Recovering $\lambda_{R}$ and $V/\sigma$ from seeing-dominated IFS data

The majority of galaxies in the Universe appear to be regular, FRs (Graham et al., 2019; van de Sande et al., 2017a; Cappellari et al., 2011b). The SAMI survey contains $\sim 10\%$ SR following aperture correction (van de Sande et al., 2017a), Atlas${}^{\text{3D}}$ (selected for early type galaxies) contains $\sim 4-11\%$ (Emsellem et al., 2011), and MaNGA contains $\sim 1-7\%$ (Graham et al., 2019). Califa (Calar Alto Legacy Integral Field Area; Sanchez et al., 2012) contains 28% SRs for stellar masses above $10^{11}$ M_⊙ (Falcón-Barroso et al., 2019). SR fractions only become high in the most massive regimes around $10^{12}$ M_⊙ where the value was shown to go up to $\sim 90\%$ by the Massive survey (Veale et al., 2017). For this reason, it seems sensible to optimise our correction to work best for the regular FR class.

We define SRs using the criteria from Cappellari (2016), as shown in Equation 3. In the alternative case that an observation is greater than this criterion for round isophotes ($\varepsilon_{e}<0.4$), or flatter than $\varepsilon_{e}=0.4$, the system is classed as an FR.²²2The boundary of $\varepsilon_{e}=0.4$ is based on observations made by Atlas${}^{\text{3D}}$ that all disk-less SR are rounder than $\varepsilon_{e}=0.4$. This has been further supported by Sami (Fogarty et al., 2015) and Califa observations (Falcón-Barroso et al., 2017). For measurements of $\lambda_{R}$ made at greater or smaller radii than R${}_{\text{eff}}$, the system still retains the classification made at R${}_{\text{eff}}$.

Using N-body simulations, we can generate a wide variety of visual morphologies (i.e. E-S0 to Sd systems) with regularly-rotating velocity structures. These systems sit in equilibrium and describe the “perfect” isolated regular-rotator case. We have generated a sample of 18 models, shown in Table 1, spanning the visual morphology parameter space. Of this sample, we expect the S0-Sd galaxies to sit within the FR regime. The three E-S0 galaxies sit closer to the SR/FR division.

We aim to apply our corrections to the full range of galaxies observed in a survey. This will include systems which have irregular kinematic morphologies, such as 2-$\sigma$ galaxies (where two dispersion maxima are seen near the centre of the galaxy in the two-dimensional LOS velocity and dispersion maps). Similarly, real galaxies in the universe may not be fully relaxed, equilibrium structures with regular velocity fields.

To validate our corrections for the variety of kinematic classes, we have selected a further seven galaxies from the cosmological, hydro-dynamical simulation, Eagle (Schaye et al., 2015; Crain et al., 2015; McAlpine et al., 2016). These galaxies are shown in Table 2. While they have been selected because they are reasonably isolated systems at the present day, these systems have grown from cosmological initial conditions via mergers and accretions, as well as experiencing interactions. This provides us with a complementary sample of simulated galaxies whose kinematics are shaped by cosmologically realistic assembly histories.

This gives us a full catalogue of 25 model galaxies. Because we have the full three-dimensional model, we can rotate and project each of these systems at a variety of different angles and measure the kinematics within various apertures. Each galaxy is observed multiple times in order to build up a comprehensive picture of the kinematic parameter space.

We have taken a series of mock observations of the 25 galaxies shown in Table 1 and Table 2 using the R-package SimSpin (Harborne et al., 2020). This code takes N-body/SPH models of galaxies and constructs kinematic data-cubes like those produced in an IFS observation. This code is registered with the Astrophysics Source Code Library (Harborne, 2019) and can be downloaded directly from GitHub³³3https://github.com/kateharborne/SimSpin. Using this framework, we have explored a large parameter space that includes kinematic measurements made at a variety of seeing conditions, projected inclinations and measurement radii.

