Astronomy

Are there neutron stars whose magnetic axis and rotating axis are the same, and if so what will happen?

Are there neutron stars whose magnetic axis and rotating axis are the same, and if so what will happen?

I know that there's probably a higher chance of having a neutron star that has its magnetic axis inclined to the rotational axis rather than having it perfectly aligned. If they are not aligned, the neutron star will create beams of radiation that sweep through space like the beams of light from a lighthouse. But what will happen if they were to be aligned?


It is believed that old pulsars may have their rotational axes closely aligned with their magnetic field. This would happen over a timescale of $ ausim10^7$ years (Lyne & Manchester (1988)). There are three sets of phenomena driving the dynamics of the alignment (Casini & Montemayor (1998)):

  • Short-term ($sim50$ days) variations caused by glitches
  • Intermediate increases of the alignment angle $ heta$, moving the magnetic dipole moment towards the equator
  • A set of long-term ($sim10^7$ years) dynamics leading to a decrease in $ heta$

Only in older pulsars does this last set of dynamics dominate. What this means is that the pulses from the neutron star would appear wider than pulses from neutron stars with significant misalignment. We also would not see interpulses, signals from the opposite magnetic pole, which we see in a number of young (or simply short-period) pulsars; we would just observe one pulse per rotation period.

I will add that Lyne & Manchester identified some young pulsars (including PSR 1800-21 and PSR 1823-13) that exhibited near alignment, a possible indication that the initial offset between the two axes may have a uniform distribution. Therefore, there should be examples of young pulsars with magnetic and rotation axes almost aligned. I would assume the intermediate-timescale mechanisms would then lead to misalignment before the long-term dynamics began to dominate.


NICER observations of PSR J0030+0451 in x-rays show hot spots clustered near one pole. The hot spots are presumed to be the termination of the active magnetic field lines, so there is really no magnetic "axis". The field is more complicated.

First surface map of a pulsar


Neutron star

A neutron star is a compact star in which the weight of the star is carried by the pressure of free neutrons. It is also called a degenerate star. The neutron is an elementary particle and one of the building blocks of atomic nuclei. Neutrons are electrically neutral (hence the name) and in contrast to protons, can be packed to form extremely large "nuclei", up to several times the mass of the Sun. Neutron stars are the first major astronomical object whose existence was first predicted from theory (1933) and later (1968) found to exist, at first as radio pulsars.

Neutron stars have a mass of the same order as the mass of the Sun. Their size (radius) is of order 10 km, about 70,000 times smaller than the Sun. So a neutron star's mass is packed in a volume 70,000 3 or approximately 10 14 times smaller than the Sun and the average mass density can be 10 14 times higher than the density in the Sun. Such densities have yet to be produced in the laboratory. In fact, the density of a neutron star is about the density of an atomic nucleus.

Due to its small size and high density, a neutron star possesses a surface gravitational field about 2×10 11 times that of Earth. One of the measures for the gravity is the escape velocity, the velocity one would need to give an object, such that it can escape from the gravitational field into infinity. For a neutron star, such velocities are typically 150,000 km/s, about 1/2 of the velocity of light. Conversely: an object falling onto the surface of a neutron star would impact the star also at 150,000 km/s. To put this in perspective, if an average human were to encounter a neutron star, he or she would impact with roughly the energy yield of a 100 megaton nuclear explosion.

Neutron stars are one of the few possible endpoints of stellar evolution, therefore sometimes called a dead star. They are formed in a supernova as the collapsed remnant of a massive star (a Type II or Ib supernova) or as the remnant of a collapsing white dwarf in a Type Ia supernova.

Neutron stars are typically about 20 km in diameter, have greater than 1.4 times the mass of our Sun (the Chandrasekhar limit, below which they'd be white dwarfs instead) and less than about 3 times the mass of our Sun (otherwise they'd be black holes), and spin very rapidly (one revolution can take anything from thirty seconds to a hundredth of a second).

The matter at the surface of a neutron star is composed of ordinary nuclei as well as ionized electrons. The "atmosphere" of the star is roughly one metre thick, below which one encounters a solid "crust". Proceeding inward, one encounters nuclei with ever increasing numbers of neutrons such nuclei would quickly decay on Earth, but are kept stable by tremendous pressures. Proceeding deeper, one comes to a point called neutron drip where free neutrons leak out of nuclei. In this region we have nuclei, free electrons, and free neutrons. The nuclei become smaller and smaller until the core is reached, by definition the point where they disappear altogether. The exact nature of the superdense matter in the core is still not well understood. Some researchers refer to this theoretical substance as neutronium, though this term can be misleading and is more frequently used in science fiction. It could be a superfluid mixture of neutrons with a few protons and electrons, other high-energy particles like pions and kaons may be present, and even sub-atomic quark matter is possible. However so far observations have not indicated nor ruled out such exotic states of matter.


Are there neutron stars whose magnetic axis and rotating axis are the same, and if so what will happen? - Astronomy

    The transition from white dwarf matter to neutron star matter begins at a density of 4x10 11 gm/cm 3 , according to theoretical calculation. It shows several phases to the transition:

Figure 08-19a Pulsar Signal [view large image]

Figure 08-19b Pulsar Model [large image]

Figure 08-19c Magnetar Pathway [view large image]

Figure 08-19d Magnetar [view large image]

1000 years. Meanwhile, the star reaches a maxiunum radius of about 200 Rsun. It becomes very luminous and is shedding mass because of the very rapid rotation approaching vrot/vcrit

Figure 08-19e HR Diagram of Merging Stars to Magnetar

Figure 08-19i GRB130427A Spectral Lags [view large image]

Figure 08-19i shows the very high energy burst (I), which is preceded by a relatively long time interval of "silence" (II), and some high energy photons farther back in time (III). The sequence seems to indicate two separate events. A single event with spectral lags would have a pattern of continuous curve in photon energy.

Using the formula for estimating the Planck structure "turn on" energy EQG,1 :
/>t = (D/c) ( />E/EQG,1),
with GRB130427A red shift z

4.6x10 27 cm), t

300 sec, E

100 Gev (from Figure 08-19i), we obtain EQG,1

0.04 EPlanck, which seems to be too low for the granular structure of space to show up.


Context for Proof Based on Cen X-3

The proof that is going to be presented is an empirical proof. It is based on the 1974-proof, which was published in the Meta Research Bulletin edited by Tom van Flandern in 1993:
https://archive.is/n1X2o
(see also: “Dark Matter, Missing Planets & New Comets” by Tom van Flandern: The Secret of the Pulsars listed on p484, as appearing in Meta Research Bulletin, Volume II, p30-38.)

Unfortunately, the above site appears to be inactive, but I will put a copy of the paper on this site in the near future. Be that as it may, we shall push forward and continue with the proof in basically a self-contained manner on this page.

First, given the context of the Crab above, we must talk a bit about what Cen X-3 is, and then put that together with the Crab.

The properties of Cen X-3 will give us our empirical proof that the NS-Creation theory falls short and leads to self-contradiction, which as described above forces us to choose the NS-Capture theory, which implies there are 25 times as many neutron stars as there are regular stars in the galaxy, which means there are 5 trillion neutron stars in the Milky Way Galaxy. Enough with the preliminaries, now, let’s get right into the meat of the proof.

