The galaxy size to halo spin relation of disk galaxies in cosmological hydrodynamical simulations

The IllustrisTNG project (Pillepich et al., 2018b; Springel et al., 2018; Nelson et al., 2018; Marinacci et al., 2018; Naiman et al., 2018) is a suite of cosmological magneto-hydrodynamic simulations, which was performed with the magneto-hydrodynamic moving mesh code AREPO (Springel, 2010). The IllustrisTNG project assume $\Omega_{m}=0.3089$, $\Omega_{b}=0.0486$, $\Omega_{\Lambda}=0.6911$,$h=0.6774$, $n_{s}=0.9667$ and $\sigma_{8}=0.8159$ (Planck Collaboration et al., 2014). In this work, we use the tng100-1 project with a box size about 110Mpc. We refer the reader for the detailed galaxy formation models of the IllustrisTNG simulations to Weinberger et al. (2017) and Pillepich et al. (2018a).

The auriga project (Grand et al., 2017) comprises a suite of 30 zoom-in cosmological simulations of Milky Way-sizes haloes and their surroundings. The parent haloes in auriga were selected from a dark matter only simulation eagle (L100N1504) (Schaye et al., 2015). Similar to the tng100-1, the auriga projects were performed with the magneto-hydrodynamic moving mesh code AREPO (Springel, 2010), but assume slightly different Cosmological parameters, $\Omega_{m}=0.307$, $\Omega_{b}=0.048$, $\Omega_{\Lambda}=0.693$, $h=0.6777$, $n_{s}=0.9611$ and $\sigma_{8}=0.829$ (Planck Collaboration et al., 2014).

The eagle project (Schaye et al., 2015; Crain et al., 2015) is a suite of cosmological hydrodynamic simulations, which were performed with a version of the N-body Tree-PM smoothed particle hydrodynamics (SPH) code GADGET-3 by Springel et al. (2005). The cosmological parameters adopted in the eagle project are $\Omega_{m}=0.307$, $\Omega_{b}=0.048$, $\Omega_{\Lambda}=0.693$,$h=0.6777$, $n_{s}=0.9611$ and $\sigma_{8}=0.829$ (Planck Collaboration et al., 2014). In this work, we use the Ref-L0100N1504 run which has a volume of $(100Mpc)^{3}$.

The apostle project (Sawala et al., 2016; Fattahi et al., 2016) performed a suite of cosmological hydrodynamic zoom-in simulations of 12 volumes selected to match the kinematics of Local Group. High-resolution regions of the apostle were selected from dark matter only simulation DOVE which evolved a cosmological volume of ($100Mpc)^{3}$. The apostle project was performed with the same code GADGET-3 as eagle, and run with three different resolutions: low(L1), medium(L2), and high(L3). Since only two volumes have been run at high-resolution (L3) in apostle, we use the medium-resolution (L2) data in this work. The cosmological parameters in apostle simulation adopt the result of WMAP-7, namely $\Omega_{m}=0.272$, $\Omega_{b}=0.0455$, $\Omega_{\Lambda}=0.728$,$h=0.704$, $n_{s}=0.967$ and $\sigma_{8}=0.81$ (Komatsu et al., 2011).

The table 1 summarize the typical individual particle mass and softening length for all the above simulations.

In all the above simulations, dark matter haloes are identified with friends-of-friends (FOF) algorithm (Davis et al., 1985) and subhaloes are subsequently identified with the SUBFIND algorithm (Springel et al., 2001; Dolag et al., 2009).

In order to reliably measure the sizes of the simulated galaxies, we include all central galaxies containing at least $250$ ²²2We have also selected galaxies that have at least $1000$ stellar particles (Tacchella et al., 2019) and found the result remains are qualitatively similar. stellar particles and host halo mass satisfy $\log{(M_{200}/{~{}\rm M_{\odot}})}<12.3$. These galaxies span almost four orders of magnitude in stellar mass and reside in a variety of environments. We further discard the galaxies contaminated by low-resolution particles in the zoom-in auriga and apostle simulations. We have excluded all the satellites but only use the central galaxies in this study. The final galaxy sample contains 19315 galaxies from the tng100-1, 282 galaxies from the auriga, 12327 galaxies from the eagle, and 408 galaxies from the apostle.

