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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Three-dimensional Simulations of Massive Stars. II. Age Dependence

We present 3D full star simulations, reaching up to 90% of the total stellar radius, for three 7 M⊙ stars of different ages: zero-age main sequence (ZAMS), mid–main sequence (midMS), and terminal-age main sequence (TAMS). A comparison with several theoretical prescriptions shows that the generation spectra for all three ages are dominated by convective plumes. Two distinct overshooting layers are observed, with most plumes stopped within the layer situated directly above the convective boundary; overshooting to the second, deeper layer becomes progressively more infrequent with increasing stellar age. Internal gravity wave (IGW) propagation is significantly impacted in the midMS and TAMS models as a result of some IGWs getting trapped within their Brunt–Väisälä frequency spikes. A fundamental change in the wave structure across radius is also observed, driven by the effect of density stratification on IGW propagation causing waves to become evanescent within the radiative zone, with older stars being affected more strongly. We find that the steepness of the frequency spectrum at the surface increases from ZAMS to the older models, with older stars also showing more modes in their spectra.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Chemical Analysis of the Brightest Star of the Cetus II Ultrafaint Dwarf Galaxy Candidate*

We present a detailed chemical abundance analysis of the brightest star in the ultrafaint dwarf (UFD) galaxy candidate Cetus II from high-resolution Magellan/MIKE spectra. For this star, DES J011740.53-173053, abundances or upper limits of 18 elements from carbon to europium are derived. Its chemical abundances generally follow those of other UFD galaxy stars, with a slight enhancement of the α-elements (Mg, Si, and Ca) and low neutron-capture element (Sr, Ba, and Eu) abundances supporting the classification of Cetus II as a likely UFD. The star exhibits lower Sc, Ti, and V abundances than Milky Way (MW) halo stars with similar metallicity. This signature is consistent with yields from a supernova originating from a star with a mass of ∼11.2 M$_{⊙}$. In addition, the star has a potassium abundance of [K/Fe] = 0.81, which is somewhat higher than the K abundances of MW halo stars with similar metallicity, a signature that is also present in a number of UFD galaxies. A comparison including globular clusters and stellar stream stars suggests that high K is a specific characteristic of some UFD galaxy stars and can thus be used to help classify objects as UFD galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

${S}^5$: The destruction of a bright dwarf galaxy as revealed by the chemistry of the Indus stellar stream

The recently discovered Indus stellar stream exhibits a diverse chemical signature compared to what is found for most other streams due to the abundances of two outlier stars, Indus$\_$0 and Indus$\_$13. Indus$\_$13, exhibits an extreme enhancement in rapid neutron-capture ($r$-)process elements with $\mathrm{[Eu/Fe]} = +1.81$. It thus provides direct evidence of the accreted nature of $r$-process enhanced stars. In this paper we present a detailed chemical analysis of the neutron-capture elements in Indus$\_$13, revealing the star to be slightly actinide poor. The other outlier, Indus$\_0$, displays a globular cluster-like signature with high N, Na, and Al abundances, while the rest of the Indus stars show abundances compatible with a dwarf galaxy origin. Hence, Indus$\_0$ provides the first chemical evidence of a fully disrupted dwarf containing a globular cluster. We use the chemical signature of the Indus stars to discuss the nature of the stream progenitor which was likely a chemically evolved system, with a mass somewhere in the range from Ursa Minor to Fornax.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Cannibals in the thick disk II. Radial-velocity monitoring of the young α -rich stars

