Spheroidal Galaxies Research Articles

Andromeda XXV – a dwarf galaxy with a low central dark matter density

ABSTRACT Andromeda (And) XXV has previously been reported as a dwarf spheroidal galaxy (dSph) with little-to-no dark matter. However, the uncertainties on this result were significant. In this study, we nearly double the number of member stars and re-derive the kinematics and mass of And XXV. We find that And XXV has a systemic velocity of νr = −107.7 ± 1.0 kms−1 and a velocity dispersion of $\sigma _{\rm \nu }=3.7^{+1.2}_{-1.1}$ kms−1. With this updated velocity dispersion and a new literature measurement of the radial surface brightness profile, we derive a mass contained within the half-light radius of $M(r\lt r_{\rm h})=4.7^{+3.0}_{-2.9}\times 10^6$ M⊙. This mass corresponds to a mass-to-light ratio of $\text{[M/L]}_{\rm r_{h}}=25^{+17}_{-16}$ M⊙/L⊙, demonstrating that And XXV is most-likely dark matter dominated. We also measure the metallicity of And XXV to be $\rm [Fe/H]=-1.9\pm 0.1$ dex, which is in agreement with previous results. Finally, we extend the analysis of And XXV to include mass modelling using GravSphere. We find that And XXV has a low central dark matter density, ρDM(150 pc) = $2.3^{+1.4}_{-1.1}\times 10^7$ M⊙ kpc−3, which makes And XXV a clear outlier when compared to other local group dSphs of the similar stellar mass. In a companion paper, we will explore whether some combination of dark matter cusp-core transformations and/or tides can explain And XXV’s low density.

Enhanced cosmic-ray antihelium production from dark matter annihilation through light mediators

Cosmic-ray (CR) antihelium is an important probe for the indirect search of dark matter (DM) annihilation in the Galaxy. However, due to stringent constraints from the measurements of CR antiprotons and γ-rays, the flux of CR antihelium from the conventional DM direct annihilation into Standard Model final states is expected to be far below the sensitivity of the current experiments. We show that the production of antihelium can be significantly enhanced if the DM particles annihilate through light mediator particles with a mass mϕ ≈ 8 GeV close to the antihelium production threshold. After taking into account the constraints from the AMS-02 antiproton data and the Fermi-LAT γ-ray data on the spheroidal dwarf galaxies, we find that in this scenario the CR antihelium flux can be enhanced by three orders of magnitude, which makes it within the sensitivity of the ongoing AMS-02 experiment.

Searching for TeV Dark Matter in Irregular Dwarf Galaxies with HAWC Observatory

We present the results of dark matter (DM) searches in a sample of 31 dwarf irregular (dIrr) galaxies within the field of view of the HAWC Observatory. dIrr galaxies are DM-dominated objects in which astrophysical gamma-ray emission is estimated to be negligible with respect to the secondary gamma-ray flux expected by annihilation or decay of weakly interacting massive particles (WIMPs). While we do not see any statistically significant DM signal in dIrr galaxies, we present the exclusion limits (95% C.L.) for annihilation cross section and decay lifetime for WIMP candidates with masses between 1 and 100 TeV. Exclusion limits from dIrr galaxies are relevant and complementary to benchmark dwarf Spheroidal (dSph) galaxies. In fact, dIrr galaxies are targets kinematically different from benchmark dSph, preserving the footprints of different evolution histories. We compare the limits from dIrr galaxies to those from ultrafaint and classical dSph galaxies previously observed with HAWC. We find that the constraints are comparable to the limits from classical dSph galaxies and ∼2 orders of magnitude weaker than the ultrafaint dSph limits.

CEERS Key Paper. III. The Diversity of Galaxy Structure and Morphology at z = 3–9 with JWST

