Mass-luminosity Relation Research Articles

Eddington's mass-luminosity relation and the laws of thermodynamics

Ever since its formulation by A. S. Eddington, the mass-luminosity relation has been viewed as a triumph for theoretical astronomy and astrophysics. The idea that the luminosity of the stars could be controlled solely by their mass was indeed a revolutionary concept. The proof involved two central aspects: (1) the belief that stars could be treated as ideal gases in hydrostatic equilibrium, and (2) that the opacity of Capella could be used as a reference mark applicable to other stars. Yet, when the mass-luminosity relation was advanced, no thought was given to the need for thermodynamic balance. Within thermodynamic expressions, not only must the dimensions (hence units) be consistent on each side of the equals sign, but the extensive nature of the properties must also balance. Namely, thermodynamic expressions must be balanced by properties which are extensive to the same degree. In this regard, mass is an extensive thermodynamic property and can be represented by a homogenous function of degree 1. Conversely, the luminosity of a star is neither extensive nor intensive, but rather can be represented by a homogenous function of degree 2/3. Consequently, the mass-luminosity expression is thermodynamically unbalanced and stands in violation of the laws of thermodynamics.

The Nearby Very Low Mass Triple Star LHS 1070

Abstract The component masses and kinematic parameters of the triple star LHS 1070 are redetermined by using the three-body model. It is shown that the discrepancy between new observations and the two-body fitting results is due to the three-body effects, which justifies the necessity of using the three-body model. Numerical integration shows that the three-body system remains integrated and its hierarchical configuration remains unchanged since the formation of the components. This implies that the triple star may not be formed by a three-body encounter. Also, the possibility that the most massive component is itself a tightly bounded binary is excluded based on the redetermined dynamical mass. The improved dynamical masses of the three components are greatly helpful to constrain at the lower end of the stellar empirical mass–luminosity relation.

Testing modified gravity with wide binaries in Gaia DR2

ABSTRACT Several recent studies have shown that very wide binary stars can potentially provide an interesting test for modified-gravity theories which attempt to emulate dark matter; these systems should be almost Newtonian according to standard dark-matter theories, while the predictions for MOND-like theories are distinctly different, if the various observational issues can be overcome. Here we explore an observational application of the test from the recent Gaia DR2 data release: we select a large sample of ∼24 000 candidate wide binary stars with distance $\lt 200 \, {\rm pc}$ and magnitudes G < 16 from Gaia DR2, and estimated component masses using a main-sequence mass–luminosity relation. We then compare the frequency distribution of pairwise relative projected velocity (relative to circular-orbit value) as a function of projected separation; these distributions show a clear peak at a value close to Newtonian expectations, along with a long ‘tail’ which extends to much larger velocity ratios; the ‘tail’ is considerably more numerous than in control samples constructed from DR2 with randomized positions, so its origin is unclear. Comparing the velocity histograms with simulated data, we conclude that MOND-like theories without an external field effect (ExFE) are strongly inconsistent with the observed data since they predict a peak-shift in clear disagreement with the data; testing MOND-like theories with an ExFE is not decisive at present, but has good prospects to become decisive in future with improved modelling or understanding of the high-velocity tail, and additional spectroscopic data.

Triage of astrometric binaries – how to find triple systems and dormant black hole secondaries in the Gaia orbits

ABSTRACT Preparing for the expected wealth of Gaia detections, we consider here a simple algorithm for classifying unresolved astrometric binaries with main-sequence (MS) primary into three classes: binaries with a probable MS secondary, with two possible values for the mass ratio; probable hierarchical triple MS systems with an astrometric secondary as a close binary, with a limited range of mass-ratio values; and binaries with a compact-object secondary, with a minimal value of the mass ratio. This is done by defining a unitless observational parameter ‘Astrometric Mass-Ratio Function’ (AMRF), $\mathcal {A}$, of a binary, based on primary-mass estimation, in addition to the astrometric parameters – the angular semimajor axis, the period, and the parallax. We derive the $\mathcal {A}$ value that differentiates the three classes by forward modelling representative binaries of each class, assuming some mass–luminosity relation. To demonstrate the potential of the algorithm, we consider the orbits of 98 Hipparcos astrometric binaries with MS primaries, using the Hipparcos parallaxes and the primary-mass estimates. For systems with known spectroscopic orbital solution, our results are consistent with the spectroscopic elements, validating the suggested approach. The algorithm will be able to identify hierarchical triple systems and dormant neutron star and black hole companions in the Gaia astrometric binaries.