Determining ages of stars for reconstructing the history of the Milky Way remains one of the most difficult tasks in astrophysics. This involves knowing when it is possible to relate the stellar mass with its age and when it is not. The young α-rich (YAR) stars present such a case in which we are still not sure about their ages because they are relatively massive, implying young ages, but their abundances are α-enhanced, which implies old ages. We report the results from new observations from a long-term radial-velocity-monitoring campaign complemented with high-resolution spectroscopy, as well as new astrometry and seismology of a sample of 41 red giants from the third version of APOKASC, which includes YAR stars. The aim is to better characterize the YAR stars in terms of binarity, mass, abundance trends, and kinematic properties. The radial velocities of HERMES, APOGEE, and Gaia were combined to determine the binary fraction among YAR stars. In combination with their mass estimate, evolutionary status, chemical composition, and kinematic properties, it allowed us to better constrain the nature of these objects. We found that stars with M < 1 M ⊙ were all single, whereas stars with M > 1 M ⊙ could be either single or binary. This is in agreement with theoretical predictions of population synthesis models. Studying their [C/N], [C/Fe], and [N/Fe], trends with mass, it became clear that many YAR stars do not follow the APOKASC stars, favoring the scenario that most of them are the product of mass transfer. Our sample further includes two likely undermassive stars, that is to say of such as low mass that they cannot have reached the red clump within the age of the Universe, unless their low mass is the signature of mass loss in previous evolutionary phases. These stars do not show signatures of currently being binaries. Both YAR and undermassive stars might show some anomalous APOGEE abundances for the elements N, Na, P, K, and Cr; although, higher-resolution optical spectroscopy might be needed to confirm these findings. Considering the significant fraction of stars that are formed in pairs and the variety of ways that makes mass transfer possible, the diversity in properties in terms of binarity, and chemistry of the YAR and undermassive stars studied here implies that most of these objects are likely not young.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Overview of oxygen opacity experiments at the National Ignition Facility and investigation of potential systematic errors

Experiments to measure oxygen opacity at stellar interior conditions have been performed at the National Ignition Facility in a Discovery Science campaign. These experiments utilize the Opacity-on-NIF platform with a sample comprised of O, Mg, and Si. The spectral data from the Opacity Spectrometer cover the 1000–2000 eV photon energy range showing bound-free continuum absorption from O and line absorption from Mg and Si. DANTE and the Gated X-ray Detector are employed to measure the sample plasma’s temperature and density, respectively. Initial data show lower transmission than expected by theoretical models, raising questions of whether potential background or data uniformity concerns could produce systematic errors in the inferred transmission. Here, we investigate three concerns thought to be important for the oxygen opacity data, including instrumental scattered background, sample self-emission non-uniformity, and backlight continuum non-uniformity. Additionally, we show the effect of a recently developed method to account for 2nd order crystal reflection. The total effect of these concerns on one experiment is found to be small compared to the observed difference between the inferred transmission and a model calculation at the inferred temperature and density. Thus, we conclude that these potential sources of systematic error cannot account for the observed difference, increasing the likelihood of a real effect due to the high temperature and density conditions. However, because this is only a single experiment, we cannot make a firm conclusion. More experiments measuring the opacity and necessary calibrations are needed to assess the reproducibility and uncertainty of this result.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

The Galactic Distribution of Phosphorus: A Survey of 163 Disk and Halo Stars*

Phosphorus (P) is a critical element for life on Earth, yet the cosmic production sites of P are relatively uncertain. To understand how P has evolved in the solar neighborhood, we measured abundances for 163 FGK stars over a range of –1.09 < [Fe/H] < 0.47 using observations from the Habitable-zone Planet Finder instrument on the Hobby–Eberly Telescope. Atmospheric parameters were calculated by fitting a combination of astrometry, photometry, and Fe I line equivalent widths. Phosphorus abundances were measured by matching synthetic spectra to a P I feature at 10529.52 Å. Our [P/Fe] ratios show that chemical evolution models generally underpredict P over the observed metallicity range. Additionally, we find that the [P/Fe] differs by ~0.1 dex between thin disk and thick disk stars that were identified with kinematics. The P abundances were compared with α-elements, iron-peak, odd-Z, and s-process elements, and we found that the evolution of P in the disk most strongly resembles that of the α-elements. We also find that molar P/C and N/C ratios for our sample match the scatter seen from other abundance studies. Finally, we measure a [P/Fe] = 0.09 ± 0.1 ratio in one low-α halo star and probable Gaia–Sausage–Enceladus member, an abundance ratio ~0.3–0.5 dex lower than the other Milky Way disk and halo stars at similar metallicities. Overall, we find that P is likely most significantly produced by massive stars in core-collapse supernovae, based on the largest P abundance survey to date.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Three-dimensional Hydrodynamic Simulations of Convective Nuclear Burning in Massive Stars Near Iron Core Collapse

