© The Authors 2024

1. Introduction

The size and morphology of galaxies provide valuable insights into the formation of galaxies and the accumulation of stellar mass. In the standard picture of disk galaxy formation (e.g., White & Rees 1978; Fall & Efstathiou 1980), disk galaxies are believed to form as baryons cool inside dark matter haloes, which grow through gravitational instability and acquire angular momentum from cosmological tidal torques (e.g., Hoyle 1951; Peebles 1969; Doroshkevich 1970; White 1984). According to this paradigm, the baryons inherit the same distribution of specific angular momentum as the dark matter, and this conservation is maintained during the cooling process, except when large spheroids form. In this picture, baryonic matter settles into an exponential disk in centrifugal equilibrium. The size of this disk is largely determined by stellar mass M_⋆ and specific angular momentum j_⋆. Analytical models based on these assumptions have been successful in producing disk sizes for a given M_⋆ that align reasonably well with observations (e.g., Dalcanton et al. 1997; Mo et al. 1998; Dutton et al. 2007; Somerville et al. 2008). The size r_e of a disk galaxy is proportional to the virial radius r_vir of its parent dark matter halo with a form as r_e ∝ λr_vir, as presented by the standard framework of Mo et al. (1998). Here the spin parameter $\lambda =j_{\mathrm{h}}/(\sqrt{2}{v}_{\mathrm{vir}}{r}_{\mathrm{vir}})$ is a dimensionless parameter that is often used to characterize the specific angular momentum of dark matter halos j_h, and v_vir is the virial velocity of the halo.

The debate surrounding the mass and structural assembly of galaxies is still highly contested. Modern advanced cosmological hydrodynamic simulations, such as EAGLE simulations (Schaye et al. 2015), IllustrisTNG (TNG hereafter), SIMBA (Davé et al. 2019), and NewHorizon (Dubois et al. 2021), have successfully produced galaxies with realistic morphologies across a wide mass range, and therefore a powerful tool for gaining profound insights into physical correlations. It is well established that galaxy sizes are related to both stellar and active galactic nucleus (AGN) feedback (summarized by Somerville & Davé 2015; Naab & Ostriker 2017). Therefore, some previous studies found either no correlation or only a weak correlation between the residuals in r_e/r_vir and the halo spin parameter (Teklu et al. 2015; Zavala et al. 2016; Zjupa & Springel 2017; Desmond et al. 2017). However, there are also clear indications that the properties of galaxies are heavily influenced by their parent dark matter halos in numerous aspects. For instance, Zavala et al. (2016) and Lagos et al. (2017) identified a noteworthy link between the specific angular momentum evolution of the dark matter and baryonic components of galaxies in EAGLE simulations. Yang et al. (2023) demonstrated that disk-dominated galaxies selected via kinematics in TNG and AURIGA (Grand et al. 2017) reproduce a correlation between galaxy size and the spin parameters of their dark matter haloes. Similarly, Desmond et al. (2017) found a weak correlation between galaxy size and the host halo spin parameter in the EAGLE simulation. Liao et al. (2019) uncovered a strong correlation between sizes and host halo spin parameters for field dwarf galaxies in the AURIGA simulation. Zanisi et al. (2020) contended that the scatter in the galaxy–halo size relation for late-type galaxies could be explained by the scatter in stellar angular momentum rather than in the halo spin parameter. Jiang et al. (2019) also identified a weak correlation between size and spin in the VELA and NIHAO zoom-in simulations, but instead found a significant correlation between size and NFW halo concentration.

In particular, Du et al. (2022) showed that the TNG simulations (Marinacci et al. 2018; Nelson et al. 2018, 2019; Naiman et al. 2018; Pillepich et al. 2018, 2019; Springel et al. 2018) have achieved significant success in replicating a $j_{\star }\propto M_{\star }^{0.55}$ relationship consistent with observations (see also e.g., Rodriguez-Gomez et al. 2022). A more thorough investigation of the disk galaxies from TNG revealed that this scaling relation arises as a consequence of three physically meaningful scaling relations, involving j_⋆, M_⋆, total mass M_tot, and total specific angular momentum j_tot: (a) the $j_{\mathrm{tot}}\propto M_{\mathrm{tot}}^{0.81}$ relation deviates notably from the prediction based on tidal torque theory that $j_{\mathrm{tot}}\propto M_{\mathrm{tot}}^{2/3}$; (b) the stellar-to-halo mass ratio consistently increases in log-log space according to $M_{\mathrm{tot}}\propto M_{\star }^{0.67}$; and (c) angular momentum is approximately conserved (with a certain factor) during galaxy formation; that is, j_tot ∝ j_⋆. Du et al. (2022) suggest that the assembly of disk galaxies in the TNG simulation follows a consistent framework akin to that proposed by Mo et al. (1998), but some adjustments, potentially attributable to baryonic processes, should be considered for a more precise understanding.

It is crucial to disentangle the influence of external factors to comprehend the formation of galaxies in terms of their properties and structure. Indeed, both galaxy size and mass growth are significantly influenced by external processes, especially major mergers. Specifically, gas-poor mergers tend to increase galaxy size, whereas gas-rich mergers lead to a decrease in size (Covington et al. 2008, 2011; Naab et al. 2009; Hopkins et al. 2009; Oser et al. 2012). Consequently, the variation in galaxy size for these entities would be expected to correlate with both the frequency of mergers experienced by a galaxy and the gas content of those mergers which means nurture. Covington et al. (2008, 2011) and Porter et al. (2014) presented findings on the size evolution of bulge-dominated galaxies by incorporating the size growth observed in binary merger simulations into a semi-analytic model, which exhibits good overall agreement with observational data (see also Shankar et al. 2013). Romanowsky & Fall (2012) argued that the Hubble sequence of galaxy morphologies is a sequence of increasing angular momentum at any fixed mass. Obreschkow & Glazebrook (2014) introduce the j_⋆ − M_⋆–morphology relation wherein morphology is quantified by the mass fraction of bulges. Rodriguez-Gomez et al. (2022) identified a j_⋆ − M_⋆–morphology relation in TNG, albeit with a notable degree of scatter. This picture suggests galaxy mergers, particularly “dry” major mergers, can give rise to the parallel trajectory observed in the j_⋆ − M_⋆ diagram by effectively disrupting the disk structures of galaxies. As a consequence, earlier-type galaxies generally exhibit weaker rotational characteristics and have massive bulges in morphology.

Moreover, extensive research indicates that the present-day profile of a galactic disk is not primarily determined by the initial conditions, even in the absence of mergers. Simulations pertaining to the formation of disk galaxies consistently reveal that the distribution of stellar birth radii often exhibits substantial deviations from an exponential profile (Debattista et al. 2006; Roškar et al. 2008, 2012; Minchev et al. 2012; Berrier & Sellwood 2015; Herpich et al. 2015). This deviation occurs because stars do not remain confined to their original orbits, but exert minimal impact on the overall angular momentum of the galaxy disk. Both analytical arguments and numerical experiments have demonstrated that the angular momenta of individual disk particles are influenced by transient nonaxisymmetric perturbations, such as spiral arms and bars, leading to a process commonly referred to as “churning” or “shuffling” (e.g., Sellwood 2014, and references therein). This phenomenon is commonly known as radial migration, and can significantly change the profile of a galactic disk.

