Strangeness oscillations

Hello again! On the off chance that you're not sick of the neutral kaons yet, here's yet another post on how awesome they are. Aside from hinting at parity violation in the weak interactions, providing the first evidence of CP violation, and generally confusing physicists, they also have this really neat behavior called strangeness oscillations.

The first hint of these is that by second-order weak interactions (a Feynman diagram for one of these is shown below. Feynman diagrams are awesome and certainly deserve a future post, but I won't go into how to read them at the moment. Note, however, that I use the convention used in my current quantum course, with time on the vertical axis), a $K^0$ can transform into a $\overline{K}^0$, and vice versa. We can identify a particle as either a $K^0$ or its antiparticle based on a couple of decay modes. For example, the $K^0$ has quark content $d\overline{s}$, so it's easy for it to decay into a negative pion and a positive lepton (like the positron or the positive muon), along with the correspondingly flavored neutrino. On the other hand, the $\overline{K}^0$ has quark content $\overline{d}s$, so it preferentially decays into a positive pion and a negative lepton, along with an antineutrino. These allow us to experimentally determine the $K^0$ content of a beam of neutral kaons.

One of the two lowest order
Feynman diagrams in the
$K^0\leftrightarrow \overline{K}^0$ oscillation.

As I've previously mentioned, neutral kaon decays are determined by the CP eigenstates $K_1$ and $K_2$. What I hadn't mentioned is that because these two states are their own antiparticles, the CPT theorem (which states that the combined operation of C, P, and T transformations should yield a universe identical to our own) allows them to have distinct lifetimes and masses. The lifetime effect we've already seen, in the form of the quick $2\pi$ decay of the $K_1$ state and the longer-lifetime $K_2$, which decays instead into $3\pi$. The mass difference, even though it ends up being absolutely tiny, still turns out to be really important to the system. It means that the two states have slightly different energies, which means that they have slightly different time evolutions (the time evolution of a system is a complex phase determined by its energy. This can be easily derived from the Schrödinger equation). If we start off with a pure $K^0$ beam, such as by interacting $K^+$ mesons with matter (the interactions are strong, so they preserve strangeness and generate only $K^0$), we can view it as a combination of $K_1$ and $K_2$. These evolve slightly differently in time, so if we switch back to the $K^0$/$\overline{K}^0$ states, we find that the probability of detecting a $K^0$ at a time $t$ oscillates a bit. This oscillation (with appropriate experimental constants like lifetime and mass differences) is shown in the graph below, and it matches experiment beautifully. The strangeness of a neutral kaon beam can change over time, and it's all predicted by elementary quantum mechanics. Neat, eh?

The strangeness evolution of an originally pure $K^0$ beam. The black
shows the probability that a given member of the beam is a $K^0$, and the
gray shows the probability that it's a $\overline{K}^0$. The units on the
time axis are $10^{-10}$ seconds.

Neutral kaons

As I've previously mentioned, the K mesons have a glorious history in messing with particle physicists. The discovery of the K mesons and $\Lambda$ baryons was a puzzle for physicists. Here were particles that were produced in strong interactions but decayed with lifetimes more than a thousand times those typical of strong decays. Instead, the decays were by weak interactions. This, of course, begs the question: why can't the K and $\Lambda$ decay strongly? To explain the phenomenon, physicists invented the concept of strangeness, and stated that strong and electromagnetic interactions had to conserve it. So once a particle was produced that had nonzero strangeness, the only way it could decay in the absence of other strange particles to interact with was by the weak force, which explained the extraordinarily long lifetimes (on the order of a nanosecond instead of a femtosecond) of the strange particles.

Strangeness is an additive quantum number, so a combined state including one particle with -1 strangeness and another with +1 strangeness (like the $K^+ + \Lambda^0$) has a total strangeness of zero. Therefore under charge conjugation, a particle's strangeness switches sign. In other words, a particle and its antiparticle have opposite strangeness.

Nowadays, with our knowledge of quarks, we say that a strange quark $s$ has strangeness -1, and an anti-strange quark $\overline{s}$ has strangeness +1. Previously, these were arbitrary designations in which the $K^+$ (quark content $u\overline{s}$) had strangeness +1 and other particles' strangeness values were determined by examination of strange production processes. Anyhow, enough about strangeness - let's get to the interesting bit!

Neutral K mesons (or just kaons) are produced in strong and electromagnetic interactions, usually by pair production of strange quarks, which produces a meson along with a strange hadron. Strangeness is conserved in this interaction, so the two possible neutral K mesons to be produced have quark contents of $d\overline{s}$ ($K^0$) and $\overline{d}s$ ($\overline{K}^0$). This is in sharp contrast to the neutral pion ($\pi^0$), which is a superposition of two states $u\overline{u}$ and $d\overline{d}$. Since these states are indistinguishable by any quantum number, the neutral pion is its own antiparticle. The distinguishing feature here is that the $K^0$ and $\overline{K}^0$ have different values for strangeness.

