Europa

Jupiter's icy moon Europa is slightly smaller than the Earth's Moon. Like the Earth, Europa is thought to have an iron core, a rocky mantle and a surface ocean of salty water. Unlike on Earth, however, this ocean is deep enough to cover the whole surface of Europa, and being far from the sun, the ocean surface is globally frozen over.

Europa orbits Jupiter every 3.5 days and is phase locked -- just like Earth's Moon -- so that the same side of Europa faces Jupiter at all times. However, because Europa's orbit is eccentric (i.e. an oval or ellipse not a circle) when it is close to Jupiter the tide is much higher than when it is far from Jupiter. Thus tidal forces raise and lower the sea beneath the ice, causing constant motion and likely causing the cracks we see in images of Europa's surface from visiting robotic probes.

This "tidal heating" causes Europa to be warmer than it would otherwise be at its average distance of about 780,000,000 km (485,000,000 miles) from the sun, more than five times as far as the distance from the Earth to the sun. The warmth of Europa's liquid ocean could prove critical to the survival of simple organisms within the ocean, if they exist.

Discovery:

Europa was discovered on 8 January 1610 by Galileo Galilei. The discovery, along with three other Jovian moons, was the first time a moon was discovered orbiting a planet other than Earth. The discovery of the four Galilean satellites eventually led to the understanding that planets in our solar system orbit the sun, instead of our solar system revolving around Earth. Galileo apparently had observed Europa on 7 January 1610, but had been unable to differentiate it from Io until the next night.

How Europa Got its Name:

Galileo originally called Jupiter's moons the Medicean planets, after the Medici family and referred to the individual moons numerically as I, II, III, and IV. Galileo's naming system would be used for a couple of centuries.

It wouldn't be until the mid-1800s that the names of the Galilean moons, Io, Europa, Ganymede, and Callisto, would be officially adopted, and only after it became apparent that naming moons by number would be very confusing as new additional moons were being discovered.

Europa was originally designated Jupiter II by Galileo because it was the second satellite of Jupiter. Europa is named for the daughter of Agenor. Europa was abducted by Zeus (the Greek equivalent of the Roman god Jupiter), who had taken the shape of a spotless white bull. Europa was so delighted by the gentle beast that she decked it with flowers and rode upon its back. Zeus seizing his opportunity rode away with her into the ocean to the island of Crete, where he transformed back into his true shape. Europa bore Zeus many children, including Minos.

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Unified Field Theory

Unified field theory, in particle physics, an attempt to describe all fundamental forces and the relationships between elementary particles in terms of a single theoretical framework. In physics, forces can be described by fields that mediate interactions between separate objects. In the mid-19th century James Clerk Maxwell formulated the first field theory in his theory of electromagnetism. Then, in the early part of the 20th century, Albert Einstein developed general relativity, a field theory of gravitation. Later, Einstein and others attempted to construct a unified field theory in which electromagnetism and gravity would emerge as different aspects of a single fundamental field. They failed, and to this day gravity remains beyond attempts at a unified field theory.

At subatomic distances, fields are described by quantum field theories, which apply the ideas of quantum mechanics to the fundamental field. In the 1940s quantum electrodynamics (QED), the quantum field theory of electromagnetism, became fully developed. In QED, charged particles interact as they emit and absorb photons (minute packets of electromagnetic radiation), in effect exchanging the photons in a game of subatomic “catch.” This theory works so well that it has become the prototype for theories of the other forces.

During the 1960s and ’70s particle physicists discovered that matter is composed of two types of basic building block—the fundamental particles known as quarks and leptons. The quarks are always bound together within larger observable particles, such as protons and neutrons. They are bound by the short-range strong force, which overwhelms electromagnetism at subnuclear distances. The leptons, which include the electron, do not “feel” the strong force. However, quarks and leptons both experience a second nuclear force, the weak force. This force, which is responsible for certain types of radioactivity classed together as beta decay, is feeble in comparison with electromagnetism.

At the same time that the picture of quarks and leptons began to crystallize, major advances led to the possibility of developing a unified theory. Theorists began to invoke the concept oflocal gauge invariance, which postulates symmetries of the basic field equations at each point in space and time (see gauge theory). Both electromagnetism and general relativity already involved such symmetries, but the important step was the discovery that a gauge-invariant quantum field theory of the weak force had to include an additional interaction—namely, the electromagnetic interaction. Sheldon Glashow, Abdus Salam, and Steven Weinberg independently proposed a unified “electroweak” theory of these forces based on the exchange of four particles: the photon for electromagnetic interactions, and two charged W particles and a neutral Z particle for weak interactions.

During the 1970s a similar quantum field theory for the strong force, called quantum chromodynamics (QCD), was developed. In QCD, quarks interact through the exchange of particles called gluons. The aim of researchers now is to discover whether the strong force can be unified with the electroweak force in a grand unified theory (GUT). There is evidence that the strengths of the different forces vary with energy in such a way that they converge at high energies. However, the energies involved are extremely high, more than a million million times as great as the energy scale of electroweak unification, which has already been verified by many experiments.

Grand unified theories describe the interactions of quarks and leptons within the same theoretical structure. This gives rise to the possibility that quarks can decay to leptons and specifically that the proton can decay. Early attempts at a GUT predicted that the proton’s lifetime must be in the region of 1032 years. This prediction has been tested in experiments that monitor large amounts of matter containing on the order of 1032 protons, but there is no evidence that protons decay. If they do in fact decay, they must do so with a lifetime greater than that predicted by the simplest GUTs. There is also evidence to suggest that the strengths of the forces do not converge exactly unless new effects come into play at higher energies. One such effect could be a new symmetry called “supersymmetry.”