Isolating the internal and external processes can uncover the underlying mechanisms of the assembly of galaxies. This duality is underscored in the studies of Du et al. (2020, 2021), where the authors employ a fully automatic kinematical method to decompose the kinematic intrinsic structures of TNG galaxies (Du et al. 2019). The mass ratio of kinematically derived stellar halos is sensitive to external impacts and thus can be used to quantify the effect of external processes in galaxies (Sect. 2). The conceptual significance of the “nature–nurture” framework – within the context of internal versus external factors – is illustrated in Sect. 2.2. Galaxies that have experienced minimal external effects adhere to the fiducial j_⋆ − M_⋆–size relation, which is detailed in Sects. 3 and 4. This scaling relation is primarily governed by universal and natural physical processes, while nurture, as described above, plays a minor role. In Sect. 5, we summarize our results.

2. Sample selection and data extraction

2.1. The IllustrisTNG simulation

The TNG Project is a suite of cosmological simulations run with the moving-mesh code AREPO (Springel 2010; Pakmor et al. 2011, 2016) that uses gravo-magnetohydrodynamics and incorporates a comprehensive galaxy model (Weinberger et al. 2017; Pillepich et al. 2018). The TNG50-1 run within the TNG suite has the highest resolution, consisting of 2 × 2160³ initial resolution elements in a comoving box of approximately 50 Mpc. This corresponds to a baryon mass resolution of 8.5 × 10⁴ M_⊙ and a gravitational softening length for stars of about 0.3 kpc at redshift z = 0. Dark matter particles are resolved with a mass of 4.5 × 10⁵ M_⊙, while the minimum gas softening length reaches 74 comoving parsecs. These resolutions enable the accurate reproduction of the overall kinematic properties of galaxies with stellar masses greater than or equal to 10⁹ M_⊙ (Pillepich et al. 2019).

The identification and characterization of galaxies within the simulations are performed using the friends-of-friends (Davis et al. 1985) and SUBFIND algorithms (Springel et al. 2001). Resolution elements including gas, stars, dark matter, and black holes that belong to an individual galaxy are gravitationally bound to its host subhalo. All galaxies in our sample are rotated to the face-on view based on the stellar angular momentum in order to accurately measure their properties.

To determine the positions of galaxies, we employ a centering method that places them at the location corresponding to the minimum gravitational potential energy in all measurements. The bulk velocity of all particles is subtracted. All quantities presented in this paper were computed using the full complement of particles associated with galaxies and subhalos, encompassing all gravitationally bound particles identified through the SUBFIND algorithm. We refrain from imposing any constraints on the radial extent of galaxies when deriving their comprehensive properties. The units of length, j, and mass used throughout the paper are kpc, kpc km s⁻¹, and M_⊙, respectively. We use “log” to denote the logarithm with base 10 throughout.

2.2. Physical meaning of kinematic structures: Data extraction and galaxy classification

We recently developed an automated method called auto-GMM to efficiently decompose the structures of simulated galaxies based on their kinematic phase-space properties (Du et al. 2019, 2020). This method uses the GaussianMixture Module (GMM) of the Python scikit-learn package to cluster the three-dimensional phase space of dimensional parameters that quantify circularity, binding energy, and nonazimuthal angular momentum (Doménech-Moral et al. 2012) into distinct structures. This kind of kinematic method has become a standard way to accurately decompose galaxies (see similar attempts in Obreja et al. 2018; Zana et al. 2022; Proctor et al. 2024). We successfully identified cold disk, warm disk, bulge, and stellar halo structures of TNG galaxies using auto-GMM (Du et al. 2020). The overall disk and spheroidal structures are obtained by summing the stars from the cold+warm disks and the bulge+stellar halo, respectively. Notably, stars within kinematically derived disks are predominantly characterized by strong rotation, exhibiting a mass-weighted average circularity ⟨j_z/j_c⟩> 0.5, where j_z and j_c are the azimuthal and circular angular momentum, respectively. Conversely, the kinematically derived stellar halos share a similar weak rotation (⟨j_z/j_c⟩< 0.5) with bulges but possess looser binding stars than bulges. It is important to emphasize that this decomposition method does not assume that the disk of a galaxy follows an exponential profile, nor does it presuppose that the bulge follows a Sérsic profile.

In Du et al. (2021), we propose the “nature–nurture” picture to understand the structures of galaxies and their evolution. In this paper, “nature” is equivalent to internal processes, while “nurture” is equivalent to external processes. By successfully identifying kinematic structures within galaxies, it becomes possible to establish connections between these structures and either nature (internal) or nurture (external) physical processes. The early phase at redshifts z > 2, which is characterized by chaotic physical processes and gas accretion in the host dark matter halo and protogalaxy, is regarded as one aspect of the nature of galaxies. In the later phase, there is no doubt that long-term evolution is one of the natural processes of galaxies in the absence of significant mergers. Consequently, the nature of galaxies substantially contributes to the formation of both kinematic bulges and disk structures, as demonstrated by Du et al. (2021). In contrast, only kinematic stellar halos are strongly linked to external events – primarily mergers but not exclusively – representing the “nurture” aspect of galaxy evolution.

In this study, our primary focus is to elucidate the influences of internal and external processes on the scaling relations of j_⋆ − M_⋆ and the size of galaxies from TNG50. We use galaxies within the stellar mass range of 10⁹ − 10^11.5 M_⊙ from the TNG50-1 simulation. Figure 1 presents the distribution of the mass fractions of kinematic structures in three stellar mass ranges from top to bottom, as detailed in Du et al. (2021)¹. We proceeded to categorize galaxies into three groups based on their stellar halo mass fractions f_halo, as follows:

Fig. 1. Distribution of the mass fraction of kinematic structures in TNG50 galaxies. The sum of the bar heights is normalized to 1. We then classify galaxies into tiny-halo galaxies, halo-subdominated galaxies, and halo-dominated galaxies by fhalo ≤ 0.2, 0.2 < fhalo ≤ 0.5, and fhalo > 0.5, respectively. The criterion fhalo = 0.2 and 0.5 are marked by the black vertical lines. The mass fraction of spheroids fspheroid is equal to fbulge + fhalo. From top to bottom, we show the galaxies with stellar mass log (M⋆/M⊙)∈[109, 1011.5]. The galaxies with log (M⋆/M⊙)∈[9.5, 10.5] are the most representative sample.

– Tiny-halo galaxies: f_halo ≤ 0.2 selects 997 galaxies. These galaxies are robustly classified as disk galaxies in terms of morphology and can be considered to have formed via internal processes largely unaffected by mergers and other environmental factors. These objects serve as the physical basis for the fiducial scaling relations.

