A Cuspy Dark Matter Halo

Our L-band observations were performed with the Karl G. Jansky Very Large Array (VLA) through two projects corresponding to the D configuration (19B-072; PI: Y. Shi) and the C and B configurations (20A-004; PI: Y. Shi). The D-configuration observations were performed on 2019 September 21 and 23, each with 1.5 hr observing time and ∼66 minutes on-source time. A total of 27 antennas were employed in both executions, with the first under overcast weather conditions and the second under clear weather conditions. The projected baselines of the D-configuration array are in the range of 34–1050 m. The flux and bandpass calibrator was 3C 295, and the gain calibrator was J1419+0628. We configured the spectrometers with a mixed setup. The spectral line window that covers the H i 21 cm line has a bandwidth of 128 MHz and a channel width of 32.3 kHz (∼6.9 km s⁻¹), while the remaining windows cover a frequency range between 963.0 and 2017.0 MHz to optimize the sensitivity for the radio continuum.

The C-configuration observations were performed on 2020 February 4, 9, and 13, each with 2 hr observing time and ∼90 minutes on-source time. Two observations were conducted with 25 antennas, while the rest were observed with 26 antennas, mostly under overcast or clear weather conditions. The projected baselines of the C configuration are in the range of 40–3200 m. In the C-configuration observations, the 21 cm emission line was observed with a spectral line window that has a channel width of 5.682 kHz (∼1.2 km s⁻¹) and a bandwidth of 32 MHz. The remaining windows cover a frequency range between 963.0 and 2017.0 MHz for optimizing the radio continuum bandwidth. The flux and bandpass calibrator was 3C 286, and the gain calibrator was J1419+0628.

The B-configuration observation was performed with five executions during 2020 July and August, each with 2 hr observing time and ∼90 minutes on-source time. In most executions, 27 antennas were employed under cloudy conditions. The projected baselines of the B configuration are in the range of 230–11,000 m. The hardware setup and calibrators were the same as those used for the C-configuration observations.

We reduced all data manually with the Common Astronomy Software Applications (CASA) package (McMullin et al. 2007) v5.6.1. Both D-configuration observations were severely affected by radio frequency interference (RFI). Therefore, we calibrated and flagged the data manually. Approximately 20% of the data have to be flagged across different frequency ranges. The data obtained on 2019 September 21, including the calibrators, have about three times higher noise than that estimated from the VLA sensitivity estimator, for unknown reasons. However, the fluxes, line profile, and spatial distributions are highly consistent with the data obtained on 2019 September 23. Whether we combine it to the final data does not change the results. All gain calibrators were checked carefully to ensure a flat bandpass, a point-source distribution, and excellent calibration solutions. The 1.4 GHz flux densities of 3C 286, 3C 295, and J1419+0628 were 15.1 ± 0.1, 22.3 ± 0.1, and 6.0 ± 0.1 Jy.

After flagging and calibration, we weighted all data sets with their noise levels using the statwt task. Then, through the uvcontsub task, we fitted and subtracted the radio continuum from the visibility data, with line-free channels on both sides of H i, using the first-order linear function. We resampled the C- and B-array data to match with the spectral resolution of the D-array data using the mstransform task. The final spectrally matched line-only data from the D, C, and B configurations were combined together with the concat task.

In the end, we inverted the visibility data to the image plane and cleaned the data cube with Briggs' robust parameter of 2.0 using the tclean task. The final D + C + B data cube has velocity coverage of 500 km s⁻¹ and a channel width of 32 kHz, corresponding to a velocity resolution of ∼7 km s⁻¹. The rms noise level reaches 0.26 mJy beam⁻¹ channel⁻¹ with a restoring synthesis beam size of 985 × 933 and a position angle of 1756.

The H i intensity map is shown in Figure 2(a). The integrated spectrum of the B + C + D configuration has a flux of 3.8 ± 0.16 Jy km s⁻¹, which is comparable to the flux of 3.4 Jy km s⁻¹ obtained with Arecibo (Leisman et al. 2017), as shown in Figure 2(b). The total H i gas mass is (8.51 ± 0.36) × 10⁸ M_⊙. A slightly higher flux as seen by the VLA in the red wing may be caused by a small offset of the Arecibo beam from the galaxy center, given the large size of the galaxy of 2' in diameter. The radial profile of the gas mass surface density is shown in Figure 2(c). The channel map is shown in Figure 3.

