Friday, September 13, 2013

New Model Could Help I.D. Potentially Habitable Alien Planets

The headlines have been coming thick and fast.

A trio of super-Earths found in the habitable zone of the star Gliese 667C, two probably rocky planets in the Goldilocks zone around Kepler-62 and possible super-Earths orbiting Tau Ceti and HD 40307 at just the right distance for liquid water to exist on their surfaces, albeit under certain conditions.
These are all just from the past twelve months. Should those exoplanet hunters who are seeking out Earth 2, a planet where life as we know it could possibly exist, start to feel excited? [The Strangest Alien Planets (Photos)]

Not yet. Our knowledge of these planets is woefully incomplete. However, the times may be changing. While we cannot yet determine whether a planet is hospitable to life, David Kipping of the Harvard–Smithsonian Center for Astrophysics has led a team of astronomers to develop a new theoretical model that can tell us with one swift glance whether a super-Earth — a world with two to 10 times the mass of our planet and up to twice the diameter — has an atmosphere that might not be suitable for life.

Consequently, we could rule such worlds out of our search for analogs to Earth. It’s all about whether a planet has an atmosphere and how that atmosphere is connected to the relationship between a planet’s mass and diameter.

The two main exoplanet detecting techniques are beautifully complementary. When a planet transits its star — that is, passes in front of its star, blocking a fraction of the starlight — we can determine the diameter of the planet from the size of the transit. Meanwhile, that orbiting planet also exerts a gravitational tug on its parent star. If we can detect that tug we can calculate the planet’s mass based on the extent by which the planet is pulling on the star.

The only problem is that not all planets orbit their star at an appropriate angle for us to see a transit, while some exoplanets and their stars are too distant and faint for us to accurately measure their "radial velocity" tug (many of the Kepler spacecraft’s candidate planets fall into this category).
However, for those worlds where we are fortunate to know both properties, we can work out a planet’s volume and then divide the mass by the calculated volume to determine the planet’s density, which tells us whether it is likely rocky, gaseous or icy.

The computer model that Kipping has developed, along with Harvard’s Dimitar Sasselov and Princeton’s David Spiegel, allows an astronomer to plug in these numbers for mass and radius and, with the knowledge of the density, figure out if a planet — in particular a super-Earth — has a light but extended atmosphere or a relatively thin, heavy atmosphere.

That’s important because Earth’s atmosphere is the latter kind — a 100 kilometer (62 mile) layer filled with the likes of nitrogen, oxygen, carbon dioxide, argon, water vapor and neon that contributes just 1.5 percent of Earth’s radius. We don’t know if an extended atmosphere of mostly hydrogen and helium — similar to Uranus’ or Neptune’s atmospheres but warmer — could support life, and so searches for Earth’s twin may want to avoid such worlds.

Solid, liquid or gas
The way Kipping, Sasselov and Spiegel’s model makes use of a graph that plots a planet’s mass against its radius, and where a world falls on that graph, tells us whether it is solid rock, partly watery or has a significant fraction of gas.

"There’s a full range of models that we think a super-Earth can be built out of," Kipping said. "You can make them out of iron, or out of silicate, or out of water, or some mixture of those things."
However, when a planet transits a star, not only does the solid body of the planet block some of the starlight, but so too does its atmosphere. By simply detecting the planet’s silhouette we cannot automatically figure out which part is solid and which part is gaseous atmosphere. The mass-radius diagram, however, offers a way around this problem. [9 Exoplanets That Could Host Alien Life (Countdown)]

Kipping and his cohorts have calculated theoretical limits — boundary conditions — for each type of planet. The lower boundary condition denotes a super-Earth made of solid rock with an iron core and lacking an atmosphere. The top boundary signifies a planet made entirely of water that, Kipping said, is probably impossible — there needs to be a solid core in there somewhere — and thus you cannot get a super-Earth less dense than a water-world (purely gaseous planets, it is thought, cannot exist as small as super-Earths and even Neptune-type worlds have a large rocky core lurking inside them).

Therefore, if you discover a planet and plot its mass against its radius only to discover that it resides on the graph above the impossible pure-water line, then the only way to explain its apparent density given its radius is that it must have a large atmosphere.