– Halo-subdominated galaxies: 0.2 < f_halo ≤ 0.5 selects 1369 galaxies. The morphological classification of these galaxies is challenging. The existence of a massive stellar halo is a sign that such a galaxy has experienced some kind of external effect(s) which may lead to a scatter in any fiducial scaling relation originating from internal processes.

– Halo-dominated galaxies: f_halo > 0.5 selects 442 galaxies, indicating elliptical galaxy morphology. For these galaxies, any fiducial scaling relation resulting from internal processes may have been disrupted or substantially altered due to the pronounced influence of external processes.

In comparison with the ex situ mass fraction measured in Rodriguez-Gomez et al. (2015), the use of f_halo is more convenient, as it eliminates the need to account for variations in the strength, orbits, and frequency of mergers and close tidal interactions. The mass fraction of spheroids f_spheroid in each galaxy can be computed simply as f_bulge + f_halo.

3. The fiducial j_⋆–M_⋆–h_R plane of bulge- and disk-dominated galaxies based on their exponential nature

It is well known that more massive galaxies are greater in size, although the scatter is large at a given M_⋆ (e.g., Shen et al. 2003; Fernández Lorenzo et al. 2013; Lange et al. 2015; Muñoz-Mateos et al. 2015). TNG has successfully reproduced the relation between stellar mass and half-mass radius R_e within observational uncertainties (e.g., Genel et al. 2018; Huertas-Company et al. 2019; Rodriguez-Gomez et al. 2019, see also the left-most panel of Fig. 2). However, the substantial scatter in galaxy sizes, ranging from 1 kpc to more than 10 kpc, and in j_⋆ (Du et al. 2022; Fall & Rodriguez-Gomez 2023) continue to pose a perplexing challenge. In this study, we isolate the influence of internal processes on the mass–size relation and j_⋆ by selecting the tiny-halo galaxies. The effect of nurture is then revealed by comparing halo-subdominated and halo-dominated galaxies with their counterparts with tiny stellar halos.

Fig. 2. Scaling relations of tiny-halo (blue dots), halo-subdominated (gray dots), and halo-dominated (red dots) galaxies. From left to right, we show the M⋆ − Re relation, the j⋆ − M⋆ relation, rotation κrot, and the mass fraction of disk structures fdisk. The shades of red and blue colors represent the log (Re/kpc) in all panels, showing that larger galaxies have larger j⋆ for a given stellar mass. We perform the linear fitting for galaxies in two mass ranges log(M⋆/M⊙): ∈[9.5, 11.5] (green dotted lines) and [9.0, 9.5] (blue dotted lines), respectively. The j⋆ − M⋆ relation of the disk galaxies from Du et al. (2022) is overlaid in black in the second panel.

3.1. The large scatter of the j_⋆–M_⋆ relation

Conducting a comparative study is crucial in order to differentiate between the effects of internal and external processes on galaxy evolution and the large scatter of the j_⋆ − M_⋆ relation. In the second panel of Fig. 2, we show the j_⋆ − M_⋆ relation for the three types of galaxies defined in Sect. 2: namely tiny-halo galaxies (blue dots), halo-subdominated galaxies (gray dots), and halo-dominated galaxies (red dots). Tiny-halo galaxies in the absence of mergers exhibit almost exactly the same j_⋆ − M_⋆ relation (the blue and green dotted lines fitted in different mass ranges) as disk galaxies selected based on the relative importance of cylindrical rotations (κ_rot ≥ 0.5) as described in Du et al. (2022), corresponding to the black solid line. Specifically, these galaxies follow a scaling relation of $j_{\star }\propto M_{\star }^{0.55}$. This finding suggests that the wide scatter observed around the $j_{\star }\propto M_{\star }^{0.55}$ relation exists regardless of whether external processes have played a significant role in their evolution. Moreover, it is evident that halo-subdominated and halo-dominated galaxies generally possess smaller j_⋆ values for a given M_⋆. It is not surprising that mergers lead to an increase in mass as well as a decrease in angular momentum by destroying disky structures.

The extensive scatter observed in the j_⋆ − M_⋆ relation of tiny-halo galaxies exhibits a distinct correlation with galaxy size, as indicated by the shade of the blue color in the plot. Furthermore, our analysis does not reveal a significant correlation between rotation and R_e in these galaxies, as demonstrated in the third and fourth panels of Fig. 2. The substantial scatter in the j_⋆ − M_⋆ relation primarily stems from the significant variation in galaxy size, which is driven by internal processes, as elucidated in Sects. 3.2 and 4. It is not surprising that both halo-subdominated and halo-dominated galaxies exhibit a clear deviation toward lower j_⋆ compared to the j_⋆ − M_⋆ relation of tiny-halo galaxies, which is consistent with the prediction of the so-called j_⋆ − M_⋆–morphology relation (Sweet et al. 2018; Obreschkow & Glazebrook 2014; Rodriguez-Gomez et al. 2022). This divergence is linked to their noticeably weaker rotation shown in the third and fourth panels. However, the scatter of j_⋆ induced by the nature of galaxies plays a more important role. We can therefore conclude that a pronounced scatter in the j_⋆ − M_⋆ relation is inherently present in the initial conditions of galaxies, shaping the observed j_⋆ − M_⋆ and mass–size relations in the local Universe. On the other hand, external influences induce a systematic offset, further contributing to this inherent variation.

3.2. The fiducial j_⋆ − M_⋆ − h_R relation in nature satisfying the exponential hypothesis

In a realistic scenario of the exponential hypothesis, galaxies consist of both spheroidal and disk components. The angular momentum of a galaxy is largely determined by the mass fraction of its disk components, denoted as f_disk, as spheroidal components exhibit little or no rotation. Here a disk structure includes thin (cold) and thick (warm) disk components. We then have

$$\begin{array}{cc}\hfill j_{\star ,\mathrm{theory}}& =f_{\mathrm{disk}}{j}_{\mathrm{disk}}\hfill \\ \hfill & =\frac{1}{M_{\star }}{\displaystyle \int }2\pi R^2{\mathrm{\Sigma }}_{\mathrm{disk}}(R)v_{\varphi }(R)\mathrm{d}R,\hfill \end{array}$$(1)

where v_ϕ and Σ_disk are the cylindrical rotation velocity and surface density at cylindrical radius R, respectively. Exponential disks exhibit a simple surface density profile described as Σ_disk = Σ_0, diskexp( − R/h_R, theory). The central surface density Σ_0, disk can be expressed as $f_{\text{disk}}{M}_{\star }/(2\pi h_{\text{R},\text{theory}}^2)$. ∫R²exp( − R/h_R, theory)dR integrates from 0 to infinity resulting in $2h_{\text{R},\text{theory}}^3$. The accuracy of estimating j_{⋆, theory} using Eq. (1) hinges on the precise characterization of disk structures by h_R, theory, f_disk, and their rotation curves. We define the factor ϵ = ⟨v_ϕ/v_flat⟩ to quantify the deviation of v_ϕ from the flat rotation curve with a velocity v_flat. Therefore, ϵ has a similar physical meaning to the circularity ⟨j_z/j_c⟩. Through a simple derivation, Eq. (1) gives

$$\begin{array}{cc}\hfill j_{\star ,\mathrm{theory}}& =\frac{2\pi ϵ{\mathrm{\Sigma }}_{0,\mathrm{disk}}{v}_{\mathrm{flat}}}{M_{\star }}{\displaystyle \int }{R}^2{e}^{-R/h_{\mathrm{R},\mathrm{theory}}}\mathrm{d}R\hfill \\ \hfill & =2ϵf_{\mathrm{disk}}{v}_{\mathrm{flat}}{h}_{\mathrm{R},\mathrm{theory}}.\hfill \end{array}$$(2)

