry gases around the object via jet streams. In other words, the reflection/emission part of BEER can be used to measure the rotation of the companion.

Like nearly all other exoplanet observation techniques, BEER works best for high-mass exoplanets orbiting very close to their host stars — a class known as the “hot Jupiters”. Kepler-76b is between 1.74 and 2.26 times the mass of Jupiter and orbits once every 1.54 days, so it certainly is a hot Jupiter. It’s also close enough to be tidally locked: presenting one face to its star the same way the Moon always presents the same face to Earth. However, a planet that large is certainly gaseous (as Jupiter is), which means processes in the atmosphere keep things churning and turbulent. As the planet rotates to keep facing the star, the jet stream — flow of gas in the atmosphere — will push the hottest spot away from its position nearest to the star. This is evidence that the atmosphere rotates faster than the planet itself, something known as superrotation (no doubt one of Superman’s weird powers from the embarrassing era of superhero comics).

BEER is a very sophisticated algorithm, relying on measuring small variations in the light output form a star system and cross-correlating them to make sure they aren’t due to something else. However, independent observations of Kepler-76b indicate the method is reliable — and that it can give us information unavailable any other way. Hot Jupiters are relatively common objects in our galaxy, we’d like to understand their structure and interactions with their host stars, not to mention how they end up so close. BEER is a promising means to that end.

Now for some reason, I’m thirsty.

• S. Faigler et al., BEER analysis of Kepler and CoRoT light curves: I. Discovery of Kepler-76b: A hot Jupiter with evidence for superrotation. ArXiV 1304.6841, accepted to Astrophysical Journal.

If you think theories about the universe are mind-bending, rest assured that many scientists feel the same way. But the question isn’t a philosophical one: it has potentially real, testable aspects. [Credit: ESA/Planck Collaboration/D. Ducros]

The Big Bang model is our best explanation for why the cosmos appears as it does. Nevertheless, it’s not able to answer some of the more challenging questions, including what – if anything – came before it? Despite how it might sound, this question isn’t a philosophical one: it has potentially real, testable aspects. But to understand any of the possible answers, we must first understand the question itself.

First of all, the language we use to describe what we know and don’t know can sometimes be muddy. For instance, the Universe may be defined as all that exists in a physical sense, but we can only observe part of that. Nobody sensible thinks the observable Universe is all there is, though. Galaxies in every direction seem similar to each other; there’s no evident special direction in space, meaning that the Universe doesn’t have an edge (or a centre). In other words, if we were to instantaneously relocate to a galaxy far, far away, we’d see a cosmos very similar to the one we observe from Earth, and it would have an effective radius of 46 billion light-years. We can’t see beyond that radius, wherever we’re located.

For many reasons, cosmologists think the early Universe underwent inflation: an incredibly rapid expansion right after the Big Bang. As the Universe expanded, it also cooled, so in the distant past, it was hotter, more dense, and opaque to all forms of light. When the cosmos became transparent, about 380,000 years after the Big Bang, it left behind a bath of photons, detectable today as the Cosmic Microwave Background (CMB). As space observatories like COBE, WMAP and more recently Planck have shown, the CMB is remarkably smooth, but not quite uniform. Within this were tiny ripples that were stretched to enormous sizes during inflation, and in turn these became the seeds for large-scale objects like galaxies and galactic clusters we see today.

While inflation comes in many possible versions, the gist is that random fluctuations in temperature and density produced by the Big Bang were smoothed out by the rapid expansion, much like a wrinkly uninflated balloon grows into a smooth object when filled with air. Inflation happened so quickly that, in many versions, the Universe has disconnected regions – parallel universes – that might even have different sets of physical laws. Inflation would have produced a lot of gravitational waves – fluctuations in the structure of space and time – which in turn should leave their mark on the CMB. Gravitational waves in the very early Universe churned up space-time, creating an environment that twisted light emission.

Pocket universes

However, none of this really tells us what, if anything, came before the Big Bang. In many models for inflation, as in some of the older Big Bang theories, this is the only Universe that exists – or at the very least, the only Universe we can observe.

A partial exception to this is a model known as eternal inflation. In this scheme, the observable Universe is part of a “pocket universe”, a bubble in a larger froth of inflation that is ongoing. In our particular bubble, inflation began and ended, but in other pocket universes – unconnected (“parallel”) and thereby unobservable to our pocket universe – inflation might have had different properties. Eternal inflation effectively emptied the regions outside of bubbles of all matter; these would have no stars, galaxies, or other familiar hallmarks.

If eternal inflation is correct, then the Big Bang is the origin of our pocket universe, but not the beginning of the whole Universe, which may have begun much earlier. The evidence for multiverses will be indirect at best, even with confirmation of inflation from Planck or other observations. In other words, eternal inflation could answer the question of what preceded the Big Bang, but still leave the question of ultimate origin out of reach.