Such mass-radius models have been around for a while, but what makes Kipping’s different is that they are based on a new understanding of the physics of materials placed under the enormous amounts of pressure that the interior of a super-Earth would impose on them. Dimitar Sasselov, along with his student Li Zeng, was able to create superior models of the interior of super-Earths using new laboratory technology that is able to simulate those pressures.

They published their work in the March 2013 issue of the Publications of the Astronomical Society of the Pacific and Kipping’s mass-radius diagram, itself to be published in the Monthly Notices of the Royal Astronomical Society, is modeled around those interior structures derived by Sasselov and Zeng.
What does the model tell us about super-Earths we have already discovered? Kipping, Spiegel and Sasselov concentrated on GJ 1214b, a world with six and a half times the mass and two and a half times the diameter of our planet that is orbiting a red dwarf star 47 light years away.
Prior to now the planet had been a puzzle — no matter what wavelength it was observed in, the size of the planet was always the same, which shouldn’t happen because an atmosphere should be more opaque to some wavelengths than others. Was its atmosphere extended and topped with thick, opaque clouds, or was its atmosphere thin enough not to be noticed? Employing the mass–radius diagram settles the matter.

"Our method says that 20 percent of this planet’s radius is pure atmosphere, which strongly favors the idea of a very light, extended hydrogen-helium atmosphere with clouds on top," Kipping said. "So we are able to come into this discussion with these two possibilities and say which is more likely, just based on the simple measurement of the mass and the radius of the planet."

Uncertain data
Another intriguing world is Kepler-22b, which was the first habitable zone planet to be discovered by NASA’s Kepler spacecraft. Around 620 light-years from Earth, it orbits a sun-like star at a distance of 0.85 astronomical units (one astronomical unit is the average distance between Earth and the sun, 149.6 million km) and has a diameter two and a half times that of our planet. [Gallery: A World of Kepler Planets]

"We tried to apply our technique to this planet but unfortunately the mass measurement is very poor because it is a very distant star," Kipping said. "What we found was that the data was unable to say one way or another what kind of planet it is; it sits right on the blue [water-world] line, so we can’t tell whether it is a rocky planet with an extended atmosphere or a water-world with very little atmosphere."

Unfortunately that’s also the story for the rest of the potentially habitable planets discovered so far, a list of which is maintained by Professor Abel Mendez of the Planetary Habitability Laboratory at the University of Puerto Rico at Arecibo, in the form of the Habitable Exoplanets Catalog.
One dozen planets currently reside on the list, meeting the criteria of being (probably) rocky and existing within their star’s habitable zone. However, as we found with Kepler-22b, in most cases either the mass or the radius is little more than an estimate, and as such the majority tend to sit on that boundary condition.

"Astronomers estimate mass or radius from the assumption that smaller planets are more rocky in composition and those larger planets close to two Earth radii are water-worlds," Mendez said. "This seems to be a good estimate for most cases but there is a lot of uncertainty; for example, Kepler-11f has just over two Earth masses but it is a gas planet, while Kepler-20b with about nine Earth masses is rocky."

Kipping’s mass–radius diagram is only half the job. Without good data the new mass-radius relationship is limited in what it can tell us. For Kepler planets, better mass measurements from radial velocities are required, but this is tricky given that most of the stars around which Kepler discovers planets are faint and distant.

For those worlds discovered by radial velocity, we need more luck in observing transits to give us their diameter. The approval of the Transiting Exoplanet Survey Satellite (TESS), which is scheduled to launch in 2017 and will systematically survey all the brightest stars in the sky for transiting planets, will be a massive boon to the field.

"The TESS mission promises to dramatically change this picture," said Heather Knutson, a planetary astronomer at the California Institute of Technology whose research is focused in the area of exoplanet atmospheres. "At the moment there are currently only three transiting super-Earths that are suitable for detailed characterization and all three have been observed with either the Spitzer or Hubble space telescopes, or both. In the era of TESS we will have far more super-Earths than we can reasonably study and Kipping’s criterion will provide a useful means to select targets that are likely to have detectable atmospheric signatures."