We begin with a sample of 53,010 observations of $\lambda_{R}$ and $V/\sigma$ for the 18 galaxies in Table 1. We remove any values for which $\lambda_{R}$ or $V/\sigma$ is less than 0.05, as well as any where the ellipticity is less than 0.05 (as explained in Section 3). This leaves us with a total sample of 41,202 and 41,325 observations for $\lambda_{R}$ and $V/\sigma$ respectively, including the three E-S0 regular rotators.

We divide our observations into FRs and SRs using the criteria of Cappellari (2016), as shown in Equation 3. As described in Section 2.3.2, these classifications are based on the measurements of $\lambda_{R}$ made at R${}_{\text{eff}}^{\text{fac}}=1$. This gives a sample of 34,039 FR and 7,163 SR observations.

Figure 7 demonstrates the effect of applying corrections to both $\lambda_{R}$ (left) and $V/\sigma$ (right) of the N-body models in Table 1. For FR observations (above), this gives 1317 individual tracks for both $\lambda_{R}$ and $V/\sigma$. As shown on the left, not only does our correction remove the negative trend with seeing for uncorrected $\lambda_{R}$, it also significantly reduces the scatter of the distribution. This is because we have included the observed ellipticity which effectively accounts for inclination in our correction; the PSF is always circular and so the effects of seeing are dependent on the projected inclination of the galaxy (for more details, see appendix C in G18). The scatter is also reduced for $V/\sigma$ on the right, but to a lesser extent.

We note here that, when plotted as tracks, some of the tracks will move from being FR to SR as the seeing grows worse causing some to appear incomplete. It is important to divide these tracks in this way, as an observation of a regularly-rotating FR made in poor seeing may be observed as a SR (Graham et al., 2019). We need to ensure that these systems can also be corrected.

In the lower panels of Figure 7, we consider the SR observations of the N-body models. We show 383 individual tracks for both $\lambda_{R}$ and $V/\sigma$. Of these, 49 observations remain within the SR regime across the full track length. Many more tracks begin at $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ > 0 in this plot, where the increased level of seeing has caused an intrinsically FR observation to drop below the criteria and appear as a SR. Nonetheless, we still bring all values back towards true while also reducing the scatter. A possible reason for this correction being effective for both FRs and SRs is that we have considered the effects of seeing in a relative parameter space. In this space the track shape is similar for both slow and fast rotation, as long as that rotation is regular.

For the plots considering $\lambda_{R}$, we compare the correction presented in this work to G18. In some respects, this comparison is unfair as the G18 correction was designed for FR measurements made within a single effective radius and for Sérsic indices between $0.5<n<6.5$; hence, the distinction is made between those observations that meet the conditions and those that do not by colour coding R${}_{\text{eff}}$ $\neq$ 1 and $n>6.5$ in yellow and valid G18 corrections in orange. This places emphasis on the fact that including a factor that fully parameterises galaxy shape is important if we wish to reduce the scatter in the $\Delta\lambda_{R}$-$\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ space.

The distribution of corrected values for the full sample of 41,202/41,325 observations are shown in Figure 8. The following statistics are presented as the median of each distribution with the \nth16 and \nth84 percentiles below and above respectively ($\nu_{16\tiny{\text{th}}}^{84\tiny{\text{th}}}$). On average, the effect of seeing conditions is to reduce the values of $\Delta\lambda_{R}^{\text{obs}}$ $\sim-0.245_{\tiny{-0.388}}^{\tiny{-0.051}}$ and $\Delta V/\sigma^{\text{obs}}$ $\sim-0.310_{\tiny{-0.463}}^{\tiny{-0.071}}$. By applying the correction presented in this work, these values become $\Delta\lambda_{R}^{\text{corr}}$ $\sim 0.000_{\tiny{-0.016}}^{\tiny{+0.017}}$ and $\Delta V/\sigma^{\text{corr}}$ $\sim 0.000_{\tiny{-0.056}}^{\tiny{+0.038}}$. The key result is that we bring the median of the distribution back to zero, within $\Delta\lambda_{R}$ and $\Delta V/\sigma$ $\sim 1\times 10^{-3}$ dex. This is well below the statistical median uncertainty of $\lambda_{R}$ and $V/\sigma$ in surveys (van de Sande et al., 2017a). We also significantly reduce the spread of the distribution in applying this correction. However, in comparing $\sigma$ for $\Delta\lambda_{R}^{\text{corr}}$($\sigma=0.02$ dex) and $\Delta V/\sigma^{\text{corr}}$($\sigma=0.06$ dex), as shown in Figure 8, we see that $\lambda_{R}$ is more effectively corrected than $V/\sigma$. We believe that this is due to the fact that the seeing conditions impact the value of LOS velocity more than the dispersion; as $\lambda_{R}$ has factors of velocity in both the numerator and the denominator, these effects are partially cancelled out, unlike in $V/\sigma$. The skew of each distribution demonstrates that we tend to under-correct our values. In comparison, the G18 correction has a more significant skew towards over-correction, where $\Delta\lambda_{R}^{\text{G18}}$ $\sim 0.006_{\tiny{-0.040}}^{\tiny{+0.129}}$.

