4.1 Fall relation for dwarf galaxies

Figure 8 shows the specific angular momentum for the SH $\alpha$ DE galaxies, including both the high-mass ‘control’ galaxies in red and the dwarf galaxies in blue. We compare these findings with the Fall relation derived from disc galaxies by Posti et al. (Reference Posti2018), with $\alpha = 0.55$ and $\beta = 0.34$ . The high-mass disc galaxies neatly follow the expected relation with little scatter. This illustrates that deriving $j_{*}$ from these H $\alpha$ emission line observations is able to recover $j_{*}$ values consistent with those of previous studies of higher-mass galaxies, and in addition, shows that the angular momentum of star-forming regions is tracing the angular momentum of the general stellar population. The dwarf galaxies in our sample, while featuring a larger degree of scatter particularly for stellar masses between $10^{7}$ M $_{\odot} \lt M_{*} \lt 10^{8}$ M $_{\odot}$ , follow the relation of high-mass disc galaxies extrapolated to lower masses. The $j_{*}$ values for each galaxy are provided in Table 1.

Following Posti et al. (Reference Posti2018), we carry out a log likelihood fit to Equation (2) accounting for the uncertainties in $j_{*}$ (as determined from Monte Carlo simulations, see Section 4.2) and intrinsic scatter orthogonal to the relation. We then employ the python package emcee (Foreman-Mackey et al. Reference Foreman-Mackey2013) and use a Monte Carlo Markov chain to sample the posterior distribution in order to determine the posterior 1 $\sigma$ range. We find a best fit over the complete mass range of $\alpha = 0.53 \pm 0.04$ and $\beta = 3.23 \pm 0.11$ , with an intrinsic scatter of 0.35 dex (where the uncertainties correspond to the 68 per cent credible interval). This is in close agreement with Posti et al. (Reference Posti2018) and shows that this relation can be extended down to stellar masses as low as $10^{6}\,{{M}}_{*} ({\rm M}_{\odot})$ , thus covering 5 orders of magnitude in stellar mass.

Given that the maximum extent of our measurements varies across our sample, we restricted the radius to which $j_{*}$ is calculated to 1, 2, and 3 effective radii in order to determine how the varying radial extent may influence our results. While the absolute $j_{*}$ values across the whole sample increase as the radial limit increases and we include more of the outskirts, the observed result of a continuation in the $j_{*}$ – $M_{*}$ relation from high-mass to low-mass galaxies remains consistent, showing that the varying radial extent of our data does not significantly impact our conclusions.

Figure 4.

$j_{*}(\lt r/R_e)$ as a function of radius for the dwarf (blue) and control (red) galaxies. The radial profiles approach a constant value towards the edge of our observed limits in most cases, showing that our observations extend far enough over the galaxies to capture the bulk of their total angular momentum.

Since our sample is selected to be star-forming, quenched dwarf galaxies may deviate or be offset from this relation (as is the case for the high-mass elliptical galaxies). As a test of whether there may be a significant difference between the populations, we look for a correlation within our sample between the star formation rate and the residuals from the fitted relation. We do indeed observe such a correlation, with dwarf galaxies above the relation tending to have higher star formation rates relative to those below (Spearman’s correlation coefficient = $0.48$ , p-value = $0.001$ ). Such a relationship is consistent with previous findings of a correlation between the neutral gas fraction and j (Hardwick et al. Reference Hardwick2022), with gas-rich galaxies having higher j values. Separating our sample into two populations above and below the median specific star formation rate log(sSFR/yr $^{-1}$ ) = $-9.45$ and carrying out a log likelihood fit on each population, we find $\alpha = 0.56 \pm 0.07$ , $\beta = 3.46 \pm 0.25$ for the population with higher star formation rates and $\alpha = 0.59 \pm 0.06$ and $\beta = 3.22 \pm 0.17$ for the population with lower star formation rates. While the normalisation of the relation is shifted to higher values for galaxies with higher star formation rates, the slopes are consistent with each other and both are consistent with the complete sample, within uncertainties. This suggests that our sample selection biases the normalisation but not necessarily the slope, however a quenched dwarf galaxy sample is needed to fully test this finding. We therefore emphasise that our results apply primarily to the star-forming dwarf galaxies, with additional observations being needed to study the quenched population.

