Stellar Shape Research Articles

Galaxy shapes in Magneticum

Context. Despite being one of the most fundamental properties of galaxies that dictate the form of the potential, the 3D shapes are intrinsically difficult to determine from observations. The improving quality of triaxial modeling methods in recent years has made it possible to measure these shapes more accurately. Aims. This study provides a comprehensive understanding of the stellar and dark matter (DM) shapes of galaxies and the connection between them. As these shapes are the result of the formation history of a galaxy, we investigate which galaxy properties they are correlated with, which will be especially useful for interpreting the results from dynamical modeling. Methods. Using the hydrodynamical cosmological simulation Magneticum Pathfinder Box4 (uhr), we computed the stellar and DM intrinsic shapes of 690 simulated galaxies with stellar masses above 2 × 1010 M⊙ at three different radii with an iterative unweighted method. We also determined their morphologies, their projected morphological and kinematic parameters, and their fractions of in situ formed stars. Results. The DM follows the stellar component in shape and orientation at three half-mass radii, indicating that DM is heavily influenced by the baryonic potential in the inner parts of the halo. The outer DM halo is independent of the inner properties such as the DM shape or galaxy morphology, however, and is more closely related to the large-scale anisotropy of the gas inflow. Overall, DM halo shapes are prolate, consistent with previous literature. The stellar shapes of galaxies are correlated with their morphology, with early-type galaxies featuring more spherical and prolate shapes than late-type galaxies out to 3 R1/2. Galaxies with more rotational support are flatter, and the stellar shapes are connected to the mass distribution, though not to the mass itself. In particular, more extended early-type galaxies have larger triaxialities at a given mass. Finally, the shapes can be used to better constrain the in situ fraction of stars when combined with the stellar mass. Conclusions. The relations between shape, mass distribution, and in situ formed star fraction of galaxies show that the shapes depend on the details of the accretion history through which the galaxies are formed. The similarities between DM and stellar shapes in the inner regions of galaxy halos signal the importance of baryonic matter for the behavior of DM in galaxies and will be of use for improving the underlying assumptions of dynamical models for galaxies in the future. However, at large radii the shapes of the DM are completely decoupled from the central galaxy, and their shapes and spin are coupled more to the large scale inflow than to the galaxy in the center.

Intimal Macrovascular Pericytes: Their Role in Vascular Biology and Atherogenesis.

Atherosclerosis remains a major challenge to global healthcare despite decades of research and constant trials of novel therapeutic approaches. One feature that makes atherosclerosis treatment so elusive is an insufficient understanding of its origins and the early stages of the pathological process, which limits our means of effective prevention of the disease. Macrovascular pericytes are cells with distinct shapes that are located in the arterial wall of larger vessels and are in many aspects similar to microvascular pericytes that maintain the functionality of small vessels and capillaries. This cell type combines the residual contractile function of smooth muscle cells with a distinct stellar shape that allows these cells to make numerous contacts between themselves and the adjacent endothelial layer. Moreover, pericytes can take part in the immune defense and are able to take up lipids in the course of atherosclerotic lesion development. In growing atherosclerotic plaques, the morphology and function of pericytes change dramatically due to phagocytic and synthetic phenotypes that are actively involved in lipid accumulation and extracellular matrix synthesis. In this review, we summarize our knowledge of this less-studied cell type and its role in atherosclerosis.

Radial velocities: Direct application of Pierre Connes’ shift-finding algorithm to cross-correlation functions

Context. Pipelines of state-of-the-art spectrographs dedicated to planet detection provide, for each exposure, series of cross-correlation functions (CCFs) built with a binary mask (BM), as well as the absolute radial velocity (RV) derived from the Gaussian fit of a weighted average CCFtot of the CCFs. Aims. Our aim was to test the benefits of the application of the shift-finding algorithm developed by Pierre Connes directly to the total CCFtot, and to compare the resulting RV shifts (DRVs) with the results of the Gaussian fits. In a second step, we investigated how the individual DRV profiles along the velocity grid derived from the shift-finding algorithm can be used as an easy tool for detection of stellar line shape variations. Methods. We developed the corresponding algorithm and tested it on 1151 archived spectra of the K2.5 V star HD 40307 obtained with ESO/ESPRESSO during a one-week campaign in 2018. Tests were performed based on the comparison of DRVs with RVs from Gaussian fits. DRV profiles along the velocity grid (DRV(i)) were scrutinized and compared with direct CCFtot ratios. Results. The dispersion of residuals from a linear fit to RVs from 406 spectra recorded within a single night, a measure of mean error, was found to be σ = 1.03 and 0.83 m s−1 for the Gaussian fit and the new algorithm, respectively, which is a significant 20% improvement in accuracy. The two full one-week series obtained during the campaign were also fitted with a three-planet system Keplerian model. The residual divergence between data and best-fit model is significantly smaller for the new algorithm than for the Gaussian fit. Such a difference was found to be associated in a large part with an increase of ≃1.3 m s−1 in the difference between the two types of RV values between the third and fourth nights. Interestingly, the DRV(i) profiles reveal at the same time a significant variation of line shape. Conclusions. The shift-finding algorithm is a fast and easy tool that provides additional diagnostics on the RV measurements in series of exposures. For observations made in the same instrumental configuration, and if line shapes are not varying significantly, it increases the accuracy of velocity variation determinations. On the other hand, departures from constancy of the DRV(i) profiles, as well as varying differences between RVs from this new method and RVs from a Gaussian fit can detect and report in a simple way line shape variations due to stellar activity.

