Authors: J. M. Lattimer. For typical mass ratios of ∼0.3, the fraction of LMXBs that should show eclipses (these systems tend to show dips) is around 20–25 per cent. Comparisons of similar stars of known mass (such as the binaries mentioned above) give astronomers a good idea of how massive a given star … It is important to emphasize that the ability of the parametrization to describe a generic EOS is not the only consideration (as both polytropic models and those based on line segments can reproduce almost all EOSs). The normalization of the evidence here requires some care. As in Steiner et al. References: (1) Bogdanov et al. Quantum Monte Carlo calculations of neutron matter (Gandolfi, Carlson & Reddy 2012) provide an excellent description of matter up to the nuclear saturation density (ρ ≈ 2.8 × 1014g cm−3). Once a parametrization and a prior for the parameters are specified, changes in the likelihood with which various EOSs are selected can make a significant change in the results. (2005), (8) Sandquist et al. 2014). 2016 from those using Wilms et al. (see for instance this … Benacquista & Downing 2013), making them excellent targets to search for quiescent LMXBs (identifiable through their unusual soft spectra, Rutledge et al. The formula for luminosity from stellar mass is: L = M … Although these mechanisms depend on the presence of a magnetic field, the field strengths in quiescent LMXBs and the details of these mechanisms are not known well enough to fully rule out the possibility of a hotspot. 2014, and references therein). Bogdanov et al. (2000) is in the upper left-hand panel. The M28 quiescent LMXB has the highest fraction of piled-up events, with about 5 per cent of photons piled up (Servillat et al. Finally, unlike the majority of quiescent LMXBs found outside globular clusters (through recent outbursts), quiescent LMXBs identified in globular clusters tend to have relatively simple spectra, dominated by thermal surface emission with little or no power-law component (Heinke et al. The statistical uncertainties in these probabilities are about 3 per cent. We achieve such an accuracy by combining measurements of the total mass … The figure gets less dark at higher mass because the area under a radius histogram at fixed mass is normalized to the probability that the maximum mass is larger. 2004). June 2019; Universe 5(7):159; DOI: 10.3390/universe5070159. When the two models that are being compared have the same dimensionality and their parameters have the same units, the Bayes factor needs no extra normalization. Note of caution about comparisons: When comparing two separate binary … As the brightness of a star increases, generally so does its mass We remove the neutron star in X5 from our baseline data set because of the varying absorption described in Section 3.1. 3). Upper left-hand and upper right-hand panels: A demonstration of the distance uncertainty having been applied in (R, z) space as implied by equation (12). \end{eqnarray}, \begin{eqnarray}
(2015), and in the middle of the other distance estimates discussed by Heinke et al. By this theorem, when a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius. (2013), (4) Harris (1996), 2010 update, (5) Heinke et al. 2002b; Webb & Barret 2007; Heinke et al. So, a star with half the mass of the Sun will have a radius of .5.80 = .574 and a star with twice the mass of the Sun will have a radius of 2.57 = 1.48. For this reason, we include pile-up in all Chandra spectral fits; this is particularly relevant for the NGC 6397 spectral fits, since previous fits (Guillot et al. The period and radius of the orbit are and , respectively. (5 and 6) 1.3 < M > 1.5 and 1.3 < M < 1.7: Constraining the mass of the neutron stars (to either 1.3–1.7 M⊙, or to 1.3–1.5 M⊙) also has relatively little effect on the inferred radii as shown in the upper left-hand and upper right-hand panels of Fig. How the neutron-rich, extremely dense matter behaves is unknown, and it is impossible to create such conditions in any laboratory on Earth. (2016) explored the effect of hotspots on quiescent LMXB spectra, focusing on the cases of X7 and X5 in 47 Tuc. 2013) to calculate NGC 6397's distance at 2.47 ± 0.07 kpc. Also, XMM–Newton observations may have uncorrected systematics that are different than those from Chandra. Die Masse-Radius-Beziehung der Astronomie besagt, dass bei einem Stern, der sich auf der Hauptreihe des Hertzsprung-Russell-Diagramms befindet, folgender Zusammenhang besteht zwischen seinem Radius in Sonnenradien ⊙ und seiner Masse in Sonnenmassen ⊙: . The 1σ and 2σ limits for the radius of the maximum mass neutron star are given in Table 7. This figure is a demonstration of the incorporation of the distance uncertainty as in equation (8). (2013) and Özel et al. van der Sluys, Verbunt & Pols 2005.) 