Mathematics of Christmas

Post-Christmas, actually. The backside of the holiday, when all that glitters, sparkles, blinks, and sings or dances at the flip of a switch, must be packed away until next year. I always aim to repack things exactly as I found them, but inevitably I fail miserably. Luckily though, in my house we have a system that has been in place for as long as I can remember for many of the ornaments, including some original 1970's Bloomingdale's packaging! You can't beat that.

You know what else you can't beat? The tendency of Christmas lights to tie themselves in knots no matter how hard you try to outsmart them.


Each year, you open the box to find a tangled mess, right? And after each Christmas, possessed with renewed determination, you scheme to store them more carefully this year, convinced you have finally come up with a fool proof system. So maybe you roll each strand in a loop around your elbow and then lay it carefully in a box, or you return each rolled strand to it's original box, or perhaps you wrap each one in tissue paper, secure them with a rubber band, twist tie, or god forbid, even a scrunchy. Once all the holiday accoutrements have been put away, you relax, rest assured that this time you have bested the Christmas lights and their confounding ways. But then next Christmas rolls around, and you retrieve the box full of lights only to open them and discover a twisted, tangled mess, taunting you.

Good news!...sort of. You can stop blaming yourself for this frustrating phenomenon as researchers at the University of San Diego have recently concluded that it's a mathematical and physical certainty. Blame the universe instead!

There are a couple reasons for this argument. The first being the Second Law of Thermodynamics (Code name: Entropy). Order tends to disorder, and your carefully packed strands obey by doing their part. The second more complicated reason is explained in great detail by UCSD physicists, Douglas Smith and Dorian Raymer, in their October 16th paper: "Spontaneous Knotting of an Agitated String."

Basically, mathematicians have studied knots forever and developed all sorts of theories and classifications of their variations, but physicists have only recently began to explore what equations govern their formation. To look into this, Smith and Raymer built a very simple experimental apparatus consisting of a clear plastic box and a motor. They put one piece of string in the box at a time and spun it around and around, then took it out and documented its final state - knotted or not. They did this 3,415 times varying: length of string, rotation speed, number of rotations, and size of the box.

Why 3,415 times, you may ask? I quote: "The scientific answer is that 3,415 was around the point where we had statistically compelling results," Smith said. "The human answer is that 3,415 times was about as much as we could stand." ('A string theory just in time for Christmas', Scott LaFee, Bend Weekly)

They concluded that with a minimum length of string (18.124 inches) and sufficient space for the string to shift around in it's container, knots formed fairly quickly, often within the first few seconds. Inputting these results into a computer model, they even managed to create a program which could identify the 'Jones polynomial' for each resulting knot, a mathematical property based on parameters such as the number of string crossings.

So think about the length of your Christmas tree lights (undoubtedly more than 18.124 inches), how much space they have to move around in their storage boxes (a fair amount I would presume), and how often they might be jostled when moved to and from their yearly resting place (not exactly a smooth ride up those attic stairs). In all likelihood, a knot has formed while the box is till in your hands. I guess you can still blame yourself! 

Another and better, in-depth write-up of this labyrinthine marvel can be found here.

Thrill-a-minute, edge-of-your-seat video lets you see the fun first hand!


Think you are immune to knots? They go deeper into your life than Christmas. Think about all the electrical cords in your house and the iPod headphones in your bag. Deeper still, knots are required for the DNA molecules in your cells to fit into the tiny spaces they are allotted. And even more fundamental than that, knots in the fabric of space-time could explain the distribution of matter we see today. ESA is even launching a satellite next year to look for them.

And...we're back to blaming the universe.

(photo credit: Eduardo Contreras, Union-Tribune)

Set Your Tree Free

Here are two amazing fun facts for you:

1. There is a National Christmas Tree Association.
   
2. Your Christmas tree is recyclable.

Click here to be green and find your nearest tree recycling center.

