Galaxy masses

Galaxy masses play a fundamental role in our understanding of structure formation models. This review addresses the variety and reliability of mass estimators that pertain to stars, gas, and dark matter. The different sections on masses from stellar populations, dynamical masses of gas-rich and gas-poor galaxies, with some attention paid to our Milky Way, and masses from weak and strong lensing methods all provide review material on galaxy masses in a self-consistent manner.

Figure 1

Time evolution of the exponential (upper panel) and 501 (lower panel) star formation histories (solid curves). The dotted curve is a Sandage-style burst of star formation in which 10% of the total mass of stars are formed. From 361.

Figure 2

The stellar bolometric $M^{*}/L$ and $M^{*}/L$ in various bands ($B$, $V$, $K$) as a function of age for SSPs (single-burst models) and the different indicated metallicities. All models for Kroupa IMF, except for the thin solid line in the top-left diagram for a Salpeter IMF. The stellar mass $M^{*}$ accounts for stellar mass losses, as in 383, 384. See also text for details. From 384.

Figure 3

Evolution of the stellar mass fractions for stellar populations with the same total initial mass (normalized to $1M_{\odot }$) and different initial mass functions. “Bottom-heavy” and “Top-heavy” are extreme cases of single-sloped IMFs with exponents 3.5 and 1 in the notation in which the Salpeter slope is 2.35. $M^{*}$ evolves because stars die progressively and leave remnants with mass lower than the initial mass. Also shown are the fractions of $M^{*}$ in remnants, namely, white dwarfs (WD), neutron stars (NS), and black holes (BH). Based on 384 models.

Figure 4

Trends of 384 SSP models with various ages and metallicities, and a Salpeter IMF. $M^{*}/L$ values in $B$ and $K$ bands (top to bottom) are shown vs ($B\mathrm{\text{−}}R$), ($I\mathrm{\text{−}}K$), and the $\mathrm{H}{\delta }_A$ line index (left to right). The latter models are from 566; colored lines connect models for the $[\alpha /\mathrm{Fe}]$ ratio of $+0.3$ and solar-scaled models. Models with the same age (in Gyr) are connected with dashed lines, models with the same metallicity with solid lines ($Z=0.001$, 0.01, 0.02, 0.04). Other colored lines highlight solar metallicity ($Z=0.02$).

Figure 5

Trends in $M^{*}/L$ ratios using exponentially declining star formation rate models. The models are the same as in Fig. 4, except that instead of a single burst an exponentially declining star formation rate is used observed at age 12 Gyr after star formation began. Models with the same $e$-folding time scale $\tau$ are connected with dashed lines, where positive $\tau$ stands for the decreasing star formation rate with time, negative $\tau$ for the increasing star formation rate, and $\tau =\infty$ is a constant star formation rate model. Solid lines connect models with equal indicated metallicity, where the colored line highlights solar metallicity. Based on 384 models.

Figure 6

$M^{*}/L$ ratios for exponentially declining star formation rate models with a different final age. The models are the same as in Fig. 5, except that the populations are now observed 12, 8.5, 6, 3, 2 Gyr (top to bottom) after star formation commenced. The top ends of the lines are the ends with the smallest positive $\tau$ values, i.e., the oldest average age. For clarity, only the solar metallicity ($Z=0.02$, thick lines) and the $1/20$th solar metallicity ($Z=0.001$, thin lines) are plotted and the dashed lines connecting the same $\tau$ values are not shown as in Fig. 5. In principle these models form similar grids with different offsets. Based on 384 models.

Figure 7

$M^{*}/L$ ratios for exponentially declining star formation rate models with an additional star burst. The starting models without star burst are the same 12 Gyr old exponentially decaying SFR models as in Fig. 5, except that for clarity only the solar metallicity ($Z=0.02$, thick lines) and the $1/20$ solar ($Z=0.001$, thin lines) are plotted. For $\tau$ values $\infty$, 8, 4, 1 a 10% final mass fraction star burst of 0.2 Gyr duration is added that occurred 1, 2, 3, 4, 6 Gyr ago (solid circles connected with dotted lines). Based on 384 models.

