Trivia - A Google a Day XL

Having begun his artistic pursuits as a graffiti artist, what did Jean-Michel Basquiat often incorporate into his paintings?
Words

As of 2012, what is the only current NFL franchise with the letter X in its name?
Houston Texans

In Carl Sandburg's poem "Fog", what is used as a metaphor for the fog?
A Cat

What is the term used to describe a model of gaming console that is able to read the latest format of games as well as previous formats?
Backwards-Compatibility

When the property owner decided to build a home, he was persuaded by family members to call it "Arlington House." Who was his then deceased adoptive father, after whom he initially planned to name the house?
George Washington

How many Oscars were won by the actress who was elected the first female president of the Academy of Motion Picture Arts and Sciences?
2

What is the sum of the jersey numbers worn by the Lakers' Black Mamba in 2011 and the Celtics' Big Shamrock?
60

What was the maiden name of the mother of the second eldest Beatles' first child?
Powell
Also known as Cynthia Lennon, wife of John Lennon.

Similar to henges and monoliths, what ancient megalithic structures, thought to be for ceremonial purposes, have survived in modern times, with more than 1000 of them found across the British Isles?
Stone Circles

In addition to the Andes, what other influence causes great climate diversity in Peru, despite the fact that it is an equatorial country?
Humboldt CurrentHaving begun his artistic pursuits as a graffiti artist, what did Jean-Michel Basquiat often incorporate into his paintings?Words

As of 2012, what is the only current NFL franchise with the letter X in its name?
Houston Texans

In Carl Sandburg's poem "Fog", what is used as a metaphor for the fog?
A Cat

What is the term used to describe a model of gaming console that is able to read the latest format of games as well as previous formats?
Backwards-Compatibility

When the property owner decided to build a home, he was persuaded by family members to call it "Arlington House." Who was his then deceased adoptive father, after whom he initially planned to name the house?
George Washington

How many Oscars were won by the actress who was elected the first female president of the Academy of Motion Picture Arts and Sciences?
2

What is the sum of the jersey numbers worn by the Lakers' Black Mamba in 2011 and the Celtics' Big Shamrock?
60

What was the maiden name of the mother of the second eldest Beatles' first child?
Powell
Also known as Cynthia Lennon, wife of John Lennon.

Similar to henges and monoliths, what ancient megalithic structures, thought to be for ceremonial purposes, have survived in modern times, with more than 1000 of them found across the British Isles?
Stone Circles

In addition to the Andes, what other influence causes great climate diversity in Peru, despite the fact that it is an equatorial country?
Humboldt Current

Trivia - A Google a Day XXXIX

What advice columnist reported that one of the most unusual reader problems she encountered had been about a man who wanted to be buried in his 1939 Dodge?
Ann Landers

A famous essay contains the line "The grave in the woods is unmarked, but Fred can direct the mourner to it unerringly and with immense good will . . . ." Who or what is buried in the grave?
A Pig

In what 1864 publication did Spencer coin the phrase "survival of the fittest"?
Principles Of Biology

What is the most important river in the country for which "Homat el Diyar" is the national anthem?
Euphrates

Prior to the 2011 Davis Cup, who held the record for the fastest men's serve?
Andy Roddick

What is the antiparticle for an electron that has the same mass as that electron but has a positive charge?
Positron

In "Gulliver's Travels", how many countries does Jonathan Swift's main traveler encounter before discovering the land of horses?
8

How many costume changes, setting a Guinness World Record, were required of the star of Alan Parker's adaptation of Andrew Lloyd Webber's musical about the wife of the president of Argentina?
85

What monarchy has no written constitution, relying instead on rigid convention to dictate the relationship between the PM and the monarch as head of state?
Britain

Who created the main art piece for the 1945 film version of Oscar Wilde's novel about corrupted souls?
Ivan Albright

Differential Equations - Power Series

In order to solve a differential equation by the use of power series, we simply find the coefficients then attempt to match the created series to a maclaurin or taylor expansion of a known function. Let us try solving for the solution of the following differential equation.

y'' - xy' + y = 0

We substitue in for the differentials with a power series that represent the differential.

Which will give us the following equation

Now we set y to 0 to solve around the origin, which is an ordinary point in this case, as well as changing the indices from m to k. For the first term, we set 

m - 2 = k

For the second and third terms we just set

m = k

Giving us the equation

Now we need to match our indices so we'll write the first term where k = 0, then write the combined sum starting from k = 1.

Which we can then solve for the recursion relation.

Now that we have the recursion relation we can then solve for the first few coefficient terms in the solution.

C₂ = -1/2 C₀
C₃ = 0
C₄ = 1/12 C_{2 =} -1/24 C₀

Which gives us the following as the solution to this power series problem

Differential Equations - Euler Cauchy Equation

The euler cauchy equation is a method for solving differential equation with variable coefficients but only in a specific form. The form is that the equation must have the variable match the order of the differential next to it, so for example x³y'''. We then plug this into a polynomial and solve in order to find the coefficients to the general solution. Depending on the solutions of the roots of the polynomial we get different general solutions. Lets find the solution to the following equation.