The launch of the James Webb Space Telescope (JWST) a year after TESS will also dramatically boost the nascent science of exoplanetary atmospheric investigations. JWST, with its 6.5-meter mirror, will extend its observations well into the near-infrared, perfect for picking up the tenuous signatures of water, methane, oxygen, carbon monoxide and carbon dioxide in atmospheres, which could be interpreted as biosignatures depending upon their concentrations. TESS will identify the planets, the mass-radius model will decide which ones we want to observe, and JWST will tell us about them. It is going to be an exciting time and the wait will be excruciating for scientists. [See a video about the JWST]

"At this point almost anything is possible!" Knutson said.

Finding a potentially habitable planet
There are many factors that go into making a planet habitable, from the presence of a magnetic field to protect its atmosphere to the question of whether it has plate tectonics to recycle carbon. A stable rotational axis, a moderate impact rate and sufficient gravity are also plausible necessities.
Yet, the possession of an atmosphere, especially one that contains some form of greenhouse gas, is one of the most crucial factors, essential for maintaining cozily warm temperatures that permit all-important liquid water to exist on its surface. That said, the range of suitable atmospheres may not be as narrow as we may think.

"I don’t think that thick hydrogen–helium atmospheres will rule out the potential for life on these planets as long as the pressure at the surface/water transition allows for liquid water," Mendez said.
So a super-Earth, with a thick envelope of hydrogen swathing a rocky core deep down could still have watery conditions at depths where the pressure, according to Mendez, drops below 10,000 atmospheres, although of course temperature will also have a say where and if this transition point occurs.

There is one more intriguing possibility. On Earth, convection currents and air flows are strongly influenced by what is on the surface, be it oceans, continents or mountains. Could a careful study of the atmosphere of a super-Earth tell us things about the terrain below that are otherwise beyond the capabilities of our telescopes?

"Yes, potentially, but the atmosphere would need to be thin enough for our observations to detect the atmosphere flows from the region close to the surface," said Knutson, who also points out that a thin atmosphere will be transparent enough for us to spectroscopically measure the surface of the planet and determine whether there are oceans, desert or even plant life.

"When we get these new super telescopes in the future [such as the Thirty Meter Telescope, the Giant Magellan Telescope and the European Extremely Large Telescope] we’ll be able to go down to sort of Earth-like atmospheres," Kipping said. "In special cases we could probably go down to these very small atmospheres that are potentially life-harboring."

But we’re getting ahead of ourselves; the new mass–radius model only provides us with a way of saying which planets don’t have a thin atmosphere. If we come to the conclusion that a super-Earth does not have an extended atmosphere, then it might be worth pointing JWST at it to measure the spectrum of any atmosphere present and see whether it is analogous to Earth’s atmosphere. [7 Ways to Discover Alien Planets (Countdown)]

"If you are really hunting Earth-like planets and our method tells you it has a big extended atmosphere, then you are probably wasting your time," Kipping said. "So it’s a way of making our searches for Earth-analogues more efficient."

With TESS and JWST and the next generation of extremely large telescopes on the horizon, Kipping’s new model is timely indeed. The way things are going, the next decade might be the decade of the super-Earth. All the hints are it is going to be an exciting time.

This story was provided by Astrobiology Magazine, a web-based publication sponsored by the NASA astrobiology program.
Follow SPACE.com @Spacedotcom, Facebook and Google+.


New Model Could Help I.D. Potentially Habitable Alien Planets

Monday, June 10, 2013

China inches closer to building its own space station - space

China will launch the Shenzhou-10 spacecraft on 11 June, lofting three astronauts on a 15-day mission to learn how to rendezvous and dock with an orbiting module. The mission is the last of three scheduled experiments designed to help astronauts master the skills for building and operating a space station.
 
If all goes to plan, the mission will mark the end of the beginning of China's slow but steady approach to human space flight. Right now, the country is not doing anything revolutionary. But progress so far suggests that more advanced plans, such as a moon base or a crewed Martian trek, may not be beyond China's reach.
 
In a press conference Monday, a spokeswoman for the Chinese human space programme, Wu Ping, announced that Shenzhou-10 will lift off at 0938 UTC from the Jinquan Satellite Launch Centre in the Gobi desert, according to the Chinese news service Xinhua. The capsule will carry two men, Nie Haisheng and Zhang Xiaoguang, and one woman, Wang Yaping.
 