The real weirdness kicks in when the particle decays. It decays in a weak interaction, so strangeness is not conserved. This means that the particle's strangeness eigenstate ($K^0$ or $\overline{K}^0$) is irrelevant to the decay mode. Instead, what matters is the particle's CP eigenstate, since CP is (mostly) conserved in weak interactions. But of course the strangeness eigenstates are not also CP eigenstates. What can we do? Examine the two superpositions of strangeness eigenstates that are CP eigenstates! We know that the charge conjugate of the $K^0$ is the $\overline{K}^0$, and vice versa. It turns out that both strangeness eigenstates have odd parity, so we find that the eigenstates of CP are superpositions of $K^0$ and $\overline{K}^0$ with either a plus or minus sign between them. The even CP eigenstate is called $K_1$ (creative name, right?), and the CP-odd eigenstate is called (wait for it...) $K_2$.

Any state involving neutral kaons can be expressed as a superposition of the strangeness eigenstates or the CP eigenstates. In particular, if a bunch of $K^0$ mesons are produced, we can view them as half
$K_1$'s and half $K_2$'s. Or a bunch of $K_2$'s can be viewed as half $K^0$'s and half $\overline{K}^0$'s. It's a matter of which basis you choose to express the state of the system.

In weak decays, the CP eigenstate is (mostly) conserved, so $K_1$ has to decay to a CP-even final state. In particular, it decays into two pions. By similar logic, $K_2$ can decay into three pions. As a result of something called phase space that determines the probabilities of various decays, the $2\pi$ decay is about a hundred times faster than the $3\pi$ decay. So the two CP eigenstates have distinct lifetimes. That means that if we start with a pure $K^0$ beam, the $K_1$ component decays away very quickly in a flurry of $2\pi$ decays, so a short while later, only $K_2$'s are left.

There are some philosophical implications to the distinction between CP and strangeness eigenstates. In particular, what is a particle? Is it the thing produced in a strong interaction, or the thing that decays away in a weak interaction? Is it the thing with a definite lifetime, or the thing with a definite production cross-section? In the end, though, these are primarily questions of philosophy rather than physics, and it makes little difference which states you define as particles.

Oh, and another thing. Not content with introducing strangeness and violating parity conservation, the kaons were also the first proof of CP violation. If CP were conserved, the strangeness and CP eigenstates of the neutral kaon would be all we needed to analyze the (somewhat bizarre) system. In that case, after all the $K_1$'s had decayed away, we would expect to see only $3\pi$ decays of $K_2$'s. Instead, Cronin and Fitch observed that a beam of pure $K^0$, after traveling over fifty feet (plenty of time for all the $K_1$ components to decay away), around one in five hundred of the decays were in fact $2\pi$ decays. That means that the long-lived neutral kaon (which gets yet another name, $K_L$, for K-long) is actually a mixture of both $K_2$ and $K_1$, with substantially more $K_2$. And that means that like parity, CP is not a perfect symmetry in weak interactions. Another score for kaons in that fascinating game of Confuse the Physicists.

CP symmetry

A natural thought upon noticing the parity-violating interactions (Co-60 beta decay, neutrino helicity, and the theta-tau puzzle, for instance) is that if instead of applying only a parity transformation, we instead apply a parity transformation followed by a charge conjugation (switch particles for antiparticles), we might get the same laws of nature. So for instance reflecting a neutrino in a mirror results in a neutrino with the opposite helicity, which we never see in nature. But if we also change it to an antineutrino, then the helicity is as needed for an antineutrino. Similarly, if we both look in a mirror and use anti-cobalt-60, then the positrons (anti-electrons) are emitted in the same direction as the nuclear spin, so it seems that CP symmetry should hold.
Unlike P-symmetry, which is maximally violated in weak interactions, CP symmetry is mostly true (and appears to be exact in strong and electromagnetic interactions), and it took a while for physicists to find an interaction in which CP was violated. The first (indirect) evidence for CP violation came from the neutral kaon system, which I'm sure I'll write extensively about in the future. Since then, neutral kaons have yielded direct CP violation, as have B and D mesons, so it's a well-established fact that CP is not an exact internal symmetry.
This has a few interesting implications. For one thing, CP violation is able to favor matter over antimatter, which can at least partially explain why the universe we live in appears to be made entirely of matter.* Basically in the Big Bang, since the forces were primarily electromagnetic and strong forces, CP should have been an exact symmetry, which means that for a while, at least, the universe had exactly as much matter as antimatter. The question then becomes what allowed the universe to evolve into predominantly matter, and CP violation says that through weak interactions, matter and antimatter can be distinguished and preferentially generated.
CP violation also means that we can finally absolutely define the positive charge, whereas previously we simply had to stick with a convention based on rubbing glass and silk together. We can say that positive is the charge of the lepton preferentially emitted in the decay of the neutral K meson (again, I'll explain more about these fascinating creatures in the nearish future).
Another interesting implication is that quantum field theory predicts that the combined CPT transformation should be an exact symmetry in all interactions. Remember that T is a transformation that reverses the direction of time. The derivation of that idea is way beyond me, but it's apparently based only on the basics; things like special relativity and the idea that only local properties can affect an interaction. And since CP is violated, if the CPT theorem holds, then T must be violated as well. T violation is a lot harder to detect than C, P, or CP violation, because we're pretty much stuck in a universe moving forwards in time, but physicists are diligently looking for things like a non-zero electric dipole moment of the neutron as we speak, which would imply T violation and allow a confirmation of the CPT theorem.