Figure 8

$M^{*}/L$ ratios vs ($B\mathrm{\text{−}}R$) for exponentially declining star formation rate models of age 12 Gyr and solar metallicity, with a Salpeter, Kroupa, and the same top-heavy and bottom-heavy IMFs as in Fig. 3.

Figure 9

$M^{*}/L$ ratios for exponential SFR models of age 12 Gyr using SSPs from different models. The models compared are GALAXEV 2003 (98) with Padova 1994 tracks, GALAXEV [2007, an updated version of 98 with different treatment of the TP-AGB phase], PEGASE 2.0 (218), and the FSPS models (140) all based on Padova stellar models and a 119 IMF, and the 384 and the 626 models using a 335. Solar metallicity ($Z=0.02$, thick lines) and $1/5$ solar metallicity ($Z=0.004$, thin lines) models are used, except for the Maraston models, where this metallicity is not available and a lower metallicity model of $Z=0.001$ is plotted. The Chabrier and Kroupa IMFs give essentially the same results for broadband colors; hence by using these IMFs all the model sets can be compared even though some models are available only with either Chabrier or Kroupa IMFs.

Figure 10

Mass profiles from HI rotation curves of NGC 45 (122), $M31$ (123), and $M33$ (Chemin et al., in preparation). Mass profiles on the left were obtained from rotation curves assuming a kinematical model with constant inclination and constant position angle of the major axis (open squares) or a full tilted-ring model (filled squares). The adopted distances are NGC 45 ($D=5.9\mathrm{Mpc}$), $M31$ ($D=785\mathrm{kpc}$), and $M33$ ($D=800\mathrm{kpc}$). The ratio of these mass profiles is shown on the right side where the radial scale has been normalized by the optical exponential disk scale length $h$.

Figure 11

Simulated velocity curves for a dwarf galaxy. The solid curve shows the spherical averaged circular velocity $(G{M}_{\mathrm{tot}}/r{)}^{1/2}$. The long-dashed curve is for the azimuthally averaged rotation velocity of gas particles. The stellar rotation velocity is shown by the dotted curve. Curves for the gas and stars are substantially below the circular velocity for $r<3h$, where $h$ is the disk scale length. From 599.

Figure 12

Mass models for three spiral galaxies with a range of peak circular velocity. The $X$ axis is in units of the optical disk scale length $h$. The velocity on the $Y$ axis is normalized by the maximum observed orbital velocity. There are four free fitting parameters: ${{\rm Y}}_R$ is the stellar $M/L_{*}$ (here in the $R$ band) and $\alpha$, $c$, and $V_{200}$ describe the NFW dark-matter halo [see 196 for details]. Variations in the fitting parameter can result in the same data-model residuals, as gauged by the model residuals $\Delta V$ (shown in the lower windows) and the overall ${\chi }^2$-square statistic (shown above each figure column). The model decompositions in each column (two per galaxy) have the same overall ${\chi }^2$-square statistic, shown at the top. Mass models are thus nonunique. See text for details.

Figure 13

Mass model for the bright spiral galaxy NGC 2403 with an adiabatically contracted halo model (long-dashed line) or not (initial model shown with dotted line). The disk and gas profiles correspond to the contracted final model. The data-fit residuals $\Delta V$ are shown in the lower part. The ratio of dark to baryonic mass within the optical disk of a galaxy can increase by almost 40% within $V_{2.2}$ if AC is invoked. The observed velocity data (black points) are from 64. See text for details about the modeling technique.

Figure 14

Rotation curve, as determined from the analysis of BHB stars by 664, the terminal velocity method by 371, and the analysis of Hii regions-reclection nebulae by 87. The upper panel shows the model predictions from 665 for the total rotation curve (solid) and the contributions from the disk (dashed), bulge (long dashed), and halo (dotted). The lower panel shows the WPD model.