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We also tested tilted-ring modeling with the combined data using the B + C configurations and found essentially no difference, except for a higher noise level. However, the better velocity resolution leads to better estimates of the pressure support, which has a very minor contribution to the decomposition of the rotation curve.

The coadded images at 3.6 and 4.5 μm were obtained from the Wide-field Infrared Survey Explorer (WISE) archive (Wright et al. 2010) with spatial resolutions of 60 and 68, respectively. The target is well detected at 3.6 μm, as presented in Figure 4(a), but shows almost no detection at 4.5 μm. To derive the radial profile of the stellar mass surface density as presented in Figure 4(b), a few nearby bright stars in the field were subtracted using the stellar point-spread functions. ⁴ The stellar mass based on the 3.6 μm image is estimated to be (1.37 ± 0.05) × 10⁸ M_⊙ for a Kroupa stellar initial mass function and a mass-to-light ratio ϒ_{3.6 μm} = 0.6 (M_⊙/L_{⊙,3.6 μm}) (see below), with a half-light radius of 3.7 kpc.

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The optical g and r images were obtained from the Dark Energy Camera Legacy Surveys (Dey et al. 2019). The far-ultraviolet image was retrieved from the Galaxy Evolution Explorer (GALEX) data archive. The integrated flux was converted to an SFR of (8.2 ± 0.4) × 10⁻³ M_⊙ yr⁻¹ for a Kroupa stellar initial mass function (Leroy et al. 2008).

Integral field unit observations of Hα were carried out with the Wide-Field Spectrograph (WiFeS) on board the Australian National University 2.3 m telescope on the nights of 2020 March 21–22, May 26, and July 20–22. An R7000 grism (5290–7060 Å) at a resolution of 7000 was adopted to cover the Hα emission line. WiFeS has a field of view of 25'' × 38''. As shown in Figure 4(a), exposures were taken at several positions to cover the whole optical extent of the galaxy, with one pointing toward a nearby bright star for astrometric calibration. Each exposure was 30 minutes, and the total on-source integration time varied from 2.0 to 3.5 hr at each pixel. For every 1–2 hr, an off-target blank sky and a standard star were observed. The seeing was between 15 and 20. Each frame was first reduced following the standard procedure by pyWiFeS (Childress et al. 2014) and then was subtracted by the median value of the sky frame at each wavelength. Individual frames were aligned with each other to produce the final mosaic image using the positions of the Hα clumps, as the continuum emission was too faint. The absolute astrometry was obtained through alignment with the brightest star in the mosaic field of view. Since Hα clumps are not perfect point sources, we estimated the final astrometric uncertainty to be about 1''. To correct the barycentric velocity offset and any possible intrinsic instrumental shift, the wavelength solution of each night was further cross-calibrated based on Hα lines of the same clumps observed during different nights. The integrated Hα flux map is presented in Figure 4(c) with the integrated spectrum shown in Figure 4(d).

We derived the rotation curve of the dark matter by first obtaining the total rotation curve from the H i and Hα data with a correction for the pressure support and then quadratically subtracting the gas and stellar gravity contributions as detailed below.

3.1.1. The Observed Rotation Curve from Tilted-ring Fitting to the H i Data Cube

We fitted the tilted-ring model to the H i 3D data cube with ^3D Barolo (Di Teodoro & Fraternali 2015) to obtain the rotation curve as listed in Table 2. The radial width of each ring is set to 9'', which is roughly the beam size, so that each ring is independent. We adopted uniform weighting (WFUNC = 0) and least-squares minimization (FTYPE = 1). All other optional parameters are set as default. The full list of the setup is included in Table A1.

