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Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: The radial profiles of the Gauss–Hermite representations (up to h 4) with 1σ errors as derived from the spectroscopic data (gray) and the corresponding fit of the best dynamical orbit model we found for each galaxy (colored). The 1σ errors are quantified by fitting Gauss–Hermites to 100 Monte Carlo realizations on the velocity-binned LOSVDs. The models fit the observed kinematics quite well considering deviations from axisymmetry in the data cannot be emulated by axisymmetric models.
Paper: ÆSOPUS 2.1: Low-temperature Opacities Extended to High Pressure
Authors: Paola Marigo, Francesco Addari, Diego Bossini, Alessandro Bressan, Guglielmo Costa, Léo Girardi, Giada Pastorelli, Michele Trabucchi, Guglielmo Volpato
Figure caption: The impact of our new Rosseland mean opacities on low-mass stellar models computed with the PARSEC v2.0 code. The left panel shows the H-R diagram for models in the mass range from 0.1 to 0.85 M ⊙ and computed at intervals of 0.05 M ⊙ (from bottom right to top left) and plotted at ages of 5 Gyr. The blue symbols indicate models computed with the previous version of the opacity tables from P. Marigo et al. (2022), while the red and green symbols are for models computed with present opacity tables. The green symbols correspond to models with a shift in their T − τ relation (see text for details). The right panel shows the same models in the mass–radius plane, now compared with the empirical data of low-mass stars in double-lined eclipsing binary catalogs from D. Ségransan et al. (2003), B. O. Demory et al. (2009), G. Torres et al. (2010), J. A. Carter et al. (2011), L. R. Doyle et al. (2011), A. L. Kraus et al. (2011), S. G. Parsons et al. (2012a), S. G. Parsons et al. (2012b), J. Southworth (2015).
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: The kinematic maps of the LOSVDs approximated by Gauss–Hermite expansion with fixed n GH = 4; i.e., this does not show potentially higher-order deviations. In a few bins, the recovered LOSVDs are essentially Gaussian because the optimum n GH was 2. This makes them appear white in the h 3,4-maps (the northernmost bin of VCC 200 was excluded due to its poor spectrum). We use a common spatial and color scale (see color bars on the right sides of the panels) to enable an easy comparison between different sample galaxies. Spatial variations within a given galaxy are more discernible in the radial profiles (Figure 4). In the velocity panels, we indicate north and east as well as the photometric 1.0r e aperture (dashed ellipse).
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: Correlations of the SSP parameters for the dE sample. Shown are the results for the spectra in the two apertures centered around r = 2.″5 (left) and r = 7.″5 (right). Top: age vs. metallicity. Bottom: [Mg/Fe] vs. metallicity.
Paper: ÆSOPUS 2.1: Low-temperature Opacities Extended to High Pressure
Authors: Paola Marigo, Francesco Addari, Diego Bossini, Alessandro Bressan, Guglielmo Costa, Léo Girardi, Giada Pastorelli, Michele Trabucchi, Guglielmo Volpato
Figure caption: The relative difference between the observed and model radii for the three sets of models and the data presented in the right panel of Figure 16 as a function of stellar mass. It can be seen that while our new opacities reduce the systematic discrepancies between the predicted and observed radii for stars in the M < 0.6 M ⊙ range, a shift in the T − τ relation (or other alternatives as discussed in the text) seems to still be necessary to eliminate it completely.
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: An imaging preview of the dE sample investigated in this study. Shown is the g, r, i image for each galaxy based on publicly available SDSS data. We obtained the images with the help of Aladin (T. Boch & P. Fernique 2014).
Paper: ÆSOPUS 2.1: Low-temperature Opacities Extended to High Pressure
Authors: Paola Marigo, Francesco Addari, Diego Bossini, Alessandro Bressan, Guglielmo Costa, Léo Girardi, Giada Pastorelli, Michele Trabucchi, Guglielmo Volpato
Figure caption: Rosseland mean opacities with the inclusion of condensed dust grains. The chemical composition is the same as in Figure 14. The curves are distributed every 1 dex in ﹩\mathrm{log}(R)﹩, in the interval ﹩-8\leqslant \mathrm{log}(R)\leqslant 6﹩.