The astronauts will rendezvous with the Tiangong 1 (Heavenly Palace 1) space module, which has been orbiting Earth since September 2011. The crew will do one automatic and one manual docking test. They will also run medical and technical tests and broadcast a science lesson to Chinese students from orbit.

Very ambitious

The launch continues the execution of an orderly programme laid out in the 1990s, says Joan Johnson-Freese of the US Naval War College in Newport, Rhode Island.
 
"It's a slow, incremental programme, but it's also very ambitious," she says. China started with the uncrewed launch of the Shenzhou 1 spacecraft in 1999 and continued with its first crewed launch in 2003. This week's lift-off will mark their fifth human mission in 10 years.
 
The ultimate goal is to build a space station by 2020. What China plans to do with the space station is still unclear, and they may need a new heavy launch vehicle called the Long March 5 in order to build it.
 
There is a good chance they can make it happen, in part because China's approach has been markedly different from the frenetic space race between the US and the USSR in the 1960s and '70s, says Johnson-Freese.
 
"There was a space race between the US and Russia because we each started at the same place. But for China there's no race [with the US], because we're at very different starting positions," she says.

Persistent effort

The two countries also have different political attitudes towards space exploration. "What we have seen more than anything else is a truly long-term commitment to space that dates back at least 25 years, and a sustained interest during those 25 years," says Dean Cheng of the Heritage Foundation, a conservative policy research group in Washington DC.
 
By contrast, NASA's human spaceflight programme has struggled under changing budgets and political whims. Plans to return to the moon under George W. Bush's administration, for instance, morphed into crewed missions to an asteroid under Barack Obama's presidency.
 
When it comes to sending humans beyond Earth orbit, China's unwavering goals may see it beat other space powers like the US to the punch, says Cheng. "So as long as the money holds out and political stability reigns, they might well get to some place like Mars or establish a lunar presence, precisely because they are persistent and willing to spend the money and make the effort."

Lisa Grossman

China inches closer to building its own space station - space - 10 June 2013 - New Scientist

Tuesday, February 26, 2013

Russia meteor's origin tracked down

Using amateur video footage, they were able to plot the meteor's trajectory through Earth's atmosphere and then reconstruct its orbit around the Sun.

As the space rock burned up over the city of Chelyabinsk, the shockwave blew out windows and rocked buildings.

The team, from Colombia, has published details on the Arxiv website.

Numerous videos of the fireball were taken with camera phones, CCTV and car-dashboard cameras and subsequently shared widely on the web. Furthermore, traffic camera footage of the fireball had precise time and date stamps.

Early estimates of the meteor's mass put it at ten tonnes; US space agency Nasa later estimated it to be between 7,000 and 10,000 tonnes. Nasa estimates the size of the object was about 17m (55ft).

Using the footage and the location of an impact into Lake Chebarkul, Jorge Zuluaga and Ignacio Ferrin, from the University of Antioquia in Medellin were able to use simple trigonometry to calculate the height, speed and position of the rock as it fell to Earth.

To reconstruct the meteor's original orbit around the Sun, they used six different properties of its trajectory through Earth's atmosphere. Most of these are related to the point at which the meteor becomes bright enough to cast a noticeable shadow in the videos.

Infographic The Chelyabinsk meteor (labelled ChM) appears to have been on elliptical orbit around the Sun before it collided with Earth

The researchers then plugged their figures into astronomy software developed by the US Naval Observatory.

The results suggest the meteor belongs to a well known family of space rocks - known as the Apollo asteroids - that cross Earth's orbit.

The BBC's Daniel Sandford says people described a ball of fire in the sky

Of about 9,700 near-Earth asteroids discovered so far, about 5,200 are thought to be Apollos. Asteroids are divided into different groups such as Apollo, Aten, or Amor, based on the type of orbit they have.

Dr Stephen Lowry, from the University of Kent, said the team had done well to publish so quickly.

"It certainly looks like it was a member of the Apollo class of asteroids," he told BBC News.

"Its elliptical, low inclination orbit, indicates a solar system origin, most likely from the asteroid belt between Mars and Jupiter.

Dr Lowry added: "Perhaps with more data, we can determine roughly where in the asteroid belt it come from."

