The Intrinsic Scatter of the Radial Acceleration Relation^*

The local observed radial acceleration ${g}_{\mathrm{obs}}$ is computed from a galaxy rotation curve (RC) with the formula

$$\begin{eqnarray}&&{g}_{\mathrm{obs}}=\displaystyle \frac{{\left({V}_{\mathrm{obs}}/\sin (i)\right)}^{2}}{\theta D},\end{eqnarray} \tag{ 2 }$$

where V_obs is the measured line-of-sight velocity, θ is the angular radius of the velocity measurement, D is the distance to the galaxy, and the inclination i is inferred from the axial ratios of the galaxy. The latter, which represents the galaxy's photometric tilt relative to the line of sight, is computed according to

$$\begin{eqnarray}&&{\cos }^{2}(i)=\displaystyle \frac{{q}^{2}-{q}_{0}^{2}}{1-{q}_{0}^{2}},\end{eqnarray} \tag{ 3 }$$

where q = b/a is the measured axis ratio of the semimajor, a, and semiminor, b, axes of the isophote, and q₀ = h_Z/h_R is a parameter representing the intrinsic flattening of a galaxy (the ratio of disk scale height, h_Z, to scale length, h_R). If q < q₀, an inclination of 90° is used. Typical values for q₀ are in the range 0.1–0.25 (Kregel et al. 2002; Dutton et al. 2005; Hall et al. 2012). While the value of q₀ for a given galaxy cannot be measured directly, some correlations with other galaxy parameters do exist. For instance, h_z scales with V_max in edge-on galaxies (Kregel et al. 2005); however, V_max is an inclination- (and thus q₀-) dependent quantity and ill-suited for this analysis. Here q₀ also correlates with morphological type (T), and we use three flattening formulations based on T-Type, q₀^[1–3] = 0.20 ± 0.03, ${q}_{0}^{[4]}=0.17\pm 0.03$, and ${q}_{0}^{[5-10]}=0.12\pm 0.02$, where the superscript represents the T-Type (Haynes & Giovanelli 1984; de Grijs 1998).

The uncertainty for each introduced variable plays a significant role in determining ${\sigma }_{{g}_{\mathrm{obs}}}$, except σ_θ, which is assumed to be negligibly small. Distance uncertainties enter into the calculation of ${g}_{\mathrm{obs}}$ only via D, as the velocity and inclination measurements are distance-independent. Using Equations (2) and (3), one may compute the first-order uncertainty on ${g}_{\mathrm{obs}}$,

$$\begin{eqnarray}&&{\left[\displaystyle \frac{{\sigma }_{{g}_{\mathrm{obs}}}}{{g}_{\mathrm{obs}}}\right]}^{2}={\left[\displaystyle \frac{2{\sigma }_{{V}_{\mathrm{obs}}}}{{V}_{\mathrm{obs}}}\right]}^{2}+{\left[\displaystyle \frac{2q{\sigma }_{q}}{1-{q}^{2}}\right]}^{2}+{\left[\displaystyle \frac{2{q}_{0}{\sigma }_{{q}_{0}}}{1-{q}_{0}^{2}}\right]}^{2}+{\left[\displaystyle \frac{{\sigma }_{{\rm{D}}}}{D}\right]}^{2},\end{eqnarray} \tag{ 4 }$$

where σ_x is the observational uncertainty on variable x. While this is formally the uncertainty on a single ${g}_{\mathrm{obs}}$ measurement, it includes D and i, which are shared variables between all ${g}_{\mathrm{obs}}$ values for a given galaxy. As the first-order calculation does not account for shared variables, it cannot yield accurate predictions. However, this simplified calculation is still useful for comparison with a full Monte Carlo model (Section 5) and the analysis performed in L17. Depending on the relative values of ${\sigma }_{{V}_{\mathrm{obs}}}$ versus σ_D and σ_i, ${\sigma }_{{g}_{\mathrm{obs}}}$ could be dominated by the local velocity measurement or shared galaxy variables (D and i). Once again, the RAR data are highly correlated, and a first-order error analysis will incorrectly predict σ_obs.

