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At least 469 records · Page 26

THE ORIGIN OF THE SPURIOUS IRON SPREAD IN THE GLOBULAR CLUSTER NGC 3201

NGC 3201 is a globular cluster suspected to have an intrinsic spread in the iron content. We re-analyzed a sample of 21 cluster stars observed with UVES-FLAMES at the Very Large Telescope and for which Simmerer et al. found a 0.4 dex wide [Fe/H] distribution with a metal-poor tail. We confirmed that when spectroscopic gravities are adopted, the derived [Fe/H] distribution spans ∼0.4 dex. On the other hand, when photometric gravities are used, the metallicity distribution from Fe I lines remains large, while that derived from Fe II lines is narrow and compatible with no iron spread. We demonstrate that the metal-poor component claimed by Simmerer et al. is composed by asymptotic giant branch stars that could be affected by non-local thermodynamical equilibrium effects driven by iron overionization. This leads to a decrease of the Fe I abundance, while leaving the Fe II abundance unaltered. A similar finding has been already found in asymptotic giant branch stars of the globular clusters M5 and 47 Tucanae. We conclude that NGC 3201 is a normal cluster, with no evidence of intrinsic iron spread.

79 ASTRONOMY AND ASTROPHYSICS↗

Confirmation and characterization of the protoplanet HD 100546 b—Direct evidence for gas giant planet formation at 50 AU

We present the first multi-wavelength, high-contrast imaging study confirming the protoplanet embedded in the disk around the Herbig Ae/Be star HD 100546. The object is detected at L′ (∼3.8 μm) and M′ (∼4.8 μm), but not at K{sub s} (∼2.1 μm), and the emission consists of a point source component surrounded by spatially resolved emission. For the point source component we derive apparent magnitudes of L′=13.92±0.10 mag, M′=13.33±0.16 mag, and K{sub s}>15.43±0.06 mag (3σ limit), and a separation and position angle of (0.457±0.014){sup ′′} and (8.4±1.4){sup ∘}, and (0.472±0.014){sup ′′} and (9.2±1.4){sup ∘} in L′ and M′, respectively. We demonstrate that the object is co-moving with HD 100546 and can reject any (sub-)stellar fore-/background object. Fitting a single-temperature blackbody to the observed fluxes of the point source component yields an effective temperature of T{sub eff}=932{sub −202}{sup +193} K and a radius for the emitting area of R=6.9{sub −2.9}{sup +2.7} R{sub Jupiter}. The best-fit luminosity is L=(2.3{sub −0.4}{sup +0.6})⋅10{sup −4} L{sub ⊙}. We quantitatively compare our findings with predictions from evolutionary and atmospheric models for young, gas giant planets, discuss the possible existence of a warm, circumplanetary disk, and note that the deprojected physical separation from the host star of (53±2) AU poses a challenge to standard planet formation theories. Considering the suspected existence of an additional planet orbiting at ∼13–14 AU, HD 100546 appears to be an unprecedented laboratory to study the formation of multiple gas giant planets empirically.

79 ASTRONOMY AND ASTROPHYSICS↗

ALMA observations of the largest proto-planetary disk in the Orion Nebula, 114–426: A CO silhouette

We present ALMA observations of the largest protoplanetary disk in the Orion Nebula, 114–426. Detectable 345 GHz (856 μm) dust continuum is produced only in the 350 AU central region of the ∼1000 AU diameter silhouette seen against the bright Hα background in Hubble Space Telescope images. Assuming optically thin dust emission at 345 GHz, a gas-to-dust ratio of 100, and a grain temperature of 20 K, the disk gas-mass is estimated to be 3.1±0.6 Jupiter masses. If most solids and ices have been incorporated into large grains, however, this value is a lower limit. The disk is not detected in dense-gas tracers such as HCO{sup +} J = 4–3, HCN J = 4–3, or CS = 7–6. These results may indicate that the 114–426 disk is evolved and depleted in some light organic compounds found in molecular clouds. The CO J = 3–2 line is seen in absorption against the bright 50–80 K background of the Orion A molecular cloud over the full spatial extent and a little beyond the dust continuum emission. The CO absorption reaches a depth of 27 K below the background CO emission at V{sub LSR} ≈ 6.7 km s{sup −1} ∼ 0.″52 (210 AU) northeast and 12 K below the background CO emission at V{sub LSR} ≈ 9.7 km s{sup −1} ∼ 0.″34 (140 AU) southwest of the suspected location of the central star, implying that the embedded star has a mass less than 1 M{sub ⊙}.