Trillion-year cycle

Many cosmologists regard inflation as being the worst model we have, except for all the alternatives. Inflation’s generic properties are pretty nice, thanks to its usefulness in solving difficult problems in cosmology, but the specifics are slippery. What caused inflation? How did it begin, and when did it end? If eternal inflation is correct, how many pocket universes could there be with similar properties to our own? Was there a Bigger Bang that started the multiverse going? Finally, since we’re scientists and not philosophers, how can we tell all of these options apart: can we test them?

There is one possible alternative to inflation, which bypasses these questions and, along the way, resolves what came before the Big Bang. In Paul Steinhardt and Neil Turok’s cyclic universe model, the observable Universe resides in a higher-dimensional void. Coupled to our universe is a parallel shadow universe that we can’t directly observe, but is connected via gravity. The Big Bang was not the beginning, but a moment when the two “branes” (short for “membrane”) collided. The Universe in the cyclic model goes between periods when the branes are moving apart, accelerated expansion, and new Big Bangs when the branes re-collide. While each cycle would take about a trillion years to complete, the whole cosmos could be infinitely old, bypassing the philosophical problems with inflationary models.

The cyclic universe is not a popular model among working cosmologists, but at least it could be ruled out by experimental observations: if the gravitational-wave signature of inflation is found, then the cyclic model is dead. The cyclic model isn’t complete: it doesn’t explain how much dark energy there is in the Universe any more than standard cosmology does, for example. In other words, the cyclic model is not complete, so at present there’s no physical evidence to distinguish it from inflationary models.

If you think all these options are fairly mind-bending, rest assured that professional scientists feel the same way. Since the observable Universe is currently accelerating with no sign of re-collapse even in the far future, why should there be a cosmos with a beginning but no similar ending? If inflation or the Big Bang erases information of what (if anything) came before, are we stuck debating over the number of angels dancing Gangnam Style on the head of a pin? Even if eternal inflation or the cyclic model is correct, it pushes the question of ultimate origin into the realm of untestability.

In another decade or century, the questions and the methods we use to answer these questions will most likely have evolved. But for now, it’s unclear how we can possibly know what preceded the Big Bang.

Postscript

I usually avoid the kinds of sexy big questions that often make cosmology books by Paul Davies or Stephen Hawking or Roger Penrose popular. The main reason for that is because those big questions may not be answerable, because they are beyond the reach of our telescopes or experiments. One such question—what, if anything, came before the Big Bang?—is cause for a great deal of speculation, and a good amount of nonsense. If memory serves, Pope John Paul II was the first pontiff to explicitly accept Big Bang cosmology, but he also forbade Catholic cosmologists from even pondering the question of whether anything came before.

However, BBC Future provided me a great opportunity to examine the meta-question: “Will we ever know what happened before the Big Bang?” That’s a question better suited to me: it’s not speculation, but pondering how can we know? Thanks again to Simon Frantz, my editor at BBC Future, who asked me to write the piece and helped turn it into something coherent, instead of Grumpy Matthew grumbling into his coffee.

xkcd as usual gets it right.

If you judged from my email inbox and Twitter feed, you might think that Albert Einstein was a stupid person, wrong about everything. You might also think that Einstein was the only physicist of the 20th century, the one against whom all our achievements are measured, one way or another. Press releases announce “Was Einstein wrong?”, “Einstein survives another test”, and the like; crackpots tell me that “Einstein was never right to begin with”.

That’s why I feel I could use the accompanying xkcd comic on many of my blog posts. Einstein was arguably the single biggest contributor to modern physics, having established or helped develop several different major theories. However, the theories stand or fall on other things than Einstein’s connection to them. If Einstein had never been born, we’d still have quantum mechanics, both theories of relativity, and his approach to the statistical treatment of photons and materials. Sure, the general theory of relativity might have taken a few more years, but it’s hubristic to think it never would have happened without Einstein.

To punish me for my contempt for authority, fate made me an authority myself.
–Albert Einstein

A famous Zen koān states, “If you meet the Buddha on the road, kill him.” In Zen, that idea means that even the great teachers can be blocks to enlightenment. In science, holding on to cherished heroes—or even making it all about people and personality, rather than evidence and its interpretation within the framework of theory.

The recent specific tests of gravity in the strong regime are in a regime that, as far as I can tell, Einstein was personally uninterested. (I deliberately wrote this article without mentioning Einstein, even though every press release and most of the other stories did.) He didn’t really accept black holes as a possibility, and was mostly unconcerned with the practical application of his theories. At best, Einstein should be considered the first word on relativity, not the last.

Even more: every theory should be considered provisional. Almost nobody thinks general relativity is the final theory of gravity we’ll ever have. It’s almost certainly going to break down somewhere, either where gravity is very strong, or at microscopic scales, or (perhaps) at very large scales, if dark energy is an aspect of gravity not previously suspected. (I kind of like that idea myself, but it’s not a given!) That’s not a problem with science—it’s simply the way things are, in a Universe where we must map our thinking onto a reality we didn’t construct.