In the lower panel of Figure 8, we break down the distributions into samples of SRs and FRs in red and blue respectively. If we consider the effect of seeing on the rotator types independently, we find that $\Delta\lambda_{R}^{\text{obs}}$ $\sim-0.230_{\tiny{-0.374}}^{\tiny{-0.038}}$ for FRs and $\Delta\lambda_{R}^{\text{obs}}$ $\sim-0.330_{\tiny{-0.433}}^{\tiny{-0.209}}$ for SRs. Similarly, $\Delta V/\sigma^{\text{obs}}$ $\sim-0.292_{\tiny{-0.462}}^{\tiny{-0.054}}$ for FRs and $\Delta V/\sigma^{\text{obs}}$ $\sim-0.362_{\tiny{-0.463}}^{\tiny{-0.228}}$ for SRs. For both kinematic measurements, we see that FRs are relatively affected by seeing a lesser amount than SRs, as concluded in Harborne et al. (2019) because the median values are much larger in the SRs. However, following correction, we bring all values back towards zero successfully. On average, these corrected values have a distribution described by $\Delta\lambda_{R}^{\text{corr}}$ $\sim 0.000_{\tiny{-0.016}}^{\tiny{+0.015}}$ and $\Delta V/\sigma^{\text{corr}}$ $\sim 0.000_{\tiny{-0.053}}^{\tiny{+0.036}}$ for the FRs; for the SRs, these values are $\Delta\lambda_{R}^{\text{corr}}$ $\sim-0.001_{\tiny{-0.020}}^{\tiny{+0.027}}$ and $\Delta V/\sigma^{\text{corr}}$ $\sim 0.000_{\tiny{-0.065}}^{\tiny{+0.045}}$. The difference between the spread for the corrected FR and SR is very small, but the skews are opposite, with SRs more over-corrected on average than the FRs. However, while the effect of seeing on the SRs is relatively larger, the corrected position in real space is close to true due to the fact that these values are by definition smaller.

Kinematic measures are often presented as a function of ellipticity, where SRs and FRs can be distinguished. Figure 9 demonstrates the effect of applying our correction to six galaxies from our sample in this spin-ellipticity plane for measurements made within 1 R${}_{\text{eff}}$. On the left we demonstrate the observations made at a variety of seeing conditions up to the limit of $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ = 0.6, similar to the cuts made for observations in SAMI (van de Sande et al., 2017a). As the conditions get poorer, we see that the measurements are shifted down towards the slow rotator regime. Following correction, these tails are significantly reduced for both $\lambda_{R}$ and $V/\sigma$.

