One situation where this rarely happens is in a binary system: two stars locked in mutual orbit. The reason is that microlensing is a small effect, usually not detectable compared with other variations (starspots, star flares, and other stellar weather). But what if one of the objects is a white dwarf rather than a star? White dwarfs are the remnants of the cores of stars like the Sun; they are as massive as stars, but only the size of Earth. That intensifies their gravitational field, making it possible for them to serve as a microlens in a binary system. I wrote about one such system for *Ars Technica*:

In the binary system known as KOI-3278, microlensing from the white dwarf boosts the light of its companion by 0.1 percent. (KOI stands for “Kepler object of interest”, meaning it was discovered using the Kepler telescope and identified as a possible exoplanet system.) Each eclipse lasts about five hours, and each orbit takes about 88 days, coincidentally the same as Mercury’s orbit around the Sun. [Read more....]

This is also an exciting discovery because white dwarfs in binaries could eventually become type Ia supernovas: the kinds of explosions astronomers use to measure the expansion of the Universe.

As a final note, I wanted to point out how hard this kind of observation is. Gravitational lensing depends on the distance between the lens — in this case, the white dwarf — and the source object. When both source and lens are in the same binary system, that number is very small compared with the distance between the binary and Earth. Maybe that’s another post in its own right!

Filed under: Astronomy ]]>

I covered this story in my column for *The Daily Beast*:

Small red stars vastly outnumber their larger cousins, and the new exoplanet is orbiting one of those. That already means Kepler-186f isn’t

quiteEarth-like: its orbit is smaller than Mercury’s in the Solar System. However, because the star is less than half the diameter of the Sun, it emits a lot less light, meaning the planet only gets around one-third of the light Earth gets. However, it’s enough warmth to place it in the star’s habitable zone. [Read more...]

There are a couple of interesting details I didn’t have space for in that piece. Astronomers have found a number of exoplanets in the habitable zone; like Kepler-186f, those systems contain red dwarf stars. These stars are much smaller and fainter than the Sun, so the habitable zone is correspondingly smaller. The inner edge, for example, is close enough to the star that an exoplanet orbiting there would be tidally locked, presenting the same face to the star, much as Earth’s Moon does. That could be problematic for life or even liquid water, because the side closest to the star would have eternal day, while the opposite side would be night forever. However, Kepler-186f is farther out, meaning it should rotate at a more reasonable rate. That’s a point in its favor for habitability — at least based on what we know about Earth.

Of course, it’s far too early to say if there’s water of any form on Kepler-186f, much less liquid oceans. We don’t know if the planet is rocky or a mixture of rock and ice, or anything about its atmosphere (if it even has one). We can only speculate about life bathed in light from a red star. However, it’s still exciting, and that much closer to an Earth-like world.

Filed under: Astronomy ]]>

I’m still a work in progress —

That’s irkingly obvious

A gift and a curse, I currently work but I’m jobless

Feel like a hot mess, but I could have a lot less

Won’t beg for change — that’s already the only constant

—Silent Knight, “Work In Progress”

I’ve mentioned Silent Knight before on this blog: he’s an indie rapper who performs under his own name and with a great group The Band Called FUSE. (The name of the band is clumsy, but don’t let that stop you from checking them out; try the haunting breakup song “Last Call” for a taste.) While he writes songs about a lot of topics, the ones that resonate the most with me deal with his life as a jobbing artist, trying to make a living and get his music out into the world.

The song quoted above (which samples an obscure Paul Williams track) is one that gets to me. Silent Knight describes his insecurity, but also his modest aspirations — not to be fabulously wealthy, but just to succeed, and to do it on his own terms. Like him, I don’t need caviar or a Caddy car, but I want my work to pay enough for me to get by, and not worry all the time. We may not have “jobs”, but we work long hours. He gets the life of a freelance writer, even if that’s not what he’s actually writing about.

What do most of us really want out of life, except to do what we love and (somehow) have enough food on the table and a roof over our heads?

(I’m also reminded of a passage in David Quammen’s excellent book, *Song of the Dodo*, in which he states his preference for Alfred Russel Wallace over the better-known Charles Darwin and Joseph Hooker:

Darwin was a more ingenious theorist, admittedly. Hooker became the preeminent botanist of his time. Wallace, much later in life, involved himself in certain crankish interests (including the Land Nationalization Society, an anti-vaccination crusade, and spiritualism) that have made it easier for historians to treat him unfairly. Still, Wallace remains the most heroically appealing, at least to my crankish taste. No doubt I’m biased by the fact that he, unlike Darwin and Hooker, was an impecunious freelancer.

This may be the only time this book is cited in the same place as a rap song.)