Nonspherical structure in massive stars at the point of iron core collapse can have a qualitative impact on the properties of the ensuing core-collapse supernova explosions and the multimessenger signals they produce. Strong perturbations can aid successful explosions by strengthening turbulence in the postshock region. Here we report on a set of 4π 3D hydrodynamic simulations of O- and Si-shell burning in massive star models of varied initial masses using MESA and the FLASH simulation framework. We evolve four separate 3D models for roughly the final 10 minutes prior to and including iron core collapse. We consider initial 1D MESA models with masses of 14, 20, and 25 M ⊙ to survey a range of O/Si-shell density and compositional configurations. Further, we characterize the convective shells in our 3D models and compare them to the corresponding 1D models. In general, we find that the angle-average convective speeds in our 3D simulations near collapse are three to four times larger than the convective speeds predicted by MESA at the same epoch for our chosen mixing length parameter of α MLT = 1.5. In three of our simulations, we observe significant power in the spherical harmonic decomposition of the radial velocity field at harmonic indices of ℓ = 1–3 near collapse. Our results suggest that large-scale modes are common in massive stars near collapse and should be considered a key aspect of presupernova progenitor models.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

The Pulsating Helium-atmosphere White Dwarfs. I. New DBVs from the Sloan Digital Sky Survey

Abstract We present a dedicated search for new pulsating helium-atmosphere (DBV) white dwarfs from the Sloan Digital Sky Survey using the McDonald 2.1 m Otto Struve Telescope. In total we observed 55 DB and DBA white dwarfs with spectroscopic temperatures between 19,000 and 35,000 K. We find 19 new DBVs and place upper limits on variability for the remaining 36 objects. In combination with previously known DBVs, we use these objects to provide an update to the empirical extent of the DB instability strip. With our sample of new DBVs, the red edge is better constrained, as we nearly double the number of DBVs known between 20,000 and 24,000 K. We do not find any new DBVs hotter than PG 0112+104, the current hottest DBV is at T eff ≈ 31,000 K, but do find pulsations in four DBVs with temperatures between 27,000 and 30,000 K, improving empirical constraints on the poorly defined blue edge. We investigate the ensemble pulsation properties of all currently known DBVs, finding that the weighted mean period and total pulsation power exhibit trends with effective temperature that are qualitatively similar to the pulsating hydrogen-atmosphere white dwarfs.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Reconstructing the Stripping History of the Sagittarius Stream with Neural Networks

The Sagittarius (Sgr) Stream is produced by the ongoing disruption of the Sgr dwarf spheroidal (dSph) galaxy and is thought to contain multiple wraps that were stripped during different pericentric passages. In this study, we introduce a neural-network–based method trained on N-body simulations to infer the stripping time of Sgr Stream stars directly from their phase-space coordinates. We combine spectroscopic data from SEGUE, APOGEE DR17, and LAMOST DR7 low-resolution spectroscopic (LRS) survey with Gaia EDR3 astrometry and distance estimates from the latest StarHorse catalog to identify high-quality Sgr Stream members. Applying our method to these stars, we measure a clear metallicity gradient with stripping time, well described by a linear relation with slope ∼0.3 dex Gyr −1 . We further predict the stripping times of globular clusters previously suggested to originate from the Sgr dSph. M 54, Terzan 7, Terzan 8, and Arp 2 exhibit stripping times consistent with being currently bound to the Sgr remnant. Pal 12, Whiting 1, and NGC 2419 are inferred to have been stripped 0.9 ± 0.1, 1.1 ± 0.2, and 2.1 ± 0.2 Gyr ago, respectively. For NGC 4147 and NGC 5634, whose membership in the Sgr system remains uncertain, our analysis suggests stripping times of 1.1 ± 0.4 and 1.1 ± 0.1 Gyr, respectively, if they are ultimately confirmed as genuine Sgr members. These results demonstrate that data-driven models of dynamical stripping histories offer a promising approach for reconstructing the formation and chemical evolution of the Sgr Stream.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