A successful GUT will still not include gravity. The problem here is that theorists do not yet know how to formulate a workable quantum field theory of gravity based on the exchange of a hypothesized graviton. See also quantum field theory.

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Electromagnetism

Magnetic Effect Of Current Or Electromagnetism

The term "magnetic effect of current" means that "a current flowing in a wire produces a magnetic field around it". The magnetic effect of current was discovered by Oersted in 1820. Oersted found that a wire carrying a current was able to deflect a magnetic needle. Now, a magnetic needle can only be deflected by a magnetic field. Thus it was concluded that a current flowing in a wire always gives rise to a magnetic field round it. The magnetic effect of current is called electromagnetism which means that electricity produces magnetism.

Tenets Of Electromagnetism:

Magnetic Field Pattern Due To Straight Current-Carrying Conductor

The magnetic lines of force round a straight conductor carrying current are concentric circles whose centers lie on the wire.

The magnitude of magnetic field produced by a straight current-carrying wire at a given point is:

1. Directly proportional to the current passing in the wire, and
2. Inversely proportional to the distance of that point from the wire.

So, greater the current in the wire, stronger will be the magnetic field produced. And greater the distance of a point from the current-carrying wire, weaker will be the magnetic field produced at that point.

Magnetic Field Pattern Due To A Circular Coil Carrying Current

We know that when a current is passed through a straight wire, a magnetic field is produced around it. It has been found that the magnetic effect of current increases if, instead of using a straight wire, the wire is converted into a circular coil. A circular coil consists of twenty or more turns of insulated copper wire closely wound together. When a current is passed through a circular coil, a magnetic field is produced around it. The lines of force are circular near the wire, but they become straight and parallel towards the middle point of the coil. In fact, each small segment of the coil is surrounded by such magnetic lines of force. At the center of the coil, all the lines of force aid each other due to which the strength of the magnetic field increases.

The magnitude of magnetic field produced by a current carrying wire at its center is:

1. Directly proportional to the current passing through the circular wire, and
2. Inversely proportional to the radius of the circular wire.

A current carrying circular wire (or coil) behaves as a thin disc magnet, whose one face is a north pole and the other face is a south pole.

The strength of magnetic field produced by a current carrying circular coil can be increased

1. By increasing the number of turns of wire in the coil
2. By increasing the current flowing through the coil
3. By decreasing the radius of the coil.

Solenoids

The solenoid is a long coil containing a large number of close turns of insulated copper wire. The magnetic field produced by a current carrying solenoid is similar to the magnetic field produced by a bar magnet. The lines of magnetic force pass through the solenoid and return to the other end. If a current carrying solenoid is suspended freely, it comes to rest pointing North and South like a suspended magnetic needle. One end of the solenoid acts like a N-pole and the other end a S-pole. Since the current in each circular turn of the solenoid flows in the same direction, the magnetic field produced by each turn of the solenoid adds up, giving a strong resultant magnetic field inside the solenoid. A solenoid is used for making electromagnets.

The strength of magnetic field produced by a current carrying solenoid is:

1. Directly proportional to the number of turns in the solenoid
2. Directly proportional to the strength of current in the solenoid
3. Dependent on the nature of "core material" used in making the solenoid. The use of soft iron rod as core in a solenoid produces the strongest magnetism.

Electromagnet:

An electric current can be used for making temporary magnets known as electromagnets. An electromagnet works on the magnetic effect of current. It has been found that if a soft iron rod called core is placed inside a solenoid, then the strength of the magnetic field becomes very large because the iron ore is magnetized by induction. This combination of a solenoid and a soft iron core is called an electromagnet. Thus, an electromagnet consists of a long coil of insulated copper wire wound on a soft iron core.

The electromagnet acts as a magnet only so long as the current is flowing in the solenoid. The moment the current is switched off the solenoid is demagnetized. The core of the electromagnet must be of soft iron because soft iron loses all of its magnetism when current in the coil is switched off. Steel is not used in electromagnets, because it does not lose all its magnetism when the current is stopped and becomes a permanent magnet.

Electromagnets can be made of different shapes and sizes depending on the purpose for which they are to be used.

Factors Affecting The Strength Of An Electromagnet:

The strength of an electromagnet is: 1) Directly proportional to the number of turns in the coil. 2) Directly proportional to the current flowing in the coil. 3) Inversely proportional to the length of air gap between the poles.

In general, an electromagnet is often considered better than a permanent magnet because it can produce very strong magnetic fields and its strength can be controlled by varying the number of turns in its coil or by changing the current flowing through the coil.