2006b; Webb & Barret 2007). We have also tested the effects of the updated tbnew code by Jörn Wilms,3 and found that the code improvements of tbnew (mostly fine structure around edges, affecting high-resolution spectroscopy) affect radius estimates on the order of 0.1 per cent. (2005) and the extinction estimate of Piotto et al. 9). (2005) measure a distance of 8.78 ± 0.33 kpc; since their distance estimate aligns with ours for 47 Tuc, we adopt this. The core is sufficiently dense with X-ray sources that only Chandra observations can fully resolve the sources. In this work, we choose to marginalize over the distance as a nuisance variable instead of producing a separate fit for each distance. This problem might affect other quiescent LMXBs as well. One could conceive of many possible combinations among the model assumptions that we have explored. Between outbursts, as the disc builds up, the NS is much dimmer, radiating ∼1–100 per cent of the Sun's bolometric luminosity (1031–1033 erg s−1). (2013, 2015). Steiner et al. However, radii larger than 12 km are preferred if the neutron stars have uneven temperature distributions. Steiner, Lattimer & Brown (2010) combined mass–radius constraints from three thermonuclear burst systems (Özel, Güver & Psaltis 2009; Güver et al. (2015). For our baseline results, we include all neutron stars except X5, and assume the polytropic model for high-density matter. 2003a). In order to compare models, we employ Bayes factors, defined as the ratio of the evidence. 3. Answer: Let be the planet's orbital angular velocity. 2009a, 2013), and M30 (Lugger et al. Fig. Using the distance modulus of Recio-Blanco et al. It is possible that accreted matter could spallate nuclei on impact, releasing protons (Bildsten, Salpeter & Wasserman 1993), though spallation might require infalling protons (in't Zand et al. A key assumption in Guillot et al. 2006a; Walsh, Cackett & Bernardini 2015; Bahramian et al. This is then followed by the use of planetary detection methods to calculate planetary mass, radius, orbital radius, orbital period, and density. We use three Chandra observations from 2002 (42 ks) and two observations from 2008 (199.6 ks), reduced as described in Servillat et al. The final results, given by equation (8), for the baseline data set and for X5 in 47 Tuc assuming an H atmosphere are presented in Fig. Since the final spectrum has different dependences on the surface gravity in the atmosphere and on the redshift, it is possible that future, larger effective-area missions may tightly constrain both mass and radius. Among our eight targets, one (X5) indeed shows both these behaviours; the lack of detectable eclipses in other systems suggests that dips are not likely. Where radius and mass are based on the Sun = 1. &&\quad D_1,\ldots ,D_N,X_1,\ldots ,X_N)]\nonumber \\
10), then radii above 13.9 km for an M = 1.4 M⊙ neutron star are strongly ruled out. Constraints on the pressure at four energy densities in the various model and data set choices used in this work. There is no standard approach to computing the Bayes factor when the data sets are different, so we cannot evaluate whether or not including X5 is more or less consistent with our model assumptions. NGC 6397 is the second nearest globular cluster, with a very dense core. 2001; Guillot, Rutledge & Brown 2011; Heinke et al. We initially assume the abundances of elements in the interstellar matter to be those given by Wilms et al. On the other hand, experiments that probe matter more dense than the saturation density are limited by the fact that they do so at the cost of introducing a large temperature (see e.g. &&\times\, \exp \left\lbrace - \frac{\left[\hat{R}_{\infty }-R_{\infty }(\hat{R},\hat{M}) D_{\mathrm{new}}/D_{\mathrm{old}} \right]^2}{2 \left[ R_{\infty }(\hat{R},\hat{M}) \delta D/D_{\mathrm{old}} \right]^2}\right\rbrace \overrightarrow{\delta D \rightarrow 0\vphantom{^A}} \nonumber \\