We define the morphology of each galaxy of the above galaxy sample by introducing the $\kappa$ parameter defined as the ratio of rotational kinetic energy $K_{rot}$ to total kinetic energy $K$ for a galaxy (Sales et al., 2012), written as

Where $\hat{\boldsymbol{L}}$ is the unity total angular momentum vector of stellar components. The $m_{i}$, $r_{i}$ and $v_{i}$ are mass, position vector to centre, velocity vector to centre for stellar particle $i$, respectively.

For each galaxy in our sample, we calculate its $\kappa$ parameter with all star particles within 2 times of its half-stellar-mass radius, $2r_{1/2}$. A galaxy is classified as a disk (or spheroidal) galaxy if its $\kappa>(<)0.5$. Note, $\kappa$ is a definition of morphology according to kinematics and correlates strongly with the axial ratios of a galaxy. In Figure 1 we show $b/c$ versus $a/b$ of our galaxy sample in different simulations. Here the axial ratios of galaxies are obtained by diagonalizing the inertia tensor matrix,

Where $x_{i}$ is the spatial position for particle $i$ and $x_{c}$ is the position with the minimal gravitational potential for the galaxy. We calculate the inertia matrix by the stellar components within twice the stellar half mass radius, $a<b<c$ are eigenvalues of the inertia tensor matrix $I_{\alpha\beta}$.

The disk galaxies are shown as blue dots and spheroidal galaxies are shown as red ones. As can be seen clearly that the classification of galaxy morphology type with $\kappa=0.5$ is reasonable. We also tested other critical values for $\kappa$(from $0.4$ to $0.5$), and it had little effect on the final correlation results. We summarize the number of various samples in the table 2. It is worth noting that our kinematic morphological classification differs from the standard photometry-based method used in observation, with the two showing a moderate correlation with considerable scatter (e.g. Abadi et al., 2003; Scannapieco et al., 2010).

For each halo in our sample, we calculate its dimensionless spin parameter $\lambda$ using the formula given by (Bullock et al., 2001).

Previous works have shown that the distribution of halo spin parameter $\lambda$ is independent of halo mass and follow a log-normal distribution with the mean value $\left<\lambda\right>\sim{0.03-0.04}$ and standard deviation $\sigma_{\log_{10}\lambda}\sim{0.2-0.3}$ (Macciò et al., 2007; Bett et al., 2007; Jiang et al., 2019). In figure 2, we show the cumulative halo spin parameter distributions of our halo sample of all simulations used in this study without morphology cut, results for different simulations are distinguished with different colors as indicated in the label. Mean values and standard deviation of spin parameters of different simulations are also shown on the label. Clearly, these results are consistent with each other and with previous works.

Compared with observations, we adopt a different morphology definition method ($\kappa$) here. In Figure 3 we present the size-stellar mass relation of all simulated galaxies at $z=0$ without morphology cutting, and then fairly compare the results with observations. The results for the auriga and tng100-1 are shown in the left panel and those for the apostle and eagle are shown in the right panel, results for different simulations are distinguished with different colors as shown in the label. The shaded areas show $1\sigma$ scatters of each simulation, and the median values are shown with dashed lines. Here the size of a simulated galaxy is defined as the half-mass radius within which the enclosed stellar mass is half of the whole galaxy. In all simulations, galaxy sizes increase with increasing galaxy stellar masses with $r_{1/2}\sim 1~{}\rm kpc$ for low-mass dwarf galaxies, and $r_{1/2}\sim 5~{}\rm kpc$ for MW-mass galaxies. We find overall a good agreement between the full-box simulations (tng100-1 and eagle) and their zoom-in counterparts (auriga and apostle) at the mass where the two simulations overlap. For masses $M_{\star}\leq 10^{10.5}{~{}\rm M_{\odot}}$, the size-mass relation of the galaxies in tng100-1 and eagle is almost flat, while the slope of the zoom-in galaxies in apostle or auriga is much steeper, the small difference in the slope is due to the effect of simulation resolution on the size of the galaxies as studied in prior literature (e.g. Ludlow et al., 2019; Pillepich et al., 2018b).