Christmas in Cambridge

My first Christmas sans family, but never fear, I had a member of my astrophysics family to lean on. My friend Stas and I are keeping misery company as we spend the holidays locked in the Cavendish in front of our respective computers. Our topics actually overlap so we are helping to push each other along up the dreaded and steep hill of completion. (In fact, watch this space for our upcoming publication on the life cycle of black holes!) In the meantime though, we took a brief break to cook an outrageous amount of food and stuff ourselves silly.

Just another day by English weather standards.

The Christmas spread...Yum!

My modest first servings.

Stas's not-so-modest first servings.

Boys have such an unbelievable capacity for food!

Stas post-dinner. He went over capacity.

Attempted Christmas self-portrait of me, Stas and the Christmas tree. (Apparently the camera was more interested in the Eiffel Tower.)

Rudolph the Red Planet

Mars overload? Never.


The previous Mars posts were in reference the close pass between Earth and Mars which occurred on Monday night. However, orbital mechanics gives us something else to celebrate this holiday season. Earth, Mars, and the Sun will all align on Christmas Eve!

This particular configuration is called "opposition" as Mars will be directly opposite from the Sun with respect to Earth, or more simply Earth is directly between the Sun and Mars. For those of you who are uber keen, "conjunction" is the other time this alignment occurs, but with Mars on the far side of the Sun or the Sun directly between Earth and Mars. The diagram in Wednesday's post is actually more pertinent to this post as it illustrates the oppositions of Earth and Mars rather than their closest encounters. Jump back there for a look and then proceed on.

It's not easy to see in the diagram, but the elliptical orbits of the planets actually allow for their closest encounter and their opposition to occur separately. If all orbits were perfectly circular, the two events would be simultaneous and inseparable.

Let's step back a second, and discuss the geometry of the Solar System.

Each planet in the Solar System orbits the Sun in an elliptical orbit, as discovered by Johannes Kepler back in the day. As opposed to a circle, and ellipse has two 'centers' - termed foci - instead of one. For our Solar System, the elliptical orbits of the planets all have the Sun at one focus. But the orbit of each planet is elliptical in a different way; ellipses come in varying degrees of eccentricity, or distortion from circular. A circle is defined by an eccentricity of 0, while ellipses can have a range of eccentricities greater than 0 and less than 1. The more eccentric the orbit, the farther apart the two foci are. At an eccentricity of 1, the closed path of an ellipse breaks and becomes a parabola instead with the second focus at a distance of infinity. But that's a lesson for another day, back to Earth and Mars.

Earth's orbit is less eccentric than Mars' orbit, so the second focus of our elliptical path is closer to the Sun than the second focus for Mars. In the diagram this exaggerated to the point where Earth's orbit looks almost circular by comparison. This, combined with the oppositions plotted on the diagram, make it harder to see how the Earth can really be closer to Mars when the two aren't directly in line with the Sun, but this can be the case. Only once in much less than a blue moon will the two events occur at the same time. More often they will happen in succession as they do this year, with the closest encounter on Monday, December 18th and opposition on Monday, December 24th.

If you feel you are sinking into a geometrical quagmire, just take my work for it.

Whew, all that academic lead-up for what is hopefully still a worthwhile payoff...

The reason for this point being that in honor of this Christmas Eve spectacular, The New York Times ran an AP piece last week that proposed a cover version of "Rudolph the Red-Nosed Reindeer", with Mars as Santa's new best friend. Check out the lyrics in their entirety below and the full article here. Feel free to sing along!

Mars is a red-tinged planet
With a very shiny glow
And if you look to see it
You will find the moon in tow.

All of the other Yuletides
Santa would have at his side
The shiny nose of Rudolph
Acting as his big sleigh's guide

But this very Christmas Eve
Santa came to say:
''Rudolph, now with Mars so bright,
You can stay at home tonight.''

Then all the reindeer teased him.
And they shouted out with glee:
''Rudolph, the red-nosed reindeer
Outsourced to astronomy.''

I Miss My Serotonin

The winter solstice comes tonight. The shortest day and the longest night of the year. I, for one, am ready for daylight to make a comeback.

Did you know that the word solstice come from the Latin for sun (sol) and standing still (sistere)? The solstice is when the Sun stands still.

