Issue 
A&A
Volume 665, September 2022



Article Number  A159  
Number of page(s)  17  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/202244143  
Published online  23 September 2022 
Kinematics and mass distributions for nonspherical deprojected Sérsic density profiles and applications to multicomponent galactic systems
^{1}
MaxPlanckInstitut für extraterrestrische Physik (MPE), Giessenbachstr. 1, 85748 Garching, Germany
email: sedona@mpe.mpg.de
^{2}
Cavendish Laboratory, University of Cambridge, 19 J.J. Thomson Avenue, Cambridge CB3 0HE, UK
^{3}
Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
^{4}
Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands
^{5}
UniversitätsSternwarte LudwigMaximiliansUniversität (USM), Scheinerstr. 1, 81679 München, Germany
^{6}
Departments of Physics and Astronomy, University of California, Berkeley, CA 94720, USA
^{7}
Physics Department, University of Alaska, Fairbanks, AK 99775, USA
^{8}
University of the Western Cape, Bellville, Cape Town 7535, South Africa
Received:
30
May
2022
Accepted:
12
July
2022
Using kinematics to decompose the mass profiles of galaxies, including the dark matter contribution, often requires parameterization of the baryonic mass distribution based on ancillary information. One such model choice is a deprojected Sérsic profile with an assumed intrinsic geometry. The case of flattened, deprojected Sérsic models has previously been applied to flattened bulges in local starforming galaxies (SFGs), but can also be used to describe the thick, turbulent disks in distant SFGs. Here, we extend this previous work that derived density (ρ) and circular velocity (v_{circ}) curves by additionally calculating the sphericallyenclosed 3D mass profiles (M_{sph}). Using these profiles, we compared the projected and 3D mass distributions, quantified the differences between the projected and 3D halfmass radii (R_{e}; r_{1/2, mass, 3D}), and compiled virial coefficients relating v_{circ}(R) and M_{sph}(< r = R) or M_{tot}. We quantified the differences between mass fraction estimators for multicomponent systems, particularly for dark matter fractions (ratio of squared circular velocities versus ratio of spherically enclosed masses), and we considered the compound effects of measuring dark matter fractions at the projected versus 3D halfmass radii. While the fraction estimators produce only minor differences, using different aperture radius definitions can strongly impact the inferred dark matter fraction. As pressure support is important in analyses of gas kinematics (particularly, at high redshifts), we also calculated the selfconsistent pressure support correction profiles, which generally predict less pressure support than for the selfgravitating disk case. These results have implications for comparisons between simulation and observational measurements, as well as for the interpretation of SFG kinematics at high redshifts. We have made a set of precomputed tables and the code to calculate the profiles publicly available.
Key words: galaxies: luminosity function, mass function / galaxies: kinematics and dynamics
© S. H. Price et al. 2022
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This article is published in open access under the SubscribetoOpen model.
This Open access funding provided by Max Planck Society.
1. Introduction
Galaxy kinematics, such as rotation curves, are a powerful tool to measure the mass of all components in a galaxy (e.g., van der Kruit & Allen 1978; Courteau et al. 2014). Notably, this technique has been used to study the dark matter content of galaxies at a wide range of epochs, including constraints on the halo profile shapes (e.g., Sofue & Rubin 2001; de Blok 2010; Genzel et al. 2020, among many others). Furthermore, by using kinematics to probe the mass and angular momentum distribution within galaxies, it is possible to constrain the processes regulating galaxy growth and evolution over time (van der Kruit & Freeman 2011; Förster Schreiber & Wuyts 2020; see also, e.g., Mo et al. 1998; Sofue & Rubin 2001; Romanowsky & Fall 2012). It is especially informative to study the kinematics of starforming galaxies (SFGs), which tend to lie on a tight “starforming main sequence” where much of cosmic star formation occurs (Speagle et al. 2014; Rodighiero et al. 2011; Whitaker et al. 2014; Tomczak et al. 2016). However, there are challenges to recovering the intrinsic mass properties of galaxies from their observed kinematics.
One such challenge is that in order to overcome degeneracies in kinematic mass decomposition (particularly when including an unseen dark component; e.g., van Albada et al. 1985), separate constraints on the baryonic (gas and stellar) component are needed, either through empirical measurements or with a choice of parameterization (e.g., Persic et al. 1996; de Blok & McGaugh 1997; Palunas & Williams 2000; Dutton et al. 2005; de Blok et al. 2008; Courteau et al. 2014). Multiwavelength imaging and spectroscopy (in emission or absorption) can constrain the distribution of gas and stars in galaxies. Such observations of individual galaxies provide projected information and not the 3D quantities needed for kinematic modeling. Consequently, it is often necessary to first parameterize the projected distributions and then make reasonable assumptions about the galaxies’ intrinsic geometries in order to deproject the surface distributions into 3D mass profiles.
Observationally, the light distributions of galaxies are often described by Sérsic (1968) profiles (e.g., Peng et al. 2002, 2010; Simard et al. 2002, 2011; Blanton et al. 2003; Wuyts et al. 2011; van der Wel et al. 2012; Conselice 2014, and numerous others). In some cases, there are distinct components within galaxies, but these are also frequently described by Sérsic profiles with distinct indices, n, and effective radii, R_{e}, (e.g., a disk and bulge for starforming galaxies; Courteau et al. 1996; Bruce et al. 2012; Lang et al. 2014). Thus, Sérsic profiles are a natural choice for the projected parameterization.
Deprojections of Sérsic profiles have been studied in numerous previous works, for spherical (e.g., Ciotti 1991; Ciotti & Lanzoni 1997; Baes & Ciotti 2019a,b), triaxial (e.g., Stark 1977; Trujillo et al. 2002), and axisymmetric geometries (e.g., Noordermeer 2008). Additionally, the dynamics for exponential surface profiles have been derived for both razorthin (Freeman 1970) and finitely thick (Casertano 1983) geometries (although these are generalizable to arbitrary Sérsic index; e.g., see Binney & Tremaine 2008). These intrinsic geometries have applications for various galaxies or galaxy components, depending on the galaxy properties and epoch.
In particular, the mass distribution geometry of SFGs changes over time. Nearby SFGs often have thin disks, particularly in the gas components (van der Kruit & Freeman 2011), while distant (massive) SFGs tend to have thick, turbulent disks (Glazebrook 2013; Förster Schreiber & Wuyts 2020, and references therein). While more observations are needed to better constrain the vertical disk structure of distant, massive SFGs, flattened (oblate) distributions are more appropriate models (as adopted by, e.g., Wuyts et al. 2016; Genzel et al. 2017, 2020), using the same geometric deprojection used by Noordermeer (2008) to describe the flattening of nearby bulges.
A second challenge is that the observed rotation must be corrected for pressure support. This correction is important for gas kinematic measurements, especially at high redshifts where disks have high gas turbulence. A number of works have considered different analytic prescriptions for correcting for the pressure support in gas kinematics (e.g., Weijmans et al. 2008; Burkert et al. 2010; Dalcanton & Stilp 2010; Kretschmer et al. 2021). In general, such corrections require measurements of the gas turbulence σ from spatially resolved spectroscopy (i.e., slit along the major axis or kinematic maps) as well as constraints on or parameterizations of the gas density profile. If deprojected Sérsic distributions are used to model the mass and v_{circ} profiles for the gas and stellar components of galaxies, then a pressure support prescription derived using the density slope can be adopted for a selfconsistent kinematic analysis (as in, e.g., Weijmans et al. 2008; Burkert et al. 2010; Dalcanton & Stilp 2010). If galaxies exhibit nonconstant dispersion, support from dispersion gradients or anisotropy can also be included (e.g., Weijmans et al. 2008; Dalcanton & Stilp 2010).
In order to further consider implications for the interpretation of the kinematics of highredshift SFGs modeled using deprojected, flattened Sérsic profiles, in this paper we revisit and extend the framework first presented by Noordermeer (2008, hereafter N08). We first present various profile derivations for deprojected, flattened Sérsic profiles, including the density and circular velocity profiles determined by N08 as well as the sphericallyenclosed 3D mass profiles (Sect. 2). Using the calculated profiles, we examine the relationship between projected 2D and 3D mass distributions, including differences between the 2D R_{e} and 3D r_{1/2, mass, 3D} (Sect. 2.4). The circular velocity and 3D mass distributions are also used to calculate virial coefficients (Sect. 3). Next, we examine the circular velocity and enclosed mass profiles for multicomponent systems for a range of realistic z ∼ 2 galaxy properties (Sect. 4). We find the composite baryonic 3D halfmass radius r_{1/2, 3D, baryons} is often smaller than the projected disk effective radius R_{e, disk}. While different dark matter fraction estimators (the ratio of the dark matter to total circular velocities squared) and (the ratio of the dark matter to total mass enclosed within a sphere) are similar when calculated at the same radius, large differences in f_{DM} can result from the use of different aperture radii (r_{1/2, 3D, baryons} vs. R_{e, disk}). We then determine the selfconsistent turbulent pressure support correction, assuming a constant σ_{0}, which is typically only half the amount predicted for a selfgravitating disk, and demonstrate the correction for a range of realistic z ∼ 2 galaxy properties (Sect. 5). Finally, we discuss these results and their implications, in particular, for comparisons between simulations and observations and for studies of disk galaxy kinematics at z ∼ 1 − 3 (Sect. 6). We highlight how typical observational and simulation “halfmass” radius estimates can lead to differences of up to ∼0.15 in measured f_{DM}, and how the lower pressure support correction derived for these mass distributions (compared to the selfgravitating disk prescription) would imply a typically lower inferred f_{DM} from the observations.
A set of tables containing precomputed profiles and values – including v_{circ}(R) (Eq. (5)), M_{sph}(< r = R) (Eq. (8)), M_{spheroid}(< m = R) (Eq. (6)), ρ(r = R) (Eq. (2)), dlnρ/dlnR (derived from Eq. (17)), r_{1/2, mass, 3D}, k_{tot}(R_{e, disk}) (Eq. (10)), and k_{3D}(R_{e, disk}) (Eq. (9)) – for a range of intrinsic axis ratios q_{0} and Sérsic indices, and the code used to compute the profiles, are made available^{1}. For reference, key variables and their definitions are listed in Table 1. We assume a flat ΛCDM cosmology with Ω_{m} = 0.3, Ω_{Λ} = 0.7, and H_{0} = 70 km s^{−1} Mpc^{−1}.
Definitions of key variables.
2. Derivation of mass profiles and rotation curves
In this section, we present the formulae for the mass profiles and rotation curves for models whose projected intensity distributions follow Sérsic profiles, but that have oblate (flattened) or prolate axisymmetric 3D density distributions (i.e., the isodensity contours follow oblate or prolate spheroids), following the deprojection derivation of N08.
