Astronomy

Are there any reliable optical measurements of the radius of a neutron star?

Are there any reliable optical measurements of the radius of a neutron star?


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I know that there is plenty of theory to predict the size of a neutron star: my question is whether or not there are any reliable size estimates based on observation. Please note that, to be reliable, such an observation would need to have also a good estimate for the neutron star's distance from us, which must necessarily be reckoned from some independent source, i.e., it must not be reckoned from the apparent size of the object, or from theory, otherwise one is guilty of circular reasoning!


Yes, the neutron star RX J1856.5-3754 has an observed* radius of 17 km. After accounting for general relativity, its actual radius is calculated as 14 km. It is not the only neutron star whose radius is known.


* It's worth mentioning how astronomers arrived at this radius. Technically, fitting the X-ray data to a blackbody arrives at a calculation of a 5 km radius, which seems unusually small, so for a time it was believed that this star may be something more exotic, like a quark star. However, after atmospheric modeling based on temperature, a more accurate estimate of the emission radius was derived to be 17 km. Once gravitational redshift was accounted for, the value of 14 km was yielded. See Magnetic Hydrogen Atmosphere Models and the Neutron Star RX J1856.5−3754 - Wynn C. G. Ho, et al.


Neutron stars

Last time, we discussed the fate of stars with initial masses at least 5 or 8 times larger than the Sun's mass. At the end of their lives, they build up an inert core of iron-group elements. Since these nuclei do not produce energy when fused together, the core is eventually unable to support itself against gravity, and collapses. Today, we raise the question, "what happens to the core after it has collapsed?"

There are two main possibilities: it forms a stable and very small object called a neutron star, or it never stops collapsing and forms a black hole. We'll focus on the first possibility today, and deal with black holes in the next class.

How big is a neutron star?

As you may recall, the degeneracy pressure due to electrons can stabilize the helium or carbon/oxygen core of a star -- as long as that core contains less than about 1.4 solar masses. That pressure occurs because electrons are members of a class of sub-atomic particles called "fermions", which resist having the same quantum properties as other particles.

It turns out that neutrons are ALSO fermions, and so they, too, exhibit a degeneracy pressure when forced into very close proximity to other neutrons. During the collapse of the iron core of a massive star, electron degeneracy pressure is insufficient to halt the compression . but, in many cases, neutron degeneracy pressure can stop it. The result is, basically, a ball of neutrons, which is why these objects are known as neutron stars.

But just how big is this ball? Are there any limits to the amount of material that this degeneracy pressure can support?

The question of "how MASSIVE are neutron stars?" can be answered reasonably well. Astronomers have found a number of binary star systems in which one or both of the members are neutron stars. We can use Kepler's Third Law to convert the easily measured period, and not-so-easily measured size, of these orbits into the masses of the components.

Kepler's Third Law relates the period of two gravitationally bound objects to the size of their orbit -- and their total mass.

  • P is the period of the orbit
  • G is Universal Constant of Gravitation = 6.67 x 10 -11 N*m 2 /kg 2
  • m1, m2 are the masses of the two bodies
  • a is the semi-major axis of the orbit

Let's apply Kepler's Third Law to a very familiar system.

This same technique has revealed the mass of a number of neutron stars.


Figure 1 taken from Kiziltan et al., ApJ, 778, 66 (2013)

Most of them have masses in the range of 1.0 - 1.5 solar masses, but some reach as high as 2 solar masses. Even though the uncertainties are rather large, it is clear that neutron degeneracy pressure can support objects with masses above the Chandrasekhar limit.

Some more recent measurements suggest that the upper limit to neutron star masses may stretch as high as 2.5 solar masses, or perhaps even higher. Because we cannot create material with these properties in laboratories on Earth, the physics of neutronium are uncertain. We understand neutron stars much less well than white dwarfs.


A portion of figure 3 taken from Shao et al., Phys. Rev. D. 102, 3006 (2020)

What about the RADII of neutron stars? These are much more difficult to measure than the masses. In a very few cases, neutron stars which are accreting material from nearby companion stars show features in their emission of X-rays which can give clues to their size. The figure below shows a range of masses and radii which are consistent with the measurements of eight neutron-star systems.


Figure 1 taken from Steiner, Lattimore and Brown, ApJ 765, L5 (2013)

It's clear that neutron stars are made of stuff with crazy properties.

Young neutron stars are hot

As you might imagine, a core-collapse supernova is a very energetic event. A newly born neutron star is heated to VERY high temperatures by the collapse of its own body -- which leads to the loss of a great deal of gravitational potential energy. On top of that (literally) is all the kinetic energy of gas from the layers just outside the core that material falls down onto the surface of the neutron star and slams into it at very high speeds.

The result is a very HOT object. How hot? In a few cases, we can estimate the temperature of young neutron stars from their thermal emission.


Panel from Figure 3 taken from Grigorian, Voskresensky and Blaschke, arXiv 1702.01342 (2017)

The X-ray region in particular, for the Crab Nebula, its blackbody emission peaks at about 1.5 nm = 15 Angstroms. Young neutron stars are among the very few solid objects which can emit appreciable amounts of blackbody radiation at such short wavelengths.

Now, they also emit blackbody radiation at longer wavelengths. Because neutron stars are so small, however, their optical luminosity is pretty small. In fact, astronomers have detected optical radiation from only a handful of neutron stars. One of them is the neutron star at the center of the Crab Nebula.


Image image taken by Adam Block with the Kitt Peak Visitor Center's 0.4 m telescope. Courtesy of Adam Block, NOAO and NSF.

The neutron star is one of the three stars in the gold box near the center of the expanding gas cloud. If one looks at this region with a high-speed camera, capable of taking sub-second exposures, one will see something peculiar:

These flashes of light arrive at regular intervals of about 30 per second.


Light curve courtesy of Nicholas Law's dissertation.

Why should a neutron star emit pulses of radiation at regular intervals?

Pulses of radio emission: pulsars

It turns out that the Crab Nebula's neutron star is not alone. Many of the neutron stars we have detected also emit electromagnetic radiation in periodic flashes. In fact, the very first observational evidence for neutron stars was the anomalous radio emission from a mysterious object. Back in 1967, a graduate student named Jocelyn Bell was examining the long strips of paper which recorded the radio waves detected by a telescope near Cambridge, England. She noticed a bit of scruff at one particular location in the sky.


Image courtesy of Billthom and Wikimedia

Subsequent observations revealed that the source was present at the same location each night, and that the radio waves came in very brief pulses, at a rate of about 30 Hertz.

You can listen to some radio recordings of pulsars (transformed into audio for the benefit of those who lack innate radio receivers) at the link below. Note the wide range of pulse frequencies.

  1. the neutron stars are spinning very rapidly
  2. their VERY strong magnetic fields are mis-aligned with their rotational axes

Click on the picture below to see how this mis-alignment of rotational and magnetic axes can cause us to observe pulses of radiation.


Click on the image to start a Shockwave Flash applet demonstrating a pulsar in action.