This equation is physically robust in cases where galaxies satisfy the exponential hypothesis. It is therefore commonly used as an approximation of j_⋆ (e.g., Fall 1983; Mo et al. 1998) by measuring f_disk in morphology and making corrections by adding asymmetric drift.

It should be noted that Eq. (2) holds when we can accurately measure the mass fraction of disks that possess strong rotation and conform to an exponential distribution. Moreover, the TF relation $M_{\star }\propto v_{\text{flat}}^{\alpha }$ is generally tightly satisfied where the gaseous component is negligible in the local Universe. Observations have consistently shown in previous studies that α varies between 3 and 4 (e.g., Noordermeer & Verheijen 2007; Avila-Reese et al. 2008; Gurovich et al. 2010; Zaritsky et al. 2014; Bradford et al. 2016; Papastergis et al. 2016; Lelli et al. 2019). The TF relation we measured in TNG50 tiny-halo galaxies (black dots in Fig. 3) gives log (v_flat/km s⁻¹) = (0.242 ± 0.002) log (M_⋆/M_⊙)−(0.257 ± 0.023), which is consistent with the observation from Lelli et al. (2016) (gray dots with error bars). We then have the fiducial j_⋆ − M_⋆–size relation based on the theory of the exponential hypothesis:

$$\begin{array}{cc}\hfill \log (h_{\mathrm{R},\mathrm{theory}}/\mathrm{kpc})\simeq & \log (j_{\star ,\mathrm{theory}}/\mathrm{kpc}\mathrm{km}{\mathrm{s}}^{-1})\hfill \\ \hfill & -0.24\log (M_{\star }/M_{\odot })+C_0.\hfill \end{array}$$(3)

Fig. 3. Tully–Fisher relation of galaxies in TNG (black and red dots) and observations. The vflat of TNG galaxies is measured by averaging the flat part (0.05 − 0.2 rvir) of the rotation curve. The observations are adopted from Lelli et al. (2016) assuming the mass-to-light ratio Γ = 0.5 M⊙/L⊙ based on the IMF suggested by Kroupa et al. (2001).

The constant part C₀ is −log (2ϵf_disk)−C_TF where C_TF = −0.26 is the zero point of the TF relation. C₀, which is estimated with f_disk ∼ 0.7 and ϵ ∼ 0.7, is almost constant, at around 0.3. After considering the correction for f_disk, the right-most panel of Fig. 2 gives log f_disk = 0.054 log (M_⋆/M_⊙)−0.674. We then have

$$\begin{array}{cc}\hfill \log (h_{\mathrm{R},\mathrm{theory}}/\mathrm{kpc})\simeq & \log (j_{\star ,\mathrm{theory}}/\mathrm{kpc}\mathrm{km}{\mathrm{s}}^{-1})\hfill \\ \hfill & -0.29\log (M_{\star }/M_{\odot })+C_1,\hfill \end{array}$$(4)

where C₁ = −log (2ϵ)+0.94. Here, ϵ varies in the range of 0.85 − 1.0 and 0.5 − 0.85 for cold and warm disks, respectively, defined by the kinematical method in Du et al. (2020); estimated from the relative mass fraction of cold and warm disks, it is roughly constant around 0.7. A dynamically hotter disk has a smaller ϵ, in which case C₁ is about 0.79. The upper and lower limits can be 0.94 and 0.64 in the cases of ϵ = 0.5 and ϵ = 1, respectively. Such a theoretical j_⋆ − M_⋆ − h_R relation relies on the assumption that disks accurately satisfy the exponential profile and bulges have zero rotation. It is worth emphasizing that the correction of log f_disk is non-negligible, which may induce a deviation of C₁ − C₀ ∼ 0.5 dex when the difference in the factor of the log M_⋆ is ignored.

We examined whether tiny-halo galaxies in TNG50 obey the theoretical j_⋆ − M_⋆ − h_R relation (Eq. (4)). We first performed a surface fitting in the three-dimensional (3D) space using j_⋆, M_⋆, and h_R, morph for tiny-halo galaxies in two mass ranges: log (M_⋆/M_⊙)∈[9, 9.5] and ∈[9.5, 11.5]. Figure 4 then shows the fitting results in 2D, which is convenient for comparison with Eq. (4). h_R, morph is extracted using a 1D two-component (Sérsic bulge + exponential disk) morphological decomposition that has been widely used in observations. Here, we use a 1D bulge–disk decomposition to simplify the analysis, as we have precise knowledge of the face-on surface density map of galaxies in the simulations.

Fig. 4. The j⋆ − M⋆ − hR, morph relation comparing simulations with observations. The units of hR, morph, j⋆, and M⋆ are kpc, kpc km s−1, and M⊙, respectively. The left and right panels show galaxies in two mass ranges log (M⋆/M⊙): ∈[9.5, 11.5] and ∈[9.0, 9.5]. The fiducial j⋆ − M⋆ − hR, morph relation of tiny-halo galaxies, where hR, morph represents the scale length of the disk component defined by morphology, is visualized using a kernel density estimation (KDE) map. For convenience, we adjust the surface fitting to align with y = x. We performed mock measurements of tiny-halo galaxies based on Eq. (2) (black dots). The gray points show the halo-subdominated galaxies for comparison. The observational data points of disk galaxies from Mancera Piña et al. (2021) are represented by green triangles. The dot-dashed lines represent the cases exhibiting offsets of ±0.3 dex.