The stellar radial acceleration, ${g}_{* }$, requires the specification of a three-dimensional spatial model for the stars, as the observed photometric values only yield a projected mass distribution. In the simplest case, the stellar mass can be assumed to lie in spherical shells, and, for a given curve of growth, ${g}_{* }$ can be represented as

$$\begin{eqnarray}&&{g}_{* }=\displaystyle \frac{G{{\rm{\Upsilon }}}_{{\rm{x}}}{L}_{{\rm{x}}}}{{\left(\theta D\right)}^{2}}=\displaystyle \frac{G{{\rm{\Upsilon }}}_{{\rm{x}}}{10}^{-({m}_{{\rm{x}}}-{M}_{\odot ,{\rm{x}}})/2.5}}{{\theta }^{2}},\end{eqnarray} \tag{ 5 }$$

where G is the gravitational constant, ϒ_x is the stellar mass-to-light ratio in a photometric band x in solar units, and L_x and m_x are the luminosity and apparent magnitude in band x, respectively, enclosed by an isophote at angular radius θ. Here M_⊙,x is the absolute magnitude of the Sun in band x. The quantity ${g}_{* }$ is fortunately independent of distance errors, as the distance dependence of the luminosity L_x is canceled by the acceleration formula. The mass-to-light ratio, ϒ_x, is either constant (for the 3.6 μm band) or determined by a color mass-to-light ratio formula and thus independent of distance. The gravitational constant G and photometric normalization M_⊙,x are considered to have negligible error. However, ϒ_x can have considerable uncertainty (Conroy 2013; Courteau et al. 2014; Roediger & Courteau 2015), and the uncertainty on m_x can be significant, especially for fainter galaxies. Using Equation (5), the first-order uncertainty on ${g}_{* }$ is computed to be

$$\begin{eqnarray}&&{\left[\displaystyle \frac{{\sigma }_{{g}_{* }}}{{g}_{* }}\right]}^{2}={\left[\displaystyle \frac{\mathrm{ln}(10)}{2.5}{\sigma }_{{m}_{x}}\right]}^{2}+{\left[\displaystyle \frac{\mathrm{ln}(10)}{2.5}{\sigma }_{{m}_{0}}\right]}^{2}+{\left[\displaystyle \frac{{\sigma }_{{\rm{\Upsilon }}}}{{\rm{\Upsilon }}}\right]}^{2}\,.\end{eqnarray} \tag{ 6 }$$

As in the ${g}_{\mathrm{obs}}$ equation (Equation (2)), some variables have shared uncertainty at all points in a single galaxy. For instance, all isophotal magnitudes share the same photometric zero-point m₀ and its uncertainty. Also, m_x is correlated with all magnitude estimates interior to it, and assuming a constant ϒ_x means that its uncertainty is shared across all points in the galaxy. This means, like ${g}_{\mathrm{obs}}$, that the photometric data points ${g}_{* }$ are not independent, and a first-order analysis will incorrectly predict ${\sigma }_{{g}_{* }}$.

Equation (6) is generated under the assumption that a galaxy's mass distribution is spherical. In reality, disk galaxies are flattened structures whose potential is calculated by solving Poisson's equation, ∇²ϕ = 4πGρ, where ρ is the three-dimensional mass density. The acceleration at each location on the disk is then computed as ${g}_{* }$ = −∇ϕ. The value of ${g}_{* }$ at a given radius depends strongly on the full mass distribution. Therefore, individual ${g}_{* }$ values for disk galaxies will be even more correlated than in the spherical case. In our analysis, we use the three-dimensional density distribution from van der Kruit & Searle (1981),

$$\begin{eqnarray}&&{\rho }_{\mathrm{disk}}(R,Z)={\rm{\Sigma }}(R)\displaystyle \frac{{{\rm{sech}} }^{2}(Z/{Z}_{0})}{2{Z}_{0}},\end{eqnarray} \tag{ 7 }$$

where Σ(R) is the radial projected surface mass density, Z is the height above the disk, and Z₀ is the sech scale height of the disk. Here h_Z can be determined from q₀ and h_R by definition (h_Z = q₀h_R; see Section 2.1), and we use Z₀ = 2h_Z to find the sech scale height (Kregel et al. 2002). With a density distribution specified, Poisson's equation is solved by galpy (Bovy 2015) using a self-consistent field method (Hernquist & Ostriker 1992). This treatment is only applied to observed galaxies and thus the observed RAR. For the Monte Carlo model described in Section 5, the mock galaxies are treated as spherically symmetric, given the difficulty in inverting Poisson's equation. In the next section, we discuss the galaxy samples used to construct and analyze the RAR.