Figure 15

Data-model comparison of six fast-rotator galaxies (previously classified as either E’s and S0’s) using the “Jeans anisotropic MGE” (JAM) method. From top to bottom: bi-symmetrized observations of $V_{\mathrm{rms}}\equiv \sqrt{V^2+{\sigma }^2}$, model of the same, bi-symmetrized observations of $V$, model of $V$. The contours show the isophotes. The models generally agree with the original nonsymmetrized data within the statistical errors. From 104.

Figure 16

Systematic variation of the stellar IMF in ETGs. The six panels show the ratio between the $(M/L{)}_{\mathrm{stars}}$ of the stellar component, determined using dynamical models, and the $(M/L{)}_{\mathrm{Salp}}$ of the stellar population, measured via stellar population models with a Salpeter IMF, as a function of $(M/L{)}_{\mathrm{stars}}$. The solid lines are a local plot smoothing (LOESS) nonparametric regression to the data. Colors indicate the galaxies’ stellar velocity dispersion ${\sigma }_e$, which is related to the galaxy mass. The horizontal lines indicate the expected values for the ratio if the galaxy had (i) a Chabrier IMF (dash-dotted line); (ii) a Kroupa IMF (dashed line); (iii) a Salpeter IMF ($x=-2.3$, solid line) and two additional power-law IMFs with (iv) $x=-2.8$ and (v) $x=-1.5$, respectively (dotted line). Different panels correspond to different assumptions for the dark-matter halos employed in the dynamical models as written in the black titles. A curved relation is clearly visible in all panels. From 109.

Figure 17

Dark-matter fraction vs physical radius in NGC 3379 [assuming $R_e={47}^{\prime \prime }$ and Sérsic index $n=4.74$, following 187]. The light upper limit and light curves, respectively, show the Jeans (PNe) and orbit-modeling solutions of 486 ($\mathrm{\text{stars}}+\mathrm{PNe}$), while the other upper limit is the DF modeling (stars) of 334. The medium-thickness dark curve shows the spherical Jeans solution ($\mathrm{\text{stars}}+\mathrm{PNe}$) with double the number of PNe (187). The vertically shaded region gives the limits of NMAGIC orbit modeling ($\mathrm{\text{stars}}+\mathrm{PNe}$) (180). The lower limits show the orbit modeling (stars) by 641 and the isotropic Jeans analysis (globular clusters) of 457. The horizontal- and oblique-shaded regions give the predictions (177) from equal-mass merger $\mathrm{SPH}+\mathrm{\text{cooling}}+\mathrm{\text{feedback}}$ simulations (154) of, respectively, gas-poor and gas-rich spirals embedded in dark-matter halos. The curve is from the satellite kinematics study of 657, adopting the mean of their third and fourth stellar mass bins for red hosts (the stellar mass of NGC 3379 is in between), dashes for the extrapolation within the smallest satellite radii analyzed, while the shaded region represents the $1\sigma$ confidence from their Monte Carlo Markov chain analysis.

Figure 18

Very large telescope and FLAMES velocity measurements for individual stars along the line of sight to the Sculptor dwarf spheroidal galaxy (dSph). Top panel: Line-of-sight velocities in the galactic standard of rest vs projected radius for probable members to the Sculptor dSph (the filled and open squares show probable members with metallicity $[\mathrm{Fe}/\mathrm{H}]>-1.5$ and $<-1.7$, respectively) and for probable nonmembers (crosses). The region of probable membership is indicated by the two horizontal dotted lines, while the dashed line indicates the systemic velocity of the galaxy. Bottom panel: Line-of-sight velocity dispersion profiles for the metal-rich ($[\mathrm{Fe}/\mathrm{H}]>-1.5$) and metal-poor stars ($[\mathrm{Fe}/\mathrm{H}]<-1.7$), as shown by the filled and open squares with error bars, respectively. The solid and dashed lines show the LOS velocity dispersion profiles for the best-fitting pseudoisothermal (cored) and NFW (cusped) dark-matter models. Adapted from 37.