Table 2. The Results of the Tilted-ring Modeling

Rad	${V}_{\mathrm{obs}}^{\mathrm{rot}}$	${V}_{\mathrm{obs}}^{\mathrm{circ}}$	σdisp	${V}_{\mathrm{gas}}^{\mathrm{circ}}$	${V}_{\mathrm{star}}^{\mathrm{circ}}$	${V}_{\mathrm{dark} \mbox{-} \mathrm{matter}}^{\mathrm{circ}}$	P.A.	Inclination
(kpc)	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(deg)	(deg)
H i
0.67	10.4 ± 1.2	10.6 ± 1.2	4.5 ± 1.3	0.2 ± 1.2	1.9 ± 0.5	10.4 ± 1.2	2.6	73.0
2.02	14.5 ± 1.3	15.5 ± 1.2	7.7 ± 1.3	4.1 ± 0.4	4.5 ± 0.3	14.2 ± 1.4	0.3	72.5
3.36	22.3 ± 1.4	23.5 ± 1.3	7.7 ± 1.0	8.2 ± 0.3	8.3 ± 0.4	20.4 ± 1.5	−0.2	73.1
4.71	29.4 ± 1.8	30.7 ± 1.7	7.0 ± 1.6	11.3 ± 0.3	10.4 ± 0.5	26.6 ± 1.9	0.2	73.2
6.06	33.8 ± 1.6	35.0 ± 1.5	6.5 ± 1.6	15.0 ± 0.3	10.6 ± 0.3	29.9 ± 1.8	1.0	72.4
7.40	36.5 ± 1.6	37.5 ± 1.6	5.8 ± 1.6	19.1 ± 0.4	10.1 ± 0.3	30.6 ± 1.9	1.7	71.2
8.75	40.4 ± 2.0	40.9 ± 2.0	4.6 ± 1.6	22.4 ± 0.5	9.6 ± 0.4	32.8 ± 2.5	1.9	70.2
Hα
1.51		16.8 ± 1.6		2.5 ± 0.7	3.5 ± 0.4	16.2 ± 1.7
2.78		22.2 ± 1.3		6.7 ± 0.3	6.9 ± 0.4	20.0 ± 1.4
3.45		24.2 ± 0.8		8.4 ± 0.3	8.4 ± 0.4	21.0 ± 1.0
4.54		29.6 ± 2.2		10.9 ± 0.3	10.3 ± 0.5	25.6 ± 2.5

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The angular resolution and signal-to-noise ratio were good enough to set the rotation velocity, velocity dispersion, position angle, and inclination angle as free parameters for each ring. Before running this setup, we first ran a model by also setting the dynamic center and the systematic velocity as free parameters and then fixed them to the mean values of all rings as listed in Table 1.

The scale height was fixed to 100 pc, independent of the radius. If the height is varied by a factor of 5 (see below), the conclusion remains unchanged. We do notice that ^3D Barolo is not able to remove effects fully due to the disk height for a thick disk (Iorio et al. 2017). When running ^3D Barolo, we adopted the "two-stage" fitting method, which allows a second fitting stage after regularizing the first-stage parameter sets. As shown in Figure 5, the radial profiles of the inclination and position angles from the first-stage fitting are regularized by fitting a Bezier function, based on which the second-stage fitting is performed. The errors of the derived rotation velocity and velocity dispersion in ^3D Barolo (Di Teodoro & Fraternali 2015) are estimated through a Monte Carlo method. We also ran ^3D Barolo with only the receding and approaching sides, respectively. The velocities from three runs are within the 1σ errors, while the velocity dispersions are also within the errors except for the innermost radius, which is shown in Figure B1 of Appendix B. As shown in Figure 5, the derived rotation velocity and velocity dispersion vary well within the 1σ error bars before and after regularization.

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As shown in Figure 6, within the outer boundary of the last ring, the residual of the H i intensity is dominated by the observed flux noise, and the residuals of the velocity and velocity dispersions show small amplitudes with medians of 1.9 and 1.8 km s⁻¹, respectively. Figure 7 further shows the position–velocity diagram along the major and minor axes. Overall, the observation matches the best-fit model well.