Using the corrected panels in the centre of Figure 9, we can see a few of the deficiencies of equations 16 and 17. For $\lambda_{R}$${}^{\text{corr}}$, we see residual seeing effects present in the highly inclined systems. A similar effect is seen in the corrected $V/\sigma$${}^{\text{corr}}$, though this is secondary to the fact that the lowest Sérsic index galaxy shown, n $=1.22$, has a large residual in comparison to the others. This is important to bear in mind when applying the $V/\sigma$ correction. We have not fully described all of the scatter within the $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ vs. $V/\sigma$ parameter space, leaving a residual that can be seen in the higher $V/\sigma$ values. This residual Sérsic dependency is far less of an issue for $\lambda_{R}$.

If we consider only valid observations (i.e. $\lambda_{R}$ and $\varepsilon>0.05$) classified by their true kinematics, we found that 2.8% of the full sample of 41,202 observations are intrinsic SR (as shown in Figure 1). Redoing our classification of FRs and SRs using the measured and corrected $\lambda_{R}$, 2.7% fall into the SR regime (corrected from a fraction of 17%). This brings us much closer to the 2.8% of the original, perfect-seeing classifications. We correct 99.9% (all but 61 observations) back to their true classification. These remaining mismatched observations are nearly all SR that have been observed to have spin parameters close to the $\lambda_{R}$ = 0.05 cut-off. These observations tend to occupy the 0.5 < $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ < 1 range, and have observed ellipticities, $\varepsilon<0.2$. The residuals between the true parameter and the corrected values are less than 0.05, but even this level of difference causes the incorrect galaxy classification to be assigned. Overall, however, this difference is very small. We show the distributions of mismatched kinematic class observations in Figure 10. By comparison, the G18 correction reduces this fraction to 1.7%.

The correction presented in this work has been derived using a set of N-body regularly-rotating models. It is important to verify that this correction is valid also for an independent set of galaxies with different assembly histories. Furthermore, while the majority of galaxies appear as FRs, as discussed in Section 2, it is important to understand the effect this correction has on the full data set i.e. including the irregular rotators. We have selected three FR and four SR galaxies that exhibit slow or irregular rotation from the Eagle simulation for this test, as outlined in Table 2, and present their analysis below.

We begin with 20,615 observations of the Eagle galaxies listed in Table 2. When we remove any observations using the cuts explained in Section 3 ($\varepsilon$ and $\lambda_{R}$ or $V/\sigma$ < 0.05), this reduces the sample size to 15090 observations. Using the criteria in Equation 3, we divide the sample of regularly rotating Eagle observations into 9722/10100 FRs and 2440/2497 SRs for $\lambda_{R}$ and $V/\sigma$ respectively. We also have two galaxies that have been identified as having irregular kinematic morphology (i.e. GalaxyIDs = 10770392 (2-$\sigma$) and 10048611 (odd)) which are analysed separately.

Plotting these in Figure 11, we have 448 FR $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ tracks and 102 SR tracks, of which 8 observations are fully SR across all seeing conditions. At first glance, this set of galaxies shows a broader range of scatter in both the observed and corrected track shapes than the N-body set shown in Figure 7.

The largest discrepancy between the N-body and Eagle samples that may cause these effects is the particle resolution of each simulation. The three regular FRs have an order of magnitude fewer particles than the full N-body catalogue of galaxies (N${}_{\text{EAGLE}}\sim 1\times 10^{5}$ vs. N${}_{\text{N-body}}=2.5\times 10^{6}$). The four galaxies chosen for their SR and irregular properties have an order of magnitude fewer particles again (N$\sim 1\times 10^{4}$). From Ludlow et al. (2019), we know that for galaxies with low particle numbers, two-body particle scattering will affect the size and very likely the velocity dispersion profile of a galaxy, causing the disks to have larger scale heights than expected. This will also be affected by the intrinsic softening size and the temperature floor within Eagle. While we have selected a conservative particle limit based on convergence tests in Lagos et al. (2017), this value may need to be even larger for generating mock observations using tools such as SimSpin. We know that the softening in the simulation alone (0.70 physical kpc) is equivalent to a level of blurring which contributes on top of the effects we have added, meaning that the value we have taken as $\Delta\lambda_{R}^{\text{true}}$ is more like observations of the $N$-body systems made at $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ > 0. This will lead to an increased level of scatter in our corrected kinematics as we are applying this formula without accounting for the added seeing contribution.