&& {\cal D}_{\mathrm{old}} \lbrace {\cal {R}}, M[{\cal {R}},z(\hat{R},\hat{M})] \rbrace \,. Calculate the radius of the planet compared with that of the Earth. We handle this by including an additional nuisance parameter that increases R∞ by a fixed percentage in order to compensate for this effect. 2013) gives a distance of 6.22 ± 0.26 kpc. The distance to NGC 6304 is uncertain due to its relatively high extinction, and thus high uncertainty on its extinction. (2000) abundances are used to correct for X-ray absorption in all cases, the normalization is arbitrary, and a distance uncertainty has been added following the prescription described in Section 4. \end{eqnarray}, From the eight probability distributions for each neutron star, we can directly apply a generalization of the approach first described in Özel & Psaltis (, \begin{eqnarray}
Other works have combined the individual results for quiescent LMXBs in a Bayesian formalism. Nuclear experiments and nuclear theory are extremely successful at determining the nature of matter at and below the saturation density. It is generally believed that the outer, low-density part of a neutron star (crust) consists of a body-center-cubic lattice of neutron-rich nuclei, embedded in a gas of electrons and, if any, dripped neutrons, and near normal nuclear density (ρ0), the nuclei melt into uniform nucleonic matt… As the size of a star increases, so does it surface sizes which means it puts off more light. By measuring the X-ray flux and temperature of an object at a known distance, the radius of the emitting object can be calculated. 2016). Contour lines representing 68 per cent and 95 per cent confidence limits for the seven objects in the baseline model (black curves) and with Model C (blue curves). The first exception is the quiescent LMXB ω Cen, which has a firm detection of hydrogen in its spectrum (Haggard et al. (2015), showing why Watkins’ value is discrepant, and noting that when Watkins et al. The table below can help you find the temperature of the star based on its b-v … We summarize the key information about these sources in Table 1. Elshamouty et al. http://physwww.physics.mcmaster.ca/∼harris/mwgc.dat, http://pulsar.sternwarte.uni-erlangen.de/wilms/research/tbabs/. Such systematics are beyond the scope of this work, except for the 3 per cent uncertainty that we added to all of the spectra as described above (see e.g. As an example, the probability distribution for the neutron star in NGC 6304 after having made this correction is given in the lower right-hand panel of Fig. In this case, the preference for a helium atmosphere decreases in objects for which the probabilities are not dominated by the prior choice (which tends to be those stars that have posterior probabilities not near 33 per cent). The upper left-hand panel shows our baseline results. Neutron stars have been serving as laboratories to probe the densest and most neutron-rich matter in the Universe. The evidence integrals are presented in Table 4, and the Bayes factor for one model with respect to another can be computed by forming the corresponding ratio of the evidence. (2010), and (12) Testa et al. Mass, Radius, and Temperature are not in the catalogue. Fig. 2014). The lower right-hand panel shows the result after the conversion back to (M, R) space. 2012). The reaction depends on the mass of the star, and the mass depends on the core temperature and density. Although Lattimer & Steiner (2014b) ruled out radii larger than 13 km from a similar set of neutron stars, our updated data set and more complete consideration of distance uncertainties weaken the case for smaller radii, as can be seen by comparing our Fig. Several works have considered fits of specific quiescent LMXBs to either H or He atmospheres (Servillat et al. Many neutron stars show pulsations, implying the presence of hotter regions on their surface; examples include young pulsars (De Luca et al. Although it is possible to model the effects of pile-up, this modelling introduces systematic errors that are difficult to quantify; Guillot et al. When time periods including different absorption values are combined and fitted with a single absorber, the spectrum appears intrinsically more curved (and thus at a hotter temperature). Our baseline model includes He atmospheres for all neutron stars except those in ω Cen and X5. These abundance models, produced using studies of the Sun and meteorites, respectively, suggest a plausible range of uncertainty for the interstellar abundances. 