We present a comprehensive analysis of the evolution of the morphological and structural properties of a large sample of galaxies at z = 3–9 using early James Webb Space Telescope (JWST) CEERS NIRCam observations. Our sample consists of 850 galaxies at z > 3 detected in both Hubble Space Telescope (HST)/WFC3 and CEERS JWST/NIRCam images, enabling a comparison of HST and JWST morphologies. We conduct a set of visual classifications, with each galaxy in the sample classified three times. We also measure quantitative morphologies across all NIRCam filters. We find that galaxies at z > 3 have a wide diversity of morphologies. Galaxies with disks make up 60% of galaxies at z = 3, and this fraction drops to ∼30% at z = 6–9, while galaxies with spheroids make up ∼30%–40% across the redshift range, and pure spheroids with no evidence for disks or irregular features make up ∼20%. The fraction of galaxies with irregular features is roughly constant at all redshifts (∼40%–50%), while those that are purely irregular increases from ∼12% to ∼20% at z > 4.5. We note that these are apparent fractions, as many observational effects impact the visibility of morphological features at high redshift. On average, Spheroid-only galaxies have a higher Sérsic index, smaller size, and higher axis ratio than disk or irregular galaxies. Across all redshifts, smaller spheroid and disk galaxies tend to be rounder. Overall, these trends suggest that galaxies with established disks and spheroids exist across the full redshift range of this study, and further work with large samples at higher redshift is needed to quantify when these features first formed.

Search for Ultraheavy Dark Matter from Observations of Dwarf Spheroidal Galaxies with VERITAS

Dark matter is a key piece of the current cosmological scenario, with weakly interacting massive particles (WIMPs) a leading dark matter candidate. WIMPs have not been detected in their conventional parameter space (100 GeV ≲M χ ≲ 100 TeV), a mass range accessible with current Imaging Atmospheric Cherenkov Telescopes. As ultraheavy dark matter (UHDM; M χ ≳ 100 TeV) has been suggested as an underexplored alternative to the WIMP paradigm, we search for an indirect dark matter annihilation signal in a higher mass range (up to 30 PeV) with the VERITAS γ-ray observatory. With 216 hr of observations of four dwarf spheroidal galaxies, we perform an unbinned likelihood analysis. We find no evidence of a γ-ray signal from UHDM annihilation above the background fluctuation for any individual dwarf galaxy nor for a joint-fit analysis, and consequently constrain the velocity-weighted annihilation cross section of UHDM for dark matter particle masses between 1 TeV and 30 PeV. We additionally set constraints on the allowed radius of a composite UHDM particle.

RR Lyrae Mid-infrared Period–Luminosity–Metallicity and Period–Wesenheit–Metallicity Relations Based on Gaia DR3 Parallaxes

We present new empirical infrared period–luminosity–metallicity (PLZ) and period–Wesenheit–metallicity (PWZ) relations for RR Lyae based on the latest Gaia Early Data Release 3 (EDR3) parallaxes. The relations are provided in the Wide-field Infrared Survey Explorer (WISE) W1 and W2 bands, as well as in the W(W1, V − W1) and W(W2, V − W2) Wesenheit magnitudes. The relations are calibrated using a very large sample of Galactic halo field RR Lyrae stars with homogeneous spectroscopic [Fe/H] abundances (over 1000 stars in the W1 band), covering a broad range of metallicities (−2.5 ≲ [Fe/H] ≲ 0.0). We test the performance of our PLZ and PWZ relations by determining the distance moduli of both galactic and extragalactic stellar associations: the Sculptor dwarf spheroidal galaxy in the Local Group (finding ), the Galactic globular clusters M4 (), and the Reticulum globular cluster in the Large Magellanic Cloud (). The distance moduli determined through all our relations are internally self-consistent (within ≲0.05 mag) but are systematically smaller (by ∼2–3σ) than previous literature measurements taken from a variety of methods/anchors. However, a comparison with similar recent RR Lyrae empirical relations anchored with EDR3 likewise shows, to varying extents, a systematically smaller distance modulus for PLZ/PWZ RR Lyrae relations.

Velocity-dependent J-factors for Milky Way dwarf spheroidal analogues in cosmological simulations

We study the impact of the dark matter velocity distribution modelling on signals from velocity-dependent dark matter annihilation in Milky Way dwarf spheroidal galaxies. Using the high resolution APOSTLE simulations, we identify analogues corresponding to Milky Way dwarf spheroidal galaxies, and from these directly determine the dark matter pair-wise relative velocity distribution, and compare to best-fitting Maxwell-Boltzmann distribution models. For three velocity-dependent annihilation models, p-wave, d-wave, and the Sommerfeld model, we quantify the errors introduced when using the Maxwell-Boltzmann parameterization. We extract a simple power-law relation between the maximum circular velocity of the dwarf spheroidal analogue and the peak speed of the Maxwell-Boltzmann distribution. We show that this relation can be used to accurately calculate the dark matter relative velocity distribution, and find that it allows us to estimate the dark matter annihilation signal without the need to directly calculate the relative velocity distribution for each galaxy. The scatter in the J-factors calculated from the analogues dominates the uncertainty obtained when compared to the J-factor as determined from the observational data for each dwarf spheroidal, with the largest scatter from d-wave models and the smallest from Sommerfeld models.