*If the universe had portions consisting of matter and portions of antimatter, we would expect that the interface would have an abundance of annihilation, which would appear as a bright layer without necessarily many stars around.

Parity!

As promised, here's a short discussion of parity transformations. As I mentioned previously, P symmetry was once thought to be a fundamental symmetry of all interactions in particle physics. In essence the parity transformation is an inversion of all three spacial coordinates (x, y, and z). That's equivalent to running a mirror image of the universe. Like with charge conjugation (C), applying the parity transformation twice gets you right back to where you started, so the only possible eigenvalues of parity are +1 and -1.
Based on just this primitive analysis, we already know some things about the operation: normal vector quantities like position are inverted under a P transformation, so they have eigenvalues of -1 under P. Other quantities like angular momentum, which is defined as a cross product of two "normal" vectors, are unchanged under parity, because both component vectors change sign. These are called pseudovectors (or axial vectors), and have eigenvalues of +1 under parity transformation. Then, just to make things weirder, there are normal scalars (which are unaffected by parity and thus have eigenvalue +1) and pseudoscalars (like the dot product of a vector and a pseudovector, which is inverted under parity transformation).
Particles can be assigned parities according to their interactions with other particles. This process is somewhat arbitrary, but by convention we choose that quarks have positive parity. Then every strong and electromagnetic interaction conserves parity, which is a multiplicative quantum number. (It turns out that apart from intrinsic parities of particles, the net angular momentum of a multi-particle state also contributes to the state's parity, but we can safely ignore that for now.) This allows us to assign each particle either positive or negative parity.
Feynman described parity symmetry as the indistinguishability of right from left - if we were communicating with some sort of extraterrestrial beings but couldn't actually meet them and parity were conserved, it would be impossible to explain what is right and what is left. Well, luckily for our pen pals, it turns out that strong and electromagnetic interactions preserve parity quite nicely, but weak interactions violate parity in a major way. There are a couple of textbook examples of this:

• Beta decay of cobalt-60: This was the first experimental test of parity conservation in weak interactions. Basically scientists (Wu et al.) polarized the spins of cobalt-60 atoms in a magnetic field. Then they watched the cobalt and measured the direction in which electrons were emitted in beta decay. Astonishingly enough, they observed more electrons emitted in the same direction as the nuclear spin than in the opposite direction. The strange thing about this is that if you look at the experiment in a mirror universe, the nucleus is spinning the opposite way, but the electrons are emitted in the same direction! The mirror universe inherently behaves differently!
• It turns out that all neutrinos are left-handed, and all anti-neutrinos are right-handed. This is a striking violation of parity symmetry: if you look in a mirror, you would see right-handed neutrinos, which don't exist on our side of the mirror! Sheesh!
• The theta-tau puzzle: Once upon a time, particle physicists found two seemingly similar particles, the $\theta^+$ and the $\tau^+$. They were produced in similar reactions, had nearly identical masses, spins, and so on, but their decays were different. The $\theta^+$ decayed into two pions (a $\pi^+$ and a $\pi^0$), while the $\tau^+$ decayed into three (either $\pi^++\pi^++\pi^-$ or $\pi^++\pi^0+\pi^0$). Well, the pions all have negative parity, so by parity conservation, it seems that the $\theta^+$ has a parity of $+1$, while the $\tau^+$ has a parity of $-1$. This baffled physicists for a while, but eventually, when they figured out that parity was violated in weak interactions, they concluded that the $\theta^+$ and $\tau^+$ were actually the same particle, now known as a positive kaon ($K^+$). 

Well, that's at least an introduction to parity. I'm still trying to wrap my head around the whole concept, but the bottom line is that parity is squarely not conserved, which led physicists to their next best guess, CP conservation - a topic for another day.

Binding energy!

First of all, apologies if this page/post takes a while to load. I've chosen to use more than my usual number of equations because I think the math is moderately enlightening in this case.

Atomic nuclei are an incredibly interesting topic. While they seem straightforward (they just have mass and charge, right?), they are in fact very difficult to describe from pure theory, because quantum field theory is hard. A particularly interesting thing to examine is the curve of binding energy per nucleon. This plots the binding energy per nucleon compared with the total number of nucleons (protons and neutrons) in each stable nucleus. What this essentially describes is how energetically favorable it is for that many nucleons to form a nucleus as compared with just existing independently. A positive value for the binding energy means that it takes less energy to stick all of the protons and neutrons together than to keep them separate. What that translates to is the bizarre fact that a nucleus weighs less than the total mass of its constituent parts!

(Thank you to Mathematica's IsotopeData for the help in plotting all this)

The elements with very high binding energies per nucleon are the stablest. The king of these is iron-56, which is located conveniently at the top (the red dot in the graph above). To the left of that, fusion is profitable from an energy perspective. To the left, fission is energetically favorable. The binding energy curve is really important for nuclear physics.