The exact mechanism by which radio (and optical, and X-ray) emission is produced remains a bit of a mystery. Everyone agrees that it has something to do with the very, very strong magnetic fields surrounding neutron stars electrons caught in these magnetic fields are accelerated as the fields whirl around in sync with the rotation of the neutron star. But scientists haven't yet converged on all the details. As proof, try searching for the phrase "pulsar mechanism" in "Abstract Text" in the Astrophysics Data System. I just did, asking for papers published between Jan, 2019, and Oct, 2020, and received a list of 283 papers. Sheesh.

One item upon which all agree is that the emission of energy across the electromagnetic spectrum via the pulsar mechanism, whatever it is, must extract energy from the rotation of the neutron star. As the pulsar ages, it must slow down, and its period must increase. Look at the example in the figure below.

If one can measure the change in the rate at which a pulsar rotates over some period of time, one can compute the so-called spindown time of the pulsar: a rough estimate of the time it will take for the object to stop spinning (or, at least, to slow down by a significant amount).

In the case of PSR 0833-45, the time interval of 4000 days corresponds to 3.456 x 10 8 seconds. We can then compute the spindown time for this object in the following manner:

To a rough approximation, one can estimate the age of a pulsar as half the spindown time. Thus, a rough estimate of the age of PSR 0833-45 would be around 11,000 years.

There are a number of other fundamental properties which can be derived using measurements of the periods of pulsars see Chapter 6 of "Essential Radio Astronomy" from NRAO for more information.

Neutron star binary provides evidence for gravitational waves

In 2016, the LIGO project announced its first direct detection of gravitational waves. Excellent news, and a triumph of physics and engineering. But that was not the first time that scientists had found strong evidence for the existence of gravitational waves. Pulsars were at the heart of the earlier effort.

In 1975, Joe Taylor (a professor at UMass) and Russell Hulse (his graduate student) published a paper describing a most peculiar radio pulsar.


Abstract of Hulse and Taylor, ApJ 195, 51 (1975)

Observations made with the giant Arecibo radio telescope showed that the pulses from this particular object did not arrive at exactly regular intervals instead, they were sometimes ahead of schedule, and then at other times behind schedule. The differences from a strictly periodic behavior showed a pattern:


Figure 1 taken from Hulse and Taylor, ApJ 195, 51 (1975)

Hulse and Taylor deduced that this pulsar, PSR 1913+16, was part of a binary system. As it moved around the center of mass of this system, its pulses sometimes arrived a little earlier than expected (when it was on the near side of the orbit), and sometimes later than expected (when it was on the far side of its orbit). Interesting, for sure, and useful in an obvious way: it allowed us to determine the mass of the neutron star.

But Hulse and Taylor also realized that this system was interesting in a not-so-obvious way: it might allow us to test the theory of general relativity! Einstein's theory of general relativity (GR) postulated that if a sufficiently massive object was accelerated with a sufficiently large amplitude, it would emit gravitational radiation. This binary neutron-star system just might do the trick -- in large part because the measurements of the arrival times of the radio pulses could be made to VERY high precision.

So, over the next few years, Taylor and his collaborators continued to monitor PSR 1913+16 closely. The prediction from GR was that gravitational waves would energy away from the binary system, causing its orbit to shrink slightly. In 1982, they released an analysis of seven years worth of data, showing that the orbit WAS shrinking, and by exactly the amount predicted by GR.


Figure 6 taken from Taylor and Weisberg, ApJ 253, 908 (1982)

The evolution of the binary's orbit has continued to follow the prediction of GR ever since -- for the next twenty-two years or so.


Figure 1 taken from Weisberg and Taylor, in "Binary Radio Pulsars," ASP Conf. Ser. 328 (2005)

For their discovery of this binary pulsar system, and for using it to test Einstein's theory of general relativity, Hulse and Taylor were awarded the Nobel Prize for Physics in 1993.


Image courtesy of nobelprize.org

Using pulsars as markers on a map

In the days before the Internet and Google Maps, people used a variety of methods to indicate their location on a map. A signpost, for example, not only told one the direction to certain other locations, but also allowed one -- with a bit of geometry -- to fix the post's location on map.


Image courtesy of The National Museum of American History

Because radio waves can penetrate clouds of gas and dust, one can detect pulsars from all parts of the Milky Way Galaxy. In addition, each pulsar has its own unique frequency. For these reasons, a number of scientists and writers have figured that pulsars would make good galactic signposts.

In 1972, the Pioneer 10 and 11 spacecraft were being prepared for their missions to Jupiter and Saturn. After their encounters with the gas giants, each ship would be flung outward into space at very high speed, away from the Sun. They would travel out into interstellar space, escaping from the Solar System and wandering off into the depths of the Milky Way. Since there was a chance -- however remote -- that the craft might be discovered someday by alien creatures as they drifted through space, NASA decided to attach a plaque to each ship, giving such salvagers information about the Pioneers and the humans who made them. The figure below is a (slightly modified) version of the plaque. Note the diagram at left, below the picture of a hydrogen atom (that's one atom making a spin-flip transition, not a molecule).


A slightly modified version of the plaque placed on the Voyager spacecraft, courtesy of Wikipedia Click on the image for the original version.

The diagram shows lines radiating away from a point -- the location of the Sun -- each one labelled with a binary number. The lines represent 14 different pulsars, and the numbers the period of each one. Each period is written in terms of the period of the 21-cm spin-flip transition of neutral hydrogen.


Closeup of the pulsar map portion of the Voyager plaque, courtesy of Wikipedia

By identifying all 14 pulsars, and then figuring out the location at which they would appear at the direction and distance indicated in the diagram, one can determine the location of the Earth. To make things easier, the long horizontal line extending from the Earth's location to the right, behind the pictures of the man and woman, all the way to the right-hand edge of the plaque, indicates the direction of the center of the Milky Way.


Are there any reliable optical measurements of the radius of a neutron star? - Astronomy

We present the set of deep Neutron Star Interior Composition Explorer (NICER) X-ray timing observations of the nearby rotation-powered millisecond pulsars PSRs J0437-4715, J0030+0451, J1231-1411, and J2124-3358, selected as targets for constraining the mass-radius relation of neutron stars and the dense matter equation of state (EoS) via modeling of their pulsed thermal X-ray emission. We describe the instrument, observations, and data processing/reduction procedures, as well as the series of investigations conducted to ensure that the properties of the data sets are suitable for parameter estimation analyses to produce reliable constraints on the neutron star mass-radius relation and the dense matter EoS. We find that the long-term timing and flux behavior and the Fourier-domain properties of the event data do not exhibit any anomalies that could adversely affect the intended measurements. From phase-selected spectroscopy, we find that emission from the individual pulse peaks is well described by a single-temperature hydrogen atmosphere spectrum, with the exception of PSR J0437-4715, for which multiple temperatures are required.


3. Observations and Radius Measurements of Individual Sources

3.1. Thermonuclear Bursters

There are five sources for which thermonuclear burst data have been previously used to measure neutron star radii using their apparent angular sizes, touchdown fluxes, and distances (see Table 1).