I’m thinking of this today because (as I often do) I’m pondering this blog and its future. Blogging is the writing I do for me: it’s not paid (and in fact costs me a little money, though hardly enough to mention), and so it has to be done when I’m not busy with other deadlines. In a sense, the blog has served its purpose well: it got me enough exposure to truly work as a writer. Professionally, I don’t “need” the blog anymore. Most of my readers are finding me elsewhere now.

But I miss the types of stories I can tell here. For that reason, I’m going to change some things around a little. First, I’m retiring the blog over at Bowler Hat Science and moving that content over here. The purpose of that blog originally was to have a complete portfolio of everything I wrote, but I have an actual portfolio page now. Moving that material here will let me blog a bit more in depth about the stories I’ve written for other publications, especially if there’s anything significant that didn’t end up in the published version. Dependent on time, I also hope to record podcasts a little more regularly, possibly in collaboration with a friend or two. (I guess with having one podcast so far, *any* more recordings will be more regular.)

So change is coming. As Silent Knight says, I know I’ve got a long way to go; that’s why they call it a process though.

Filed under: Metablog, Personal Musings ]]>

It’s not the Devil’s land, you know it’s not that kind

Every devil I meet becomes a friend of mine

Every devil I meet is an angel in disguise

—”Jonas and Ezekiel“, The Indigo Girls

I visited Devils Tower on a cold March Sunday at the malingering end of winter. The roads were mostly clear, but a few icy patches left from prior snowfalls made driving in a minor adventure. I was in the region to visit a physics lab in the Black Hills of South Dakota, but I had a day before I was expected and chose to spend it in a small bit of tourism. (Before anyone asks, I’ve visited Mount Rushmore before and felt like seeing something different on this trip. That, and I’d rather see an amazing geological formation than a mountain blasted into human likeness, but that’s just me.)

Devils Tower isn’t terribly remote by modern transportation: it’s off the Interstate highway, but less than two hours’ drive from where I was staying. However, you have to try to get there: it’s not on the way anywhere else, and there’s no substantial town close by. But then as you drive closer, it appears across the valley: a tall, narrow, steep-sided protrusion of rock in the midst of low mountains. While not quite otherworldly, Devils Tower is evocative; no wonder it features heavily in the religions of the native peoples of the region.

Up close, the tower is even more interesting: it consists of a bundle of long rock columns. Most of these are hexagonal, with a few pentagonal and rectangular columns as well. This fascinating structure — and the shape of Devils Tower in general — is of course a function of its geological history. While the details are still in question, the big picture is straightforward: some 50 to 60 million years ago, hot magma forced itself into layers of softer rock deep underground. As it cooled, the former magma fragmented into the columns of the modern Devils Tower. Then, over the millennia, the land around the formation eroded away, while the igneous rock remained.So, the Devils Tower we see today was born deep underground. To me, it’s a beautiful example of something that looks inexplicable, yet we can understand through science — yet the marvel of seeing it is undiminished by comprehension. Far from losing a sense of wonder, scientific knowledge leads us to greater wonder. Every devil I meet becomes a friend of mine, indeed.

Filed under: Astronomy, Physics, and Related Fields, Personal Musings ]]>

The Music of the Spheres turns out to have some incredibly terrifying lyrics.

— Katie Mack (@AstroKatie) April 6, 2014

As deaths go, our Sun’s will be gentle: no supernova, no black hole, no cataclysm. Roughly 5 billion years from now, our host star will exhaust the hydrogen fuel in its core and expand into a red giant. As its core collapses under intense gravity, it will pass through several more stages over the course of millions of years before ultimately shedding most of its outer layers. The remnant will be a planetary nebula, surrounding what was once the Sun’s core and is now a white dwarf.

The image above is of such a planetary nebula. We don’t know what our Sun’s eventual nebula will look like — planetary nebulas come in an amazing variety of shapes. But like a religious *memento mori* (a portrait including a skull, or some such token) is a symbolic reminder of our individual deaths, the Helix Nebula could stand as a *memento mori* for all stars like our Sun.

One lesson we learn from studying the Universe: much that happens is destructive, at least in a sense. But that destruction is phenomenally slow on the human scale. A process that lasting a blink of the eye in cosmic terms — the birth or death of a star, the collisions of galaxies — can take places over millions of years, or longer. And often something of beauty is born from destruction. All stars must die, but their deaths are necessary to spread the atomic seeds for new stars and planets. Each new generation of stars is born from the remnants of previous generations, gradually changing the chemistry of the cosmos. The deaths of stars enabled us to exist, and any newborn planet will likewise bear the chemical history of the dead stars that came before it.

Five billion years is a significant amount of time even in cosmic terms: it’s more than one-third of the Universe’s current age of 13.8 billion years. Smaller stars — which are far more abundant even than Sun-like stars — may shine for trillions of years before burning out, and new stars are born all the time. The process can’t go on forever, but the death of stars is still a relative blip compared to the ongoing lives of stars.