It’s a common mistake to think that theories are a binary, right or wrong. The reason we can say the geocentric model is wrong is because, though it describes the motion of some things in the Solar System, it breaks down too quickly in the face of accumulated evidence. However, while Newtonian physics gives incorrect predictions for small things, high speeds, and strong gravity, it still works remarkably well on the scale of the everyday. More than that, it still has explanatory power: forces, momentum, and even Newton’s law of gravity give us a very good—albeit provisional—way of understanding a lot of things on the human scale. Sure, relativistic physics and quantum mechanics are more “accurate”, but it would be foolish and unnecessary to use them to describe the trajectory of a baseball. I could (and may) write another blog post about how quantum mechanics shows us why Newtonian physics works. We won’t abandon Newton’s laws, even as we understand their applicability isn’t universal.

Einstein was wrong about many things, and right about others—from the perspective of today. In the next few centuries, we’ll no doubt refine, extend, and otherwise restrict the validity of Einstein’s work, showing him to be wrong about more things. Scientists of the far future may mock us for holding what seems to them as ridiculous views of things, but they’d be mistaken to do so. All we can do is our best, like Newton, like Einstein.

“Cul de Sac” comic by Richard Thompson

The elephant secret burial ground is an extreme example of a type of wishful thinking that can lead us all into mental traps. We humans want to believe, and seem to be attracted to certain notions, because they feel right in some way. There’s also a tendency toward wanting to give ideas a fair trial, even if those ideas are a bit on the fringe: if they sound plausible to us, we’ll listen or even promote them. Thus, otherwise reasonable people will promote concepts like the non-biological origin of petroleum (which would mean oil is actually not a finite resource after all), cold fusion, or the “aquatic ape” hypothesis.

Elephant graveyards aren’t a damaging myth (at least as far as I can tell), but those others have real-world consequences. Energy policy in the light of global climate change requires understanding where our fuel comes from, so we can make long-term plans. (See Maggie Koerth-Baker’s excellent book Before the Lights Go Out for an in-depth discussion of this issue.) Despite the lack of repeatable experimental results or evidence that nuclear reactions are actually taking place, low-energy nuclear reactions (LENR, also known as “cold fusion”) is still supported by some institutions, including a few people at NASA. I’m sure many of us wish LENR was a legitimate avenue for research, since it would solve so many problems, but cold fusion has yet to fulfill its promise, despite years of effort.

Did our ancestors live in the water?

The “aquatic ape” hypothesis (AAH) is another type of idea, one more common (I think) in physics than in biology. In this model, human ancestors went through an aquatic stage of evolutionary development. On the face of it, AAH explains things such as our relative hairlessness, large breasts, upright stature, and big brains, compared to our closest primate relatives. However, the paleontological evidence used to support AAH works much better as evidence of a dry-land life for our ancestors. In fact, it’s likely many people haven’t even heard of the aquatic ape idea; I think I first ran across it as a footnote in Larry Gonick’s Cartoon History of the Universe, rather than a science magazine or newspaper.

I asked Twitter this morning for a term describing a scientific theory that was once viable, but has since fallen out of use thanks to too much evidence against it. The term that I thought best came from Reed Roberts: “obsolete theory”.

@drmrfrancis “Obsolete” might be a nice way to say it. Implied usefulness at the time.

— Reed Roberts (@ReedRoberts) April 29, 2013

However, Antonia Hamilton proposed another term that describes AAH well: a “zombie theory”. It should be dead by all rights, but still lurches on. If I may extend the metaphor, it feeds on the brains of those who are unwary.

@drmrfrancis know several zombie theories – once viable, ruled out but refuse to die.

— Antonia Hamilton (@antoniahamilton) April 29, 2013

Lest you think this is a strawman argument, set up only to be knocked down, the Guardian ran an article in their Sunday edition (the Observer) giving aid and comfort to AAH, as well as promoting a conference devoted to the topic. Among the speakers: David Attenborough, beloved host of many science programs on television. If you read the article, you’d come away with the impression that AAH is a lot more widespread than it is.

I’m sympathetic—AAH is an interesting idea, with a grand scope to explain a lot of diverse features in human anatomy—but it simply doesn’t hold up under the evidence. (The biggest problem is probably time: whales evolved from terrestrial mammals, lost their body hair, and evolved big brains, but took a lot longer than human ancestry allows for AAH.) If you want some fun with the hypothesis, try this series on the “idea” that humans actually evolved in space, showing how many of the same features AAH explains could have evolved in that rather unlikely environment.

Attenborough aside, I have yet to run into a science writer or communicator who supports AAH, and I certainly have never met an evolutionary biologist, paleontologist, or anthropologist who accepts the hypothesis. Admittedly, that’s anecdotal evidence, but I trust my colleagues and their judgment. I’m no biologist—my last formal biology training was from Mr. Cool (his actual name) in my sophomore year of high school back in 1933. However, that’s why I rely on the biologists and paleontologists and anthropologists: they are in a position to evaluate the evidence, just as I am in a position to help them evaluate the evidence for dark matter or the Big Bang. They act as my gatekeepers, allowing me to say with confidence that the AAH is almost certainly wrong: among the strongest statements one can make in science.