PPMXL and Gaia morphological analysis of Melotte 22 (Pleiades) and Melotte 25 (Hyades)

In this article, we present a morphological study of two extended open clusters in the sky, Melotte 22 and Melotte 25 using J, H, and Ks photometry with Gaia DR2 and PPMXL proper motions. Various photometric and kinematic astrophysical parameters have been determined.Based on proper motions of these clusters, we extract our member candidates and hence fit isochrones with Solar metallicity Z = 0.019, log (age) = 8.2 ± 0.05 for Melotte 22, and metallicity Z = 0.024, log (age) = 8.9 ± 0.1 for Melotte 25. Some photometric parameters are estimated, e.g. cluster heliocentric distances are 135 ± 4 and 48 ± 2 pc respectively, luminosity and mass functions with mass-luminosity relation using a second order polynomial, and we have estimated masses, i.e. 662 ± 25 and 513 ± 22 M⊙. The relaxation time of these clusters are smaller than the estimated cluster ages, which indicates that these clusters are dynamically relaxed.The second part is devoted to some kinematics of these clusters using the internal and/or external motion of members. We find the apex 96° ± 0°.1, −48° ± 0°.1, and 97° ± 1°, 6° ± 0°.1 for Melotte 22 and Melotte 25 respectively using the AD-chart method, also we have determined the components of space velocity and the cluster centers in pc as well as elements of the Solar motion. The results are in good agreement with those obtained previously.

A 3D view of the Hyades stellar and sub-stellar population

Aims. Our scientific goal is to provide a 3D map of the nearest open cluster to the Sun, the Hyades, combining the recent release ofGaiaastrometric data, ground-based parallaxes of sub-stellar member candidates and photometric data from surveys which cover large areas of the cluster.Methods. We combined the secondGaiarelease with ground-basedH-band parallaxes obtained with the infrared camera on the 2 m robotic Liverpool telescope to astrometrically identify stellar and sub-stellar members of the Hyades, the nearest open cluster to the Sun.Results. We find 1764 objects within 70° radius from the cluster centre from theGaiasecond data release, whose kinematic properties are consistent with the Hyades. We limit our study to 30 pc from the cluster centre (47.03 ± 0.20 pc) where we identify 710 candidate members, including 85 and 385 in the core and tidal radius, respectively. We determine proper motions and parallaxes of eight candidate brown dwarf members and confirm their membership. Using the 3D positions and a model-based mass-luminosity relation we derive a luminosity and mass function in the 0.04–2.5M⊙range. We confirm evidence for mass segregation in the Hyades and find a dearth of brown dwarfs in the core of the cluster. From the white dwarf members we estimate an age of 640−49+67Myr.Conclusions. We identify a list of members in the Hyades cluster from the most massive stars down to the brown dwarfs. We produce for the first time a 3D map of the Hyades cluster in the stellar and sub-stellar regimes and make available the list of candidate members.