3.1.1. M92

3.1.2. M96

3.1.3. C97

3.1.4. Courteau et al. (2000)

3.1.5. L16

3.1.6. Ouellette et al. (2017)

A number of data quality criteria were implemented in our compilation. Only galaxies with inclinations greater than 30° were considered; rotational velocities are corrected by sin(i)⁻¹ and would therefore diverge at lower inclination. This study focuses solely on late-type galaxies, and only galaxies with morphological types from Sa to Im are considered (T-Type of 1–10). Only a few other types were available in each survey and were not beneficial to this analysis. Individual RC data points were discarded if σ_V/V > 0.1, as in L17. Individual SB data points were discarded if σ_μ > 0.1 mag arcsec⁻² or μ > 25. Data points within 5'' of the center of a galaxy were also discarded to avoid seeing effects.

We now present the RAR for each survey in Figure 1. Each panel shows a fit to the data using Equation (1); the fit uses an orthogonal distance regression  (Jones et al. 2001). The axes are calculated using the analysis described in Sections 2.1 and 2.2 plus an extra 0.33 dex added to the ${g}_{* }$ axis to account for the missing gas mass to the measured stellar mass values. This is done largely for aesthetic and fitting reasons (properly positioned and reasonable fitted ${g}_{\dagger }$ values) and plays no role in our study of the RAR scatter. The RAR scatter, σ_RAR, is measured relative to a running median in a window chosen to include ∼100 data points. A parametric assessment of the scatter would be biased by the arbitrariness of the RAR functional form and suffer from an artificial increase to the scatter due to any misalignment with the data. It is thus avoided here.

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The values for ${g}_{\dagger }$ in Table 1 differ by many standard deviations. It is, however, important to note that that table only includes bootstrap random errors. The large systematic uncertainties that affect variables such as the mass-to-light ratios, which could exceed a factor of 2, are not included in this treatment. Accounting for them would easily reconcile all of the values of ${g}_{\dagger }$ with each other (Courteau et al. 2014; Roediger & Courteau 2015). Therefore, strong statements regarding the universality of ${g}_{\dagger }$ based on our six surveys cannot be made. As in Figure 1 and Table 1, each survey is analyzed separately throughout the paper. Systematic differences between the surveys that could artificially inflate the observed scatter are thus minimized.

Table 1.  Parameterization of the RAR Fit to Each PROBES Data Set

1	2	3	4
Survey	${g}_{\dagger }$	σRAR	N
	(10−10 m s−1)	(dex)	No.
SV	0.813 ± 0.044	0.166 ± 0.007	1380
SP	0.839 ± 0.020	0.131 ± 0.003	2561
SF	0.558 ± 0.008	0.139 ± 0.001	16,648
C97	5.224 ± 0.042	0.182 ± 0.001	17,626
M92	1.790 ± 0.016	0.173 ± 0.001	18,355
M96	1.345 ± 0.012	0.170 ± 0.001	21,460

Note. Column 1 indicares the survey (as in Figure 1). In column (2), ${g}_{\dagger }$ and its bootstrap uncertainty emerge from Equation (1). Column (3) gives the 16%–84% interval scatter of the forward residuals, σ_RAR, and its bootstrap uncertainty. Column (4) gives the number of data points in the RAR.

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Depending on the method, distances may carry significant uncertainties and represent an important source of scatter in the RAR. Most galaxies in the SHIVir and SPARC surveys are close enough to get distance measurements with variable stars, SB fluctuations, cluster distance, tip of the red giant branch, or supernovae. Their different uncertainties are reported in each survey. For the more distant galaxies in M92, M96, C97, and Shellflow, Hubble flow distances are used  (Riess et al. 2016). A primary source of distance uncertainty when using this method is due to peculiar motions. Here we assume the peculiar motion to be of order σ_pec = 300 km s⁻¹, where σ_pec is the peculiar velocity dispersion. Many galaxies in the surveys are far enough that this σ_pec value is relatively small, so a lower relative uncertainty bound of σ_D = 0.15D is applied. Thus, for distance uncertainties, we have σ_D = max(0.15D, 300) for Hubble flow distances.