2.1. Deprojected Sérsic density profile
We assume that the mass density of the 3D spheroid can be written as ρ(x, y, z) = ρ(m), where (m/a)^{2} = (x/b)^{2} + (y/a)^{2} + (z/c)^{2} specifies the isodensity surfaces for a given set of semiaxis lengths a, b, c. For an axisymmetric system, this is simplified to , where is the distance in the plane of axisymmetry, z is the distance from the midplane, the semiaxes a = b, and q_{0} = c/a is the intrinsic axis ratio of the spheroid. To project the intrinsic galaxy coordinates to the observer’s frame, we adopt the transformation from (x, y, z) to (ζ, κ, ξ) from Eq. (1) of N08, where ζ lies along the line of sight, κ, and ξ lie along the galaxy major and minor axes (as viewed in the sky plane, for oblate geometries^{2}; i.e., κ = a), respectively, and i is the inclination of the system relative to the observer (see also Fig. 1 of N08). The observed axis ratio of the ellipsoid is then .
Within the observer’s coordinate frame, the relationship between the 3D mass density profile and the projected light intensity along the major axis of the galaxy is (ξ = 0; from Eq. (8) of N08^{3}):
where Υ(m) is the masstolight ratio of the galaxy and . For simplicity, we assume a constant masstolight ratio, Υ(m)≡Υ.
The deprojected density profile is found by inverting this Abel integral, with (c.f. Eq. (9), N08):
We express the Sérsic profile as (c.f. Eq. (11), N08):
where R_{e} is the effective radius, n is the Sérsic index, I_{e} is the surface brightness at R_{e}, and b_{n} satisfies , where Γ(a) and γ(a, x) are the regular and lower incomplete gamma functions, respectively (e.g., Graham & Driver 2005). The derivative is then:
By inserting Eq. (4) into Eq. (2), we can numerically integrate these equations to obtain the deprojected density profile ρ(m). For the adopted convention here, where the projected κ lies along a in the midplane (so this is the usual projected major axis for oblate cases but is the projected minor axis for prolate cases), we have R_{e} = a as the projected effective radius.
2.2. Rotation curves
Next, we determine the circular rotation curve for this class of density profiles, following the derivation of Binney & Tremaine (2008, Eq. (2.132); also Eq. (10), N08). The circular rotation curve at the midplane of the galaxy is thus:
As noted by N08, this equation is valid for any observed intensity profile I(κ). Here, we combine Eqs. (2) and (5), which can be numerically integrated to yield v_{circ}(R).
2.3. Enclosed 3D mass
We next derive the enclosed mass for models with the density profiles given above. Given the modified coordinate m, the mass enclosed within a spheroid with intrinsic axis ratio, q_{0}, can be expressed as:
Integrated to infinity, this is equivalent to the total luminosity of the Sérsic profile times the constant assumed masstolight ratio, or M_{tot} = ΥL_{tot}, so the intensity normalization for a flattened Sérsic profile with observed axis ratio, q_{obs}, is:
However, there may be situations where we wish to compute the mass enclosed within a sphere of radius instead of within a flattened (or prolate) spheroid. We thus use a change of coordinates to calculate the spherical enclosed mass:
using ρ(m) from Eq. (2), with . This integral can be numerically evaluated to find the 3D spherical enclosed mass profile corresponding to the deprojected, axisymmetric Sérsic profile. We note that when q_{0} ≠ 1, then (with from Eq. (5)); however, the enclosed mass and circular velocity can be related through the introduction of a nonunity, radially varying virial coefficient (see Sect. 3).
Finally, we note that the mass enclosed within an ellipsoidal cylinder of axis ratio q_{obs} (and infinite length) is equivalent to the enclosed luminosity for the 2D projected Sérsic profile times the masstolight ratio, , with x = b_{n}(R/R_{e})^{1/n} (e.g., Graham & Driver 2005).
2.4. Properties of enclosed mass and circular velocity curves for nonspherical deprojected Sérsic profiles
The 3D spherical enclosed mass profiles for models with a range of Sérsic indices (n = 0.5, 1, …, 8) and different intrinsic axis ratios q_{0} = 1, 0.4, 0.2) are shown in Fig. 1. The 3D spherical halfmass radius (where r_{1/2, mass, 3D} satisfies M_{sph}(< r_{1/2, mass, 3D}) = M_{tot}/2) is r_{1/2, mass, 3D} ∼ 1.3R_{e} when q_{0} = 1 (as shown by Ciotti 1991). However, from the q_{0} = 0.2, 0.4 enclosed mass profiles, we see that the ratio r_{1/2, mass, 3D}/R_{e} varies with the model intrinsic axis ratio.
Fig. 1. Fractional mass enclosed within a sphere of radius r = R for deprojected Sérsic models of different intrinsic axis ratios. From left to right: enclosed M_{sph} is plotted as a function of log radius (relative to the projected 2D effective radius, R_{e}), assuming intrinsic axis ratios of q_{0} = 1, 0.4, 0.2, respectively. The colored curves denote the enclosed mass profiles for Sérsic indices from n = 0.5 to n = 8 (yellow to purple). The vertical lines denote R = R_{e} (grey dashed) and R = 1.3R_{e} (≈r_{1/2, mass, 3D} for q_{0} = 1; grey dashdotted), and the horizontal colored lines denote the fraction of the mass enclosed within r = R_{e} for n = 1, 4 (lime, teal dashed, respectively) and 50% of the total mass (grey dashed dotted). For q_{0} = 1, the halfmass 3D spherical radius is indeed r_{1/2, mass, 3D} ≈ 1.3R_{e} regardless of n, as in Ciotti (1991). For flattened (i.e., oblate) systems, the halfmass 3D spherical radius is smaller, and approaches R_{e} as q_{0} decreases. See also Fig. 2. 
We quantify the dependence of the ratio between the 3D spherical halfmass radius and the projected effective radius, r_{1/2, mass, 3D}/R_{e}, in Fig. 2, as a function of Sérsic index, n, and intrinsic axis ratio, q_{0}^{4}. van de Ven & van der Wel (2021) make a similar comparison for both axisymmetric and triaxial systems using an approximation for ρ, but show r_{1/2, mass, 3D} relative to the projected major axis, so for the axisymmetric, prolate systems our ratio differs from theirs. The 3D spherical halfmass radius is larger than the 2D projected effective radius enclosing half of the total light (and half of the total mass, assuming constant M/L and an optically thin medium). There is a larger dependence of the ratio r_{1/2, mass, 3D}/R_{e} on q_{0} than on n, where r_{1/2, mass, 3D}/R_{e} ∼ 1.3 − 1.36 when q_{0} = 1 for all n = 0.5 − 8, but as q_{0} decreases the ratio decreases towards r_{1/2, mass, 3D}/R_{e} ∼ 1 for all n.
Fig. 2. Comparison between the 3D spherical halfmass radius, r_{1/2, mass, 3D}, and the projected 2D effective radius, R_{e}, for a range of Sérsic indices n and intrinsic axis ratios q_{0} (left, colored by q_{0}; right, colored by n). For oblate cases, R_{e} is the projected major axis, while for prolate cases R_{e} is the projected minor axis. For all cases, r_{1/2, mass, 3D} > R_{e}. However, as q_{0} decreases (i.e., flatter Sérsic distributions), the 3D halfmass radius approaches the value of R_{e}. Overall, the systematic difference between r_{1/2, mass, 3D} and R_{e} highlights that while half of the model mass is enclosed within a projected 2D ellipse of major axis R_{e} (e.g., an infinite ellipsoidal cylinder), less than half the total mass is enclosed within a sphere of radius R_{e} (ignoring any M/L gradients or optically thick regions, which would change R_{e, light}/R_{e, mass}). 
Next, we examine how the relation between the mass and circular velocity profiles deviates from the relation that holds for spherical symmetry, where . In Fig. 3 we show computed fractional enclosed mass (top) and circular velocity profiles (bottom) for n = 1, 4 (top and bottom rows, respectively), and for q_{0} = 1, 0.4, 0.2 (left, center, and right columns, respectively).
Fig. 3. Example fractional enclosed mass (top) and circular velocity (bottom) profiles computed or inferred under different assumptions. The top and bottom rows show the profiles for Sérsic indices n = 1, 4, respectively, while the columns show intrinsic axis ratios q_{0} = 1, 0.4, 0.2 (from left to right). For the top panels, we show the edgeon 2D projected mass enclosed within ellipses of axis ratio q_{0} (orange solid line), the 3D mass profile enclosed within a sphere (red dashed line), the 3D mass profile enclosed within ellipsoids of intrinsic axis ratio q_{0} (purple dashdotted line), and the mass profile inferred from the flattened deprojected Sérsic model circular velocity under the simplifying assumption of spherical symmetry (i.e., q_{0} = 1; black dotted line). In the bottom panels, we then compare the flattened deprojected Sérsic model circular velocity (black dotted line) to the inferred velocity profiles computed from the 3D spherical (red solid line) and the 3D ellipsoidal (purple dashdotted) mass profiles under the simple assumption of spherical symmetry. The same total mass M_{tot} = 5 × 10^{10} M_{⊙} is used for all cases. The vertical lines denote R = R_{e} (grey dashed) and R = 1.3R_{e} (≈r_{1/2, mass, 3D} for q_{0} = 1; grey dashdotted). These enclosed mass and velocity profiles demonstrate that when q_{0} ≠ 1, M_{sph}(< r = R)≠v_{circ}(R)^{2}R/G. The nonspherical potentials for q_{0} < 1 even result in (v_{circ}(R)^{2}R/G)/M_{tot} > 1 between R ∼ 1 − 10R_{e} (i.e., potentially leading to ≳15% overestimates in the system mass). We also see that as q_{0} decreases, M_{sph} approaches the 2D projected mass profile, as the mass enclosed in a sphere versus an infinite ellipsoidal cylinder are equivalent for infinitely thin mass distributions. 
For the spherically symmetric (q_{0} = 1) cases, the numerical evaluation of M_{sph}(< r = R) (red dashed line; Eq. (8)) and v_{circ}(R) (black dotted line; Eq. (5)) follow the expected relation (), and the isodensity spheroids are spherical, so there is no difference between the enclosed spherical and spheroidal profiles. Echoing the previous figures, we also see that the enclosed 3D mass profile increases more slowly as a function of R than the 2D projected profile (solid orange line; Eq. (3)).
In contrast, for flattened deprojected models with q_{0} < 1, the deviation of the M_{sph}(< r = R) and v_{circ}(R) profiles from the spherical relation become more pronounced for lower intrinsic axis ratios. Also, (purple dasheddotted line; Eq. (6)) does not match the correct v_{circ}(R) curve. As q_{0} decreases, M_{sph}(< r = R) approaches the projected 2D ellipse curve, because for flatter deprojected models there is less additional mass outside the sphere along the remaining lineofsight collapse.
3. Virial coefficients for enclosed 3D and total masses
We now quantify the relationship between mass and velocityderived quantities for different Sérsic indices and intrinsic axis ratios. By including a “virial” coefficient k_{3D}(R) which depends on the geometry and mass distribution (Binney & Tremaine 2008), the spherical enclosed mass and circular velocity can be related via:
This virial coefficient is evaluated by combining Eqs. (5) and (8).