How to Constrain Your M Dwarf. II. The Mass–Luminosity–Metallicity Relation from 0.075 to 0.70 Solar Masses

Abstract The mass–luminosity relation for late-type stars has long been a critical tool for estimating stellar masses. However, there is growing need for both a higher-precision relation and a better understanding of systematic effects (e.g., metallicity). Here we present an empirical relationship between and M * spanning 0.075 M ⊙ < M * < 0.70 M ⊙. The relation is derived from 62 nearby binaries, whose orbits we determine using a combination of Keck/NIRC2 imaging, archival adaptive optics data, and literature astrometry. From their orbital parameters, we determine the total mass of each system, with a precision better than 1% in the best cases. We use these total masses, in combination with resolved K S magnitudes and system parallaxes, to calibrate the –M * relation. The resulting posteriors can be used to determine masses of single stars with a precision of 2%–3%, which we confirm by testing the relation on stars with individual dynamical masses from the literature. The precision is limited by scatter around the best-fit relation beyond measured M * uncertainties, perhaps driven by intrinsic variation in the –M * relation or underestimated uncertainties in the input parallaxes. We find that the effect of [Fe/H] on the –M * relation is likely negligible for metallicities in the solar neighborhood (0.0% ± 2.2% change in mass per dex change in [Fe/H]). This weak effect is consistent with predictions from the Dartmouth Stellar Evolution Database, but inconsistent with those from MESA Isochrones and Stellar Tracks (at 5σ). A sample of binaries with a wider range of abundances will be required to discern the importance of metallicity in extreme populations (e.g., in the Galactic halo or thick disk).

Dynamic Cosmological Tests

A new class of cosmological tests is proposed which examines the dependence on redshift z of the dispersion in peculiar velocities, virial mass, and mass-luminosity ratios of galactic clusters. Data on galactic clusters with redshifts up to z ≈ 2 are accessible to modern instruments in the visible and x-ray ranges. Clusters of galaxies are less susceptible to evolutionary effects than other objects at large z, so that the proposed tests offer promise for analysis of cosmological models.

Revisiting the mass-luminosity relation with an effective temperature modifier

The mass-luminosity relation (MLR) is commonly used to estimate the stellar mass. The classical MLR can hardly fit data of all the stellar mass range, thus researchers have generally adopted piecewise MLRs based on the classical MLR with different exponents for different mass ranges. However, varying turning points for the piecewise MLRs and for the exponent of each segment were used, and the estimated stellar masses are not always as good as those obtained by dynamical methods. We suggest an alternative way to improve the mass estimation accuracy: adding an effective temperature modifier to modify every segment MLR. We use a corresponding estimating equation for G- and K-type main-sequence stars, and verify this equation on two eclipsing binary catalogs. We compare the estimated results with those from a classical MLR and several piecewise MLRs. We find that the new estimates are significantly more accurate than those from the classical MLR and some piecewise MLRs, and they are not inferior to the stellar masses from other piecewise MLRs. This indicates that the temperature modifier can effectively help improve the estimation accuracy. In addition, we discuss the effect of adding the temperature modifier on the practicability of estimating stellar masses.

Interrelated main-sequence mass–luminosity, mass–radius, and mass–effective temperature relations

Absolute parameters of 509 main-sequence stars selected from the components of detached-eclipsing spectroscopic binaries in the Solar neighbourhood are used to study mass-luminosity, mass-radius and mass-effective temperature relations (MLR, MRR and MTR). The MLR function is found better if expressed by a six-piece classical MLR ($L \propto M^{\alpha}$) rather than a fifth or a sixth degree polynomial within the mass range of $0.179\leq M/M_{\odot}\leq 31$. The break points separating the mass-ranges with classical MLR do not appear to us to be arbitrary. Instead, the data indicate abrupt changes along the mass axis in the mean energy generation per unit of stellar mass. Unlike the MLR function, the MRR and MTR functions cannot be determined over the full range of masses. A single piece MRR function is calibrated from the radii of stars with $M\leq1.5M_{\odot}$, while a second single piece MTR function is found for stars with $M>1.5M_{\odot}$. The missing part of the MRR is computed from the MLR and MTR, while the missing part of the MTR is computed from the MLR and MRR. As a result, we have interrelated MLR, MRR and MTR, which are useful in determining the typical absolute physical parameters of main-sequence stars of given masses. These functions are also useful to estimate typical absolute physical parameters from typical $T_{eff}$ values. Thus, we were able to estimate the typical absolute physical parameters of main-sequence stars observed in the Sejong Open Cluster survey, based on that survey's published values for $T_{eff}$. Since typical absolute physical parameters of main sequence stars cannot normally be determined in such photometric surveys, the interrelated functions are shown to be useful to compute such missing parameters from similar surveys.

