# Neutron star size in different reference frames

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What is the estimated size of neutron stars observed in their reference frame and in our reference frame?

That is, how bent is space-time around neutron stars?

If we ignore the rotation issue (as Pela pointed out, the correction is <1%), then we can approximate the spacetime curvature inside the neutron star using the interior Schwarzschild metric and use the exterior one for the outside. The $$r$$ coordinate is not the radial distance as one would expect, but defined in terms of $$r=$$ constant circles having circumference $$2pi r$$.

A neutron star of radius $$R$$ in these metrics has a circumference measured on its surface as $$2pi R$$ as expected. But the distance to the core (that is, the length of a hole from the surface to the core) is: $$d(R) = int_0^{R} frac{dr}{1-Kr^2} = frac{1}{sqrt{K}} anh^{-1}left(sqrt{K}R ight).$$ where $$K=r_s/R^3$$ and $$r_s$$ is the Schwarzschild radius of the star (all of this assumes constant density in the interior). If we plot this for stars of different $$R$$ (but same mass, say one solar mass, producing a constant $$r_s$$) we get the following:

As the neutron star approaches becoming a black hole it gets "deeper": there is more volume than one would expect. Less dense neutron stars have depths that scale linearly with their radius… except that it is $$r_s/3$$ larger! This odd result comes from the assumption that we look at stars of the same mass even though we make them much larger. A big constant mass object will curve spacetime too, and this produces this effect. (Yes, this means that the sun is $$r_s/3 = 984.73$$ meters deeper than it looks!)

If we instead decide that we use a constant density (say nuclear density) and plot the depth, the collapse to a black hole instead happens to the right as the neutronium sphere becomes too large. Here I still use the solar Schwarzschild radius as a length scale to keep things comparable. Now, for small spheres the coordinate and measured depths converge:

In actual neutron stars things are complicated by rotation, the core pressure diverging as one approaches $$(9/8)r_s$$, and of course that it is hard to dig a hole through superfluid neutrons.

The inferred radius by a distant observer is given by $$R_{infty} = frac{R}{sqrt{1- R_s/R}},$$ where $$R_s = 2GM/c^2$$, $$M$$ is the gravitational mass and $$R$$ is the radius measured at the surface.

The fact that $$R_{infty} > R$$ is because an observer can see more than 50 per cent of the neutron star surface. See https://physics.stackexchange.com/questions/350805/seeing-something-from-only-one-angle-means-you-have-only-seen-what-of-its-su/350814#350814

## Neutron star size in different reference frames - Astronomy

1. Illustrations of Neutron Merger and Black Hole
A new study using Chandra data of GW170817 indicates that the event that produced gravitational waves likely created the lowest mass black hole known. The first artist's illustration (left) shows a key part of the process that created this new black hole, as the two neutron stars spin around each other while merging. (The purple material depicts debris from the merger.) The second artist's illustration (right) shows the black hole that resulted from the merger, along with a disk of infalling matter and a jet of high-energy particles. (Credit: NASA/CXC/M.Weiss)

2. GW170817 Schematic
Following the merger of two neutron stars, strong gravitational waves (not shown here) produced the source GW170817. This was followed by a burst of gamma-rays generated by a narrow jet or beam, of high-energy particles, depicted in red. Initially the jet was narrow (shown on the left) with Chandra viewing it from the side, giving an X-ray signal that is too weak to be detected. However, as time passed the material in the jet slowed down and widened (shown on the right) as it slammed into surrounding material, causing the X-ray emission to rise as the jet came into direct view by Chandra. This jet & its counterpart pointed in the opposite direction are likely generated by material falling onto a black hole created by the merger of the two neutron stars. The black hole is located in the white source at the base of the jets. (Credit: NASA/CXC/K.DiVona)

3. Illustration of a Magnetar
These illustration show that a slowly rotating neutron star with an ordinary surface magnetic field is giving off bursts of X-rays and gamma rays. (Illustration: NASA/CXC/M.Weiss)

4. Illustration of a Magnetar
These illustrations show how an extremely rapidly rotating neutron star, which has formed from the collapse of a very massive star, can produce incredibly powerful magnetic fields. These objects are known as magnetars. (Illustration: NASA/CXC/M.Weiss)

5. Illustration of a Magnetar
These illustrations show how an extremely rapidly rotating neutron star, which has formed from the collapse of a very massive star, can produce incredibly powerful magnetic fields. These objects are known as magnetars. (Illustration: NASA/CXC/M.Weiss)