For comparison with integrated galaxy quantities, it is also useful to define a “total” virial coefficient k_{tot}(R) which relates the total system mass to the circular velocity at a given radius:
Figure 4 shows k_{tot}(R = R_{e}) and k_{3D}(R = R_{e}) versus Sérsic index n for a range of q_{0}. For the spherical case (q_{0} = 1), k_{3D}(R = R_{e}) = 1, as expected by spherical symmetry. However, as R_{e} < r_{1/2, mass, 3D} for spherical deprojected Sérsic models, k_{tot}(R = R_{e}, q_{0} = 1)≠2, instead it exceeds 2 for all n (i.e., 2.933 when n = 1). For oblate flattened Sérsic deprojected models (i.e., q_{0} < 1), the value for k_{tot}(R = R_{e}) is lower than the q_{0} = 1 case for all n, while prolate cases (q_{0} > 1) have larger k_{tot}(R = R_{e}). For k_{3D}(R = R_{e}), the trends are more complex, but for n ≳ 2, the oblate (prolate) models all have k_{3D}(R = R_{e}) < 1 (> 1). For reference, we also present values of k_{tot}(R) and k_{3D}(R) for a range of R, n, and q_{0} in Table 2. These total virial coefficients, in particular, allow for a more precise comparison between the dynamical M_{tot} and projectionderived quantities, such as M_{*} or M_{gas}, particularly when full dynamical modeling is not possible (e.g., the approach used in Erb et al. 2006; Miller et al. 2011; Price et al. 2016, 2020, and numerous other studies).
Fig. 4. Total k_{tot}(R_{e}) (left) and 3D enclosed k_{3D}(R_{e}) (right) virial coefficients as a function of Sérsic index, n, and intrinsic axis ratio q_{0}. The solid lines denote q_{0} = 0.2 (orange) to q_{0} = 1 (black) and two prolate cases are shown with dashed lines (q_{0} = 1.5, 2 in dark, light grey, respectively). 
Virial coefficients for select profiles and radii.
4. Mass distributions of multicomponent galactic systems
4.1. Mass and velocity distributions of systems including both flattened and spherical components
While the virial coefficients derived in the previous section allow for the conversion from circular velocities to enclosed masses for a single nonspherical mass component, observed galaxies tend to have multiple mass components of varying intrinsic shapes and profiles. Here, we explore the enclosed mass and circular velocity distributions for galaxies with multiple mass components, focusing on how the nonspherical components impact the mass fraction distributions inferred from velocity profile ratios.
We calculated profiles for a “typical” z = 2 mainsequence starforming galaxy of stellar mass log_{10}(M_{*}/M_{⊙}) = 10.5 consisting of a bulge, disk, and halo, over a range of bulgetototal ratios, B/T. We thus adopted a total M_{bar} = 6.6 × 10^{10} M_{⊙} (using the gas fraction scaling relation of Tacconi et al. 2020). We assumed a thick, flattened disk modeled as a deprojected Sérsic distribution with q_{0, disk} = 0.25, and adopt n_{disk} = 1, R_{e, disk} = 3.4 kpc (from observed trends and scaling relations; Wuyts et al. 2011; van der Wel et al. 2014). The bulge is modeled as a deprojected Sérsic component with n_{bulge} = 4, R_{e, bulge} = 1 kpc, and q_{0, bulge} = 1. We also included a NFW halo without adiabatic contraction, assuming conc_{halo} = 4.2 and M_{halo} = 8.9 × 10^{11} M_{⊙} (following observed halo concentration and stellar mass to halo mass relations, e.g., Dutton & Macciò 2014; Moster et al. 2018)^{5}. In Fig. 5, for each of the B/T ratios (left to right), we show the enclosed mass (top row) and circular velocity profiles (middle row) as a function of radius. The impact of shifting the baryonic mass from entirely in the disk (the only oblate, nonspherical mass component; B/T = 0) to entirely in the bulge (only spherical mass components; B/T = 1) can be seen in both the M_{sph} and v_{circ} profiles. The lower Sérsic index and larger R_{e} of the disk (blue dashed line) relative to the bulge (red dashdotted) result in more slowly rising mass and v_{circ} curves for the baryonic component (green dashdotdot) at low B/T, with the curves rising more quickly as the bulge contribution increases. The total galaxy mass and v_{circ} curves (solid black) are dominated by the baryonic components in the inner regions, but at large radii (R ≳ 10 kpc) where the halo begins dominating the mass and v_{circ} profiles, the curves are similar for all B/T.
Fig. 5. Enclosed mass (3D spherical, top), circular velocity (middle), and dark matter (bottom) profiles for different components of an example galaxy as a function of projected major axis radius, for bulgetototal ratios of B/T = 0, 0.25, 0.5, 0.75, 1 (left to right). For all cases, we compute the mass components assuming values for a typical z = 2 massive mainsequence galaxy with log_{10}(M_{*}/M_{⊙}) = 10.5: M_{bar} = 6.6 × 10^{10} M_{⊙}, R_{e, disk} = 3.4 kpc, n_{disk} = 1, q_{0, disk} = 0.25, R_{e, bulge} = 1 kpc, n_{bulge} = 4, q_{0, bulge} = 1, and a NFW halo with M_{halo} = 8.9 × 10^{11} M_{⊙} and c = 4.2. Shown are the M_{sph}(< r = R) and v_{circ}(R) profiles for the disk (dashed blue), bulge (dashdotted red), total baryons (disk+bulge; dashdotdot green), halo (dotted purple), and composite total system (solid black). Vertical lines mark R = R_{e, disk} (solid grey) and the 3D spherical halfmass radii r_{1/2, mass, 3D} for the disk (dashed blue), bulge (dashdotted red), and total baryons (dashdotdot green). Two dark matter fraction definitions are shown in the bottom panels, and , with long dashed grey and long dashtripledotted dark grey lines, respectively. We note that the and curves are also shown in the top and middle panels, respectively, with the scale at the right axis of each panel. When a disk component is present, the system is no longer spherically symmetric, so M_{DM, sph}/M_{tot, sph} and differ. This deviation is larger when the disk contribution is large (i.e., lower B/T), although even at low B/T the difference is relatively modest (see also Fig. 6). Additionally, while the ratio r_{1/2, mass, 3D}/R_{e} for a single component (e.g., the disk or bulge) is generally modest (see Fig. 2), for a composite disk+bulge system, the total baryon r_{1/2, 3D, baryon} becomes much smaller relative to R_{e, disk} with increasing B/T (vertical green dashdotdot and solid grey lines). If such disparate “half” radii definitions are used to define f_{DM} apertures (i.e., versus , horizontal solid grey and green dashdotdot lines), this leads to increasingly large offsets between the f_{DM} values towards higher B/T (see also Fig. 7). 
We also show the radial variation of the 3D enclosed halo to total mass ratio M_{DM, sph}/M_{tot, sph} (long lightgrey dashed line) and the squared halo to total circular velocity ratio (darkgrey dashedtriple dotted line) in the bottom row (and the M_{sph} and v_{circ} panels, respectively). As expected when B/T < 1, these two ratios are not equivalent, although they get closer as B/T → 1 and more of the total galaxy mass is found in spherical components. For B/T = 1, the galaxy is spherically symmetric, so the two ratios are equal.
4.2. Defining dark matter fractions
As illustrated in Fig. 5, the approximation deviates from the enclosed spherical mass fraction M_{DM, sph}(< r = R)/M_{tot, sph}(< r = R) for galaxies with a nonspherical disk component, particularly when the B/T ratio is ≲0.5. This deviation thus leads to differences in inferred dark matter fractions, depending on how the fraction is defined.
If the dark matter fraction is defined as the ratio of the dark matter to total mass enclosed within a sphere of a given radius, we have . This approach is often adopted for simulations, where it is easy to determine mass within a given radius. However, observations cannot directly probe the mass distributions, so generally the fraction is defined based on the circular velocity ratio, . If a galaxy has only spherically symmetric components, these two definitions are equivalent (as seen in the right column of Fig. 5); however, as noted in Fig. 5, the two definitions are no longer equivalent with nonspherical components, where is generally larger than .
To further quantify how much these definitions can vary, we compare the value of the ratio between and at R_{e, disk} over a range of B/T ratios and intrinsic disk thicknesses q_{0, disk} in Fig. 6. For this example case (using a massive galaxy with M_{bar} = 6.6 × 10^{10} M_{⊙} at z = 2, as in Fig. 5), we see that can be as low as ∼85% of – in the extreme case with q_{0, disk} = 0.01. For more typical expected disk thicknesses for galaxies at z ∼ 1 − 3, q_{0, disk} ∼ 0.2 − 0.25, we find a minimum of (for B/T = 0). While this deviation is fairly small in this example, using consistent definitions of f_{DM} when comparing simulations and observations would avoid introducing systematic shifts between the values.
Fig. 6. Ratio between the dark matter fraction at R_{e, disk} calculated from the circular velocity and from the 3D spherical enclosed mass, , versus bulgetodisk ratio B/T, for a range of different disk intrinsic axis ratios (colored lines, from q_{0, disk} = 0.01 to 1) for an example massive galaxy at z = 2. The ratio between the two dark matter fraction measurements is lower for lower B/T (i.e., higher disk contributions) and lower q_{0, disk} (i.e., more flattened disks), with for low values of both q_{0, disk} and B/T. The limiting case of a Freeman (infinitely thin) exponential disk has . As B/T increases for fixed q_{0, disk}, and likewise for increasing q_{0, disk} at fixed B/T, the ratio of the two fraction measurements approaches 1 because the composite system becomes more spherical. Overall, the discrepancy between the f_{DM} estimators measured at the same radius is relatively minor. 
4.3. Impact of aperture effects on dark matter fractions
While the fractional differences between the 3D half mass radii r_{1/2, mass, 3D} and the projected 2D effective radii R_{e} for a single component and between the and definitions are generally small for expected galaxy thicknesses, measuring f_{DM} (of either indicator) at different radii – such as the easily measurable r_{1/2, mass, 3D} for simulations versus R_{e} for observations – can lead to very large discrepancies in the f_{DM} values. We demonstrate this issue in Fig. 7.
Fig. 7. Ratio between the composite disk+bulge 3D halfmass radius and the 2D projected disk effective radius (r_{1/2, 3D, baryons}/R_{e, disk}; left) and the difference between the dark matter fraction estimators at these radii (; right), as a function of B/T, for a range of disk intrinsic axis ratios (colored lines, from q_{0, disk} = 0.05 to 2) and ratio between the bulge and disk R_{e} (solid, dashed, and dotted lines, for R_{e, bulge}/R_{e, disk} = 0.2, 0.5, 1, respectively). The adopted galaxy values are the same as in Fig. 6, except R_{e, disk} is now determined by R_{e, bulge}/R_{e, disk}. With a nonzero bulge contribution, r_{1/2, 3D, baryons}/R_{e, disk} deviates from the singlecomponent ratio (Fig. 2), decreasing with increasing B/T for R_{e, disk} = 2, 5 kpc for all q_{0, disk} (increasing, however, with B/T when q_{0} < 1, R_{e, disk} = R_{e, bulge} = 1 kpc). For large B/T and R_{e, disk} = 5 kpc, the composite r_{1/2, 3D, baryons} is less than 50% of R_{e, disk}. If the dark matter fractions are measured at different radii, the mismatch of the aperture sizes will lead to much larger f_{DM} differences than those found for the simple estimator mismatch ( vs at the same radius; Fig. 6). Here, we show , as might be adopted for modeling of observations, and , representing a simple option for simulations (where spherical curves of growth separating gas, star, and DM particles could be used to find both the composite baryon r_{1/2, 3D, baryons} within, e.g., R_{vir} and then ). For small B/T, is larger than , but for large B/T, the trend reverses (excepting the R_{e, disk} = R_{e, bulge} = 1 kpc case), and can be up to 50%–400% larger than as B/T → 1 (for R_{e, disk} = 2, 5 kpc, respectively). This example illustrates how, depending on galaxy structures, quoted “halfmass” f_{DM} values can be very different – but that this is primarily driven by the aperture radii definitions and not by estimator mismatches. 