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Renormalization

1. The Game Called "Renormalization"

Okay, let's see.... let's consider a quantum field theory whose Lagrangian has a few free parameters — masses and charges and so. Just to sound cool, let's call all of these numbers "coupling constants". Now to get finite answers from this theory, we need to impose a "frequency cutoff". We do this by simply ignoring all waves in our fields that have a a frequency higher than some fixed value. This works best after we replace "t" by "it" everywhere in our equations, so let's do that — this is called a "Wick rotation" by the experts. Now we're working with a theory on Euclidean spacetime, and the frequency cutoff can also be thought of as a distance cutoff. In other words, it amounts to ignoring effects that involve fields varying on distance scales shorter than some distance D.
In what follows, you have to keep your eye on the parameters in the theory: I'm gonna keep shuffling them around, so to check that I'm not conning you, you have to make sure there's always the same number of 'em around — sort of like watching a magician playing a shell game. So make sure you see what we're starting with! Our Lagrangian has some numbers in it called "coupling constants", but our theory really has one more parameter: the cutoff scale D.
Now our Lagrangian has some coupling constants in it, but it's hard to measure these directly. Even though they have names like "mass", "charge" and so on, these parameters aren't what you directly measure by colliding particles in an accelerator. In fact, if you try to measure the charge of the electron (say) by smashing two electrons into each other in an accelerator, seeing how much they repel each other, and naively using the obvious formula to determine their charge, the answer you get will depend on their momenta in the center-of-mass frame — or in other words, how hard you smashed them into each other. The same is true for the electron mass and any other coupling constants there are in the Lagrangian of our theory. They have a "bare" value — the value that appears in the Lagrangian — and a "physical" value — the value you measure by doing an experiment and an obvious naive sort of calculation. The "physical" values depend on the "bare" values, the cutoff D, and a momentum scale p.
(Of course, we could cleverly try to use a less naive formula to determine the bare values of the coupling constants from experiment, but let's not do that — let's just use the stupid obvious formula that neglects the funky quantum effects that are making the physical values differ from the bare values! By being deliberately "naive" here, we're actually being very smart here — as you'll eventually see.)
There are all sorts of games we can play now. The simplest, oldest game is this. We can measure the physical coupling constants at some momentum scale p, and then figure out which bare coupling constants would give these physical values — assuming some cutoff D. Then we can try to take a limit as D → 0, adjusting the bare coupling constants as we take the limit, in order to keep the predicted physical coupling constants at their experimentally determined values. This "continuum limit", if it exists, will be a theory without any shortest distance scale in it. That's very important if you think spacetime is a continuum!
This game is called "renormalization".
Sometimes you win this game — and sometimes you lose. The main thing to worry about is this: even if certain bare coupling constants are zero, the corresponding physical coupling constants may be nonzero. For example, if you start with a Lagrangian in which the mass of some particle is zero, you might not have bothered to include that mass among your bare coupling constants. But its physical mass (measured at some momentum scale) can still be nonzero. In this case, we say the particle "acquires a mass through its interactions with other particles". This sort of thing happens all the time.
What this means is that to succeed in adjusting the bare coupling constants to fit the experimentally observed physical coupling constants, we need to start with a Lagrangian that has enough bare coupling constants to begin with. You can't expect to fit N numbers with fewer than N numbers!
So, if someone hands you a Lagrangian, you may have to stick in some extra terms with some extra bare coupling constants before playing the renormalization game. If you can succeed with only finitely many extra terms, you say your theory is "renormalizable". If you need infinitely many terms, you throw up your hands in despair and say the theory is "nonrenormalizable". A nonrenormalizable Lagrangian is like a hydra-headed monster that keeps needing more extra terms to be added the more you add.
Note: when we try to take the continuum limit, we don't care if the bare coupling constants do something screwy like go to infinity. All we care about is whether the experimental predictions of our theory converge. If the bare coupling constants converge we say our theory is "finite". But truly realistic theories usually aren't this nice.

2. The "Renormalization Group" Game
Okay, now I want to talk about the renormalization group. I'm deliberately going to simplify things to the point where I'm verging on inaccuracy, but hopefully I won't actually say anything actually false.
So, let's recall what we've got. We have a quantum field theory described by a Lagrangian with a bunch of coupling constants in it — let's call them "bare" coupling constants. We can write all these bare coupling constants in a list and think of it as a vector: call it C. But to do calculations with this theory we need one more number, too: we need to ignore effects going on at length scales smaller than some distance D, called the "cutoff".
Now, starting from these numbers, we can compute the "physical" coupling constants at any momentum scale. For example, the measured charge of the electron depends on the momentum with which we collide two electrons. Another way to put it is that the physical coupling constants depend on a distance scale: for example, the measured charge of the electron depends on the distance at which you measure its charge. These two ways of thinking about it are equivalent, since using hbar and c we can freely convert between momentum and inverse distance.
Let's work with distance instead of momentum, and call the distance at which we measure the physical coupling constants D'.
So: if we know the "bare" coupling constants C and the cutoff D, we can compute the "physical" coupling constants C' at any distance scale D'. In short:
C' = f(C,D,D')
Now let's play the "renormalization group" game. In this game, we fix the bare coupling constants and the cutoff, and see how the physical coupling constants C' change as we vary the distance scale D' at which we measure them. It's fun to imagine turning a dial to adjust the distance scale D' and watching the physical coupling constants C' move around like a little dot in n-dimensional space, where n is the number of coupling constants. People draw pictures of this and speak of "running coupling constants" or the "renormalization group flow".
Note that we can play this game whether or not our field theory is renormalizable! In the last section I talked about a different game, called "renormalization". That game was all about letting the cutoff D go to zero. For "renormalizable" theories there's a nice way to do it, while for "nonrenormalizable" ones it's a real mess. But here we aren't letting D go to zero.
So what happens if we start with a nonrenormalizable theory and play this "renormalization group" game? Our Lagrangian will typically have a bunch of terms in it: some nasty ones that are making the theory nonrenormalizable, and some nice ones that would give a renormalizable theory if we just threw out the nasty ones. Each of these terms is multiplied by a coupling constant. Now let's look at the corresponding physical coupling constants as we crank up the distance scale D'.
As we do this, the physical coupling constants in front of the nasty nonrenormalizable terms get smaller and smaller, approaching zero! At large distances, nonrenormalizable interactions become irrelevant!
This is an incredibly important fact, because it may explain why the quantum field theory that seems to describe our world — the Standard Model — is renormalizable. There may be all sorts of strange quantum gravity stuff going on at very short distance scales — perhaps spacetime is not even a continuum! But if at larger scales we assume that ordinary quantum field theory on flat spacetime is a reasonably accurate approximation to what's going on, then this renormalization group stuff assures us that at still larger scales, nonrenormalizable interactions are going to look very weak.
In fact, this may explain why gravity is so weak! If we treat quantum gravity perturbatively as a quantum field theory on flat spacetime, it's nonrenormalizable. If we assume the gravitational constant is reasonably large near the Planck scale, and we follow the renormalization group flow, we find that it's very small at macroscopic distance scales. In fact, we even get the right order of magnitude. But this isn't surprising: it's really just the magic of dimensional analysis.
This sort of idea goes back to Kenneth Wilson who won the Nobel prize in physics in 1982, for work he did around 1972 on the renormalization group and critical points in statistical mechanics. His ideas are now important not only in statistical mechanics but also in quantum field theory. For a nice short summary of the "Wilsonian philosophy of renormalization", let me paraphrase Peskin and Schroeder:

In Chapter 10 we took the philosophy that the distance cutoff D should be disposed of by taking the limit D → 0 as quickly as possible. We found that this limit gives well-defined predictions only if the Lagrangian contains no coupling constants with dimensions of length^d with d > 0. From this viewpoint, it seemed exceedingly fortunate that quantum electrodynamics, for example, contained no such coupling constants since otherwise this theory would not yield well-defined predictions.Wilson's analysis takes just the opposite point of view, that any quantum field theory is defined fundamentally with a distance cutoff D that has some physical significance. In statistical mechanical applications, this distance scale is the atomic spacing. In quantum electrodynamics and other quantum field theories appropriate to elementary particle physics, the cutoff would have to be associated with some fundamental graininess of spacetime, perhaps the result of quantum fluctuations in gravity. We discuss some speculations on the nature of this cutoff in the Epilogue. But whatever this scale is, it lies far beyond the reach of present-day experiments. Wilson's arguments show that this this circumstance explains the renormalizability of quantum electrodynamics and other quantum field theories of particle interactions. Whatever the Lagrangian of quantum electrodynamics was at the fundamental scale, as long as its couplings are sufficiently weak, it must be described at the energies of our experiments by a renormalizable effective Lagrangian.

3. Ultraviolet and Infrared Fixed Points

In the last section I described the "renormalization group" game. Now I want to explain "ultraviolet and infrared fixed points" of the renormalization group, but first let me summarize what I already said. We have a quantum field theory described by a Lagrangian with a bunch of terms multipled by numbers called "bare" coupling constants — we call the list of all of them C. We ignore effects going on at length scales smaller than some distance D called the "cutoff". And now we can compute stuff....
In particular, we can compute the so-called "physical" coupling constants C' as measured at any given length scale D'. And we can watch how C' changes as we slowly crank D' up. This is called the "renormalization group flow".
Various things can happen. I already said a bit about this: I said that for nonrenormalizable terms in the Lagrangian, the physical coupling constants shrink as we increase D'.
In fact we can say more: they scale roughly like D' to some negative power. If you're smart, you can even guess what this power is by staring at the term in question and doing some dimensional analysis! Using Planck's constant and the speed of light you can express all units in terms of length. If a particular bare coupling constant c in front of some term in the Lagrangian has dimensions of length to the power d, then the corresponding physical constant c' will scale roughly like D' to the power -d. More precisely:
c'/c ~ (D'/D)^-d
In particular, this term will be nonrenormalizable if d is greater than zero.
Of course, another way to put this is that for nonrenormalizable theories, the physical coupling constants grow as we decrease D'. This is another way to see why nonrenormalizable theories are "bad" — they involve interactions that get ridiculously strong at short distance scales. Why is this bad? Well, it's certainly bad if you're trying to do perturbation theory and think of the interaction as a small perturbation. It may not always be bad in any more profound sense, because there arenonrenormalizable theories that are perfectly consistent, mathematically speaking.
On the other hand, if d is less than zero we say our term in the Lagrangian is "superrenormalizable". In this case the physical coupling constant scales roughly like D' to some positive power. In the same sense that nonrenormalizable theories are not nice, superrenormalizable theories are super-nice.
Finally, for "renormalizable" theories, the physical coupling constants scale roughly like D to the zeroth power — i.e., they're roughly constant. They are right on the brink between nasty and nice. We actually have to do a more careful analysis to see if they are nasty or nice. For example, quantum electrodynamics is renormalizable, but it turns out to be nasty: at first the charge of the electron looks almost constant as we decrease D', but it actually grows — logarithmically at first, but then faster and faster. On the other hand, lots of nonabelian gauge theories are nice: the coupling constant slowly shrinks to zero as we decrease D'. We say they are "asymptotically free".
Now, to get ready for my explanation about what all this has to do with 2nd-order phase transitions, let's just introduce some concepts to help us tie all these ideas together. We've seen that sometimes when we keep making D' smaller and smaller, the physical coupling constants C' approach some particular value. I've just talked about the case when they approach zero, but other cases are important too! Whenever this sort of thing happens, we say the limiting value of C' is an "ultraviolet fixed point of the renormalization group". Here "ultraviolet" refers to the fact that we are looking at ever smaller distance scales.
Similarly, if C' approaches some value when D' keeps getting larger, we say that value is an "infrared fixed point".
For example, suppose we have a superrenormalizable or asymptotically free theory with just one coupling constant. Then as we keep making D' smaller, the physical coupling constant approaches zero, so zero is an ultraviolet fixed point. Of course "zero" here corresponds to a free field theory with no interaction at all. So free theories are ultraviolet fixed points of superrenormalizable or asymptotically free theories. Similarly, free theories are infrared fixed points of nonrenormalizable theories, and certain renormalizable but nasty theories like quantum electrodynamics.