Now lets take the information we need from this and plug it into the following equation which is the characteristic equation for an Euler Cauchy equation thats second order.

Where we get b and c knot from

So this means that if x² has some coefficient or didn't match this form, we'd have to adjust the equation to make the first term x². In this case though its fine, and we just have to find b(x) and c(x), which are

b(x) = -3

c(x) = 4

which means that our equation will be 

Which after factoring becomes 

So r is equal to 2 twice, which means its a repeated result, one of the possibilities. Now the general solution to the Euler Cauchy equation is x to the n where n is the solution, but in this case we have a repeated result. This means we need to add an additional factor of ln(x) into our solution giving us the following.

Which is our final result assuming there's no initial conditions.

Trivia - A Google a Day XXXVIII

What event in 1911 resulted in the demise of the Qing Dynasty, China's last imperial dynasty, and the creation of the Republic of China?
Chinese Revolution

Whose passing inspired Merle Kilgore to say "It's a sad day in Tennessee, but a great day in Heaven"?
Johnny Cash

What is the technical label of a plastic that can be remolded each time it is heated?
Thermoplastic

What is the name of the sense possessed by all humans that allows for the perception of pain from body organs, skin or joints?
Nociception

What is the oldest written masterpiece of English literature?
Beowulf

What is the birthplace of the only man who has won league MVP honors in both the National and American Leagues, as of 2012? (City)
Beaumont, Texas
The birthplace of Frank Robinson

One of the Massachusetts' signers of the Declaration of Independence had daughters name Nabby and Susanna. What was his alma mater?
Harvard
The signer is John Adams.

Which NBA player shares his name with a founding member of New Edition and a song on the album "Sheik Yerbouti"?
Bobby Brown

Who is the competitive eater who has been nicknamed after one of two poisonous spiders found in the U.S?
Sonya Thomas

Who gave up baseball's 1951 "Shot Heard 'Round The World"?
Ralph Branca

Differential Equations - Constant Coefficients

This will go over the process of solving simple differential equations with constant coefficients by using the characteristic equation. The basic idea is to replace the order of the differential with a polynomial that has the same power and coefficients as the differential. The solution of this polynomial, depending on the sign or type of number the solution is gives us the coefficients to the general solutions we already know for this type of differential equation. Lets go over the following example

We can replace this with

r³ - 3 r² + 6 r - 18 = 0

We then solve this equation in order to find the coefficients to a general solution.

r² (r - 3) + 6( r - 3) = 0
(r - 3) ( r² + 6) = 0
r = 3, +- (6)^1/2i

So now that we solved this equation, we can note that theres a 3. This means that the coefficient of an exponential is 3

y = c₁e^3x

The complex solutions result in a different way, resulting in sin and cosine solutions. This can be proven using complex analysis and substituting in using euler's equation.

Resulting in a final combined solution of this

Which is in a form we can use in order to completely solve if we have initial conditions to find the coefficients c₁, c₂ and c₃. Lets use the example that at y(0) = 1, y'(0) = 1, y''(0) = 1. This would mean that

After taking the derivative, then the double derivative, and plugging in for x = 0. This gives us a system of equations which we can solve by taking the reduced row echelon form of it.

Giving us a solution for all the coefficients and a final answer of

Trivia - Google a Day XXXVII

At the conclusion of the 2010-2011 NFL season, what team had made the most Super Bowl appearances, taking home six wins?
Pittsburgh Steelers

In 1928, a bacteriologist made a chance discovery from a contaminated Petri dish that had been discarded. What did he discover that would change the world?
Penicillin

In November 2011, French police seized 300 tons of illegal souvenirs, most of which were miniature versions of what structure originally built as an entrance to the 1889 World's Fair?
Eiffel Tower

The only place in the world where you'll find Emperor penguins also boasts some of the driest spots on earth. What kind of winds, named for the Greek "going down", are responsible for the condition?
Katabatic Winds

Who convinced Nathaniel Hawthorne to develop the cautionary tale of Hester Prynne into a novel instead of the originally intended short story?
James Thomas Fields

In 1980 who was Atari's biggest rival due to the release of the Intellivision platform?
Mattel

What novel by Fyodor Dostoyevsky is considered by many to be the first existentialist novel?
Notes From Underground

A major U.S. airport is named for the attorney who was instrumental in the victory in Brown v. Board of Education, and who then went on to serve in the nation's highest court. By what three-letter moniker is this airport known?
BWI
Baltimore-Washington International Thurgood Marshall Airport

Though the NBA won't confirm or deny whose silhouette was used in its logo, of what team was this player assumed to be the model a member?
Los Angeles Lakers
The player in the picture is Jerry West.