Paul.Rincon-INTERNET@bbc.co.uk

Astronomers have traced the origin of a meteor that injured about 1,000 people after breaking up over central Russia earlier this month.



Russia meteor's origin tracked down

Thursday, February 07, 2013

LibreOffice, a Free Alternative to Office, Is New and Improved

 
libreoffice
Jared Newman / TIME.com
I understand that some people need Microsoft Office – that for the sake of compatibility, familiarity and features, nothing else will do.

But anyone who doesn’t feel that way should consider trying LibreOffice, a free, open-source alternative. The new version, LibreOffice 4, offers better compatibility and more features than the previous version, along with lots of under the hood improvements.

I’ve been a happy LibreOffice 3 user for about a year, and I wouldn’t say the new version is a drastic change — at least not on its face. Perhaps the most significant new feature is the ability to attach comments to a range of text, not just a single point, which will help improve compatibility with Office documents.

But the lack of flashy changes is okay, I think. While Microsoft seems to make a point of shaking up the look and feel of each new version of its Office suite, part of LibreOffice’s allure is how it stays the same. (In fact, if you hated the Ribbon layout of Office 2007 and beyond, I’d argue that LibreOffice is just the respite you’re looking for.)

Like I said, not everyone will be able to work with LibreOffice. But in my experience it handles basic compatibility very well. It supports all Office file formats, has all the major features you might expect, and gets the job done for typical document and spreadsheet editing. Give it a shot if your office software needs don’t justify Microsoft’s $140-and-up asking price.


Read more: http://techland.time.com/2013/02/07/libreoffice-a-free-alternative-to-office-is-new-and-improved/#ixzz2KGF1a6hq



LibreOffice, a Free Alternative to Office, Is New and Improved | TIME.com

Tuesday, January 29, 2013

The 11 Most Beautiful Mathematical Equations | Beauty of Math | LiveScience

General Relativity Equation


The equation for general relativity formulated by Albert Einstein.
CREDIT: Shutterstock/R.T. Wohlstadter


Mathematical equations aren't just useful — many are quite beautiful. And many scientists admit they are often fond of particular formulas not just for their function, but for their form, and the simple, poetic truths they contain.

While certain famous equations, such as Albert Einstein's E = mc^2, hog most of the public glory, many less familiar formulas have their champions among scientists. LiveScience asked physicists, astronomers and mathematicians for their favorite equations; here's what we found:

General relativity
The equation above was formulated by Einstein as part of his groundbreaking general theory of relativity in 1915. The theory revolutionized how scientists understood gravity by describing the force as a warping of the fabric of space and time.

"It is still amazing to me that one such mathematical equation can describe what space-time is all about," said Space Telescope Science Institute astrophysicist Mario Livio, who nominated the equation as his favorite. "All of Einstein's true genius is embodied in this equation." [Einstein Quiz: Test Your Knowledge of the Genius]

"The right-hand side of this equation describes the energy contents of our universe (including the 'dark energy' that propels the current cosmic acceleration)," Livio explained. "The left-hand side describes the geometry of space-time. The equality reflects the fact that in Einstein's general relativity, mass and energy determine the geometry, and concomitantly the curvature, which is a manifestation of what we call gravity." [6 Weird Facts About Gravity]

"It's a very elegant equation," said Kyle Cranmer, a physicist at New York University, adding that the equation reveals the relationship between space-time and matter and energy. "This equation tells you how they are related — how the presence of the sun warps space-time so that the Earth moves around it in orbit, etc. It also tells you how the universe evolved since the Big Bang and predicts that there should be black holes."

The Standard Model Lagrangian
The Standard Model Lagrangian represents the main set of equations describing the fundamental particles that make up our universe.
CREDIT: Shutterstock/R.T. Wohlstadter
Standard model
Another of physics' reigning theories, the standard model describes the collection of fundamental particles currently thought to make up our universe.
The theory can be encapsulated in a main equation called the standard model Lagrangian (named after the 18th-century French mathematician and astronomer Joseph Louis Lagrange), which was chosen by theoretical physicist Lance Dixon of the SLAC National Accelerator Laboratory in California as his favorite formula.

"It has successfully described all elementary particles and forces that we've observed in the laboratory to date — except gravity," Dixon told LiveScience. "That includes, of course, the recently discovered Higgs(like) boson, phi in the formula. It is fully self-consistent with quantum mechanics and special relativity."