An Electron-scattering Time Delay in Black Hole Accretion Disks

Abstract Universal to black hole X-ray binaries, the high-frequency soft lag gets longer during the hard-to-intermediate state transition, evolving from ≲1 to ∼10 ms. The soft lag production mechanism is thermal disk reprocessing of nonthermal coronal irradiation. X-ray reverberation models account for the light-travel time delay external to the disk, but assume instantaneous reprocessing of the irradiation inside the electron-scattering-dominated disk atmosphere. We model this neglected scattering time delay as a random walk within an α -disk atmosphere, with approximate opacities. To explain soft lag trends, we consider a limiting case of the scattering time delay that we dub the thermalization time delay, t th ; this is the time for irradiation to scatter its way down to the effective photosphere, where it gets thermalized, and then scatter its way back out. We demonstrate that t th plausibly evolves from being inconsequential for low mass accretion rates m ̇ characteristic of the hard state, to rivaling or exceeding the light-travel time delay for m ̇ characteristic of the intermediate state. However, our crude model confines t th to a narrow annulus near peak accretion power dissipation, so cannot yet explain in detail the anomalously long-duration soft lags associated with larger disk radii. We call for time-dependent models with accurate opacities to assess the potential relevance of a scattering delay.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

The kinematic properties of Milky Way stellar halo populations

ABSTRACT In the Gaia era, stellar kinematics are extensively used to study Galactic halo stellar populations, to search for halo structures, and to characterize the interface between the halo and hot disc populations. We use distribution function-based models of modern data sets with 6D phase space data to qualitatively describe a variety of kinematic spaces commonly used in the study of the Galactic halo. Furthermore, we quantitatively assess how well each kinematic space can separate radially anisotropic from isotropic halo populations. We find that scaled action space (the ‘action diamond’) is superior to other commonly used kinematic spaces at this task. We present a new, easy to implement selection criterion for members of the radially anisotropic Gaia–Enceladus merger remnant. Assuming a 1:1 ratio of Gaia–Enceladus stars to more isotropic halo, we find our selection achieves a sample purity of 86 per cent in our models with respect to contamination from the more isotropic halo. We compare this criterion to literature criteria, finding that it produces the highest purity in the resulting samples, at the expense of a modest reduction in completeness. We also show that selection biases that underlie nearly all contemporary spectroscopic data sets can noticeably impact the E−Lz distribution of samples in a manner that may be confused for real substructure. We conclude by providing recommendations for how authors should use stellar kinematics in the future to study the Galactic stellar halo.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

ARMADA. I. Triple Companions Detected in B-type Binaries α Del and ν Gem

Ground-based optical long-baseline interferometry has the power to measure the orbits of close binary systems at ∼10 μas precision. This precision makes it possible to detect “wobbles” in the binary motion due to the gravitational pull from additional short-period companions. We started the ARrangement for Micro-Arcsecond Differential Astrometry (ARMADA) survey with the Michigan Infra-Red Combiner (MIRC)/MIRC-X instrument at the Center for High Angular Resoloution Astronomy (CHARA) array for the purpose of detecting giant planets and stellar companions orbiting individual stars in binary systems. We describe our observations for the survey, and introduce the wavelength calibration scheme that delivers precision at the tens of microarcseconds level for <02 binaries. We test our instrument performance on a known triple system, κ Peg, and show that our survey is delivering a factor of 10 better precision than previous similar surveys. We present astrometric detections of tertiary components to two B-type binaries: a 30 day companion to α Del, and a 50 day companion to ν Gem. We also collected radial velocity data for α Del with the Tennessee State University Automated Spectroscopic Telescope at Fairborn Observatory. We are able to measure the orbits and masses of all three components in these systems. We find that the previously published radial velocity orbit for the inner pair of ν Gem is not consistent with our visual orbit. The precision achieved for these orbits suggests that our ARMADA survey will be successful at discovering new compact triple systems to A/B-type binary systems, leading to better statistics of hierarchical system architectures and formation history.