What is the nickname of the area located east of Coney Island Beach and west of Manhattan Beach in Brooklyn?
Little Odessa

Series - Radius of Convergence

The radius of convergence is the radius of the largest disk in which the series converges. There are a few methods for finding the radius of convergence. The first condition for convergence is that the last term in the series must go to 0. We could test this by just taking the lim u_n  as n goes to infinity, but this isn't enough to determine convergence.

The easiest test is to make sure that the series converges fast enough and to do that we can use the ratio test.

This limit is the absolute value of the last term in the series over the second last term of series, and if this value is < 1 it converges, if its > 1 it diverges, and if this limit is equal to 0 then the test fails. 

We can use the ratio test to determine the radius of convergence in the following fashion, by equating it to the ratio test to the radius of convergence and solving it.

For the following sum

We just start by plugging into the equation

And this solves to 4 when we take n going to infinity, giving us the radius of convergence. If the absolute value of x is greater than this, it doesn't converge.

Trivia - A Google a Day XXXVI

What island kingdom, a British protectorate until 1971, is located off the coasts of Saudi Arabia and Qatar?
Bahrain

What did Southern political leaders in Washington, D.C. threaten if Abraham Lincoln were elected president in 1860?
Secession

What World Series-bound team had players dubbed "The Nasty Boys"?
Cincinnati Reds

Who tried to assassinate the man who would eventually become the driving force behind the Alaska Purchase?
Lewis Powell

The actor who played Sean Parker in "Social Network" rose to fame as the lead singer of a band that was formed in Orlando and launched where?
Germany

The confluence of the largest tributary and the largest river in the U.S. occurs just south of what city?
Cairo, Illinois

What form of torture do the aliens who blow up Earth use on their victims in the first book of Douglas Adams' six volume series?
Reading Poetry

A professional tennis player set a record in by serving up 1477 aces during the 1996 season. What is the capital of the country where he was born?
Zagreb
The holder of the record is Goran Ivanisevic who was born in croatia.

Which one of the Beach Boys became involved in Transcendental Meditation, one of the first pop musicians to do so?
Mike Love

On what body of water did Ernest McSorley lose the "Mighty Fitz"?
Lake Superior

Paper - Magnetic Fields in Star Formation

Magnetic Fields in Star Formation

A star’s evolution begins and ends in the interstellar medium, the dust and gas that exists between the stars. At the beginning of this cyclic process of star formation where dust and gas collapse and compress together due to gravity. However, if gravity always draws things together, why isn’t everything a star? Why isn’t the universe mostly stars and planets rather than dust and empty space? This is because on average the distance between two particles in the interstellar medium is too far, and so on average the dust is too diffuse for star formation to occur. Only through turbulence and random chance can the dust group up enough to begin coagulating into a star.

The first requirement for star formation is if there is enough mass in a given area to form a star. This minimum mass over an area is called Jean’s criterion, which models the interstellar medium as a uniform, stationary cloud of gas that acts under only gravity and ideal gas pressure. Given these assumptions we can use the Virial theorem to model the collapse of the gas. The virial theorem tells us that for a closed system in equilibrium, the following is true.

So we can see that on average the total potential energy of every single particle in the system is equal to twice the kinetic energy in the system. So from this stability assumption we can assume that if the potential energy of the system exceeds twice the kinetic energy, it will begin to collapse in on itself as gravity links more and more mass together. With a bit of substitution we get the equations

M is the minimum mass to trigger collapse, and R is the minimum radius for that this mass must be contained in. Now we can envision a new process that has to occur just from thinking about this equation. As the density increases the jean’s mass decreases, which means as the object collapses less mass is needed to collapse, and greater density increases the potential energy of gravity. This gives rise to the process of fragmentation where, the collapse of a dust cloud is a runaway process in which denser areas of gas will collapse faster. This means that not all the material in an area gets used, the more dense and massive a cloud, the smaller fragments and thus smaller stars are formed from the gas cloud.

Another property we can deduce from the above equations is the average velocities of the particles in the cloud. <T> represents average kinetic energy which is equal to ½ m <v>2. This gives us

Which is the root mean squared speed of the particle, which gives us an average speed of a collapsible gas cloud of about 500 m/s. So just by making the simplest of assumptions we already have a lot of information and approximations for star formation. We have a rough idea of the density and size necessary to form a star, and we have some idea of the structure of it by knowing the velocity of the particles within it. However, it obviously lacks many other important aspects which contribute significantly and change the criteria of this basic model including: the initial velocity of the the cloud, radiation transport through the cloud, ionization, rotation, and magnetic effects. In particular the magnetic field plays a very important effect in preventing or triggering collapses.