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: Left panel: the relative differences Δ* of stellar mass-to-light ratios derived from single stellar population analysis ϒKroupa and dynamical modeling as a function of age of the SSP. Dots with error bars: the differences of the two mass-to-light ratios we derived from the VIRUS-W spectra binned at r = 2.″5 and r = 7.″5 (Section 2.3). Gray dots: the differences when ϒKroupa values are based on literature stellar population results (see Appendix B). The dotted, horizontal line indicates identical population and dynamical results. Assuming the dynamical results are correct, galaxies that lie above this horizontal line would require a super-Kroupa IMF, while galaxies below it would require a sub-Kroupa IMF. Right panel: similar to the left panel, but instead of comparing ϒKroupa with the decomposed stellar component ϒdyn, the population ratio is compared to the total dynamical mass-to-light ratio M tot/L; i.e., the dark matter component is included in the mass. Using VCC 2048 as an example, the diagonal arrow symbolizes how an (unrealistically) large error in the SSP age would shift Δ* and Δtot. We measure VCC 2048's population to be young, but if it were actually twice as old (∼8 Gyr), ϒKroupa would be higher, and the galaxy should actually be at the arrow’s head if we measured its age correctly.
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: Comparison of the local mass-to-light ratio ϒ as a function of radius r[″] for all Virgo dEs as obtained from different models. Each panel shows one dE, with the box showing its VCC-id and the band in which the mass-to-light ratio is measured (see Figure 2). The colored lines show the mass-to-light ratios we derived using dynamical modeling. Dashed lines: the projected ϒ* of the dynamically decomposed stellar component. Solid lines: the total 3D dynamical mass-to-light ratio ϒtot (i.e., including the dark matter). Note that ϒtot is not a projected quantity, as this would include all of the dark matter along the LOS, i.e., even the poorly constrained DM far outside a galaxy’s FOV. This can make the projected ϒ* appear to be higher than the 3D ϒtot values at some radii, even though the latter is by definition the upper bound for the 3D stellar mass-to-light ratios. The gray, dotted horizontal line shows the mass-to-light ratio of the best mass-follows-light model (i.e., a dynamical orbit model with no dark matter and stellar ϒ-gradient). The black dots with error bars show the stellar mass-to-light ratio from the population analysis (Section 2.3) in the two annuli centered around r = 2.″5 and 7.″5. We also draw a connecting line between the two points to highlight a population gradient if it exists.
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: Left: the dynamically recovered stellar mass-to-light ratios ϒ*(r) evaluated at radii r = 2.″5 and r = 7.″5 vs. SSP age in the corresponding annulli. Right: the same but for the total mass-to-light ratio (including dark matter). For this comparison, we calibrated all mass-to-light ratios to the z band. The mass-to-light ratios appear to be anticorrelated with age. If the IMF is assumed to be universal across formation epochs, then this trend is counterintuitive, because one would expect that mass-to-light ratio is positively correlated with the time passed since star formation. The dotted line indicates the expected present-day z-band mass-to-light ratio for a single stellar population with a universal Kroupa IMF and a metallicity of [Z/H] = −0.3 (C. Maraston 2005) as a function of its age.
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: The average cylindrical anisotropies β  z  (left) and γ (right) vs. the intrinsic, average stellar ellipticity ﹩{\epsilon }_{\mathrm{intr}}﹩. The anisotropies were averaged within the sphere of radius r ≤ 0.8 kpc (∼10″) because values outside the FOV are poorly constrained by the data. Squares: the massive COMA ETGs of J. Thomas et al. (2009a). Stars: the ETGs from M. Cappellari et al. (2007). The dashed diagonal line indicates the empirical relation ﹩{\beta }_{z}=0.7\cdot {\epsilon }_{\mathrm{intr}}﹩ from M. Cappellari et al. (2007), where flatter galaxies have higher anisotropy β  z .
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: Comparison of an optimized LOSVD model parameterized by a Gauss–Hermite series (left) and a nonparametric description (right) for the same Voronoi Bin of VCC 200 (S/N = 26). Both are broadly consistent with one another. In the case of this specific Voronoi Bin, the Gauss–Hermite representation is slightly preferred by AIC p . The error suppression by a Gauss–Hermite parameterization at high velocities is substantial, which is why we readjusted the errors as described in Section 2.2.