Cen X-3 was first discovered to be an X-ray source in 1967 as a result of a sounding rocket study, which simply determined that there were X-rays coming from a specific location in the constellation Centaurus.

Then, in May 1971, this X-ray source was examined by the Uhuru satellite, and discovered to be pulsing in X-rays, once every 4.8 seconds. This represented the first time a pulsar had been discovered at wavelengths other than radio waves. X-rays are much stronger than radio waves, in fact, X-rays contain 100,000 times as much energy as visible light waves, which, in turn, contain 1 million or more times as much energy as radio waves. Therefore, an X-ray pulsar is a much more powerful emitter of radiation than a normal radio pulsar.

Then, in December 1971, an even more remarkable discovery was made about Cen X-3. It was indisputably disputably to be in a close binary system, such that it was determined to be a neutron star orbiting in the upper atmosphere of a companion star every 2.1 days. The source of X-rays was therefore determined to be the result of material from the atmosphere of the companion star being accelerated in the huge magnetic and gravitational fields of the neutron star 4.8 second X-ray pulsar. It was known that Cen X-3 was right in its companion’s atmosphere, because it pulsar was eclipsed of 90 degrees or one quarter of the orbit, plus the spectrum showed absorption when the pulsar was just entering and just leaving the eclipse.

In addition, one more amazing fact was determined about the Cen X-3 4.8 second pulsar in Dec 1971:

  • All the radio pulsars that had been discovered, as isolated pulsars were not in binary systems and their spin rate was slowing down.
  • However, measurements on the Cen X-3 pulsar determined that its spin rate was speeding up! And, ongoing measurements for the last 48 years have shown that it continues to spin up.

It was determined that the reason the Cen X-3 pulsar was speeding up was that the material falling into Cen X-3’s huge gravitational field was exerting a torque on it, causing it to spin faster, much the way when an figure skater spins faster when they pull their arms in.

To complete the Cen X-3 contextual framework, there was one more discovery made in 1974, the companion star to the Cen X-3 pulsar was determined to be an O-type supergiant, called Krzeminski’s star. This O-type supergiant is 20.5 times the mass of the Sun, and 12 times the radius of the Sun.

That means that the circumference of the Cen X-3 orbit around the supergiant is approximately 30 million miles, which means that for it to have a 2-day orbit, that it must be travelling about 200 miles per second (calculated to be 339 km/sec).


Like water in a cup

Pulsars form when a star with a mass anywhere from one and a half to three times that of the sun runs out of hydrogen fuel and collapses under its own weight. The collapse fuses the outer layers of the star's core star and creates an explosion — a supernova. But much of the star's mass remains. The atoms themselves are crushed, and the electrons (which typically orbit the atom's central nucleus) either collide with protons (turning them into neutrons), or escape. What's left is a ball of neutrons, surrounded by a crust of neutrons and protons. Neutron stars are less than 10 miles (16 kilometers) across but are so dense that a teaspoon of its substance would weigh hundreds of times as much as the Egyptian pyramids.

When rotating objects (like stars) collapse, they speed up (think of an ice-skater pulling in her arms to spin faster). That's why pulsars spin so fast — stars have a lot of mass and start out millions of miles across.

Neutron stars, including pulsars, have intense magnetic fields that beam radio waves out into space along their two poles. Because these poles don't always line up with the pulsar's rotational axis, the radio waves are like the beams of a lighthouse, moving across the sky as the pulsar spins. When astronomers see the beam with a radio telescope, the signal appears to pulse on and off. Pulsars rotate at very regular intervals, only slowing down by about 1 second every million years, on average (hence why they are known as cosmic time keepers). They slow down because they lose energy through radio waves and particles emitted from the surface. So scientists can't explain why pulsars occasionally "glitch" or speed up, but the angular momentum to make this happen must come from somewhere.

That's where the idea of superfluids comes in. Most astronomers think the crust of a neutron star is like a rigid crystal lattice, but below the surface, the increasing pressure would make the material more and more malleable, until it becomes a fluid. In the new study, Wynn Ho, a lecturer in mathematical sciences at the University of Southampton in England, and his colleagues say the fluid is a superfluid, and can store angular momentum because it has zero viscosity. A fluid's viscosity is similar to its thickness, so water has a much lower viscosity than honey.

"If you have a cup of water on a table and spin the cup, the water will spin up," Ho told Space.com. "The cup will slow down because of friction from the table, but the water will keep going." Ordinary water will eventually stop as it gives up energy to the cup via friction, but superfluid water wouldn't, he added.

The cup, in this case, is like the surface of the neutron star. The surface will slow down, because it is radiating energy. But the superfluid made of neutrons keeps going. As the difference between the two speeds increases, the superfluid interacts with the crust, and gives it a jerk, spinning it faster for a short time. That's the glitch — which, in turn, releases energy and changes the radio signal the pulsar emits. [Inside a Neutron Star (Infographic)]


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Answers and Replies

They don't emit rotation to earth as such, they transfer angular momentum to any bits of dust and gas around them, but since they are formed from a supernova which tends to remove most of the matter in their system there isn't much of this left.
It would be very difficult to detect non rotating NS, it's only from the pulsar jets from rotating ones we get any signal.

The slow down rate is incredibly low, 10^-10 to 10^-20 /s, or in other words if a N. star had a period of 1s after 1 million years it would rotate at 1.03 seconds!

Thanks for the reply.
1. Are the pulsar jets connected to the rotational speed? I guess so. in that case - here's another detection method gone bad when the NS (thanks for the initials :) stops rotating.
2. Even though the slowing-down rate is incredibly low, there are probably millions of NS out there, so some should be rotating really slowly, no?
3. What does, theoreticaly, happen to a NS that loses all it's rotational energy?

1, I don't think the strength of the jets are connected to rotation speed but the rotating magentic field is needed for them.
2, Some neutron stars will have almost stopped rotating, absolutely zero rotation speed is difficult to reach of course.

3, Apart from being a traffic hazard to UFOs not much! It just sits there essentially for ever occasionally emitting X rays if any matter lands on it.

I was wrong, you can detect non-pulsar neutron stars wether rotating or not if a companion survived the initial supernova, you can then detect the wobble in the orbit of the other star as it rotates around the invisible neutron star. You could also detect x-rays from any gas being pulled off the companion and dumped onto the surface of neutron star. If it is rotating and has a magnetic field (neutron stars have strong fields) you will also get pulsar jets from this material.

It is impossible for a stellar nova to NOT induce angular momentum on a core element during its gravitational collapse phase, nor can an Equation of State be demonstrated that can NOT induce angular momentum.

Non-rotational neutron stars are a violation of conservation of angular momentum, they do not exist.

Also, a neutron stars companion would induce angular momentum on the neutron star.

(conservation of angular momentum)

The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars (indeed, decreasing the size of object 10^4 times results in increase of its angular velocity by the factor 10^8).

Reducing the angular momentum of a neutron star from 10^8 to 0 with a binary companion in the lifetime of the Universe is highly improbable. However, anyone is welcome to perform the GR calculation.