Who is custodian of the custard custodians?

(Joke courtesy of Terry Pratchett.) My ability to trust other scientists and writers in fields that are not mine is a big time-saver: I don’t have to recreate every experiment, read every paper, and become an expert in every branch of science. However, U.S. Representative Lamar Smith (R-TX) has decided that the Congress should put itself in the stead of experts when it comes to awarding grants from the National Science Foundation (NSF). Mr. Smith has looked over several NSF grant awards, and concluded they are not worth receiving government money.

Frankly, I would hesitate to evaluate the value of a lot of grant applications in physics, and I have a PhD in physics. However, Mr. Smith thinks that the Congress is in a better position to know whether a grant is scientifically valid than other scientists: the process of peer review currently used. Derek Lowe has a lot more on why this idea is bad, but suffice to say that Mr. Smith has a poor understanding of how science works. We don’t always know in advance what research will be valuable, what ideas will be right, or what experiments will prove applicable in areas that are not directly connected to their original field. Particle colliders are used in medical and materials research, studies of the statistical behavior of particles end up in helping ease traffic on major roads, and the history of science is full of cases where pure research ended up with unexpected technological applications.

U.S. Representative Eddie Bernice Johnson (D-TX) gets it, as her letter to her fellow Texan Rep. Smith states in detail (PDF format). In other words, this isn’t a problem with Congress (or Texas) per se. And I’m certainly not saying no oversight should exist: the grant process is imperfect, with peer review subject to personality conflicts or—yes—politics and trends. The solution, however, is not to put the decisions in the hands of those who are even more subject to personality conflicts, political winds, and (very likely) a poor understanding of how science works. The real solution must include more openness from scientists, better public science education, and frankly a wider cultural recognition that science isn’t just a point of view among many that can be gainsaid by asserting an opposing opinion. Update: U.S. President Barack Obama gave a short speech today strongly in favor of peer review, untainted by politics.

In this era of austerity measures, huge budget cuts to research and education, and continuing attempts to remove evolution and climate change from schools, it’s all the more important to keep the importance of basic research in the public eye. Otherwise, we may see ideas like cold fusion, aquatic apes, and other scientific versions of the elephant’s graveyard as policy because they are expedient to certain people in power.

The Horsehead Nebula, in infrared light. [Credit: NASA, ESA, and the Hubble Heritage Team (STScI/AURA)]

Little round planet in a big universe
Sometimes it looks blessed,
Sometimes it looks cursed.
Depends on what you look at, obviously
But even more it depends on the way that you see.
– Bruce Cockburn *

The past week was not a healthy one: it was marked by violence, natural disasters, and the more subtle hatreds surrounding race, gender, and ethnicity that masquerade as “civilized” discourse. Sadly, the main reason many in the United States paid attention is because many of those events happened here, instead of at a safe remove in another country. I can only hope that the violence in Boston would lead to increased sympathy on the part of Americans toward those in other nations for whom such things are a common occurrence.

However, this week also marked the 23rd anniversary of the Hubble Space Telescope, an occasion marked by a new observation of a familiar object. Many of us no doubt have seen photos of the Horsehead Nebula, a large cloud of gas and dust in the constellation Orion; some of us may even have seen it through a telescope, as it’s a relatively easy thing to spot. In visible light, the cloud takes the form of a dark shape similar to—yes—a horse’s head, in front of a somewhat brighter background of red light from hydrogen atoms.

The dark shape is made of dust: molecules and aggregates of molecules, containing carbon and other atoms. Such dust clouds are frequently opaque to visible light, but in the infrared they can glow, thanks to the presence of young stars hiding within. That’s true for the Horsehead Nebula: the “horse” hides a creche of newborn stars, which the Hubble’s infrared Wide Field Camera revealed in the image above.

The Horsehead Nebula in both infrared (left) and visible light. [Credit: NASA, ESA, and the Hubble Heritage Team (AURA/STScI); ESO]

The infrared also shows details in the dust that aren’t obvious in visible light, even with high-resolution images. What is a shadow in one form of light is a beautiful glowing cloud in another. The darkness hides new birth, but we can see the renewal if we know how to see it.

* I know, I’ve used this quote before. So sue me. I’ll probably use it again.

Filed under: Astronomy ]]> http://galileospendulum.org/2013/04/20/shadows-and-light/feed/ 2 matthewfrancisrmc Horsehead nebula in infrared The Horsehead Nebula in both infrared (left) and visible light. [Credit: NASA, ESA, and the Hubble Heritage Team (AURA/STScI); ESO] http://galileospendulum.org/2013/04/16/of-dark-matter-and-hope/ http://galileospendulum.org/2013/04/16/of-dark-matter-and-hope/#comments Tue, 16 Apr 2013 21:06:32 +0000 Matthew R. Francis http://galileospendulum.org/?p=4096 ]]> Perhaps hope lies deep underground. Specifically, hope of detecting dark matter: the stuff making up 80 percent of the total mass of the Universe.