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: A comparison of the IMF parameter (left) and the single stellar population age (right) of dEs with those of the “ordinary” ETGs from the ATLAS3D-survey (M. Cappellari et al. 2011). All galaxies are shown vs. their central velocity dispersion (measured within r e/8), which is essentially a proxy for their total mass. ATLAS3D ETGs that reside in Virgo are highlighted using squares, while the others are shown as circles. We calculated the Δ* for a Kroupa IMF by transforming the mass-to-light ratios from M. Cappellari et al. (2013a), which were given for a Salpeter IMF. The SSP equivalent age of the ATLAS3D galaxies was obtained from R. M. McDermid et al. (2015) and is stated within a 1r e aperture. For our dEs, we show the results for both apertures that we analyzed. M. Cappellari et al. (2013a) and R. M. McDermid et al. (2015) flag their galaxies depending on the quality of the data and analysis. For details on the definition of these flags, see the corresponding studies. We show galaxies flagged with the best quality in black and “bad” galaxies that may not be as reliable in red.
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: Top:  example of a typical Voronoi-binned spectrum (one of the bins of VCC 200 within the central 5″). Black: the observed VIRUS-W spectrum. The green band around the data indicates the 1σ  flux uncertainty. Red: the corresponding fit model. Orange:  the continuum modeled by a second-order multiplicative polynomial. The gray regions indicate the parts that were masked before the fit. Bottom panel: the residuals between fit and data.
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: The Hβ index vs. the SSP age derived from the two spectra at r = 2.″5 (left) and r = 2.″5 (left). The measurement of the strong age-indicator Hβ appears to be a strong predictor of the age, which suggests the age we derived is a direct result of the measured spectra and not a random modeling result.
Paper: The VIRUS-dE Survey. I. Stars in Dwarf Elliptical Galaxies—3D Dynamics and Radially Resolved Stellar Initial Mass Functions
Authors: Mathias Lipka, Jens Thomas, Roberto Saglia, Ralf Bender, Maximilian Fabricius, Gary J. Hill, Matthias Kluge, Martin Landriau, Ximena Mazzalay, Eva Noyola, Taniya Parikh, Jan Snigula
Figure caption: The abundance ratio [Mg/Fe] (a proxy for extended star formation) vs. the single-population age. Especially in their center (r = 2.″5), dEs that are young also have lower solar-like [Mg/Fe], which suggests they had a more extended star formation history (SFH) than older ones that (presumably) were quenched earlier.
Paper: The ALMA-CRISTAL Survey: Spatially Resolved Star Formation Activity and Dust Content in 4 < z < 6 Star-forming Galaxies
Authors: Juno Li, Elisabete Da Cunha, Jorge González-López, Manuel Aravena, Ilse De Looze, N. M. Förster Schreiber, Rodrigo Herrera-Camus, Justin Spilker, Ken-ichi Tadaki, Loreto Barcos-Munoz, Andrew J. Battisti, Jack E. Birkin, Rebecca A. A. Bowler, Rebecca Davies, Tanio Díaz-Santos, Andrea Ferrara, Deanne B. Fisher, Jacqueline Hodge, Ryota Ikeda, Meghana Killi, Lilian Lee, Daizhong Liu, Dieter Lutz, Ikki Mitsuhashi, Thorsten Naab, Ana Posses, Monica Relaño, Manuel Solimano, Hannah Übler, Stefan Anthony van der Giessen, Vicente Villanueva
Figure caption: The spatially rSFMS of our CRISTAL galaxies. Each panel (except the last) shows the surface density of SFR (ΣSFR) plotted against the surface density of stellar mass (﹩{{\rm{\Sigma }}}_{{M}_{* }}﹩) for an individual galaxy (ID on the top-left corner). The black crosses correspond to our spatial bins (∼0.5–1 kpc). For each spatial bin, the values are derived from the median-likelihood MAGPHYS estimates using the full UV-to-IR SED fitting, and the error bars are based on the 16th–84th percentiles of the posterior likelihood distributions. The red lines show the best-fit power laws; for each one, we indicate the power-law slope α in the bottom-right corner. The yellow hexagons correspond to using a hexagonal tessellation with spatial bin sizes comparable to the ALMA beam, such that adjacent bins are less likely to be correlated due to flux blending. The last panel shows the joint distribution of the whole sample (hexagon bins only), which we use to measure the best-fit power-law slope to the rSFMS at z ≃ 5.