4. Second-Order Phase Transitions

Okay, now I'm going to finish by describing Wilson's ideas relating renormalization to 2nd-order phase transitions. First of all, what's a 2nd-order phase transition?
Actually, first of all, what's a first-order phase transition?
The most familiar examples are when ice melts or liquid water boils: we have two phases of matter, and the internal energy changes discontinuously as we go from one phase to another. But look at this phase diagram, which I borrowed from Scott Lanning:

           \              
            \ liquid   X  (critical point)
   |         \       /
^  | solid    \    /
|  |           \ /
P  |          /
R  |        /
E  |      /
S  |    /
S  |  /          gas
U  |/  
R  |
E  |_______________________________________
   
              TEMPERATURE --→

We see something interesting: the sharp boundary between liquid and gas phases fizzles out at a point called the "critical point". Above this point there is no real difference between a liquid and gas! This critical point is a "2nd-order phase transition", because while the internal energy doesn't change discontinuously there, its first derivative becomes infinite there.
Right at the critical point, something very cool happens: the system transforms in a simple way under scaling! What does this mean? Well, if you get some water right at the critical point, it looks "opalescent" like a moonstone. If you stare at it carefully, you'll see a bunch of liquid water droplets of all different sizes floating around in steam. However, if you look closely at any of these droplets, you see they are full of bubbles of steam, and if you look closely at the steam, you see it's full of little droplets of liquid! It's like a random fractal: no matter how closely you look, you see the same thing. You can't tell if you're looking at water droplets in steam or bubbles of steam in water, and there is no distinguished length scale... at least until you get down to the scale of atoms, that is.
Building on insights due to Landau, Kadanoff and others, Wilson realized that you could come up with a very precise theory of critical points by taking advantage of this symmetry under change of scale. In particular, this theory lets us understand so-called "critical exponents".
To explain this, let me switch to a simpler example of a critical point. Consider a ferromagnet like a crystal of iron. At temperatures above a certain point called the Curie temperature, the iron will not be magnetized. But as we cool it below the Curie point the spins of certain electrons in the atoms will line up and the iron will become magnetized. If there is an external magnetic field around when we cool the iron below the Curie temperature, the spins will line up with this magnetic field. Suppose the magnetic field points along the z axis - either up or down. Then we have the following phase diagram:

^ |
| |
M |        
A |    magnetized up 
G |
N |
E |                                 
T | ----------TEMPERATURE-->---------X           unmagnetized 
I |                              (critical 
C |                                point)
  |
F |    magnetized down
I |
E |
L |
D |

The sharp boundary between the "up" and "down" magnetized phases fizzles out at the Curie temperature. The Curie temperature is a critical point! Right at this critical point the magnet displays symmetry under scaling. If we look at the atoms in the crystal lattice and see which ones are "spin-up" and which ones are "spin-down", at the critical point we see regions of spin up and regions of spin down, but all these regions are speckled with smaller regions of the opposite type, and so on... on down to the length scale set by the crystal lattice itself.
To describe this scaling symmetry a bit more mathematically, let's simplify things a bit and imagine that for each point x in the crystal lattice we have a variable s(x) which equals 1 if that atom is spin-up and -1 if it's spin-down. When the crystal is in thermal equilibrium this variable keeps randomly flipping sign, so we can think of it as a random variable. This means we can talk about its mean, standard deviation and stuff like that.
When the external magnetic field is zero, the mean of s(x) is zero:
<s(x)> = 0
because each atom has a 50-50 chance of being spin-up or spin-down. This isn't particularly interesting. What's interesting is the mean of the product of s(x) and s(y) for two different points in the lattice, x and y:
<s(x)s(y)>
This is called a "2-point function". It measures the correlation of spins at different points in the lattice, since it equals 1 if the two spins always point the same way and 0 if they are completely uncorrelated.
The 2-point function only depends on the distance between x and y. Away from the critical point it decays exponentially with distance (at least when the external magnetic field is zero), and this exponental decay determines a special length scale called the "correlation length":
<s(x)s(y)> ~ exp(-|x-y|/L)
But as we approach the critical point, the correlation length goes to infinity, and right at the critical point, the 2-point function decays like some power of distance:
<s(x)s(y)> ~ 1/|x-y|^d
The number d is an example of what we call a "critical exponent".
A similar thing is true for all the higher "n-point functions", at least if we define them correctly, which I won't bother to do here. They all satisfy nice power laws at the critical point. This is what people mean when they say that a system at a critical point transforms simply under scaling.
Now, I'm oversimplifying something important here, so I'd better explain it. These power laws like
<s(x)s(y)> ~ 1/|x-y|^d
are really only approximate! Actually this is obvious, because the left hand side can't get bigger than 1, while the right hand side goes to infinity as |x-y| goes to zero. In reality, the the 2-point function behaves in a very complicated way when the distance between our two atoms is very small. It's only when the distance gets big that things simplify and the power law becomes a better and better approximation.
Does this remind you of anything?
It should: this is where the renormalization group comes in! We can imagine "zooming out" on our crystal, looking at it from ever larger distance scales. As we do, things simplify: we can forget about individual atoms and approximate the situation by a field theory defined in the continuum. In fact, we can try to use one of the field theories that we've been talking about in the previous sections! Remember, quantum field theory in Euclidean space is just the same as statistical mechanics. Quantum field theory needs a cutoff, but we've got one: the distance between atoms in our crystal. So we're all set: we can write down some Lagrangian and start playing the renormalization group game to see what happens as we zoom out.
You may be suspicious here: how are we ever going to guess which Lagrangian corresponds to our original problem involving a crystal of iron? After all, iron is complicated stuff!
Luckily, it's not so bad. At short distance scales, to get a decent approximation to our original problem, we may need to start with a really complicated Lagrangian. However, suppose we do this. Then as we zoom out to large distance scales, the renormalization group game says that the Lagrangian will simplify. For example, we've already seen that nonrenormalizable terms in the Lagrangian become "irrelevant" as we go to large distance scales: the physical coupling constants in front of them go to zero!
More generally, we shouldn't be at all surprised if our physical coupling constants approach an infrared fixed point as we zoom out, letting the distance scale approach infinity. This is exactly what infrared fixed points are all about! Even better, all sorts of theories with different bare coupling constants can approach the same infrared fixed point. We say two different theories, or two different physical systems, are in the same "universality class" if they approach the same infrared fixed point as we crank up the distance scale.
For example, when we're studying what happens at the Curie temperature, lots of different ferromagnets lie in the same universality class. Indeed, it turns out that you can study a lot of them using slight variations of one of the simplest quantum field theories of all: the φ⁴ theory.
There is a lot more to say, and I'm too tired to say most of it, but there's one thing I must tell you, just to wrap up some loose ends. Wilson's real triumph was to calculate critical exponents like the number d in the power law for the 2-point function:
<s(x)s(y)> ~ 1/|x-y|^d
How did he do it? Well, Landau already had one way to do this, which gives just the results you would guess using dimensional analysis. But that method didn't always give the right answers. To get the right answers, it helps to realize that n-point functions are closely related to physical coupling constants. In fact, while I never actually defined the physical coupling constants, they are really just a way of extracting some information about n-point functions. So if we calculate the "running of coupling constants" using the renormalization group game, we can work out the critical exponents.