We also include in the present analysis 4U 1724 − 307, for which Suleimanov et al. (2011) reported a radius measurement based on the spectral evolution during the cooling tail of one long burst observed from this source. As discussed in Güver et al. (2012b), the spectra from that long burst used in the Suleimanov et al. (2011) study are significantly different from blackbodies and from model atmosphere spectra, resulting in χ 2 / d.o.f. in the range 1-8 in the spectral fits (see also in’t Zand & Weinberg 2010). This indicates significant contamination of the surface emission, either by the accretion flow or by atomic lines from the ashes of the burst that have been brought up to the photosphere, which makes the results unreliable. Instead, we make use of the cooling tails of the two normal bursts observed from 4U 1724 − 207 to determine the apparent angular size (see Güver et al. 2012b). The spectra from these two bursts show the expected thermal shape and result in acceptable values for χ 2 /d.o.f. We also make use of the touchdown flux measured from these bursts (Güver et al. 2012a) when determining the neutron star radius.

Since the earliest measurements, Güver et al. (2012a,b) conducted studies on the entire RXTE burst dataset and found ∼ 10 % systematic uncertainties in the apparent angular sizes and the touchdown fluxes in the most prolific bursters. In addition, Güver et al. (2015) placed an upper limit of ∼ 1 % on the systematic differences in the flux calibration between RXTE and Chandra , which, in principle, can affect the measured burst fluxes.

We reanalyze the data on these six sources uniformly, following the procedures used and described in Güver et al. (2012 a,b). Specifically, (i) we apply the appropriate deadtime correction to the observed countrates, which leads to a small increase in the angular sizes and touchdown fluxes for the brightest sources. (ii) We employ a Bayesian Gaussian-mixture model to quantify the intrinsic scatter in the measurements of the angular sizes and the touchdown fluxes. This is typically larger than the formal uncertainties in the measurements and increases the uncertainties in the inferred radii. (iii) When the number of Eddington-limited bursts of an individual source is too small to assess the scatter in the touchdown flux, we take an 11% systematic uncertainty in this quantity following the analysis of Güver et al. (2012a) on the sample of sources with limited number of bursts. (iv) We add an uncertainty of 1% in the apparent angular sizes and the touchdown fluxes to account for the flux calibration uncertainties. We summarize all the measurements in Table 1 and discuss the additional details of the source distances and atmospheric compositions for individual sources below. Note that the uncertainties in Table 1 do not include the 1% flux calibration uncertainties, which we add in quadrature when inferring the radii. We also list the ID numbers of the bursts used in this study in Table A2 of the Appendix, following the numbering system used in Galloway et al. (2008a).

Figure 3.— (Top Left) The evolution of the flux and temperature measured for all the spectra in the cooling tails of bursts from 4U 1820 − 30. The diagonal red lines show the best-fit blackbody normalization and its 1 σ uncertainty. (Top Right) The 1- and 2 σ confidence contours over the blackbody normalization and temperature measured at the touchdown moment in the PRE bursts. The black diagonal lines correspond to contours of constant touchdown flux. (Bottom Left) The black solid curve shows the 68% confidence contour over the mass and radius of 4U 1820 − 30 obtained by combining all the measurements and priors the filled circle marks the location of the highest likelihood. The solid blue and green lines denote the mass-radius curves obtained from the most likely values of the apparent angular size and Eddington flux, respectively, while the dashed curves denote the 1 σ uncertainties of these measurements. (Bottom right) The posterior likelihood over the neutron star radius after the two-dimensional likelihoods are marginalized over mass. The dashed line assumes a flat prior over mass while the solid line assumes as a prior the observationally inferred mass distribution of recycled pulsars, as discussed in the text.
Figure 4.— Same as in Figure 3 but for SAX J1748.9 − 2021.

3.1.1 4u 1820 − 30

4U 1820 − 30 is an ultracompact binary in the metal-rich globular cluster NGC 6624. Güver et al. (2010) discussed two distance measurements performed in the optical (Kuulkers et al. 2003) and in the near-IR bands (Valenti et al. 2007). The first gives a distance of 7.6 ± 0.4 kpc and the second gives 8.4 ± 0.6 kpc. Harris et al. (1996 2010 revision) find a compatible distance estimate of 7.9 kpc in the optical band, with an uncertainty of 0.4 kpc 1 1 1 see http://physwww.physics.mcmaster.ca/ ∼ harris/mwgc.ref . Without any further information to choose between the optical and the near-IR measurements, Güver et al. (2010b) combined them into a single boxcar likelihood between 6.8 to 9.6 kpc, which placed more than warranted likelihoods at the shortest and intermediate distances. Here, we instead opt to use a double Gaussian likelihood with means and standard deviations that reflect the individual measurements of Kuulkers et al. (2003) and Valenti et al. (2007) and give equal integrated likelihood to each.

The fact that the neutron star is in a 11.4 minute binary (Stella et al. 1987) requires that it is fed by a degenerate dwarf companion that is free of hydrogen. For this reason, when inferring the radii, we set the hydrogen abundance to X = 0 . No burst oscillations or persistent pulsations have been observed from this source. Because of this, when applying spin corrections to the apparent angular size, we assume a flat prior in spin between 250 and 650 Hz, as discussed in Section 2.3.

The top left panel of Figure 3 shows the evolution of the flux and temperature during the cooling tails of five bursts observed from 4U 1820 − 30, while the top right panel shows the 68% and 95% confidence contours in the measurement of the blackbody normalization vs. temperature during the touchdown phases in the five Eddington-limited bursts. Because the intrinsic scatter in the touchdown flux of this source is very small, we assign an 11% uncertainty to this measurement as discussed above.

The lower left panel shows the 68% and 95% confidence contours over the mass and radius of 4U 1820 − 30 inferred within the Bayesian framework discussed in Section 2.3, along with the contours of constant apparent angular size (blue) and touchdown flux (green) obtained for this source. The lower right panel shows the likelihood over the radius when we marginalized the two-dimensional likelihood over mass. We do this for a flat prior on mass between 0 and 3 M ⊙ as well as for the observed mass distribution of fast radio pulsars, which are the descendants of the low-mass X-ray binaries that make up our sample. As discussed in Özel et al. (2012b), the latter mass distribution can be represented by a Gaussian with a mean of 1.46 M ⊙ and a dispersion of 0.21 M ⊙ . The difference in the result between using the two different priors over the mass is minor. (Note that we use the full two-dimensional likelihoods without these observational priors on mass when inferring the parameters of the equation of state.)

Figure 5.— Same as in Figure 3 but for EXO 1745 − 248.

3.1.2 Sax j1748.9 − 2021

The transient neutron star X-ray binary SAX J1748.9 − 2021 is located in the globular cluster NGC 6440, which is a massive and old cluster in the Galactic bulge. Two optical and one near-IR studies give consistent and well-constrained distances to NGC 6440: Kuulkers et al. (2003) reported 8.4 + 1.5 − 1.3 kpc, Harris et al. (2010) found 8.5 kpc, while Valenti et al. (2007) found 8.2 ± 0.6 kpc using near-IR data. In this last study, the distance uncertainty is improved and takes into account the systematic errors introduced by the method of comparing the properties of NGC 6440, including its metallicity and age, to the reference cluster. Because the central values of the two measurements differ by less than the 1 σ uncertainty of either, we adopt here the latter distance and its uncertainty.