Well...that isn't exactly what happens, but I can see how the ancients might have seen it that way. The Sun actually moves a lot in our sky. In addition to tracing out a path across the sky from sunrise to sunset, the height of the Sun at midday also changes throughout the year. The winter solstice, in the general sense, marks the day when the Sun is the lowest in the sky during midday. However, in astronomical terms, the winter solstice is the exact moment when Earth's axis is tilting farthest away from the sun.

From Astronomy 101, you may remember that our axis is tilted 23.5 degrees from vertical with respect to the plane of the Solar System. You may also be familiar with the North Star, right? Well the North Star got it's name for a reason - it's where our axis appears to point if you were to extend a line straight out into space from the north pole. And that line always points there, regardless of the season. This is because while the Earth rotates on its axis and orbits around the Sun, the tilt of our axis doesn't change. So for the winter solstice, the Earth is at the exact spot in its orbit where the our axis tilts the farthest away from the Sun and for the summer solstice, the axis is tilting towards the Sun.


So from now until June 21st, the Sun will only rise to a higher and higher point during its daily travels. However, right now it's killing me by only getting up to a little over 16.5 degrees above the horizon. Here in Cambridge, I am at 52 degrees latitude. For comparison, back in the states, New York City is around 40 degrees latitude and will see the sun reach 25.7 degrees at its highest. Those 9-ish degrees in latitude may not seem like much, but they translate to 1.5 less hours of daylight for me versus you New Yorkers out there. Given England's penchant for grey skies, I want all the Sun I can get. I may have to overcompensate and move the equator after my tenure here so I can benefit from the Sun wandering back and forth across the zenith. So while the Sun never exactly stands still in our sky, on the solstice it appears to stop at the high or low extreme of its midday elevation. I can see where the ancients where going with that.



These two figures show the range of elevations for the sun at 50 degrees latitude (think England) and 0 degrees latitude (on the Equator). Actually, the illustration for the equator above helps to correct a common misconception. The Sun does not shine directly overhead every day at noon on the equator - it only does so on the vernal and autumnal equinoxes. The equinoxes are the half-way points between the solstices in the spring and fall where the hours of day and night are equal. Don't forget that the equinoxes and solstices are switched below the equator in the southern hemisphere!

Another myth that I would like to dispel is that the Earth is closest to the Sun on the summer solstice and consequently farthest from the Sun this time of year. To that I say: first, props for remembering that Earth's orbit is elliptical - I'm glad you are paying attention, but second, that is unfortunately an incorrect conclusion. Turns out the Earth's distance from the Sun makes less of a difference than the tilt of its axis. When the Earth is at perihelion (its closest point to the sun, peri for close, helio for sun) the heat from the Sun's radiation doesn't differ drastically enough as compared to at apehelion (its farthest point). However, when the northern hemisphere is tilted 23.5 degrees away from the Sun, less radiation strikes the atmosphere in the north and therefore causes the cold temperatures we have come to call winter. The opposite being true for summer.

Lastly, I feel I should warn you about a dangerous trend regarding the winter solstice. It precesses, along with Earth's axis. In fact, one day in the future, Christmas will be in July and Independence Day will be in December...so to speak. My point being that the solstices will be opposite to how they are now, with the shortest day being June 21st and the longest day being December 21st. That will make for quite the change in holiday traditions! Of course, I'm scaring you for no reason, because this won't happen for tens of thousands of years. Breathe easy and go make a snowman.

(N.B. For you those of you who prefer an exact science, the solstice actually occurs on December 22nd at approximately 06:09 UT, also known as Greenwich Mean Time. That's AM here in England. East coasters are 5 hours behind at 1:09 am- West coasters are 8 hours behind at 10:09 pm, back on the 21st.)

(image credits: http://www.physicalgeography.net/fundamentals/6h.html)

Mars Returns

A follow up post to yesterday's, but with a bit more detail about what's happening with Mars and why - for those of you looking to expand your science literacy (and your minds).