The headframe of the Soudan Iron Mine, which still operates the lift cage descending a half-mile below ground, where the CDMS experiment is located. I visited Soudan last May. [Credit: moi]

The new announcement is a different beast. Rather than looking for elevated positron levels from dark matter annihilation as AMS-02 does, CDMS is a set of solid geranium germanium [update: and silicon] crystals, cooled by liquid helium to 40 millikelvins—0.04° C above absolute zero. If a dark matter particle known as a WIMP (weakly interacting massive particle) hits a nucleus inside the crystal, it will set up a small vibration: a quantum sound wave known as a phonon.

Thanks to the cryogenic temperatures, CDMS doesn’t vibrate that much on its own, but other particles could create false positive signals. And in fact, there’s a good chance of such deceptive detections, just by random fluctuations: three fake-out signals from jerky Mama Nature have a little more than 5 percent chance of happening. That might not sound like a lot, but CDMS has a lot of data. (For example, an earlier possible detection of 2 particles at CDMS was probably stray electrons leaking from the crystals, rather than phonons.)

However, these three particle candidates seemed to correspond to the WIMP regime, and that lowers the probability of this being the jerky cosmos being jerky again. The technical term for the level of detection is “three sigma” (3σ), which means the detection is 99.7 percent likely to be real. While that might seem like it’s good enough to count as a positive detection, we must remember that there are a lot of background signals to confuse things. To put it another way: if we had 1000 background particles, we might get 3 fake signals by random chance. CDMS has a lot more than 1000 events to sift through.

You’ll have to take my word for it that CDMS sits inside the room behind the glass in this photo. [Credit: moi]

If these results are indeed real, then they correspond to a WIMP of mass around 8.6 billion electron-volts (8.6 GeV). For comparison, a proton’s mass is about 0.9 GeV, and the Higgs boson detected at the LHC is about 125 GeV. Even though it’s high mass compared to ordinary matter particles, it’s at the lower end of the WIMP mass regime. The pending upgrade of CDMS, known creatively as SuperCDMS, will use Interestingly, the WIMP signal was found in silicon crystals, which are more sensitive to lower mass WIMPs. As always, time and more data will tell. Optimism doesn’t always pay in particle physics, but these days, I’ll take what I can get.

A Green Pea galaxy. [Credit: SDSS]

Green Pea galaxies were discovered by citizen scientists working with the GalaxyZoo project. GalaxyZoo is an ongoing collaborative effort to sift through Sloan Digital Sky Survey (SDSS) and Hubble Space Telescope (HST) data to classify galaxies according to their shape and other properties. As Alice Shepherd described on the GalaxyZoo blog, the people who first spotted the Green Peas were puzzled: they didn’t really fit into any neat categories. The strong oxygen emission required to make them appear so green meant they were pumping out a lot of very energetic light, which points to aggressive star formation. However, they were definitely galaxies, not something in the Milky Way.

Why green is weird

Emission spectra of light for objects with various surface temperatures. Our Sun is most like the the hottest one, labeled with the green curve. [Credit: moi]

The reason no star appears green, then, isn’t because they don’t emit green light, but because register light across the entire visible light spectrum. If we could imagine a hypothetical critter whose vision was primarily based in the ultraviolet, incapable of seeing reds, yellows, and oranges, said critter could  perceive our Sun—and many other stars with cooler surfaces—as green. (How it would perceive ultraviolet light is, of course, something we will never quite grasp.)

However, colors don’t just come from surface temperature. For gases or solids, they also can be the result of emission or absorption by atoms or ions: atoms with one or more missing electrons. (“Ion” can also denote atoms with extra electrons, but those aren’t common in astronomy.) Each type of atom or ion has its own unique spectrum, as described by quantum mechanics, so astronomers can identify the chemical composition of gas clouds in distant galaxies—even if the cloud itself can’t be seen through a telescope.

Most atoms in the Universe are hydrogen, and most of the remainder are helium. The most common (visible light) color we see emitted by hydrogen is red, for example. Of the heavier elements, oxygen and carbon are common enough to show up on occasion, especially in the form of molecules. Many of these molecules we see through their absorption rather than emission, at least in visible wavelengths.

A galaxy the Hulk would love

The Green Pea galaxies are noteworthy in that they have strong emission in the green part of the spectrum, corresponding to oxygen with two missing electrons. That kind of emission is only possible if there’s a lot of energetic ultraviolet light stripping electrons away, and if the gas is relatively diffuse. (The reasons for this are a little hard to explain without doubling the length of this post, but suffice to say it has to do with the rules of quantum physics again: what sorts of transitions are allowed within the oxygen ions.)