79 ASTRONOMY AND ASTROPHYSICS↗

A chemical TROMPE-L’ŒIL: No iron spread in the globular clusteR M22

We present the analysis of high-resolution spectra obtained with UVES and UVES-FLAMES at the Very Large Telescope of 17 giants in the globular cluster (GC) M22, a stellar system suspected to have an intrinsic spread in iron abundance. We find that when surface gravities are derived spectroscopically (in order to obtain the same iron abundance from Fe i and Fe ii lines) the [Fe/H] distribution spans ∼0.5 dex, according to previous analyses. However, the gravities obtained in this way correspond to unrealistically low stellar masses (0.1–0.5 M{sub ⊙}) for most of the surveyed giants. Instead, when photometric gravities are adopted, the [Fe ii/H] distribution shows no evidence of spread, in variance with the [Fe i/H] distribution. This difference has been recently observed in other clusters and could be due to non-local thermodynamical equilibrium effects driven by over-ionization mechanisms that mainly affect the neutral species (thus providing lower [Fe i/H]), but leave [Fe ii/H] unaltered. We confirm that the s-process elements show significant star-to-star variations and their abundances appear to be correlated with the difference between [Fe i/H] and [Fe ii/H]. This puzzling finding suggests that the peculiar chemical composition of some cluster stars may be related to effects able to spuriously decrease [Fe i/H]. We conclude that M22 is a GC with no evidence of intrinsic iron spread, ruling out that it has retained the supernovae ejecta in its gravitational potential well.

79 ASTRONOMY AND ASTROPHYSICS↗

Searches for continuous gravitational waves from nine young supernova remnants

We describe directed searches for continuous gravitational waves (GWs) in data from the sixth Laser Interferometer Gravitational-wave Observatory (LIGO) science data run. The targets were nine young supernova remnants not associated with pulsars; eight of the remnants are associated with non-pulsing suspected neutron stars. One target's parameters are uncertain enough to warrant two searches, for a total of 10. Each search covered a broad band of frequencies and first and second frequency derivatives for a fixed sky direction. The searches coherently integrated data from the two LIGO interferometers over time spans from 5.3–25.3 days using the matched-filtering F-statistic. We found no evidence of GW signals. We set 95% confidence upper limits as strong (low) as 4 × 10{sup −25} on intrinsic strain, 2 × 10{sup −7} on fiducial ellipticity, and 4 × 10{sup −5} on r-mode amplitude. These beat the indirect limits from energy conservation and are within the range of theoretical predictions for neutron-star ellipticities and r-mode amplitudes.

79 ASTRONOMY AND ASTROPHYSICS↗

WHITE DWARF-RED DWARF SYSTEMS RESOLVED WITH THE HUBBLE SPACE TELESCOPE. II. FULL SNAPSHOT SURVEY RESULTS

Results are presented for a Hubble Space Telescope Advanced Camera for Surveys high-resolution imaging campaign of 90 white dwarfs with known or suspected low-mass stellar and substellar companions. Of the 72 targets that remain candidate and confirmed white dwarfs with near-infrared excess, 43 are spatially resolved into two or more components, and a total of 12 systems are potentially triples. For 68 systems where a comparison is possible, 50% have significant photometric distance mismatches between their white dwarf and M dwarf components, suggesting that white dwarf parameters derived spectroscopically are often biased due to the cool companion. Interestingly, 9 of the 30 binaries known to have emission lines are found to be visual pairs and hence widely separated, indicating an intrinsically active cool star and not irradiation from the white dwarf. There is a possible, slight deficit of earlier spectral types (bluer colors) among the spatially unresolved companions, exactly the opposite of expectations if significant mass is transferred to the companion during the common envelope phase. Using the best available distance estimates, the low-mass companions to white dwarfs exhibit a bimodal distribution in projected separation. This result supports the hypothesis that during the giant phases of the white dwarf progenitor, any unevolved companions either migrate inward to short periods of hours to days, or outward to periods of hundreds to thousands of years. No intermediate projected separations of a few to several AU are found among these pairs. However, a few double M dwarfs (within triples) are spatially resolved in this range, empirically demonstrating that such separations were readily detectable among the binaries with white dwarfs. A straightforward and testable prediction emerges: all spatially unresolved, low-mass stellar and substellar companions to white dwarfs should be in short-period orbits. This result has implications for substellar companion and planetary orbital evolution during the post-main-sequence lifetime of their stellar hosts.