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As gas is collisional, pressure support is a driver of gas motion in addition to gravity (Bureau & Carignan 2002; Oh et al. 2015; Iorio et al. 2017; Pineda et al. 2017). A gas disk in equilibrium satisfies

$$\begin{eqnarray}&&{V}_{\mathrm{circ}}^{2}={V}_{\mathrm{rot}}^{2}+{V}_{{\rm{P}}}^{2},\end{eqnarray} \tag{ 1 }$$

where the circular velocity V_circ = $\sqrt{R\tfrac{\partial {\rm{\Phi }}}{\partial R}}$ solely reflects the effect of the gravitational potential, V_rot is the observed rotation velocity, and V_P is the velocity driven by the pressure. Here V_P is related to the gas density (ρ) and velocity dispersion (σ_v) following

$$\begin{eqnarray}&&{V}_{{\rm{P}}}^{2}=-R{\sigma }_{{\rm{v}}}^{2}\displaystyle \frac{\partial \mathrm{ln}(\rho {\sigma }_{{\rm{v}}}^{2})}{\partial R}.\end{eqnarray} \tag{ 2 }$$

Assuming that the scale height is independent of the radius and that the disk is thin, the above equation becomes

$$\begin{eqnarray}&&{V}_{{\rm{P}}}^{2}=-R{\sigma }_{{\rm{v}}}^{2}\displaystyle \frac{\partial \mathrm{ln}({\sigma }_{{\rm{v}}}^{2}{{\rm{\Sigma }}}_{\mathrm{obs}}\cos i)}{\partial R},\end{eqnarray} \tag{ 3 }$$

where Σ_obs is the observed gas mass surface density, and i is the inclination angle. Following Equation 3, the pressure support correction is implemented in ^3D Barolo (Iorio et al. 2017). For AGC 242019, the correction is small, with ∣V_circ − V_rot∣ being smaller than ∼1.0 km s⁻¹ across the radius. In this case, only the error of V_rot is propagated to V_circ while ignoring the error of V_A. The radial profile of V_circ is referred to as the observed rotation curve.

Note that our H i data have a spectral resolution of 7 km s⁻¹, which is comparable to or even lower than the derived velocity dispersion shown in Figure 5. To check the effect of the limited spectral resolution, we also performed tilted-ring fitting on the H i data cube from the B and C configurations that has a spectral resolution of 2.4 km s⁻¹. The difference in the velocity dispersion between the two data is within the 1σ error, and the difference in the derived rotation curve is within the 1σ error too.

3.1.2. The Observed Rotation Curve from the Hα Map

3.1.3. The Contribution to the Rotation Curve from the Gravity of Gas

3.1.4. The Contribution to the Rotation Curve from the Gravity of Stars

The models of the cuspy dark matter halo are motivated by numerical simulations of dark matter, including an NFW model (Navarro et al. 1997) and an Einasto model (Einasto 1965). The models of the cored dark matter halo are observationally motivated, including the ISO model (Begeman et al. 1991) and Burkert model (Burkert 1995).

(1) An NFW halo model (Navarro et al. 1997) has a density profile of

$$\begin{eqnarray}&&{\rho }_{\mathrm{NFW}}(R)=\displaystyle \frac{{\rho }_{c}{\delta }_{\mathrm{char}}}{(R/{R}_{S}){\left(1+R/{R}_{S}\right)}^{2}},\end{eqnarray} \tag{ 4 }$$

where ρ_c = 3H²/(8πG) is the present critical density, δ_char is dimensionless density contrast, and R_S is the scale radius. Here R_S is related to R₂₀₀ through the concentration c = R₂₀₀/R_S , where R₂₀₀ is the radius within which the halo average density is 200 times the present critical density. Here ${\delta }_{\mathrm{char}}=\tfrac{200{c}^{3}g}{3}$ with $g=\tfrac{1}{\mathrm{ln}(1+c)-c/(1+c)}$. The halo mass with R₂₀₀ is ${M}_{200}\,=\tfrac{100{H}^{2}{R}_{200}^{3}}{G}$. The inner density profile of the NFW model shows a cusp with ρ ∝ R⁻¹. The corresponding rotation velocity of the NFW model is

$$\begin{eqnarray}&&{V}_{\mathrm{NFW}}(R)={V}_{200}\sqrt{\displaystyle \frac{\mathrm{ln}(1+{cx})-{cx}/(1+{cx})}{x\left[\mathrm{ln}(1+c)-c/(1+c)\right]}},\end{eqnarray} \tag{ 5 }$$

where V₂₀₀ is the circular velocity at R₂₀₀ with V₂₀₀ = R₂₀₀ h, and x = R/R₂₀₀.