Troland and Heiles in 1986, found that the magnetic field strength in the beginning stages of collapse is on the same order as the gravitational and kinetic energy of the molecular cloud. Although we might be able to guess that it has a strong effect because em fields are stronger than gravitational ones, we have to realize that electromagnetic field strength goes down as r2 which should be extremely significant due to the large distances we deal with in molecular clouds. So its not inherently obvious that em fields can play such a huge role. However, it does, and we know this because of the Zeeman effect, which allows us to detect the strength of magnetic fields from a distance.

The Zeeman effect is the splitting of electron energy levels in an atom due to the presence of a magnetic field. This separates a solid emission line into 3 distinct lines, and the distance between these lines is proportional to the strength of the field. In particular the 21 cm line produced by hydrogen, and the 18 cm line produced by OH are often measured because their wavelengths are large enough to penetrate out through the clouds of interstellar dust. Troland and Heiles measured these lines using a method called stellar spectropolarimetry, which takes advantage of the polarization of e-m fields in order to measure the strength of the electric field. These measurements led to the discovery of the magnetic field strength in molecular clouds.

Now that we know the strength of the field, we can now try to characterize what they do. It serves two main functions, the first aids in supporting or collapsing the cloud, the second is changing the distribution of mass in a cloud. Lets start by analyzing the first function. We can think of the magnetic field lines as distributed among the particles in the gas. Since the fields are linked to the particles, when the gas tries to collapse in on itself, the field lines come together. We know from lenz’s law that in e-m fields resist changes, so this increases the magnetic pressure in the system which resists compression due to gravity. If we include this to the Jean’s mass criterion it rises to

Where cB is 380 N1/2m-1T-1. We call clouds that are above this new Jean’s mass magnetically supercritical, and clouds that are below this mass magnetically subcritical.

However, this idea has some issues. If we use this to predict star formation, the initial mass function would be much higher than it is observed to be. The initial mass function is the distribution of masses of newly formed stars since, and since the magnetic field here is treated in a way where it only resists collapse, the mass function rises.This means we most likely have a problem with our assumption that the magnetic field lines are linked to the position of the particles in the clouds. This is supported by the fact that newly formed stars would contain a very high magnetic field as the contributions add up. However, recent theories indicate that there is a way to account for this magnetic field loss, so the assumption may not be entirely untrue.

One of these theories is ambipolar diffusion. The logic behind this is that although the interstellar medium is mostly neutral, there are still ions. These ions must travel along the path of the magnetic field lines in the ISM. So the ions can be thought of as a stationary grid which slows down the neutral particles that collide into it, letting the magnetic field absorb the energy transferred to the ionic particles instead of acting as an elastic collision. This means that more mass will coagulate around magnetic field lines.

In addition since the position of the ions are fixed in space, even though the neutral mass is collapsing down, the ions star relatively close to where they are and are linked to the magnetic field. This means that the magnetic field doesn’t get drawn in nearly as much as we thought in the original model where the field can gets dragged in uniformly with the collapse of the dust cloud. And accounts for the weaker magnetic fields of new stars, since the density can increase without drastically increasing magnetic flux. This effect also helps explain the high field strength in the interstellar medium because its a function of all the ions left over after star formation, so star formation doesn’t reduce the magnetic field strength in the surrounding area.

So now we have a good explanation right? Unfortunately it still has some issues, one of the major problems is that the effectiveness of this process would be dependent on the metallicity of the medium, since the process is dependent on the number of ions that can link up to the magnetic field. This does not match observations, which seem to be independent of metallicity. It was also shown by Troland and Heiles in 1986 that ambipolar diffusion is too slow of a process to remove the magnetic field from a gas if the gas is diffuse, and Shu et. al 2006 showed that ambipolar diffusion was too slow for dense gases, so it doesn’t account for either extreme, only mid range cases.

However, since the results match for many general cases, ambipolar diffusion is still a reasonable theory to follow. It could mean that it is a main process for mid range molecular clouds, and that we only need to change the model for fringe cases, or the data could be matching out of chance and we require a different explanation.

Another explanation that has become more popular recently is magnetic reconnection. In this model, we take into account that the gas is turbulent. So the magnetic field lines are not stationary but are warping and twisting. The field lines are simply an indication of flux, but it is a good idea to think about them as actual lines in this case because magnetic reconnection occurs when field lines “touch”. This means that huge amounts of magnetic flux are passing through the same area which results in a huge release of energy. This is the same process that happens in a more familiar event, the solar flare. In either case, magnetic reconnection that happens in a short time frame is capable of quickly releasing a lot of energy from the magnetic field associated with those field lines, which can account for the field loss rate depending on the rate of reconnections or touches that occur due to turbulence.

Lazarian and Vishniac were the ones who calculated the timescale for this process by solving the incompressible magnetohydrodynamic equations using fourier methods. They got the solution for the magnetic field by solving for the field in a box sized to the fundamental wavelength of the B-field, then repeat that over all space, while meeting boundary conditions. The equations are listed as follows.