A closed binary system may alternate between spin and orbital momentum, however either probably not absolutely to total or zero in the classical sense, the system would either collapse or fly apart.

Reference:
http://en.wikipedia.org/wiki/Angular_momentum#Conservation_of_angular_momentum

Well, why assume a binary system?

Is the slowing-down rate of a neutron star decreasing with time? If so, it's probably right - a neutron star will never really get to "zero" angular speed.
But if the slowing-down rate is pretty much a constant - I still don't see why theoratically a neutron star couldn't slow down to zero.
Unless, of course, the star itself can't exist (as a whole) with a small ang. velocity - therefore it "dies" (again!) before slowing more.

Most stars are in binary systems - so assuming that the companion survived the SN the NS would be in a binary.
The slow down rate is constant so it would eventually stop ( for very large values of eventually - as Orion says) unlike say cooling where the rate is proportional to temperature and so something never entirely cools.
The slowest radio emitting NS have a period of under a day, of course there may be slower ones that we can't detect - the fastest spin at nearly 1000/s !

There is no theoretical reason why a NS cannot exist with zero rotation, it's just going to take an impossibly long time to get there.

1000/s :-
Makes you want to.
See it.

The fastest spinning NSs are probably the youngest and have a discus or oblate spheroid geometry, while the slowest spinning NSs are probably the oldest and evolve from discus or oblate spheroid to spheroid then to sphere while experiencing periodic seismic activity to its changing stellar core geometry.

There also appears to be alot of dynamics involved with its angular momentum and Equation of State, such as its dynamic composition, mass, radius, crust thickness, and magnetic field strength as a few dynamic factors.

Meaning, a NS is going to really shake and shudder and even explode before its angular momentum shuts down.

A group of scientists has announced that recent observations of one particularly tempestuous neutron star seems to bolster the theory that irregular gamma and x-ray bursts are caused by starquakes.

Last year, the gamma repeater known as SGR 1900+14 suddenly flashed to life after 20 years of relative dormancy.

The neutron star was identified as a source of gamma ray bursts in 1979, but it was only seen to flash seven times between then and 1998. Last May, though, the star flared up, and it has done so more than 200 times since. Each burst releases as much energy as the Sun does in a year.

"These flashes are really the starquakes," said Robert Duncan, a research astronomer at the University of Texas

Neutron stars are the only stars (along with some white dwarfs) that have a solid surface. They are thought to be the collapsed cores of stars, the remnants of immense explosions called supernovae. About 12 miles (20 kilometers) in diameter, neutron stars contain the mass of the Sun and have a superfluid interior covered by a metallic crust. That crust is thought to be slightly more than a half mile (about 1 kilometer) thick.

A starquake can be thought of as a displacement or rupture of the crust, similar to the tear between two of Earth's tectonic plates along a fault. A displacement as small as a few millimeters could create a typical gamma ray burst, Duncan said.

But the two-hundred-plus typical bursts were impotent flickers compared to the explosion of August 27. On that day the star erupted with a flash more than 1,000 times as powerful as any yet observed. The gamma ray flash lasted almost six minutes. It exploded as much energy during the first second as the sun releases in 1,000 years.

What most amazed Woods, though, was a dramatic deceleration in the star's spin rate -- what looked like the sudden breaking of the whirling magnetic star.

Scientists classify SGRs as magnetars, a 10-member family of neutron stars that has the peculiar characteristic of slowing in rotation very rapidly. The rate of this deceleration -- a few milliseconds per year -- is called the spindown rate. Theorists believe that a strong magnetic field is responsible for applying the breaks.

"During the summer of 1998, we observed a rapid change in the spindown, and it happened in a period when the burst source was very active," Woods said. After that 80-day period, it (the spindown) was about 2 times faster than it was previously."

In May 1998, the rate was measured to be about 2 milliseconds per year. It then increased to about 4.5 milliseconds per year, and today is back to 2 milliseconds per year, Woods said. A rotation now takes almost one two-hundredth of a second longer to complete than it did last year. This is an important difference in a star that spins once every 5.16 seconds.


Neutron star cooling

For a few tens of isolated NSs, the detected X-ray spectra show a clear thermal contribution directly originated from a relatively large fraction of the star surface. For the cases in which an independent estimate of the star age is also available, one can study how temperatures correlate with age, which turns out to be an indirect method to test the physics of the NS interior. The evolution of the temperature in a NS was theoretically explored even before the first detections, in the 1960s (Tsuruta 1964). Today, NS cooling is the most widely accepted terminology for the research area studying how temperature evolves as NSs age and their observable effects. We refer the interested reader to the introduction in a recent review (Potekhin et al. 2015b) for a thorough historical overview of the foundations of the NS cooling theory.

According to the standard theory, a proto-NS is born as extremely hot and liquid, with (Tgtrsim 10^<10>) K, and a relatively large radius, (sim 100) km. Within a minute, it becomes transparent to neutrinos and shrinks to its final size, (Rsim 12) km (Burrows and Lattimer 1986 Keil and Janka 1995 Pons et al. 1999). Neutrino transparency marks the starting point of the long-term cooling. At the initially high temperatures, there is a copious production of thermal neutrinos that abandon the NS core draining energy from the interior. In a few minutes, the temperature drops by another order of magnitude to (T sim 10^<9>) K, below the melting point of a layer where matter begins to crystallize, forming the crust. Since the melting temperature depends on the local value of density, the gradual growth of the crust takes place from hours to months after birth. The outermost layer (the envelope, sometimes called the ocean) with a typical thickness (<<>>>(10^2

mathrm)) , remains liquid and possibly wrapped by a very thin (<<>>(mathrm)>) gaseous atmosphere. In the inner core, a mix of neutrons, electrons, protons and plausibly more exotic particles (muons, hyperons, or even deconfined quark matter), the thermal conductivity is so large that the dense core quickly becomes isothermal.

The central idea of NS cooling studies is to produce realistic evolution models that, when confronted with observations of the thermal emission of NSs with different ages (Page et al. 2004 Yakovlev and Pethick 2004 Yakovlev et al. 2008 Page 2009 Tsuruta 2009 Potekhin et al. 2015a), provide useful information about the chemical composition, the magnetic field strength and topology of the regions where this radiation is produced, or even the properties of matter at higher densities deeper inside the star. Two interesting examples are the low temperature (and thermal luminosity) shown by the Vela pulsar, arguably a piece of evidence for fast neutrino emission associated to higher central densities or exotic matter, or the controversial observational evidence for fast cooling of the supernova remnant in Cassiopeia A (Heinke and Ho 2010 Posselt and Pavlov 2018), proposed to be a signature of the core undergoing a superfluid transition (Page et al. 2011 Shternin et al. 2011 Ho et al. 2015 Wijngaarden et al. 2019). We now review the theory of NS cooling, beginning with a brief revision of the stellar structure equations and by introducing notation.