79 ASTRONOMY AND ASTROPHYSICS↗

INFRARED SPECTRA AND OPTICAL CONSTANTS OF NITRILE ICES RELEVANT TO TITAN's ATMOSPHERE

Spectra and optical constants of nitrile ices known or suspected to be in Titan's atmosphere are presented from 2.0 to 333.3 {mu}m ({approx}5000-30 cm{sup -1}). These results are relevant to the ongoing modeling of Cassini CIRS observations of Titan's winter pole. Ices studied are: HCN, hydrogen cyanide; C{sub 2}N{sub 2}, cyanogen; CH{sub 3}CN, acetonitrile; C{sub 2}H{sub 5}CN, propionitrile; and HC{sub 3}N, cyanoacetylene. For each of these molecules, we also report new cryogenic measurements of the real refractive index, n, determined in both the amorphous and crystalline phases at 670 nm. These new values have been incorporated into our optical constant calculations. Spectra were measured and optical constants were calculated for each nitrile at a variety of temperatures, including, but not limited to, 20, 35, 50, 75, 95, and 110 K, in both the amorphous phase and the crystalline phase. This laboratory effort used a dedicated FTIR spectrometer to record transmission spectra of thin-film ice samples. Laser interference was used to measure film thickness during condensation onto a transparent cold window attached to the tail section of a closed-cycle helium cryostat. Optical constants, real (n) and imaginary (k) refractive indices, were determined using Kramers-Kronig analysis. Our calculation reproduces the complete spectrum, including all interference effects.

79 ASTRONOMY AND ASTROPHYSICS↗

THE ULTRAVIOLET UPTURN IN ELLIPTICAL GALAXIES AND ENVIRONMENTAL EFFECTS

It is suspected that the ultraviolet (UV) upturn phenomenon in elliptical galaxies and extended horizontal-branch stars in globular clusters have a common origin. An extremely high abundance of helium (Y {approx} 0.4) allows for a working hypothesis, but its origin is unclear. Peng and Nagai proposed that primordial helium sedimentation in dark halos over cosmic timescales may lead to extreme helium abundances in galaxy cluster centers. In this scenario, UV upturn should be restricted to brightest cluster galaxies (BCGs) only. This is a clear and testable prediction. We present tests of this hypothesis using galaxy clusters from Yoon et al. that were detected by both the Sloan Digital Sky Survey and the Galaxy Evolution Explorer Medium Imaging Survey. Using a new UV classification scheme based on far-UV, near-UV, and optical photometry we found that only 5% of cluster elliptical galaxies show a UV upturn, while 27% and 68% are classified as 'recent star formation' and 'UV-weak' ellipticals, respectively. The data reveal a modest positive dependence of the UV upturn fraction on galaxy velocity dispersion, which is in agreement with the earlier findings of Burstein et al. and possibly with the helium sedimentation theory. However, we do not see any dependency on rank or luminosity of galaxies. Besides, BCGs do not show any marked difference in UV upturn fraction or strength, which is inconsistent with the prediction. We conclude that the aforementioned helium sedimentation theory and its inferred environmental effects are not supported by the available data.

79 ASTRONOMY AND ASTROPHYSICS↗

BAR EFFECTS ON CENTRAL STAR FORMATION AND ACTIVE GALACTIC NUCLEUS ACTIVITY

Galactic bars are often suspected to be channels of gas inflow to the galactic center and to trigger central star formation and active galactic nucleus (AGN) activity. However, the current status on this issue based on empirical studies is unsettling, especially regarding AGNs. We investigate this question based on the Sloan Digital Sky Survey Data Release 7. From the nearby (0.01 < z < 0.05) bright (M{sub r} < -19) database, we have constructed a sample of 6658 relatively face-on late-type galaxies through visual inspection. We found 36% of them to have a bar. Bars are found to be more common in galaxies with earlier morphology. This makes sample selection critical. Parameter-based selections would miss a large fraction of barred galaxies of early morphology. Bar effects on star formation or AGNs are difficult to understand properly because multiple factors (bar frequency, stellar mass, black hole mass, gas contents, etc.) seem to contribute to them in intricate manners. In the hope of breaking these degeneracies, we inspect bar effects for fixed galaxy properties. Bar effects on central star formation seem higher in redder galaxies. Bar effects on AGNs on the other hand are higher in bluer and less massive galaxies. These effects seem more pronounced with increasing bar length. We discuss possible implications in terms of gas contents, bar strength, bar evolution, fueling timescale, and the dynamical role of supermassive black hole.