47 OTHER INSTRUMENTATION↗
DOE OSTI2022

Measuring Chemical Likeness of Stars with Relevant Scaled Component Analysis

Identification of chemically similar stars using elemental abundances is core to many pursuits within Galactic archeology. However, measuring the chemical likeness of stars using abundances directly is limited by systematic imprints of imperfect synthetic spectra in abundance derivation. We present a novel data-driven model that is capable of identifying chemically similar stars from spectra alone. We call this relevant scaled component analysis (RSCA). RSCA finds a mapping from stellar spectra to a representation that optimizes recovery of known open clusters. By design, RSCA amplifies factors of chemical abundance variation and minimizes those of nonchemical parameters, such as instrument systematics. The resultant representation of stellar spectra can therefore be used for precise measurements of chemical similarity between stars. We validate RSCA using 185 cluster stars in 22 open clusters in the Apache Point Observatory Galactic Evolution Experiment survey. We quantify our performance in measuring chemical similarity using a reference set of 151,145 field stars. We find that our representation identifies known stellar siblings more effectively than stellar-abundance measurements. Using RSCA, 1.8% of pairs of field stars are as similar as birth siblings, compared to 2.3% when using stellar-abundance labels. We find that almost all of the information within spectra leveraged by RSCA fits into a two-dimensional basis, which we link to [Fe/H] and α-element abundances. We conclude that chemical tagging of stars to their birth clusters remains prohibitive. However, using the spectra has noticeable gain, and our approach is poised to benefit from larger data sets and improved algorithm designs.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Comparative Analysis of TRGBs (CATs) from Unsupervised, Multi-halo-field Measurements: Contrast is Key

The tip of the red giant branch (TRGB) is an apparent discontinuity of the luminosity function (LF) due to the end of the red giant evolutionary phase and is used to measure distances in the local universe. In practice, tip localization via edge detection response (EDR) relies on several methods applied on a case-by-case basis. It is hard to evaluate how individual choices affect a distance estimation using only a single host field while also avoiding confirmation bias. To devise a standardized approach, we compare unsupervised, algorithmic analyses of the TRGB in multiple halo fields per galaxy. We first optimize methods for the lowest field-to-field dispersion, including spatial filtering, smoothing, and weighting of LF, color band selection, and tip selection based on the number of likely RGB stars and the ratio of stars below versus above the tip (R). We find R, which we call the tip contrast, to be the most important indicator of the quality of EDR measurements; higher R selection can decrease field-to-field dispersion. Further, since R is found to correlate with the age or metallicity of the stellar population based on theoretical modeling, it might result in a displacement of the detected tip magnitude. We find a tip-contrast relation with a slope of -0.023 ± 0.0046 mag/ratio, an ~5σ result that can be used to correct these variations in the detections. When using TRGB to establish a distance ladder, consistent TRGB standardization using tip-contrast relation across rungs is vital to make robust cosmological measurements.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

The predicted properties of helium-enriched globular cluster progenitors at high redshift

Globular cluster progenitors may have been detected by Hubble Space Telescope, and are predicted to be observable with James Webb Space Telescope (JWST) and ground-based extremely large telescopes with adaptive optics. This has the potential to elucidate the issue of globular cluster formation and the origins of significantly helium-enriched subpopulations, a problem in Galactic astronomy with no satisfactory theoretical solution. Given this context, we use model stellar tracks and isochrones to investigate the predicted observational properties of helium-enriched stellar populations in globular cluster progenitors. We find that, relative to helium-normal populations, helium-enriched (ΔY = +0.12) stellar populations similar to those inferred in the most massive globular clusters, are expected, modulo some rapid fluctuations in the first ~30 Myr, to be brighter and redder in the rest frame. At fixed age, stellar mass, and metallicity, a helium-enriched population is predicted to converge to being ~0.40 mag brighter at $\lambda \approx 2.0\, {\mu \rm m}$, and to be 0.30-mag redder in the JWST–NIRCam colour (F070W - F200W), and to actually be fainter for $\lambda \lesssim 0.50 \, {\mu \rm m}$. Separately, we find that the time-integrated shift in ionizing radiation is a negligible $\sim \!5{{\ \rm per\ cent}}$, though we show that the Lyman-α escape fraction could end up higher for helium-enriched stars.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Relating galaxies across different redshift to study galaxy evolution