Green Peas must therefore be pretty extreme environments, bursting with star formation. As a matter of fact, they have roughly three times the star formation rate the Milky Way has, in a galaxy with less than 1 percent the mass! Newborn stars produce a lot of ultraviolet radiation. If some of those stars are really massive, they live short lives before exploding as supernovas—another source of ionizing radiation. The green light is a sign of Hulk-like powers of smashing atoms.

Here’s where things get fun: if the Green Pea galaxies allow enough of that ionizing radiation out into intergalactic space, and if these galaxies are of the same type as some we see in the early Universe, then they might be the solution to a cosmological conundrum. After the Universe became transparent, forming the first stable atoms, something reionized the gas, undoing the work and stripping electrons back off nuclei. Most cosmologists are pretty sure aggressive stars in young galaxies had something to do with reionization, but demonstrating that is another challenge, since those galaxies are very distant and—if the Green Peas are any guide—pretty small.

I wrote about the possible connection between Green Peas and reionization for Ars Technica this week. The story isn’t done yet, of course: we need more data on both the Green Peas and the early galaxies they might resemble. However, I love the idea that citizen scientists discovered a galaxy type we didn’t know about before, and would love even more if that type could solve one of the mysteries of modern cosmology.

A panel from “Saturday Morning Breakfast Cereal”. The comic behind the link is very NSFW and very nerdy, so don’t click if you’re offended by either. [Credit: Zach Weiner]

People often ask the same question when they learn about imaginary numbers: if i is the square root of -1, what’s the square root of i? Is it a new kind of imaginary number, or can you write it in terms of “regular” imaginary numbers, or is it even a meaningful question? As you might expect, the question is not only meaningful, but leads to some interesting geometrical insights (even if most of us won’t find it as exciting as Zach Weiner’s character in his comic strip).

Complex numbers in boyshorts brief

Illustration of the complex plane: the connection between complex numbers and points in two dimensions. Four points are plotted so you can see the correspondence between x and y coordinates and the real and imaginary parts of the complex numbers.

Complex numbers are the combination of real and imaginary numbers. In general, we write complex numbers as the sum of the real part and the imaginary part, and often find it useful to plot them on the complex plane, as shown at the left. All real (“ordinary”) numbers lie on the horizontal axis, and all imaginary numbers (which are just multiples of i) lie along the vertical axis.

An equivalent but often more useful way of writing complex numbers uses the exponential function:

where r and φ are both real numbers. Additionally, r is a positive number that represents the magnitude of the complex number: the distance from the origin of the complex plane to the point represented by z. The relative weight to the real and imaginary portions are represented by φ (the Greek letter “phi”, pronounced either “fye” or “fee”, depending on how pedantic you want to be), which is known as the phase. Finally, e is the exponential number, roughly equal to 2.71828 (but going on to infinite number of digits). The exponential number, like π, is a fundamental geometric quantity.

The exponential part of the equation is what we most care about, since r is just a scaling factor. We can write it as a sum of trigonometry functions:

Think of it like this: the complex number is like one point on a right triangle, whose hippopotamus hypotenuse connects the number to the origin. The legs of the triangle are the real and imaginary parts of the complex number.

Relating a circle to complex numbers. The radius line (in blue) has a length of 1, and we’ll use that as the hypotenuse of a triangle. Then the x- and y-coordinates of the end of the line are given by the cosine and sine of the angle as shown.

Now we can see that the phase φ represents an angle, but we need to use radians instead of degrees. One full circle is 2π radians (360°), a half-circle is π radians (180°), and a quarter circle is π/2 radians (90°). With that, we can see that positive real numbers are just complex numbers where the phase φ = 0, and negative real numbers correspond to complex numbers with φ = π. We also get Euler’s formula, which I wrote about in a previous blog post:This is one of those really interesting formulas, since it takes two irrational numbers (e and π) with the imaginary unit i, and the result is…a negative integer.

The phase is like the hour hand on a clock: there’s a bit of redundancy built in. If you add or subtract 2π from any phase, you get the same complex number!In fact, you can do the same with 4π, 6π, or any other even number multiplying π —they all give you the same complex number. We’ll use that redundancy shortly.

Totally radical, man

Now let’s get back to the original definition of i:The notation in the last expression might not be familiar, but it’s pretty straightforward. The square root reverses the action of squaring something:(Savvy people have noticed that I’m neglecting the negative square root solution, but hold on: we’re getting there.) In fact, that’s one particular version of a general rule: if you raise a number to an exponent, then follow that with another exponent, that’s the same as raising the first number to the multiple of both exponents. For example,
If the exponent is a whole number, then its meaning is fairly obvious: x² means you multiply the number x by itself, x³ means you multiply the number x by x², and so forth. If the exponent is a fraction, the meaning is a little less clear, but still perfectly manageable.