The particle resolution is also very important to the final track shape that is observed, as highlighted in Figure 11. In the top and centre panels, we have highlighted the higher resolution models in a darker, thicker colour than the low resolution set. The irregular tracks shown in the lower panels are also of low resolution particle models. Noticeably, the tracks for the low resolution models follow a different shape to the higher resolution models, appearing to reduce rapidly at very small seeing effects and plateau to a maximum $\Delta$ earlier in the track. The convolution of any size PSF with the sparser particle numbers will make it much easier for atmospheric effects to blur out any rotation.

This also manifests in the shape of the track for the smaller apertures (R${}_{\text{eff}}$-factor = 0.5) which, by definition, contain less than half the total number of particles in the model. We see that these tracks show a greater departure from the expected “S”-shaped track, even for the FR selected simulations with $1\times 10^{5}$ particles.

Because these low resolution models have the highest particle numbers available from the SR sample extracted from the Eagle RefL0100N1504 box, we cannot fully evaluate how effective this correction is for irregular systems. Nevertheless, we show the 35 $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ tracks for 395/503 observations of $\lambda_{R}$ and $V/\sigma$ respectively in the lower panels of Figure 11 for completeness. Interestingly, we see that using the criteria in Equation 3, there are several of the observations of the irregular galaxies that would have been classified as FRs at low seeing (as can be seen by tracks beginning between $0<$ $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ $<1$ in Figure 11). This may suggest that looking for irregular systems within the SR category may not be sufficient. In the future, we would like to test this on our own cosmological zoom models of irregular systems built at a much higher particle resolution, but this is beyond the scope of this paper.

We consider the average statistics of the distributions using the median and \nth16 and \nth84 percentiles below and above respectively. For the FR observations, $\Delta\lambda_{R}^{\text{obs}}$ $\sim-0.275_{\tiny{-0.413}}^{\tiny{-0.055}}$ reduces to $\Delta\lambda_{R}^{\text{corr}}$ $\sim-0.013_{\tiny{-0.127}}^{\tiny{+0.091}}$; SR observations, $\Delta\lambda_{R}^{\text{obs}}$ $\sim-0.370_{\tiny{-0.517}}^{\tiny{-0.181}}$ reduces to $\Delta\lambda_{R}^{\text{corr}}$ $\sim-0.027_{\tiny{-0.239}}^{\tiny{+0.079}}$; and for the irregular galaxies, $\Delta\lambda_{R}^{\text{obs}}$ $\sim-0.500_{\tiny{-0.651}}^{\tiny{-0.263}}$ reduces to $\Delta\lambda_{R}^{\text{corr}}$ $\sim-0.208_{\tiny{-0.442}}^{\tiny{-0.055}}$. The overall effect of the correction does move the kinematic measurement up and closer to true, but does not do much to reduce the spread. Values are also often skewed towards under correction; this is understandable if we consider that the effect of particle softening in Eagle effectively moves what we have taken as the intrinsic value along the track towards $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ > 0. However, even in the worst case of the irregular systems, we reduce the effects of seeing by a factor of 2. For the majority of the regularly rotating systems drawn from Eagle, we are reducing the effects by a factor of 4.

By applying our correction to this data set, the SR fraction is reduced from 19% to 11% (where the fraction for the intrinsic data is 1%). In this case, the G18 correction reduces the SR fraction to 7%. In Figure 12, we consider the observations that have been labelled incorrectly following correction. We find that the distributions for this work and G18 are fairly similar to one another. From Figure 8, we know that the G18 correction tends to over-correct systems pushing them further toward the FR regime. Given that all mismatched observations have been labeled as SRs after correction implies that neither our correction nor G18 is pushing these observations far enough, but that G18 goes further, as can also be seen in the residual histogram in the upper right panel of Figure 12. While we incorrectly categorise more observations as SR, we are getting the corrected $\lambda_{R}$ values much closer to true than G18, which has a tail of observations that are strongly over-corrected.