SAX J1748.9 − 2021 has a spin frequency of 420 Hz, detected during intermittent pulsations observed in the persistent emission (Altamirano et al. 2008). The same study also found a binary orbital period of 8.7 hr. Because there is no specific information about the evolutionary state of the donor, we take a flat prior in the hydrogen mass fraction between 0 and 0.7.

The top left panel of Figure 4 shows the evolution of the flux and temperature during the cooling tails of four bursts observed from SAX J1748.9 − 2021, while the top right panel shows the 68% and 95% confidence contours in the measurement of the blackbody normalization vs. temperature during the touchdown phases in its two Eddington-limited bursts.

The lower panels of Figure 4 show the 68% confidence contour in mass and radius as well as the posterior likelihood marginalized over mass using the Bayesian framework and priors discussed above.

Figure 6.— Same as in Figure 3 but for KS 1731 − 260.

3.1.3 Exo 1745 − 248

EXO 1745 − 248 is located in Terzan 5, one of the most metal-rich globular clusters in the Galaxy. The distance to Terzan 5 was obtained using HST/NICMOS data (Ortolani et al. 2007). The sources of uncertainty in the distance measurement were discussed in detail in Özel et al. (2009). We adopt here the same flat likelihood over distance centered at 6.3 kpc with a width of 0.63 kpc.

No burst oscillations or persistent pulsations have ever been observed from EXO 1745 − 248. As before, we adopt a flat prior over its spin frequency between 250 and 650 Hz when calculating the spin corrections to the apparent angular size. The nature of the companion of EXO 1745 − 248 is ambiguous (Heinke et al. 2003). While the empirical comparison of its spectrum to those of ultracompact sources suggested an ultracompact binary with a hydrogen-poor companion, the identification of a possible infrared counterpart leaves open the possibility of a hydrogen-rich donor. To account for both possibilities, we take a flat prior over the hydrogen mass fraction in the range X = 0 − 0.7 .

The top panels of Figure 5 show the evolution of the flux and temperature during the cooling tails of two bursts (left) and the 68% and 95% confidence contours in the measurement of the blackbody normalization vs. temperature during the touchdown phases of these two Eddington-limited bursts (right).

The lower panels of Figure 5 show, as before, the 68% confidence contour in mass and radius (left) derived in the Bayesian framework from the measurements of the apparent angular size, touchdown flux, and the distance, as well as the posterior likelihood marginalized over mass (right).

3.1.4 Ks 1731 − 260

KS 1731 − 260 is a binary in the Galactic bulge, lying in the direction of Baade’s window. Özel et al. (2012a) derived a distance prior to this source based on the stellar density along the line of sight. We use the same numerical prior in the current study, which places KS 1731 − 260 at a distance of approximately 7 − 9 kpc.

KS 1731 − 260 has a spin frequency of 524 Hz based on the detection of burst oscillations (Smith et al. 1997). Its optical counterpart has been identified (Zurita et al. 2010) and the duration and the energetics of some of its X-ray bursts point to accreted fuel that is hydrogen-rich. Nevertheless, because there is no conclusive evidence on the hydrogen content of the bursts we analyze here, we allow for a flat distribution in the hydrogen mass fraction X between 0 and 0.7.

We show in the top panels of Figure 6 the flux vs. temperature observed during the cooling tails of twenty four X-ray bursts used for the measurement of the apparent angular size and the 68% and 95% confidence contours in the blackbody normalization and temperature measured during the touchdown phases of two Eddington limited bursts.

The lower left panel of Figure 6 shows the 68% confidence contours over the mass and radius of KS 1731 − 260 inferred within the Bayesian framework, along with the contours of constant apparent angular size (blue) and touchdown flux (green) obtained for this source. The lower right panel shows the likelihood over the radius when we marginalized the two-dimensional likelihood over mass.

Figure 7.— Same as in Figure 3 but for 4U 1724 − 207.

3.1.5 4u 1724 − 207

4U 1724 − 207 lies in the globular cluster Terzan 2. Early studies of the distance to this cluster by Ortolani et al. (1997) obtained a distance of 5.3 or 7.7 kpc, depending on whether the selective extinction, R, was set to 3.1 or 3.6. The most recent study by Valenti et al. (2012) used near-IR observations of red giant branch stars and led to a distance of 7.4 kpc. Independent of the reddening or color-magnitude measurements, one can statistically argue that the distance of Terzan 2 should be the same as the distance to the Galactic center (Racine & Harris 1989). This is because the whole system of globular clusters is centrally concentrated around the Galactic center and, given the fact that the direction of Terzan 2 is within the Galactic bulge region, it is likely that its distance is close to 8.0 kpc (Reid 1993). Based on these arguments and in order to avoid using a measurement that depends strongly on the assumed extinction, we adopt the recent measurement of Valenti et al. (2012) of 7.4 ± 0.5 kpc. The error primarily reflects the systematic uncertainty in the measurements of the distances to globular clusters, as estimated by using 47 Tuc as a reference.

There have been no studies on the composition of the companion to 4U 1724 − 207 and no detected burst oscillations or persistent pulsations from this source. For this reason, when inferring its radius, we use a flat distribution in hydrogen abundance between X=0 and X=0.7 and a flat distribution in spin frequency between 250 and 650 Hz.

In the top left panel of Figure 7 , we show the flux vs. temperature diagram during the cooling tails of the bursts for which the blackbody model provides an acceptable fit to the data (see discussion in Güver et al. 2010b). In the top right panel, we show the 68% and 95% confidence contours in the measured blackbody normalization vs. temperature during the touchdown phases of two Eddington-limited bursts. Finally, in the lower two panels of the same figure, we show (left) the 65% confidence contours in the inferred mass and radius of 4U 1724 − 207 and (right) the posterior likelihood over radius, after we marginalize over the mass of the neutron star.

Figure 8.— Same as in Figure 3 but for 4U 1608 − 52. In the upper left panel, the blue points correspond to the main cooling tail and the black points denote the outliers, as discussed in the Appendix.

3.1.6 4u 1608 − 52

4U 1608 − 52 lies in the Galactic disk. The distance to this source was measured in Güver et al. (2010a) by comparing the extinction obtained from the red clump stars along the line of sight to the extinction to the binary inferred from the high energy-resolution X-ray observations. We repeat this analysis in the Appendix, utilizing a new Chandra observation and the latest relation between the optical extinction A V and the hydrogen column density N H obtained in a new study (Foight et al. 2015). The new results place a lower limit of 3 kpc on the distance and give the highest likelihood at ∼ 4 kpc.

4U 1608 − 52 is the fastest spinning source in the current sample, with a spin frequency of 620 Hz (Hartman et al. 2003). As with the other sources whose companions do not have known compositions, we take a boxcar prior for the hydrogen mass fraction between X = 0 and X = 0.7 .

In Güver et al. (2012b), we developed a Bayesian Gaussian-mixture method for outlier detection and measuring the systematic uncertainties in the apparent angular size during the cooling tails of the bursts. 4U 1608 − 52 is the only source in the present sample for which such outliers were detected in the observed bursts. We discuss these in the Appendix. The top left panel of Figure 8 uses different color symbols to distinguish the main sequence of the cooling tail from these outliers.