Star formation histories of dwarf spheroidal and dwarf elliptical galaxies in the local Universe

ABSTRACTWe present the star formation histories (SFHs) of early-type dwarf galaxies, dSphs and dEs, in the local Universe within z = 0.01. The SFHs of early-type dwarf galaxies are characterized by pre-enriched, metal-poor old stellar populations, absence of moderately old stars that have ages of a few Gyr. There are some differences in the SFHs of dSphs and dEs. In particular, dSphs formed old (≳ 10 Gyr old) metal-poor stars ∼2 times more than dEs. The effects of reionization and feedback from supernova explosions are thought to be strong enough to remove the gas left, which prevent moderately old stellar populations in dSphs. In contrast, the ejected gases are not completely removed from dEs and fall back to ignite burst of star formation at a few Gyr after the first period of violent bursts of star formation, showing a suppression of star formation at lookback time ≈9.6 Gyr. The second peak of star formation at lookback time ≈4.5 Gyr in dEs produces moderately old stellar populations. Distinction between dSphs and dEs is useful to examine the SFHS of the early-type dwarfs since the cumulative SFHs are most closely related to their morphology. The stellar mass plays an important role in the SFHs of the early-type dwarfs as a driver of star formation, especially in galaxies with primordial origin.

Forecasts on the Dark Matter Density Profiles of Dwarf Spheroidal Galaxies with Current and Future Kinematic Observations

We forecast parameter uncertainties on the mass profile of a typical Milky Way dwarf spheroidal galaxy (dSph) using the spherical Jeans equation and Fisher matrix formalism. For a Draco-like system we show that radial velocity measurements for 1000 individual stars can constrain the mass contained within the effective radius of a dSph to within 5%. This is consistent with constraints extracted from current observational data. We compare two systems, a cusp and core, and demonstrate that a minimum sample of 100,000 (10,000) stars with both radial and proper motions measurements is required to disentangle their inner slopes at the 2σ (1σ) level. If using the log-slope measured at the half-light radius as a proxy for differentiating between a core or cusp slope, only 1000 line-of-sight and proper motions measurements are required; however, we show this choice of radius does not always unambiguously differentiate between core and cusped profiles. Once observational errors are below half the value of the intrinsic dispersion, improving the observational precision yields little change in the density profile uncertainties. The choice of priors in our profile shape analysis plays a crucial role when the number of stars in a system is less than 100 but does not affect the resulting uncertainties for larger kinematic samples. Our predicted 2D confidence regions agree well with those from a full likelihood analysis run on a mock kinematic data set taken from the Gaia Challenge, validating our Fisher predictions. Our methodology is flexible, allowing us to predict density profile uncertainties for a wide range of current and future kinematic data sets.

Molecular Gas Properties in Young Stellar Clusters with a Suppressed Star Cluster Wind

In compact and dense star-forming clouds a global star cluster wind could be suppressed. In this case stellar feedback is unable to expel the leftover gas from the cluster. Young massive stars remain embedded in a dense residual gas and stir it by moving in the gravitational well of the system. Here we present a self-consistent model for the molecular gas distribution in such young, enshrouded stellar clusters. It is assumed that the cloud collapse terminates and the star formation ceases when a balance between the turbulent pressure and gravity and between the turbulent energy dissipation and regeneration rates is established. These conditions result in an equation that determines the residual gas density distribution that, in turn, allows one to determine the other characteristics of the leftover gas and the star formation efficiency. It is shown that our model predictions are in good agreement with several observationally determined properties of cloud D1 in the nearby dwarf spheroidal galaxy NGC 5253 and its embedded cluster.