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BTW MERRY CHRISTMAS !!!!!!!

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Gliese 581c

Gliese 581 c or Gl 581 c is a planet orbiting the red dwarf Gliese 581. It is the second planet discovered in the system and the third in order from the star. With a mass at least 5.6 times that of the Earth, it is a super-Earth (a planet of 1 to 10 Earth masses). It was the smallest known extrasolar planet around a main-sequence star, but on April 21, 2009, another planet orbiting Gliese 581, Gliese 581 e, was announced with an approximate mass of 1.9 Earth masses.

Gliese 581 got interest because it was reported to be the first potentially Earth-like planet in the habitable zone of its star, with a temperature right for liquid water on its surface, and by extension, potentially capable of supporting extremophile forms of Earth-like life. However, further research casts doubt upon the habitability of Gliese 581 c. The (unconfirmed) fourth planet in the system, Gliese 581 g, is a better candidate for habitability.

In astronomical terms, the Gliese 581 system is relatively close to Earth, at 20.3 light years (192 trillion km or 119 trillion miles) in the direction of the constellation of Libra. This distance, along with the declination and right ascension coordinates, give its exact location in our galaxy. It is identified as Gliese 581 by its number in the Gliese Catalogue of Nearby Stars; it is the 89th closest known star system to the Sun.

Discovery

The team released a paper of their findings dated April 27, 2007, published in the July, 2007 journal Astronomy and Astrophysics. In the paper they also announced the discovery of another planet in the system, Gliese 581 d, with a minimum mass of 7.7 Earth masses and a semi-major axis of 0.25 astronomical units. (A reanalysis of the radial velocity data has since reduced the minimum possible mass of Gliese 581 d to 5.6 Earth masses.)

Physical characteristics

Mass

The existence of Gliese 581 c and its mass have been measured by the radial velocity method of detecting extrasolar planets. The mass of a planet is calculated by the small periodic movements around a common centre of mass between the host star Gliese 581 and its planets. When all six planets are fit with a Keplerian solution, the minimum mass of the planet is determined to be 5.6 Earth masses. The radial velocity method cannot by itself determine the true mass, but it cannot be very much larger than this or the system would be dynamically unstable. Dynamical simulations of the Gliese 581 system which assume the orbits of the planets are coplanar indicate that the planets cannot exceed approximately 1.6 – 2 times their minimum masses or the planetary system becomes unstable (this is primarily due to the interaction between planets e and b). For Gliese 581 c, the upper bound is 10.4 Earth masses.

Radius

Since Gliese 581 c has not been detected directly, there are no measurements of its radius. Furthermore, the radial velocity methodused to detect it only puts a lower limit on the planet's mass, which means theoretical models of planetary radius and structure can only be of limited use. However, assuming a random orientation of the planet's orbit, the true mass is likely to be close to the measured minimum mass.

Assuming that the true mass is the minimum mass, the radius may be calculated using various models. For example, if Gliese 581 c is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth, according to Udry's team. Gravity on such a planet's surface would be approximately 2.24 times as strong as on Earth. However, if Gliese 581 c is anicy and/or watery planet, its radius would be less than 2 times that of Earth, even with a very large outer hydrosphere, according to density models compiled by Diana Valencia and her team for Gliese 876 d. Gravity on the surface of such an icy and/or watery planet would be at least 1.25 times as strong as on Earth. They claim the real value of the radius may be anything between the two extremes calculated by density models outlined above.

Some modelled radii of Gliese 581 c, compared with Earth and Neptune.

Other scientists' views differ. Sara Seager at MIT has speculated that Gliese 581 c and other five-Earth-mass planets could be:

• "rock giants" mostly of silicate.
• "cannonball" planets of solid iron.
• "gas dwarfs" mostly of helium and hydrogen.
• carbon-rich "diamond worlds"
• purely hot "ice VII worlds".
• purely "carbon monoxide worlds".

If the planet transits the star as seen from our direction, the radius should be measurable, albeit with some uncertainty. Unfortunately, measurements made with the Canadian-built MOST space telescope indicate that transits do not occur.

The new research suggests that the rocky centres of super-Earths are unlikely to evolve into terrestrial rocky planets like the inner planets of our Solar System because they appear to hold on to their large atmospheres. Rather than evolving to a planet composed mainly of rock with a thin atmosphere, the small rocky core remains engulfed by its large hydrogen-rich envelope.

Orbit

The orbits of the Gliese 581 planetary system, as per the 2009 analysis excluding planets g and f. In the picture, Gliese 581 c is the third planet from the star.