79 ASTRONOMY AND ASTROPHYSICS↗

HERSCHEL AND SPITZER OBSERVATIONS OF SLOWLY ROTATING, NEARBY ISOLATED NEUTRON STARS

Supernova fallback disks around neutron stars have been suspected to influence the evolution of the diverse neutron star populations. Slowly rotating neutron stars are the most promising places to find such disks. Searching for the cold and warm debris of old fallback disks, we carried out Herschel PACS (70 μm, 160 mu m) and Spitzer IRAC (3.6 μm, 4.5 μm) observations of eight slowly rotating (P ≈ 3-11 s) nearby (<1 kpc) isolated neutron stars. Herschel detected 160 μm emission (>5σ) at locations consistent with the positions of the neutron stars RX J0806.4-4123 and RX J2143.0+0654. No other significant infrared emission was detected from the eight neutron stars. We estimate probabilities of 63%, 33%, and 3% that, respectively, none, one, or both Herschel PACS 160 μm detections are unrelated excess sources due to background source confusion or an interstellar cirrus. If the 160 μm emission is indeed related to cold (10-22 K) dust around the neutron stars, this dust is absorbing and re-emitting ∼10% to ∼20% of the neutron stars' X-rays. Such high efficiencies would be at least three orders of magnitude larger than the efficiencies of debris disks around nondegenerate stars. While thin dusty disks around the neutron stars can be excluded as counterparts of the 160 μm emission, dusty asteroid belts constitute a viable option.

79 ASTRONOMY AND ASTROPHYSICS↗

RECURRENT NOVAE IN M31

The reported positions of 964 suspected nova eruptions in M31 recorded through the end of calendar year 2013 have been compared in order to identify recurrent nova (RN) candidates. To pass the initial screen and qualify as a RN candidate, two or more eruptions were required to be coincident within 0.′1, although this criterion was relaxed to 0.′15 for novae discovered on early photographic patrols. A total of 118 eruptions from 51 potential RN systems satisfied the screening criterion. To determine what fraction of these novae are indeed recurrent, the original plates and published images of the relevant eruptions have been carefully compared. This procedure has resulted in the elimination of 27 of the 51 progenitor candidates (61 eruptions) from further consideration as RNe, with another 8 systems (17 eruptions) deemed unlikely to be recurrent. Of the remaining 16 systems, 12 candidates (32 eruptions) were judged to be RNe, with an additional 4 systems (8 eruptions) being possibly recurrent. It is estimated that ∼4% of the nova eruptions seen in M31 over the past century are associated with RNe. A Monte Carlo analysis shows that the discovery efficiency for RNe may be as low as 10% that for novae in general, suggesting that as many as one in three nova eruptions observed in M31 arise from progenitor systems having recurrence times ≲100 yr. For plausible system parameters, it appears unlikely that RNe can provide a significant channel for the production of Type Ia supernovae.

79 ASTRONOMY AND ASTROPHYSICS↗

PHYSICAL PROPERTIES OF THE CURRENT CENSUS OF NORTHERN WHITE DWARFS WITHIN 40 pc OF THE SUN

We present a detailed description of the physical properties of our current census of white dwarfs within 40 pc of the Sun, based on an exhaustive spectroscopic survey of northern hemisphere candidates from the SUPERBLINK proper motion database. Our method for selecting white dwarf candidates is based on a combination of theoretical color–magnitude relations and reduced proper motion diagrams. We reported in an earlier publication the discovery of nearly 200 new white dwarfs, and we present here the discovery of an additional 133 new white dwarfs, among which we identify 96 DA, 3 DB, 24 DC, 3 DQ, and 7 DZ stars. We further identify 178 white dwarfs that lie within 40 pc of the Sun, representing a 40% increase of the current census, which now includes 492 objects. We estimate the completeness of our survey at between 66% and 78%, allowing for uncertainties in the distance estimates. We also perform a homogeneous model atmosphere analysis of this 40 pc sample and find a large fraction of massive white dwarfs, indicating that we are successfully recovering the more massive, and less luminous objects often missed in other surveys. We also show that the 40 pc sample is dominated by cool and old white dwarfs, which populate the faint end of the luminosity function, although trigonometric parallaxes will be needed to shape this part of the luminosity function more accurately. Finally, we identify 4 probable members of the 20 pc sample, 4 suspected double degenerate binaries, and we also report the discovery of two new ZZ Ceti pulsators.