Figure 19

Ensemble averaged tangential shear plotted as a function of distance from the lens using $334{\mathrm{deg}}^2$ of $r^{\prime }$ data from the RCS2 (622). The lenses are selected to have apparent magnitudes $19.5<m_r<21.5$ and the sources $22<m_r<24$. For reference the best-fit NFW profile is also drawn, which shows that on scales beyond $1^{\prime }$ ($\sim 300\mathrm{kpc}$) the clustering of the lenses becomes important.

Figure 20

Joint constraints on the scale radius $r_s$ and mass $M_{200}$ (and corresponding rotation velocity $V_{200}$ for a galaxy with an NFW profile and a fiducial luminosity $L_B={10}^{10}{h}^{-2}{L}_{B\odot }$). The contours indicate the 68%, 95%, and 99.7% confidence on two parameters jointly. The dotted lines indicate the predictions from the numerical simulations by NFW (272). Our updated version of this figure shows expectations from works by 430 and 191.

Figure 21

A subsample of SLACS lenses in false color. Each panel shows the data on the left and a model of the system on the right. From Adam Bolton.

Figure 22

The logarithmic density slopes of 58 SLACS early-type galaxies (thin solid curves). The filled curve is the joint posterior probability distribution of the average density slope of the sample. The histogram indicates the distribution of median values of the density slopes. The dotted Gaussian curve indicates the intrinsic scatter in ${\gamma }_{\mathrm{LD}}^{\prime }$ (see text). We assume a Hernquist luminosity-density profile and no anisotropy (i.e., ${\beta }_r=0$). The small dashes indicate the shift in average density slope for ${\beta }_r=+0.50$, $+0.25$, $-0.50$, and $-0.25$ (left to right), respectively. Note the reversal of the ${\beta }_r=-0.50$ and $-0.25$ dashes. The vertical solid line and gray region indicates the best-fit value and 68% C.L. interval, respectively, of the average density derived from scaling relations.

Figure 23

Top: Comparison between stellar mass in the cylinder of radius equal to the Einstein radius as inferred from lensing and dynamical models ($x$ axis) and that inferred from fitting stellar population synthesis models to the observed spectral energy distribution ($y$ axis). The solid line indicates the identity. Stellar population synthesis models by 98 are assumed together with an informative metallicity prior to 230. Bottom: Template mismatch parameter $\alpha \equiv M_{*,\mathrm{Ein}}^{\mathrm{LD}}/M_{*,\mathrm{Ein}}^{\mathrm{SPS}}$ for Salpeter IMF as a function of lensing velocity dispersion (left), stellar velocity dispersion (center), and $V$-band luminosity corrected to $z=0.2$. A tentative positive trend with velocity dispersion is observed (solid line). The dashed line represents the trend expected for a universal Salpeter IMF.

Figure 24

Relations between the projected dark-matter fraction within half of the effective radius and $M_{*}/r_e$. Gray points are for a Salpeter IMF and black points are for a Chabrier IMF.

Figure 25

Overview of the lens system the “Clone.” This false- color image was created from HST and WFPC2 images through filters F450W, F606W, and F814W.

Figure 26

Illustration of pixelized reconstruction of the source surface brightness and lens potential corrections of the lens system shown in Fig. 25 using the gravitational imaging technique. The top-left panel shows the data, consisting of the surface brightness of a highly distorted lensed source. The top-middle panel shows the model surface brightness, while the top-right one shows the image residuals (data model). The bottom-left panel shows the source surface brightness distribution reconstructed in the source plane (i.e., after “delensing”; note the zoomed-in angular scale). The bottom-middle panel shows the corrections to the gravitational potential, with respect to a smooth simply parametrized mass distribution. The bottom-right panel shows the inferred convergence, i.e., the projected surface mass distribution. Note the peak at the lower-right corner of the image corresponding to the satellite responsible for the curvature in the arc.

Figure 27

Gravitational imaging analysis of the Jackpot gravitational lens system. The panels are as in Fig. 25. Note the convergence peak in the top-right portion of the bottom-right panel, corresponding to the detected substructure.