The fitting result of the fiducial j_⋆ − M_⋆ − h_R relation of tiny-halo galaxies with log (M_⋆/M_⊙)∈[9.5, 11.5] in the TNG50 simulation gives

$$\begin{array}{cc}\hfill \log h_{\mathrm{R},\mathrm{morph}}=& (0.970\pm 0.020)[\log j_{\star }\hfill \\ \hfill & -(0.293\pm 0.015)\log M_{\star }+(0.770\pm 0.123)],\hfill \end{array}$$(5)

where h_R, morph, j_⋆, and M_⋆ are in units of kpc, kpc km s⁻¹, and M_⊙, respectively. The left and right parts of this equation are used as the y and x axes, respectively, in Fig. 4. The fitting result matches Eq. (4) perfectly, suggesting that galaxies evolve in a natural way, obeying the exponential hypothesis. It is worth emphasizing that the kinematic disk structures indeed show some noticeable deviations from simple exponential profiles (see Fig. 12 in Du et al. 2022), while the exponential hypothesis is still valid and explains the overall properties well. This result is not sensitive to stellar mass for massive galaxies but has a clear deviation in less-massive galaxies. The right panel of Fig. 4 gives the fiducial j_⋆ − M_⋆ − h_R relation of dwarf galaxies with log (M_⋆/M_⊙)∈[9, 9.5]:

$$\begin{array}{cc}\hfill \log h_{\mathrm{R},\mathrm{morph}}=& (0.870\pm 0.024)[\log j_{\star }\hfill \\ \hfill & -(0.400\pm 0.043)\log M_{\star }+(1.825\pm 0.387)].\hfill \end{array}$$(6)

The deviation with respect to the more massive galaxies is largely due to the fact that log f_disk (blue dashed line) has a steeper slope and a smaller intercept, as shown in the right-most panel of Fig. 2.

Moreover, the halo-subdominated galaxies (gray dots) exhibit a similar j_⋆ − M_⋆ − h_R relation, albeit with a noticeable degree of scatter. These galaxies clearly deviate toward the left in comparison to the fiducial j_⋆ − M_⋆ − h_R relation depicted in Fig. 4. This deviation is likely due to the reduced j_⋆ from external influences.

In summary, disk structures of TNG galaxies indeed conform to the exponential hypothesis. The significant scatter in the j_⋆ − M_⋆ relation, as seen in Fig. 2, which is inherent in protogalaxies or their host dark matter halos, effectively explains the underlying physical basis for the fiducial j_⋆ − M_⋆ − h_R relation. Consequently, this scatter results in a wide range of galaxy sizes, indicating that the evolution of these galaxies has only been minimally impacted by external influences. External factors play a relatively minor role in shaping the overall mass–size and j_⋆ − M_⋆ scaling relationships in disk galaxies. Moreover, this result suggests that the effect of stellar migrations (e.g., Debattista et al. 2006; Roškar et al. 2012; Berrier & Sellwood 2015; Herpich et al. 2015) also has a minor effect on the overall properties of disk galaxies.

3.3. Deviation of the j_⋆ − M_⋆ − h_R relation between the TNG50 simulation and observations

The fiducial j_⋆ − M_⋆ − h_R relation on the basis of the exponential hypothesis provides a reference point for galaxies that are primarily rotation-dominated. It provides valuable constraints on the physical model used to explain the galaxy’s size. In Fig. 4, we compare the observational data with the fiducial j_⋆ − M_⋆ − h_R relation derived from TNG50. The observational data for j_⋆ used in the present study are sourced from Mancera Piña et al. (2021, green triangles). It is worth emphasizing that all observed galaxies here are in close proximity, allowing relatively reliable measurements of M_⋆ and h_R, morph and in turn enabling a meaningful comparison. The estimation of the mass fraction and h_R, morph of disks is based on the 2D bulge–disk decomposition conducted by Fisher & Drory (2008). This decomposition combines high-resolution Hubble Space Telescope imaging with wide-field ground-based imaging, which helps to minimize uncertainties on h_R, morph and f_disk.

Galaxies in TNG50 follow a similar trend to those in observations, but there is indeed a notable discrepancy between the observed galaxies of Mancera Piña et al. (2021) and the galaxies in TNG50, as illustrated in the left panel of Fig. 4. This discrepancy is smaller in less massive galaxies, as seen in the right panel. It is evident that many galaxies exhibit an offset of approximately > 0.2 dex to the right of the fiducial j_⋆ − M_⋆ − h_R relation. We first investigate the uncertainty on the observations to figure out the potential source of the observed offset. The determination of j_⋆ in Mancera Piña et al. (2021) involves measuring the rotation curve from gas after applying a stellar-asymmetric drift correction. Part of the measurement of j_⋆ (e.g., Romanowsky & Fall 2012) employs slit spectroscopy of both starlight and ionized gas. According to Sweet et al. (2018), the typical uncertainty on j_⋆ is given by |Δj_⋆|/j_⋆ = 0.05 − 0.1, reaching a maximum of 0.32 (∼0.15 dex) for the data from Romanowsky & Fall (2012). Moreover, the uncertainty on M_⋆ is about 0.2 dex, which is estimated based on the uncertainty on the mass-to-light ratio adopting a factor of ∼1.5. To estimate the overall uncertainty on log j_⋆ − 0.3 log M_⋆, we can use the formula $\sqrt{0.{15}^2+{(0.3\times 0.2)}^2}\approx 0.16$. This uncertainty can partially explain the inconsistency between the observational results using Eq. (2) and simulations.

To quantify any potential uncertainty on the measurement of j_⋆, we performed mock measurements based on Eq. (2) (black dots in Fig. 4). The v_flat is measured by the average value of the flat part of the outer edge (0.05 − 0.2 r_vir) of a rotating curve. We do not make any asymmetric drift correction, which will only lead to a negligible offset toward the left side. It cannot therefore explain the deviation between observations and TNG simulations. We confirmed that halo-subdominated galaxies measured using Eq. (2) follow a similar j_⋆ − M_⋆ − h_R relation. This is likely because of the fact that the transformation from disks to stellar halos due to mergers generally induces a minor change in both v_flat and h_R, morph measured in morphological decomposition. It is clear that halo-subdominated galaxies (red dots) follow a similar TF relation to tiny-halo galaxies (black dots), as shown in Fig. 3. Thus, j_⋆ of halo-subdominated galaxies are likely to be significantly overestimated using this method, although this does not affect the results in our study.

The observation may not be able to reflect the physics due to poor statistics and large uncertainty. The 3D surface fitting of observational data from Mancera Piña et al. (2021) gives

$$\begin{array}{cc}\hfill \log h_{\mathrm{R},\mathrm{morph}}=& (0.028\pm 0.089)\log j_{\star }\hfill \\ \hfill & +(0.345\pm 0.130)\log M_{\star }-(0.844\pm 0.661).\hfill \end{array}$$(7)

This result suggests that $h_{\mathrm{R},\mathrm{morph}}\propto M_{\star }^{1/3}$, which is nearly independent of j_⋆. The strong correlation that implies h_R, morph ∝ j_⋆ in the fiducial j_⋆ − M_⋆ − h_R relation disappears when only the observational data are considered. However, there is indeed a substantial uncertainty of ±0.661 dex on the constant part on the right side of Eq. (7), indicating that no satisfactory 3D fitting results can be obtained. Moreover, it is worth mentioning that the deviation between simulations and an integral field spectroscopic (IFS) measurement (Sweet et al. 2018) is even larger. Obreschkow & Glazebrook (2014) noted that j_⋆ measured by Eq. (2) shows systematic variations in comparison to the IFS observations. Many observational issues should be examined in detail. We thus do not compare with the result of IFS measurements in this work. Consequently, we still cannot reach a robust conclusion due to the poor statistics and large uncertainty on the observational data.