There's this great thing called the semi-empirical mass formula, which is physicists' attempt to fit the curve of binding energy to a nice function. Then they go back after the fact and explain each term in terms of the physics it represents. It's a little hokey, to be sure, but who am I to talk? The formula is this:
$B.E.=a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{\left(Z-\frac{A}{2}\right)^2}{A}+\frac{a_5}{\sqrt{A}}\left(\begin{array}{c}1\\0\\-1\end{array}\right)$
Before you panic, that fifth term isn't really a vector. It's a sort of piecewise function that I'll explain in due course. $Z$ is the total number of protons in the nucleus, and $A$ is the total number of nucleons (protons and neutrons - an excellent example of isospin, which I'll have to save for another post).

The first term, $a_1 A$, is the strong force volume term. Basically nuclei are held together by the strong nuclear force, without which they seriously wouldn't exist. It has a positive contribution to the binding energy, because the greater the volume, the more the strong force pulls inwards. The volume is proportional to $A$, because of course the more nucleons you have, the greater the volume, right? In this case, that model (called the liquid drop model) seems to work quite well, but with quantum mechanics and field theory in play, it's important to remember that nothing is ever exactly what you think it should be.

The second term, $a_2A^{2/3}$, is the strong force surface term. In essence, it's a correction to the first term, which assumed that all nucleons were in the middle of the nucleus and experienced forces all around them. Instead the nucleons on the outside (think surface area here) experience less strong force, since it's adjacent to fewer other nucleons, so this term has a negative contribution to the binding energy.

The third term, $a_3\frac{Z^2}{A^{1/3}}$, is called the electrostatic term. It has a negative contribution from packing all that positive charge into a small space (seriously, nuclei are tiny!). The form of the term (proportional to charge squared and inversely proportional to the radius of the sphere) is derived from basic electricity and magnetism.

Once you get to the fourth term, quantum mechanics kicks in. What this term expresses is the desire of a stable nucleus to have around the same number of protons and neutrons. This is as a result of the way in which protons and neutrons fit into "orbitals," just like electrons. They're fermions, so two protons can fit in a single energy level, and two neutrons can fit in a single energy level. Because protons and neutrons are identical to the strong nuclear force (again, an isospin discussion will have to happen at some point), their energy levels are approximately the same, so it is energetically favorable to have close to the same number of protons and neutrons, or the unhappy neutrons (if there are more of them) are forced into higher energy levels. What the fourth term describes is a negative contribution to the binding energy as the balance between protons and neutrons is skewed.

Now we get to that crazy fifth term, the one that looks like a vector. As promised, it's a piecewise function. The column-vector like part is one if there are an even number of protons and neutrons in the nucleus (an even-even nucleus). It is zero for an even-odd or odd-even nucleus, and negative one for an odd-odd nucleus. This one seems like a real mystery at first; why should the nucleus care whether the number of protons or neutrons is divisible by two? The answer, as before, comes from the spin of the proton and neutron. Because there are two spin states (up and down), two protons fit into each energy level. The same goes for neutrons. And it turns out it's a lot more energetically favorable to have full energy levels than half-full ones. So when there are an even number of both protons and neutrons in the nucleus, the fifth term increases the binding energy (thus decreasing the mass of the nucleus). The opposite occurs for odd-odd nuclei.

There's a sixth term that accounts for gravity. Its coefficient is so tiny that it can be quite safely ignored when dealing with chemical elements, but it is really handy for things like neutron stars, which are essentially enormous nuclei held together by the strong force and gravity.

Particle decay

I mentioned resonances in a previous post. They're really neat, so I wanted to write a little more about them. For one thing, the particles observed solely as resonances almost invariably decay by the strong force, which has a shorter range and faster decay time than the electromagnetic or weak interactions. In other words, we can often observe electromagnetic or weak decays as particle tracks inside a detector, because the particle has time to travel a measurable distance before it decays.
I mentioned in my previous post that based on certain properties of the scattering peak, we can determine the particle's lifetime. In particular, the peak is at a range of energies, which means there's a certain amount of uncertainty in the energy of the particle, which is called the decay width. Then by using the energy-time uncertainty relation, we can calculate the lifetime of the particle (the little uncertainty in time in which it is permitted to exist).
When a particle can decay in a variety of ways, a number called the total decay width describes the probability that the particle will decay in any unit of time. In particle physics, the probability that a particle decays along a certain path is called the branching ratio. These phrases are bandied about quite a bit, and it took me until just recently to finally figure out what they really meant. It's neat how much the knowledge of just a couple of phrases can deepen the understanding of a vast field like particle decays.