We live on a tiny, fragile planet orbiting a nondescript star in a large but otherwise unremarkable galaxy. Our Milky Way contains many stars like our Sun, and possibly an abundance of planets similar to Earth (at least in size and composition, if not habitable). The Universe can seem vast, empty, and meaningless, when viewed from our small perspective. However, I don’t believe that makes *our* lives meaningless. Just as my inevitable death doesn’t imply I should give up on living a meaningful life in the interim, the eventual death of Earth and the Sun doesn’t mean our collective existence is meaningless.

Eventually all stars will die, and the Universe will go dark. But that event is so far in the future that it is a tragedy only from the standpoint of eternity. The beauty of the cosmos is now.

Filed under: Cosmology, I Was an English Minor ]]>

Grandfather, grandmother – children of the Depression

I am theirs, as much as they are mine;

Teachers demanding fealty to the life of learning,

A loyalty I eagerly, willingly gave.

They bought my books, remembering old struggles -

Their own poverty in payment for the life of learning

The payment of energy to bring them asymptotic (academic) freedom

Unbound from family failure – unreasonable destiny of drink or despair.

They bought my books, paying for my own freedom of mind.

I brought their gifts to them, proud and shy

Telling them what I learned – embryonic physicist

Full of relativity to share with relatives

But my frame of reference is mine, theirs was theirs;

We are symmetrical (not identical).

Now they are both ash – my own writing

Will never lie in their laps as my big (proud) textbooks did.

But I am theirs, as much as they are mine:

Symmetry – relative(ity).

[For April, I will be posting a poem each week. An earlier version of this poem originally appeared on my Tumblr.]

Filed under: I Was an English Minor, Poetry ]]>

One very exciting image went by almost too quickly. It didn’t look like much — just a red blob of pixels — but what it represented is greater than its nondescript appearance would indicate. Those pixels are an image of one of the earliest galaxies in the Universe, and we’re able to see it because its light was magnified by the gravitation of a closer cluster of galaxies. This phenomenon is known as *gravitational lensing*, which is the topic of our next CosmoAcademy class!

In the course, we’ll explore how lensing works from Einstein’s general theory of relativity (without delving into the math). Then we’ll describe how astronomers use it to reveal information about both the object doing the lensing and the imaged entity, whether these systems are stars, galaxies, galaxy clusters, or light from the early Universe. It’s an exciting and important subject, so please enroll today!

The course begins next week on Tuesday, April 8. For more information, see the class page.

Filed under: Astronomy, Cosmology ]]>

Filed under: Cosmology, Personal Musings ]]>

Rumors have percolated throughout the cosmology community, surrounding today’s press conference at the Harvard-Smithsonian Center for Astrophysics (CfA). So far, the rumors appear to be substantiated: researchers with the BICEP2 (Background Imaging of Cosmic Extragalactic Polarization) observatory at the South Pole are announcing a potentially very exciting measurement, albeit one that involves some complicated concepts.

In brief, BICEP2 is designed to measure the polarization of the Cosmic Microwave Background, light left over from when the Universe became transparent, about 380,000 years after the Big Bang. However, that light conveys information from an even earlier epoch, and could show signs of inflation: the hypothetical rapid expansion of the cosmos during its first moments. If it happened — and we aren’t sure it did, though we have some strong hints — inflation would have left gravitational waves, which polarized light in a characteristic fashion, known as B-mode polarization. That’s why today’s announcement is very intriguing: if BICEP2 data contains the right kind of polarized light, it’s a sign of those gravitational waves, which in turn could be a clue about inflation. (For a lot more information, see Sean Carroll’s blog post.)

Now I’m using all the weasel words for a reason! Measuring B-mode polarization is hard, and plagued with a number of systematic difficulties. Trust me when I say that cosmologists will be scrutinizing the BICEP2 paper when it’s released in about two hours, probing all its weaknesses, and trying to determine if its claims are as strong as hinted. Obviously the BICEP2 researchers believe it’s significant, or else they wouldn’t be holding a big press conference and building up this much excitement, but (to quote Carl Sagan), extraordinary claims require extraordinary evidence. That’s not a bad thing — it’s how science works, at least ideally.

Given that I’m already seeing headlines proclaiming that this means we’ve detected gravitational waves (no, and we know they exist already) or that Einstein is vindicated (Einstein died long before inflation was predicted), I realize I’m already “behind” on this story. However, I would much rather wait until I can read the paper and see what the scientists say before I write my article. I suspect that the story will be more complicated than it appears on the surface, but again: that’s the way of science. When my actual article appears later today, I’ll try to to explain what’s going on to the best of our knowledge in a clear way.

In the meantime, if you want to watch the press conference, it begins at 11:55 AM, US Eastern Time. You can watch at this link.