Figure 9 shows the sample categorised by visual morphology; those that resembled discs are shown in Figure 9(a), compact galaxies in Figure 9(b), and clumpy irregular galaxies in Figure 9(c). Disc galaxies follow the global $j_{*}$ – $M_{*}$ relation with a smaller degree of scatter (median offset $= 0.035$ , $\sigma = 0.30$ ) relative to the compact (median offset $= 0.002$ , $\sigma = 0.42$ ) and irregular (median offset $= 0.003$ , $\sigma = 0.43$ ) galaxies. Restricting to disc galaxies below $10^{8.5}\,{{M}}_{*} ({\rm M}_{\odot})$ , the median offset becomes 0.093 with $\sigma = 0.35$ . Together, these results show that despite the more disordered structure of irregular galaxies and the lack of a clear disc in compact galaxies, these galaxy types continue to follow the $j_{*}$ – $M_{*}$ relation to below $10^{6}\,{{M}}_{*} ({\rm M}_{\odot})$ , albeit with increased scatter.

Figure 5.

Convergence of the cumulative $j_{*}(\lt r/R_e)$ –R profiles for our observational sample. (a) displays the cumulative $j_{*}$ profile for a converging (ID: 1, filled points) and non-converging (ID: 322, open points) galaxy, along with their respective best-fit exponential curve. The exponential function provides a good description of the profile and demonstrates how well each profile has converged on the total $j_{*}$ value. (b) shows the distribution of $j_{*,\max}/j_{*,\infty}$ , illustrating that the majority of our galaxies are approaching convergence.

Figure 6.

Optical images of example simulated dwarf and low-mass spiral galaxies created using the radiative transfer code SKIRT. The images demonstrate the range of morphologies in the simulated galaxies, which resemble the variety of morphologies in our observational sample (see Figure 1).

Figure 7.

Examples of observationally-equivalent flux and kinematic maps for the simulated galaxies. The maps were made to mimic the SH $\alpha$ DE survey, showing the H $\alpha$ flux and line-of-sight velocity over a $14\times 22$ spaxel window resembling the FLAMES footprint. These maps were then analysed using the same methods as for the observational sample, resulting in the modelled kinematic maps shown in the right-hand column. The optical images on the left-hand side were created using the radiative transfer code SKIRT.

Figure 8.

The $j_{*}$ – $M_{*}$ relation for our observational sample, showing the SAMI control galaxies (magenta) and the dwarf galaxies (purple). The dark blue line and associated shaded region display the fit of the entire sample to Equation (2) along with its 68% credible interval. Fits for above and below $10^{8.5}\,{{M}}_{*} ({\rm M}_{\odot})$ (corresponding to the control and dwarf galaxy samples) are shown by the dashed purple and magenta lines, respectively. The corresponding values and intrinsic orthogonal scatter ( $\sigma$ ) for all three fits are given in the bottom right of the figure. The black dotted line and associated grey shaded region display the relation found by Posti et al. (Reference Posti2018). Non-converging galaxies ( $j_{*\rm tot}\lt0.8j_{\infty}$ ) are shown with an open circle. Example HSC r-band images of a large disc galaxy, a small disc galaxy, and an irregular dwarf galaxy are shown. The control galaxies closely follow the relation found by Posti et al. (Reference Posti2018), and our dwarf sample extends this relation down to below 10 $^{6}\,{{M}}_{*} ({\rm M}_{\odot})$ .

Figure 8. Long description

A scatter plot representing the relationship between the logarithm of specific angular momentum (log(j_*/kpc km s^-1)) and the logarithm of stellar mass (log(M_*/M_sun)). The plot includes several data points, with magenta circles representing control galaxies and purple circles representing dwarf galaxies. The dark blue line with a shaded region shows the fit for the entire sample, while the dashed purple and magenta lines represent fits for the control and dwarf galaxy samples, respectively. The black dotted line with a grey shaded region indicates the relation found by Posti et al. (2018). Non-converging galaxies are marked with open circles. Example images of different types of galaxies are also included. The control galaxies closely follow the relation found by Posti et al. (2018), and the dwarf sample extends this relation down to below 10.

Figure 9.

The $j_{*}$ – $M_{*}$ relation with galaxies divided by their visually classified morphology, illustrating how the deviation from the previous relationship is morphology-dependent. The top panel shows disc galaxies, the middle panel shows compact galaxies and the lower panel shows clumpy and irregular galaxies.