Gliese 581 c has an orbital period ("year") of 13 Earth days and its orbital radius is only about 7% that of the Earth, about 11 million km, while the Earth is 150 million kilometres from the Sun. Since the host star is smaller and colder than the Sun—and thus less luminous—this distance places the planet on the "warm" edge of the habitable zone around the star according to Udry's team. Note that in astrophysics, the "habitable zone" is defined as the range of distances from the star at which a planet could support liquid water on its surface: it should not be taken to mean that the planet's environment would be suitable for humans, a situation which requires a more restrictive range of parameters. A typical radius for an M0 star of Gliese 581's age and metallicity is 0.00128 AU, against the sun's 0.00465 AU. This proximity means that the primary star should appear 3.75 times wider and 14 times larger in area for an observer on the planet's surface looking at the sky than the Sun appears to be from Earth's surface.

Tidal lock

Because of its small separation from Gliese 581, the planet has been generally considered to always have one hemisphere facing the star (only day), and the other always facing away (only night), or in other words being tidally locked. Although a recent orbital fit to the Gliese 581 system uses a circular orbit for Gliese 581 c,older fits use an eccentricity between 0.10 and 0.22. If the orbit of the planet were eccentric, it would undergo violent tidal flexing. Because tidal forces are stronger when the planet is close to the star, eccentric planets are expected to have a rotation period which is shorter than its orbital period, also called pseudo-synchronization. An example of this effect is seen in Mercury, which is tidally locked in a 3:2 resonance, completing three rotations every two orbits. In any case, even in the case of 1:1 tidal lock, the planet would undergo libration and the terminator would be alternatively lit and darkened during libration.

Models of the evolution of the planet's orbit over time suggest that heating resulting from this tidal locking may play a major role in the planet's geology. Models proposed by scientists predict that tidal heating could yield a surface heat flux about three times greater than the Jupiter's moon Io's, which could result in major geological activity such as volcanoes and plate tectonics.

Habitability and climate

The study of Gliese 581 c by the von Bloh et al. team has been quoted as concluding "The super-Earth Gl 581c is clearly outside the habitable zone, since it is too close to the star." The study by Selsis et al. claims even "a planet in the habitable zone is not necessarily habitable" itself, and this planet "is outside what can be considered the conservative habitable zone" of the parent star, and further that if there was any water there then it was lost when the red dwarf was a strong X-ray and EUV emitter, it could have surface temperatures ranging from 700 K to 1000 K (430 to 730 °C), like Venus today. Temperature speculations by other scientists were based on the temperature of (and heat from) the parent star Gliese 581 and have been calculated without factoring in the margin of error (96 °C/K) for the star's temperature of 3432 K to 3528 K, which leads to a large irradiance range for the planet, even before eccentricity is considered.

Effective temperatures

Using the measured stellar luminosity of Gliese 581 of 0.013 times that of our Sun, it is possible to calculate Gliese 581 c's effective temperature a.k.a. black body temperature. (note: this probably differs from its surface temperature). According to Udry's team, the effective temperature for Gliese 581 c, assuming an albedo (reflectivity) such as Venus' (0.64), would be −3 °C (27 °F), and assuming an Earth-like albedo (0.296), then it would be 40 °C (104 °F), a range of temperatures which overlaps with the range that water would be liquid at a pressure of 1 atmosphere. However, the effective temperature and actual surface temperature can be very different due to the greenhouse properties of the planetary atmosphere: for example, Venus has an effective temperature of 34.25 °C (307.40 K; 93.65 °F), but a surface temperature of 463.85 °C (737.00 K; 866.93 °F) (mainly due to a 96.5% carbon dioxide atmosphere), a difference of about 430 °C (770 °F). Studies of the habitability (i.e. liquid water for extremophile forms of life) conclude that Gliese 581 c is likely to suffer from a runaway greenhouse effect similar to that found on Venus, as such, is highly unlikely to be habitable. Nevertheless, this runaway greenhouse effect could be prevented by the presence of sufficient reflective cloud cover on the planet's day side. Alternatively, if the surface were covered in ice, it would have a high albedo (reflectivity), and thus could reflect enough of the incident sunlight back into space to render the planet too cold for habitability, although this situation is expected to be very unstable except for very high albedos greater than about 0.95 (i.e. ice): release of carbon dioxide by volcanic activity or of water vapor due to heating at the substellar point would trigger a runaway greenhouse effect.

Liquid water

Gliese 581 c is likely to lie outside the habitable zone. No direct evidence has been found for water to be present, and it is probably not present in the liquid state. Techniques like the one used to measure the extrasolar planet HD 209458 b may in the future be used to determine the presence of water in the form of vapor in the planet's atmosphere, but only in the rare case of a planet with an orbit aligned so as to transit its star, which Gliese 581 c is not known to do.

Tidally-locked models

Theoretical models predict that volatile compounds such as water and carbon dioxide, if present, might evaporate in the scorching heat of the sun-ward side, migrate to the cooler night side, and condense to form ice caps. Over time, the entire atmosphere might freeze into ice caps on the night side of the planet. However, it remains unknown if water and/or carbon dioxide are even present on the surface of Gliese 581c. Alternatively, an atmosphere large enough to be stable would circulate the heat more evenly, allowing for a wider habitable area on the surface. For example, although Venus has a small axial inclination, very little sunlight reaches the surface at the poles. A slow rotation rate approximately 117 times slower than Earth's produces prolonged days and nights. Despite the uneven distribution of sunlight cast on Venus at any given time, polar areas and the night side of Venus are kept almost as hot as on the day side by globally circulating winds.