A.C.I.D – an improved LSD technique for accurate line profile retrieval

ABSTRACT Stellar activity and planetary effects induce radial velocity (RV) offsets and cause temporal distortions in the shape of the stellar line profile. Hence, accurately probing the stellar line profile offers a wealth of information on both the star itself and any orbiting planets. Typically, cross-correlation functions (CCFs) are used as a proxy for the stellar line profile. The shape of CCFs, however, can be distorted by line blending and aliasing limiting the stellar and planetary physics that can be probed from them. Least-squares deconvolution (LSD) offers an alternative that directly fits the mean line profile of the spectrum to produce a high-precision profile. In this paper, we introduce our novel method ACID (Accurate Continuum fItting and Deconvolution) that builds on LSD techniques by simultaneously fitting the spectral continuum and line profile as well as performing LSD in effective optical depth. Tests on model data revealed ACID can accurately identify and correct the spectral continuum to retrieve an injected line profile. ACID was also applied to archival High Accuracy Radial-velocity Planet Searcher (HARPS) data obtained during the transit of HD189733b. The application of the Reloaded Rossiter–McLaughlin technique to both ACID profiles and HARPS CCFs shows ACID residual profiles improved the out-of-line root mean square (RMS) by over 5 per cent compared to CCFs. Furthermore, ACID profiles are shown to exhibit a Voigt profile shape that better describes the expected profile shape of the stellar line profile. This improved representation shows that ACID better preserves the stellar and planetary physics encoded in the stellar line profile shape for slow rotating stars.

Formula omitted]-SGP: Scaled Gradient Projection with [formula omitted]-divergence for astronomical image restoration

Image restoration in astronomy has been considered a vital step in many ground-based observational programs that often suffer from sub-optimal seeing due to atmospheric turbulence, distortion of stellar shapes due to instrumental aberrations, trailing, and other issues. It holds importance for various tasks: improved astrometry, deblending of overlapping sources, faint source detection, and identification of point sources near bright extended objects, such as galaxies, to name a few. We conduct an empirical study by applying the Scaled Gradient Projection (SGP) iterative image deconvolution algorithm to restore distorted stellar shapes in our observed data. We investigate using a more flexible divergence measure, the β-divergence, which contains the commonly-used Kullback–Leibler (KL) divergence as a special case and allows automatic adaptation of the parameter β to the data. An extensive set of experiments comparing the performance of SGP and its β-divergence variant (β-SGP) is carried out on extracted star stamps and on images containing multiple stars (both crowded and relatively sparser fields). We show a consistent enhancement in the flux conservation across all considered scenarios using β-SGP compared to SGP. Using a few quantifiable metrics such as the Full-Width-at-Half-Maximum (FWHM) and ellipticity of stars, we observe that β-SGP improves restoration quality, compared to the SGP, in many cases and still preserves restoration quality in others. We conclude that generalized versions of image restoration algorithms are more robust due to their enhanced flexibility and could be a promising modification for astronomical image restoration.