So let’s combine the exponent rules with Euler’s formula:That’s a great consistency check, but it doesn’t tell us anything we didn’t already know. Or does it? Let’s go back to the complex plane and see what the square root did in a geometrical sense:

Taking the square root of -1 is the equivalent of rotating the blue dot on the edge of the circle by 90 degrees, or (more appropriately) π/2 radians.

That’s very interesting: the square root acts like a rotation, moving the dot from its location at -1 to a new place at i along the circumference of the circle. (Update: see note at the end of the post.)

Now we’re finally ready to answer the question from the beginning of the post: what is the square root of i?In other words, the square root of i doesn’t need a new mathematical concept: it’s just another complex number. Using the complex plane again, we can see that the square root again just rotates things around.

The square root of i is just another clockwise rotation, halving the phase angle.

So, at its root, the square root function is a rotation. (Another name for “root” is “radical”, from “radix”, from which we also get the word “radish”. So the next time you think of political radicals, think of giant walking radishes reciting square roots. Or something.)

One number enters, two numbers leave

Now let’s turn to an easy and obvious question: what’s the square root of 1? One answer is quick:However, we determined earlier that we can add 2π to any phase angle and get back the same complex number, so let’s repeat the process with that knowledge in hand:Obviously 1 and -1 aren’t the same number, so there are two distinct answers for the square root of 1, as we expect. Complex numbers show us another way to see that, and as before, we can think of it as a rotation.

We can do the same trick for any complex number, as you might guess:andThese second square roots are all a rotation by π away from the other solutions we found previously! The square root of every complex number will have two different solutions, similarly separated by π radians. Since positive real numbers and negative real numbers are also separated by π radians in the complex plane, they also obey the same rule.

Extending everything we’ve learned to an arbitrary complex number is really easy: Since r is always a positive real number, we can calculate its square root using a calculator (or using the binomial series, which is how I learned to do it in geometry class back in 1933). The act of taking the square root ends up rescaling and rotating any complex number. If I remember to get around to it in my copious spare time, there’s some deep stuff going on there, relating to something known as conformal geometry. We can also  keep going with cube roots (found by using 1/3 instead of 1/2, and getting three solutions instead of 2), or any exponent we want.

Now let’s look at the mathematical expression that got Zach Weiner’s protagonist all hot and bothered: raising i to the power of i. As his ladyfriend pointed out, it’s a real number:to three decimal places (it’s an irrational number). That won’t be true for just any complex number, since the real number r raised to the power of i will be complex. However, that’s a story for another day. You wouldn’t want me to use up all the mathy talk in one post, would you?

Postscript update

As a commenter noted and as I said above, the operation of taking a power is not a rotation in the usual sense. For hints on how to define rotations in a more rigorous way, see my earlier posts on quaternions, Clifford algebras, and complex numbers. I was attempting to use an analogy to clarify how we can understand the phase of a complex number, and as a lead-in to a possible future post on conformal geometry.

Which brings us up to yesterday, with big press build-up by NASA and CERN, and no availability of the research paper or even where the paper would be published until right before the seminar announcing the results. That’s the kind of secrecy involved in big announcements like the Higgs boson or the Planck cosmic microwave background data. However, when the AMS-02 results were revealed, they seemed a bit…anticlimactic. You wouldn’t necessarily gather that from the press releases (or subsequent press coverage, which generally takes its tone from press releases), but to many of us watching, it seemed that this announcement was not quite worthy of the major build-up and suggestive hints.

I wrote about the results and some of their implications at Ars Technica, but here’s the brief summary. AMS-02 is a multipurpose particle detector, so it’s able to measure the relative amounts of different kinds of particles. Specifically, one set of dark matter (DM) models predicts excess positron emission due to annihilation of two DM particles. (Positrons are the antimatter partner of electrons. They have identical mass and opposite electric charge to electrons, but they are far less common.) Earlier data from the PAMELA and Fermi telescopes hinted at elevated numbers of positrons, but things were pretty unclear—certainly nothing you could point to and say, “This is from dark matter!”

This plot shows the excess of positrons as a function of their energy. The red dots are from AMS-02, while the other colors represent data from other detectors. [Credit: AMS collaboration]

Don’t get me wrong: the AMS-02 results are very good looking, and provide a lot of details not present in earlier data. The experiment itself appears (to this non-particle physicist) well-designed. I applaud the goal of looking for DM annihilation signatures, and hope someone finds them. However, yesterday’s AMS-02 announcement doesn’t provide “hints” of dark matter: it provides refined measurements of the positron excess, and that’s it. Interesting, potentially exciting, possibly associated with DM annihilation—but nothing you could call a hint, and nothing to justify the hype.

Annihilate! Annihilate!

Hunting for dark matter particles is an inherently complicated thing; see my piece for BBC Future and an earlier blog post for some of the challenges. However, many physicists suspect DM belongs to a particle type predicted by supersymmetry (abbreviated as SUSY, pronounced as SOO-see) called neutralinos. SUSY is too complicated to try summarizing in one blog post, but its details aren’t really important right now anyway. Suffice to say that neutralinos are widely considered to be a good DM candidate, and they have one nice feature: they are their own antiparticle. If a neutralino meets a neutralino coming through the rye, they annihilate, producing (among other things) a positron.