First, while we show the ratio of r_{1/2, mass, 3D}/R_{e} for a single component in Fig. 2, the ratio for a multicomponent system is not selfsimilar, but depends also on the ratio of effective radii for the components. For a disk + bulge system, a number of observational studies use the disk effective radius as the dark matter fraction aperture. We thus determine the 3D halfmass radius for the composite disk+bulge system, and plot the ratio r_{1/2, 3D, baryons}/R_{e, disk} as a function of B/T in the left panel of Fig. 7, for a range of ratios R_{e, bulge}/R_{e, disk} (line style) and disk intrinsic thicknesses (q_{0, disk}, assuming a spherical bulge). Depending on the R_{e, bulge}/R_{e, disk} and q_{0, disk} values, this ratio can range from ∼0.3 − 1.3 (for oblate or spherical disk geometries, or up to ∼1.7 for prolate disks), with the lowest values arising from the combination of a low R_{e, bulge}/R_{e, disk} and a high B/T.
We then demonstrate the effects of measuring f_{DM} at these different aperture radii in the right panel of Fig. 7. Here we plot the absolute difference as a function of B/T, calculated for the same R_{e, bulge}/R_{e, disk} and q_{0, disk} values. For consistency, the f_{DM} estimators for each are chosen to reflect the typical definitions from observations and simulations, respectively, in line with the “halfmass” radii choices (although, as seen in Fig. 6, using versus contributes very little to the differences seen in this figure). For very low B/T, most cases produce (e.g., larger ). For most practical cases with a larger disk than bulge (R_{e, bulge}/R_{e, disk} < 1), the difference increases towards larger B/T, with by B/T ∼ 0.2 − 0.5 (for R_{e, bulge}/R_{e, disk} = 0.2, 0.5, respectively). This difference can be very large, up to ∼0.2 for large B/T and low R_{e, bulge}/R_{e, disk}, as might be expected for massive galaxies that simultaneously have massive bulges (i.e., high B/T) but also large effective radii for the disk (i.e., low R_{e, bulge}/R_{e, disk}).
We extended these test cases to consider how f_{DM} for the different definitions and apertures change with redshift and stellar mass for a “typical” starforming galaxy, as shown in Fig. 8. We used empirical scaling relations or other estimates to determine R_{e, disk}, q_{0, disk}, f_{gas}, B/T, log_{10}(M_{halo}/M_{⊙}), and c_{halo} for a range of z and log_{10}(M_{*}/M_{⊙}) (assuming the disk and bulge follow deprojected Sérsic models, with fixed n_{disk} = 1, n_{bulge} = 4, and R_{e, bulge} = 1 kpc; top panels). These toy models predict (Fig. 8, center left) to increase over time at fixed stellar mass (in part because of the increasing c_{halo} and R_{e, disk} over time) and that lower M_{*} galaxies have higher at fixed redshift, with relatively low for the most massive galaxies (∼20%) at z ∼ 1 − 3. This is qualitatively in agreement with recent observations (e.g., Genzel et al. 2020; Price et al. 2021; Bouché et al. 2022), although these recent studies also provide evidence for nonNFW halo profiles (in particular, cored profiles), which would produce lower for the same M_{halo} than our toy models. As a further example, we consider how the predictions would change over time for a Milky Way and M31mass progenitor. In both cases, the predicted decrease from z = 3 to a minimum at z ∼ 1.5, and then increase until the present day. For these toy values, the difference in dark matter fraction definitions measured (specifically) at the same radius (R_{e, disk}; Fig. 8, center right) typically differ by only ∼ − 0.005 to ∼ − 0.025 (typically ∼4 − 6% of the measured values), with a larger typical offset at lower redshifts and for lower masses.
Fig. 8. Toy model of how (upper left), (upper right), r_{1/2, 3D, baryons}/R_{e, disk} (lower left), and (lower right) vary with redshift for a range of fixed log_{10}(M_{*}/M_{⊙}), using “typical” galaxy sizes, intrinsic axis ratios, gas fractions, B/T ratios, halo masses, and halo concentrations (from empirical scaling relations or other estimates; Dutton & Macciò 2014; Lang et al. 2014; van der Wel et al. 2014; Moster et al. 2018; Übler et al. 2019; Tacconi et al. 2020; Genzel et al. 2020). The assumed (interpolated, extrapolated) property profiles as a function of redshift for each of the fixed log_{10}(M_{*}/M_{⊙}) are shown in the top panels. Using abundancematching models (inferred from Fig. 4 of Papovich et al. 2015, based on the models of Moster et al. 2013), we show the path of a Milky Way (MW, M_{*} = 5 × 10^{10} M_{⊙} at z = 0; black stars) and M31 progenitor (M_{*} = 10^{11} M_{⊙} at z = 0; grey squares) over time in each of the panels, assuming the progenitors are “typical” at all times. This inferred “typical” evolution would predict an increase in f_{DM}(R_{e, disk}) with time at fixed M_{*}, with lower masses having higher f_{DM} at all z. The evolution of the structure and relative masses of the disk, bulge, and halo predict an increase (M_{*} ≳ 10^{10.25} M_{⊙}) or “dip” (M_{*} ≲ 10^{10.25} M_{⊙}) in the difference between z ∼ 0 and z ∼ 0.75, and then an increase until z ∼ 2 when the difference flattens (largely reflecting the flat q_{0, disk} estimate for z ≳ 2). The difference is minor, between ∼ − 0.025 and −0.005 for the stellar masses shown. The ratio of the composite r_{1/2, 3D, baryons}/R_{e, disk} increases with redshift for all masses, with more massive models predicting smaller ratios at each z. The MW and M31 progenitors have f_{DM}(R_{e, disk}) evolving from ∼0.33 and ∼0.25 (respectively) at z = 3, decreasing to ∼0.25 and ∼0.2 at z ∼ 1.5, and then increasing to roughly same value ∼0.45 at z = 0. The and values are relatively similar down to z ∼ 1.5, but at lower redshifts (where r_{1/2, 3D, baryons}/R_{e, disk} ≲ 0.9) the difference increases up to ∼0.065 (MW) and ∼0.14 (M31) at z ∼ 0. While this “typical” case predicts f_{DM} offsets of only 0.035 at z = 2 and increasing to 0.14 at z ∼ 0 for the most massive case, objects with even larger bulges (B/T > 0.4) or radii above the masssize relation will have even more discrepant f_{DM} values when adopting these radii definitions (see Fig. 7). 
We find the ratio of the 3D baryonic half mass radius to the disk effective radius (r_{1/2, 3D, baryons}/R_{e, disk}; Fig. 8, bottom left) for these models decreases towards lower redshifts and with increasing stellar mass, from ∼1 at z ∼ 2 − 3 to ∼0.94 at z = 0 for the lowest M_{*}, and from ∼0.8 at z = 3 down to ∼0.6 at z = 0 for the highest M_{*}. For the MW and M31 progenitors, this ratio decreases from ∼1, 0.96 at z = 3 down to ∼0.72, 0.59 at z = 0, respectively. The difference between the two dark matter fraction definitions measured at these different radii apertures (; Fig. 8, bottom right) for these toy models is typically much larger than the difference for the definitions alone, and tends to increase towards lower redshifts and with stellar mass. The difference changes from ∼ − 0.015, 0.,0.025 at z = 3 to ∼ − 0.01, 0.015, 0.14 at z = 0 for log_{10}(M_{*}/M_{⊙}) = 9, 10, 11, respectively. The MW and M31 progenitors have offsets increasing from ∼ − 0.013, −0.003 at z = 3 to ∼0.065, 0.14 at z = 0, respectively.
Given the very modest offsets for the f_{DM} definition differences alone, these offsets are nearly entirely driven by the differences between the aperture radii. Indeed, although the differences for these toy calculations – driven almost entirely by the aperture mismatches – do not reach the extreme differences of seen in Fig. 7 for parts of the parameter space (in part because the maximum toy model B/T is ∼0.4), we still predict absolute differences up to almost ∼0.15 at z = 0, and ∼0.03 − 0.07 at z ∼ 1 − 2. This offset is on par with the current observational uncertainties at z ≳ 1 (∼0.1 − 0.2; e.g., Genzel et al. 2020). To ensure the most direct comparison between observations and simulations – particularly as observational constraints on f_{DM} at higher redshifts continue to improve – it will be important to account for such aperture differences (either by measuring in equivalent apertures, or by applying an appropriate correction factor) in order to better determine if, and how, observation and simulation predictions differ.
5. Turbulent pressure support effects on rotation curves
5.1. Derivation of pressure support for a single component
As many dynamical studies of highredshift, turbulent disk galaxies use gas motions as the dynamical tracer, here, we consider how turbulent pressure support will modify the rotation curves if the gas is described by a deprojected Sérsic model. We followed the derivation of Burkert et al. (2010) and also assumed the pressure support is due only to the turbulent gas motions (i.e., the thermal contribution is negligible).
We thus begin from Eq. (2) of Burkert et al. (2010), where the pressurecorrected gas rotation velocity is:
where v_{circ} is the circular velocity in the midplane of the galaxy determined from the total system potential (including all mass components: stars, gas, and halo; i.e., the rotational velocity if there is no pressure support), ρ_{g} is the gas density, and σ is the (onedimensional) gas velocity dispersion. While the gas has the same circular velocity as the total system, the pressure support correction term from the turbulent gas motions only applies to the gas rotation and only depends on the gas density distribution and the gas velocity dispersion.
This relation can be generally rewritten as:
where . If we assume the velocity dispersion σ = σ_{0} is constant, then this can be simplified to:
For a selfgravitating exponential disk, as assumed in Burkert et al. (2010), , which yields:
Burkert et al. (2016) generalized this result to a selfgravitating disk with an arbitrary Sérsic index, where α_{self − grav, n}(R) = 2b_{n}(R/R_{e})^{1/n}.