The Gaia-ESO survey: Placing constraints on the origin of r-process elements

Context. A renewed interest in the origin of r-process elements has been stimulated by the multi-messenger observation of the gravitational event GW170817, with the detection of both gravitational waves and electromagnetic waves corresponding to the merger of two neutron stars. Such a phenomenon has been proposed as one of the main sources of the r-process. However, the origin of the r-process elements at different metallicities is still under debate. Aims. We aim at investigate the origin of the r-process elements in the Galactic thin-disc population. Methods. From the sixth internal data release of the Gaia-ESO, we have collected a large sample of Milky Way (MW) thin- and thick-disc stars for which abundances of Eu, O, and Mg are available. The sample consists of members of 62 open clusters (OCs), located at a Galactocentric radius between ∼5 kpc and ∼20 kpc in the disc, in the metallicity range [ − 0.5, 0.4], and covering an age interval from 0.1 to 7 Gy, and about 1300 Milky Way disc field stars in the metallicity range [ − 1.5, 0.5]. We compare the observations with the results of a chemical evolution model, in which we varied the nucleosynthesis sources for the three elements considered. Results. Our main result is that Eu in the thin disc is predominantly produced by sources with short lifetimes, such as magneto-rotationally driven SNe. There is no strong evidence for additional sources at delayed times. Conclusions. Our findings do not imply that there cannot be a contribution from mergers of neutron stars in other environments, as in the halo or in dwarf spheroidal galaxies, but such a contribution is not needed to explain Eu abundances at thin-disc metallicities.

Boundary Conditions in Hydrodynamic Simulations of Isolated Galaxies and Their Impact on the Gas-loss Processes

Three-dimensional hydrodynamic simulations are commonly used to study the evolution of the gaseous content in isolated galaxies, besides its connection with galactic star formation histories. Stellar winds, supernova blasts, and black hole feedback are mechanisms usually invoked to drive galactic outflows and decrease the initial galactic gas reservoir. However, any simulation imposes the need of choosing the limits of the simulated volume, which depends, for instance, on the size of the galaxy and the required numerical resolution, besides the available computational capability to perform it. In this work, we discuss the effects of boundary conditions on the evolution of the gas fraction in a small-sized galaxy (tidal radius of ∼1 kpc), like classical spheroidal galaxies in the Local Group. We found that open boundaries with sizes smaller than approximately 10 times the characteristic radius of the galactic dark-matter halo become unappropriated for this kind of simulation after ∼0.6 Gyr of evolution, since they act as an infinity reservoir of gas due to dark-matter gravity. We also tested two different boundary conditions that avoid gas accretion from numerical frontiers: closed and selective boundary conditions. Our results indicate that the later condition (that uses a velocity threshold criterion to open or close frontiers) is preferable since it minimizes the number of reversed shocks due to closed boundaries. Although the strategy of putting computational frontiers as far as possible from the galaxy itself is always desirable, simulations with a selective boundary condition can lead to similar results at lower computational costs.

Semiannihilation of fermionic dark matter

The continued nonobservation of events emanating from dark matter annihilations in various direct and indirect detection experiments calls into question the mechanism for determining the relic density of a weakly interacting massive particle. However, if the relic density is determined primarily by a semiannihilation process, as opposed to the usual annihilation, this tension can be ameliorated. Here, we investigate a ${Z}_{3}$ symmetric effective field theory incorporating a fermionic dark matter that semiannihilates to right-handed neutrinos. The dynamics of the right-handed neutrinos and the impact of its late decays are also scrutinized while obtaining the correct dark matter relic. Finally, indirect detection bounds on the semiannihilation cross sections are drawn from the gamma-ray observations in the direction of dwarf spheroidal galaxies (Fermi-LAT), and including the projections obtained for the H.E.S.S. and the CTA detectors.

Reading the tea leaves in the Mbh–M*,sph and Mbh–Re,sph diagrams: dry and gaseous mergers with remnant angular momentum