Inclination uncertainties are modeled in multiple steps. First, a per-galaxy axis ratio uncertainty is computed using the scatter in isophotal ellipticity of the outer regions of the SB profile (ellipticities beyond R_e). These cover a large range of isophotal fitting uncertainties, as shown in Figure 2. Second, the intrinsic flattening parameter q₀ is selected with a Gaussian uncertainty, truncated to the range [0.05–0.25], representing typical acceptable values for the intrinsic flattening of a disk (Kregel et al. 2002).

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These are combined through Equation (3) to determine the adopted inclinations. Figure 3 demonstrates the results of this model. For a given axis ratio q, a vertical slice in Figure 3 gives the corresponding uncertainty distribution in inclination i.

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Rotational velocity uncertainties are reported for each survey except M92 and M96. For these, a uniform velocity uncertainty of σ_V = 6 km s⁻¹ is used, representing the median uncertainty for the other surveys. Figure 4 demonstrates the distribution of relative rotational velocity uncertainties for each survey. The black dashed line indicates the adopted cutoff threshold of 10% (L17), where velocity measurements are not considered in the analysis. Velocity uncertainties account for measurement errors but ignore noncircular motions (see Section 6.2).

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As with rotational velocities, magnitude uncertainties are reported for each survey except M92 and M96. The reported uncertainties are depicted in Figure 5 as a function of the SB values. We find that magnitude uncertainties correlate more tightly with SB than magnitude measurements, hence this choice of variable. The dashed red line in the Shellflow panel represents the SB uncertainties used for M92 and M96. This is done by fitting σ_SB = a + e^{b(μ − c)} to the values that meet the data quality threshold (black dashed lines). The Shellflow distribution is adopted, as it relies on the same photometric band as M92 and M96; the resulting parameterization for (a, b, c) is (0.00075, 0.52, 31.97). Magnitude uncertainties are small compared to other sources of uncertainty; however, they are retained for consistency.

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Stellar mass-to-light ratios also carry a significant model uncertainty. The exact normalization of the mass-to-light ratio is inconsequential to our analysis; only the intrinsic variability about the relation matters. Roediger & Courteau (2015) found the random uncertainty of the stellar mass-to-light ratio to be of order 0.09–0.13 dex for optical mass-to-light ratios. A representative conservative uncertainty of 0.13 dex is therefore used. Systematic uncertainties (e.g., model-to-model differences) for the stellar mass-to-light ratio of the order of 0.3 dex (Conroy 2013; Courteau et al. 2014; Roediger & Courteau 2015) are of course larger; however, this uncertainty is not used for our analysis, as a systematic shift (constant factor in log space) does not impact scatter measurements. For the uncertainty in the 3.6 μm stellar mass-to-light ratio, an uncertainty of 0.11 dex is used, as suggested by Meidt et al. (2014). The effect of choosing even larger stellar mass-to-light ratio uncertainties is partially explored in Section 6.2.

To assess the scatter induced by observational uncertainties, one must eliminate the effects of intrinsic scatter (if any) in the RAR. As the observational and intrinsic scatter are mixed, let us first eliminate all scatter. To this end, all data points in the ${g}_{* }$ ${g}_{\mathrm{obs}}$ space are projected onto the fitted RAR to impose a zero-scatter assumption. To project a data point onto the RAR, its final scatter-free location must first be identified. This is equivalent to asking what new values of g_*,zs and g_obs,zs, where "zs" means zero scatter (constrained to the RAR), should be chosen. One option is to project along the ${g}_{* }$ or ${g}_{\mathrm{obs}}$ axis; however, this means artificially selecting an axis. A second option is to adopt an orthogonal projection, where each g_*,zs and g_obs,zs is the minimum distance (in ${g}_{* }$, ${g}_{\mathrm{obs}}$ space) from the measured ${g}_{* }$, ${g}_{\mathrm{obs}}$. However, this assumes that the uncertainties in each axis are identical, which may not be true either. Instead, a third and favored option is to consider the full uncertainty distributions from Section 4 for each parameter, construct a likelihood function by multiplying the probabilities, and choose the location that maximizes the likelihood while imposing a scatter-free RAR (see Equation (1)).