Alternatively, as derived by Dalcanton & Stilp (2010) (their Eqs. (16) and (17)), for a disk with turbulent pressure P_{turb} ∝ Σ^{0.92}, where the authors infer the exponent using results from hydrodynamical simulations of turbulence in stratified gas by Joung et al. (2009) combined with a Schmidt law of slope N = 1.4 (Kennicutt 1998), the pressure support is described by:
for arbitrary σ_{R}(R) (not only constant σ_{0} as considered here). Further forms of the pressure support have also been explored, as compared and discussed by Bouché et al. (2022), including the case for constant disk thickness (Meurer et al. 1996; Bouché et al. 2022), or when accounting for the full Jeans equation (Weijmans et al. 2008).
For gas following a deprojected Sérsic model, we find α(R) = α(R, n) by differentiating ρ_{g} = ρ(m = R, n)^{6}. After combining Eqs. (2) and (4), performing a change of variable, and applying the Leibniz rule, we have:
This expression for dρ(m)/dm can be evaluated numerically, and together with the numerical evaluation of ρ(m), we have dlnρ/dlnm = (m/ρ)(dρ/dm)^{7}. Alternatively, the log density can be differentiated numerically. (A similar derivation of the pressure support for spherical deprojected Sérsic profiles is presented in Sect. 2.2.3 of Kretschmer et al. 2021, who also showed α(n) versus n at select radii in their Fig. 6, and gave an approximate equation for α(n) at select radii in Sect. 3.5)^{8}.
Figure 9 (left panel) shows the α(R, n) derived for the deprojected Sérsic models as a function of radius for a range of Sérsic index n (colored lines). For comparison, we also show the selfgravitating disk case α_{self − grav}(R) as presented in Burkert et al. (2010) (black dashed line), as well as α determined following Dalcanton & Stilp (2010), and as measured from simulations in Kretschmer et al. (2021). In the latter, the density is determined from the smoothed cumulative mass profile of the cold gas and R_{e} of the cold gas is the halfmass radius measured within 0.1R_{vir} (c.f. Sects. 2.3 and 3.2 of Kretschmer et al. 2021). The right panel additionally shows the ratio α/α_{self − grav}. We find that α(n) is lower than α_{self − grav} at R ≳ 0.2 − 0.8R_{e} for n ≳ 1. However, at small radii (R ≲ R_{e}) we find α(n) > α_{self − grav} for n ≳ 1 (with the crossover radius varying with n). In contrast, we find the inverse for n = 0.5: α(n = 0.5) is lower than α_{self − grav} up to R ∼ 2.4R_{e}, but then α(n) exceeds α_{self − grav} at larger radii. In comparison to the selfgravitating disk case, we find the deprojected Sérsic α(n) are in better agreement with the pressure support for an exponential distribution from Dalcanton & Stilp (2010), as well as with the simulationderived pressure support by Kretschmer et al. (2021, and similar findings by Wellons et al. 2020), with roughly half as much pressure support as the selfgravitating case.
Fig. 9. Pressure support correction, α(R), versus R/R_{e} for a selfgravitating exponential disk and deprojected Sérsic models. The left panel directly compares α_{self − grav}(r) = 3.36(R/R_{e}) for the selfgravitating disk (as in Burkert et al. 2010; black dashed line) to α(R, n) = − dlnρ(R, n)/dlnR determined for a range of Sérsic indices n (colored lines). The ratio α/α_{self − grav}(R) is shown in the right panel. For n ≥ 1, α(n) is smaller than α_{self − grav} when R ≳ 0.2 − 0.8R_{e}; however, α(n ≥ 1) does exceed α_{self − grav} at the smallest radii. This implies that for most radii, there is less asymmetric drift correction (and thus higher v_{rot}) for the deprojected Sérsic models (e.g., n = 1) than for the selfgravitating disk. However, for n = 0.5, α(n) is greater than α_{self − grav} at R ≳ 2.4R_{e}, so at large radii the n = 0.5 deprojected Sérsic model predicts a larger pressure support correction than for the selfgravitating disk case. The lower pressure support predicted for α(n ≳ 1) than for α_{self − grav} is in agreement with recent predictions from simulations by Kretschmer et al. (2021) (red circles; with the vertical grey bars denoting the 1σ distribution), as well the relation by Dalcanton & Stilp (2010) for a power law relationship between the gas surface density and the turbulent pressure (orange dashed line). 
Furthermore, as demonstrated by Bouché et al. (2022) (using an example v_{circ} with n = 1.5 and an NFW profile), the Dalcanton & Stilp (2010) correction produces pressure support that is very similar to the constant scale height (ρ(R)∝Σ(R), Meurer et al. 1996; Bouché et al. 2022) and Weijmans et al. 2008 cases (assuming constant dispersion), which all predict lower support corrections than for the selfgravitating disk case. This difference arises because these three cases assume constant scale height or a thin disk approximation, resulting in a correction of approximately dlnΣ(R)/dlnR. In contrast, the selfgravitating disk case explicitly assumes a constant vertical dispersion, so predicts ρ(R)∝Σ(R)^{2}, yielding a correction term that is roughly twice that of the other cases.
These differences between α predict different pressure supportcorrected v_{rot}(R) for the same circular velocity profile and intrinsic velocity dispersion. We demonstrate these differences for α for deprojected Sérsic models and a selfgravitating disk in Fig. 10, over a range of Sérsic indices (n = 0.5, 1, 2, 4; left to right) and intrinsic axis ratios (q_{0} = 1, 0.2; top and bottom, respectively)^{9}. For all cases, we determine the circular velocity v_{circ} (solid black line) assuming the mass distribution follows a single deprojected Sérsic model of M_{tot} = 10^{10.5} M_{⊙} (i.e., a pure gas disk, or gas+stars where both components follow the same density distribution). We then calculate v_{rot} using both α(n) and α_{self − grav} (dashed and dotted lines, respectively). As implied by Fig. 9, for n ≳ 1 the rotation curves v_{rot} computed with α(n) are higher than with α_{self − grav} at R ≳ R_{e} (i.e., smaller correction from v_{circ}). The difference between the two v_{rot} curves becomes more pronounced towards larger radii, in line with the continued decrease of α(n)/α_{self − grav} with increasing radius. We also see the opposite behavior in the n = 0.5 case, where v_{rot} computed in the selfgravitating case is higher than for α(n) at R ≳ 2.4R_{e} (but the v_{rot} computed with α(n) is higher than with α_{self − grav} at smaller radii).
Fig. 10. Comparison between determined using the deprojected Sérsic model α(n) and the selfgravitating exponential disk α_{self − grav} (as shown in Fig. 9), for a range of Sérsic indices n, intrinsic axis ratios q_{0}, and velocity dispersions σ_{0}. For all cases, we consider a single deprojected Sérsic mass distribution with M_{tot} = 10^{10.5} M_{⊙}. The columns show curves for n = 0.5, 1, 2, 4 (left to right, respectively), while the rows show the case of spherical (q_{0} = 1; top) and flattened (q_{0} = 0.2; bottom) Sérsic distributions. For each panel, the solid black line shows the circular velocity v_{circ} (determined following Eq. (5)). The colored lines show v_{rot} determined using α(n) (dashed) and α_{self − grav} (dotted), with the colors denoting σ_{0} = [30, 60, 90] km s^{−1} (purple, turquoise, orange, respectively). As expected by the α(R) trends shown in Fig. 9, for n ≥ 1, we see that for most radii, the pressure support implied by α_{self − grav} results in lower v_{rot} than for α(n) (although at the smallest radii, the inverse holds). In some cases, the magnitude of σ_{0} combined with the form of α(R) additionally predict disk truncation within the range shown, although truncation generally occurs at smaller radii for α_{self − grav} than for α(n). 
The amplitude of the intrinsic dispersion further impacts the v_{rot} profiles by causing disk truncation for sufficiently high σ_{0} relative to v_{circ}, as previously discussed by Burkert et al. (2016). For the highest dispersion case (σ_{0} = 90 km s^{−1}; orange), the pressure support correction predicts disk truncation (i.e., ) within R ≲ 5R_{e} for both α(n) and α_{self − grav}. With medium dispersion (σ_{0} = 60 km s^{−1}; turquoise), we still find disk truncation at R ≲ 5R_{e} for all n when using α_{self − grav}, but only α(n = 0.5, 1) produce truncation within this radial range. Finally, α_{self − grav} does not produce truncation within 5R_{e} in any case at the lowest dispersion (σ_{0} = 30 km s^{−1}; purple), and only α(n = 0.5) predicts truncation at R ∼ 5R_{e} (for both the spherical and flattened cases).
5.2. Pressure support for multicomponent systems
Generally, however, the gas in galaxies may be distributed in more than one component, which would modify the pressure support correction term^{10}. We can then derive the composite α_{tot}(R) using the α(R, n) of the individual gas components. For example, if the composite system includes gas in both a bulge and a disk, we have the total ρ_{tot} = ρ_{disk} + ρ_{bulge}. As dlnρ/dlnR = (R/ρ)(dρ/dR), we have:
As discussed in Sect. 5.1, this composite gas pressure support term is applicable to the gas velocity curve regardless of the distribution of the other, nongas mass components.
We demonstrate an example composite pressure support term for a galaxy with gas distributed in both a disk and bulge over a range of B/T and R_{e, bulge}/R_{e, disk} values in Fig. 11. Here, we assume n_{disk} = 1 and n_{bulge} = 4, that is, an exponential disk and de Vaucouleurs spheroid bulge, as adopted for recent bulge and disk decompositions at z ∼ 1 − 3, given the current observation spatial resolution limitations (Bruce et al. 2012; Lang et al. 2014), with a range of B/T (from disk to bulgeonly; black to light red colors) and R_{e, bulge}/R_{e, disk} (from R_{e, disk} = 5R_{e, bulge} down to R_{e, disk} = R_{e, bulge}; solid to dotted line styles). As expected, the composite α_{tot} is lower than α_{self − grav} at large radii, but can be larger than α_{self − grav} at R/R_{e, disk} ≲ 1 when there is nonzero bulge contribution (see Fig. 9).
Fig. 11. Composite pressure support correction, α_{tot}(R), for gas distributed in a composite disk+bulge system (with n_{disk} = 1, n_{bulge} = 4), for a range of B/T (colors) and R_{e, bulge}/R_{e, disk} ratios (dash length). For the limiting cases, we recover the profiles shown in Fig. 9: B/T = 0 has α_{tot} = α(n = 1) (black solid line), while B/T = 1 has α_{tot} = α(n = 4) but with different radial scaling, owing to the different adopted R_{e, bulge}/R_{e, disk} ratios (colored lines). For the cases with 0 < B/T < 1, the bulge contribution modifies the α(n = 1) profile at both small and large radii, leading to larger α_{tot} in inner regions and smaller α_{tot} in the outskirts (R/R_{e, disk} ≲ , ≳ 1 − 2). At fixed B/T, the deviation from the disk α(n = 1) in the center (R/R_{e, disk} ≲ 1 − 2) is larger for smaller R_{e, bulge}/R_{e, disk}, while at large radii the deviation is larger for larger R_{e, bulge}/R_{e, disk}. For reference, we mark R_{e, bulge}/R_{e, disk} with vertical light grey lines, and also show α_{self − grav} (grey dashdotted line). 