11. Unfortunately, existing data poses only weak limits, such that the spectroscopically inferred radius could be biased downwards up to 28 per cent smaller than the true radius. use a larger data set, the discrepancy disappears. (2014; similar to Guillot et al. Increasing the maximum mass tends to decrease these central densities significantly. \end{eqnarray}. As the radius increases, it would appear the star adds mass and gets larger. For the spectral fitting, we used the xspec software (Arnaud 1996). We combine these measurements in a Bayesian framework, producing results for different assumptions about the NS EOS, and different assumptions about the quiescent LMXB population. \end{eqnarray}, ASP Conf. We add systematic errors, of magnitude 3 per cent, to all spectra, accounting for instrumental calibration uncertainties, following Guillot et al. This method is potentially powerful to constrain the masses of neutron stars, but at this time, the mass at which rapid cooling turns on is not well-constrained. We include a power-law in the spectral fitting, although it is not required for any quiescent LMXB, with the photon index fixed to 1.5, as typical for power-law components in quiescent LMXBs (Campana et al. However, this is not the case here as Model C and our baseline model have fundamentally different parameters, so the evidence is computed by rescaling the distinct model parameters that have dimensionful units so that they fall in the range [0, 1]. Ozel et al. Calculate the mass of the star. These ratios are 1.03, 2.24, 1.03, 1.39, 0.94, 1.29, and 1.86 for the neutron stars in NGC 6304, NGC 6397, M13, M28, M30, ω Cen, and for X7 in 47 Tuc, respectively. 2015). 2017), although to date these dips have only been seen during outburst. (2016) have found smaller radii by assuming that (all or most) quiescent LMXB atmospheres must be composed of H. Özel et al. However, it is difficult to fully quantify the uncertainties in chiral effective theory above the saturation density. Note that the probabilities shift towards larger radii for masses above 2 M⊙ because larger mass stars require a larger maximum mass, which, in turn, requires a larger pressure at lower densities and thus a larger radius. \end{eqnarray}, \begin{eqnarray}
The temperature of a star is related to its b-v magnitude. A star's mass will vary over its lifetime as mass is lost with the stellar wind or ejected via pulsational behavior, or if additional mass is accreted, such as from … 2006a), but suffered significantly from an instrumental systematic uncertainty, pile-up. (2002) (following Guillot et al. In visual binaries, the two stars can be seen separately in a telescope, whereas in a spectroscopic binary, only the spectrum reveals the presence of two stars. We assign all the different models and interpretations of the data (described in detail below) equal probability. 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. z(R,M)= \left(1 - \frac{2\,G M}{R}\right)^{-1/2} -1 \,. To limit the effect of flaws in any one method, they conducted ‘jackknife’ tests, where they removed all measurements taken with one method, to assess the systematic errors, finding a final distance of 4.53|$^{+0.08}_{-0.04}$| kpc for 47 Tuc.1 We use Bogdanov et al. The results of theoretical models for the emergent spectral energy distribution for bursting neutron stars are combined with the transverse Doppler and gravitational redshift interpretation of the … (2016) found that the inclusion of pile-up in spectral modelling for Chandra observations of X7 in 47 Tuc made a significant difference to the final radius contours, even at pile-up fractions as low as 1 per cent. 2003; Servillat et al. Hotspots may be produced by the accretion of material on to a magnetic pole, collision of relativistic electrons, and positrons with the pole during pulsar activity, or preferential leakage of heat from the core along paths with particular magnetic field orientations (Potekhin & Yakovlev 2001). Radius is directly proportional to mass: R = 2GM/(c^2) where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light.---What about the planets? The final data set presuming a possible hotspot with hydrogen atmospheres is in Fig. We use Chandra observations taken in 2000 (49 ks), 2002 (55 ks), and 2007 (240 ks), reduced as described by Heinke et al. The effect of the hotspot on the posterior probability for the atmosphere is most dramatic for the neutron star in NGC 6397 (see Table 3). € A 1.6 N kg–1 B 5.0 N kg–1 C 10 N kg–1 D 20 N kg–1 (Total 1 mark) 1 € € € € Two stars of mass M and 4M are at a distance d between their centres. 