As every parameter is given a probability distribution in Section 4, it is possible to construct a likelihood that is a function of each variable but constrained to lay on the RAR. This removes one degree of freedom for each point on the RAR. The equations from Sections 2.1 and 2.2 can be used to connect ${g}_{\mathrm{obs}}$ and ${g}_{* }$ to their respective measurements,

$$\begin{eqnarray}&&V=\sin (i)\sqrt{{g}_{\mathrm{obs}}\theta D},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{L}_{x}=\displaystyle \frac{{g}_{* }{\left(\theta D\right)}^{2}}{G{{\rm{\Upsilon }}}_{x}},\end{eqnarray} \tag{ 9 }$$

where ${g}_{\mathrm{obs}}$ is determined using Equation (1), thus constraining it to lay on the RAR. Some of the variables are shared between multiple observations in the same galaxy (distance, axis ratio, intrinsic thickness, and mass-to-light ratio), so the optimization must be performed for all observations simultaneously. For a galaxy with N observations on the RAR, the likelihood function can be represented as

$$\begin{eqnarray}&&\begin{array}{l}{ \mathcal L }({g}_{* ,1}^{{\prime} },{g}_{* ,2}^{{\prime} },\ldots ,{g}_{* ,{\rm{N}}}^{{\prime} },q^{\prime} ,{q}_{0}^{\prime} ,{{\rm{\Upsilon }}}_{x}^{\prime} ,D^{\prime} )\\ \ =\ P(q^{\prime} | q,\sigma q)\cdot P({q}_{0}^{\prime} | {q}_{0},\sigma {q}_{0})\cdot P({{\rm{\Upsilon }}}_{x}^{\prime} | {{\rm{\Upsilon }}}_{x},\sigma {{\rm{\Upsilon }}}_{x})\cdot \\ \ P(D^{\prime} | D,\sigma D)\cdot {{\rm{\Pi }}}_{i\,=\,1}^{N}P({V}_{i}^{{\prime} }| V,\sigma V)\cdot P({L}_{i}^{{\prime} }| {L}_{i},\sigma {L}_{i}),\end{array}\end{eqnarray} \tag{ 10 }$$

where the primed quantities are the new zero-scatter values and the unprimed quantities are the measurements and their uncertainties. Here ${V}_{i}^{{\prime} }$ and ${L}_{i}^{{\prime} }$ are computed from Equations (8) and (9), respectively, using the primed quantities; ${g}_{\mathrm{obs}}$ is computed from Equation (1). Ultimately, this process finds the values for all quantities that are most consistent with the original measurements while remaining constrained on the RAR. The values for each variable are now taken as the "true" values, upon which the observational uncertainties can scatter measurements off the RAR. This process is described in the next section.

Using the uncertainty models from Section 4, new "observations" can be sampled about the zero-scatter values as described in Section 5.1. To visualize the effect of this Monte Carlo model, Figure 6 shows how the different uncertainties affect the final location of a single point off of the RAR. New data points are sampled for each galaxy and compiled into the same data structure as the original measurements. As the RAR is a local scaling relation, each galaxy places multiple data points on the relation; however, some variables are global for the whole galaxy. For example, a single distance value is resampled for each galaxy. Thus, it is clear that the data points in the RAR are not independent, as multiple measured values with large uncertainties are shared.

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To match the statistics of the original observations, each point is resampled only once. Thus, the simulated data have exactly the same number of initial entries as the measured data. To include the effects of data quality cuts, all measured values are resampled before the data quality cuts are applied. This means that some points in the RAR that were initially cut may be newly introduced, for example, if a galaxy had an inclination below 30° but was resampled above that value.