Compared to the diskonly α(n = 1) (solid black line), the inclusion of the bulge component leads to larger α_{tot} at small radii (R/R_{e, disk} ≲ 1 − 2) and lower α_{tot} at large radii (R/R_{e, disk} ≳ 1 − 2). This is the result of a steeper inner density slope together with a shallower decline at large radii for n = 4 compared to an exponential deprojected Sérsic model, so the bulge component becomes more important at very small and very large radii. When varying R_{e, bulge}/R_{e, disk}, we find the most pronounced changes to α_{tot} at small radii when the R_{e, bulge}/R_{e, disk} ratio is smallest (solid lines). This effect is less pronounced for larger R_{e, bulge}/R_{e, disk} values, as the bulge density profile is more extended and the disk profile becomes important at smaller R/R_{e, disk}. For larger radii, the opposite holds: the largest changes with B/T are found for the largest R_{e, bulge}/R_{e, disk} (dotted lines), as the bulge component becomes important at smaller R/R_{e, disk} owing to the larger R_{e, bulge}.
6. Discussion and implications
In this paper, we present properties and implications when using deprojected, axisymmetric Sérsic models to describe mass density distributions or kinematics, over a wide range of possible galaxy parameters. Some of these effects will be more important for certain galaxy populations and epochs than others (as initially hinted in Fig. 8). Here, we discuss the implications for the models presented in this work, focusing on which aspects are most important for interpreting observations and for comparing observations to simulations as a function of cosmic time and galaxy mass.
6.1. Low redshift
Nearby, presentday starforming galaxies (that are not dwarf galaxies) typically host fairly thin disk components and some also host a bulge. The disks of such galaxies would generally be characterized by geometries with small q_{0} – relatively similar to the infinitely thin exponential disk case (Freeman 1970). Thus, when modeling the circular velocity curves of these disks, the choice of adopting the infinitely thin disk versus deprojected oblate Sérsic models has a relatively small impact. The thin gas disks of these local galaxies also have relatively low intrinsic velocity dispersions, with relatively little pressure support. The exact pressure support correction formulation therefore has less of an impact on the interpretation of the dynamics.
However, the low q_{0} and typically large disk effective radii R_{e, disk} in z ∼ 0 starforming galaxies, when coupled with a nonnegligible bulge component, do result in ratios of r_{1/2, 3D, baryons}/R_{e, disk} less than 1. This deviation of the 2D and 3D halfmass radii can lead to large aperture effects when interpreting projected versus 3D quantities, such as when comparing observational or simulation quantities (e.g., f_{DM}). This aperture mismatch would be most severe for higher mass lowz galaxies, as these will tend to have larger values of R_{e, disk} and B/T (since a more prominent bulge will decrease r_{1/2, 3D, baryons} relative to R_{e, disk}). For example, aperture differences can lead to discrepancies of up to Δf_{DM} = f_{DM}(R_{e, disk})−f_{DM}(r_{1/2, 3D, baryons})∼0.15 at M_{*} ∼ 10^{11} M_{⊙} for typical values of R_{e, disk} and B/T (Fig. 8, lower right). In contrast, lower mass lowz galaxies are generally less impacted by aperture mismatches, owing to the lower typical B/T and smaller R_{e, disk} of these galaxies.
Compared to the impact of aperture mismatches, definition differences in f_{DM} (as might be measured from observations and simulations) lead to only minor discrepancies. However, for lower stellar mass lowz galaxies where the aperture mismatch is relatively minor, the typically low B/T and thus more prominent thin disk leads to a larger relative impact of f_{DM} estimator differences, as these galaxies are overall less spherically symmetric (see Figs. 6 and 8).
Overall, for starforming galaxies at low redshift, the most important effect to consider is to correct for – or avoid – any aperture mismatches when comparing measurements between simulations and observations of, for instance, f_{DM}, particularly for high stellar masses. The impact of other aspects (use of infinitely thin disks vs. finite thickness, pressure support correction formulation, or f_{DM} estimator definition) are all relatively minor and can be ignored for most purposes.
6.2. High redshift
In contrast to the local universe, at high redshift (e.g., z ∼ 1 − 3), relatively massive starforming galaxies generally exhibit thick disks, with increasing bulge contributions towards higher masses. These thick disks would be reasonably well described by elevated q_{0} ∼ 0.2 − 0.25. As the derived circular velocity curve for such a geometry is fairly different from that of an infinitely thin exponential disk (e.g., N08), the choice of rotation curve parameterization (i.e., adopting v_{circ} based o a deprojected profile such as those presented here versus using an infinitely thin exponential disk) is important at highz.
The thick geometries of highz disks are coupled with relatively high intrinsic velocity dispersions, which implies that the overall amount of pressure support is expected to be much higher than for the dynamicallycold, thin disks at lowz. Thus, not only is it more important to account for pressure support, but the choice of adopted pressure support correction matters much more for interpreting kinematics at highz than for nearby galaxies.
In this paper, we derive the log density slopedriven pressure support correction α(n) as a function of radius R for the deprojected Sérsic models, and compare this correction term to other formulations, particularly the correction for a selfgravitating exponential disk, α_{self − grav} (as in Burkert et al. 2010, 2016). A key implication of the differences between these pressure support corrections is that, for the same v_{rot}(R) and σ_{0}, α(n) predicts a lower v_{circ} than would be inferred when applying α_{self − grav} (i.e., the inverse of the demonstration in Fig. 10). Furthermore, the shape of the inferred v_{circ} profile can also differ (particularly when considering a composite disk+bulge gas distribution; Fig. 11). Both effects can impact the results of mass decomposition from modeling of galaxy kinematics, which have important implications for the measurement of dark matter fractions.
Although the smaller disk sizes of highz galaxies help to alleviate the diskhalo degeneracy that strongly impacts kinematic fitting at z ∼ 0, there are nonetheless often degeneracies between mass components when performing kinematic modeling at z ∼ 1 − 3 (see e.g., Price et al. 2021, Sect. 6.2 and Fig. 5). The strong pressure correction from α_{self − grav} can further complicate the reduced but still present diskhalo degeneracy at highz. When combined with high σ_{0}, modest variations in σ_{0} (allowed within the uncertainties) can extend the degeneracy between galaxyscale dark matter fractions and total baryonic masses – in the most extreme cases, allowing the 1σ region to extend from 0% to 50+% dark matter fractions.
However, the strength of this added degeneracy effect depends not only on σ_{0}, but also on the pressure support prescription. The large correction from α_{self − grav} can result in a falling v_{rot} even for a flat or rising v_{circ} (with a large halo contribution; Fig. 5b of Price et al. 2021). Alternatively, if α(n) were adopted, the comparable correction to v_{circ} would produce a less steeply dropping (or potentially flat) v_{rot} profile. Thus, to match the observed v_{rot} profile, the intrinsic v_{circ} would be limited to lower amplitudes (i.e., implying lower dynamical masses) with less shape modification than when using α(n) instead of α_{self − grav}. This in turn implies partial breaking of the added pressure support impact to the diskhalo degeneracy, restricting the higher likelihood regions towards a lower value for f_{DM}. While adopting α(n) would have the greatest impact on the objects with high σ_{0} (where the pressure support has the largest impact), the change in prescription should impact the inferred mass distribution for all objects to some extent. The choice of pressure support formulation is thus an important factor in the interpretation of dynamics of highredshift galaxies and has direct implications for the interpretation of mass fractions. Overall, this will have the largest impact for galaxies with low v_{rot}/σ_{0} (and the smallest impact for high v_{rot}/σ_{0}), as this will lead to the largest fractional change in v_{rot} relative to v_{circ}. Since there is currently no observed trend of σ_{0} with M_{*} at highz (e.g., Übler et al. 2019; although the dynamic range of M_{*} is currently limited), the correlation of v_{rot} with M_{*} would then cause this effect to generally be most important for lowmass galaxies.
On the other hand, the higher q_{0} and lower R_{e, disk} of highz disk galaxies implies that aperture effects arising from deviations of r_{1/2, 3D, baryons} versus R_{e, disk} are less important than for lowz galaxies, as the disk and bulge sizes are more similar. Still, there can be up to ∼20% radii aperture differences in the 2D and 3D halfmass radii (although with only ∼2.5% f_{DM} differences), so depending on the particular measurement quantity and accuracy required, this effect could still be important. As is the case for the local limit, the aperture radii difference (and the resulting impact on inferred f_{DM}) typically has a larger impact for higher mass objects, since these tend to have higher B/T and R_{e, disk} than lower mass objects. Finally, as with the lowz case, the f_{DM} estimator differences are relatively minor compared to the other effects and can be generally ignored. We note, however, that the same comments on trends with B/T and necessary comparison accuracy from the lowz discussion apply in this case.
In conclusion, for highredshift starforming galaxies, the most important effects to consider are:

Adopting circular velocity curves that account for the finite, thickdisk geometry;

Including a reasonable pressure support correction when interpreting rotation curves.
In this limit (higher q_{0}, lower R_{e, disk}, high σ_{0}), the other aspects (2D vs. 3D halfmass radii apertures, f_{DM} estimator definitions) have relatively small impacts and can typically be ignored.
7. Summary
We present a number of properties for 3D deprojected Sérsic models with a range of values for the intrinsic axis ratio q_{0} = c/a (i.e., flattened or oblate, spherical, or prolate). We followed the derivation of N08, who presented the deprojection of the 2D Sérsic profile to a 3D density distribution ρ(m), as well as the midplane circular rotation curve v_{circ}(R) for such a mass distribution. We then extended this work by numerically deriving spherical enclosed mass profiles M_{sph}(< r = R) and the log density slope dlnρ/dlnR.
Using these profiles, we determined a range of properties of these mass models. Specifically, we examined the differences between the 2D projected effective radius, R_{e}, and the 3D sphericallyenclosed halfmass radius, r_{1/2, mass, 3D}, over a range of intrinsic axis ratios q_{0} and Sérsic indices n, and find r_{1/2, mass, 3D} > R_{e}, with the ratio approaching unity as q_{0} → 0, in agreement with previous results. We also calculated virial coefficients that relate the circular velocity to either the total mass (k_{tot}) or the enclosed mass within a sphere (k_{3D}).
Furthermore, we calculated derived properties for example composite galaxy systems (consisting of both flattened deprojected Sérsic and spherical components), to consider how varying galaxy properties (i.e., B/T, R_{e, disk}, z) impacts these properties, such as r_{1/2, 3D, baryons}/R_{e, disk}. We also examined the impact of different methods of inferring f_{DM}(< R) and the compounding effects from measuring f_{DM} within different aperture radii. We find that using different apertures, such as r_{1/2, 3D, baryons} versus R_{e, disk}, can lead to very large differences in the measured f_{DM}, particularly for high B/T and low R_{e, bulge}/R_{e, disk}. In contrast, using different f_{DM} definitions, such as and , only produces minor differences when measured at the same radius. Using toy models, we estimated how r_{1/2, 3D, baryons}/R_{e, disk} and the f_{DM} estimators (measured both at R_{e, disk} and with mismatched r_{1/2, 3D, baryons} vs. R_{e, disk} apertures) change as a function of redshift and stellar mass and find increasing offsets towards higher M_{*} and lower z.