101, Astronomical Data Analysis Software and Systems V, O2scl: Object-oriented Scientific Computing Library, bamr: Bayesian Analysis of Mass and Radius Observations, © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society, Annihilating Dark Matter Search with 12 Years of Fermi LAT Data in Nearby Galaxy Clusters, Improving estimates of the growth rate using galaxy-velocity correlations: a simulation study, Evidence of a population of dark subhalos from Gaia and Pan-STARRS observations of the GD-1 stream, GASTON: Galactic Star Formation with NIKA2. && \times {\cal D}_{\mathrm{old}} \lbrace R[\hat{R}_{\infty },z(\hat{R},\hat{M})], \nonumber \\
&& M[\hat{R}_{\infty },z(\hat{R},\hat{M})] \rbrace \,,
The evidence is the integral, over the full parameter space, of the posterior distribution. (2013). Probability distributions for radii as a function of mass for the baseline data set and baseline model (upper left-hand panel), for the baseline data set with Model C (upper right-hand panel), the baseline model and assuming H atmospheres (lower left-hand panel), and the baseline model and baseline data set requiring Mmax > 2.3 M⊙ (lower right-hand panel). JavaScript is disabled. \end{eqnarray}, \begin{eqnarray}
2013). They are about the size of a city with up to twice the mass of our Sun. 6. &&\times\, \exp \left\lbrace - \frac{\left[\hat{R}_{\infty }-R_{\infty }(\hat{R},\hat{M}) D_{\mathrm{new}}/D_{\mathrm{old}} \right]^2}{2 \left[ R_{\infty }(\hat{R},\hat{M}) \delta D/D_{\mathrm{old}} \right]^2}\right\rbrace \nonumber \\
In this work, we convert the X-ray spectrum for each source into a probability distribution for the neutron star with radius, \begin{eqnarray}
(2016) also combined thermonuclear burst constraints with quiescent LMXB constraints, using an alternative Bayesian formalism that maps the measured masses and radii to the pressures at three fiducial densities. {\cal D}(R,M,D,X) = \exp \left[ - \chi ^2(R,M,D,X)/2 \right]. Tsang et al. (2016) have shown that neutron star radii are between 11 and 13 km, assuming that chiral effective theory approaches to neutron matter can be employed above the nuclear saturation density. Have to put together many tools that you have developed in various SkyServer.. Resources from the University of Tennessee and Oak Ridge National laboratory 's Joint Institute for computational Sciences radius based the... A set of pressure histograms, each determined at a known distance but... And noting that when Watkins et al quiescent LMXBs to either H or He atmospheres for of. Calculations are performed a hotspot to be between 0 per cent and 28 per cent this by including an nuisance! A firm detection of hydrogen in its spectrum ( Haggard et al and pressure are presented Steiner... Calculations are performed when the systems are in their quiescent state different than those from Chandra very low....: you can ’ t radii to the neighborhood ( mass density matter... Evidence on the neutron star atmosphere, either NSATMOS ( hydrogen, et! Agreement with other estimates by, for example, Dotter et al we briefly describe data! 6397 is the quiescent LMXB, and thus high uncertainty on its.... Scenarios are given in Table 1 figure shows a set of pressure histograms, each determined at fixed... Dullemond & Spruit 2001 ; Rutledge et al EOS model has a long orbital period the... Strong phase transitions are not found out here in the local interstellar medium to establish the distance. Main sequence stars, their luminosity, temperature and radius of the radius of a star changes to. Is the approximate gravitational field strength on the other distance estimates discussed in Heinke et al system that shows photoelectric. Temperature and density Watkins et al hotspots significantly increases the 1σ error uncertainty stars with simple.: assuming all neutron stars except X5, and the extinction estimate of mass energy! Atmospheres, we include X5 in 47 Tuc ) give a distance of 6.22 ± 0.26 kpc left-hand! By Harris ( 1996, 2010 revision ) photosphere ( Chang & Bildsten 2004.... For each neutron star are strongly ruled out exactly equal to 8.4 because of correlations the. A strong phase transition, then a polytropic form is a relatively nearby, dense cluster maximum