The gravitational mass of Proxima Centauri measured with SPHERE from a microlensing event

Proxima Centauri, our closest stellar neighbour, is a low-mass M5 dwarf orbiting in a triple system. An Earth-mass planet with an 11 day period has been discovered around this star. The star's mass has been estimated only indirectly using a mass-luminosity relation, meaning that large uncertainties affect our knowledge of its properties. To refine the mass estimate, an independent method has been proposed: gravitational microlensing. By taking advantage of the close passage of Proxima Cen in front of two background stars, it is possible to measure the astrometric shift caused by the microlensing effect due to these close encounters and estimate the gravitational mass of the lens (Proxima Cen). Microlensing events occurred in 2014 and 2016 with impact parameters, the closest approach of Proxima Cen to the background star, of 1\farcs6 $\pm$ 0\farcs1 and 0\farcs5 $\pm$ 0\farcs1, respectively. Accurate measurements of the positions of the background stars during the last two years have been obtained with HST/WFC3, and with VLT/SPHERE from the ground. The SPHERE campaign started on March 2015, and continued for more than two years, covering 9 epochs. The parameters of Proxima Centauri's motion on the sky, along with the pixel scale, true North, and centering of the instrument detector were readjusted for each epoch using the background stars visible in the IRDIS field of view. The experiment has been successful and the astrometric shift caused by the microlensing effect has been measured for the second event in 2016. We used this measurement to derive a mass of 0.150$^{\textrm{+}0.062}_{-0.051}$ (an error of $\sim$ 40\%) \MSun for Proxima Centauri acting as a lens. This is the first and the only currently possible measurement of the gravitational mass of Proxima Centauri.

SAFARI – I. A SPHERE discovery of a super metal-rich M-dwarf companion to the star HD 86006

We report the direct detection of a fully convective, early-to-mid M-dwarf companion orbiting the star HD 86006, using ESO-SPHERE during Science Verification as part of the SAFARI program. HARPS+CORALIE radial velocity measurements first indicated a possible companion. Such work highlights the synergies that are now possible between these two observing methods. We studied the companion by comparing our observed spectra with BT-Settl models and template spectra, measuring spectral indices to obtain a spectral type, and used a joint radial velocity and astrometric fit to simulate the companion's orbit. The companion was found to be 4.14 mag fainter than the primary in the H2 band, residing at a physical separation of $\sim$ 25 AU, with a $T_\mathrm{eff}$ and spectral type of 3321 $\pm$ 111 K and M 4.1 $\pm$ 1.1, respectively. We note that the age derived from BT-Settl models for such a star is too low by over two orders of magnitude, similar to other known field mid-M stars. We searched for the radial velocity companion to HD 90520 without any clear detection, however we reached a low contrast level of $\Delta$H2 = 10.3 mag (or $1.3 * 10^{-4}$) at 0.2$''$ and 12.6 mag (or $10^{-5}$) at 0.5$''$, allowing us to rule out any low-mass companions with masses of 0.07 and 0.05 M$_{\odot}$ at these separations. This discovery provides us with the exciting opportunity to better constrain the mass-luminosity relation for low-mass stars in the super metal-rich domain, expanding our understanding of the most-common types of stars and substellar objects.

The mass-lifetime relation

In a recent “AstroNote,” I described a simple exercise on the mass-luminosity relation for main sequence stars as an example of exposing students in a general education science course of lower mathematical level to the use of quantitative skills such as collecting and analyzing data. Here I present another attempt at a meaningful experience for such students that again involves both the gathering and analysis of numerical data and comparison with accepted result, this time on the relationship of the mass and lifetimes of main sequence stars. This experiment can stand alone or be used as an extension of the previous mass-luminosity relationship experiment.