4.2. Monte Carlo model fitting

To better characterise the uncertainties in our calculations of $j_{*}$ , we performed a Monte Carlo simulation of our observations. For each galaxy, we created 100 random realisations where the velocity maps, mass, and redshift were all randomly altered based on the measurement uncertainties in each respective property. For the velocity maps, the velocity value of each individual spaxel was replaced by a random number from a normal distribution centred on the observed value and with standard deviation equal to the uncertainty in the measurement for that spaxel. The redshift and mass of the galaxy was likewise replaced by a random number taken from a normal distribution centred on the original measurement and with the appropriate standard deviation.

The resulting 100 realisations were then analysed in the same way as the original observations, including the fitting of a disc model and the subsequent calculation of $j_{*}$ from the deprojected maps. The distributions of $j_{*}$ from this process are displayed in Figure 10. There is great variety in the shapes and sizes of the resulting distributions. Many highly disc-like galaxies have a very small range of $j_{*}$ values, showing that in these cases the resulting disc model is very consistent between different realisations, and hence that it is robust. In other cases there is a large range of $j_{*}$ values, often for more irregular galaxies, showing that there is greater uncertainty in the modelling of these galaxies and reflecting the fact that it is difficult to model them with a disc. There are also cases where there are two favoured solutions in the model fitting, creating two regions in the $j_{*}$ – $M_{*}$ plane; an example of this is shown in Figure 10(b). In this case the higher- $j_{*}$ realisations occur when the galaxy is modelled as being closer to face-on, meaning that its intrinsic rotational velocity has to be higher in order to match the value observed along the line-of-sight.

4.3. Testing the thin-disc assumption

As dwarf galaxies often have an irregular or thick-disc morphology, the method of deprojecting a thin kinematic disc, as used here, may not lead to realistic modelling of their intrinsic kinematics. We therefore test an alternative method for deprojecting the line-of-sight velocities for each dwarf galaxy based on the optically observed ellipticity (derived from multi-Gaussian fitting to SDSS imaging, see Barat et al. Reference Barat2020) and systematically varying the assumed intrinsic disc thickness ( $q_0$ ) from 0 to 0.35. For higher $q_0$ values, our sample size becomes too small to carry out a meaningful fit due to the loss of galaxies (such as nearly edge-on discs) which cannot physically have a $q_0$ value at the set quantity. Using these deprojections, we then calculated $j_{*}$ as before. For low values of $q_0$ , we found close agreement between the two methods (for $q_0=0.05$ : median offset $= -0.044$ , $\sigma = 0.062$ dex), showing consistency between the optically-derived and kinematically-derived disc deprojections. For each value of $q_0$ we fitted Equation (2) for $\alpha$ and $\beta$ to the resulting $j_{*}$ – $M_{*}$ plane using the same method as before, and compared it to our original result for the dwarf galaxy population. We found these relations with varying $q_0$ to be consistent with our original result for the dwarf galaxies ( $M_{*} \leq 10^{8.5}\,{\rm M}_{\odot}$ ) within uncertainties. This suggests that for the sample presented here, our derived slope remains consistent when varying the assumed disc thickness up to $q_0 = 0.35$ .

4.4. Comparison with SAMI and the effect of the PSF

Since the 20 control galaxies in our sample were taken from the SAMI survey, we can also use the SAMI velocity maps as an observationally independent way to measure their angular momentum. The SAMI survey derived velocity maps from both their stellar component (using the stellar absorption lines) and from their ionised gas component (measured from the gas emission lines). With H $\alpha$ emission being a component of the gas emission, we expect the gas velocity field to more closely match what was observed in SH $\alpha$ DE. We found that the angular momentum measured from the SAMI velocity maps was consistently lower than SH $\alpha$ DE for both the stellar and gas component, and with the SAMI stellar component being lower than the SAMI gas component in most cases. To test whether this may be due to the different field of views of the two instruments, we overlaid the Argus footprint onto the SAMI maps, with its position and orientation matched to the SH $\alpha$ DE observation. The resulting $j_{*}$ – $M_{*}$ relation was tighter and more consistent with the SH $\alpha$ DE relation, however the downward offset remained. This is likely due to the SAMI survey having a larger PSF (median full width at half-maximum (FWHM) of 2.04 arcsec, Croom et al. Reference Croom2021) relative to the SH $\alpha$ DE survey (median FWHM of 0.88 arcsec). The larger PSF acts to blur the velocity field, reducing the velocity gradient across the field and therefore reducing its apparent rotation (while increasing the apparent velocity dispersion).