Neutron star structure

The first NS cooling studies (and most of the recent works too) considered a spherically symmetric 1D background star, in part for simplicity, and in part motivated by the small deviations expected. The matter distribution can be assumed to be spherically symmetric to a very good approximation, except for the extreme (unobserved) cases of structural deformations due to spin values close to the breakup values ( (P lesssim 1) ms) or ultra-strong magnetic fields ( (B gtrsim 10^<18>) G, unlikely to be realized in nature). Therefore, using spherical coordinates ((r, heta ,varphi )) , the space-time structure is accurately described by the Schwarzschild metric

where ( lambda (r) = - frac<1> <2>ln left[ 1- frac<2 G> frac ight] ) accounts for the space-time curvature,

is the gravitational mass inside a sphere of radius r, ( ho ) is the mass-energy density, G is the gravitational constant, and c is the speed of light. The lapse function (e^<2 u (r)>) is determined by the equation

with the boundary condition (mathrm ^<2 u (R)>=1-2GM/c^2R) at the stellar radius (r=R) . Here, (Mequiv m(R)) is the total gravitational mass of the star. The pressure profile, P(r), is determined by the Tolman–Oppenheimer–Volkoff equation

Throughout the text, we will keep track of the metric factors for consistency, unless indicated. The Newtonian limit can easily be recovered by setting (mathrm ^< u >=mathrm ^=1) in all equations.

To close the system of equations, one must provide the equation of state (EoS), i.e., the dependence of the pressure on the other variables (P=P( ho ,T, Y_i)) ( (Y_i) indicating the particle fraction of each species). Since the Fermi energy of all particles is much higher than the thermal energy (except in the outermost layers) the dominant contribution is given by degeneracy pressure. The thermal and magnetic contributions to the pressure, for typical conditions, are negligible in most of the star volume. Besides, the assumptions of charge neutrality and beta-equilibrium uniquely determine the composition at a given density. Thus, one can assume an effective barotropic EoS, (P=P( ho )) , to calculate the background mechanical structure. Therefore, the radial profiles describing the energy-mass density and chemical composition can be calculated once and kept fixed as a background star model for the thermal evolution simulations.

Structure and composition of a (1.4,M_odot ) NS, with SLy EoS. The plot shows, as a function of density from the outer crust to the core, the following quantities: mass fraction in the form of nuclei (X_h) (blue dot-dashed line), the fraction of electrons per baryon (Y_e) (black dashes), the fraction of free neutrons per baryon (Y_n) (red dashes), the atomic number Z (dark green triple dot-dashed), the mass number A (cyan long dashes), radius normalized to R (pink solid), and the corresponding enclosed mass normalized to the star mass (green solid)

In Fig. 1 we show a typical profile of a NS, obtained with the EoS SLy4 (Douchin and Haensel 2001), which is among the realistic EoS supporting a maximum mass compatible with the observations, (M_sim 2.0) – (2.2,M_odot ) (Demorest et al. 2010 Antoniadis et al. 2013 Margalit and Metzger 2017 Ruiz et al. 2018 Radice et al. 2018 Cromartie et al. 2019). We show the enclosed radius and mass, and the fractions of the different components, as a function of density, from the outer crust to the core. For densities ( ho gtrsim 4 imes 10^

ext < g > ext< cm >^<-3>) , neutrons drip out the nuclei and, for low enough temperatures, they would become superfluid. Note that the core contains about 99% of the mass and comprises 70–90% of the star volume (depending on the total mass and EoS). Envelope and atmosphere are not represented here. For a more detailed discussion we refer to, e.g., Haensel et al. (2007) and Potekhin et al. (2015b).

Heat transfer equation

Spherical symmetry was also assumed in most NS cooling studies during the 1980s and 1990s. However, in the 21st century, the unprecedented amount of data collected by soft X-ray observatories such as Chandra and XMM-Newton, provided evidence that most nearby NSs whose thermal emission is visible in the X-ray band of the electromagnetic spectrum show some anisotropic temperature distribution (Haberl 2007 Posselt et al. 2007 Kaplan et al. 2011) . This observational evidence made clear the need to build multi-dimensional models and gave a new impulse to the development of the cooling theory including 2D effects (Geppert et al. 2004, 2006 Page et al. 2007 Aguilera et al. 2008a, b Viganò et al. 2013). The cooling theory builds upon the heat transfer equation, which includes both flux transport and source/sink terms.

The equation governing the temperature evolution at each point of the star’s interior reads:

where (c_mathrm ) is specific heat, and the source term is given by the neutrino emissivity Q (accounting for energy losses by neutrino emission), and the heating power per unit volume H, both functions of temperature, in general. The latter can include contributions from accretion and, more relevant for this paper, Joule heating by magnetic field dissipation. All these quantities (including the temperature) vary in space and are measured in the local frame, with the metric (redshift) corrections accounting for the change to the observer’s frame at infinity. Footnote 2

The heat flux density (mathbf ) is given by

with (>) being the thermal conductivity tensor. In Fig. 2 we show the different contributions to the specific heat by ions, electrons, protons and neutrons, for (T= <10,5,1,0.5> imes 10^8) K, respectively, computed again with SLy EoS. For the superfluid/superconducting gaps we use the phenomenological formula for the momentum dependence of the energy gap at zero temperature employed in Ho et al. (2012), in particular their deep neutron triplet model.

Contributions to the specific heat from neutrons (red dashes), protons (green dot-dashed), electrons (blue dots), and ions (black solid line) as a function of density, from the outer crust to the core, and for different temperatures in each panel (as indicated). The superfluid gaps employed are the same as in Ho et al. (2012)

The bulk of the total heat capacity of a NS is given by the core, where most of the mass is contained. The regions with superfluid nucleons are visible as deep drops of the specific heat. The proton contribution is always negligible. Neutrons in the outer core are not superfluid, thus their contribution is dominant. The crustal specific heat is given by the dripped neutrons, the degenerate electron gas and the nuclear lattice (van Riper 1991). The specific heat of the lattice is generally the main contribution, except in parts of the inner crust where neutrons are not superfluid, or for temperatures (lesssim 10^8) K, when the electron contribution becomes dominant. In any case, the small volume of the crust implies that its heat capacity is small in comparison to the core contribution. For a detailed computation of the specific heat and other transport properties, we recommend the codes publicly available at http://www.ioffe.ru/astro/EIP/, describing the EoS for a strongly magnetized, fully ionized electron-ion plasma (Potekhin and Chabrier 2010).

The second ingredient needed to solve the heat transfer equation is the thermal conductivity (dominated by electrons, due to their larger mobility). For weak magnetic fields, the conductivity is isotropic: the tensor becomes a scalar quantity times the identity matrix. Since the background is spherically symmetric, at first approximation, the temperature gradients are essentially radial throughout most of the star. In this limit, 1D models are accurately representing reality, at least in the core and inner crust. However, for strong magnetic fields (needed to model magnetars), the electron thermal conductivity tensor becomes anisotropic also in the crust: in the direction perpendicular to the magnetic field the conductivity is strongly suppressed, which reduces the heat flow orthogonal to the magnetic field lines.