Symmetries and conservation laws

Symmetries and conservation laws lie at the heart of all physics. Newtonian mechanics can be summarized by conservation of energy, conservation of momentum, and $\vec{F}=m\vec{a}$, which describes how individual objects interact. But what is a symmetry, and where do conservation laws come from, anyway?
A symmetry can be viewed as a transformation that doesn't affect the way the world works. For instance, translational symmetry states that if you pick up the whole universe and move it one foot to the left, all the laws of physics remain the same. Another way of saying the same thing is that there is no such thing as absolute position; we can only measure the position of one object with respect to another. As a direct result of this symmetry, it can be (relatively) easily shown that linear momentum must be conserved.
Other similar symmetry-conservation correspondences include invariance under translation in time (start the whole universe five seconds later (kind of), and it works the same way) and the conservation of energy and invariance under rotation, which corresponds to the conservation of angular momentum.
These are all linked together by Noether's theorem, one of the most beautiful constructs in modern physics, which states that every symmetry corresponds to a conserved quantity.
Then there are the approximate symmetries. You find a lot of these in particle physics. They are symmetries that are mostly true. Often strong and electromagnetic interactions are symmetric, while weak interactions don't fully obey the symmetry. The most commonly mentioned ones are charge conjugation symmetry (C), parity transformation (P), and time reversal (T). All of these are discrete symmetries, which means they can be applied an integral number of times, in contrast to continuous symmetries like rotation and translation.
Charge conjugation is a transformation which when applied to a system causes all charges in the system  to invert. Just to confuse you, the charges that are reversed include not just electric charge but also things like strangeness, baryon number, and lepton number. Therefore under charge conjugation, every particle is transformed into its antiparticle. Note that applying a C transformation twice transforms the system back to its original state. In math speak, this means that the eigenvalues of charge transformation (which are called charge parity) are +1 and -1. The so-called eigenstates of C transformation are particles which are unchanged under a C transformation. These are their own antiparticles. All other particles are not eigenstates of C, and have antiparticles distinct from themselves. Other consequences of charge conjugation include the reversal of the directions of electric and magnetic fields, since these are caused by the presence and motion of electric charges, respectively.
I'm still a little confused by parity and time reversal symmetry, so I'll plan to write a bit about those sometime in the near future.

Awesome experimental physics

Once upon a time, there was a theory called Einstein's theory of general relativity. It made a lot of predictions about how the world works, and some of them were a little tricky to verify. So a group of physicists set out on what can only be described as an extraordinary experimental journey. They called it Gravity Probe B, and it was designed to measure two general relativistic effects: the warping of spacetime around the Earth, and the dragging of nearby spacetime along with the Earth's rotation. These required an unprecedented degree of accuracy (the frame-dragging effect apparently resulted in a precession of less than 40 milliarcseconds over the course of a year!), and produced some fantastic detectors. 

The essence of the experiment is that a telescope and four gyroscopes (for redundancy - theoretically only one is required) are aligned on a guide star. As the spacecraft (did I mention this was an experiment in orbit? Talk about complicated mission control) orbited the Earth, the telescope stayed aligned on the guide star, and the gyroscopes rotated happily onward without changing their alignment - relative to the spacetime around them. Then by measuring the angle between the telescope and gyroscopes' alignments, the experimenters could deduce properties of the warped spacetime around the Earth.

One of the many tricky bits of this experiment is that the gyroscopes had to be pretty much perfect. Any imperfections would cause them to precess differently, which would result in systematic errors in the final analysis. They also needed to be suspended stably without any friction, while still allowing for readout. The solution to the first problem was to manufacture the most perfectly spherical manmade objects ever - fused quartz balls with less than 25 nanometers of variation from the tallest peak to the deepest valley. The group doing the experiment actually had to create a whole new polishing method in order to attain this, and the gyroscopes still hold the record for the most spherical manmade objects in the world. The solution to the suspension problem was to coat these quartz balls in a thin layer of superconducting niobium. This was suspended by an electrical casing that never actually contacted the gyros, which continuously adjusted the power to six electrodes according to their simultaneous readout to keep the gyro centered in the casing. 

Gyro readout is another interesting issue the group had to deal with. First off, there's the issue of reading out the direction of the spin axis of a perfectly uniform sphere with no distinguishing markings. Then there's the issue of doing so without interfering with the sphere's rotation. Both of these are nontrivial technical challenges, and both are solved by strange properties of superconductivity. There's this phenomenon called the London moment, in which a spinning superconductor produces a magnetic moment along its spin axis. Gravity Probe B needed to measure that without generating a magnetic field to cause a torque on the magnetic moment, and they needed to do it with the remarkable precision mentioned above. Once again, superconductivity came to the rescue. There's this fancy thing called a SUperconducting Quantum Interference Device (because who doesn't like SQUIDs?) which is able to measure tiny magnetic fields over time. And certain feedback mechanisms in the circuit were able to cancel any magnetic field change caused by the readout, so the gyros could spin happily away completely undisturbed. 

In the end, Gravity Probe B was a successful mission, and it came up with spectacular agreement with theory. Another win for Einstein.