Overall, this independent test confirms that this correction is applicable to a broad number of regularly-rotating galaxies. The issue of particle resolution makes it difficult to conclude how effective this correction is for irregularly-rotating systems. However, for the systems shown in Figure 11, we see that the net effect is to move them closer to true.

It seems reasonable to assume that the $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ track shape with seeing is different for a galaxy with embedded, irregular kinematic features. However, if the impact of seeing is large enough to have washed out these features, we expect to move the kinematics on a similar track, given that the tracks for FRs and SRs are similar. In applying this correction to irregulars, at least in the cases shown in Figure 11, the result is moving the value towards the correct value.

We have demonstrated that these corrections are applicable to a whole range of different simulated data sets, thus it seems prudent to explore the effect of this correction on real observations. Here, we apply our corrections to SAMI DR2 observations (Scott et al., 2018). Following the methodology of van de Sande et al. (2017b, 2019), we begin by applying the standard quality cuts to the sample. From the initial sample of 960 galaxies, we exclude 11 observations for which the radius out to which the stellar kinematics can be accurately measured is less than the half-width half-maximum of the PSF (HWHM${}_{\text{PSF}}$), and a further 95 observations which are poorly resolved for which kinematics can not be obtained. Galaxies are removed that have a stellar mass less than $10^{9.5}$ M_⊙ due to the reliability of kinematics below this mass, removing 89 systems from the sample. We further remove 24 galaxies with $\varepsilon$, $\lambda_{R}$ or $V/\sigma$ < 0.05 and $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ > 1, in line with our empirical correction requirements. This leaves us with 741 observations from DR2. In order to demonstrate the effect of the correction, we compare the SR fraction of this sample before and after correction. We find, using the FR/SR criteria of Cappellari (2016) in Equation 3 that this sample of 914 galaxies contains $9\%$ SR. Using Equations 16 and 17, we calculate the corrected kinematic values, $\lambda_{R}$ and $V/\sigma$, and re-calculating the SR fraction, we find that this has dropped to $5\%$.

In Figure 13, we show these observations before and after correction in the spin-ellipticity plane. The grey empty points show the original observed kinematic parameter, while the full, coloured points show the value after correction. Each pair of points is joined by a line to show how far that single observation has moved following correction. The sizes of these lines vary across the parameter space, which we have shown is dependent on a combination of galaxy shape, ellipticity, measurement radius and seeing conditions.

The range of colours in this figure demonstrates that such corrections are important for comparing survey data. While the predominant $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ value is $\sim 0.3$, this value varies from $0.08-0.82$. The length of lines connecting uncorrected and corrected points does not scale linearly with $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$, so simple adjustments made using the seeing conditions alone are not sufficient to make consistent comparisons. This justifies the requirement in the community for the corrections presented in this work. We further find that the corrections presented in this paper are important for quantifying the presence of a bimodality in the $\lambda_{R}$-$\varepsilon$ plane (van de Sande et al., in prep).

Finally, we use our idealised data set, as shown in Table 1, to examine how seeing conditions and measurement radius affect the relationship between $\lambda_{R}$ and $V/\sigma$. In Emsellem et al. (2007), they introduce a simple approximation that allows conversion from one kinematic parameter to the other:

This $\kappa$-parameter has been calculated and compared across different surveys (Sauron $\sim 1.1$ (Emsellem et al., 2007), Atlas${}^{\text{3D}}\sim 1.06$ (van de Sande et al. 2017a, Emsellem et al. 2011 found a value of $\kappa\sim 1.1$ originally with mixed aperture sizes), SAMI $\sim 0.97$ (van de Sande et al., 2017a)), but the value and the level of scatter in the plotted distribution is not consistent. The SAMI distribution of $\lambda_{R}$ vs. $V/\sigma$ is a much tighter distribution than that seen in Atlas${}^{\text{3D}}$, for example.⁶⁶6The SAMI value is also smaller due to the different definition of $\lambda_{R}$ that is used in this survey. For more details on this difference, see Appendix A.