In the relaxation time approximation, the ratio of conductivities parallel ( (kappa ^parallel ) ) and orthogonal ( (kappa ^perp ) ) to the magnetic field is

Here we have introduced the so-called magnetization parameter (Urpin and Yakovlev 1980), (omega _B au _e) , where ( au _e) is the electron relaxation time and (omega _B = eB/m^*_ec) is the gyro-frequency of electrons with charge (-e) and effective mass (m^*_e) moving in a magnetic field with intensity B. Equation (6) is only strictly valid in the classical approximation (see Potekhin and Chabrier 2018 for a recent discussion of quantizing effects), but this dimensionless quantity is always a good indicator of the suppression of the thermal conductivity in the transverse direction. We will see later that this is also the relevant parameter to discriminate between different regimes for the magnetic field evolution.

Figure 3 shows the thermal conductivity including the contributions of all relevant carriers, for two different combinations of temperatures and magnetic field, roughly corresponding to a recently born magnetar ( (T=10^9) K, (B=10^<15>) G), or after (sim 10^4) yr ( (T=10^8) K, (B=10^<14>) G). Note that the thermal conductivity of the core is several orders of magnitude higher than in the crust, which results in a nearly isothermal core. Thus, the precise value of the core thermal conductivity becomes unimportant, and thermal gradients can only be developed and maintained in the crust and the envelope. In the crust, the dissipative processes responsible for the finite thermal conductivity include all the mutual interactions between electrons, lattice phonons (collective motion of ions in the solid phase), impurities (defects in the lattice), superfluid phonons (collective motion of superfluid neutrons) or normal neutrons. The mean free path of free neutrons, which is limited by the interactions with the lattice, is expected to be much shorter than for the electrons, but a fully consistent calculation is yet to be done (Chamel 2008). Quantizing effects due to the presence of a strong magnetic field become important only in the envelope, or in the outer crust for very large magnetic fields ( (B gtrsim 10^<15>) G). For comparison, we also plot the (B=0) values. The quantizing effects are visible as oscillations around the classical (non-magnetic) values, corresponding to the gradual filling of Landau levels. More details about the calculation of the microphysics input ( (>, c_v, Q) ) can be found in Sect. 2 of Potekhin et al. (2015b).

Thermal conductivity in the directions parallel (solid lines) and perpendicular (dashes) to the magnetic field, including quantizing effects. We show the cases (T=10^9) K, (B=10^<15>) G (left panel) and (T=10^8) K, (B=10^<14>) G (right panel). For comparison, the (B=0) values are shown with green lines in both figures

We can understand how and where anisotropy becomes relevant by considering electron conductivity in the presence of a strong magnetic field (and for now, ignoring quantizing effects). The heat flux is then reduced to the compact form (Pérez-Azorín et al. 2006):

where (mathbf equiv mathbf /B) is the unit vector in the local direction of the magnetic field. The heat flux is thus explicitly decomposed in three parts: heat flowing in the direction of the redshifted temperature gradient, (> (e^ u T)) , heat flowing along magnetic field lines (direction of (mathbf ) ), and heat flowing in the direction perpendicular to both.

In the low-density region (envelope and atmosphere), radiative equilibrium will be established much faster than the interior evolves. The difference by many orders of magnitude of the thermal relaxation timescales between the envelope and the interior (crust and core) makes computationally unpractical to perform cooling simulations in a numerical grid including all layers up to the star surface. Therefore, the outer layer is effectively treated as a boundary condition. It relies on a separate calculation of stationary envelope models to obtain a functional fit giving a relation between the surface temperature (T_s) , which determines the radiation flux, and the temperature (T_b) at the crust/envelope boundary. This (T_s - T_b) relation provides the outer boundary condition to the heat transfer equation. The radiation from the surface is usually assumed to be blackbody radiation, although the alternative possibility of more elaborated atmosphere models, or anisotropic radiation from a condensed surface, have also been studied (Turolla et al. 2004 van Adelsberg et al. 2005 Pérez-Azorín et al. 2005 Potekhin et al. 2012). A historical review and modern examples of such envelope models are discussed in Sect. 5 of Potekhin et al. (2015b). Models include different values for the curst/envelope boundary density, magnetic field intensity and geometry, and chemical composition (which is uncertain).

The first 2D models of the stationary thermal structure in a realistic context (including the comparison to observational data) were obtained by Geppert et al. (2004, 2006) and Pérez-Azorín et al. (2006), paving the road for subsequent 2D simulations of the time evolution of temperature in strongly magnetized NS (Aguilera et al. 2008b, a Kaminker et al. 2014). In all these works, the magnetic field was held fixed, as a background, exploring different possibilities, including superstrong ( (Bsim 10^<15>) – (10^<16>) G) toroidal magnetic fields in the crust to explain the strongly non-uniform distribution of the surface temperature. Only recently (Viganò et al. 2013), the fully coupled evolution of temperature and magnetic field has been studied with detailed numerical simulations. In the remaining of this section, we focus on the main aspects of the numerical methods employed to solve Eq. (4) alone, and we will return to the specific problems originated by the coupling with the magnetic evolution in the following sections.

Numerical methods for 2D cooling

There are two general strategies to solve the heat equation: spectral methods and finite-difference schemes. Spectral methods are well known to be elegant, accurate and efficient for solving partial differential equations with parabolic and elliptic terms, where Laplacian (or similar) operators are present. However, they are much more tedious to implement and to be modified, and usually require some strong previous mathematical understanding. On the contrary, finite-difference schemes are very easy to implement and do not require any complex theoretical background before they can be applied. On the negative side, finite-difference schemes are less efficient and accurate, when compared to spectral methods using the same amount of computational resources. The choice of one over the other is mostly a matter of taste. However, in realistic problems with “dirty” microphysics (irregular or discontinuous coefficients, stiff source-terms, quantities varying many orders of magnitude, etc), simpler finite-difference schemes are usually more robust and more flexible than the heavy mathematical machinery normally carried along with spectral methods, which are often derived for constant microphysical parameters. For this last reason, here we will discuss the use of finite-difference methods to solve our particular problem.

Let us consider the energy balance equation (4), with the flux given by Eq. (7). We first note that, in axial symmetry, the (varphi -) component of the flux is generally non-zero but need not to be evaluated since it is independent of (varphi ) , so that its contribution to the flux divergence vanishes. For example, in the case of a purely poloidal field (only (r, heta ) components), we can ignore the last term in Eq. (7) because it does not result in the time variation of the temperature. However, in the presence of a significant toroidal component (B_varphi ) , the last term gives a non-negligible contribution to the heat flux in the direction perpendicular to (> (e^ u T)) (it acts as a Hall-like term).

In Aguilera et al. (2008a, b) and Viganò et al. (2013) and related works, they assume axial symmetry and adopt a finite-differences numerical scheme. Values of temperature are defined at the center of each cell, where also the heating rate and the neutrino losses are evaluated, while fluxes are calculated at each cell-edge, as illustrated in Fig. 4. The boundary conditions at the center ( (r=0) ) are simply (mathbf =0) , while on the axis the non-radial components of the flux must vanish. As an outer boundary, they consider the crust/envelope interface, (r=R_b) , where the outgoing radial flux, (F_>) , is given by a formula depending on the values of (T_b) and (mathbf ) in the last numerical cell. For example, assuming blackbody emission from the surface, for each outermost numerical cell, characterized by an outer surface (varSigma _r) and a given value of (T_b) and (mathbf ) , one has (F_mathrm=sigma _B varSigma _r T_s^4) where (sigma _B) is the Stefan–Boltzmann constant, and (T_s) is given by the (T_s - T_b) relation (dependent on (mathbf ) ), as discussed in the previous subsection.