The Mass–Luminosity Relation for a Refined Set of Late-K/M Dwarfs

The Mass–Luminosity Relation for a Refined Set of Late-K/M Dwarfs

Close binary evolution

Context. Massive stars with solar metallicity lose important amounts of rotational angular momentum through their winds. When a magnetic field is present at the surface of a star, efficient angular momentum losses can still be achieved even when the mass-loss rate is very modest, at lower metallicities, or for lower-initial-mass stars. In a close binary system, the effect of wind magnetic braking also interacts with the influence of tides, resulting in a complex evolution of rotation. Aims. We study the interactions between the process of wind magnetic braking and tides in close binary systems. Methods. We discuss the evolution of a 10 M⊙ star in a close binary system with a 7 M⊙ companion using the Geneva stellar evolution code. The initial orbital period is 1.2 days. The 10 M⊙ star has a surface magnetic field of 1 kG. Various initial rotations are considered. We use two different approaches for the internal angular momentum transport. In one of them, angular momentum is transported by shear and meridional currents. In the other, a strong internal magnetic field imposes nearly perfect solid-body rotation. The evolution of the primary is computed until the first mass-transfer episode occurs. The cases of different values for the magnetic fields and for various orbital periods and mass ratios are briefly discussed. Results. We show that, independently of the initial rotation rate of the primary and the efficiency of the internal angular momentum transport, the surface rotation of the primary will converge, in a time that is short with respect to the main-sequence lifetime, towards a slowly evolving velocity that is different from the synchronization velocity. This “equilibrium angular velocity” is always inferior to the angular orbital velocity. In a given close binary system at this equilibrium stage, the difference between the spin and the orbital angular velocities becomes larger when the mass losses and/or the surface magnetic field increase. The treatment of the internal angular momentum transport has a strong impact on the evolutionary tracks in the Hertzsprung-Russell Diagram as well as on the changes of the surface abundances resulting from rotational mixing. Our modelling suggests that the presence of an undetected close companion might explain rapidly rotating stars with strong surface magnetic fields, having ages well above the magnetic braking timescale. Our models predict that the rotation of most stars of this type increases as a function of time, except for a first initial phase in spin-down systems. The measure of their surface abundances, together, when possible, with their mass-luminosity ratio, provide interesting constraints on the transport efficiencies of angular momentum and chemical species. Conclusions. Close binaries, when studied at phases predating any mass transfer, are key objects to probe the physics of rotation and magnetic fields in stars.

Multiwavelength Light-Curve Analysis of Cepheid Variables

AbstractThis poster presented results from a detailed analysis of observed and theoretical light-curves of classical Cepheid variables in the Galaxy and the Magellanic Clouds. The theoretical light-curves were based on non-linear convective hydrodynamical pulsation models; the observational data were taken from ongoing wide-field variability surveys. The variation which we found in theoretical and observed light-curve parameters as a function of period, wavelength and metallicity was used to constrain the input physics to the pulsation models, such as the mass–luminosity relations obeyed by Cepheid variables. We also accounted for the variation in the convective efficiency as entered into the stellar pulsation models and its impact on the theoretical amplitudes and Period-Luminosity relations for Cepheid variables.