To determine how the PSF of our observations may influence the derived $j_{*}$ – $M_{*}$ relation, we tested for a correlation in the scatter of the $j_{*}$ – $M_{*}$ relation with both the number of resolved elements in each galaxy and with their effective radii as measured in arcseconds on the sky. In relation to the number of resolved elements, we found no significant trend with the scatter from the relation (Spearman’s correlation p-value $= 0.48$ ). We do however find a weak but significant trend between the scatter and effective radius in arcseconds (Spearman’s correlation coefficient = $-$ 0.11, p-value $= 0.026$ ). This is in the opposite sense to what one may expect from PSF blurring, and may reflect the increased reliance on model extrapolation to 3 effective radii.

4.5. Model Fall relation

We then attempted to reproduce the Fall relation using our sample of galaxies from the IllustrisTNG-50 simulation. To be consistent with our observational sample, $j_{*}$ was calculated from the velocity maps made to resemble the SH $\alpha$ DE observations (deriving the maps from the H $\alpha$ emitting gas); the resulting relation is shown in Figure 11, along with the SH $\alpha$ DE observations from Figure 8. Above a stellar mass of $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ , the simulated galaxies closely follow the relation found by Posti et al. (Reference Posti2018), and are in good agreement with the high-mass SH $\alpha$ DE control sample. This agreement of IllustrisTNG with previous observational studies in this mass range has been seen previously by Genel et al. (Reference Genel2015) (who used the original Illustris simulation), and Rodriguez-Gomez et al. (Reference Rodriguez-Gomez2022), Du et al. (Reference Du, Ho, Yu and Debattista2022) and Hardwick et al. (Reference Hardwick2023) (all using IllustrisTNG). However, all of these studies were restricted to masses above $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ . Below this mass there is an increased scatter towards lower $j_{*}$ values, with the upper edge continuing to follow the observed relation. Alternatively this could be seen as a drop and flattening of the relation, as reflected in the residual plot in the bottom panel of Figure 11. The low- $j_{*}$ dwarf galaxies may be missing from our observational sample if they are faint or have low surface brightness, or if they reside in higher-density environments not probed by our nearby dwarf galaxy sample.

We compared our derived $j_{*}$ values to the intrinsic values of the galaxies as calculated from their particles directly. We found relatively close agreement in $j_{*}$ between our observation-equivalent values and the star-forming particles, with the residuals (observed $-$ intrinsic) having a median bias of 0.08 and a scatter (one standard deviation) of 0.39. We found our observed values to overestimate the intrinsic stellar j (median bias = 0.44, scatter = 0.43), highlighting that the star-forming component has elevated angular momentum relative to the broader stellar component. We also find the relation between our observed values and the intrinsic $j_{*}$ to deviate from a one-to-one relation, finding a gradient m of $0.80\pm0.02$ for $j_{\rm observed}$ = $mj_{\rm intrinsic}$ , largely driven by an increase in scatter in the intrinsic stellar j at low mass. This suggests the difference in j between the star-forming component and stellar component may be greater for dwarf galaxies relative to high-mass galaxies, though again we stress the potential resolution limitations of the simulation at lower masses.

In the process of calculating the intrinsic angular momentum, we found the total angular momentum vector of all stellar particles within each galaxy, allowing us to determine the inclination angle as projected onto the observed x–y plane. We found no significant correlation between the inclination angle and the residuals of the observed $-$ intrinsic j (Spearman’s coefficient $= 0.01$ , p-value $= 0.76$ ), implying minimal bias in our observed values due to inclination. As a kinematic measure of the disc thickness, we calculated a simplified intrinsic $v/\sigma$ value (where $\sigma$ is the velocity dispersion) by taking the ratio of the sum of particle velocity components within the plane of rotation to the sum of velocity components outside this plane, and found there to be no significant correlation with the residuals (Spearman’s coefficient $= -0.02$ , p-value $= 0.62$ ) We also find no significant correlation with effective radius (Spearman’s coefficient $= -0.06$ , p-value $= 0.24$ ). However as stated through this paper, when applying these findings to our observational results, potential resolution effects within the simulation should be kept in mind.