We additionally used the deprojected Sérsic models to derive selfconsistent pressure support correction terms, with α(R, n) = − dlnρ_{g}(R, n)/dlnR for constant gas velocity dispersion. At R ≳ R_{e}, we find that α(R, n) typically predict a smaller pressure support correction than is inferred for a selfgravitating disk (as in Burkert et al. 2010, 2016) and are more similar to predictions derived for thin disks with ∼constant scale heights under various assumptions (e.g., Dalcanton & Stilp 2010; Meurer et al. 1996; Bouché et al. 2022; Weijmans et al. 2008) and from simulations (e.g., Kretschmer et al. 2021; also Wellons et al. 2020). The effect of a lower pressure support with α(n) implies larger v_{rot} for the same v_{circ} and σ_{0} (or lower v_{circ} for the same v_{rot} and σ_{0})than if assuming α_{self − grav}, and would predict any disk truncation (where v_{rot} → 0, as in Burkert et al. 2016) at larger radii than for the selfgravitating case.
Finally, we discuss implications of this work for future studies of galaxy mass distributions and kinematics. Lowz starforming disk galaxies typically have thin disks with small q_{0} and low intrinsic velocity dispersion, so the most important effect to consider is aperture mismatches when comparing measurements – such as measuring f_{DM} within 2D and 3D apertures, as typically adopted for observations and simulations, respectively. In contrast, the thick disks in highz starforming galaxies are characterized by large q_{0} and high intrinsic velocity dispersion, so adopting circular velocity curves accounting for this finite thickness and accounting for the pressure support correction are the most important aspects. The high σ_{0} of these highz galaxies can produce large pressure support corrections, in some cases causing greaterthanKeplerian falloff in outer rotation curves (e.g., Genzel et al. 2017). In this limit of relatively large correction amplitudes, the choice of the adopted pressure support correction is also important and can impact constraints of the diskhalo mass decomposition, as lower correction amplitudes (e.g., using α(n) versus the larger correction of α_{self − grav}) will tend to lead to lower inferred dark matter fractions, particularly for high σ_{0}. Furthermore, while differences in quantity estimators (e.g., vs. ) have only modest effects at both low and highz, as measurements improve it would be worth correcting for, or avoiding, estimator differences to improve the accuracy of comparisons between different studies.
The deprojected Sérsic profile models presented here can be used to aid comparisons between observations and simulations and help convert between simulation quantities that are typically determined within spherical shells and observational constraints based on 2D projected quantities. As demonstrated in this work, commonly adopted apertures for simulations (3D halfmass) versus observations (2D projected halflight or halfmass) can probe different physical scales, impacting observationsimulation comparisons, particularly for dark matter fractions. The precomputed profiles and values (or similar calculations) can help in shifting towards more direct, applestoapples comparisons between the two, without resorting to the more direct but complex step of constructing and analyzing mock observations based on simulated galaxies (as in, e.g., Übler et al. 2021; but see also Genel et al. 2012; Teklu et al. 2018; Simons et al. 2019). The code used to compute these profiles, as well as precomputed profiles and other quantities for a range of Sérsic index, n, and intrinsic axis ratio, q_{0}, have been made publicly available.
The python package deprojected_sersic_models used in this paper and the precomputed tables are both available for download from sedonaprice.github.io/deprojected_sersic_models/downloads.html; the full code repository is publicly available at github.com/sedonaprice/deprojected_sersic_models. The code also includes functions for scaling and interpolating the profiles from the precomputed tables to arbitrary total masses and R_{e} as a function of radius.
For prolate geometries, the projected major axis lies parallel to the long intrinsic axis, c. Here, however, we use a geometry definition where κ is parallel to a for all cases, for a consistent convention relative to the rotation axis (z; parallel to c) – so technically κ is parallel to the major axis as usual for oblate geometry, but lies along the minor axis for prolate geometry.
In N08, ρ(m) denotes the 3D luminosity density distribution, while we define ρ(m) as the 3D mass density. Thus, we instead write the 3D luminosity density as ρ(m)/Υ(m) in the projection integral.
However, many massive SFGs at these redshifts exhibit lower f_{DM} that suggest more cored halo profiles; see e.g., Genzel et al. (2020).
Note that the pressure correction term α(n) discussed here is the same as α_{ρ} as defined in Kretschmer et al. (2021). However, we emphasize that it is not directly comparable to the α_{v} derived by Kretschmer et al. for their simulations. Kretschmer et al. determine circular velocities from mass enclosed within a sphere, , and instead fold the effects of nonspherical potentials into the correction term Δ_{Q}. Here we explicitly consider v_{circ} determined for nonspherical deprojected Sérsic profiles, so α(n) does not need such a correction. Of course, the total α considered here would be modified by terms incorporating variable σ(R) or anisotropic velocity dispersion, but these terms vanish as we assume a constant σ_{0}.
Only the gas density distribution that impacts α(R), regardless of other (e.g., stellar or halo) components (see Sect. 5.1).
Acknowledgments
We thank Michael Kretschmer for sharing the values of α_{ρ} derived from the simulated galaxies presented in Fig. 4 of Kretschmer et al. (2021), Taro Shimizu for helpful discussions, and Dieter Lutz for comments on the manuscript. We also thank the anonymous referee for their comments and suggestions that improved this manuscript. HÜ gratefully acknowledges support by the Isaac Newton Trust and by the Kavli Foundation through a NewtonKavli Junior Fellowship. This work has made use of the following software: Astropy (http://www.astropy.org; Astropy Collaboration 2013, 2018), dill (McKerns et al. 2011; McKerns & Aivazis 2010), IPython (Pérez & Granger 2007), Matplotlib (Hunter 2007), Numpy (Van Der Walt et al. 2011; Harris et al. 2020), Scipy (Virtanen et al. 2020)
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All Tables
All Figures
Fig. 1. Fractional mass enclosed within a sphere of radius r = R for deprojected Sérsic models of different intrinsic axis ratios. From left to right: enclosed M_{sph} is plotted as a function of log radius (relative to the projected 2D effective radius, R_{e}), assuming intrinsic axis ratios of q_{0} = 1, 0.4, 0.2, respectively. The colored curves denote the enclosed mass profiles for Sérsic indices from n = 0.5 to n = 8 (yellow to purple). The vertical lines denote R = R_{e} (grey dashed) and R = 1.3R_{e} (≈r_{1/2, mass, 3D} for q_{0} = 1; grey dashdotted), and the horizontal colored lines denote the fraction of the mass enclosed within r = R_{e} for n = 1, 4 (lime, teal dashed, respectively) and 50% of the total mass (grey dashed dotted). For q_{0} = 1, the halfmass 3D spherical radius is indeed r_{1/2, mass, 3D} ≈ 1.3R_{e} regardless of n, as in Ciotti (1991). For flattened (i.e., oblate) systems, the halfmass 3D spherical radius is smaller, and approaches R_{e} as q_{0} decreases. See also Fig. 2. 

In the text 
Fig. 2. Comparison between the 3D spherical halfmass radius, r_{1/2, mass, 3D}, and the projected 2D effective radius, R_{e}, for a range of Sérsic indices n and intrinsic axis ratios q_{0} (left, colored by q_{0}; right, colored by n). For oblate cases, R_{e} is the projected major axis, while for prolate cases R_{e} is the projected minor axis. For all cases, r_{1/2, mass, 3D} > R_{e}. However, as q_{0} decreases (i.e., flatter Sérsic distributions), the 3D halfmass radius approaches the value of R_{e}. Overall, the systematic difference between r_{1/2, mass, 3D} and R_{e} highlights that while half of the model mass is enclosed within a projected 2D ellipse of major axis R_{e} (e.g., an infinite ellipsoidal cylinder), less than half the total mass is enclosed within a sphere of radius R_{e} (ignoring any M/L gradients or optically thick regions, which would change R_{e, light}/R_{e, mass}). 

In the text 
Fig. 3. Example fractional enclosed mass (top) and circular velocity (bottom) profiles computed or inferred under different assumptions. The top and bottom rows show the profiles for Sérsic indices n = 1, 4, respectively, while the columns show intrinsic axis ratios q_{0} = 1, 0.4, 0.2 (from left to right). For the top panels, we show the edgeon 2D projected mass enclosed within ellipses of axis ratio q_{0} (orange solid line), the 3D mass profile enclosed within a sphere (red dashed line), the 3D mass profile enclosed within ellipsoids of intrinsic axis ratio q_{0} (purple dashdotted line), and the mass profile inferred from the flattened deprojected Sérsic model circular velocity under the simplifying assumption of spherical symmetry (i.e., q_{0} = 1; black dotted line). In the bottom panels, we then compare the flattened deprojected Sérsic model circular velocity (black dotted line) to the inferred velocity profiles computed from the 3D spherical (red solid line) and the 3D ellipsoidal (purple dashdotted) mass profiles under the simple assumption of spherical symmetry. The same total mass M_{tot} = 5 × 10^{10} M_{⊙} is used for all cases. The vertical lines denote R = R_{e} (grey dashed) and R = 1.3R_{e} (≈r_{1/2, mass, 3D} for q_{0} = 1; grey dashdotted). These enclosed mass and velocity profiles demonstrate that when q_{0} ≠ 1, M_{sph}(< r = R)≠v_{circ}(R)^{2}R/G. The nonspherical potentials for q_{0} < 1 even result in (v_{circ}(R)^{2}R/G)/M_{tot} > 1 between R ∼ 1 − 10R_{e} (i.e., potentially leading to ≳15% overestimates in the system mass). We also see that as q_{0} decreases, M_{sph} approaches the 2D projected mass profile, as the mass enclosed in a sphere versus an infinite ellipsoidal cylinder are equivalent for infinitely thin mass distributions. 

In the text 
Fig. 4. Total k_{tot}(R_{e}) (left) and 3D enclosed k_{3D}(R_{e}) (right) virial coefficients as a function of Sérsic index, n, and intrinsic axis ratio q_{0}. The solid lines denote q_{0} = 0.2 (orange) to q_{0} = 1 (black) and two prolate cases are shown with dashed lines (q_{0} = 1.5, 2 in dark, light grey, respectively). 