X-ray Binaries

I've previously mentioned binary stellar systems. They're pretty neat. But like good physicists, let's make them even neater by taking one of the stars to its logical extreme. Say it's massive - more than a few solar masses. It goes through the normal stellar evolution process and eventually becomes either a neutron star or a black hole. If we look in the rotating frame of the binary, the potential energy (from a combination of gravitational potential and centrifugal potential as a result of doing physics in a noninertial frame) behaves roughly as we would expect. There are deep wells where the stars live, and the potential energy drops off as you move further away. The interesting bit is the saddle point in between the two stars. It's a sort of tipping point - a little bit of mass near the saddle falls into the closer well.
Interesting things start to happen when a star extends past this saddle point. It's called exceeding its Roche lobe, and it means that the material past the saddle point starts to spiral in towards the other star (in our case, a very compact object). Because the mass already has some angular momentum, it is forced to spiral inwards instead of just falling. This creates an accretion disk, with lots of mass swirling around the compact object. As it does so, it falls in slowly, just based on viscous effects as particles run into each other.
The radiative efficiency of accretion disks is stunning. For comparison, the efficiency of hydrogen fusion is 0.7% (that's 0.007). That means that 0.7% of the mass energy of the hydrogen atoms is converted to heat/radiated energy. But for accretion disks, the efficiency can be as high as 40%, depending on rotational speed, inner radius of the disk, and the nature of the compact object (neutron stars can have way higher-efficiency accretion disks because of surface effects). That's almost two orders of magnitude more efficient than fusion. Wow.

Rotating black holes

I've previously written a little about black holes - they're pretty strange. It turns out that rotating black holes are even weirder.

First of all, there's a hard upper limit on how fast a black hole can rotate (more specifically on how much angular momentum it can have), past which the equations stop making physical sense. At this angular momentum, the event horizon surface of the rotating black hole is substantially smaller than the event horizon for a correspondingly massive non-rotating black hole, which is a strange concept on its own. 

Also, while non-rotating black holes have a point-like singularity at their heart,* in order to allow any sort of rotation, the singularity at the center of a rotating black hole is instead an infinitesimally thin ring with finite radius.

Then there's the ergosphere, which is an ellipsoidal region outside the Schwartzchild radius in which material is forced to rotate at the same rate as the black hole itself. This is because the so-called 'fabric' of spacetime is being dragged along with the black hole's rotation, at a rate faster than the speed of light to an outside observer! That means that at the outer edge of the ergosphere, it is possible for a photon to be traveling along the edge at the speed of light relative to spacetime, but appear stationary to an observer watching from outside. Weird, eh?

* This is a result of general relativity (as are most things to do with black holes). Basically inside the Schwartzchild radius of a black hole, space becomes timelike. There are a few ways of looking at that. The most simplistic one (not to say it's not the most correct - I'm not claiming to be a GR expert) is that time in the normal universe goes only one way. You don't have any choice in what direction you go through time, or how fast you do so - you are guaranteed to experience one second per second (in your local coordinates, in any case). Similarly, once you pass the event horizon of a black hole, the extreme curvature of spacetime means that there's only one way you can go: inwards. And with the amount of mass black holes contain, it is believed to be impossible to maintain enough outward pressure to support any spacial extent of the mass inside. It has to all condense into a single point at the very center, which is called a singularity. At this point, quantum gravity hits, and we have no clue what is going on.

Black holes

Black holes, apart from being a favorite topic of sci-fi writers, are physically very cool objects. The basic principle is that it's a blob with an escape velocity so high that even light can't escape it. In other words, since no information can travel faster than the speed of light, we cannot possibly know anything about the interior of a black hole. The 'interior' is determined by what's called the Schwarzschild radius, or the event horizon. This is the point of no return - once you hit that point, whether you're a person or a bit of information, you're in the black hole and not coming out.
A fascinating consequence of the lack of information exiting a black hole is what's called the 'no hair theorem', which says that black holes have only three observable properties: mass, angular momentum, and charge. Since any sort of charge imbalance out in space inevitably balances out very quickly, we can ignore that case. So black holes really just have mass and angular momentum.

Oh, and another interesting fact about them:
 Nonrotating black holes have this thing called a photon sphere, around fifty percent further away from their centers than the event horizon, at which photons can orbit the black hole at the speed of light. That's a direct result of general relativity, which states that mass and energy bend spacetime. Light takes the straight path in this curved space time, so heavy objects can noticeably bend light. Imagine that, an orbiting photon!

Three Quarks for Muster Mark

I'll be the first to admit this isn't a physics fact as much as historical trivia, but I think it's interesting nonetheless.
I recently checked out a book called The Quark Model, edited by J. J. J. Kokkedee. I didn't realize until I opened it and read the preface that it was published in 1969, when quarks were still a cute idea rather than a confirmed part of the Standard Model. In any case, this book also has Murray Gell-Mann's original paper proposing quarks. I was surprised to find it was less than two pages long, at least half of which was entirely intelligible to the moderately-educated reader (minimal quantum field theory knowledge necessary for a general understanding, in any case). Furthermore, I'd previously heard that quarks were named after a phrase from Finnegan's Wake (three quarks for Muster Mark), but it's not every day that you see a work of literature cited in a particle physics paper. Sure enough, though, reference six of eight in this groundbreaking article is to James Joyce's Finnegan's Wake. It's nice to see that physicists (and the people who publish their papers) have a sense of humor. It made me smile, in any case.