The decrease in $j_{*}$ for low-mass galaxies, and the apparent flattening of the relation below $10^{8}\,{{M}}_{*} ({\rm M}_{\odot})$ may also be a result of the simulation’s limited resolution. Galaxies with stellar masses around $10^{8}\,{{M}}_{*} ({\rm M}_{\odot})$ consist of a few thousand stellar particles, while $10^{7}\,{{M}}_{*} ({\rm M}_{\odot})$ consist only of a few hundred stellar particles, meaning the galaxies’ structures are less resolved which may lead to the loss of coherent motions. Similarly in the mass-radius relation, as discussed above, there is a flattening below $10^{8}\,{{M}}_{*} ({\rm M}_{\odot})$ . As a test of whether the observed low $j_{*}$ values among the dwarf galaxies are due to resolution effects, we carried out the same analysis using the lower-resolution TNG100 and TNG300 simulations. While an increasing degree of scatter is seen as we go from TNG50 to TNG300, the distribution of galaxies on the $j_{*}$ – $M_{*}$ plot remains consistent. In particular, we do not see a change in the mass at which galaxies begin to scatter below the expected relationship – if this was purely a resolution effect, we may expect to see the downward scatter in $j_{*}$ begin at higher galaxy masses as the simulation’s resolution decreases. This suggests that the measured $j_{*}$ seen among the dwarfs may be due to physical processes within the simulation rather than the limited resolution. We also note that the H $\alpha$ kinematic maps for dwarf galaxies around $10^{6}\,{{M}}_{*} ({\rm M}_{\odot})$ remain physical, with coherent rotation clearly visible in well-ordered systems.

4.6. Evolution of $j_{*}$

Figure 12 displays the evolution of the specific angular momentum for a sample of galaxies in our simulation sample. The evolution is plotted in the $j_{*}$ – $M_{*}$ plane, showing the stellar mass and specific angular momentum at each simulation snapshot for every galaxy. Considering the computational burden of generating mock observations, here we use the true $j_{*}$ calculated from the individual particles in each galaxy. The particles used here are the gas particles weighted by their star formation rate, in order to match the H $\alpha$ component measured in our observation sample. The combined evolution of all galaxies forms a flat distribution at the low-mass end, a large dip to low angular momentum around $10^{7} - 10^{8.5}\,{{M}}_{*} ({\rm M}_{\odot})$ and an upward slope at higher masses. Galaxies initially form with low masses and with a relatively significant and highly fluctuating amount of angular momentum, creating the broad low-mass arm of the plot. As they grow in mass, most galaxies either fluctuate around a relatively constant value or lose angular momentum, creating the downward dip with increasing mass. At around $10^{7} - 10^{8}\,{{M}}_{*} ({\rm M}_{\odot})$ there is a transition to a rising relation between $j_{*}$ and $M_{*}$ , where the galaxies appear to be gaining angular momentum as their mass further increases. This creates the upward-sloping right-hand side of the figure. From this pattern, it appears that galaxies first lose angular momentum and then either maintain or begin gaining angular momentum once they grow past a certain mass range. We note that here we are only considering star-forming galaxies, leaving out quenched elliptical galaxies which likely have differing histories. In contrast, lower-mass galaxies appear to always be losing angular momentum while gaining mass and never reach this transition point. Finally, a few galaxies are seen to dip down in $j_{*}$ and to move to the left, suggesting that they are being stripped of their outer rotating disc.

Figure 10.

Distributions in $j_{*}$ after 100 Monte Carlo simulations for each galaxy. The top panel shows the MC distributions of each galaxy plotted together on the $j_{*}$ – $M_{*}$ relation, with the colours as before; contours show 1 and 2 $\sigma$ from the centre of each distribution. The lower panels show example galaxies: left – a galaxy where the MC simulations led to a bimodal distribution in $j_{*}$ ; middle – a more typical dwarf galaxy; and right – a typical high-mass disc galaxy.

Figure 10. Long description

Panel A: A scatter plot showing the relationship between the logarithm of specific angular momentum (log(j*/kpc km/s^-1)) and the logarithm of stellar mass (log(M*/M⊙)). The horizontal axis represents the logarithm of stellar mass, and the vertical axis represents the logarithm of specific angular momentum. The plot includes hundreds of data points with visible clusters and a positive correlation trend. A regression line is present, indicating the overall trend of the data. Panel B: Three smaller scatter plots showing specific examples of galaxies. Each plot has the same axes as Panel A. The left plot shows a galaxy with a bimodal distribution, the middle plot shows a typical dwarf galaxy, and the right plot shows a typical high-mass disc galaxy. Each plot includes data points and contours representing the distribution of the data.