The initial masses of the red supergiant progenitors to Type II supernovae

There are a growing number of nearby supernovae (SNe) for which the progenitor star is detected in archival pre-explosion imaging. From these images it is possible to measure the progenitor's brightness a few years before explosion, and ultimately estimate its initial mass. Previous work has shown that II-P and II-L SNe have red supergiant (RSG) progenitors, and that the range of initial masses for these progenitors seems to be limited to ≲ 17 M⊙. This is in contrast with the cut-off of 25–30 M⊙ predicted by evolutionary models, a result that is termed the ‘red supergiant problem’. Here we investigate one particular source of systematic error present in converting pre-explosion photometry into an initial mass, which of the bolometric correction (BC) used to convert a single-band flux into a bolometric luminosity. We show, using star clusters, that RSGs evolve to later spectral types as they approach SN, which in turn causes the BC to become larger. Failure to account for this results in a systematic underestimate of a star's luminosity, and hence its initial mass. Using our empirically motivated BCs we reappraise the II-P and II-L SNe that have their progenitors detected in pre-explosion imaging. Fitting an initial mass function to these updated masses results in an increased upper mass cut-off of Mhi = |$19.0^{+2.5}_{-1.3}$| M⊙, with a 95 per cent upper confidence limit of <27 M⊙. Accounting for finite sample size effects and systematic uncertainties in the mass–luminosity relationship raises the cut-off to Mhi = 25 M⊙ (<33 M⊙, 95 per cent confidence). We therefore conclude that there is currently no strong evidence for ‘missing’ high-mass progenitors to core-collapse SNe.

Exclusion of Stellar Companions to Exoplanet Host Stars

Abstract Given the frequency of stellar multiplicity in the solar neighborhood, it is important to study the impacts this can have on exoplanet properties and orbital dynamics. There have been numerous imaging survey projects established to detect possible low-mass stellar companions to exoplanet host stars. Here, we provide the results from a systematic speckle imaging survey of known exoplanet host stars. In total, 71 stars were observed at 692 and 880 nm bands using the Differential Speckle Survey Instrument at the Gemini-north Observatory. Our results show that all but two of the stars included in this sample have no evidence of stellar companions with luminosities down to the detection and projected separation limits of our instrumentation. The mass–luminosity relationship is used to estimate the maximum mass a stellar companion can have without being detected. These results are used to discuss the potential for further radial velocity follow-up and interpretation of companion signals.

Individual Dynamical Masses of Ultracool Dwarfs* † ‡

Abstract We present the full results of our decade-long astrometric monitoring programs targeting 31 ultracool binaries with component spectral types M7–T5. Joint analysis of resolved imaging from Keck Observatory and Hubble Space Telescope and unresolved astrometry from CFHT/WIRCam yields parallactic distances for all systems, robust orbit determinations for 23 systems, and photocenter orbits for 19 systems. As a result, we measure 38 precise individual masses spanning 30–115 . We determine a model-independent substellar boundary that is ≈70 in mass (≈L4 in spectral type), and we validate Baraffe et al. evolutionary model predictions for the lithium-depletion boundary (60 at field ages). Assuming each binary is coeval, we test models of the substellar mass–luminosity relation and find that in the L/T transition, only the Saumon &amp; Marley “hybrid” models accounting for cloud clearing match our data. We derive a precise, mass-calibrated spectral type–effective temperature relation covering 1100–2800 K. Our masses enable a novel direct determination of the age distribution of field brown dwarfs spanning L4–T5 and 30–70 . We determine a median age of 1.3 Gyr, and our population synthesis modeling indicates our sample is consistent with a constant star formation history modulated by dynamical heating in the Galactic disk. We discover two triple-brown-dwarf systems, the first with directly measured masses and eccentricities. We examine the eccentricity distribution, carefully considering biases and completeness, and find that low-eccentricity orbits are significantly more common among ultracool binaries than solar-type binaries, possibly indicating the early influence of long-lived dissipative gas disks. Overall, this work represents a major advance in the empirical view of very low-mass stars and brown dwarfs.

Revising the Evolutionary Stage of HD 163899: The Effects of Convective Overshooting and Rotation

Abstract We revise the evolutionary status of the B-type supergiant HD 163899 based on the new determinations of the mass–luminosity ratio, effective temperature, and rotational velocity, as well as on the interpretation of the oscillation spectrum of the star. The observed value of the nitrogen-to-carbon abundance fixes the value of the rotation rate of the star. Now, more massive models are strongly preferred than those previously considered, and it is very likely that the star is still in the main-sequence stage. The rotationally induced mixing manifests as the nitrogen overabundance in the atmosphere, which agrees with our analysis of the HARPS spectra. Thus, HD 163899 probably belongs to a group of evolved nitrogen-rich main-sequence stars.