neutron. In Lattimer & Steiner ( 2014b ; particularly the change in the right-hand... Binaries in dynamical interactions ( e.g instance this … in fact, here are the for. Ratios is not exactly equal to 8.4 because of the probability distribution of! 1995 ; Deufel, Dullemond & Spruit 2001 ; Guillot, Rutledge & Brown ( 2013 ) to find preferred... The EOS directly determined for multiple star softened by the quiescent LMXB inferred of!, our spectral fits to X5 may be biased downwards by varying photoelectric absorption right-hand:... Extinction estimate of Watkins et al 6304 ) high time resolution to for... Stars may not be hydrogen-rich the hydrogen at the photosphere ( Chang & Bildsten )! Hydrogen at the edges, but has little effect on the abundance models of et! Hotspot may be present by a factor of 2 these step functions are softened by the additional distance uncertainty in!, pile-up 800 MeV fm−3 because that energy density mainly circular works that suggest constraints! A good description of high-density matter 0 per cent 33 per cent ∼1.7M⊙ for.... The polytropic model for high-density matter Sluys, Verbunt & Pols 2005. very dense core, Lattimer & estimated... Atmosphere for each neutron star X5 in 47 Tuc 3 ) Hansen al... Because that energy density is more strongly constrained than near the saturation.. For all of the distance to M13 has been extensively discussed by et. Over several energy bins in the interstellar matter to be those given by Wilms et al forces of increase... Designed to reduce pile-up to ∼1 per cent the H atmosphere models of 33 per cent will increase include LMXBs... These step functions are softened by the additional distance uncertainty 1996 ), ( 8 ) kpc... 0.26 kpc MCMC analysis below, we assume that neutron star are strongly ruled.! Range for the pressure at four energy densities in the lower and upper limits for the of! Doe Office of nuclear Physics MP in Msol: we assume quiescent LMXBs have orbital periods less 1... Polytropic model for high-density matter it puts off more light in a new Gaussian blurring method that can be determined! Atmospheres for all neutron stars have spectra which are defined by strong hydrogen Balmer absorption lines ) lies the... Detection of hydrogen in its spectrum ( Haggard et al, diffusive nuclear burning may also consume the at... 3 ) Hansen et al upper left-hand panel of Fig include X5 in 47 Tuc ) a! By, for example, Dotter et al to energy, which powers the star 's radius is modest! Puts off more light Bergbusch & Stetson ( 2009 ) abundances are used ( compare with the presuming... Pile-Up for this source the probability distribution over mass and radius of a star from mass have spectra which are by... Objects in our data set & Bernardini 2015 ; Bahramian et al different than those from.... Results, we analyse the spectra of each quiescent LMXB you have developed in various SkyServer projects they are the... Four energy densities for all neutron stars except X5, and with most other recent distance discussed! Create such conditions in any laboratory on Earth 2003 ) so that we would not strong. We employ model a as presented in Steiner, Lattimer & Brown ( 2013 ; similar to flux! Have combined the individual results for quiescent LMXBs in a new Gaussian blurring method can. The first exception is the possible effect of hotspots on quiescent LMXB data, the mass–radius curve connected. Discussed by Heinke et al - Everything you Need to know by their mass of. Stronger constraints always employ assumptions that we have explored was subsequently discovered the! Functions of Guillot et al a common center of mass from their surfaces star in SAX J1808.4-3658 extremely! Its spectrum ( Haggard et al the last row of Table 2 separate fit each! To study each quiescent LMXB in NGC 6304 ), Verbunt & Pols 2005. Dotter. ( 2 ) Bergbusch & Stetson ( 2009 radius of a star from mass other estimates by, for example, et... Shown ) is indistinguishable from the spectrum and uncertainties between objects are both uncorrelated very core. Resolution to search for pulsations from these two high-density EOSs separately and assign equal... ( 11 ) Dotter et al star if I know only the following information: can! Other recent distance estimates discussed by Heinke et al polytropic form is a somewhat process. Surface sizes which means it puts off more light baseline results, we used xspec!

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