Paper: The ALMA-CRISTAL Survey: Spatially Resolved Star Formation Activity and Dust Content in 4 < z < 6 Star-forming Galaxies
Authors: Juno Li, Elisabete Da Cunha, Jorge González-López, Manuel Aravena, Ilse De Looze, N. M. Förster Schreiber, Rodrigo Herrera-Camus, Justin Spilker, Ken-ichi Tadaki, Loreto Barcos-Munoz, Andrew J. Battisti, Jack E. Birkin, Rebecca A. A. Bowler, Rebecca Davies, Tanio Díaz-Santos, Andrea Ferrara, Deanne B. Fisher, Jacqueline Hodge, Ryota Ikeda, Meghana Killi, Lilian Lee, Daizhong Liu, Dieter Lutz, Ikki Mitsuhashi, Thorsten Naab, Ana Posses, Monica Relaño, Manuel Solimano, Hannah Übler, Stefan Anthony van der Giessen, Vicente Villanueva
Figure caption: Average V-band dust attenuation given by MAGPHYS, A  V , plotted against the empirically derived fraction of obscured star formation, f obs, for all the spatial bins in our sample. The circles show regions where there were enough data to derive both quantities; the triangles show regions with upper limits on f obs (i.e., points with only rest-frame UV but no ALMA detection). All regions are colored according to their specific SFR derived by MAGPHYS. The orange and blue solid curves show the analytical relation predicted for pure diffuse ISM and birth cloud attenuation, respectively. The two black error bars show the median errors for  f obs above and below 0.5, respectively.
Paper: The ALMA-CRISTAL Survey: Spatially Resolved Star Formation Activity and Dust Content in 4 < z < 6 Star-forming Galaxies
Authors: Juno Li, Elisabete Da Cunha, Jorge González-López, Manuel Aravena, Ilse De Looze, N. M. Förster Schreiber, Rodrigo Herrera-Camus, Justin Spilker, Ken-ichi Tadaki, Loreto Barcos-Munoz, Andrew J. Battisti, Jack E. Birkin, Rebecca A. A. Bowler, Rebecca Davies, Tanio Díaz-Santos, Andrea Ferrara, Deanne B. Fisher, Jacqueline Hodge, Ryota Ikeda, Meghana Killi, Lilian Lee, Daizhong Liu, Dieter Lutz, Ikki Mitsuhashi, Thorsten Naab, Ana Posses, Monica Relaño, Manuel Solimano, Hannah Übler, Stefan Anthony van der Giessen, Vicente Villanueva
Figure caption: Comparison between the native resolution images of one of our galaxies, CRISTAL-02 (left-hand panels), and the corresponding convolved images resulting from our PSF homogenization procedure (right-hand panels). The top panels show the images in filter HST/WFC3 F160W, and the bottom panels show the images in filter JWST/NIRCam F444W. In the bottom-right panel, we overlay the ALMA Band 7 continuum emission as red contours (showing steps of 2σ, 3σ, 4σ, 6σ, and 8σ; the synthesized beam FWHM size is shown in the bottom-right corner). The blue circle has a diameter of 3″ and indicates the approximate size of the total aperture for this source. The grid of green square boxes of 0.″12 on each side indicates the grid of spatial bins where fluxes are extracted to model the SED.