In the text 
Fig. 5. Enclosed mass (3D spherical, top), circular velocity (middle), and dark matter (bottom) profiles for different components of an example galaxy as a function of projected major axis radius, for bulgetototal ratios of B/T = 0, 0.25, 0.5, 0.75, 1 (left to right). For all cases, we compute the mass components assuming values for a typical z = 2 massive mainsequence galaxy with log_{10}(M_{*}/M_{⊙}) = 10.5: M_{bar} = 6.6 × 10^{10} M_{⊙}, R_{e, disk} = 3.4 kpc, n_{disk} = 1, q_{0, disk} = 0.25, R_{e, bulge} = 1 kpc, n_{bulge} = 4, q_{0, bulge} = 1, and a NFW halo with M_{halo} = 8.9 × 10^{11} M_{⊙} and c = 4.2. Shown are the M_{sph}(< r = R) and v_{circ}(R) profiles for the disk (dashed blue), bulge (dashdotted red), total baryons (disk+bulge; dashdotdot green), halo (dotted purple), and composite total system (solid black). Vertical lines mark R = R_{e, disk} (solid grey) and the 3D spherical halfmass radii r_{1/2, mass, 3D} for the disk (dashed blue), bulge (dashdotted red), and total baryons (dashdotdot green). Two dark matter fraction definitions are shown in the bottom panels, and , with long dashed grey and long dashtripledotted dark grey lines, respectively. We note that the and curves are also shown in the top and middle panels, respectively, with the scale at the right axis of each panel. When a disk component is present, the system is no longer spherically symmetric, so M_{DM, sph}/M_{tot, sph} and differ. This deviation is larger when the disk contribution is large (i.e., lower B/T), although even at low B/T the difference is relatively modest (see also Fig. 6). Additionally, while the ratio r_{1/2, mass, 3D}/R_{e} for a single component (e.g., the disk or bulge) is generally modest (see Fig. 2), for a composite disk+bulge system, the total baryon r_{1/2, 3D, baryon} becomes much smaller relative to R_{e, disk} with increasing B/T (vertical green dashdotdot and solid grey lines). If such disparate “half” radii definitions are used to define f_{DM} apertures (i.e., versus , horizontal solid grey and green dashdotdot lines), this leads to increasingly large offsets between the f_{DM} values towards higher B/T (see also Fig. 7). 

In the text 
Fig. 6. Ratio between the dark matter fraction at R_{e, disk} calculated from the circular velocity and from the 3D spherical enclosed mass, , versus bulgetodisk ratio B/T, for a range of different disk intrinsic axis ratios (colored lines, from q_{0, disk} = 0.01 to 1) for an example massive galaxy at z = 2. The ratio between the two dark matter fraction measurements is lower for lower B/T (i.e., higher disk contributions) and lower q_{0, disk} (i.e., more flattened disks), with for low values of both q_{0, disk} and B/T. The limiting case of a Freeman (infinitely thin) exponential disk has . As B/T increases for fixed q_{0, disk}, and likewise for increasing q_{0, disk} at fixed B/T, the ratio of the two fraction measurements approaches 1 because the composite system becomes more spherical. Overall, the discrepancy between the f_{DM} estimators measured at the same radius is relatively minor. 

In the text 
Fig. 7. Ratio between the composite disk+bulge 3D halfmass radius and the 2D projected disk effective radius (r_{1/2, 3D, baryons}/R_{e, disk}; left) and the difference between the dark matter fraction estimators at these radii (; right), as a function of B/T, for a range of disk intrinsic axis ratios (colored lines, from q_{0, disk} = 0.05 to 2) and ratio between the bulge and disk R_{e} (solid, dashed, and dotted lines, for R_{e, bulge}/R_{e, disk} = 0.2, 0.5, 1, respectively). The adopted galaxy values are the same as in Fig. 6, except R_{e, disk} is now determined by R_{e, bulge}/R_{e, disk}. With a nonzero bulge contribution, r_{1/2, 3D, baryons}/R_{e, disk} deviates from the singlecomponent ratio (Fig. 2), decreasing with increasing B/T for R_{e, disk} = 2, 5 kpc for all q_{0, disk} (increasing, however, with B/T when q_{0} < 1, R_{e, disk} = R_{e, bulge} = 1 kpc). For large B/T and R_{e, disk} = 5 kpc, the composite r_{1/2, 3D, baryons} is less than 50% of R_{e, disk}. If the dark matter fractions are measured at different radii, the mismatch of the aperture sizes will lead to much larger f_{DM} differences than those found for the simple estimator mismatch ( vs at the same radius; Fig. 6). Here, we show , as might be adopted for modeling of observations, and , representing a simple option for simulations (where spherical curves of growth separating gas, star, and DM particles could be used to find both the composite baryon r_{1/2, 3D, baryons} within, e.g., R_{vir} and then ). For small B/T, is larger than , but for large B/T, the trend reverses (excepting the R_{e, disk} = R_{e, bulge} = 1 kpc case), and can be up to 50%–400% larger than as B/T → 1 (for R_{e, disk} = 2, 5 kpc, respectively). This example illustrates how, depending on galaxy structures, quoted “halfmass” f_{DM} values can be very different – but that this is primarily driven by the aperture radii definitions and not by estimator mismatches. 

In the text 
Fig. 8. Toy model of how (upper left), (upper right), r_{1/2, 3D, baryons}/R_{e, disk} (lower left), and (lower right) vary with redshift for a range of fixed log_{10}(M_{*}/M_{⊙}), using “typical” galaxy sizes, intrinsic axis ratios, gas fractions, B/T ratios, halo masses, and halo concentrations (from empirical scaling relations or other estimates; Dutton & Macciò 2014; Lang et al. 2014; van der Wel et al. 2014; Moster et al. 2018; Übler et al. 2019; Tacconi et al. 2020; Genzel et al. 2020). The assumed (interpolated, extrapolated) property profiles as a function of redshift for each of the fixed log_{10}(M_{*}/M_{⊙}) are shown in the top panels. Using abundancematching models (inferred from Fig. 4 of Papovich et al. 2015, based on the models of Moster et al. 2013), we show the path of a Milky Way (MW, M_{*} = 5 × 10^{10} M_{⊙} at z = 0; black stars) and M31 progenitor (M_{*} = 10^{11} M_{⊙} at z = 0; grey squares) over time in each of the panels, assuming the progenitors are “typical” at all times. This inferred “typical” evolution would predict an increase in f_{DM}(R_{e, disk}) with time at fixed M_{*}, with lower masses having higher f_{DM} at all z. The evolution of the structure and relative masses of the disk, bulge, and halo predict an increase (M_{*} ≳ 10^{10.25} M_{⊙}) or “dip” (M_{*} ≲ 10^{10.25} M_{⊙}) in the difference between z ∼ 0 and z ∼ 0.75, and then an increase until z ∼ 2 when the difference flattens (largely reflecting the flat q_{0, disk} estimate for z ≳ 2). The difference is minor, between ∼ − 0.025 and −0.005 for the stellar masses shown. The ratio of the composite r_{1/2, 3D, baryons}/R_{e, disk} increases with redshift for all masses, with more massive models predicting smaller ratios at each z. The MW and M31 progenitors have f_{DM}(R_{e, disk}) evolving from ∼0.33 and ∼0.25 (respectively) at z = 3, decreasing to ∼0.25 and ∼0.2 at z ∼ 1.5, and then increasing to roughly same value ∼0.45 at z = 0. The and values are relatively similar down to z ∼ 1.5, but at lower redshifts (where r_{1/2, 3D, baryons}/R_{e, disk} ≲ 0.9) the difference increases up to ∼0.065 (MW) and ∼0.14 (M31) at z ∼ 0. While this “typical” case predicts f_{DM} offsets of only 0.035 at z = 2 and increasing to 0.14 at z ∼ 0 for the most massive case, objects with even larger bulges (B/T > 0.4) or radii above the masssize relation will have even more discrepant f_{DM} values when adopting these radii definitions (see Fig. 7). 

In the text 
Fig. 9. Pressure support correction, α(R), versus R/R_{e} for a selfgravitating exponential disk and deprojected Sérsic models. The left panel directly compares α_{self − grav}(r) = 3.36(R/R_{e}) for the selfgravitating disk (as in Burkert et al. 2010; black dashed line) to α(R, n) = − dlnρ(R, n)/dlnR determined for a range of Sérsic indices n (colored lines). The ratio α/α_{self − grav}(R) is shown in the right panel. For n ≥ 1, α(n) is smaller than α_{self − grav} when R ≳ 0.2 − 0.8R_{e}; however, α(n ≥ 1) does exceed α_{self − grav} at the smallest radii. This implies that for most radii, there is less asymmetric drift correction (and thus higher v_{rot}) for the deprojected Sérsic models (e.g., n = 1) than for the selfgravitating disk. However, for n = 0.5, α(n) is greater than α_{self − grav} at R ≳ 2.4R_{e}, so at large radii the n = 0.5 deprojected Sérsic model predicts a larger pressure support correction than for the selfgravitating disk case. The lower pressure support predicted for α(n ≳ 1) than for α_{self − grav} is in agreement with recent predictions from simulations by Kretschmer et al. (2021) (red circles; with the vertical grey bars denoting the 1σ distribution), as well the relation by Dalcanton & Stilp (2010) for a power law relationship between the gas surface density and the turbulent pressure (orange dashed line). 

In the text 
Fig. 10. Comparison between determined using the deprojected Sérsic model α(n) and the selfgravitating exponential disk α_{self − grav} (as shown in Fig. 9), for a range of Sérsic indices n, intrinsic axis ratios q_{0}, and velocity dispersions σ_{0}. For all cases, we consider a single deprojected Sérsic mass distribution with M_{tot} = 10^{10.5} M_{⊙}. The columns show curves for n = 0.5, 1, 2, 4 (left to right, respectively), while the rows show the case of spherical (q_{0} = 1; top) and flattened (q_{0} = 0.2; bottom) Sérsic distributions. For each panel, the solid black line shows the circular velocity v_{circ} (determined following Eq. (5)). The colored lines show v_{rot} determined using α(n) (dashed) and α_{self − grav} (dotted), with the colors denoting σ_{0} = [30, 60, 90] km s^{−1} (purple, turquoise, orange, respectively). As expected by the α(R) trends shown in Fig. 9, for n ≥ 1, we see that for most radii, the pressure support implied by α_{self − grav} results in lower v_{rot} than for α(n) (although at the smallest radii, the inverse holds). In some cases, the magnitude of σ_{0} combined with the form of α(R) additionally predict disk truncation within the range shown, although truncation generally occurs at smaller radii for α_{self − grav} than for α(n). 

In the text 
Fig. 11. Composite pressure support correction, α_{tot}(R), for gas distributed in a composite disk+bulge system (with n_{disk} = 1, n_{bulge} = 4), for a range of B/T (colors) and R_{e, bulge}/R_{e, disk} ratios (dash length). For the limiting cases, we recover the profiles shown in Fig. 9: B/T = 0 has α_{tot} = α(n = 1) (black solid line), while B/T = 1 has α_{tot} = α(n = 4) but with different radial scaling, owing to the different adopted R_{e, bulge}/R_{e, disk} ratios (colored lines). For the cases with 0 < B/T < 1, the bulge contribution modifies the α(n = 1) profile at both small and large radii, leading to larger α_{tot} in inner regions and smaller α_{tot} in the outskirts (R/R_{e, disk} ≲ , ≳ 1 − 2). At fixed B/T, the deviation from the disk α(n = 1) in the center (R/R_{e, disk} ≲ 1 − 2) is larger for smaller R_{e, bulge}/R_{e, disk}, while at large radii the deviation is larger for larger R_{e, bulge}/R_{e, disk}. For reference, we mark R_{e, bulge}/R_{e, disk} with vertical light grey lines, and also show α_{self − grav} (grey dashdotted line). 

In the text 
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