The standard model theory has not yet, however, been united with general relativity, which is why it cannot describe gravity. [Infographic: The Standard Model Explained]
Fundamental Theorem of Calculus forms the backbone of the mathematical method known as calculus.
The Fundamental Theorem of Calculus
CREDIT: Shutterstock/agsandrew
Calculus
While the first two equations describe particular aspects of our universe, another favorite equation can be applied to all manner of situations. The fundamental theorem of calculus forms the backbone of the mathematical method known as calculus, and links its two main ideas, the concept of the integral and the concept of the derivative.

"In simple words, [it] says that the net change of a smooth and continuous quantity, such as a distance travelled, over a given time interval (i.e. the difference in the values of the quantity at the end points of the time interval) is equal to the integral of the rate of change of that quantity, i.e. the integral of the velocity," said Melkana Brakalova-Trevithick, chair of the math department at Fordham University, who chose this equation as her favorite. "The fundamental theorem of calculus (FTC) allows us to determine the net change over an interval based on the rate of change over the entire interval."

The seeds of calculus began in ancient times, but much of it was put together in the 17th century by Isaac Newton, who used calculus to describe the motions of the planets around the sun.
Pythagorean Theorem
The Pythagorean Theorem is credited to the the Greek mathematician Pythagoras, who lived in the sixth century B.C.
CREDIT: Shutterstock/ igor.stevanovic
Pythagorean theorem
An "oldie but goodie" equation is the famous Pythagorean theorem, which every beginning geometry student learns.

This formula describes how, for any right-angled triangle, the square of the length of the hypotenuse, c, (the longest side of a right triangle) equals the sum of the squares of the lengths of the other two sides (a and b). Thus, a^2 + b^2 = c^2

"The very first mathematical fact that amazed me was Pythagorean theorem," said mathematician Daina Taimina of Cornell University. "I was a child then and it seemed to me so amazing that it works in geometry and it works with numbers!" [5 Seriously Mind-Boggling Math Facts]

1=0.999999...
This simple equation states that the quantity 0.999, followed by an infinite string of nines, is equivalent to one.
CREDIT: Shutterstock/Tursunbaev Ruslan
1 = 0.999999999….
This simple equation, which states that the quantity 0.999, followed by an infinite string of nines, is equivalent to one, is the favorite of mathematician Steven Strogatz of Cornell University.
"I love how simple it is — everyone understands what it says — yet how provocative it is," Strogatz said. "Many people don't believe it could be true. It's also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity."

Special Relativity Equation
This equation of special relativity describes time dilation.
CREDIT: Shutterstock/optimarc
Special relativity
Einstein makes the list again with his formulas for special relativity, which describes how time and space aren't absolute concepts, but rather are relative depending on the speed of the observer. The equation above shows how time dilates, or slows down, the faster a person is moving in any direction.
"The point is it's really very simple," said Bill Murray, a particle physicist at the CERN laboratory in Geneva. "There is nothing there an A-level student cannot do, no complex derivatives and trace algebras. But what it embodies is a whole new way of looking at the world, a whole attitude to reality and our relationship to it. Suddenly, the rigid unchanging cosmos is swept away and replaced with a personal world, related to what you observe. You move from being outside the universe, looking down, to one of the components inside it. But the concepts and the maths can be grasped by anyone that wants to."

Murray said he preferred the special relativity equations to the more complicated formulas in Einstein's later theory. "I could never follow the maths of general relativity," he said.

Euler's Equation
Euler's Equation
CREDIT: Shutterstock/Jezper
Euler's equation
This simple formula encapsulates something pure about the nature of spheres:
"It says that if you cut the surface of a sphere up into faces, edges and vertices, and let F be the number of faces, E the number of edges and V the number of vertices, you will always get V – E + F = 2," said Colin Adams, a mathematician at Williams College in Massachusetts.

"So, for example, take a tetrahedron, consisting of four triangles, six edges and four vertices," Adams explained. "If you blew hard into a tetrahedron with flexible faces, you could round it off into a sphere, so in that sense, a sphere can be cut into four faces, six edges and four vertices. And we see that V – E + F = 2. Same holds for a pyramid with five faces — four triangular, and one square — eight edges and five vertices," and any other combination of faces, edges and vertices.