To overcome the strong limitation on the time step in the heat equation, (varDelta t propto (varDelta x)^2) , the diffusion equation can be discretized in time in a semi-implicit or fully implicit way, which results in a linear system of equations described by a block tridiagonal matrix (Richtmyer and Morton 1967). The “unknowns” vector, formed by the temperatures in each cell, is advanced by inverting the matrix with standard numerical techniques for linear algebra problems, like the lower-upper (LU) decomposition, a common Gauss elimination based method for general matrices, available in open source packages like LAPACK . However, this is not the most efficient method for large matrices. A particular adaptation of the Gauss elimination to the block-tridiagonal systems, known as Thomas algorithm (Thomas 1949) or matrix-sweeping algorithm, is much more efficient, but its parallelization is limited to the operations within each of the block matrices. A new idea that has been proposed to overcome parallelization restrictions is to combine the Thomas method with a different decomposition of the block tridiagonal matrix (Belov et al. 2017).

Schematic illustration of the allocation of temperatures (cell centers) and fluxes (cell interfaces) in a typical grid in polar coordinates

A word of caution is in order regarding the treatment of the source term. The thermal evolution during the first Myr is strongly dominated by neutrino emission processes, which enter the evolution equation through a very stiff source term, typically a power-law of the temperature with a high index ( (T^8) for modified URCA processes, (T^6) for direct URCA processes). These source terms cannot be handled explicitly without reducing the time step to unacceptable small values but, since they are local rates, linearization followed by a fully implicit discretization is straightforward and results in the redefinition of the source vector and the diagonal terms of the matrix. A very basic description to deal with stiff source terms can be found in Sect. 17.5 of Press et al. (2007). This procedure is stable, at the cost of losing some precision, but it can be improved by using more elaborated implicit-explicit Runge–Kutta algorithms (Koto 2008).

Temperature anisotropy in a magnetized neutron star

An analytical solution that can be used to test numerical codes in multi-dimensions is the evolution of a thermal pulse in an infinite medium, embedded in a homogeneous magnetic field oriented along the z-axis, which causes the anisotropic diffusion of heat. Assuming constant conductivities, and neglecting relativistic effects, the following analytical solution for the temperature profile can be obtained for (t>t_0) :

where (T_0) is the central temperature at the initial time (t_0) . In Fig. 5 we show the comparison between the analytical (solid) and numerical (stars) solution for a model with (t_0=10^<-4>) , (T_0=1) , (kappa ^perp = 10^2) and (omega _B au _e = 3) . The boundary conditions employed are (F=0) at the center and the temperature corresponding to the analytical solution at the surface ( (r=1) ). Pérez-Azorín et al. (2006) found deviations from the analytical solution to be less than 0.1% in any particular cell within the entire domain, even with a relatively low grid resolution of 100 radial zones and 40 angular zones.

Temperature profiles at different times comparing the analytic solution (solid) and the numerical evolution (stars) of a thermal pulse in a medium embedded in a homogeneous magnetic field. The left (right) panel shows four different times during the evolution of polar (equatorial) profiles in arbitrary units. The simulation has been done with a fully implicit scheme and the linear system is solved with the Thomas algorithm. Image reproduced with permission from Pérez-Azorín et al. (2006), copyright by ESO

Temperature anisotropy induced in the NS crust by the presence of a strong magnetic field confined into the crust. The projections of the poloidal field lines are shown with solid lines in the left and right panels, and dashed lines in the central panel. The left panel corresponds to a model without toroidal field, the central panel to a force-free configuration (toroidal magnetic flux contours and poloidal magnetic field lines are aligned), and the right panel shows a model with a toroidal component confined to a narrow region of the crust represented by dashed lines. Image reproduced with permission from Pérez-Azorín et al. (2006), copyright by ESO

To conclude this section, the induced anisotropy in a realistic NS reported by Pérez-Azorín et al. (2006) is shown in Fig. 6. The figure shows equilibrium thermal solutions, in the absence of heat sources and sinks. The core temperature is kept at (5 imes 10^7) K, and the surface boundary condition is given by the (T_s-T_b) relation, assuming blackbody radiation. The poloidal component is the same in all models ( (B_p = 10^<13>) G). The effect of the magnetic field on the temperature distribution can be easily understood by examining the expression of the heat flux (7). When (omega _B au _e gg 1) , the dominant contribution to the flux is parallel to the magnetic field and proportional to (mathbf cdot abla (e^ u T)) . Thus, in the stationary regime (i.e., ( abla cdot (e^<2 u >mathbf )=0) if no sources are present), the temperature distribution must be such that (mathbf perp abla (e^ u T)) : magnetic field lines are tangent to surfaces of constant temperature. This is explicitly visible in the left panel, which corresponds to the stationary solution for a purely poloidal configuration with a core temperature of (5 imes 10^7) K. Only near the surface, the large temperature gradient can result in a significant heat flux across the magnetic field lines. When we add a strong toroidal component, the Hall term (proportional to (omega _B au _e) ) in Eq. (7), activates meridional heat fluxes which lead to a nearly isothermal crust. The central panel shows the temperature distribution for a force-free magnetic field with a global toroidal component, present in both the crust and the envelope. The right panel shows a third model with a strong toroidal component confined to a thin crustal region (dashed lines). It acts as an insulator maintaining a temperature gradient between both sides of the toroidal field.


Pulsars

What is left-over from such a catastrophic event, is a black hole or a neutron star, formed from the matter of the star's central region. In a neutron star, the iron nuclei are squeezed together to form what is essentially a plasma of neutrons. Like a figure scater doing a pirouette spins faster when (s)he brings the arms close to the body, due to the conservation of angular momentum, the collapse of the slowly rotating progenitor star results in the neutron star to spin with periods of 1 second and faster.

During the final collapse, the magnetic fields of the star are also squeezed into a small volume, thus the field strength at the surface of the neutron star is extremely high: about 10 10 Tesla, while our Sun has a field of about 10 -4 Tesla! Because of the rapid rotation the rapidly changing strong magnetic fields induce a huge electric field of about 10 11 V/m, which easily overcomes the gravitational pull on charged particles, and electrons and positrons are pulled away from the neutron star's surface. As charged particles can only move in a spiral path around and along the curved magnetic field lines, they are pulled into higher regions of lower density. Their curved path also causes them to radiate electromagnetic waves which comes out in a narrow cone parallel to the magnetic field lines. Thus the radiation is concentrated in a direction of the magnetic axis.

If the magnetic axis is inclined with respect to the rotation axis, the neutron star sends out beams of radiation, from its magnetic north and south poles. Due to the rotation, the beams point to two large circles in space, just like the light beam of a lighthouse.