ABSTRACT We recently revealed that bulges and elliptical galaxies broadly define distinct, superlinear relations in the (black hole mass, Mbh)–(spheroid stellar mass, M*,sph) diagram or Mbh−M*,sph diagram, with the order-of-magnitude lower Mbh/M*,sph ratios in the elliptical galaxies due to major (disc-destroying, elliptical-building) dry mergers. Here, we present a more nuanced picture. Galaxy mergers, in which the net orbital angular momentum does not cancel, can lead to systems with a rotating disc. This situation can occur with either wet (gas-rich) mergers involving one or two spiral galaxies, e.g. NGC 5128, or dry (relatively gas-poor) collisions involving one or two lenticular galaxies, e.g. NGC 5813. The spheroid and disc masses of the progenitor galaxies and merger remnant dictate the shift in the Mbh−M*,sph and Mbh−Re,sph diagrams. We show how this explains the (previously excluded merger remnant) Sérsic S0 galaxies near the bottom of the elliptical sequence and core-Sérsic S0 galaxies at the top of the bulge sequence, neither of which are faded spiral galaxies. Different evolutionary pathways in the scaling diagrams are discussed. We also introduce two ellicular (ES) galaxy types, explore the location of brightest cluster galaxies and stripped ‘compact elliptical’ galaxies in the Mbh−M*,sph diagram, and present a new merger-built Mbh−M*,sph relation which may prove helpful for studies of nanohertz gravitational waves. This work effectively brings into the fold many systems previously considered outliers with either overly massive or undermassive black holes relative to the near-linear Mbh−M*,sph ‘red sequence’ patched together with select bulges and elliptical galaxies.

Dynamics of dwarf galaxies in f(R) gravity

ABSTRACT We use the kinematic data of the stars in eight dwarf spheroidal galaxies to assess whether f(R) gravity can fit the observed profiles of the line-of-sight velocity dispersion of these systems without resorting to dark matter. Our model assumes that each galaxy is spherically symmetric and has a constant velocity anisotropy parameter β and constant mass-to-light ratio consistent with stellar population synthesis models. We solve the spherical Jeans equation that includes the Yukawa-like gravitational potential appearing in the weak field limit of f(R) gravity, and a Plummer density profile for the stellar distribution. The f(R) velocity dispersion profiles depends on two parameters: the scale length ξ−1, below which the Yukawa term is negligible, and the boost of the gravitational field δ > −1. δ and ξ are not universal parameters, but their variation within the same class of objects is expected to be limited. The f(R) velocity dispersion profiles fit the data with a value $\xi ^{-1}= 1.2^{+18.6}_{-0.9}$ Mpc for the entire galaxy sample. On the contrary, the values of δ show a bimodal distribution that picks at $\overline{\delta }=-0.986\pm 0.002$ and $\overline{\delta }=-0.92\pm 0.01$. These two values disagree at 6σ and suggest a severe tension for f(R) gravity. It remains to be seen whether an improved model of the dwarf galaxies or additional constraints provided by the proper motions of stars measured by future astrometric space missions can return consistent δ’s for the entire sample and remove this tension.

Indirect signals of dark matter can change depending on where you look

We propose that the nature of indirect signals of dark matter (DM) can depend on the Galactic environment they originate from. We demonstrate this possibility in models where DM annihilates into light mediators whose branching fractions depend on a long-range force sourced by ordinary matter. In particular, electromagnetic signals of DM may only arise near the centers of galaxies where the ordinary matter densities, and hence astrophysical background levels, are high. We briefly discuss how our model could explain the Galactic Center gamma ray excess, without leaving much of a trace in baryon-poor environments, like dwarf spheroidal galaxies. Similar spatial dependence of indirect signals can also apply to models featuring metastable DM decay into light mediators.

The metallicity distribution in the core of the Sagittarius dwarf spheroidal: Minimising the metallicity biases

We present the metallicity and radial velocity for 450 bona fide members of the Sagittarius dwarf spheroidal (Sgr dSph) galaxy, measured from high-resolution spectra (R ≃ 18 000) obtained with FLAMES at the VLT. The targets were carefully selected (a) to sample the core of the main body of Sgr dSph while avoiding contamination from the central stellar nucleus, and (b) to prevent any bias on the metallicity distribution by selecting targets based on theirGaiaparallax and proper motions. All the targets selected in this way were confirmed as radial velocity members. We used this sample to derive the first metallicity distribution of the core of Sgr dSph, which is virtually unaffected by metallicity biases. The observed distribution ranges from [Fe/H] ≃ −2.3 to [Fe/H] ≃ 0.0, with a strong, symmetric, and relatively narrow peak around [Fe/H] ≃ −0.5 and a weak and extended metal-poor tail, in which only 13.8 ± 1.9% of the stars have [Fe/H] < −1.0. We confirm previous evidence of correlations between chemical and kinematical properties of stars in the core of Sgr. In our sample, stars with [Fe/H] ≥ −0.6 display a lower velocity dispersion and a higher rotation amplitude than those with [Fe/H] < −0.6, confirming previous suggestions of disk/halo structure for the progenitor of the system.