The mock data are then processed with the same code as the original data, thus duplicating any data quality cuts, interpolations, and conditional statements. This critical element is missing from a first-order analysis (e.g., L17) and can potentially impact the final result, as will be seen in Section 6.2. An element of L17's analysis that is not reproduced here is the flat-disk model for the baryonic acceleration. Inverting Poisson's equation adds unnecessary complexity without impacting the scatter measurements that are the primary focus of this paper. In Section 5.1, the projection onto the RAR is made under the assumption that the galaxies are spherical; thus, the calculations remain internally consistent. In both the flat-disk analysis and spherical assumption, the baryonic acceleration depends linearly on luminosity and mass-to-light ratio, which are the two dominant sources of uncertainty in ${g}_{\mathrm{bar}}$. The uncertainty model for these quantities should behave identically for the two baryon distributions.

A first-order calculation of the average observational uncertainty for the SPARC data set was presented by L17. Here we reproduce this calculation for the full PROBES sample. Equations (4) and (6) present the propagation of uncertainties through the calculation of ${g}_{* }$ and ${g}_{\mathrm{obs}}$. In L17, the assumed luminosity uncertainty is σ_L = 0.04 dex, and the assumed mass-to-light ratio uncertainty is σ_ϒ = 0.1 dex, meaning that the baryonic acceleration uncertainty is always ${\sigma }_{{g}_{b}}$/${g}_{b}$ = 0.11 dex. In this paper, we consider the same luminosity uncertainty and more conservative values for the mass-to-light ratio uncertainty of ${\sigma }_{{\rm{\Upsilon }}}^{[3.6\,\mu {\rm{m}}]}=0.11\,\mathrm{dex}$ (Meidt et al. 2014). For optical wavelengths, we use ${\sigma }_{{\rm{\Upsilon }}}^{\mathrm{optical}}=0.13\,\mathrm{dex}$ (Roediger & Courteau 2015; Zhang et al. 2017). While both ${g}_{* }$ and ${g}_{\mathrm{obs}}$ contribute to the scatter in the forward direction, the stellar acceleration uncertainty is modified by the RAR as the slope changes across the relation. The average scatter in the forward residuals takes the form

$$\begin{eqnarray}&&{\sigma }_{1\mathrm{stOrder}}=\displaystyle \frac{1}{N}\displaystyle \sum ^{N}\displaystyle \frac{1}{{g}_{\mathrm{obs}}}\sqrt{{\sigma }_{{g}_{\mathrm{obs}}}^{2}+{\left(\displaystyle \frac{\partial { \mathcal F }}{\partial {g}_{* }}{\sigma }_{{g}_{* }}\right)}^{2}},\end{eqnarray} \tag{ 11 }$$

where σ_1stOrder ≈ σ_obs is an estimate of the scatter in the residuals due to observational uncertainties, N is the number of observations, and ${ \mathcal F }$ is the functional form of the RAR (Equation (1)). Table 2 presents the results of this calculation for each survey, where the first column represents the survey, the second column reports the observed scatter of the RAR, the third column gives the first-order scatter result from Equation (11), and the fourth column reports the intrinsic scatter resulting from the difference in quadrature (${\sigma }_{\mathrm{int}}^{2}={\sigma }_{\mathrm{RAR}}^{2}-{\sigma }_{1\mathrm{stOrder}}^{2}$). Each survey does indeed have different observational uncertainties, as expected for surveys with such a range of properties. The median value for the intrinsic scatter is then 0.13 dex; this can be compared to the results found in ΛCDM simulations, which range from 0.06 to 0.08 dex (Keller & Wadsley 2017; Ludlow et al. 2017). The value determined from first-order calculations is substantially larger than expected from ΛCDM simulations and certainly not consistent with zero.

The values in Table 2 are close to those from L17. Our respective values of σ_RAR are essentially identical. For σ_1stOrder, we find 0.1 dex, while L17 quoted 0.12 dex. This discrepancy originates in slightly different choices in uncertainty values for some parameters. We used larger uncertainty values than L17 for the stellar mass-to-light ratios; for the inclination, we typically found smaller uncertainties with our model from Section 4.2. These differences account for most of the 0.02 dex discrepancy.