79 ASTRONOMY AND ASTROPHYSICS↗

Electric field determination in the plasma-antenna boundary of a lower-hybrid wave launcher in Tore Supra through dynamic Stark-effect spectroscopy

Interaction of radio-frequency (RF) waves with the plasma in the near-field of a high-power wave launcher is now seen to be important, both in understanding the channeling of these waves through the plasma boundary and in avoiding power losses in the edge. In a recent Letter a direct non-intrusive measurement of a near antenna RF electric field in the range of lower hybrid (LH) frequencies ($E_{LH}$) was announced (Phys. Rev. Lett., 110:215005, 2013). The measurement was achieved through the fitting of Balmer series deuterium spectral lines utilizing a time dependent (dynamic) Stark effect model. In this article, the processing of the spectral data is discussed in detail and applied to a larger range of measurements and the accuracy and limitations of the experimental technique is investigated. We find through an analysis of numerous Tore Supra pulses that good quantitative agreement exists between the measured and full-wave modeled $E_{LH}$ when the launched power exceeds 0.5MW. For low power the measurement becomes formidable utilizing the implemented passive spectroscopic technique because the spectral noise overwhelms the effect of the RF electric field on the line profile. Additionally, effects of the ponderomotive force are suspected at sufficiently high power.

47 OTHER INSTRUMENTATION↗

First-principles calculations of niobium hydride formation in superconducting radio-frequency cavities

Niobium hydride is suspected to be a major contributor to degradation of the quality factor of niobium superconducting radio-frequency (SRF) cavities. In this study, we connect the fundamental properties of hydrogen in niobium to SRF cavity performance and processing. We modeled several of the niobium hydride phases relevant to SRF cavities and present their thermodynamic, electronic, and geometric properties determined from calculations based on density-functional theory. We find that the absorption of hydrogen from the gas phase into niobium is exothermic and hydrogen becomes somewhat anionic. The absorption of hydrogen by niobium lattice vacancies is strongly preferred over absorption into interstitial sites. A single vacancy can accommodate six hydrogen atoms in the symmetrically equivalent lowest-energy sites and additional hydrogen in the nearby interstitial sites affected by the strain field: this indicates that a vacancy can serve as a nucleation center for hydride phase formation. Small hydride precipitates may then occur near lattice vacancies upon cooling. Vacancy clusters and extended defects should also be enriched in hydrogen, potentially resulting in extended hydride phase regions upon cooling. We also assess the phase changes in the niobium-hydrogen system based on charge transfer between niobium and hydrogen, the strain field associated with interstitial hydrogen, and the geometry of the hydride phases. The results of this study stress the importance of not only the hydrogen content in niobium, but also the recovery state of niobium for the performance of SRF cavities.

Ford, Denise C.↗

Black hole-neutron star binaries near neutron star disruption limit in the mass regime of event GW230529