We then compare the size-mass relations of simulated galaxies with an observational study by Somerville et al. (2018) (blue dots with error bars ). The conversion of the observational projected semimajor half light radius $r_{e,2d}$ into $r_{1/2}$ involves two factors,

The projection correction factor $f_{p}=1(0.68)$ and the light to mass weighting factor $f_{k}=1.2(1.15)$ for disk (spheroidal) galaxies in Somerville et al. (2018). The observational results clearly also show an increasing slope for galaxies with stellar mass. At masses, $M_{\star}\geq 10^{9.5}{~{}\rm M_{\odot}}$, the agreement between tng100-1 and observations is rather good (see the left panel in Fig. 3), consistent with the study by Genel et al. (2018). auriga uses the same hydrodynamics solver and similar subgrid physics with tng100-1. Thus, the sizes of auriga central galaxies that have stellar mass, $M_{\star}\geq 10^{10.5}{~{}\rm M_{\odot}}$, also agree with the observations.

For galaxies in eagle and apostle (right panel in Fig. 3), the sizes are systematically about $0.17$ dex larger than the observed values, whereas Furlong et al. (2017) who uses the same eagle data found a better agreement with observation than our result. The discrepancy could be due to the fact that Furlong et al. (2017) selected galaxies in a redshift bin of $\Delta z=0.5$ while we only make use of galaxies at $z=0$, and the inclusion of high-redshift galaxies could reduce the median of the galaxy sizes. A secondary effect is a different definition of disk galaxies. The final effect is that our definition of size $r_{1/2}$ is typically larger than if we take into account the stellar particles within a specific spherical aperture. Furlong et al. (2017) adopted an aperture of radius, $r=100$ kpc, to exclude the stellar particles that belong to the galaxy by the subhalo but are located far out. However, the aperture measurements only affect very high-mass galaxies with $M_{\star}\geq 10^{10.5}{~{}\rm M_{\odot}}$ which is larger than most of our selected galaxies. In addition, the results for passive and active galaxies was separated when Furlong compared eagle with observations.

In Figure 4, we show $r_{1/2}/R_{200}$ of each simulated galaxy ($\kappa>0.5$) of our sample versus spin parameter $\lambda$ of its host halo. The upper panels show results for Milky way sized galaxies and the bottom show results for dwarfs, results from different simulations are shown with different symbols as indicated in the label. Here we follow (e.g. Wang et al., 2015a, 2020) to define Milky way sized galaxies as the galaxies whose halo masses are in the range, $M_{\rm 200}\in[0.5,2]\times 10^{12}{~{}\rm M_{\odot}}$, and dwarfs as those with halo masses ranging from $M_{\rm 200}{\leq}1.5{\times}10^{11}{~{}\rm M_{\odot}}$. The thick black dots indicate the median values of each $\lambda$ bin in the tng100-1 and eagle simulations. Error bars show the 16th to 84th percentile of the size ratios. There is a strong correlation between the $r_{1/2}/R_{200}$ and $\lambda$ for $\lambda\geq 0.01$ in the tng100-1 and auriga, while there is very weak or no correlation below the value. Interestingly, the $\lambda$-size relation is much weaker in the eagle and apostle simulations. Table 3 summarizes the values of the Spearman correlation coefficient $\rho_{s}$ of all simulations.

The purple and blue dashed lines in all panels show predictions of Mo et al. (1998) models, assuming $f_{j}=1$ and $f_{j}=0.5$, respectively. Interestingly, MMW98 model with $f_{j}=0.5$ agrees reasonably with tng100-1 and auriga for the MWs samples, while the agreement is worse for dwarfs, even though the auriga dwarfs seems to agree with MMW98 with $f_{j}=1$. It is noticeable that the $r_{1/2}/R_{200}$ of the auriga disk dwarfs tends to be overall larger than those of the tng100-1. The reason may be due to that the size of auriga sample is smaller than the tng100-1. For dwarf-sized dark matter haloes in the eagle and apostle, the size–$\lambda$ relation is almost null. We have to note that the slope and amplitude of the size–$\lambda$ relation may also rely on the halo concentration and angular momentum retention factor $f_{j}$ (see appendix A and B respectively). Galaxies with lower concentration have larger size in MW-mass sample and galaxies with higher $f_{j}$ have larger size in all sample.