Crazy magnetic fields

I'm feeling a little under the weather at the moment, but here's a random fun physics fact.

Both the Earth and the Sun have magnetic fields of around a hundred microtesla. Refrigerator magnets have a slightly higher magnetic field of around a millitesla. The strongest maintainable magnetic field created in a lab is around 45 tesla. In contrast, a neutron star creates a magnetic field on the order of $10^7$ tesla. That is a very strong magnetic field.

White Dwarfs and the Chandrasekhar Mass

I've previously written a bit about evolution of main sequence stars once they leave the main sequence, but what I didn't mention is that basically any star lighter than a few solar masses will eventually peter off into a white dwarf star. These are 'stars' that no longer produce heat. They are entirely at equilibrium between gravitational contraction pulling in and electron degeneracy pressure pushing out. The only reason we can see them at all is that they still radiate some of the thermal energy left over from the days when they were still undergoing fusion. As they do this, they cool and dim, until eventually they are (pretty much) unobservable. These compact bodies will exist pretty much forever, unless other objects interact with them.
We know that these stars are entirely supported by degeneracy pressure, which we can quantify in terms of density and composition of the star. We can also calculate the necessary pressure in the center of the star to avoid collapse, which primarily depends on the radius and density of the star. Setting these equal allows us to derive the mass-radius relation for white dwarfs, which describes the preferred radius of a white dwarf with a given mass. It turns out that in general, for stars in which the electrons are nonrelativistic, $MR^3$ is a constant. That seems a little strange at first - normally we expect stars to increase in size when they get more mass. What it's saying is that if the mass of the white dwarf increases, its radius has to decrease in order to increase the density enough to provide the necessary supporting degeneracy pressure. If we keep adding mass, the electrons pack in tighter and tighter, which means they have higher and higher energies. Eventually our assumption that the electrons are nonrelativistic starts to break down, and we end up with a new expression for degeneracy pressure in ultrarelativistic white dwarfs. (This means the electrons are moving close to the speed of light, not that the star itself is careening off through space.) In this expression, the degeneracy pressure depends less on the density of the electrons in the star. When we try to use it to find a new and improved mass-radius relation, we find that with a given mass, a star has no preferred radius! The star is now unstable, and if we compress it a little, the extra degeneracy pressure isn't enough to combat the compression. It instead starts an unstable collapse until the protons in the nuclei start to capture electrons. As a result, the whole thing becomes a neutron star.
Luckily this doesn't happen for all white dwarfs, just the ultrarelativistic ones. And a white dwarf can only go ultrarelativistic if it has a sufficiently high mass, called the Chandrasekhar mass, of around 1.4 solar masses.
Okay...we now have a neutron star supported by neutron degeneracy pressure. The pressure equation takes the same form as for electrons, so the mass-radius relation (for nonrelativistic neutron stars, anyway) is the same - $MR^3$ is a constant, albeit a somewhat different constant than in the electron case. But once again, if the neutron star passes a certain mass (probably around 3 solar masses), the neutrons start to go ultrarelativistic, and we have an unstable collapse again, this time into a black hole.
We have some (very) fuzzy ideas about what the inside of a neutron star is like - I've heard rumors of neutrons being simultaneous superfluids and superconductors, but it's still very much an active area of research. Keep your eyes peeled!

Optical fibers

I discussed total internal reflection briefly yesterday, and aside from allowing us to see quantum tunneling on a macroscopic scale, I also mentioned that a neat application is optical fibers. They turn out to be both interesting and practical. In essence, the way they work is by having a thin core surrounded by a layer of cladding material, which has a lower index of refraction than the core. As long as the light is mostly directed along the fiber, when it encounters the core-cladding boundary, it undergoes total internal reflection and stays inside the core. It bounces back and forth along the inside all the way from one end to the other, with pretty high efficiency. An interesting issue in the most easily-manufactured type of optical fiber is that different photons come into the fiber at slightly different angles, and thus end up with different path lengths as they travel through the fiber. This produces diffusion of the transmitted signal. But there's another form of fiber optic cable, called single-mode optical fibers, in which the core is small even compared to the wavelength of the propagating light. In this case, quantum mechanics comes into play, and only certain modes of light are permitted to propagate, which reduces or eliminates the diffusion.

And look, physics is useful! Optical fibers can be used for a variety of purposes. One you hear about a lot is the use of optical fibers for long-distance communication. That's because they transmit light, which tends to travel fast (though from a quick Google search it looks like the speed of light is around 35% lower in optical fibers than in vacuum), with very high efficiency and can operate over long distances. This is used in telephone communication, internet connection, etc. There's also a whole field based on the technology, with textbooks and conferences and papers up the wazoo. Outside of that, those cool light toys that people play with are based on optical fibers, and that's why it seems like the light is only present at the very end. But of course any damage to the fiber can produce a point at which the total internal reflection doesn't entirely work, so you'll see a spot of light there. They're also really useful as detectors, especially because they require no power and aren't affected by strong electric or magnetic fields. It's often a convenient way to get a measurement and propagate it away from the experiment site before data processing.