6. Neutron Star Illustration
This artist's conception illustrates 1E 1207.4-5209, a neutron star with a polar hot spot and a strong magnetic field (purple lines). (Illustration: NASA/CXC/M.Weiss)

7. Closeup of a Neutron Star
Closeup of neutron star, showing how matter falls, or accretes, from accretion disk onto the neutron star. (Illustration: CXC/S. Lee)

8. Size Comparison of RX J1856 to Neutron and Quark Stars
This artist's rendition shows the diameter of RX J1856.5-3754, determined by data from NASA's Chandra X-ray Observatory, is too small to be a neutron star. The data are consistent with predicted size for a strange quark star, an object never before seen in nature. (Illustration: NASA/CXC/M.Weiss)

9. Neutron Star/Quark Star Interior
In a neutron star (left), the quarks that comprise the neutrons are confined inside the neutrons. In a quark star(right), the quarks are free, so they take up less space and the diameter of the star is smaller. (Illustration: NASA/CXC/M.Weiss)

10. Illustration of relative sizes of Grand Canyon, neutron star and quark star
The Grand Canyon is 18 miles rim to rim. A neutron star is about 12 miles in diameter, and a quark star is about 7 miles in diameter. (Illustration: CXC/D. Berry)

## Team obtains the best measurement of neutron star size to date

A typical neutron star with a radius of eleven kilometres is about as large as a medium-sized German city. Credit: NASA's Goddard Space Flight Center

An international research team led by members of the Max Planck Institute for Gravitational Physics (Albert Einstein Institute AEI) has obtained new measurements of how big neutron stars are. To do so, they combined a general first-principles description of the unknown behavior of neutron star matter with multi-messenger observations of the binary neutron star merger GW170817. Their results, which appeared in Nature Astronomy today, are more stringent by a factor of two than previous limits and show that a typical neutron star has a radius close to 11 kilometers. They also find that neutron stars merging with black holes are in most cases likely to be swallowed whole, unless the black hole is small and/or rapidly rotating. This means that while such mergers might be observable as gravitational-wave sources, they would be invisible in the electromagnetic spectrum.

"Binary neutron star mergers are a gold mine of information!" says Collin Capano, researcher at the AEI Hannover and lead author of the Nature Astronomy study. "Neutron stars contain the densest matter in the observable universe. In fact, they are so dense and compact, that you can think of the entire star as a single atomic nucleus, scaled up to the size of a city. By measuring these objects' properties, we learn about the fundamental physics that governs matter at the sub-atomic level."

"We find that the typical neutron star, which is about 1.4 times as heavy as our Sun has a radius of about 11 kilometers," says Badri Krishnan, who leads the research team at the AEI Hannover. "Our results limit the radius to likely be somewhere between 10.4 and 11.9 kilometers. This is a factor of two more stringent than previous results."

Binary neutron star mergers as astrophysical treasure trove

Neutron stars are compact, extremely dense remnants of supernova explosions. They are about the size of a city with up to twice the mass of our Sun. How the neutron-rich, extremely dense matter behaves is unknown, and it is impossible to create such conditions in any laboratory on Earth. Physicists have proposed various models (equations of state), but it is unknown which (if any) of these models correctly describe neutron star matter in nature.

Mergers of binary neutron stars—such as GW170817, which was observed in gravitational waves and the entire electromagnetic spectrum in August 2017—are the most exciting astrophysical events when it comes to learning more about matter at extreme conditions and the underlying nuclear physics. From this, scientists can in turn determine physical properties of neutron stars such as their radius and mass.

The research team used a model based on a first-principles description of how subatomic particles interact at the high densities found inside neutron stars. Remarkably, as the team shows, theoretical calculations at length scales less than a trillionth of a millimeter can be compared with observations of an astrophysical object more than a hundred million light years away.

"It's a bit mind boggling," says Capano. "GW170817 was caused by the collision of two city-sized objects 120 million years ago, when dinosaurs were walking around here on Earth. This happened in a galaxy a billion trillion kilometers away. From that, we have gained insight into sub-atomic physics."

How big is a neutron star?

The first-principles description used by the researchers predicts an entire family of possible equations of state for neutron stars, which are directly derived from nuclear physics. From this family, the authors selected those members that are most likely to explain different astrophysical observations they picked models

• which agree with gravitational-wave observations of GW170817 from public LIGO and Virgo data,
• which produce a short-lived hyper-massive neutron star as result of the merger, and
• which agree with known constraints on the maximum neutron star mass from electromagnetic counterpart observations of GW170817.

This not only allowed the researchers to derive robust information on dense-matter physics, but also to obtain the most stringent limits on the size of neutron stars to date.