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Dark Energy and Dark Matter

In the early 1990's, one thing was fairly certain about the expansion of the Universe. It might have enough energy density to stop its expansion and recollapse, it might have so little energy density that it would never stop expanding, but gravity was certain to slow the expansion as time went on. Granted, the slowing had not been observed, but, theoretically, the Universe had to slow. The Universe is full of matter and the attractive force of gravity pulls all matter together. Then came 1998 and the Hubble Space Telescope (HST) observations of very distant supernovae that showed that, a long time ago, the Universe was actually expanding more slowly than it is today. So the expansion of the Universe has not been slowing due to gravity, as everyone thought, it has been accelerating. No one expected this, no one knew how to explain it. But something was causing it.

Eventually theorists came up with three sorts of explanations. Maybe it was a result of a long-discarded version of Einstein's theory of gravity, one that contained what was called a "cosmological constant." Maybe there was some strange kind of energy-fluid that filled space. Maybe there is something wrong with Einstein's theory of gravity and a new theory could include some kind of field that creates this cosmic acceleration. Theorists still don't know what the correct explanation is, but they have given the solution a name. It is called dark energy.

What Is Dark Energy?

Universe Dark Energy-1 Expanding Universe-

This diagram reveals changes in the rate of expansion since the universe's birth 15 billion years ago. The more shallow the curve, the faster the rate of expansion. The curve changes noticeably about 7.5 billion years ago, when objects in the universe began flying apart as a faster rate. Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart.

More is unknown than is known. We know how much dark energy there is because we know how it affects the Universe's expansion. Other than that, it is a complete mystery. But it is an important mystery. It turns out that roughly 68% of the Universe is dark energy. Dark matter makes up about 27%. The rest - everything on Earth, everything ever observed with all of our instruments, all normal matter - adds up to less than 5% of the Universe. Come to think of it, maybe it shouldn't be called "normal" matter at all, since it is such a small fraction of the Universe.

One explanation for dark energy is that it is a property of space. Albert Einstein was the first person to realize that empty space is not nothing. Space has amazing properties, many of which are just beginning to be understood. The first property that Einstein discovered is that it is possible for more space to come into existence. Then one version of Einstein's gravity theory, the version that contains a cosmological constant, makes a second prediction: "empty space" can possess its own energy. Because this energy is a property of space itself, it would not be diluted as space expands. As more space comes into existence, more of this energy-of-space would appear. As a result, this form of energy would cause the Universe to expand faster and faster. Unfortunately, no one understands why the cosmological constant should even be there, much less why it would have exactly the right value to cause the observed acceleration of the Universe. 

Dark Matter Core Defies Explanation-

This image shows the distribution of dark matter, galaxies, and hot gas in the core of the merging galaxy cluster Abell 520. The result could present a challenge to basic theories of dark matter.

Another explanation for how space acquires energy comes from the quantum theory of matter. In this theory, "empty space" is actually full of temporary ("virtual") particles that continually form and then disappear. But when physicists tried to calculate how much energy this would give empty space, the answer came out wrong - wrong by a lot. The number came out 10120 times too big. That's a 1 with 120 zeros after it. It's hard to get an answer that bad. So the mystery continues.

Another explanation for dark energy is that it is a new kind of dynamical energy fluid or field, something that fills all of space but something whose effect on the expansion of the Universe is the opposite of that of matter and normal energy. Some theorists have named this "quintessence," after the fifth element of the Greek philosophers. But, if quintessence is the answer, we still don't know what it is like, what it interacts with, or why it exists. So the mystery continues.

A last possibility is that Einstein's theory of gravity is not correct. That would not only affect the expansion of the Universe, but it would also affect the way that normal matter in galaxies and clusters of galaxies behaved. This fact would provide a way to decide if the solution to the dark energy problem is a new gravity theory or not: we could observe how galaxies come together in clusters. But if it does turn out that a new theory of gravity is needed, what kind of theory would it be? How could it correctly describe the motion of the bodies in the Solar System, as Einstein's theory is known to do, and still give us the different prediction for the Universe that we need? There are candidate theories, but none are compelling. So the mystery continues.

The thing that is needed to decide between dark energy possibilities - a property of space, a new dynamic fluid, or a new theory of gravity - is more data, better data.

What Is Dark Matter?

Abell 2744: Pandora's Cluster Revealed-

One of the most complicated and dramatic collisions between galaxy clusters ever seen is captured in this new composite image of Abell 2744. The blue shows a map of the total mass concentration (mostly dark matter).

By fitting a theoretical model of the composition of the Universe to the combined set of cosmological observations, scientists have come up with the composition that we described above, ~68% dark energy, ~27% dark matter, ~5% normal matter. What is dark matter?

We are much more certain what dark matter is not than we are what it is. First, it is dark, meaning that it is not in the form of stars and planets that we see. Observations show that there is far too little visible matter in the Universe to make up the 27% required by the observations. Second, it is not in the form of dark clouds of normal matter, matter made up of particles called baryons. We know this because we would be able to detect baryonic clouds by their absorption of radiation passing through them. Third, dark matter is not antimatter, because we do not see the unique gamma rays that are produced when antimatter annihilates with matter. Finally, we can rule out large galaxy-sized black holes on the basis of how many gravitational lenses we see. High concentrations of matter bend light passing near them from objects further away, but we do not see enough lensing events to suggest that such objects to make up the required 25% dark matter contribution.

However, at this point, there are still a few dark matter possibilities that are viable. Baryonic matter could still make up the dark matter if it were all tied up in brown dwarfs or in small, dense chunks of heavy elements. These possibilities are known as massive compact halo objects, or "MACHOs". But the most common view is that dark matter is not baryonic at all, but that it is made up of other, more exotic particles like axions or WIMPS (Weakly Interacting Massive Particles). 

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