∂v/∂t= (∇ × v) × v − (∇ × B) × B + ν∇2v + f + ∇P′ (1)

∂B/∂t= ∇ × (v × B) + η∇2B (2)

∇ · v = ∇ · B = 0 (3)

where f is the driving force, P′ ≡ P/ρ + v · v/2, v is the velocity over r.m.s. velocity, and B is the Alfven speed over the r.m.s. velocity. The real space components of time t is in units of the large eddy turnover time (∼ L/v) and the length in units of L, the inverse wavenumber of the fundamental box mode.

The magnetic reconnection model better accounts for the loss of magnetic field in a molecular cloud mainly because its much faster in those fringe cases of high and low density, and is not dependent on metallicity, rather on turbulence, and there is enough random motion in the ISM to account for the number of reconnections necessary. This gives us a more accurate inital mass function. However, this model has a major problem in that it predicts tangled field lines in the cores of molecular clouds, which contradicts observations that show that the B field in molecular clouds are regular, thats how we were able to polarize and observer the additive effect.

There now exists research into models that try to combine the two effects together, in which both ambipolar diffusion and magnetic reconnection are accounted for. However, these models are difficult to create because magnetic reconnection by itself already accounts for the reduction rate of the magnetic field in this stage of stellar evolution. Also by definition these two processes counteract one another since the stronger the ambipolar diffusion effect, the less turbulence can occur since the ionic grid slows down the particles, which makes it difficult to model due to feedback loops.

In conclusion, more research into this field is necessary in order to create a model that matches observations. Ambipolar diffusion explains many cases, and matches observations of regular field lines. Magnetic reconnection models match initial mass functions more accurately and over a wider range of star formation scenarios. Some new formulation, either involving both processes or a new process altogether must be found in order to accurately match all the data gathered.

Trivia - A Google a Day XXXV

In 2009 Tareq Salahi and his wife crashed a White House dinner hosted by the Obamas. In whose honor was the dinner given?
Prime Minister Of India

So hush, little baby, don't you cry is a line from an American standard from what opera?
Porgy And Bess

Despite its small size and relatively simple composition, which of Leonardo da Vinci's paintings is among the most popular and recognizable in the world?
Mona Lisa

What constellation houses a star in its northern region that shares a name with a satellite radio company?
Canis Major

What adaptation of a classic play disturbs The Savage with its violent and salacious interpretation in Huxley's "Brave New World"?
Shakespeare's Othello

Josephine Goldman and Edward Marr were indicted in Philadelphia in May 1937 for abducting a man for Ms. Goldman to marry. According to an article in the Gettysburg Times on May 27, under what new law were Marr and Goldman charged?
Lindbergh Law

This law lets federal authorities step in and pursue kidnappers once they had crossed state lines with their victim

What river is the only non-tidal river that crosses the U.S. mountain range that is divided into a Northern, Central and Southern section, which includes the prolongation of the Blue Ridge?
New River

What is the nickname of the man who coached the Miami Heat's 2011-2012 #31 on a college team?
Coach K

Through the 2010-11 season, how many of the NHL teams that have won La Coupe Stanley are now defunct?
5

What batting honors did an AL right fielder earn in 1966?
Triple Crown

Paper - Pulsar Formation and Structure

Pulsar Formation and Structure

            Even a star cannot live forever. Over time, the fuel of the hydrogen core fuses into helium, the helium can be fused into carbon, and even heavier elements. These changes in the fuel of the star, causes drastic changes in the star’s structure and is called the process of stellar evolution. The most common end stage of this process is the white dwarf, a low mass, high density star that emits very little radiation. These stars no longer undergo nuclear fusion and are supported by electron degeneracy and will eventually radiate away its energy and cool down until it can no longer be detected, vanishing into the darkness of space.

How does degeneracy support a white dwarf from collapsing under its own mass? Degeneracy pressure is caused by the Pauli exclusion principle which says that two fermions cannot occupy the same quantum state, meaning they can’t have the same quantum number. Some examples of fermions include protons, neutrons and electrons which have antisymmetric wavefunctions. This property limits the number of particles you can stuff into a given volume because they cannot share an energy state, imposing a limit to the proximity of the particles. This means that white dwarfs and other objects in a degenerate state cannot be compacted down any farther.

This paper will describe the formation of another type of degenerate object, pulsars, which are a type of neutron star. As the name suggests neutron stars are neutron degenerate objects, so instead of being packed so tightly that the electrons can’t be packed any closer they are limited by the proximity of their neutrons. These stars can be formed by certain types of supernova.

The first way a supernova can form neutron stars is by having a white dwarf find a companion star that’s less dense than itself, such as a main sequence star. Due to its greater density, it can draw mass away from the less dense companion star. We can explain this ability to remove mass from another object by both Newtonian and Relativistic physics. Overall it boils down to the greater density more sharply curving spacetime around the white dwarf, or by fringe material on the companion star escaping its original gravitational pull.