The upper right panel in the same figure shows the confidence contours in the blackbody normalization and temperature in the touchdown moments of the three Eddington limited bursts, which includes the newly detected burst during the simultaneous RXTE and Chandra observations (see Appendix A2). Finally, the lower panels of Figure 8 show the 68% confidence contour in mass and radius (left) as well as the posterior likelihood marginalized over mass (right).

As with the other sources, these Eddington-limited bursts were selected using the robust photospheric radius expansion (PRE) criteria outlined in Güver et al. (2012a) and form a much smaller sample than those initially identified as potential PRE events by Galloway et al. (2008a). The earlier selection criteria of Galloway et al. (2008) admitted a large number of non-PRE bursts into the sample, because they were based primarily on the non-monotonic evolution of the inferred apparent radii after the peak of each burst. As discussed in Güver et al. (2012a), a careful scrutiny of these bursts clearly demonstrates that the inferred touchdown fluxes (had they been PRE events) are much smaller than the peak fluxes seen in the brightest (true) PRE bursts, in a way that cannot be accounted for by the change in the general relativistic redshift 2 2 2 This is the same argument used in Galloway et al. (2008b) to reject bursts from high inclination sources. . In addition, the inferred photospheric radii during these misidentified PRE events are comparable to the asymptotic radii of the same bursts. For these reasons, they are not PRE events and do not pass the criteria of Güver et al. (2012a).

Figure 9.— A comparison between the prior and the posterior likelihoods over the blackbody normalization, the touchdown flux, and the distance to 4U 1820 − 30. When the measured systematic uncertainties in the two spectroscopic measurements as well as the full prior likelihood over the distance are taken into account, there is no evidence for inconcistencies between the observables.

3.1.7 Comparison with Previous Work

There are some differences in the mass-radius contours presented here compared to our earlier studies of the same sources. The primary reasons for these differences were discussed at the end of section 3.1 and include applying appropriate deadtime corrections to the observed countrates, incorporating the measured intrinsic scatter in the measurements (beyond the statistical uncertainties), applying spin and temperature corrections to the apparent angular sizes and touchdown fluxes, and using a Bayesian method to infer the masses and radii from the observables that does not suffer from the biases of the earlier frequentist approach. All of these improvements in the analysis methods lead to most likely values for the radii that are ≲ 1 k m larger than before (compare with Özel et al. 2010 Özel et al. 2012a Güver & Özel 2013) but with 68% contours that encompass the most likely radii of the earlier studies.

Our analysis and results are different from those of Steiner et al. (2010, 2013) and especially on the upper range of likely values of radii. Steiner et al. (2010) used the measurements reported in Özel et al. (2009) and Güver et al. (2010a,b) but explored a number of different possibilities, including varying the location of the photosphere at the point we identify as the touchdown in a PRE burst. Their analysis favored the assumption that the photospheric radius at that point is much larger than the neutron star radius such that the general relativistic redshift is negligible. They followed this approach because they argued that the two spectroscopic measurements from each source are otherwise inconsistent with each other. In Özel & Psaltis (2015), we demonstrated that this potential inconsistency is alleviated when the true systematic scatter in the measurements is taken into account. Moreover, we showed in Figure 1 the role that the rotational correction to the angular size and the temperature correction to the Eddington flux play in determining the consistency of observables. When these corrections are not taken into account, two highly accurate measurements of these quantities will appear to be inconsistent with each other and will not lead to a solution for the neutron star mass and radius.

As the lower left panels of Figures 3-8 show, taking these effects into account makes the two spectroscopic observables in all sources consistent with each other at the 68% level even when only the most likely value of the distance and the central value of the color correction factor are considered. This is further illustrated in Figure 9 , which compares the prior and posterior likelihoods of the blackbody normalization, the touchdown flux, and the distance for 4U 1820 − 30 this is the source for which Steiner et al. (2010) made the argument that the solutions were the least consistent. As is evident from this figure, combining the two observables lead to posterior likelihoods that are well within the prior likelihoods, indicating a high level of consistency. We report in Table A3 of the Appendix the posterior likelihoods over each of the three measured quantities for all of the six thermonuclear burst sources used in this study. In all of the cases, the central values of the posterior likelihoods are within the 68% range of the prior likelihoods shown in Table 1, pointing again to a high degree of consistency between the measurements that are used to infer the neutron star radii.

Because the new analysis eliminates the concern over the consistency of solutions, it does not force us into the astrophysically unreasonable assumption of Steiner et al. (2010) that the photosphere at what we identified as the touchdown point is much larger than the neutron star radius. This was problematic for two reasons. First, in order for the blackbody normalization to remain small at that point while the photospheric radius is still extremely large, the color correction factor needs to be unphysically large i.e, larger by factors of three or more than what the atmosphere models predict. Second, within 1-2 time bins, as the photosphere settles onto the neutron star, the color correction factor would need to evolve in such a way that it exactly cancels out the change in the photospheric radius, keeping a constant blackbody normalization. We do not need to make these implicit assumptions in the present study.

Our radii are significantly smaller than those reported by Suleimanov et al. (2011) and Poutanen et al. (2014) who selected bursts and obtained radius measurements using the evolution of the blackbody normalization during the cooling tails of 4U 1724 − 207 and 4U 1608 − 52, respectively. In the case of Suleimanov et al. (2011), the selection criteria identified one burst. Unfortunately, as shown in Güver et al. (2012b) and discussed earlier, the spectra from these bursts are inconsistent with the atmosphere models, leading to reduced χ 2 values in the 2 − 8 range, rendering them unsuitable for radius measurements.

Poutanen et al. (2014) selected bursts from 4U 1608 − 52 by requiring the bursts to follow the trends expected from the bursting neutron star atmosphere models of Suleimanov et al. (2012) at near-Eddington fluxes. In Özel et al. (2015), we showed that this criterion is not useful for burst selection from RXTE data for three reasons. First, the spectral evolution at the end of a photospheric radius expansion episode occurs too rapidly to be resolved with the current data, because over the typical 0.25 s time bin used to extract spectral parameters, the flux evolves by ∼ 10 % . This is exactly the range of fluxes near the Eddington limit that one needs to resolve in order to see the expected evolution of the color correction factor. Second, the scatter in the blackbody normalization due to even a mild change in the emitting area (due to, e.g., uneven burning or an evolving photosphere) masks the theoretical trends. Finally, the correlated measurement uncertainties between the blackbody normalization and temperature further smear any trends. By not taking these data limitations into account, Poutanen et al. (2014) selected a set of bursts that are not actual PRE bursts, contrary to the implicit assumption in their method.

As discussed above, in both the Suleimanov et al. (2011) and Poutanen et al. (2014) studies, applying these theoretically motivated criteria led to selection of bursts that are inconsistent with the framework of the method: in the former by selecting spectra that are clearly not described by their atmopsheric models and, in the latter, by comparing models of the color correction factor evolution near the Eddington limit to bursts that have not reached it. Kajava et al. (2014) tried to generalize this selection procedure to several more sources (without reporting any additional radius measurements) but also did not consider the limitations of the data. We conclude that with the present data, the application of this procedure motivated by the spectral models leads neither to unbiased data selection, nor to reliable radius measurements.