3.4. No strong evidence of incorrect galaxy properties in IllustrisTNG simulations

In the previous section, we demonstrate that the disparity between simulations and observations is predominantly due to measurement uncertainties. As of now, we cannot rule out the possibility that this inconsistency is a result of errors in the simulated galaxy properties generated by the IllustrisTNG simulations. Generating galaxies with a faster, flat rotation curve, a smaller disk size, or a larger mass fraction of disk structures in simulations may solve their inconsistency with observations. There is no clear inconsistency in the TF relation, as shown in Fig. 3. Though the observational findings by Lelli et al. (2016, represented by open dots with error bars) are marginally lower compared to the disk galaxies in the TNG50 simulation, this small discrepancy only accounts for a negligible 0.05 dex deviation in the fiducial j_⋆ − M_⋆ − h_R relation. In this section, we further investigate the specific properties of disk galaxies in TNG50 to elucidate the discrepancies in the j_⋆ − M_⋆ − h_R relation observed.

In Fig. 5, we show the mass–h_R relation, comparing the tiny-halo galaxies from TNG50 (represented by blue shaded regions) with those observed (triangles). Additionally, we overlay the late-type galaxies from the SDSS DR7 dataset in black. The selection criterion for late-type galaxies is based on a color threshold of g − r < 0.7, as suggested by Blanton et al. (2003). We adopt the scale length approximated by Simard et al. (2011) and the stellar mass provided by Mendel et al. (2014). It is evident that galaxies in SDSS observations (black histogram and shaded regions) demonstrate a consistent trend with galaxies in TNG50 simulations (blue histogram and shaded regions). The galaxies used in Mancera Piña et al. (2021) include a group of compact massive galaxies as we can see by comparing the green with the black and blue histograms on the right side of Fig. 5. Such galaxies generally have stellar masses of greater than 10^10.2 M_⊙, but compact disks with h_R < 2 kpc. Such galaxies are uncommon. It suggests that the galaxies selected for j_⋆ measurements may be biased towards compact galaxies, and therefore may not be sufficiently representative to draw definitive conclusions.

Fig. 5. Mass–hR, morph diagram. Disk galaxies from TNG50 are represented by the blue-shaded area, while observed disk galaxies from SDSS DR7 are shown in the gray-shaded region. The solid profiles are the median. The shaded areas denote the 1σ envelope, representing the 16th and 84th percentiles. For the SDSS data, we used scale-length values from Simard et al. (2011) and stellar-mass data from Mendel et al. (2014). It is apparent that SDSS galaxies are consistent with TNG50 disk galaxies, and the difference between halo-subdominated galaxies and tiny-halo galaxies is minimal. The green triangles and the histograms on the right illustrate that the data utilized in Mancera Piña et al. (2021, green) are biased towards smaller-sized galaxies.

Moreover, the significance of rotation in disks quantified by ϵf_disk here is hard to accurately approximate in observations. As shown in Fig. 6, the mass fraction of morphologically decomposed disk structures f_{disk, morph} are generally slightly larger by about 0 − 0.2 (median at ∼0.1) in tiny-halo galaxies (blue) than those measured using the kinematical method f_{disk, kinem} from Du et al. (2019, 2020). The difference is larger in halo-subdominated galaxies (red), reaching about 0.05 − 0.3 (red, median at ∼0.2). Taking the potential overestimation of ϵf_disk from 0.5 to 0.72 in observations into account leads to an offset toward the right side of the fiducial j_⋆ − M_⋆ − h_R relation of about 0.15 dex. The inconsistency between Mancera Piña et al. (2021) and TNG50 is understandable.

Fig. 6. Relative mass fraction of disk structures of tiny-halo (blue) and halo-subdominated (red) galaxies, measured using morphological fdisk, morph and kinematic fdisk, kinem methods. The morphologically defined disks are generally 10% larger than those defined in the kinematical method from Du et al. (2019, 2020). The envelopes of shaded regions represent the 16th and 84th percentiles and the solid profile is the median value.

In conclusion, we do not find any clear evidence for incorrect properties in the size and rotations of galaxies in TNG50. The inconsistency in the j_⋆ − M_⋆ − h_R relation between observations and the TNG simulations is likely attributable to several factors: (1) the substantial uncertainty in the measurement of j_⋆; (2) overestimation of ϵf_disk; (3) the limited statistical quality of the data sample; (4) the bias toward compact galaxies; and (5) the contamination from halo-subdominated galaxies. This outcome may be linked to the weak dependence of the mass–size relation on j_⋆.

4. The j_⋆ − M_⋆ − R_e relation: Origin of the mass–size relation and its scatter

4.1. The j_⋆ − M_⋆ − R_e relation

A similar fiducial j_⋆ − M_⋆ − R_e relation can be derived, assuming that the half-mass radius of galaxy R_e is proportional to h_R. Figure 7 shows the surface fitting result of the 3D space of j_⋆, M_⋆, and R_e using two mass ranges. The fitting result of tiny-halo galaxies with M_⋆ ∈ [10^9.5, 10^11.5] M_⊙ (red KDE map in the left panel of Fig. 7) gives

$$\begin{array}{cc}\hfill \log R_{\mathrm{e}}=& (1.019\pm 0.015)[\log j_{\star }\hfill \\ \hfill & -(0.331\pm 0.012)\log M_{\star }+(1.206\pm 0.093)].\hfill \end{array}$$(8)

Fig. 7. Fiducial j⋆ − M⋆ − Re relation of tiny-halo galaxies (red KDE contours) from the TNG50 simulation. This tight relation is obtained by a surface fitting of the 3D space of j⋆, M⋆, and Re using two mass ranges: log (M⋆/M⊙)∈[9.5, 11.5] and ∈[9.0, 9.5]. Halo-subdominated galaxies (gray dots) are also shown for comparison.

It is clear that R_e follows a very similar correlation to the fiducial j_⋆ − M_⋆ − h_R relation in Fig. 4. This equation can also be written as

$$\begin{array}{c}\hfill \log R_{\mathrm{e}}\simeq \log j_{\star }-0.29\log M_{\star }+(1.2-0.04\log M_{\star }),\end{array}$$(9)

which can be directly compared with the fiducial j_⋆ − M_⋆ − h_R relation. The constant term 1.2 − 0.04log(M_⋆/M_⊙) is approximately 0.8 for the sample of galaxies whose log(M_⋆/M_⊙)≃10. Remarkably, the tiny-halo galaxies exhibit a tight correlation between j_⋆ − M_⋆ and galaxy size, which closely resembles the j_⋆ − M_⋆ − h_R relation. We verified that the half-mass radius of kinematically derived disk structures, denoted R_e, disk, is 1.4−1.8 (median at ∼1.6) times larger than h_R, morph (left panel of Fig. 8). This result aligns well with the expected behavior for disk structures with exponential profiles. Furthermore, the ratio between R_e and h_R, morph varies roughly from 0.9 to 1.5 (median at ∼1.2) in both TNG50 and observations shown in Fig. 8.