In van de Sande et al. (2017a), they demonstrate that the scatter in the distribution is affected by systematics such as the difference in spatial resolution and seeing conditions. Atlas${}^{\text{3D}}$ has a much higher resolution, such that complex internal dynamical features are not washed out by atmospheric blurring. It follows that there is a larger scatter in the $\kappa$-relation for this survey, as $\lambda_{R}$ is designed to better distinguish internal kinematic structures than $V/\sigma$ (Emsellem et al., 2007, 2011).

In Cortese et al. (2019), it was suggested that conversion between $\lambda_{R}$ and $V/\sigma$ may not be trivial if $\kappa$ has a strong dependence on the size of the PSF. We can use our data set to investigate this. We measure $\kappa$ in our idealised data set using regression, minimising the sum of least squares for the uncorrected values of $\lambda_{R}$ and $V/\sigma$. We use the uncorrected values because we want to examine how $\kappa$ is affected by seeing. The error on this fitted value can be found by,

where $Q$ is the remaining sum of the square residuals from the best fit $\kappa$ value, $n$ is the number of observations in the data set, and $p$ is the number of parameters being fit (i.e. $p=1$ for the one parameter, $\kappa$). In the case of the full 41,231 observations, $\kappa=1.05\pm 0.016$. This is shown in Figure 14 by the black curve.

Our idealised observations have been generated using the same resolution as the SAMI survey and so we cannot investigate the effect of spatial resolution with this data alone. Furthermore, we have only considered regular rotators in this analysis. When we add our sample of observations from Eagle to the distribution, we find they do add to the overall scatter of the distribution as shown by the grey points in Figure 14. When fitting $\kappa$ to the full idealised plus Eagle data set, we find that the value decreases slightly, but with larger scatter, to $\kappa=1.04\pm 0.018$. This is shown by the grey curve in Figure 14. This indicates that $\kappa$ value may also see variations due to the number of irregular systems in the data-set.

We then bin our idealised data set of 41,202 observations by the five R${}_{\text{eff}}$-factors and by seeing conditions, with bin widths of $\Delta$ $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ = 0.1. For each bin, we fit Equation 19. The inbuilt panels on the left demonstrate the trends of $\kappa$ with these variables independently.

As in van de Sande et al. (2017a), we find that there is a slight effect due to seeing, $\Delta\kappa\sim\pm 0.02$, though there is a stronger dependence on aperture size, where we see a positive correlation with measurement radius. Looking at the seeing conditions within each measurement radius bin, we see that the correlations with seeing conditions change with the measurement radius being considered. These trends are shown in Figure 14 on the right.

The gradient change can be understood by considering the trends seen in Figure 2. We know that, overall, the effect of seeing on $V/\sigma$ is slightly greater than that of $\lambda_{R}$. We also know this effect is consistent across R${}_{\text{eff}}^{\text{fac}}$, but on average we measure higher values of $\lambda_{R}$ and $V/\sigma$ at larger measurement radii. At the lower spin end of the parameter space, a more rapid change in $V/\sigma$ is going to scatter points towards higher $\kappa$. Moving points in a similar direction at higher spin, the effect causes $\kappa$ to be pulled down. This is illustrated by the annotations in Figure 14, where each arrow has exactly the same gradient. The absolute change in $\kappa$ seen in Figure 14 is very small, with the maximum variation $\Delta\kappa\sim\pm 0.02$ in all figures on the right-hand side. This may be due to the fact that there is opposing trends with Sérsic index and ellipticity for $\lambda_{R}$ and $V/\sigma$ that we see in Figure 3. The potential variation in the gradient change with $\sigma_{\text{PSF}}$/R${}_{\text{maj}}$ at each R${}_{\text{eff}}^{\text{fac}}$ is important. If a given survey contains a larger portion of one R${}_{\text{eff}}$-factor than another, the average fit to the full data set will be biased. This also confirms the results of van de Sande et al. (2017a), in so much as reiterating the need for such aperture corrections.