Schematic sketch of a neutron star rotating about the &Omega-axis, but with an inclined magnetic axis, indicated by the field lines B near the magnetic poles. Electrons and positrons move along the field lines and produce beams of radiation in both directions of the magnetic axis. The star's rotation takes these beams around (adapted after Ruderman and Sutherland, 1975).

As the charged particles move into regions high above the neutron star's surface where the density is lower, their radiation is centered at lower frequencies. In this way they produce radiation over a wide range of frequencies (from the X-rays to radio waves), as they travel through the neutron star's magnetosphere.

The frequency of the radiation which is produced in a zone depends on the height above the neutron star's surface: &gamma and X rays are produced close to the surface, radio waves at greater heights, where the density is lower (adapted after Ruderman and Sutherland, 1975). The radio flux density of the pulsar B0329+54 decreases with frequency (from Backer and Fisher, 1974 1 f.u. (flux unit) = 1 Jy).

If our Earth happens to be in a direction towards which this beam may point, we observe a periodic flash of radiation: a pulsar. A famous pulsar is in the centre of the Crab nebula:

The Crab nebula, the gaseous left-over of a supernova explosion. This is both gas ejected by the dying star and gas filaments formed by the collision with the ambient interstellar gas. The composite image shows the emission in the X-rays from synchrotron radiation (blue) and by emission lines in the optical (green) and infrared (red). In the centre of the nebula there is the Crab pulsar, which emits radiation pulses in the whole range from &gamma-rays and X-rays to radio waves, with a period of 0.0333 s

Jodrell Bank Observatory has a very nice collection of the Sounds of Pulsars.

Observations: B0329+54

This is the strongest pulsar in the northern skies with a (true) period of 0.714 518 663 98 s and a flux density of 200 mJy at 1.4 GHz.
Phase diagram (or folded light curve) from the last 8 min part from an observation on 2 nov 2014 with a sampling rate of 1 kHz. The 'old' instrument configuration with the HP437B power meter has a response time of 50 ms, which broadens the pulse. Phase diagram from 6.5 min of data, averaged over 10 samples taken at 30 kHz rate. This was done on 30 sep 2015, with the 'new' instrument configuration of a EP441 power meter. With the better time resolution the main pulse is sharper, and the subpulses before and behind the main pulse are discernible.
Waterfall plot of the data of 2 nov 2014. Waterfall plot of the data of 30 sep 2015.
In a 2 hour section with high signal to noise from 26 oct 2015, the period of B0239+54 is found to 0.714492 s. Data was taken with 40 kHz sampling rate and the averages were recorded at 500 Hz. The phase diagram shows the weak sub-pulses before and after the main pulse.

Observations: B0950+08

The second brightest is B0950+08 with a (true) period of 0.253 065 068 19 s and a flux density of 85 mJy at 1.4 GHz. The best result was obtained on 30 sep 2015 during a 40 min observation, averaging over 10 samples taking at a rate of 30 kHz. This gives about 7.2 million data points.
Searching for the period of B0950+08: The faint signal requires many data. The search for the period needs to be confined in a narrow range, as only the exact period will make a reasonable phase diagram. The best fit is found with the period of 0.25305 s. Phasediagram of B0950+08 for a period of 0.25305 s.
In the waterfall plot of this data set the pulsar signal appears only clearly, if one choses a time intervall of 300 s from which phase diagrams are computed. When the pulses are made to form a vertical line, i.e. if the pulses are made to appear at the same phase, one gets the period of 0.253049 s period

Observations: B1933+16

The third brightest is B1933+16 with a (true) period of 0.358 736 248 270 s and a flux density of 40 mJy at 1.4 GHz. On 25 sep 2015 a 30 min observation with averaging over 10 samples at 50 kHz rate yields this result:
Search for the period of B1933+16: Again, the search range needs to be rather small around the expected value. The quite prominent peak at 0.358388 s does not show a pulsar signal, but 0.35877 s is the pulsar Phase diagram of B1933+16 with a period of 0.35877 s
In the waterfall plot of this data set the pulsar signal appears only clearly, if one choses a time intervall of 300 s from which phase diagrams are computed. When the pulses are made to form a vertical line, i.e. if the pulses are made to appear at the same phase, one gets the period of 0.35877 s period

Observations: Scintillations of B0329+54

This causes the signal from the normally bright pulsar B0329+54 to become much fainter for times, which makes testing and optimising the receiving system quite challenging. Here is a waterfall showing what happened during 10 hours on 26 oct 2015: for one hour around UT 15:00 the pulsar B0329+54 was exceptionally strong. At other times it changed between a reasonably good signal and being buried in the noise. Although this diagram is done with too short a period, which causes the line of the pulses to drift to the right, one can easily follow the coming and going of the signal:

B0329+54 can be used to study scintillation, by continously measuring its signal, and observing the time variation of the strength of the pulse with respect to the noise. This is done for several days in autumn 2015, during which all instrumental details are kept unchanged. The data is sampled at 40 kHz, but averaged over 80 data points, so that the data are recorded with a rate of 500 Hz, which makes smaller file lengths. From 1 minute sections of the raw data, phase diagrams are computed and the ratio of the height of the highest peak to the standard deviation of the background around this maximum is taken as the signal to noise ratio.
From the entire data set the strong variation of the signal to noise ratio is apparent. It can vary between about 20 to close to 1.

During a time span of 12 hours, the pulse may be present for a few minutes or as long as an hour, only to suddenly disappear in the noise.


Are there neutron stars whose magnetic axis and rotating axis are the same, and if so what will happen? - Astronomy

In the 1980s, while working at Berkeley Lab's Bevatron/Bevalac, Norman Glendenning of the Nuclear Sciences Division found his thoughts turning to neutron stars.

Norman Glendenning

Researchers were using the Bevalac to study nuclear "equations of state" — the way nuclear matter changes when subjected to extremes of pressure, density, or temperature. Some hoped the Bevalac could create nuclear densities great enough to free quarks from their imprisonment within protons, neutrons, and other hadrons (an accomplishment that hasn't been claimed even yet, although researchers at Brookhaven's Relativistic Heavy Ion Collider may be close). Although they failed to liberate quarks, the Bevalac investigators did observe fleeting states of matter with up to three times the density of the nucleus.

"But when I considered what one could learn from the interior of a neutron star, with ten times nuclear density, I rather lost interest in the accelerators of that era," says Glendenning. Speculating on what forms the densest matter in the universe might take, Glendenning soon began raising startling questions and proposing equally startling answers. Theorists and observers are still grappling with his ideas today.

The nativity of neutron stars

In 1934 astronomers Walter Baade and Fritz Zwicky coined new terms for the brightest exploding stars, suggesting that supernovae were powered by the collapse of the cores of large ordinary stars into neutron stars — objects only 10 to 20 kilometers across, made entirely of neutrons, and so dense that gravity at their surface would be 100 billion times greater than Earth's.

Jocelyn Bell with the Cambridge University radiotelescope that discovered the first pulsar.

This was a bold proposal, the neutron itself having been discovered only two years earlier. The existence of neutron stars wasn't proved until 1967, when Jocelyn Bell (Burnell), a graduate student of Cambridge radio astronomer Antony Hewish, discovered the self-advertising, spinning neutron stars called pulsars.