Fig. 8. Ratio between the effective radius (Re) and the morphologically measured scale length (hR, morph) as a function of stellar mass (M⋆). Here, Re represents the half-mass radii of entire galaxies, while Re, disk specifically denotes the half-mass radii of the disk structures derived using the kinematic method. Tiny-halo and halo-subdominated galaxies are shown in blue and red, respectively. The envelopes of shaded regions represent the 16th and 84th percentiles and the solid profile is the median value. The black profile shows the measurement from SDSS r-band adopted from Simard et al. (2011) for comparison.

Equation (8) can also be written as $R_{\mathrm{e}}\propto \lambda M_{\star }^{0.212}$ where the spin parameter of galaxies $\lambda \propto \frac{j_{\star }}{M_{\star }^{0.543}}$ is almost constant according to the second panel of Fig. 2. This equation thus gives the overall mass–size relation $R_{\mathrm{e}}\propto M_{\star }^{0.225}$ of disk galaxies with M_⋆ ∈ [10^9.5, 10^11.5] M_⊙ shown in the first panel of Fig. 2. The large scatter originates from the scatter of λ. Therefore, the fiducial j_⋆ − M_⋆ − R_e relation explains well both the mass–size relation of disk galaxies and its large scatter. This result suggests that the disk structure of galaxies, while displaying a broad range of sizes, does not deviate significantly from the exponential hypothesis. Stellar migration (e.g., Debattista et al. 2006; Roškar et al. 2012; Berrier & Sellwood 2015) does not dramatically alter the fiducial j_⋆ − M_⋆–size relation. It is also worth emphasizing that halo-subdominated galaxies (gray dots in Fig. 7) adhere to a scaling relation that is nearly identical to that of tiny-halo galaxies, despite the large scatter. We can conclude that nature shapes the overall j_⋆ − M_⋆ and mass–size relations and that the effect of external factors plays a minor role.

4.2. Relation between galaxy size and dark matter halo virial radius

We adopt log j_⋆ ≃ log j_tot and log M_⋆ ≃ (log M_tot − 4.8)/0.67 from Du et al. (2022), and thus Eq. (8) can be written as

$$\begin{array}{c}\hfill \log R_{\mathrm{e}}\simeq \log j_{\mathrm{tot}}-0.50\log M_{\mathrm{tot}}+3.60.\end{array}$$(10)

Such a correlation does exist for galaxies with stellar mass 10^9.5 − 10^11.5 M_⊙ but has a relatively large scatter, as shown in Fig. 9. We note that we did not run a surface fitting of j_tot, M_tot, and R_e due to the poor statistics and large scatter. When considering r_vir/kpc ≃ 0.02(M_tot/M_⊙)^1/3 and the constant spin parameter defined as ${\lambda }_{\mathrm{tot}}=({10}^{3.60}/0.02)j_{\mathrm{tot}}/M_{\mathrm{tot}}^{0.81}=0.08$ adopting $j_{\mathrm{tot}}/M_{\mathrm{tot}}^{0.81}\sim {10}^{-6.37}$ from Du et al. (2022, Eq. (8)), Eq. (10) is written as

$$\begin{array}{c}\hfill R_{\mathrm{e}}\sim {\lambda }_{\mathrm{tot}}{r}_{\mathrm{vir}}\sim 0.08r_{\mathrm{vir}}.\end{array}$$(11)

Fig. 9. Relation between galaxy size and dark matter halo virial radius in the TNG50 simulation. Central tiny-halo galaxies (black dots) and central halo-subdominated galaxies (gray dots) are shown here. The x-axis represents λrvir. This correlation is given by Eq. (10) instead of running a 3D surface fitting.

As a result, TNG50 predicts that the ratio of galaxy size to halo virial radius is R_e/r_vir ∼ 0.08. This finding is in agreement with results derived from pure N-body simulations, where R_e ∝ λR_vir (Mo et al. 1998; Dalcanton et al. 1997; Somerville et al. 2008), while also considering the adjustment for the offset from j ∝ M^2/3 and the correction of the stellar-to-halo mass relation for disk galaxies. It is important to emphasize that defining λ ∝ j/M^2/3 in a conventional manner introduces an additional dependency on the mass. The size–size relation is very sensitive to the scaling factor of log (M_tot/M_⊙). A deviation of 0.02 log (M_tot/M_⊙) can lead to an uncertainty on R_e by factor 2. Additionally, we verified that the correlation does not hold for less massive galaxies, which is consistent with the findings of Karmakar et al. (2023). The halo-subdominated galaxies (represented by gray dots) also exhibit a relatively large scatter. Hence, such a size–size correlation is not always apparent, explaining the somewhat conflicting conclusions shown in Yang et al. (2023) and Karmakar et al. (2023). Thus, we propose the galaxy size–virial radius relation should not be viewed as conclusive evidence when deciding whether or not the characteristics of galaxies are dependent on their parent dark matter halos.

5. Summary

In this study, we elucidate the influence of internal and external processes on the scaling relations of the specific angular momentum j_⋆, the mass M_⋆, and the size of galaxies from TNG50. We employed a fully automatic kinematical method to decompose the kinematic structures of IllustrisTNG galaxies. Galaxies with more massive kinematic stellar halos have, in general, been more strongly influenced by external factors, such as mergers or close tidal interactions with neighboring galaxies.

Our analysis verifies the crucial role played by the inherent scatter in j_⋆ arising from internal (natural) processes, including but not limited to the properties of protogalaxies, secular processes, and host dark matter halos of galaxies. We selected galaxies that have tiny kinematic stellar halos with a mass ratio of f_halo ≤ 0.2 in order to isolate the effect of internal physical processes. Such galaxies are widely distributed over the observed j_⋆ − M_⋆ relation and the mass–size relation in the local Universe. We confirm that the disk structures of tiny-halo galaxies in IllustrisTNG adhere to the exponential hypothesis. The substantial scatter in the j_⋆ − M_⋆ relation then provides a robust explanation for the fiducial j_⋆ − M_⋆ − h_R relation. This further leads to the mass–size relation and the large scatter of galaxy size. Additionally, our findings indicate that stellar migrations, as suggested by previous studies, play a minor role in determining the overall properties of galaxies. The companion piece to this paper will explore the evolutionary process of galaxies of different sizes (Ma et al., in prep.).