However, despite giving us an acceptable reason Newtonian physics fails to give us the appropriate rate of mass loss and gain from the companion star to the white dwarf. This process is called mass accretion and allows a white dwarf to destabilize itself by exceeding the Chandrasekhar mass limit set by:

where ω03 is a constant derived from a from the lame-emden equation, µe is the average molecular weight of the star, mH is the mass of hydrogen, and mp is the planck mass. This formula gives us a maximum weight of approximately 1.4 times the mass of the sun, which is the upper stability limit for a white dwarf. Normally the white dwarf doesn’t have an ongoing process that increases its mass and can destabilize itself. However, if a companion star is present, this cannot occur, and the white dwarf will collapse. Electron degeneracy pressure will no longer be able to support it and so the star pretty much implodes as gravity wins out over its outward pressure. This happens so fast that the mass gets superheated and undergoes runaway nuclear fusion resulting in a supernova.

Another way a supernova could occur is if the star was simply extremely massive. High mass stars burn hotter and can progress through enough cycles of stellar evolution that it can fuse much heavier elements. This allows them to naturally exceed the Chandrasekhar limit at the end of their stellar lifetimes instead of reaching a proper white dwarf phase and undergo a supernova without the aid of a companion star.

Either way, a supernova has the potential to form a new celestial body. The leftover matter at the center of these explosions is extremely condensed, which can form a neutron star. It can reach this only if the mass lies between the Chandrasekhar mass and the Tolman - Oppenheimer - Volkoff mass limit, the upper bound of a white dwarf and the lower bound of a black hole.

Now we shall delve into the structure of these interesting stars, starting with the equation of state. These tell us the pressure, the density, and the temperature of the star as a function of radius. Currently, there aren’t consistent formulations for the equations of states. Some models include APR EOS, UU, EOS FPS, and L, all of which come up with different mass predictions. An example of one of the possible equations of state for pressure is

The integral over the mass is a volume integral adjusted by the central density rho to give the mass as a function of radius. The factor nabla shows that this is a general relativity equation as the factor 2Gm/c2 factor is known as the Schwarzschild radius, the radius of a black hole.

This term is commonly used in general relativity, and we can use the Schwarzschild metric here to show us some interesting properties that occur on the surface of a neutron star.The Schwarzschild metric is

By taking some average values for a neutron star, 1.5 solar masses for its mass, and 10 km for its radius, we get the Schwarzschild radius to be roughly 4.4 km. This makes the factor 1-rs/r roughly .66 which we can see will have a significant factor in warping the space around the neutron star. This means that at the surface of the neutron star, the actual distance is ~1.2 times the coordinate distance! This means that if you were able to walk on the surface of a neutron star, in order to walk 10 meters from an observer’s reference frame, such as a scientist floating in a spaceship, the walker would need to walk 12 meters in his own frame!

The exact way in which to incorporate this spacetime curvature into an equations of state for a neutron star is currently unknown and warrants further research. The main contention between the different models, is on the application of general relativistic corrections, so its clear that we can gain a greater understanding of quantum gravity from the creation of more accurate neutron star models through data gathering and analysis.

Another structural feature to look at is the neutron star’s temperature. In particular it is important to model the cooling process. Immediately after a supernova, a neutron star must be immensely hot, on the order of 10^11 kelvins. We know from observation that it can’t dissipate this heat simply through simple radiation of heat, so there needs to be explanations on methods to quicken this rate of heat loss. In addition, neutron stars often undergo periods of quiescence where it turns on/off its x-ray emission spectrum. The speed at which it turns off also requires explanation, and one of the most common explanations is the Direct URCA process. This is when the star emits neutrinos via the processes:

This cools the star because neutrinos don’t interact very strongly with matter, they can simply pass through and leave the core of the neutron star, carrying away with it large amounts of energy very quickly. This process occurs at a rate many orders of magnitude quicker than cooling by radiation. This means that a neutron star actually cools from the inside out rather than from outside in, which is a very interesting side effect of this process. Another cooling method is the modified URCA process in which 

helps to explain the timeframe of long term changes in temperature of a neutron star, its overall cooling and eventual darkening. It was shown by Lattimer et. al in 1991 that both of these methods matched up with gathered data and so these processes are heavily looked at in neutron star research. However, the strength of the contribution depends on the possible temperatures at the core of the neutron star during formation. The direct urca process predicts a cooler core than external temperature which is a very odd process to have happen and goes against the typical star structure in any sequence of stellar evolution. In addition not only do we not know about the temperature, we don’t even know for certain what kinds of matter exists at the core of a neutron star.

For now the most accepted structure of a neutron star is as follows. The surface of the star is a solid lattice formed of degenerate electrons and heavy nuclei such as iron. Below this is an inner crust that contains a lattice of heavier nuclei such as krypton, and superfluid neutrons and electrons. It is around here that neutron drip occurs. Neutron drip is when a neutrons tunnel out of the nuclei to become free neutrons. Normally a free neutron would decay into a proton, however, since the star is neutron degenerate, there is no lower energy state to occupy, so instead of decaying into a proton and its antiparticle, it stays a neutron and tunnels out of the energy well, becoming a free floating neutron, then undergoes its decay. This forms a layer in the star of free neutrons and protons located below the inner crust, and below this is the unknown core.