Frustrated Internal Reflection

If we have a large piece of material like glass and light (or any sort of electromagnetic radiation, really) is traveling through it, if the beam is close to perpendicular to the surface of the material, it refracts off of the surface on exiting. This is the phenomenon that causes the distortion you see of a straw inside a glass of water.
If, on the other hand, the light is far enough from perpendicular to the surface, it undergoes what's called total internal reflection, and none of the light gets out of the material. It's a well-known phenomenon, and among other things, it's what makes optical fibers work so nicely. But that's not the interesting physics, at least not for now. Rather, it turns out there's a way to frustrate the internal reflection by a quantum effect.
Basically the way this works is that the electromagnetic waves propagating through the material don't end exactly at the surface of the material. This is a result of how particles behave when they encounter a barrier - it's a quantum effect. Instead, there is a non-negligible amount of this radiation within a few wavelengths of the surface outside. If you bring some of the same material close enough to the surface, the electromagnetic wave residing there see it and can start propagating through the other material. It's amazing, if you think about it - normally, no light escapes the first block of material. But if you have some more glass nearby, the light can tunnel through the gap and continue propagating!
Unfortunately this is a little tricky to observe with glass and visible light, as the wavelength of visible light is so small that the second hunk of glass has to practically be in contact with the first, which makes the tunneling somewhat difficult to observe. But if you instead use microwaves instead of visible light and polyethylene instead of glass, the gap can be as large as a couple of centimeters. Quantum tunneling on a macroscopic scale - awesome, eh?

Stellar clusters

So it turns out this week is a little crazy for me, so here's another short post on stellar clusters.

In general, a stellar cluster is a group of stars that are close together. There are two kinds - open and globular clusters, and while they may look similar, they're actually completely different. Open clusters are groups of young stars like the Pleiades. They formed from the same cloud of the interstellar medium, and since they're relatively young, they're also pretty high in metals (remember, for astronomers, any atom heavier than helium is a metal). It turns out, though, that because of the way fragmentation and the resulting star formation work, open clusters are only clusters for a short period of time. They're not gravitationally bound, so after a few million years, they spread out, and there's no more evidence that a cluster even existed. Chances are good that the sun started out as a member of an open cluster, the stars of which spread out into their current configuration.
Globular clusters, on the other hand, seem to have formed around the same time as galaxies, so the stars in them are very old (on the order of billions of years) and therefore very low in metals. They're gravitationally bound, as well, which is why we still see them as clusters today. And where open clusters consist of ten to ten thousand stars, globular clusters have more like a million stars in them.
The handy thing about both kinds of clusters is that the stars in them formed around the same time, which allows researchers to get very accurate information on their composition and age.

Helioseismology

So apparently helioseismology is actually a thing. Basically the sun, a large blob of gaseous material, has modes of oscillation. The first is called solar breathing, and involves the sun expanding and contracting periodically. In essence, if the sun is slightly larger than its equilibrium size, then gravity can overwhelm the pressure forces, and it contracts. But then of course it overshoots and contracts too much, at which point pressure pushes it outwards again. Similarly, the second mode involves the sun starting slightly pancake-shaped. It then expands towards a sphere, overshoots into a moderately conical blob, and then rebounds to a pancake shape. There are a ton of these modes, and helioseismology is the study of what the modes can tell us about the internal structure of the sun. It's a really neat field, and has a seriously cool name to boot.

Supernovae, continued

Whoops, I meant to post this yesterday. Here's a belated post about the awesomeness of supernovae.

When last we saw our friendly massive star, its core had just collapsed into a perfectly spherical blob of neutrons. The question remains, however - what has been happening to the rest of the it? A chunk of stellar material, in general, is kept in equilibrium by a balance of gravity and higher pressure towards the star's center. Now that the core is suddenly orders of magnitude smaller, the nearby layers suddenly 'see' no supporting pressure. As a result, they begin to collapse inward, falling faster and faster as they near the center.
Apart from being very spherical and incredibly dense, the neutron core of the star is also pretty much incompressible. Push on it as hard as you like, and the degeneracy pressure of the neutrons will keep you from making a dent. So when the outer layers fall inwards and hit the core, there's nothing for them to do but bounce back outward. They are assisted in their outward journey by the immense flux of neutrinos generated in the electron capture phase of the core's development.
As the core collapses into a perfect sphere of neutrons, it liberates around $1.6\cdot 10^{46}$ Joules. That's more than a hundred times the amount of energy the sun produces in its entire life. Wow. Furthermore, the majority of this energy is in the form of neutrinos, which are some of the least-interacting particles around. Just to give you a sense of how little they react, a standard neutrino can be expected to pass through more than a light year of lead without interacting with anything in it.
Around 5% of the neutrinos are absorbed by the material in the star. Around 4% are absorbed by the core (which just goes to show again how ridiculously dense it is), leaving just 1% of the neutrinos to create the entire supernova explosion we see.
Even so, supernovae are incredible. A typical brightness for a supernova is the same as the brightness of an entire galaxy. The last supernova in our galaxy, which reached our eyes in the early 1600s, shone so bright that it was visible in the daytime, and outshone all the visible stars and planets except Venus.