3.2. Quiescent Low-Mass X-ray Binaries

The second group of sources on which radius measurements have been performed are the accreting neutron stars in low-mass X-ray binaries during their quiescent epochs (qLMXBs). It is thought that, in quiescence, neutron stars reradiate the heat stored in the deep crust during the accretion phases through a light element atmosphere (Brown et al. 1998). This allows interpreting the observed thermal spectra as surface emission from atmospheres in radiative equilibrium, while allowing for the presence of a weak power-law spectral component at higher energies due to residual accretion. Because of the very short settling time of heavier elements in a neutron-star atmosphere, the photospheres of such neutron stars in quiescence are expected to be composed of hydrogen, unless the companion star is hydrogen poor. In that case, they will be composed of helium.

A number of qLMXBs in globular clusters has been observed with Chandra and XMM- Newton . Because they are very faint and are located in crowded fields, the high angular resolution and low background of these instruments were crucial for obtaining spectroscopic constraints of their apparent angular sizes (e.g., Heinke et al. 2006 Webb & Barret 2007 Guillot et al. 2011).

Guillot et al. (2013) and Guillot & Rutledge (2014) performed a uniform analysis of six sources in this category, which are the neutron stars located in the globular clusters M13, M28, M30, ω Cen, NGC 6397, and NGC 6397. These observations include those summarized in Table 1 of Guillot et al. (2013) as well as the Chandra observations of the qLMXB in M30 (ObsID 2679 Lugger et al. 2007) and of the qLMXB in ω Cen (ObsIDs 13726 and 13727), as described in Guillot & Rutledge (2014). These two studies fit the extracted spectra with hydrogen atmosphere models to measure the apparent angular sizes for these neutron stars. They explored the dependence of the results on different hydrogen model atmosphere spectra used. They also allowed for a Gaussian distribution of errors in distances (albeit narrower than the uncertainties we assigned above to bursters in globular clusters) when fitting all of the sources simultaneously.

Source N H a a NGC6397 was fitted with a Helium atmosphere model ( nsx in XSPEC). k T e f f P.L. Norm. b b “p” indicates that the posterior distribution did not converge to zero probability within the hard limit of the model. Distance c c References: 1. Harris et al. (1996, 2010 revision) 2. Servillat et al. (2012) 3. Carretta et al. (2000) 4. Lugger et al. (2007) 5. Watkins et al. (2013) 6. see also the discussion in Heinke et al. (2014) 7. Guillot et al. (2013) and references therein 8. Heinke et al. (2014) Radius d d The radius and its 68% uncertainty obtained by marginalizing the mass-radius likelihood of each source over the observed mass distribution, as in Figure 12 .
( 10 22 c m − 2 ) (eV) ( 10 − 7 k e V − 1 s − 1 c m − 2 ) (kpc) (km)
M13 0.02 + 0.04 − 0.02 p 81 + 27 − 12 4.2 + 3.6 − 3.1 p 7.1 ± 0.4 1 1 footnotemark: 10.9 ± 2.3
M28 0.30 + 0.03 − 0.03 128 + 35 − 13 8.3 + 4.9 − 4.7 p 5.5 ± 0.3 2 2 footnotemark: 8.5 ± 1.3
M30 0.02 + 0.03 − 0.02 p 96 + 30 − 13 9.3 + 5.4 − 5.3 p 9.0 ± 0.5 3,4 3,4 footnotemark: 11.6 ± 2.1
ω Cen 0.15 + 0.04 − 0.04 80 + 24 − 10 0.8 + 1.3 − 0.7 p 4.59 ± 0.08 5,6 5,6 footnotemark: 9.4 ± 1.8
NGC 6304 0.49 + 0.15 − 0.13 100 + 33 − 17 2.4 + 2.7 − 1.9 p 6.22 ± 0.26 7 7 footnotemark: 10.7 ± 3.1
NGC 6397 0.14 + 0.02 − 0.02 66 + 17 − 7 3.3 + 1.8 − 1.8 2.51 ± 0.07 8 8 footnotemark: 9.2 ± 1.8

Table 2 Properties of Quiescent LMXBs

There are several additional sources of systematic uncertainties that can affect the radius measurements that have been addressed to various degrees: the composition of the atmosphere, the composition and modeling of the interstellar medium that gives rise to the low-energy extinction, and the modeling of the power-law spectral component that is due to residual accretion. The majority of qLMXBs for which optical spectra have been obtained show evidence for H α emission (Heinke et al. 2014), indicating a hydrogen rich companion. Most of these qLMXBs are in the field and not in globular clusters as the ones we are using here. Assuming that sources in both types of environments have similar companions supports, in general, the use of hydrogen atmospheres when modeling quiescent spectra. The one source among those used here for which there is evidence to the contrary is the qLMXB in NGC 6397. Heinke et al. (2014) obtained only an upper limit on the H α emission using HST observations and, because of this, they applied a helium atmosphere model to the Chandra /XMM- Newton data sets described above.

Heinke et al. (2014) also explored the effect of assuming different spectral indices in modeling the power-law component. Even though the low counts preclude an accurate measurement of this parameter, the specific value has a small effect on the radius measurement, which can be folded in as a sytematic uncertainty. Finally, because of the low temperature of the surface emission from qLMXBs, the spectral modeling is affected significantly by the assumed model of the interstellar medium to account for the low-energy extinction. Heinke et al. (2014) explored different models for the interstellar extinction in their analysis of the qLMXBs in ω Cen and NGC 6397 and found statistically consistent results, with small differences in the central values but larger differences in the uncertainties. In the left panel of Figure 10 , we show the effect of different assumptions on the power-law index, the distance, and the interstellar extinction model on the inference of the mass and radius of the neutron star in ω Cen. In particular, one of the larger effects arises from the use of the two common interstellar extinction models they consider (the earlier Morrison & McCammon 1983 model with solar abundances, referred to as wabs in the spectral fitting software XSPEC, and the more recent Wilms et al. 2000 model, with ISM abundances from the same paper, referred to as tbabs with wilms in XSPEC). The wabs model (employed by Guillot et al. 2013) leads to somewhat larger radii for the same distance.

Figure 10.— The 68% confidence contours in mass and radius for the quiescent neutron star in ω Cen, inferred by Heinke et al. (2014 H14) and by Guillot & Rutledge (2014 G14) using different assumptions regarding the interstellar extinction (wabs: Morrison & McCammon 1983 tbabs: Wilms et al. 2000), the presence of a power-law spectral component, and for different distances to the globular cluster (4.8 kpc vs. 5.3 kpc)

Figure 11.— The combined constraints at the 68% confidence level over the neutron star mass and radius obtained from (Left) all neutron stars with thermonuclear bursts (Right) all neutron stars in low-mass X-ray binaries during quiescence.