Paper: The ALMA-CRISTAL Survey: Spatially Resolved Star Formation Activity and Dust Content in 4 < z < 6 Star-forming Galaxies
Authors: Juno Li, Elisabete Da Cunha, Jorge González-López, Manuel Aravena, Ilse De Looze, N. M. Förster Schreiber, Rodrigo Herrera-Camus, Justin Spilker, Ken-ichi Tadaki, Loreto Barcos-Munoz, Andrew J. Battisti, Jack E. Birkin, Rebecca A. A. Bowler, Rebecca Davies, Tanio Díaz-Santos, Andrea Ferrara, Deanne B. Fisher, Jacqueline Hodge, Ryota Ikeda, Meghana Killi, Lilian Lee, Daizhong Liu, Dieter Lutz, Ikki Mitsuhashi, Thorsten Naab, Ana Posses, Monica Relaño, Manuel Solimano, Hannah Übler, Stefan Anthony van der Giessen, Vicente Villanueva
Figure caption: Fraction of dust-obscured SFR, f obs, plotted against the surface density of stellar mass, ﹩{{\rm{\Sigma }}}_{{M}_{* }}﹩, for each ∼0.5–1 kpc spatial bin in our galaxies. The vertical errors are given by propagating the observational uncertainties in Equation (2); the horizontal errors are given by MAGPHYS. Each point is color-coded to reflect its distance to the brightest pixel in NIRCam F444W (which we use as an empirical proxy for the center of stellar mass).
Paper: Anisotropic Magnetized Asteroseismic Waves
Authors: B. Tripathi, Dhrubaditya Mitra
Figure caption: Theory and numerics are compared. Our theory predicts that ﹩g(\theta )={\rm{\Delta }}{{\rm{\Omega }}}^{2}/({{\boldsymbol{\epsilon }}}^{2}{{\rm{\Omega }}}_{\mathrm{hydro}}^{2})+\gamma /2﹩ depends only on θ as given in Equation (43). The function g(θ) is independent of the wavenumber K, Alfvénic Mach number  ϵ  −1, the polytropic index n, the adiabatic index γ, and the eigenstate index m. Here, we plot g(θ) obtained from our numerical solutions for several different values of m, ranging from 10 to 35, and for several different values of K (0.6 ≤ ∣K∣ ≤ 2), and for two values of  ϵ  and two values of γ (different symbols). All data points collapse on the same universal curve. The function g(θ) from our asymptotic theory, Equation (43), is also plotted here. The asymptotic curve is indistinguishable from the exact numerical solutions.
Paper: Anisotropic Magnetized Asteroseismic Waves
Authors: B. Tripathi, Dhrubaditya Mitra
Figure caption: An inhomogeneous magnetic field  B  0(z), oriented orthogonal to a constant vertical gravity  g , is considered where the wave is allowed to propagate in an arbitrary direction, shown with  K , making an angle θ with  B  0. The gradient in the color map of the box schematically represents increasing functions with depth z of the magnetic field, fluid density, pressure, and sound speed.
Paper: Anisotropic Magnetized Asteroseismic Waves
Authors: B. Tripathi, Dhrubaditya Mitra
Figure caption: Ω2(K  X , K  Y ) for hydrodynamic and magnetized polytropes. The two plotted surfaces are visually indifferentiable because the difference (ΔΩ2) between them is much smaller than Ω2; see Figure 3. The parameters used are  ϵ  = 0.1, m = 20, n = 2.5, and γ = 5/3. For variations in these parameters, the surface plot of Ω2(K  X , K  Y ) remains the same qualitatively.
Paper: Anisotropic Magnetized Asteroseismic Waves
Authors: B. Tripathi, Dhrubaditya Mitra
Figure caption:                    ﹩{\rm{\Delta }}{{\rm{\Omega }}}^{2}={{\rm{\Omega }}}^{2}-{{\rm{\Omega }}}_{\mathrm{hydro}}^{2}﹩ is plotted for a magnetized polytrope, with  ϵ  = 0.1, m = 20, n = 2.5, and γ = 5/3. This surface plot shows the difference between the two surface plots in Figure 2.
Paper: Anisotropic Magnetized Asteroseismic Waves
Authors: B. Tripathi, Dhrubaditya Mitra
Figure caption: Relative difference of the squared frequency depends on the wavevector propagation angle (spanned by the black circle), but the relative difference is insensitive to the wavevector magnitude. The parameters chosen are  ϵ = 0.1, m = 20, n = 2.5, and γ = 5/3.
Paper: Anisotropic Magnetized Asteroseismic Waves
Authors: B. Tripathi, Dhrubaditya Mitra
Figure caption: Comparison of different expressions to assess their individual contribution. The innermost (red) curve is the universal curve on which all numerical data collapse.