We propose a general framework leveraging the galaxy–halo connection to link galaxies observed at different redshift in a statistical way, and use the link to infer the redshift evolution of the galaxy population. Our tests based on hydrodynamic simulations show that our method can accurately recover the stellar mass assembly histories up to z ∼ 3 for present star-forming and quiescent galaxies down to 10 10 h −1 M ⊙ . Applying the method to observational data shows that the stellar mass evolution of the main progenitors of galaxies depends strongly on the properties of descendants, such as stellar mass, halo mass, and star formation states. Galaxies hosted by low-mass groups/haloes at the present time have since z ∼ 1.8 grown their stellar mass ∼2.5 times as fast as those hosted by massive clusters. This dependence on host halo mass becomes much weaker for descendant galaxies with similar star formation states. Star-forming galaxies grow about 2–4 times faster than their quiescent counterparts since z ∼ 1.8. Both TNG and EAGLE simulations overpredict the progenitor stellar mass at z > 1, particularly for low-mass descendants.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Substructure at High Speed. II. The Local Escape Velocity and Milky Way Mass with Gaia eDR3

Abstract Measuring the escape velocity of the Milky Way is critical in obtaining the mass of the Milky Way, understanding the dark matter velocity distribution, and building the dark matter density profile. In Necib & Lin, we introduced a strategy to robustly measure the escape velocity. Our approach takes into account the presence of kinematic substructures by modeling the tail of the stellar distribution with multiple components, including the stellar halo and the debris flow called the Gaia Sausage (Enceladus). In doing so, we can test the robustness of the escape velocity measurement for different definitions of the “tail” of the velocity distribution and the consistency of the data with different underlying models. In this paper, we apply this method to the Gaia eDR3 data release and find that a model with two components is preferred, although results from a single-component fit are also consistent. Based on a fit to retrograde data with two bound components to account for the relaxed halo and the Gaia Sausage, we find the escape velocity of the Milky Way at the solar position to be v esc = 445 − 8 + 25 km s −1 . A fit with a single component to the same data gives v esc = 472 − 12 + 17 km s −1 . Assuming a Navarro−Frenck−White dark matter profile, we find a Milky Way concentration of c 200 = 19 − 7 + 11 and a mass of M 200 = 4.6 − 0.8 + 1.5 × 10 11 M ⊙ , which is considerably lighter than previous measurements.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Photometric Recalibration of the SDSS Stripe 82 to a Few Millimagnitude Precision with the Stellar Color Regression Method and Gaia EDR3

By combining spectroscopic data from the LAMOST DR7, Sloan Digital Sky Survey (SDSS) DR12, and corrected photometric data from the Gaia EDR3, we apply the stellar color regression (SCR) method to recalibrate the SDSS Stripe 82 standard stars catalog of Ivezic' et al. With a total number of about 30,000 spectroscopically targeted stars, we have mapped out the relatively large and strongly correlated photometric zero-point errors present in the catalog, ~2.5% in the u band and ~1% in the griz bands. Our study also confirms some small but significant magnitude dependence errors in the z band for some charge-coupled devices. Various tests show that we have achieved an internal precision of about 5 mmag in the u band and about 2 mmag in the griz bands, which is about five times better than previous results. We also apply the method to the latest version of the catalog (v4.2), and find modest systematic calibration errors of up to ~1% along the R.A. direction and smaller errors along the decl. direction. The results demonstrate the power of the SCR method when combining spectroscopic data and Gaia photometry in breaking the 1% precision barrier of ground-based photometric surveys. Our work paves the way for the recalibration of the whole SDSS photometric survey and has important implications for the calibration of future surveys. Future implementations and improvements of the SCR method under different situations are also discussed.

79 ASTRONOMY AND ASTROPHYSICS↗