Halo-subdominated galaxies with 0.2 < f_halo ≤ 0.5 are moderately influenced by external processes. Such galaxies closely align with a scaling relation similar to that of tiny-halo galaxies, but have a large scatter and are systematically offset toward the low j_⋆ side. This result underscores the dominant role of internal factors in shaping the overall j_⋆ − M_⋆ and mass–size relations, with external effects playing a minor role. Additionally, we examined the correlation between galaxy size and the virial radius of the dark matter halo after adjusting for the offset from j ∝ M^2/3 and accounting for the correction of the stellar-to-halo mass relation for disk galaxies. We propose that the galaxy size-virial radius relation should not be viewed as conclusive evidence when deciding whether or not the characteristics of galaxies are dependent on their parent dark matter halos.

Acknowledgments

The authors acknowledge the support by the Natural Science Foundation of Xiamen, China (No. 3502Z202372006), the Fundamental Research Funds for the Central Universities (No. 20720230015), and the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (No. 12221003). We also acknowledge constructive comments and suggestions from H. Mo and D.Y. Zhao. L.C.H. acknowledges the support by the National Science Foundation (NSFC) of China (11721303, 11991052, 12011540375, 12233001), the National Key R&D Program of China (2022YFF0503401), and the China Manned Space Project (CMS-CSST-2021-A04, CMS-CSST-2021-A06). S.L. acknowledges the support by the NSFC grant (No. 11988101) and the K.C. Wong Education Foundation. Y.J.P. acknowledges the support by the National Science Foundation of China (NSFC) Grant Nos. 12125301, 12192220, 12192222, and the science research grants from the China Manned Space Project with NO. CMS-CSST-2021-A07. The TNG50 simulation used in this work, one of the flagship runs of the IllustrisTNG project, has been run on the HazelHen Cray XC40-system at the High Performance Computing Center Stuttgart as part of project GCS-ILLU of the Gauss centers for Supercomputing (GCS). This work is also strongly supported by the Computing Center in Xi’an.

References

All Figures

Fig. 1. Distribution of the mass fraction of kinematic structures in TNG50 galaxies. The sum of the bar heights is normalized to 1. We then classify galaxies into tiny-halo galaxies, halo-subdominated galaxies, and halo-dominated galaxies by fhalo ≤ 0.2, 0.2 < fhalo ≤ 0.5, and fhalo > 0.5, respectively. The criterion fhalo = 0.2 and 0.5 are marked by the black vertical lines. The mass fraction of spheroids fspheroid is equal to fbulge + fhalo. From top to bottom, we show the galaxies with stellar mass log (M⋆/M⊙)∈[109, 1011.5]. The galaxies with log (M⋆/M⊙)∈[9.5, 10.5] are the most representative sample.

In the text

Fig. 2. Scaling relations of tiny-halo (blue dots), halo-subdominated (gray dots), and halo-dominated (red dots) galaxies. From left to right, we show the M⋆ − Re relation, the j⋆ − M⋆ relation, rotation κrot, and the mass fraction of disk structures fdisk. The shades of red and blue colors represent the log (Re/kpc) in all panels, showing that larger galaxies have larger j⋆ for a given stellar mass. We perform the linear fitting for galaxies in two mass ranges log(M⋆/M⊙): ∈[9.5, 11.5] (green dotted lines) and [9.0, 9.5] (blue dotted lines), respectively. The j⋆ − M⋆ relation of the disk galaxies from Du et al. (2022) is overlaid in black in the second panel.

In the text

	Fig. 3. Tully–Fisher relation of galaxies in TNG (black and red dots) and observations. The vflat of TNG galaxies is measured by averaging the flat part (0.05 − 0.2 rvir) of the rotation curve. The observations are adopted from Lelli et al. (2016) assuming the mass-to-light ratio Γ = 0.5 M⊙/L⊙ based on the IMF suggested by Kroupa et al. (2001).
In the text

Fig. 4. The j⋆ − M⋆ − hR, morph relation comparing simulations with observations. The units of hR, morph, j⋆, and M⋆ are kpc, kpc km s−1, and M⊙, respectively. The left and right panels show galaxies in two mass ranges log (M⋆/M⊙): ∈[9.5, 11.5] and ∈[9.0, 9.5]. The fiducial j⋆ − M⋆ − hR, morph relation of tiny-halo galaxies, where hR, morph represents the scale length of the disk component defined by morphology, is visualized using a kernel density estimation (KDE) map. For convenience, we adjust the surface fitting to align with y = x. We performed mock measurements of tiny-halo galaxies based on Eq. (2) (black dots). The gray points show the halo-subdominated galaxies for comparison. The observational data points of disk galaxies from Mancera Piña et al. (2021) are represented by green triangles. The dot-dashed lines represent the cases exhibiting offsets of ±0.3 dex.

In the text

Fig. 5. Mass–hR, morph diagram. Disk galaxies from TNG50 are represented by the blue-shaded area, while observed disk galaxies from SDSS DR7 are shown in the gray-shaded region. The solid profiles are the median. The shaded areas denote the 1σ envelope, representing the 16th and 84th percentiles. For the SDSS data, we used scale-length values from Simard et al. (2011) and stellar-mass data from Mendel et al. (2014). It is apparent that SDSS galaxies are consistent with TNG50 disk galaxies, and the difference between halo-subdominated galaxies and tiny-halo galaxies is minimal. The green triangles and the histograms on the right illustrate that the data utilized in Mancera Piña et al. (2021, green) are biased towards smaller-sized galaxies.

In the text

Fig. 6. Relative mass fraction of disk structures of tiny-halo (blue) and halo-subdominated (red) galaxies, measured using morphological fdisk, morph and kinematic fdisk, kinem methods. The morphologically defined disks are generally 10% larger than those defined in the kinematical method from Du et al. (2019, 2020). The envelopes of shaded regions represent the 16th and 84th percentiles and the solid profile is the median value.

In the text

	Fig. 7. Fiducial j⋆ − M⋆ − Re relation of tiny-halo galaxies (red KDE contours) from the TNG50 simulation. This tight relation is obtained by a surface fitting of the 3D space of j⋆, M⋆, and Re using two mass ranges: log (M⋆/M⊙)∈[9.5, 11.5] and ∈[9.0, 9.5]. Halo-subdominated galaxies (gray dots) are also shown for comparison.
In the text

Fig. 8. Ratio between the effective radius (Re) and the morphologically measured scale length (hR, morph) as a function of stellar mass (M⋆). Here, Re represents the half-mass radii of entire galaxies, while Re, disk specifically denotes the half-mass radii of the disk structures derived using the kinematic method. Tiny-halo and halo-subdominated galaxies are shown in blue and red, respectively. The envelopes of shaded regions represent the 16th and 84th percentiles and the solid profile is the median value. The black profile shows the measurement from SDSS r-band adopted from Simard et al. (2011) for comparison.

In the text

	Fig. 9. Relation between galaxy size and dark matter halo virial radius in the TNG50 simulation. Central tiny-halo galaxies (black dots) and central halo-subdominated galaxies (gray dots) are shown here. The x-axis represents λrvir. This correlation is given by Eq. (10) instead of running a 3D surface fitting.
In the text