In the present study, we repeat the analysis of Guillot et al. (2013) individually for all the sources in M13, M28, NGC 6304, NGC 6397, M30, and ω Cen. (Note that for the last two sources, the observations were reported in Guillot & Rutledge 2014). In all of the spectral fits, we allow for a power-law component with a fixed photon index Γ = 1 but a free normalization. We use the Wilms et al. (2001) ISM abundances in all of the analyses for a uniform treatment of all qLMXBs. We leave the hydrogen column density as a free parameter in the fits and when calculating the posterior likelihoods over mass and radius. We use a hydrogen atmosphere model for all of the sources except the one in NGC 6397, for which we use a helium atmosphere model (see also Heinke et al. 2014). The best-fit spectral parameters for each source are shown in Table 2. We also fold in distance uncertainties using a Gaussian likelihood for the distance to each source with a mean and standard deviation given in Table 2.

We show the resulting posterior likelihoods over the mass and radius for all of the qLMXBs in the right panel of Figure 11 and compare them to the combined constraints from the X-ray bursters discussed earlier. There is a high level of agreement between all of these measurements. Note that the larger widths of the 68% confidence contours of each source compared to those presented in Guillot et al. (2013, their figures 3 − 7 ) are due to the fact that in the present work, we leave the hydrogen column density as a free parameter. 3 3 3 While the simultaneous “Constant Rns” fits of Guillot et al. (2013) were performed by leaving the column density N H as a free parameter, Figures 3 − 7 in that study display results from fits performed with fixed N H . Furthermore, note that the present study also includes additional X-ray data for ω Cen, which refined the M-R contours.


What is inside a Neutron Star?

Everything we can touch is made of atoms. Subatomic particles called protons and neutrons make up the nucleus, and electrons orbit around them. But at this atomic level, even the densest materials in our world, such as gold, lead and uranium, are mainly made up of empty space.

Their nuclei are very, very small compared to the size of the atom as a whole, measured out to the orbit of electrons.

Now imagine squeezing a lump of gold, uranium or lead so hard that the nuclei come close together to the point they touch, and you'll estimate the empty space. You would have something incredibly dense, similar to a giant atomic nucleus. That is what neutron stars are. The only force up to the task of squeezing come to this point is gravity.

A neutron star forms when a star reaches the end of its life and no longer has enough fuel to keep it from collapsing. Loosely speaking, the core contracts until the nuclei have fused rather violently and are touching. At that point there is a bounce back the entire core recoils and drives off the outer layers, producing a brilliant supernova. Just a rapidly spinning cinder core remains - the densest thing you can have without it becoming a black hole. It is so dense that a teaspoon of the material on earth would weigh around one billion tonnes.

Schematic view of a pulsar. The sphere in the middle represents the neutron star, the curves indicate the magnetic field lines and the protruding cones represent the emission zones.

How does matter behave at such incredibly high densities? Do the nucleus of each atom stay separate? Do the atoms become an indistinguishable mix of protons, neutron and electrons floating around each other? Or does it go still further? Do the protons and electrons merge so that we end up with a soup of neutrons? Do they merge into a giant soup of quarks and gluons? Or does it go further than that?

Basically, the nuclear physics community doesn't know what happens at these densities. The question boils down to something called the equation "f" state, a term that describes the nature of matter at a fundamental level at that relates density to pressure.

If the star has a hardcore - a stiff equation of state - you would have a larger star, with a bigger radius. If you have a soft equation of state, where the neutrons give up their identities and form a soup of quarks, you would have a soft, squishy core and a smaller radius. So if we can just measure the radius of the star we will know what the interior composition must be. But we cannot measure directly. These stars are only thought to be between 10km and 15km in radius - the size of a city - which is far smaller than Hubble could ever hope to image. So measurements must be made in indirect ways.

We believe that the best method is to infer the radius by looking at X-Ray emissions and to do that we have to observe neutron stars from space. Neutron stars that flash rapidly as they turn their glowing magnetic poles towards us like cosmic lighthouses are called pulsars.

Multiwavelength X-ray, infrared, and optical compilation image of Kepler's Supernova Remnant, SN 1604.

From accurate measurements, we will be able to learn how neutron stars warp space-time from the shapes of these pulses. How sharply do then rise and fall? Is there a residual glow? These characteristics carry further information about the star and should enable us to determine the mass and radius of several neutron stars and so deduce what lies within.

This post is part of the series Astronomical Objects. Use the links below to advance to the next tutorial in the couse, or go back and see the previous in the tutorial series.


Title: VLT/FORS2 observations of the optical counterpart of the isolated neutron star RBS 1774

40-100 eV), long X-ray pulsations (P=3-12 s), and appear to be endowed with relatively high magnetic fields, (B

10d13-14 G). RBS 1774 is one of the few XDINSs with a candidate optical counterpart, which we discovered with the VLT. We performed deep observations of RBS 1774 in the R band with the VLT to disentangle a non-thermal power-law spectrum from a Rayleigh-Jeans, whose contributions are expected to be very much different in the red part of the spectrum. We did not detect the RBS 1774 candidate counterpart down to a 3 sigma limiting magnitude of R

27. The constraint on its colour, (B-R)<0.6, rules out that it is a background object, positionally coincident with the X-ray source. Our R-band upper limit is consistent with the extrapolation of the B-band flux (assuming a 3 sigma uncertainty) for a set of power-laws F_nu

nu^alpha with spectral indeces alpha<0.07. If the optical spectrum of RBS 1774 were non-thermal, its power-law slope would be very much unlike those of all isolated neutron stars with non-thermal optical emission, suggesting that it is most likely thermal. For instance, a Rayleigh-Jeans with temperature T_O = 11 eV, for an optically emitting radius r_O=15 km and a source distance d=150 pc, would be consistent with the optical measurements. The implied low distance is compatible with the 0.04 X-ray pulsed fraction if either the star spin axis is nearly aligned with the magnetic axis or with the line of sight, or it is slightly misaligned with respect to both the magnetic axis and the line of sight by 5-10 degrees


Shapiro Delay: Measuring the mass of neutron stars

As there are no real observational biases against spotting higher mass neutron stars, even with this discovery suggests we suspect that such objects are less common than their smaller mass compatriots. Despite this, there are still many know neutron stars for which the masses are not well defined.

Cromartie goes on to explain the difficulty that astronomers and astrophysicists face with mass measurements of neutron stars. The method that the team employed relies on Shapiro delay, which is only detectable in a small subset of edge-on systems. This is because the pulsar must pass behind the companion for light pulses to be delayed. “That’s the best way we have right now to make mass measurements, though other teams have used optical photometric studies as well,” she adds.

The method the team used relies on the fact that this neutron star is a pulsar or ‘pulsing stars’— an object that emits twin radio wave beams from its magnetic poles much like a lighthouse. J0740+6620 is a millisecond pulsar, one which spins rapidly making hundreds of revolutions per second.

Shapiro delay is a time delay experienced by light as it passes a massive object caused by gravitational time dilation, as put forward by Einstein in his theory of general relativity.

In this case, the mass of the neutron star’s dense white dwarf companion warps spacetime around it. This distortion causes a delay in the light pulses from the pulsar of tens of millionths of a second as the pulsar passes behind the white dwarf. The length of this delay can be used to calculate the mass of the white dwarf. The team then take this measurement and combine it with observations of the white dwarf and pulsar’s orbits around each other, to give them an estimate of the pulsar’s mass.