Future gravitational-wave and multi-messenger observations

"These results are exciting, not just because we have been able to vastly improve neutron star radii measurements, but because it gives us a window into the ultimate fate of neutron stars in merging binaries," says Stephanie Brown, co-author of the publication and a Ph.D. student at the AEI Hannover. The new results imply that, with an event such as GW170817, the LIGO and Virgo detectors at design sensitivity will be able to easily distinguish, from gravitational waves alone, whether two neutron stars or two black holes have merged. For GW170817, observations in the electromagnetic spectrum were crucial to make that distinction.

The research team also finds that for mixed binaries (a neutron star merging with a black hole), gravitational-wave merger observations alone will have a hard time distinguishing such events from binary black holes. Observations in the electromagnetic spectrum or gravitational waves from after the merger will be crucial to tell them apart.

However, it turns out that the new results also imply that multi-messenger observations of mixed binary mergers are unlikely to happen. "We have shown that in almost all cases the neutron star will not be torn apart by the black hole and rather swallowed whole," explains Capano. "Only when the black hole is very small or rapidly spinning, can it disrupt the neutron star before swallowing it and only then can we expect to see anything besides gravitational waves."

A bright future ahead

In the next decade, the existing gravitational-wave detectors will become even more sensitive, and additional detectors will begin observing. The research team expects more very loud gravitational-wave detections and possible multi-messenger observations from merging binary neutron stars. Each of these mergers would provide wonderful opportunities to learn more about neutron star and nuclear physics.

## Neutron star size in different reference frames - Astronomy

We present a method to calculate solutions to the initial value problem in (3 + 1) general relativity corresponding to binary neutron-star systems (BNS) in irrotational quasi-equilibrium orbits. The initial value equations are solved using a conformally flat spatial metric tensor. The stellar fluid dynamics corresponds to that of systems with zero vorticity in the inertial reference frame. Irrotational systems like the ones analyzed in the present work are likely to resemble the final stages of the evolution of neutron-star binaries, thus providing insights on the inspiral process. The fluid velocity is derived from the gradient of a scalar potential. A numerical program was developed to solve the elliptic equations for the metric fields and the fluid velocity potential. We discuss the different numerical techniques employed to achieve high resolution across the stellar volume, as well as the methods used to find solutions to the Poisson-like equations with their corresponding boundary conditions. We present sequences of quasi-stable circular orbits which conserve baryonic mass. These sequences mimic the time evolution of the inspiral and are obtained without solving the complex evolution equations. They also provide sets of initial value data for future time evolution codes, which should be valid very close to the final merger. We evaluate the emission of gravitational radiation during the evolution through multipole expansions methods.

## Neutron Star Size

A team led by researchers at the Albert Einstein Institute (AEI) in Germany took those observations and then combined them with models of how subatomic particles behave in the extremely dense conditions inside neutron stars. While it’s impossible to recreate such conditions in labs on Earth, the physicists showed that they could use existing theory to extrapolate their calculations from the tiniest scales out to what’s happening in distant neutron stars.

Their results suggest that neutron stars must be between 13 and 15 miles across. And a typical neutron star should be about 13.7 miles wide. The estimates place tighter constraints on neutron star size than previous studies.

“Neutron stars contain the densest matter in the observable universe,” AEI researcher and study author Collin Capano said in a media release . “In fact, they are so dense and compact that you can think of the entire star as a single atomic nucleus, scaled up to the size of a city. By measuring these objects’ properties, we learn about the fundamental physics that governs matter at the subatomic level.”

## Neutron stars in frames of R 2 -gravity and gravitational waves

The realistic models of neutron stars are considered for simple R + α R 2 gravity and equivalent Brance–Dicke theory with dilaton field in Einsein frame. For negative values of α we have no acceptable results from astrophysical viewpoint: the resulting solution for spherical stars doesn’t coincide with Schwarzschild solution on spatial infinity. The mass of star from viewpoint of distant observer tends to very large values. For α > 0 it is possible to obtain solutions with required asymptotics and well-defined star mass. The mass confined by stellar surface decreases with increasing of α but we have some contribution to mass from gravitational sphere appearing outside the star. The resulting effect is increasing of gravitational mass from viewpoint of distant observer. But another interpretation take place in a case of equivalent Brance–Dicke theory with massless dilaton field in Einstein frame. The mass of star increases due to contribution of dilaton field inside the star. We also considered the possible constraints on R 2 gravity from GW 170817 data. According to results of Bauswein et al. the lower limit on threshold mass is 2 . 7 4 − 0 . 0 1 + 0 . 0 4 M ⊙ . This allows to exclude some equations of state (EoS) for dense matter. But in R 2 gravity the threshold mass increases for given EoS with increasing of α . In principle it can helps in future discriminate between General Relativity and square gravity (of course one need to know EoS with more accuracy rather than now).