Figure 11.

Comparing the $j_{*}$ – $M_{*}$ relation for our observational sample with that from the IllustrisTNG sample. The central panel shows the same observational data points as Figure 8, with the simulation sample added as the blue points. The distribution of the simulated galaxies closely matches the observed galaxies, including the agreement with Posti et al. (Reference Posti2018) at higher masses and the extension to lower values of $j_{*}$ at lower masses. However the simulated galaxies have a greater range of $j_{*}$ values than the observed sample, with the lower-mass dwarf galaxies having an increased scatter towards lower $j_{*}$ . This is highlighted in the bottom panel, which shows the median difference between $j_{*}$ and the expected relation from Posti et al. (Reference Posti2018) within mass bins centred on each point. The shaded regions show the standard error about each point. As an indication of how well resolved the simulated galaxies are, particularly for the low-mass end, the top panel displays the average particle number for each galaxy subhalo as a function of mass. The number of stellar particles decreases below 100 at around 10 $^{6.2}\,{{M}}_{*} ({\rm M}_{\odot})$ while the number of gas particles is more than 1 000 over the whole observed mass range.

Figure 12.

The $j_{*}$ – $M_{*}$ relation for IllustrisTNG galaxies over their evolutionary history. To reduce clutter, we only show a random sub-sample. The mass and $j_{*}$ of each galaxy at every snapshot in the simulation is shown by a point, coloured by the time of that snapshot. The final positions of each galaxy are shown by the blue points. Galaxies initially forming with low masses gradually increase in mass and lose angular momentum; then, while some continue to lose momentum, others experience an increase in momentum as they continue to increase in mass.

Figure 12. Long description

Panel A: A scatter plot showing the relationship between galaxy mass and angular momentum over time. The horizontal axis represents the logarithm of galaxy mass in solar masses, ranging from 5 to 11. The vertical axis represents the logarithm of specific angular momentum in kiloparsec kilometers per second, ranging from -1 to 3. Data points are colored by the universe age in gigayears, with a color gradient from light blue to dark blue. The plot includes hundreds of data points and several lines representing different galaxies’ evolutionary tracks. There is a visible trend where galaxies initially forming with low masses gradually increase in mass and lose angular momentum, while some continue to lose momentum and others experience an increase in momentum as they continue to increase in mass. The plot also includes a regression line indicating the overall trend.

Evidently, despite the apparent simplicity and tightness of the Fall relation as observed in this and various other studies in the present day, particularly for higher masses, the evolution of $j_{*}$ leading up to the present-day values is very complex and dynamic. We emphasise here that we are looking at galaxies which are selected as star-forming disc galaxies at the present day; high-mass galaxies which undergo more disruptive mergers and evolve into dispersion-dominated ellipticals would experience different evolutionary histories in $j_{*}$ and fall below the $j_{*}$ – $M_{*}$ relation at the present day.

Figure 13 displays the same information as in Figure 12, but instead we are plotting $j_{*}$ as a function of time for each galaxy. The galaxy sample is divided into those which have a final mass below $10^{8}\,{{M}}_{*} ({\rm M}_{\odot})$ and above $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ . This more clearly illustrates that there appears to be two main evolutionary paths for the angular momentum of these galaxies, though we note that there is a continuum between these paths rather than a clear separation.

Figure 13.

Evolution of $j_{*}$ as a function of time for all galaxies, divided into those with a final mass below $10^{8}\,{{M}}_{*} ({\rm M}_{\odot})$ on the left and above $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ on the right. The low- $j_{*}$ galaxies, after an initial rapid decrease in $j_{*}$ , continue to lose angular momentum at a steady rate, with high fluctuations in $j_{*}$ for each individual galaxy. The high- $j_{*}$ galaxies, meanwhile, go through the same initial loss of angular momentum before experiencing a rapid increase, eventually reaching a steady higher- $j_{*}$ state.