“We used pulsar timing, which is accounting for every rotation from a pulsar far into the future in order to detect deviations from expected pulse times of arrival,” Cromartie explains. “Pulsars are brightest at radio wavelengths, so we used the Green Bank radio telescope in WV for this work. The mass measurement was only made possible because this is an edge-on system that exhibits an effect called general relativistic Shapiro delay.

“This is a delay in the arrival of a pulsar’s pulses as it passes behind its companion along our line of sight, whose gravitational field delays the pulses. This allows the measurement of both the pulsar and white dwarf’s companions masses individually.”


Neutron Star going BH

You've answered your own question. The "two realities" are inconsistent, so there's something wrong with your reasoning. There's only one reality, so at least one of your descriptions is wrong.

In general relativity the source of gravity is the stress-energy tensor, not merely mass or density. This is a rather complex entity, and does not lead to an expectation that a neutron star will collapse into a black hole just because it is moving.

No. The Schwarzschild radius only has that interpretation when the massive object is not moving. If it's moving, the source term of the gravitational field has elements that do not depend solely on the mass, and you can't interpret the Schwarzschild radius the way you are doing.

An approximate way to think of things is this: the neutron star is length contracted, but so is its gravitational field.

Before someone rips me apart for that statement (length contraction is a phenomenon in flat spacetime only), imagine a cube around the neutron star large enough to enclose its entire gravitational field to a given accuracy this will length contract and must continue to enclose the gravitational field.

The poster grant hutchison on that forum is correct. Length contraction of the gravity field does not correctly describe the maths it doesn't correctly describe what happens to the neutron star either. The reason for this is that length contraction is a phenomenon in flat spacetime. The global Lorentz transforms you use to define it do not apply in the curved spacetime near a strongly gravitating mass.

That was why I was careful to mention a large cube enclosing the mass, where spacetime is flat to a good enough approximation. The cube will be measured to length contract, and will always contain the volume where there is significant gravity. Essentially, I've hidden all the difficult stuff inside the cube. Consistency requires that there be no significant spacetime curvature outside the box for any observer, and one could make rigorous claims about the angular separation of light rays grazing the neutron star and meeting at points on the cube as measured by stationary and moving observers.

At this point, I'll just summon @PeterDonis to criticise.

If you want to know what's going on deeper in the gravitational field then you'll have to solve Einstein's field equations. As far as I'm aware there's a known solution when the mass is stationary (the Schwarzschild solution) and one where it's moving at near lightspeed (the Aichelberg-Sexl ultraboost), but only numerical solutions for masses moving at less extreme speeds.

Note that the second paragraph of grant hutchison's quote from Baez's "If you go too fast, do you become a black hole?" says what I said in the first paragraph of my last post, in a bit more detail.

Special relativity is based on the concept of filling spacetime with a regular grid of clocks at rest with respect to you. I do the same. If we're in relative motion then we'll measure length contraction, time dilation and relativity of simultaneity. But add a gravitational field and your inertial clocks tick at different rates and keep crashing into each other and the neutron star. You can't build the conceptual framework you need for special relativity, so you can't borrow results from it without great care.

That's what my giant cube with observers stationary and in motion with respect to it are about. I'm hiding the bits of spacetime where my clocks keep crashing, and I'm cutting my experiment short before I can notice that there's gravity even outside my cube. That's why SR is good enough at a distance. (Note I don't need an actual cube - I just define a volume of spacetime, write "here be dragons" and don't send any clocks there).

But I can't do this close to the neutron star. My clocks tick at different rates and rapidly crash into each other and/or the star. I can't even approximately construct SR over a patch of spacetime including the star, so I have to use the full machinery of GR.

I recommend Sean Carroll's lecture notes and Ben Crowell's book on GR.

Trying to understand as much as I can. So let's go one step at a time.
Let's define volumetric mass density as mass/volume
https://en.m.wikipedia.org/wiki/Density

Would the volumetric mass density of a body such as our sun be higher than that measured in our frame of reference, than one measured in an inertial frame of reference where the sun would be observed as contracted in length?
Another way of asking the same question is, is relativistic length contraction associated with reduced volume our not?
Thanks in advance.

Note that the stationary, spherically symmetric character of a neutron star solution is global, invariant feature, irrespective of whether an observer is moving very fast relative to it (which is obviously the same as the neutron star moving fast relative to some observer). Presence or absence of a horizon is also a global, invariant feature. It is perfectly feasible to come up with an exact coordinate description of a moving 'neutron star model'. By model, I mean some simple representation of the neutron star as a classical fluid. There are numerous published exact solutions of perfect fluid balls with exterior vacuum being the Schwarzschild metric greater than some r. Any of these is easily cast to a cartesian style metric instead of spherical. Then, apply some transform with the general character of a Lorentz boost, computing metric in the new coordinates. Clearly, invariant features cannot change in this procedure. However, these coordinates would not be very physically meaningful near the neutron star (but, of course, all physics computed in them would be fine).

To answer physical questions about a neutron star as measured by a flyby observer near lightspeed, a different procedure would be better and simpler. Use vanilla Schwarzschild coordinates, and just compute fly by geodesic with initial 4-velocity that is near lightlike in these coordinates. Then, in principle, you can compute any observable for such an observer. In particular, you can deduce without computation that this observer can send and receive light signals to the surface of the ball. Thus, trivially, there is no horizon and no BH.


Acknowledgements

We thank B. Allen, W. Kastaun, J. Lattimer and B. Metzger for valuable discussions. This work was supported by US National Science Foundation grants PHY-1430152 to the JINA Center for the Evolution of the Elements (S.R.), PHY-1707954 (D.A.B. and S.D.) US Department of Energy grant DE-FG02-00ER41132 (S.R.) NASA Hubble Fellowship grant number HST-HF2-51412.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555 (B.M.) and the US Department of Energy, Office of Science, Office of Nuclear Physics, under contract DE-AC52-06NA25396, the Los Alamos National Laboratory (LANL) LDRD programme and the NUCLEI SciDAC programme (I.T.). D.A.B., S.D. and B.M. thank the Kavli Institute for Theoretical Physics (KITP) where portions of this work were completed. KITP is supported in part by the National Science Foundation under grant number NSF PHY-1748958. Computational resources have been provided by Los Alamos Open Supercomputing via the Institutional Computing (IC) programme, by the National Energy Research Scientific Computing Center (NERSC), by the Jülich Supercomputing Center, by the ATLAS Cluster at the Albert Einstein Institute in Hannover, and by Syracuse University. GWOSC is a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes. B.M. is a NASA Einstein Fellow.


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Keywords: black holes, neutron stars, gravitational waves, kilonovae, gamma-ray bursts, numerical relativity

Citation: Foucart F (2020) A Brief Overview of Black Hole-Neutron Star Mergers. Front. Astron. Space Sci. 7:46. doi: 10.3389/fspas.2020.00046

Received: 18 February 2020 Accepted: 18 June 2020
Published: 28 July 2020.

Rosalba Perna, Stony Brook University, United States

Vyacheslav Ivanovich Dokuchaev, Institute for Nuclear Research (RAS), Russia
Kenta Hotokezaka, The University of Tokyo, Japan

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