## Most Massive Neutron Star Breaks Cosmic Record

Astronomershave found the most massive neutron star yet measured ? one nearly twice themass of our sun. The discovery indicates that, as their name suggests, these stellarremnants really are made mostly of neutrons, as opposed to more exoticparticles.

Neutronstars are fast-spinning remnants left behind in the aftermath ofsupernovas: huge star explosions where protons are crushed together withelectrons to form neutrons. They are typically small, with diameters of about12 miles (19.3 km) or so, but yet so massive they weigh as much as the sun.

Butthe new, precise neutron star measurements have revealed an object more massivethan any neutron star yet observed. At nearly twice the mass of our sun, thestar is about 20 percent more massive than the last neutron star record-holderof 1.67 solar masses. [Top 10Star Mysteries]

"We didn't really know for sure that neutron stars could getquite this massive until we made this measurement ? it was very surprising andexciting," researcher Paul Demorest, an astronomer at the National RadioAstronomy Observatory, told SPACE.com. "The typical thinking was that mostneutron stars clustered pretty tightly around 1.4 solar masses."

Whilestars come in all sizes and can be dozens or hundreds of times the mass of thesun, neutron stars ? because of their properties ? are unique in thatastronomers have long-thought they were limited to masses around 1.4 times solarmasses.

The record-breaking neutron star is called PSR J1614-2230 and isroughly 3,000 light-years from Earth.

What's it really made of?

Neutronstars are made of ultra-dense matter. A chunk of a neutron star the size of asugar cube can weigh about 100 million tons. This extraordinary density makes neutronstars ideal ways to study the densest and exotic states of matter known tophysics that require far too much energy to replicate in stable form here onEarth.

Whileastronomers have long thought that neutron stars are composed solely ofneutrons, some scientists have recently proposed they might also contain moreexotic subatomic particles as well, such as hyperons and kaon condensates,which possess so-called "strangequarks."

Althoughquarks ? the building blocks of protons and neutrons ? are generally thought toalways be confined atomic nuclei in nature, some researchers had also suggestedneutron stars might contain unbound "free quarks."

Mostmassive neutron star

Tolearn more about neutron stars, investigators focused on PSR J1614-2230, whichis a millisecond pulsar, a neutron star that emits radio pulses and spinscompletely around roughly every three thousandths of a second. Millisecondpulsars spin very reliably, serving as verystable timekeepers ? changes of even a few millionths of a second can bedetected.

Thispulsar is a binary, in mutual orbit with a companion star, a white dwarf.

Todetermine the neutron star's mass, researchers measured a delay in the traveltime of its radio pulses resulting from them getting distorted by the companionstar's gravitational field. This effect, called the Shapiro delay, variessystematically as the paired stars orbit each other, and precise analysis of itallowed scientists to determine the white dwarf's mass.

Sincethe investigators know the orbital characteristics of the binary system as awhole, knowing the companion star's mass enabled them to calculate the pulsar'smass as well.

"Wegot very lucky with this system," said researcher Scott Ransom, anastronomer at the National Radio Astronomy Observatory in Charlottesville, Va.

Thepaired stars are in an orbit almost perfectly edge-on from Earth, making thevariation in the distortions of the radio pulses more pronounced, researcherssaid. Also, the white dwarf is unusually massive for a star of its type,meaning its gravitational field had an especially profound effect on thepulses.

"Thisunique combination made the Shapiro delay much stronger and thus easier tomeasure," Ransom added.

Thescientists narrowed the pulsar's mass to 1.97 times the mass of the sun, giveor take 0.04 solar masses.

Thishigh mass rules out nearly all currently proposed models for neutron starmatter that involve exotic particles such as hyperons and kaon condensates,Demorest explained. Those exotic particles are more essentially squishier thanneutrons, and if a neutron star that massive did possess those particles, it could squeeze together so much that it would collapseinto a black hole.

Althoughthe matter in the neutron stars could be made of quark matter, it could onlysupport a star this massive if they strongly interact with each other as theydo in normal atomic nuclei and not if they were free, added researcher FeryalOzel of the University of Arizona.