Wet compaction to a blue nugget: a critical phase in galaxy evolution

ABSTRACT We utilize high-resolution cosmological simulations to reveal that high-redshift galaxies tend to undergo a robust ‘wet compaction’ event when near a ‘golden’ stellar mass of $\sim \!\!10^{10}\, \rm M_\odot$ . This is a gaseous shrinkage to a compact star-forming phase, a ‘blue nugget’ (BN), followed by central quenching of star formation to a compact passive stellar bulge, a ‘red nugget’ (RN), and a buildup of an extended gaseous disc and ring. Such nuggets are observed at cosmic noon and seed today’s early-type galaxies. The compaction is triggered by a drastic loss of angular momentum due to, e.g. wet mergers, counter-rotating cold streams, or violent disc instability. The BN phase marks drastic transitions in the galaxy structural, compositional, and kinematic properties. The transitions are from star forming to quenched inside-out, from diffuse to compact with an extended disc or ring and a stellar envelope, from dark matter to baryon central dominance, from prolate to oblate stellar shape, from pressure to rotation support, from low to high metallicity, and from supernova to AGN feedback. The central black hole growth, first suppressed by supernova feedback when below the golden mass, is boosted by the compaction, and the black hole keeps growing once the halo is massive enough to lock in the supernova ejecta.

Merger Effects on the Spin and Shape Alignments of Galaxy Stellar, Cold Gas, Hot Gas, and Dark Matter Components

We present a numerical evidence supporting the scenario that the peculiar alignments of the galaxy stellar spins with the major principal axes of the local tidal tensors are produced during the quiescent evolution period when the galaxies experience no recent merger events. Analyzing the merger tree from the TNG300-1 simulation of the IllustrisTNG project, we find the latest merger epochs, a(z m ), of the galaxies, and create four a(z m )-selected samples that are controlled to share the identical mass and density distributions. For each sample, we determine the spin and shape vectors of the galaxy stellar, cold and hot gas, and dark matter components separately, and compute the average strengths of their alignments with the principal directions of the local tidal fields as well as their mutual alignment tendencies. It is found that the stellar (cold gas) spin axes of the galaxies whose latest merger events occur at earlier epochs are more strongly aligned (weakly antialigned) with the major principal axes of the tidal fields. It is also shown that, although the mass-dependent transition of the galaxy DM spins have little connection with the merger events, the morphologies, spin–shape, and shape-shear alignment strengths of the four components of the galaxies sensitively depend on a(z m ). Noting that the stellar components of the galaxies that undergo long quiescent evolution have distinctively oblate shapes and very strong spin–shape alignments, we suggest that the local tidal field might be traced by using the stellar shapes of galaxies without signatures of mergers as a proxy of their stellar spins.

A Creative Statistical Model of Geometric Area Index Number for Adequate Estimation of ESG, DESG Goals Achievement, and Other Macroeconomic (Im)balances Dynamics

This paper can be considered as a brief answer to an old research question. Why do researchers need new statistical instruments based on geometric science? In the introduction, one can find some detailed significances, questions, premises, and hypotheses of the article’s research redistributed in the logic of the section, as only a reminder can do it. There are no mandatory access conditions equivalent to “Let no one ignorant of geometry enter here” from the portal of Plato’ s Academy during the literature review of index numbers (IN) or the index numbers method (INM). The originality of a geometric area index number (GAIN) and the simple description of a statistical instrument are the main characteristics of the next section, which is dedicated to its methodology and data. Therefore, this original statistical model of a classical index number is placed in a continuous changing of geometric coordinates and surfaces, caused by alternate states of disequilibrium and equilibrium in macroeconomic life or existence. Whenever the statistical index method has been considered a dead or obsolete solution, this false conclusion has been because it was conceived without its innovative history and multidimensional mathematical perspective. The complex structure of the article also required a section dedicated to practical approaches through macroeconomic indices systems. In the context of dynamic imbalances, this aspect limits the selective options of the new concept of a statistical model of a geometric area index to several inscribable but also regular polygons (“n-gons”) at the same time. These polygons take on stellar shapes as soon as they capture real dynamics by ensuring a significant degree of coverage of simultaneous evolutions. In the pragmatic section of results and discussions, the new statistical model of the geometric area index is built essentially not only as a new type of statistical-mathematical approach to complex and associated or correlated aggregative variations but also from the idea of practically replacing concrete variants with their functions. The final remarks underline some instrumental specificities concerning both constructive advantages and disadvantages but especially to the limitations and perspectives, which are closely related to some symmetries and asymmetries characteristic of the investigated phenomena.