Here we present the Monte Carlo simulation of each survey with only observational uncertainties about the RAR. With the full data set resampled, any scatter metric (such as the orthogonal scatter) could be chosen to compare with the original data. For consistency with other studies and the first-order analysis in Section 6.1, the forward residuals will be used as the scatter metric. Figure 7 presents the simulated data with the same analysis as in Section 3.3. Many qualitative elements from Figure 1 are retained; however, the distributions in Figure 7 are clearly smoother and tighter. This visual impression is confirmed in Table 3, which lists the same parameters as Table 1, where the scatter measured for each simulated survey is below that of the observed one. Table 4 presents the scatter measurements for the mock data in the same format as Table 2.

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Table 3.  Parameterization for the RAR Fit to Each Mock Data Set

1	2	3	4
Survey	${g}_{\dagger }$	σsim	N
	(10−10 m s−1)	(dex)	No.
SV	0.768 ± 0.020	0.134 ± 0.004	1685
SP	1.051 ± 0.021	0.125 ± 0.003	2416
SF	0.611 ± 0.004	0.117 ± 0.001	17,173
C97	5.626 ± 0.024	0.100 ± 0.001	18,065
M92	1.810 ± 0.009	0.130 ± 0.001	18,401
M96	1.340 ± 0.007	0.121 ± 0.001	21,137

Note. Column (1) indicates the survey (as in Figure 1). Column (2) is the parameterization for Equation (1) and its bootstrap uncertainty. Column (3) is the 16%–84% interval scatter reported with its bootstrap uncertainty. Column (4) gives the number of data points in the RAR.

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As with our first-order scatter calculations in Section 6.1, one may compute the intrinsic scatter by taking the difference in quadrature with the observed scatter. Using the mock data, the median intrinsic scatter is 0.11 ± 0.02, where the uncertainty is the weighted (by points in the RAR) standard deviation of the intrinsic scatter values. These results are in closer agreement with ΛCDM expectations (Keller & Wadsley 2017; Ludlow et al. 2017; Dutton et al. 2019) than the first-order analysis. Dutton et al. (2019) noted that the baryonic RAR scatter depends on galaxy mass. We observe the same effect with PROBES galaxies but not our mock data. Appendix B explores the RAR as a function of mass bins. This sheds light on the range of scatter measurements among surveys that sample different mass regimes.

Several factors may potentially cause the Monte Carlo simulated scatter to underrepresent the full observational uncertainties. For instance, noncircular motions, which might increase the observed scatter, especially in the inner disk, are not modeled here. We find that the median intrinsic scatter for RAR points beyond one R_e, i.e., for regions relatively free of noncircular motions, is 0.10±0.01 dex, still in excellent agreement with ΛCDM simulations (Keller & Wadsley 2017; Ludlow et al. 2017; Dutton et al. 2019).

It is also possible that some of the observational uncertainties have been underestimated. To assess that possibility, we consider a scaling factor that is applied to all uncertainty values for a given parameter. We consider one parameter x at a time, with its uncertainty being scaled as ${\sigma }_{x}^{{\prime} }$ = σ_x and as the scale factor, while all other uncertainties are kept at their fiducial value. The scaling factor is tuned until the Monte Carlo simulated scatter is in agreement with the observed scatter (indicating zero intrinsic scatter). The final scaling factor value is reported in Table 5.

Table 5.  Scaling Factors for the Uncertainty in Each Observed Parameter

Parameter	V	ϒ	L	q	q0	D	m0
	3.0	1.7	>10	7	>10	2.1	5

Note. When scaled by one of these factors, the uncertainties reported here would be large enough to produce a median intrinsic scatter equal to zero. The parameters (left to right) are rotational velocity, stellar mass-to-light ratio, total luminosity, disk axial ratio, intrinsic disk flattening, distance, and magnitude zero-point.

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This exercise suggests that, in order to explain away all intrinsic scatter in the RAR, the parameter errors would need to be inflated by factors of 2 (stellar mass-to-light ratio) to 10 (total luminosity and disk flattening ratio). Considering the already conservative values adopted for the uncertainty in each variable, it seems unlikely that the scaled uncertainties represent a reasonable path to explain the nonzero scatter. We reiterate that the nonzero scatter is most likely a true feature of the data, in agreement with ΛCDM simulations.