The biannual date between Mars and Earth happens because both planets take a different amount of time to orbit the Sun. Earth takes 365+ days and Mars takes approximately 687 days. Therefore, Earth moves faster in it's orbit and gradually "laps" Mars in the race around the Sun. The approach and subsequent passing of Mars is what occurs every two years. However, not every meeting generates such great views of the Red Planet.

As mentioned in the previous post, both planets have elliptical orbits and as such, this complicates the occurrence of these close encounters. The diagram below best illustrates what's going on here. It show how two elliptical orbits have a point where their paths come closest together and where they are farthest apart. Also shown in the diagram, to an exaggerated effect, is that Mars' orbit is more eccentric than Earth's, i.e. it's orbit is more elliptical and in fact over five times more so. If you combine this with the different speeds of each planet in their orbit, they won't always be nearest to each other at the same point in space each time Earth is about to lap Mars.


You may or may not recall a huge fuss about Mars from 2003. This was because Earth lapped Mars that year very close to the point where their orbits are the closest they ever get, approximately 34.6 million miles. This happens roughly every 15-17 years as the lapping point precesses around the Sun over time; it takes 15-17 years to occur back at the same point in space. Each biannual meeting of the planets since a closest encounter like in 2003 will produce less and less spectacular viewing opportunities of the Red Planet until the lapping point comes around again. However, while the planets are still within a certain range of each other, the viewing can still be quite exciting. After this we will have to wait another nine years before the fun begins anew. Compared to 2003, Mars will be 20 million miles farther away this year, at a distance of 55 million miles as Earth overtakes it. But surprisingly, 20 million miles makes less of a difference than you might think. There is still a good show to be had.

(photo credit: Our Dark Skies, http://www.ourdarkskies.com)

Close Encounters of the Mars Kind

At 11:45pm tonight, Mars and Earth will be the closest they've been since October 2005. This planetary rendezvous is due to the elliptical orbits of both planets around the Sun and the point at which these two ellipses are nearest to each other. It happens once every two years for Mars and Earth and it's tonight!

So wrap yourself up in your warmest winter coat(s) and get your bum outside to check it out. No excuses for even you city dwellers as Mars easily outshines any urban light pollution (except for maybe in Times Square) and is well above the horizon. It can be best seen looking south in the middle of the night, moving westward till dawn. And with its red-orange color and steady glow, it is also larger than any star you're used to seeing in the sky - in fact, it is 1/100th the size of the full moon. That may not sound like much, but I assure you it's enough to grab your attention. For reference the full moon takes up half a degree of the sky and a typical star in the night sky is on average less than 1/1000th of the the full moon. So Mars is over 10 times bigger than the stars you are used to seeing! And to put it all in perspective, as many of you may know from seeing Apollo 13, the size of the full moon is comparable to the width of your thumb at arms length. You could say then that Mars is 1/100th the width of your thumb, but that's besides the point.

It's big, it's bright, it's red, and it's awesome.

(photo credits: NASA and the Hubble Space Telescope)

New York City Star Power

The Washington Post did a feature article in the Science section this weekend about the guy I want to be when I grow up. I've probably already talked the ears off of most of you about him because he is so stellar.

Yes, I'm not ashamed to go there.

His name is Neil de Grasse Tyson and he is the head of the Hayden Planetarium. Now I know when most of you think of a planetarium, you perhaps have flashes of elementary school field trips to a dark, grungy auditorium with tilt back seats where you fell asleep while some boring old guy lectured you on the constellations. Let me assure you, the Hayden smashes all those stereotypes to bits...to bits.

It is housed in the Rose Center for Earth and Space, a wing of the American Museum of Natural History in New York City. If you are anywhere within the radius of a reasonable day trip, this is a must. And it's 'must visit' status is in large part due to Dr. Tyson's overhaul of the Hayden back in 2000. Not only is the building itself fabulous - a cavernous glass atrium housing a massive spherical planetarium known as the Hayden sphere - the museum also offers an endless series of fantastic public programs for every age. And if neither of those get you, Dr. Tyson will. I could wax on and on about it, but instead I'm just going to challenge you to prove me wrong.