Investigating the effect of Milky Way dwarf spheroidal galaxies extension on dark matter searches with Fermi-LAT data

Satellite galaxies of the Milky Way with high mass-to-light ratios and little baryon content, i.e., dwarf spheroidal galaxies (dSphs), are among the most promising targets to detect or constrain the nature of dark matter (DM) through its final annihilation products into high-energy photons. Previously, the assumption that DM emission from dSphs is pointlike has been used to set strong constraints on DM candidates using data from the Fermi Large Area Telescope (LAT). However, due to their high DM densities and proximity, dSphs actually have sufficient angular extension to be detected by the Fermi-LAT. Here, we perform a comprehensive analysis about the impact of accounting for angular extension in the search for gamma-ray DM signals toward known dSphs with Fermi-LAT. We show that, depending on the dSph under consideration, limits on the DM cross section can be weakened by up to a factor of 2--2.5, while the impact on the stacked, i.e., combined, limits is at most 1.5--1.8 depending on the annihilation channel. This result is of relevance when comparing dSph limits to other multimessenger DM constraints and for testing the DM interpretation of anomalous ``excesses.''

A probabilistic deep learning model to distinguish cusps and cores in dwarf galaxies

ABSTRACT Numerical simulations within a cold dark matter (DM) cosmology form haloes whose density profiles have a steep inner slope (‘cusp’), yet observations of galaxies often point towards a flat central ‘core’. We develop a convolutional mixture density neural network model to derive a probability density function (PDF) of the inner density slopes of DM haloes. We train the network on simulated dwarf galaxies from the NIHAO and AURIGA projects, which include both DM cusps and cores: line-of-sight velocities and 2D spatial distributions of their stars are used as inputs to obtain a PDF representing the probability of predicting a specific inner slope. The model recovers accurately the expected DM profiles: $\sim 82{{\ \rm per\ cent}}$ of the galaxies have a derived inner slope within ±0.1 of their true value, while $\sim 98{{\ \rm per\ cent}}$ within ±0.3. We apply our model to four Local Group dwarf spheroidal galaxies and find results consistent with those obtained with the Jeans modelling based code GravSphere: the Fornax dSph has a strong indication of possessing a central DM core, Carina and Sextans have cusps (although the latter with large uncertainties), while Sculptor shows a double peaked PDF indicating that a cusp is preferred, but a core cannot be ruled out. Our results show that simulation-based inference with neural networks provide a innovative and complementary method for the determination of the inner matter density profiles in galaxies, which in turn can help constrain the properties of the elusive DM.

Search for primordial black hole dark matter with x-ray spectroscopic and imaging satellite experiments and prospects for future satellite missions

Ultra-light primordial black holes (PBHs) in the mass range of 10$^{16}$ - 10$^{22}$ g are allowed by current observations to constitute a significant fraction, if not all, of the dark matter in the Universe. In this work, we present limits on ultra-light, non-rotating PBHs which arise from the non-detection of the Hawking radiation signals from such objects in the keV-MeV energy band. Namely, we consider observations from the current-generation missions XMM-Newton and INTEGRAL/SPI and discuss the observational perspectives of the future missions Athena, eXTP, and THESEUS for PBH searches. Based on 3.4 Msec total exposure time XMM-Newton observations of Draco dwarf spheroidal galaxy, we conclude that PBH with masses $\lesssim 10^{16}$ g can not make all dark matter at 95% confidence level. Our ON-OFF-type analysis of $>100$ Msec of INTEGRAL/SPI data on the Milky Way halo puts significantly stronger constraints. Only $\lesssim 10$% dark matter can be presented by PBHs with masses $\lesssim 3\cdot 10^{16}$ g while the majority of dark matter can not be represented by PBHs lighter than $7\cdot 10^{16}$ g at 95% confidence level. We discuss the strong impact of systematic uncertainty related to the variations of instrumental and astrophysical INTEGRAL/SPI background on the derived results and estimate its level. We also show that future large-field-of-view missions such as THESEUS/X-GIS will be able to improve the constraints by a factor of 10-100 depending on the level of control under the systematics of these instruments.