That fermi sea though, causes many of the interesting properties associated with neutron stars, including the ability for it to become a pulsar. Not only are the particles in that area a superfluid they are also superconductors. Superfluidity is a state of matter where a liquid behaves as though it has no viscosity, and has infinite thermal conductivity. Due to the property of infinity thermal conductivity a superfluid is also isothermal. What this means is that since it has an infinite ability to distribute heat, its not possible to locally heat a superfluid, so it always has the same temperature throughout the entire liquid. Superfluidity also allows a liquid to maintain equilibrium in containers regardless of gravity, allowing for fluids to crawl up or down the outside surfaces of its container. Also turbulence that is self contained, like a vortex last indefinitely inside a superfluid since it has no friction.

The neutron star’s sea of protons, neutrons and electrons eventually behave like a superfluid because the degenerate neutrons can pair together to form a boson. Boson’s do not have to obey the pauli exclusion principle and can be in the same state at the same time, which allows them to pass freely through one another. This gives it even more freedom than a gas as it has a viscosity of 0, but is still considered a liquid due to the distance and bonds between the particles as a whole.

In addition a neutron star’s sea of particles is also superconductive which means that it has no electrical resistance, and can maintain currents indefinitely. Only lenz’s law can slow down the current in a superconductive material.

These properties heavily affect the structure of the neutron star. Since the sea of particles has some charge due to protons and electrons, once the movement of this sea becomes uniform, the magnetic field lines they form will become constant, making the surface which is also charged solid. The particles on the surface are degenerate electrons and heavy nuclei, which means it is purely ionic and its position is heavily fixed due to the magnetic field lines created by the rotating superfluid.

This effect seems to be akin to the effect of ambipolar diffusion in star formation. In that case, the dust of the Interstellar medium has ionic particles that couple to the magnetic field lines of a molecular cloud, which slow down the movement and helps in the accretion of mass around those field lines. This lends credence to the idea of a solid surface on a neutron star because the effect of the superfluid causes a magnetic field many times stronger than that of ambipolar diffusion so the increasing of density and slowing of movement should also increase.

When these fields are strong enough, they can force the electromagnetic radiation of the star to exit only at the poles of the star resulting in a pulsar. The only way for them to get this strong is if the superfluid has enough angular momentum through formation, but where does it come from?

It relates all the way back to the formation of the original star. In the giant molecular clouds that formed the original star. In order to shrink some of that gas down gravitationally, assuming there was some initial rotation, or turbulence in that cloud, angular momentum has to be conserved. This is why stars rotate, such as our sun, and why they have a magnetic field. The ambipolar diffusion previously mentioned is a process that explains how the rotation can be slowed by clumping of material and removal of momentum through magnetic fields. Though again ambipolar diffusion is its own topic with its own controversies.

However what holds true is that as you decrease the size of the object and pull in more particles together, the rate at which it spins has to go up. Once the rest of the star is blown away and only the compacted matter is left, the speed at which it rotates is enormous because it went from a huge cloud of dust to a few kilometers compounded with the moment of inertia calculated as r2. Neutron stars have been recorded to have a period of rotation on the order of a millisecond. Not only that it will maintain this rotation very accurately due to the superfluid nature underneath the crust, no friction to slow down the process.

However there are some strange events that can disrupt this regularity. Sometimes a pulsar will “ glitch” or “kick”. Some speculate that this is caused by asymmetric supernova, others think that there are starquakes in which the solid crust compacts, as deformities caused by landing meteorites or crust compacting as pressure towards the surface goes down, causing quakes on the surface of the star. That quake can then severely disrupt the period of rotation for that star as the motion will remain in  the superfluid. Forces continue endlessly as there is no friction disrupting its motion, so even the smallest events could have major consequences.

In conclusion there are still a lot of unknowns pertaining to the pulsar. In particular we don’t know the composition of the core, or have a definitive explanation for anomalies or “kicks” in the pulsar, or a reliable equation of state for a neutron star, which makes it very difficult to model. However, further research into this area will help further our understanding of both general relativity and quantum mechanics and possibly how to reconcile the two.