Read all about Dr. Tyson and his mad astrophysics skills here through the first link and start planning your visit to the Hayden Planetarium.

And remember all you just read when you come to visit me in New York and I show you my awesome office at the Hayden, down the hall from Dr. Tyson.

22 Down...

How many more to go? I've gotten to the bottom of the troublesome 22 sources just uncovered and that's enough for tonight. But as a famous southern belle once said: tomorrow is another day. And I'm sure it will be filled with digging into the details of the other 1,886 sources. For now though, I need to get out of this office!!!

Ich bin uberfragt!

Oh, what have I done?!?...While innocently chugging along at work today, I started uncovering error after error. Error might be too strong a word, but there are a myriad of inconsistencies in my database rearing their ugly little heads. Not cool, not cool at all. It's like when I first created the database I was on crack or something! (No offense to any crackheads out there). I am seriously baffled by this and now unsure how to proceed. I only found the mistakes I did by looking further into 22 of my 1908 radio sources. Who's to say that this problem doesn't pervade the whole database? Does this mean I should go over all 1908 entries with a fine tooth comb? Or is that a waste of time? Do I instead just wait until other potential problems rise to the surface? But if that happens enough times in the near future, then wouldn't it have just been better and more time efficient to review the full database now?

Too many questions swirling around my already full postgraduate student head.

Was the Wait Worth It?

A piece of the 10% has finally been produced! It is an overlay of histograms to show the degree to which NVSS sources are polarized, broken down into groups of total brightness. The y-axis is the fraction of total sources (1 = all sources ) while the x-axis is the percentage of polarization (1 = 100%).

It may not look like much, but it actually tells me that only the dimmest NVSS radio sources are likely to be highly polarized. Which is thankfully a useful result. Yippy skippy.

To see this, note the striking difference between the neon green histogram and the black histogram. The neon green represents the dimmest group of radio sources (less than 10 mJy) and the black represents the brightest (over 1000 mJy). The units of brightness (mJy for milliJansky) are the standard when talking about radio emission. A Jansky is the basic unit, named after Mr. Jansky, naturally, but most of my sources are ten to a thousand times dimmer - hence the milli (or 10^-3 for those of you who partial to scientifc notation). Anyway, the plot clearly shows that the dimmer sources are more likely to be more polarized, whereas very few of the brighter sources are more than 10% polarized.

Now onto further investigations...

Friday Night Lights

Oh my god. I just spent the past four - yes four - hours trying to pair two lists of sources and retain on the ones that didn't match up. I am both ecstatic that I got it to work and defeated that it took this long. Seriously, four hours. I'm just saying.

And the payoff for generating this list of unmatched sources? I now get to pour over all 181 of them to try and figure out why they didn't match up, because they darn well should have. The lights will be on in this office for a bit longer tonight...

Word.

Yes, this is another non-thesis related post, but it's for my, and your, edification. I just learned two new words that are both fun to say and not used nearly enough.

• taradiddle

1. a petty falsehood; a fib.
2. silly pretentious speech or writing, twaddle.

• twaddle

1. trivial, feeble, silly, or tedious talk or writing.
   
2. taradiddle

How fun are these? I hope this blog doesn't turn into a taradiddle. That would make me such a twaddler.

My Fifteen Minutes

In reference to the previous post, I actually first heard about the dent in the solar system on my favorite podcast, The Bryant Park Project. The BPP is a new morning show on NPR with a fresher, hipper vibe, and as such it's hosted by the absolutely awesome duo of Luke Burbank and Alison Stewart. Some of you may remember Alison from MTV Choose or Lose fame. I listen to the BPP podcast every weekday morning over here to keep me entertained, informed, and connected to the homeland. If you are at all inclined to listen to NPR but you still like your radio a little funky, you must check out the BPP.

So on the broadcast for Wednesday morning (which I was listening to on my Thursday morning), one of the newscasters, Korva Coleman, brought up the story of the dent in the solar system. Only problem is, she kept referring to it as the dent in the universe. Being the astute listener and astrophysicist that I am, I immediately took offense to this and was compelled to email the show and say as much! I then went on with my day.