Optimal shape of stellarators for magnetic confinement fusion

We are interested in the design of stellarators, devices for the production of controlled nuclear fusion reactions alternative to tokamaks. The confinement of the plasma is entirely achieved by a helical magnetic field created by the complex arrangement of coils fed by high currents around a toroidal domain. Such coils describe a surface called “coil winding surface” (CWS). In this paper, we model the design of the CWS as a shape optimization problem, so that the cost functional reflects both optimal plasma confinement properties, through a least square discrepancy, and also manufacturability, thanks to geometrical terms involving the lateral surface or the curvature of the CWS.We completely analyze the resulting problem: on the one hand, we establish the existence of an optimal shape, prove the shape differentiability of the criterion, and provide the expression of the differential in a workable form. On the other hand, we propose a numerical method and perform simulations of optimal stellarator shapes. We discuss the efficiency of our approach with respect to the literature in this area.

EMPRESS. VII. Ionizing Spectrum Shapes of Extremely Metal-poor Galaxies: Uncovering the Origins of Strong He ii and the Impact on Cosmic Reionization

Strong high-ionization lines such as He ii of young galaxies are puzzling at high and low redshift. Although recent studies suggest the existence of nonthermal sources, whether their ionizing spectra can consistently explain multiple major emission lines remains a question. Here we derive the general shapes of the ionizing spectra for three local extremely metal-poor galaxies (EMPGs) that show strong He ii λ4686. We parameterize the ionizing spectra composed of a blackbody and power-law radiation mimicking various stellar and nonthermal sources. We use photoionization models for nebulae and determine seven parameters of the ionizing spectra and nebulae by Markov Chain Monte Carlo methods, carefully avoiding systematics of abundance ratios. We obtain the general shapes of ionizing spectra explaining ∼10 major emission lines within observational errors with smooth connections from observed X-ray and optical continua. We find that an ionizing spectrum of one EMPG has a blackbody-dominated shape, while the others have convex downward shapes at >13.6 eV, which indicate a diversity of the ionizing spectrum shapes. We confirm that the convex downward shapes are fundamentally different from ordinary stellar spectrum shapes, and that the spectrum shapes of these galaxies are generally explained by the combination of the stellar and ultraluminous X-ray sources. Comparisons with stellar synthesis models suggest that the diversity of the spectrum shapes arises from differences in the stellar age. If galaxies at z ≳ 6 are similar to the EMPGs, high-energy (>54.4 eV) photons of the nonstellar sources negligibly contribute to cosmic reionization due to relatively weak radiation.

A Method to Characterize the Wide-angle Point-Spread Function of Astronomical Images

Abstract Uncertainty in the wide-angle point-spread function (PSF) at large angles (tens of arcseconds and beyond) is one of the dominant sources of error in a number of important quantities in observational astronomy. Examples include the stellar mass and shape of galactic halos and the maximum extent of starlight in the disks of nearby galaxies. However, modeling the wide-angle PSF has long been a challenge in astronomical imaging. In this paper, we present a self-consistent method to model the wide-angle PSF in images. Scattered light from multiple bright stars is fitted simultaneously with a background model to characterize the extended wing of the PSF using a Bayesian framework operating on a pixel-by-pixel level. The method is demonstrated using our software elderflower and is applied to data from the Dragonfly Telephoto Array to model its PSF out to 20′–25′. We compare the wide-angle PSF of Dragonfly to that of a number of other telescopes, including the SDSS PSF and show that, on scales of arcminutes, the scattered light in the Dragonfly PSF is markedly lower than that of other wide-field imaging telescopes. The energy in the wings of the Dragonfly PSF is sufficiently low that optical cleanliness plays an important role in defining the PSF. This component of the PSF can be modeled accurately, highlighting the power of our self-contained approach.

Direct construction of optimized stellarator shapes. Part 3. Omnigenity near the magnetic axis – ERRATUM

Direct construction of optimized stellarator shapes. Part 3. Omnigenity near the magnetic axis – ERRATUM

The Rossiter–McLaughlin effect revolutions: an ultra-short period planet and a warm mini-Neptune on perpendicular orbits