Trivia - A Google a Day XXXIV

After immigrating to England from Ireland as a young boy, what author recalled "the strange English accents with which I was surrounded seemed like the voices of demons"?
C. S. Lewis

At the end of the NBA 2010-2011 season, who still held the record for the most rebounds in an NBA game?
Wilt Chamberlain

Who directed the film in which Jane Fonda's dad received his only Academy Award for Best Actor?
Mark Rydell

What game was featured in the 1989 film for which Jessica Tandy won an Oscar?
Mahjong

The fictional town of Macondo is inspired by the birthplace of the author. In which Colombian city was this author born?
Aracataca

What town eventually sprang up as the result of the 1866 discovery of a "pretty pebble" along the banks of the Orange River?
Kimberley

The Canal Street Ferry will take you over to Algiers Point. Since when has this ferry been in service?
1827

What was the birthplace of the U.S. president featured on the $.08 stamp issued by the postal service in 1973?
Stonewall, Texas

The inner ring of the Schrodinger basin is the same distance across as which South Dakota glacial lakes and prairie region trail?
The Northeast Trail

According to the publisher's review of Diane Preston's book, "Lusitania", the event was both a pivotal moment in history and a remarkable what?
Human Drama

Trivia - A Google a Day XXXIII

What 4000 mile river is not crossed by a bridge at any point?
Amazon

What is often the subject of paintings by Mary Cassatt?
Mother & Child

The Yeomanettes of World War I were the first women to serve in the U.S. military. In what branch did they serve?
Navy

The remnants of what pluvial lake form the greatest part of the largest saltwater lake in the Western Hemisphere?
Lake Bonneville

What is the name of the sense possessed by all humans that allows for body awareness and body position?
Proprioception

In 1906, Charles Henry Baker hired a cook for his family at their rented summer home on Long Island. After an ongoing investigation by the health officials, by what nickname did the cook come to be known?
Typhoid Mary

Because of its incredibly slow metabolism, two-thirds of what animal's body weight is likely the contents of its stomach, which can take up to a month to completely process through the digestive system?
Sloth

The star on the Hollywood Walk of Fame located at 6325 Hollywood Boulevard belongs to the aunt of what Academy Award-winning actor?
George Clooney

The annual NHL trophy that is the oldest and considered the most prestigious is named for the father of a former coach of what NHL team?
Montreal Canadiens

Covering more than 2200 acres in a national forest in the U.S. Pacific Northwest, what is the Latin name of this organism, which is considered to be one of the largest in the world?
Armillaria solidipes

Gaming - League of Legends - Map Control

Importance of Map Control

One of the easiest ways that to differentiate the overall skill of a team in League of Legends is the amount of map control and presence they are able to establish. When you start the game this is an example of what you are able to see on your minimap.

Obviously if you started in the upper right hand corner your vision would flipped, but the amount of vision you have is the same. The goal of the game is to extend your vision beyond that of your opponents, more important than gold, more important than kills, more important than buffs, vision is what determines the current state of the game assuming the players on each team are of roughly equal skill level. Although those other things are definitely important, they determine the potential you have to change the state of the game in your favor, rather than an indication of who is winning. 

What Gives you Map Control

The reason for this is because those towers not only represent map vision but a threat. Map control we can define as a secure visible location, and a location is secure if it poses a significant threat to your enemy. In the beginning of the game, a champion cannot do much to a tower so the map control is very solid. If a level 1 champion tries to attack your tower, or you underneath the range of protection of your tower, in the first 2 minutes of the game where there aren't any minions supporting them, they're going to die. However, as the game goes on longer and longer, these towers become less and less of a threat to enemy champions, which means as time goes on the map control the towers provide gets weaker. 

The tower always poses a significant threat against minion waves though, and more importantly stops them from advancing as long as its there. This means turrets also deny your opponents map presence.

Now what else is a threat that give you vision? Each one of the champions you and your team controls are a form of map control. They give you vision and pose a threat to the enemy champions. Thus your control over an area is determined by how strong your champions are in relation to the enemy champions.

Map Presence

Map presence we'll define simply as an area we have vision of, and can quickly convert into map control. The speed at which you can convert it into map control. Lets take an example of what I mean by speed of conversion to map control. 

If you notice in this scenario, the champions are all spread out to their respective lanes with a roaming jungler. Although in this case this it is the very early stages of a game, this picture will look pretty similar for most of the laning phase. This is because by having the each person split up you can get more things done as well as convert as much of your map presence to map control, by converging on a location. The time to convergence tells us how much of a map presence is available in that area, so in this stage of the game everything is spread out evenly and held in check. 

This is different from a later stage of that same game, where the champions group up to take objectives or make plays. In the following picture you can see that the champions are grouped up in top and bottom lane. 

Because there are team fights going on in top and bottom lane, although they have vision of the mid lane, there isn't much of a presence there. The strength of your map presence is determined by the proximity of your movable threats to the location of visible areas. So even though there is still vision, its not exuding a presence or a force.

The same thing occurs when there is a team fight going on. If lets say the above situation was going on except there was one champion from the lower left team in mid. Because the entire opposing team is occupied, they can no longer convert their map vision into map control, which means they don't have map presence in the center of the map.

Although this may seem basic, it gives us a way to categorize the advantages we seize in game more accurately since we will be talking about how to gain map control or presence.