The energy output from annihilation depends on the mass of the DM particles, thanks to Einstein’s E = mc². Any positron emitted from annihilation will then have a maximum energy, connected to the DM particle mass. The positron spectrum, such as the one in the plot above, will have a steep drop-off at that maximum energy. The AMS-02 data lacks such a drop-off, and the results are too good for one to be hiding. Of course, it could be at a higher energy than yesterday’s data release shows; according to the press conference I listened to yesterday, the AMS-02 researchers are working on that regime next.

However, let’s put this in perspective. If dark matter is neutralinos or some other self-annihilating particle, and the product includes positrons, then the excess positron flux might be due to DM annihilation. Without a drop-off, though, we can’t say anything much. (We also have good reason to doubt the simplest versions of SUSY, but that’s another story.) In other words, some of the excess positrons could be from DM annihilation, those particles may or may not be neutralinos…or if DM is some other type of particle, we’re barking up the wrong tree.
Why the hype?

Credit: Zach Weiner. Thanks to Gabrielle Rabinowitz for reminding me about this comic.

My colleague Ethan Siegel guessed a while back that the AMS-02 results were being oversold, and he was right. After the press events yesterday, I suspect two things are involved: the personal ego of one of the lead researchers, and the need for the various entities running the International Space Station to show that it’s scientifically useful. I’m sympathetic to the latter point: this after all is the era of “austerity measures” and budget sequesters, whose purpose is to push the blame for financial problems onto the poor and onto programs that take up relatively little of a country’s expenditures. (NASA’s entire budget is a mere drop in the bucket of the US budget, but you rarely hear that in discussions of belt-tightening.) However, overhyping results doesn’t help their case, and the outcome is regrettable. The ego of the researcher, which led him to oversell the results starting several months ago, may have forced NASA’s hand, however.

Either way (as Ethan wrote) this is not how science operates. Let’s say what these results really are: an interesting refinement of earlier observations, showing an excess of positrons we need to explain. We should make ordinary science interesting and exciting, but it benefits none of us to make such results sound more revolutionary than they really are.

Pi for equality.

No one can rival a beginning teacher for focusing on details at the expense of the big picture, much to the frequent chagrin of their students. One of my office mates in graduate school used to take points off of his students’ assignments if they wrote things like “2 π = 6.28″, since (as he said) that meant they were saying that π = 3.14 exactly. He wanted them to use the symbol for approximate equality, which is two wiggly lines: ≈.

Of course, as several of us pointed out to him, the students didn’t really mean that π = 3.14 to all levels of precision. They were using the equals sign in a common sense, as a way to assign a number to a symbol. The truth is that π is very close to 3.14 in value; if you leave out all those extra digits, your numerical result will still be correct to much greater than 1% accuracy. My office mate was holding them to a level of mathematical rigor that wasn’t really necessary for the environment in which they used it. To that standard, we can never write “π = ” and finish the expression, since the number for which the symbol π stands is irrational, requiring an infinite string of digits.

(By the way, I was guilty of similar focusing on details, though a different set, so I’m not saying I was better than he was. Graduate students in physics often earn their keep by teaching introductory physics classes, but they’re frequently given little or no training on good practices, and little oversight to make sure they’re doing their jobs properly.)

The truth is that almost nothing is equal in the mathematical sense, once real numbers, real physical systems, and real life comes into play. Even the number π, ubiquitous and important as it is, stands as a kind of mathematical extension of realistic things: no matter how perfect your circle or sphere is, it will always have little imperfections such that the ratio of circumference to diameter isn’t always going to be π. It will be close enough for many things—the shapes of stars, the orbits of planets, interactions in quantum electrodynamics—all involve π, and their deviations from perfection aren’t strong enough that we stop using the equals sign.

When we speak of equality in society, we use the realistic rather than the formal mathematical version of =. The problem is that many opponents of (for example) same-sex marriage seem to think that the formal mathematical version is what applies. Two men or two women can’t be equal to a man and a woman in marriage, in this way of thinking, because the “numbers” aren’t the same to arbitrary precision. However, no marriages—no relationships—are ever equivalent in that sense.

Among my family, friends, and acquaintances, there are a variety of relationships, whether recognized by law or not. In a certain sense, though, these relationships have a symmetry. No two of them are alike to all decimal places, but who would really expect them to be? From the point of view of the law—and, you know, human decency—these relationships should be considered equal. The people within those relationships should also, as individuals, be considered equal.

The mathematical symbol “=” evokes two parallel lines, which in Euclidean geometry will never meet, no matter how far you extend them. The Greek letter π as I drew it above is a equals sign turned on its side and connected with a curve. That’s a good symbol of equality for me, in a real-world sense.