(2) An ISO halo model (Begeman et al. 1991) is observationally motivated to describe the presence of a central core,

$$\begin{eqnarray}&&{\rho }_{\mathrm{ISO}}(R)=\displaystyle \frac{{\rho }_{0}}{1+{\left(R/{R}_{C}\right)}^{2}},\end{eqnarray} \tag{ 6 }$$

where ρ₀ is the central density and R_C is the core radius of the halo. The rotation velocity is

$$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{ISO}}(R) & = & \sqrt{4\pi G{\rho }_{0}{R}_{C}^{2}\left[1-\displaystyle \frac{{R}_{C}}{R}\arctan \left(\displaystyle \frac{R}{{R}_{C}}\right)\right]}\\ & = & \ {V}_{\mathrm{norm}}\sqrt{{\left(\displaystyle \frac{{R}_{C}}{{R}_{1}}\right)}^{2}\left[1-\displaystyle \frac{{R}_{C}}{R}\arctan \left(\displaystyle \frac{R}{{R}_{C}}\right)\right]},\end{array}\end{eqnarray} \tag{ 7 }$$

where V_norm = ${R}_{1}\sqrt{4\pi G{\rho }_{0}}$ and R₁ = 1 kpc.

(3) The Burkert density profile (Burkert 1995) is described by

$$\begin{eqnarray}&&{\rho }_{\mathrm{Bk}}(R)=\displaystyle \frac{{\rho }_{0}{R}_{C}^{3}}{(R+{R}_{C})({R}^{2}+{R}_{C}^{2})},\end{eqnarray} \tag{ 8 }$$

where ρ₀ and R_C are the central core density and core radius, respectively.

The corresponding circular velocity is given by

$$\begin{eqnarray}&&{V}_{\mathrm{Bk}}(R)={V}_{C}\sqrt{\displaystyle \frac{2\,\mathrm{ln}(1+x)+\mathrm{ln}(1+{x}^{2})-2\,\arctan (x)}{x\left[3\,\mathrm{ln}(2)-2\,\arctan (1)\right]}},\end{eqnarray} \tag{ 9 }$$

where x = R/R_c and V_C is the circular velocity at the core radius.

(4) With the rotation curve of a dark matter halo, the density profile of the halo can be derived through (de Blok et al. 2001)

$$\begin{eqnarray}&&\rho (R)=\displaystyle \frac{1}{4\pi G}\left[\displaystyle \frac{2V}{R}\displaystyle \frac{\partial V}{\partial R}+{\left(\displaystyle \frac{V}{R}\right)}^{2}\right].\end{eqnarray} \tag{ 10 }$$

We fitted the above three dark matter models to the rotation curve of dark matter through Bayesian inference with the Python code PyMC3 (Salvatier et al. 2016). We adopted the No-U-Turn Sampler (NUTS) with 5000 samples, 2000 tunes, four chains, and target_accept = 0.99.

As listed in Table 3, the prior of each parameter of a dark matter model is described with a bounded normal distribution, whose lower-limit bound is set to zero. The mean of the distribution is set with some prior knowledge as detailed below, and the standard deviation is set to twice the mean.

The prior R₂₀₀ of the NFW model is set to have a mean of 75 kpc, corresponding to a halo mass of 5 × 10¹⁰ M_⊙, as expected by the stellar mass versus halo mass relationship given our stellar mass of (1.37 ± 0.05) × 10⁸ M_⊙ (Santos-Santos et al. 2016). At this halo mass, the halo concentration is about 10 in simulations (Macciò et al. 2007), giving a mean prior R_S of 7.5 kpc.