In May 2023, the LIGO-Virgo-KAGRA (LVK) Collaboration reported the likely black hole-neutron star (BHNS) merger GW230529_181500. The signal was observed with high significance in only one detector, limiting constraints on the black hole spin and motivating our study of disruption in this mass regime. That event is expected to be the merger of a 2.5–4.5 M ⊙ primary with a secondary compact object of mass between 1.2–2.0 M ⊙ . This makes it the first BHNS merger with a significant potential for the production of electromagnetic (EM) counterparts, and provides further evidence for compact objects existing within the suspected lower mass gap. To produce post-merger EM transients, the component of the black hole spin aligned with the orbital angular momentum must be sufficiently high, allowing the neutron star to be tidally disrupted. The disrupting BHNS binary may then eject a few percent of a solar mass of matter, leading to an observable kilonova driven by radioactive decays in ejecta, and/or a compact-binary gamma-ray burst (cbGRB) resulting from the formation of an accretion disk and relativistic jet. Determining which mergers lead to disruption of the neutron star is necessary to predict the prevalence of EM signals from BHNS mergers, yet most BHNS simulations so far have been performed far from the minimum spin required for tidal disruption. Here, we use the Spectral Einstein Code to explore the behavior of BHNS mergers in a mass range consistent with GW230529_181500 close to that critical spin, and compare our results against the mass remnant model currently used by the LVK Collaboration to predict the probability of tidal disruption. Furthermore, our numerical results reveal the emergence of non-zero accretion disks even below the predicted NS disruption limit, of low mass but capable of powering cbGRBs. Our results also demonstrate that the remnant mass model underpredicts the disk mass for the DD2 equation of state, while they are within expected modeling errors for SFHo. The disruption limit itself, however, is not found to significantly differ from the predictions of the analytical model, unless remnant masses M rem ≲ 0.001 M ⊙ prove interesting observationally. In all of our simulations, any kilonova signal would be dim and most likely dominated by post-merger disk outflows.

GW230529_181500↗

High-temperature strain monitoring of stainless steel using fiber optics embedded in ultrasonically consolidated nickel layers

Fiber optic sensors have long been considered for use in structural health monitoring of components because of their small size, high accuracy, and their ability to perform spatially distributed strain measurements when embedded within a component or on its surface. The typical method for transferring strain from the component to the fiber is to use an epoxy, which may not survive extended exposure to high temperatures, thus necessitating a more elaborate technique to embed the fibers. This work represents a first step towards using additive manufacturing techniques to embed fiber optic sensors on stainless steel components for high-temperature strain monitoring, including quantification of the differential thermal strains that develop between the fiber and the steel substrate. Copper-coated optical fibers were embedded in nickel layers on top of a stainless steel substrate using an ultrasonic additive manufacturing technique. The embedded fibers showed minimal signal attenuation and clear compressive strain after embedding. Heating the embedded fibers in steps to temperatures of 300 °C–400 °C resulted in measured strains with values between the expected thermal strain in stainless steel and nickel. Finite element simulations confirm the measured strain values and show that the thermal strain depends on the thickness of the nickel layers deposited on top of the stainless steel substrate. While the fibers failed before reaching temperatures of 500 °C, it is suspected that these failures occurred due to a combination of (1) the lack of strain relief, (2) the high-temperature oxidation of the fiber's copper coating, and (3) improper sizing of the machined channel in which the fiber is placed prior to embedding. If proper coating selection and sizing of the channel can prevent the failures observed in this work, the next step would be to monitor strain during mechanical loading at high temperatures.

36 MATERIALS SCIENCE↗

Visible spectroscopy of highly charged tungsten ions with the JET charge exchange diagnostic

Since the installation of the ITER-like wall on the JET tokamak, visible emission from a broad range of tungsten charge states has complicated plasma ion temperature and toroidal rotation measurements. A plethora of charge exchange transitions, from ions up to W 56+ , and 21 suspected magnetic dipole transitions have been observed. In particular W 39+ and W 46+ charge exchange emission contaminates the long-established carbon measurement at 529 nm. Fortunately the wavelength and relative intensity of the tungsten charge exchange lines is predictable and their influence can be mitigated when they are included in the spectral fit. Neon based charge exchange measurements at 525 nm are now preferred on JET since the ITER-like wall, however in this case an unidentified tungsten magnetic dipole transition contaminates the spectrum when the electron temperature is low.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Hyper-entropic gravitational fireballs (grireballs) with firewalls

Recently there has been much discussion as to whether old black holes have firewalls at their surfaces that would destroy infalling observers. Though I suspect that a proper handling of nonlocality in quantum gravity may show that firewalls do not exist, it is interesting to consider an extension of the firewall idea to what seems to be the logically possible concept of hyper-entropic gravitational hot objects (gravitational fireballs or grireballs for short) that have more entropy than ordinary black holes of the same mass. Here some properties of such grireballs are discussed under various assumptions, such as assuming that their radii and entropies both go as powers of their masses as the one independent parameter, or assuming that their radii depend on both their masses and their entropies as two independent parameters.

79 ASTRONOMY AND ASTROPHYSICS↗