Although their net charge must be neutral, however, neutron stars aren't made solely of neutrons. The binding of gravity inside a neutron star is many times greater than the nuclear binding that holds atomic nuclei together pressure and density vary with depth, and neutron stars depend on many kinds of particles to cope with these extreme and changeable conditions: neutrons, of course, but also protons, electrons, and other, weirder species. How they arrange themselves depends on a number of variables including the star's mass, its diameter, and how fast it's spinning.

Moving toward the center, density increases matter is crushed ever closer together, until at some point quarks become "deconfined," popping out of their little hadron bags to form a soup of free quarks and gluons (gluons are the bosons that carry the strong force and normally keep quarks stuck together).

Theorists were long in the habit of thinking of this phase change — from the confined-quark to the deconfined-quark stage — as analogous to phase changes in water, for example from liquid water to ice. In the case of a neutron star, it was assumed, pressure increases smoothly with depth, until at some point neutron matter makes a smooth transition to quark matter.

If phase changes in water occurred in a system with two oppositely charged components, instead of freezing from the top down, spheres of ice would form.

But phase changes in water, says Glendenning, "are first-order transitions with only one independent component" — namely water itself. "In the real world, this kind of transition is far from typical. The situation is much more interesting for substances with two or more components."

Going through a phase

The stuff of a neutron star, for example. One of the two components that vary in neutron-star phase changes is electric charge. While neutron stars are globally neutral, local regions could have excesses of positive or negative charge.

A second component is baryon number, which must also be conserved. Hadrons have positive baryon numbers, while their antiparticles have negative baryon numbers.

There is no simple correspondence between electric charge and baryon number. A neutron has a positive baryon number but no electric charge up and down quarks both have a fractional baryon number (plus 1/3), but an up quark's electric charge is plus 2/3, and a down quark's is minus 1/3.

Because of this two-component system — the hadronic matter and the quark matter — the stuff of a neutron star can make trade-offs locally to maintain overal global electrical neutrality and conserve baryon number. Between the star's outer, quark-confined regions and its innermost, quark-deconfined regions, there will be mixed phases, mixed hadronic and quark matter that take on fantastic geometries.

Hadron regions like to maintain an equal number of neutrons and protons. This is not possible globally but is approachable locally, where hadronic matter begins to mix with quark matter, because under extreme pressure neutrons can become protons by transferring electric charge to quarks — changing up quarks into down or strange quarks, for example. The result is a region of positively charged nuclear matter with negatively charged quark matter embedded in it.

Glendenning uses a vivid metaphor to describe how different a two-component phase change is from the one-component phase changes of water: "Suppose water had two independent components and one of them was electric charge, with opposite charge on the ice than on the water. Then a lake would not freeze over starting with a sheet of ice on top, but ice spheres would form throughout the volume of the lake, of slightly different size and spacing from top to bottom, because of the pressure gradient."

Glendenning theorizes that in a neutron star of the right mass and density, a crystal of hadronic and quark matter in mixed geometric configurations occupies the region between outer nuclear matter and inner quark matter.

Likewise, where hadronic matter and quark matter are mixed, if quark matter is in the minority the quarks are segregated as droplets in a crystalline array, each droplet at a lattice point. As the pressure increases the proportion of quark matter increases and the droplets elongate to rods still more pressure means still more free quarks, and the rods join into slabs.

As pressure continues to increase, quark matter becomes the dominant phase, and the hadrons inside it form slabs, rods, and finally droplets, just before the system turns to pure quark matter. Glendenning jokingly refers to this as a pasta model: "Drops like orzo, rods like spaghetti, slabs like lasagna."

The picture of neutron star interiors based on two-component phase transitions is not intuitive (nothing about neutron stars is), and while Glendenning says he's astonished everybody before him missed it, he admits it took him five years to realize it himself.

But is there any way this theoretical understanding can ever be tested experimentally?

Putting a spin on the ball

In the early 1970s, nuclear physicists at Berkeley Lab and elsewhere observed that when certain rapidly spinning rare-earth nuclei like erbium and holmium are created in accelerator experiments, there is a moment when they temporarily slow down before spinning faster again. The explanation of this "backbending" spin seemed to be that while individual spinning protons or neutrons in the nucleus like to pair with others, forces induced by the spin of the nucleus as a whole break up some of these pairs. Inertia momentarily increases until the spins realign.

Tiny atomic nuclei aren't much like neutron stars, but Glendenning found the analogy striking: a change of state in one part of a rotating mixed system, the nucleon pairs, had a noticeable if temporary effect on the spin rate of the whole nucleus. He realized that the mixed states of hadronic matter and quark matter in a neutron star offered a comparable way for changes in part of the star to affect the spin rate of the whole.

Two conditions that can effect the spin of a neutron star, once its initial spin rate has been established in the collapse of the star that formed it, are drag and mass accretion. A pulsar is a neutron star with a strong magnetic field not aligned with its rotation axis the moving magnetic field creates a broad band of electromagnetic radiation, including radio waves, which led to the discovery of the first pulsar in 1967. Radiation drags on the pulsar and gradually slows it down.

On the other hand, a neutron star that slowly sucks matter from a companion star becomes more massive and, like a spinning ice skater who pulls in her arms, spins increasingly faster.

If a spinning neutron star's magnetic field is not aligned with its rotation axis, the drag of electromagnetic radiation slows it down (foreground). A neutron star that accretes matter from a companion star becomes more massive and spins ever faster (background).

Because of centrifugal forces, the faster a star spins, the less the pressure in its interior, and vice versa. Phase change boundaries — where hadronic matter mixes with quark matter — migrate as the star's rotation rate changes.

If a pulsar spins slower, pressure increases, and pure quark matter may form or increase at its center. Quark matter is incredibly dense. So if the star as a whole is slowing down, like a spinning skater extending her arms, then adding more quark matter at its center would be like a massive tiny skater inside the big one, a set of spinning Russian dolls in which the innermost is pulling in her arms and forcing the whole cluster to spin up again — temporarily.

The same glitch happens in reverse. A star that's accreting matter and spinning faster will relieve internal pressure, which will cause a quark-matter core to move to a less dense mixed phase — temporarily causing the star to slow down.

"Neutron stars are relativistic objects in which extremes of gravity exaggerate these effects through frame-dragging," says Glendenning. "The massive spinning core causes the rest of the star to spin faster than it otherwise would."

"Backbending" as neutron stars spin faster or slower should be detectable as an increase in the number of neutron stars spinning at certain rates, among a population of neutron stars whose spin rates otherwise vary smoothly. A preliminary catalog of x-ray neutron stars showed just such a spike unfortunately the compiler of the catalog later withdrew this result. Until more catalogs of neutron-star spin are compiled, and include more kinds of neutron stars, the question remains open.

How the matter in a neutron star arranges itself through phase changes is a subject of continuing lively interest. Astronomers are using new kinds of measurements to determine the mass and radius of neutron stars by observation, and theorists argue over what exotica may be found within them. Norman Glendenning's studies are an inevitable part of the continuing intellectual ferment.


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