Thesenew findings could also shed light on the origins of gamma ray bursts, the mostpowerful explosions in the universe. A leading explanation for the cause of onetype of gamma-rayburst ? the "short-duration" bursts ? is that they are caused bycolliding neutron stars. The fact that neutron stars can be as massive as PSRJ1614-2230 hints these collisions would be powerful enough to generate thesebursts.

"Pulsarsin general give us a great opportunity to study exotic physics, and this systemis a fantastic laboratory sitting out there, giving us valuable informationwith wide-ranging implications," Ransom said. "It is amazing to methat one simple number ? the mass of this neutron star ? can tell us so muchabout so many different aspects of physics and astronomy."

Theresearch is detailed in the Oct. 28 issue of the journal Nature. They will alsodetail their calculations regarding neutron stars and free quarks inAstrophysical Journal Letters.

## Neutron star size in different reference frames - Astronomy

In this paper we present a compilation of results from our most advanced neutron star merger simulations. Special aspects of these models were refered to in earlier publications (Ruffert & Janka cite Janka et al. cite), but a description of the employed numerical procedures and a more complete overview over a large number of computed models are given here. The three-dimensional hydrodynamic simulations were done with a code based on the Piecewise Parabolic Method (PPM), which solves the discretized conservation laws for mass, momentum, energy and, in addition, for the electron lepton number in an Eulerian frame of reference. Up to five levels of nested cartesian grids ensure higher numerical resolution (about 0.6 km) around the center of mass while the evolution is followed in a large computational volume (side length between 300 and 400 km). The simulations are basically Newtonian, but gravitational-wave emission and the corresponding back-reaction on the hydrodynamic flow are taken into account. The use of a physical nuclear equation of state allows us to follow the thermodynamic history of the stellar medium and to compute the energy and lepton number loss due to the emission of neutrinos. The computed models differ concerning the neutron star masses and mass ratios, the neutron star spins, the numerical resolution expressed by the cell size of the finest grid and the number of grid levels, and the calculation of the temperature from the solution of the entropy equation instead of the energy equation. The models were evaluated for the corresponding gravitational-wave and neutrino emission and the mass loss which occurs during the dynamical phase of the merging. The results can serve for comparison with smoothed particle hydrodynamics (SPH) simulations. In addition, they define a reference point for future models with a better treatment of general relativity and with improvements of the complex input physics. Our simulations show that the details of the gravitational-wave emission are still sensitive to the numerical resolution, even in our highest-quality calculations. The amount of mass which can be ejected from neutron star mergers depends strongly on the angular momentum of the system. Our results do not support the initial conditions of temperature and proton-to-nucleon ratio needed according to recent work for producing a solar r-process pattern for nuclei around and above the A

130 peak. The improved models confirm our previous conclusion that gamma-ray bursts are not powered by neutrino emission during the dynamical phase of the merging of two neutron stars.

## Neutron star size in different reference frames - Astronomy

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Vision Across the Full Spectrum: The Crab Nebula, from Radio to X-ray

The Crab Nebula (Messier 1) is the remnant of a supernova that exploded in the year 1054 AD. This mysterious “new star,” as early sky watchers called it, was observed around the world and most notably recorded by Chinese astronomers. The supernova was triggered when the progenitor star abruptly collapsed onto its iron core, and rebounded to expel most of its layers of gas into a blast wave. This wave is seen as an optical and infrared set of filaments that continues to impact surrounding material. This material was expelled from the dying red giant progenitor star 20,000 years prior to the supernova. The ultra-dense remnant core, called a neutron star, is crushed to the size of a city. Spinning furiously, the neutron star sends out twin beams of radiation, like a lighthouse. A lot of this energy comes from the neutron star’s intense magnetic fields.

The initial radio image (from the Very Large Array) shows the cool gas and dust blown out by the supernova winds. The infrared (Spitzer) image shows synchrotron radiation, an unusual form of light produced by electrons trapped in magnetic fields. The infrared image also shows hot gas. The visible-light image (Hubble) shows the detailed filamentary structure of the blast wave as it impacts the surrounding material. The ultraviolet image (XMM-Newton) shows hot, ionized gas. Finally, the X-ray emission (Chandra) from high-energy particles ejected from the pulsar shows the expanding nebula. The bipolar structure represents a powerful jet of material funneled along the neutron star’s spin axis.

NASA, ESA, and G. Bacon (STScI)
Radio image: VLA/NRAO/AUI/NSF
Infrared image: NASA/Spitzer/JPL-Caltech
Optical image: NASA, ESA, and Hubble (STScI)
UltraViolet image: XMM-Newton/ESA
X-ray image: NASA/Chandra/CXC