"A very cool fact! The combinatorics of the vertices, edges and faces is capturing something very fundamental about the shape of a sphere," Adams said.

Lagrangian
Lagrangian
CREDIT: Shutterstock/Marc Pinter

Euler–Lagrange equations and Noether's theorem
"These are pretty abstract, but amazingly powerful," NYU's Cranmer said. "The cool thing is that this way of thinking about physics has survived some major revolutions in physics, like quantum mechanics, relativity, etc."

Here, L stands for the Lagrangian, which is a measure of energy in a physical system, such as springs, or levers or fundamental particles. "Solving this equation tells you how the system will evolve with time," Cranmer said.

A spinoff of the Lagrangian equation is called Noether's theorem, after the 20th century German mathematician Emmy Noether. "This theorem is really fundamental to physics and the role of symmetry," Cranmer said. "Informally, the theorem is that if your system has a symmetry, then there is a corresponding conservation law. For example, the idea that the fundamental laws of physics are the same today as tomorrow (time symmetry) implies that energy is conserved. The idea that the laws of physics are the same here as they are in outer space implies that momentum is conserved.

Symmetry is perhaps the driving concept in fundamental physics, primarily due to [Noether's] contribution."

Callan-Symanzik equation
Callan-Symanzik equation
CREDIT: Shutterstock/R.T. Wohlstadter

The Callan-Symanzik equation
"The Callan-Symanzik equation is a vital first-principles equation from 1970, essential for describing how naive expectations will fail in a quantum world," said theoretical physicist Matt Strassler of Rutgers University.

The equation has numerous applications, including allowing physicists to estimate the mass and size of the proton and neutron, which make up the nuclei of atoms.

Basic physics tells us that the gravitational force, and the electrical force, between two objects is proportional to the inverse of the distance between them squared. On a simple level, the same is true for the strong nuclear force that binds protons and neutrons together to form the nuclei of atoms, and that binds quarks together to form protons and neutrons. However, tiny quantum fluctuations can slightly alter a force's dependence on distance, which has dramatic consequences for the strong nuclear force.

"It prevents this force from decreasing at long distances, and causes it to trap quarks and to combine them to form the protons and neutrons of our world," Strassler said. "What the Callan-Symanzik equation does is relate this dramatic and difficult-to-calculate effect, important when [the distance] is roughly the size of a proton, to more subtle but easier-to-calculate effects that can be measured when [the distance] is much smaller than a proton."

The minimal surface equation
The minimal surface equation
CREDIT: Shutterstock/MarcelClemens




The minimal surface equation
"The minimal surface equation somehow encodes the beautiful soap films that form on wire boundaries when you dip them in soapy water," said mathematician Frank Morgan of Williams College. "The fact that the equation is 'nonlinear,' involving powers and products of derivatives, is the coded mathematical hint for the surprising behavior of soap films. This is in contrast with more familiar linear partial differential equations, such as the heat equation, the wave equation, and the Schrödinger equation of quantum physics."

The Euler Line
The Euler Line
CREDIT: Patrick Ion/Mathematical Reviews/AMS
The Euler line
Glen Whitney, founder of the Museum of Math in New York, chose another geometrical theorem, this one having to do with the Euler line, named after 18th-century Swiss mathematician and physicist Leonhard Euler.

"Start with any triangle," Whitney explained. "Draw the smallest circle that contains the triangle and find its center. Find the center of mass of the triangle — the point where the triangle, if cut out of a piece of paper, would balance on a pin. Draw the three altitudes of the triangle (the lines from each corner perpendicular to the opposite side), and find the point where they all meet. The theorem is that all three of the points you just found always lie on a single straight line, called the 'Euler line' of the triangle."

Whitney said the theorem encapsulates the beauty and power of mathematics, which often reveals surprising patterns in simple, familiar shapes.

Follow Clara Moskowitz on Twitter @ClaraMoskowitz or LiveScience @livescience. We're also on Facebook & Google+.

Clara Moskowitz, LiveScience Senior Writer

The 11 Most Beautiful Mathematical Equations | Beauty of Math | LiveScience