Filed under: Cosmology ]]>

Nobody can write about the history of science and mathematics without eventually bringing up Leonhard Euler (1707-1783). (Most Americans end up pronouncing his name like “oiler”.) So many important findings in math and physics that it’s hard to list them all, so I won’t try. I don’t really want to write a biography of him anyway: I just want to focus on one profound equation he discovered, and follow where it leads into some other interesting math…and of course physics.

Without further ado, here’s one of several formulas known as “Euler’s equation”:

The numbers 0, 1, and π are hopefully familiar to you; the *exponential number e* may be less well-known. It’s another geometrical number like π, and it has a value approximately equal to 2.71828… (but since it goes on forever without repeating, I’ll spare you any more digits). There are several ways to get at *e*, but we don’t need to worry about them for now. The main thing is that it’s standard, built into scientific calculators, and well-understood. The imaginary unit *i* also appears in the equation: recall that

So maybe you can see why Euler’s equation is a bit unexpected: *e* and π are real irrational numbers. (Irrational means these numbers can’t be expressed as the ratio of integers; integers are the whole numbers 0, 1, -1, 2, -2, 3, -3, etc.) Including *i* in the mix naïvely should yield a complex number, but it doesn’t: combining *e*, π, and *i* gives the negative integer -1.

In case you’re thinking I’m making a big deal out of nothing, type “exp(pi)” (without the quotation marks) into Wolfram Alpha. You should get

and so on, which is significantly larger than 1, and a positive number to boot. In fact, you won’t ever get a negative number by raising *e* to a real number. Try these in Wolfram Alpha: exp(-10), exp(-1), exp(0), exp(1), exp(10). Something weird and cool happens when you include an imaginary number in the exponent, and Euler realized after some careful computation what was going on.

First of all, the number π is special: if you plug another number in its place (say 1), you won’t get a real number out:

(rounding to two digits for conciseness). That’s a complex number: the sum of a real number with an imaginary number; refer to the figure on the right for a refresher on how to interpret complex number using coordinates. However, watch what happens when we square the real part and the imaginary part and add them together:

Rounding makes these numbers not quite equal, but if you include more and more digits while running , reaching ever higher levels of precision, you find that the sum is exactly equal to 1. You’ll also find this is true for other similar expressions like : you get a complex number, but if you square the real and imaginary parts, they add up to 1. Pi and its integer multiples are special, though: yields a real integer if is any integer: 0, -1, 1, -2, 2, etc.

Euler determined that you can split the exponential into real and imaginary parts like this:

where “cos” is the cosine function, “sin” is the sine function, and θ is any real number. If you’re like me, you first learned about sine and cosine in the context of triangles, and that’s a pretty useful way to think of them here too. To start, let’s draw a circle with radius equal to 1. All points on the circle will be the same distance—1—from the center; after all, that’s really what a circle means. Take an arbitrary point on the circle and map its *x*- and *y*-coordinates; draw a line from the center of the circle to your point. Now complete the triangle by connecting your point with the *x*-axis and the center, as shown in the figure below:

The standard trick in math is to associate the *x*- and *y*-axes with the real and imaginary parts of a complex number, so that the angle between the blue line and the *x*-axis, marked by θ, is the same as in Euler’s formula. The *x*-coordinate is the cosine of the angle, and the *y*-coordinate is the sine of the angle. Properly, we need to write the angle in *radians*, not degrees: one circle (360°) is 2π radians, so a half-circle (180°) is π radians. Looking at the circle above, you can see that an angle of π corresponds to *x* = -1 and *y = *0, *exactly* what Euler’s formula predicts!

I won’t prove Euler’s formula (since it really needs calculus to do properly), but with a little more work we can see how useful it is. By drawing a larger or smaller circle, we can represent *any* complex number using a variation on Euler’s formula:

where *r* is a real number representing the radius of the new circle. Instead of writing the real and imaginary parts of the complex number like *x*- and *y*-coordinates, we can use *r* and θ, which are known as *polar coordinates*. The angle θ is also known as the *phase* of the complex number, and *r* is its *magnitude*. That’s a very simple formula, and makes doing many calculations with complex numbers quite easy.

For example: if you want to mathematically represent rotations in two dimensions, you can do it using complex numbers. Take the coordinates of (say) the corners of a square, and write them as four complex numbers:

To rotate the square by angle θ, multiply the number for each corner of the square by

and you’ll get the coordinates of the rotated square:

For the example image, I used radians (or 60° if you prefer, but it’s *Pi DAY* so use the friggin’ radians already).

Therefore,

and so forth, as in the picture.

It’s endlessly fascinating how many places π shows up where you don’t necessarily expect it. However, once you’re attuned, you realize its ubiquity is because geometry hides in many places: the spiral of shells, the strength of gravity, the fluctuations of waves. Pi is an interesting number, encoding a deep connection between many apparently disparate areas of the natural world.

Filed under: Astronomy, Physics, and Related Fields ]]>