No evidence for Keplerian taper of far-out galactic rotation

Abstract

We present a statistical analysis of the 175 SPARC galactic rotation curves to test the hypothesis of whether the Keplerian velocity tapering at large radii ($V(r)\propto 1/\sqrt{r}$) germane to a convergent mass distribution in typical spherical halo models agrees with observational data. The null hypothesis is Rubin's flat-rotation curve, $V(r)=\text{constant}$ -such as can be obtained from a spherical, isothermal-like density profile, or alternatively with a very prolate halo-. To decide whether we adopt the null (Rubin behaviour) or alternative (Keplerian behaviour) hypothesis, we evaluate the derivative in each galaxy of $V(r)$ with its last data points. The test is model independent inasmuch we are testing for the \emph{slope} of the dark matter rotation curve, whether it is or not compatible with zero. We conclude that the data is presently compatible with the null hypothesis -no taper off, no decline of $V(r)$ is seen. Separately, beyond SPARC, our own Milky Way galaxy, for which recent data sets have been reported, does show clear $V(r)$ fall-off at the level of 20\%.

Figures (7)

• Figure 1: Left: Dark-Matter induced rotation curve within a supposedly spherical NFW halo (with the parameters of the Milky Way to fix the units, though the shape is universal) At distances from the center doubling that of the local maximum, the velocity curve clearly displays the Keplerian $1/\sqrt{r}$ fall-off. Right: Example galaxy with well measured rotation curve that does not fall-off but rather flattens out (shaded area) at large radius (even past three to four times the would-be maximum) in contrast to the NFW prediction. This apparent shape disagreement motivates our study.
• Figure 2: The distribution of terminal slopes $V'(r)$ is not symmetrically distributed around $A=0$, and does not seem to have a preference for negative slope values. The black box represents the interquartile range (IQR) for the distribution of the slopes fit to the rotation data, from the first (Q1) to the third (Q3) quartiles, and they are comprised within the small range of 2 km/(s kpc) (units of the OX axis). The black line inside that box lies at the median of the $V'(r)$ distribution and is not far from 0. The whiskers extend to 1.5$\times$IQR above and below Q1 and Q3, and outside these values we find the occasional outlier (empty circle).
• Figure 3: Left: Histogram representation of the galactic end slopes together with its smoothing obtained by a Kernel Density Estimation (KDE, blue line) and compared to a normal distribution (red line). The symmetric shape centered near 0, as the statistics tests suggest, is clearly visible by eyeball. Right: Quantile-quantile representation to assess whether the slopes are normally distributed as a Gaussian. The data points are sorted and represented against theoretical quantiles from a normal distribution. the red line is the best fit through the points. When the data points are close to the line the data are approximately normal, and this is reasonably the case, lending credibility to the parametric $Z$ test of Subsection \ref{['subsec:hypotest']}.
• Figure 4: Milky Way's total rotation curve as compiled by Sofue:2020rnl (the uppermost data set with $V(r)$ not yet falling), as well as the latest measurements reported by ouwangZhou_2023Jiao_2023. These data show a steady decline of the rotation curve for $r>15$ kpc. The lines underneath show the contribution of ordinary baryonic matter to the rotation curves, modelling the galaxy components: disk, bulge, gas and dust. At these distances, baryonic contributions are still important. The inferred DM contribution is dominant in Sofue's compilation at large distance and does seem to flatten out.
• Figure 5: SPARC rotation curves with slightly positive $V'(r)_{DM}$ slope at large $r$.