Galaxies with final masses below $10^{8}\,{{M}}_{*} ({\rm M}_{\odot})$ initially form with higher $j_{*}$ , then experience a steady loss of $j_{*}$ with the passage of time. Galaxies with final masses above $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ also experience an initial loss of angular momentum, then in many cases undergo a rapid transition or jump to a state of higher angular momentum. The fluctuations in $j_{*}$ also decrease significantly after this transition, and once it is completed the galaxies are able to maintain a very steady value of $j_{*}$ , resulting in a relatively constant mean and dispersion in $j_{*}$ through time for this group. Most of these galaxies have entered a steady state by 6 Gyrs after the Big Bang (corresponding to a redshift of 1), which is consistent with the lack of significant evolution in the $j_{*}$ – $M_{*}$ relation in observations over this redshift range for high-mass galaxies. The jump up to higher $j_{*}$ does not occur at a particular critical mass; rather, it occurs once the galaxy has grown into the stellar mass range between $10^{8}$ and $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ . Galaxies that never reach this mass never experience this transition, and so continue to lose angular momentum. However, as time goes on galaxies of lower and lower mass experience this jump in $j_{*}$ , and so perhaps some lower-mass galaxies haven’t yet had enough time to experience this transition.

It remains possible that this transition in $j_{*}$ corresponds to a resolution effect, for example, the crossing of a threshold in the number of particles within the galaxy resulting in an alteration of internal dynamics. As a test on whether the jump in $j_{*}$ is a result of resolution limitations, we trace the evolution of $j_{*}$ for high-mass galaxies within the lower-resolution TNG100 simulation and find similar behaviour, with a transition to a high- $j_{*}$ disc-dominated galaxy occurring from $10^{8}$ – $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ ; an example from TNG100 is provided in Figure A1 in the Appendix. This consistency in the $j_{*}$ transition between TNG50 and TNG100 suggests that this may not be due to resolution limitations. We note also that within TNG50, the timing of this transition does not correspond to galaxies reaching a particular particle number as they grow in mass.

To determine how the Fall relation evolves over time within our simulation sample, we performed a fit of Equation (2) for each snapshot over the complete sample and for galaxies above and below $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ . The evolution of $\alpha$ and $\beta$ over time is displayed in Figure 14. We find that despite the varied behaviour of individual galaxies, the value of $\alpha$ over the full sample reaches the present-day value at around $z = 1.5$ and becomes stable by $z = 0.6$ .

Figure 14.

Evolution of the $j_{*}$ – $M_{*}$ relation with the age of the Universe. The left panel shows the slope of the relation while the right panel shows the intercept. Light blue lines show the evolution of galaxies which have a final mass below $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ , the dark blue lines shows the evolution of those which masses above $10^{9}\,{{M}}_{*} ({\rm M}_{\odot})$ and the purple line shows the evolution for the complete sample. The overall slope is shallower at high redshifts and increases with time until it reaches the present-day value at a redshift of around 1.5, and becomes stable by $z = 0.6$ . The slope amongst the high-mass galaxies is seen to continually increase with time, while the slope amongst the dwarfs increases at a slower rate before decreasing towards the present day.

Figure 14. Long description

Two line graphs depict the evolution of the slope and intercept of a relation with the age of the Universe. Panel A: The left panel shows the slope of the relation. The x-axis represents the Universe Age in Gyrs, ranging from 0 to 13.5, and the top x-axis shows the corresponding redshift values from 4 to 0. The y-axis represents the slope (α), ranging from 0.0 to 0.6. Light blue lines show the evolution of galaxies with a final mass below 10^10 solar masses, dark blue lines show the evolution of galaxies with masses above 10^10 solar masses, and the purple line shows the evolution for the complete sample. The overall slope is shallower at high redshifts and increases with time until it reaches the present-day value at a redshift of around 1.5, and becomes stable by a redshift of 0. The slope amongst the high-mass galaxies continually increases with time, while the slope amongst the dwarfs increases at a slower rate before decreasing towards the present day. Panel B: The right panel shows the intercept of the relation. The x-axis represents the Universe Age in Gyrs, ranging from 0 to 14, and the top x-axis shows the corresponding redshift values from 4 to 0. The y-axis represents the intercept (β), ranging from 0.5 to 1.5. Light blue lines show the evolution of galaxies with a final mass below 10^10 solar masses, dark blue lines show the evolution of galaxies with masses above 10^10 solar masses, and the purple line shows the evolution for the complete sample.

Figure 15.

Examples of a galaxy with a low final value of $j_{*}$ (right) and with a high final value (left). The top panels show the evolution of $j_{*}$ with time, with the blue sequence showing the stellar component and the pink sequence showing the gas component. The bottom two rows show the optical image as modelled by SKIRT (top row) and the distribution of the gas component (bottom row) at three different times.