Since the core radius of a cored dark matter halo is on a kiloparsec scale (Oh et al. 2015), the mean prior R _C of the ISO model is set to 2 kpc. The mean prior V_norm of the ISO model is thus set to 25 km s⁻¹, so that the galaxy is on the baryonic Tully–Fisher relationship (Mancera Piña et al. 2020) given its stellar mass of (1.37 ± 0.05) × 10⁸ M_⊙ and an H i mass of (8.51 ± 0.36) × 10⁸ M_⊙. Similarly, the mean prior R _C of the Burkert model is set to 2 kpc, and the mean prior V_C of the model is set to 35 km s⁻¹.

The fitting is convergent. By exploring the full posterior distribution, the best-fit parameter is given as the mean of the distribution, and the error is given by the standard deviation, as well as the 3% and 97% highest posterior density (HPD). The percentage of small Pareto shape diagnostic values (k < 0.7) is also listed. A larger diagnostic value indicates that the model is incorrectly specified.

Figure 9(a) shows the observed rotation curve of AGC 242019. The H i measurements cover a radial extent of 9 kpc, while the Hα data cover the central 5 kpc in radius. Two data sets are overall consistent with each other in the overlap region. The rotation curve rises all the way up to the last measurable radius.

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Figure 9(b) shows the rotation curve of dark matter after quadratically subtracting the gas and stellar gravity contributions. The sampling of the H i curve is at the beam size, while the Hα curve has a radial bin of 1.0 kpc. As shown in the figure, the Hα curve is overall well consistent with the H i curve over the spatial extent covered by both data sets. As mentioned before, since the Hα emission is sporadic and the errors of the Hα rotation curve are only based on the detected regions, its rotation curve is only used for the sanity check of the H i curve. We fitted the H i curve with two spherical halo models, namely, the NFW model (Equation (5), Navarro et al. 1997) and the ISO model (Equation (7), Begeman et al. 1991), through the Python code PyMC3 (Salvatier et al. 2016) to represent the cuspy and cored profiles, respectively. The ISO model is too steep at the inner radii, whereas the NFW model matches the observed data in the overall fitting, which is also illustrated by the fitting residuals as shown in Figure 10. To quantitatively discriminate between the two models, we performed a leave-one-out (LOO) predictive check (Vehtari et al. 2015). We found that the estimated effective number of parameters (p_loo = 1.6) of the NFW model is smaller than the real number of free parameters, i.e., two, while that of the ISO model has p_loo = 6.3, significantly larger than two (see Table 4). This quantitatively indicates that the NFW model is valid, while the ISO model is ruled out. The corresponding difference in the reduced χ² is 1.9 for dof = 5, equivalent to a Gaussian significance of 3.0σ. As listed in Table 4, some Pareto diagnostic value k of the ISO model is larger than 0.7, further indicating that the model is incorrectly specified. We also ran the fitting by including the Hα and obtained a similar difference in p_loo values and Δ(χ²/dof) = 2.8 for dof = 9 between the two models, equivalent to a Gaussian significance of 5.0σ.

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The Burkert model is another observationally motivated formula to describe a cored dark matter halo (Equation (9), Burkert 1995). Its density profile is flat toward the center, which is like the ISO model, but it has a power-law index of −3.0 toward infinity, which is like the NFW model. As shown in Figure 9(b) and listed in Table 5, the Burkert model gives a bad fitting too, with p_loo = 4.4, much larger than (1.6) that of the NFW model.

The best-fit NFW model has a halo scale radius of ∼33 kpc. Such a large cusp is well spatially resolved given that its size relative to the H i beam is 24. This suggests that the presence of the cusp is not an artifact caused by a limited spatial resolution (de Blok et al. 2001; Oh et al. 2015). The dark matter halo has a mass of (3.5 ± 1.2) × 10¹⁰ M_⊙ within R₂₀₀, the radius at which the average halo density is 200 times the average cosmic density. Due to the fact that the rotation curve does not reach the flat part, the constraints on the R₂₀₀ (or M_halo) and R _S do not reach a small error. But a small halo concentration of only 2.0 ± 0.36 is conclusive. This is much smaller than the median concentration of 15 at the same halo mass in the local universe (Macciò et al. 2007). The implication of this small concentration on the formation of the galaxy is discussed in Section 5.3.