Comparisons of the alignment of exoplanets with a common host star and each other can be used to distinguish among concurrent evolution scenarios for the star and the planets. However, multi-planet systems usually host mini-Neptunes and super-Earths, whose sizes make orbital architecture measurements challenging. We introduce the Rossiter-McLaughlin effect Revolutions (RMR) technique, which can access the spin-orbit angle of small exoplanets by exploiting the full extent of information contained in spectral transit time series. We validated the technique through its application to published HARPS-N data of the mini-Neptune HD 3167c (P = 29.8 days), refining its high sky-projected spin-orbit angle (−108.9−5.5+5.4°), and we applied it to new ESPRESSO observations of the super-Earth HD 3167 b (P = 0.96 days), revealing an aligned orbit (−6.6−7.9+6.6°). Surprisingly different variations in the contrast of the stellar lines occulted by the two planets can be reconciled by assuming a latitudinal dependence of the stellar line shape. In this scenario, a joint fit to both datasets constrains the inclination of the star (111.6−3.3+3.1°) and the 3D spin-orbit angles of HD 3167b (29.5−9.4+7.2°) and HD 3167c (107.7−4.9+5.1°). The projected spin-orbit angles do not depend on the model for the line contrast variations, and so, with a mutual inclination of 102.3−8.0+7.4°, we can conclude that the two planets are on perpendicular orbits. This could be explained by HD 3167b being strongly coupled to the star and retaining its primordial alignment, whereas HD 3167c would have been brought to a nearly polar orbit via secular gravitational interactions with an outer companion. Follow-up observations of the system and simulations of its dynamical evolution are required to search for this companion and explore the likelihood of this scenario. HD 3167 b (R = 1.7 REarth) is the smallest exoplanet with a confirmed spectroscopic Rossiter-McLaughlin signal. The RMR technique opens the way to determining the orbital architectures of the super-Earth and Earth-sized planet populations.

An analytic treatment of magnetically deformed neutron stars

AbstractIn this study, we develop an analytic treatment, which would allow us to model magnetic deformation of neutron stars in the quasi‐Newtonian framework. We assume non‐magnetic, spherical relativistic stars as backgrounds and regard the magnetic fields as the perturbation. We analyze the problem by taking density profiles from various analytic approaches. The first density profile is the n = 1 solution of the Lane‐Emden equation, the second one is a parabolic ansatz for density and the last density distribution is a fourth‐order polynomial with a free parameter fit to a relativistic solution obtained from a microphysical equation of state (EoS). In this article, the EoS of neutron matter is obtained from the lowest order constraint variational (LOCV) method. Furthermore, both purely poloidal and purely toroidal magnetic field configurations are obtained by solving the Euler's magnetohydrodynamic equations. We almost incorporate all the magnetic field configurations assessed from different density profiles. In order to study the structure of the neutron stars, the magnetic pressure, and density should be treated as perturbations of non‐magnetized stars. We find out that the stellar shape is prolate for positive distortion (purely toroidal magnetic field) and oblate for negative distortion (purely poloidal magnetic field). The magnetic effect on the gravitational mass is also explained.

Magnetic well and Mercier stability of stellarators near the magnetic axis

We have recently demonstrated that by expanding in small distance from the magnetic axis compared with the major radius, stellarator shapes with low neoclassical transport can be generated efficiently. To extend the utility of this new design approach, here we evaluate measures of magnetohydrodynamic interchange stability within the same expansion. In particular, we evaluate the magnetic well, Mercier's criterion, and resistive interchange stability near a magnetic axis of arbitrary shape. In contrast to previous work on interchange stability near the magnetic axis, which used an expansion of the flux coordinates, here we use the ‘inverse expansion’ in which the flux coordinates are the independent variables. Reduced expressions are presented for the magnetic well and stability criterion in the case of quasisymmetry. The analytic results are shown to agree with calculations from the VMEC equilibrium code. Finally, we show that near the axis, Glasser, Greene and Johnson's stability criterion for resistive modes approximately coincides with Mercier's ideal condition.

The stellar halos of ETGs in the IllustrisTNG simulations: The photometric and kinematic diversity of galaxies at large radii

Context. Early-type galaxies (ETGs) are found to follow a wide variety of merger and accretion histories in cosmological simulations. Aims. We characterize the photometric and kinematic properties of simulated ETG stellar halos, and compare them to the observations. Methods. We selected a sample of 1114 ETGs in the TNG100 simulation and 80 in the higher-resolution TNG50. These ETGs span a stellar mass range of 1010.3 − 1012 M⊙ and they were selected within the range of g − r colour and λ-ellipticity diagram populated by observed ETGs. We determined photometric parameters, intrinsic shapes, and kinematic observables in their extended stellar halos. We compared the results with central IFU kinematics and ePN.S planetary nebula velocity fields at large radii, studying the variation in kinematics from center to halo, and connecting it to a change in the intrinsic shape of the galaxies. Results. We find that the simulated galaxy sample reproduces the diversity of kinematic properties observed in ETG halos. Simulated fast rotators (FRs) divide almost evenly in one third having flat λ profiles and high halo rotational support, a third with gently decreasing profiles, and another third with low halo rotation. However, the peak of rotation occurs at larger R than in observed ETG samples. Slow rotators (SRs) tend to have increased rotation in the outskirts, with half of them exceeding λ = 0.2. For M* > 1011.5 M⊙ halo rotation is unimportant. A similar variety of properties is found for the stellar halo intrinsic shapes. Rotational support and shape are deeply related: the kinematic transition to lower rotational support is accompanied by a change towards rounder intrinsic shape. Triaxiality in the halos of FRs increases outwards and with stellar mass. Simulated SRs have relatively constant triaxiality profiles. Conclusions. Simulated stellar halos show a large variety of structural properties, with quantitative but no clear qualitative differences between FRs and SRs. At the same stellar mass, stellar halo properties show a more gradual transition and significant overlap between the two families, despite the clear bimodality in the central regions. This is in agreement with observations of extended photometry and kinematics.