To my surprise, around 2pm, I received an email from Korva Coleman herself with the best. subject. line. ever. - "sorry about the universe". Not only did she get my comment and correct the mistake on air, but best of all, she gave me a shout out during the broadcast for bringing the matter to her attention. I MUST find a way to isolate that segment of the podcast to post later! It is the greatest fifteen seconds of my astrophysics career to date. And for that, the BPP is totally getting an acknowledgment in my thesis.

How Did That Dent Get There?

Second digression of the day, but this time relate to astrophysics, is the news that the two Voyager spacecraft, launched in 1977 mind you, have both now passed through the heliopause. This is the somewhat fluid boundary between the edge of our solar system as defined by the meeting of the solar wind with the interstellar medium in our neighborhood of the Milky Way.

Think of it like this...

The sun generates energy in the form of radiation that blasts out into space in all directions (like an expanding sphere or bubble with the sun, and with it the solar system, at its center). But our solar system is surrounded by other stars and gas and dust in our galaxy. So the outward expansion of the bubble created by our sun burning is balanced at somepoint by the pressure from not only all the other bubbles around stars in our neighborhood, but also from the existence of gas and dust between the stars in our galaxy (the aforementioned interstellar medium). The balancing point between these two forces is termed the heliopause, helio for sun and pause as in break.

The scientific result of this event is that the two Voyager spacecraft left our solar system, crossing the heliopause at two different points of the bubble. They have since reported back to us and it turns out that our bubble is dented! Have a read if you want to know more:

Scientists find the dent in our solar system
http://www.msnbc.msn.com/id/22186054/

(photo credit: Opher et al. 2006, 2006AIPC..858...45O)

But I Digress...Episode 1

The first digression to get my attention was in the New York Times today. It's biology, biomechanical engineering, and physics all rolled into one! Are you ready for this?:

Why Pregnant Women Don't Tip Over
http://tinyurl.com/2ylpp3

Hurry Up and Wait

Turns out a lot of an astrophysics Ph.D. can be grunt work. The most frustrating thing is spending 90% of your time writing code and formatting plots for the 10% of the time when you can actually uncover some meaningful scientific result. Apparently that's the way it goes, which I guess makes sense when you think about it; data doesn't exactly organize itself, otherwise breakthroughs would be a dime a dozen.

So 90% of my day today was spent running programs on a catalog of radio sources from the NRAO VLA Sky Survey (NVSS for short). This is a radio survey done at 1.4GHz with the Very Large Array (VLA) in Socorro, New Mexico. For any of you that have seen the movie Contact, this is the telescope she was at in the scene where she sits on the hood of her car with her headphones on - supposedly listening to the telescope signals, but that's a beef for another post. Anyway, the NVSS catalog has 1810672 radio source in it, and by radio sources, I mean galaxies outside our own that emit in the radio part of the spectrum.

I've been sorting these 1.8 million galaxies into files according to how bright they look to us so that I can dig deeper and find out if their brightness correlates to any other parameters - specifically polarization. I have measurements for how polarized each source is and my goal is to make plots galore to see if this correlation exists. Stay tuned for details...

The Inaugural Post

This is an experiment.

I am in the throes of the end stages of my Ph.D. and I need something creative to maintain my sanity up until the final hour. I figured if I blog (and blag) my way through my thesis, it might stimulate me to produce something to blog (and blag) about. Just a theory.

So gentle readers, of which there are none as I write this (...yet), what follows will be a sequence of reports on my daily progress. I dare say it could bore you to tears but it also might serve to enlighten and perhaps even educate you on the wonderful world of radio astronomy. Of course hard work always comes with a healthy dose of procrastination and digression so there will be plenty of non-sequiturs to keep it interesting. Comments, words of encouragement, and blatant stroking of my ego will be welcome, appreciated and required from time to time, respectively.

I now entreat you to come along for the academic ride of your life...well, at least of mine.