Understanding the Fundamental Plane and the Tully Fisher Relation

The relation between early type galaxy size, surface brightness and velocity dispersion, "the fundamental plane", has long been understood as resulting from equilibrium in their largely pressure supported stellar dynamics. The dissipation and feedback involved in reaching such an equilibrium through merger formation of these galaxies over cosmic time can be responsible for the orientation of the plane. We see a correlation between surface brightness enhancement and youth in the 6dF Galaxy Survey. Correlations of this `tilt' with stellar mass, age, concentration, shape and metallicity now point the direction for further work on the resolved kinematics and structure of these nearby galaxies and on their initial mass function and dark matter component. On the face of it, the Tully Fisher relation is a simpler one dimensional scaling relation. However, as late type galaxies have bulges as well as disks, and, as the surface density of disks is only standard for the more massive galaxies, additional parameters are involved.

ODUSSEAS: a machine learning tool to derive effective temperature and metallicity for M dwarf stars

Aims. The derivation of spectroscopic parameters for M dwarf stars is very important in the fields of stellar and exoplanet characterization. The goal of this work is the creation of an automatic computational tool able to quickly and reliably derive the Teff and [Fe/H] of M dwarfs using optical spectra obtained by different spectrographs with different resolutions. Methods. ODUSSEAS (Observing Dwarfs Using Stellar Spectroscopic Energy-Absorption Shapes) is based on the measurement of the pseudo equivalent widths for more than 4000 stellar absorption lines and on the use of the machine learning Python package “scikit-learn” for predicting the stellar parameters. Results. We show that our tool is able to derive parameters accurately and with high precision, having precision errors of ~30 K for Teff and ~0.04 dex for [Fe/H]. The results are consistent for spectra with resolutions of between 48 000 and 115 000 and a signal-to-noise ratio above 20.

Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis

A direct construction of equilibrium magnetic fields with toroidal topology at arbitrary order in the distance from the magnetic axis is carried out, yielding an analytical framework able to explore the landscape of possible magnetic flux surfaces in the vicinity of the axis. This framework can provide meaningful analytical insight into the character of high-aspect-ratio stellarator shapes, such as the dependence of the rotational transform and the plasma beta limit on geometrical properties of the resulting flux surfaces. The approach developed here is based on an asymptotic expansion on the inverse aspect ratio of the ideal magnetohydrodynamics equation. The analysis is simplified by using an orthogonal coordinate system relative to the Frenet–Serret frame at the magnetic axis. The magnetic field vector, the toroidal magnetic flux, the current density, the field line label and the rotational transform are derived at arbitrary order in the expansion parameter. Moreover, a comparison with a near-axis expansion formalism employing an inverse coordinate method based on Boozer coordinates (the so-called Garren–Boozer construction) is made, where both methods are shown to agree at lowest order. Finally, as a practical example, a numerical solution using a W7-X equilibrium is presented, and a comparison between the lowest-order solution and the W7-X magnetic field is performed.

Direct construction of optimized stellarator shapes. Part 3. Omnigenity near the magnetic axis

The condition of omnigenity is investigated, and applied to the near-axis expansion of Garren & Boozer (Phys. Fluids B, vol. 3 (10), 1991a, pp. 2805–2821). Due in part to the particular analyticity requirements of the near-axis expansion, we find that, excluding quasi-symmetric solutions, only one type of omnigenity, namely quasi-isodynamicity, can be satisfied at first order in the distance from the magnetic axis. Our construction provides a parameterization of the space of such solutions, and the cylindrical reformulation and numerical method of Landreman & Sengupta (J. Plasma Phys., vol. 84 (6), 2018, 905840616); Landreman et al. (J. Plasma Phys., vol. 85 (1), 2019, 905850103), enables their efficient numerical construction.