To constrain the innermost density slope α_innermost of the dark matter halo, we derived the density profile of dark matter from the rotation curve (de Blok et al. 2001) as shown in Figure 11. Following de Blok et al. (2001) and Oh et al. (2015), we measured α_innermost by carrying out the least-squares fitting to the density profile within the break radius and defined the error of α_innermost as the difference between the result including the first data point outside the break radius and the result including only data points within the break radius. Here the break radius is the radius where the density profile shows a maximum change in the slope. If adopting the core radius of the best-fit ISO model as the break radius, α_innermost = −(0.96 ± 0.12). If adopting the scale radius of the best-fit NFW model as the radius, α_innermost = −(0.90 ± 0.08). Here the error is given by the least-squares fitting, since the scale radius of the NFW model is larger than the last ring. Two measurements indicate a statistical significance of about 10σ that the dark matter halo of AGC 242019 is cuspy down to a radius of 0.67 kpc.

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The identification of a cuspy profile in AGC 242019 is robust against systematic uncertainties from several aspects.

(1) Position and inclination angles. Since these two angles have been set as free parameters during the fitting, their uncertainties have already been included in the derived rotation curve. For comparison, we further estimated the photometric position and inclination angles based on the r-band image. The galaxy in the r-band is somewhat asymmetric with clumpy features, but outside the radius of 8'', the position and inclination angles converge to (2 ± 2)° and (67 ± 3)°, respectively. These results are consistent with the values derived from the dynamic fitting of the H i data, as shown in Figure 5 and listed in Table 2.

(2) Mass-to-light ratio in 3.6 μm. The rotation velocity due to stellar gravity is proportional to the square root of the stellar mass-to-light ratio. The overall stellar contribution to the observed rotation curve is small, so our result is not sensitive to the mass-to-light ratio ϒ_{3.6 μm}, as detailed here. The mass-to-light ratio in a broad band is obtained by fitting a synthetic spectrum to the observed spectrum or a broadband spectral energy distribution. The uncertainties of the stellar population synthesis model, the star formation history, the dust extinction, the metallicity, and the initial mass function all result in the variation of the derived mass-to-light ratio.

With the measured stellar mass and SFR of AGC 242019, we assumed a low metallicity with [Fe/H] = −1 and suppressed asymptotic giant branch (AGB) stars, as seen in some low surface brightness dwarfs (Schombert et al. 2019), to derive the ϒ_{3.6 μm} of our target to be 0.6 (Schombert et al. 2019). The preceding assumption about AGB stars causes the mass-to-light ratio to be ∼30% larger than that in normal galaxies. The ϒ_{3.6 μm} is also color-dependent (Bell et al. 2003; Zibetti et al. 2009; Jarrett et al. 2013; Meidt et al. 2014; Shi et al. 2018; Schombert et al. 2019; Telford et al. 2020). As the object is not detected in 4.5 μm, we used the radial variation in g − r color with a median of 0.3 and standard deviation of 0.03. Converting to B − V = 0.98*(g − r) + 0.22 (Jester et al. 2005), the variation in ϒ_{3.6 μm} is expected to have a standard radial deviation as small as 0.02 dex (Schombert et al. 2019); thus, its effects on the rotation curve of dark matter are negligible.

The systematic uncertainties due to the differences in star formation histories, initial mass functions, and stellar population models among different studies are much larger, even at a fixed color (Bell et al. 2003; Zibetti et al. 2009; Jarrett et al. 2013; Meidt et al. 2014; Schombert et al. 2019; Telford et al. 2020). As a result, we adopted 0.1 dex as the 1σ error to encompass the results in different studies. We then varied ϒ_{3.6 μm} by ±3σ to investigate its effect on the rotation curve of dark matter. As listed in Table 4 and shown in Figure 12, the cusp-like NFW model is a more reasonable model than the core-like ISO model for both ϒ_{3.6 μm} values.

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(3) Distance. The velocities due to baryonic gravity vary with the square root of the distance, while the observed rotation curve is independent of the distance. As the object is an isolated galaxy with no close-by companion that is brighter than m_r = 17.7 (or M_r = −14.8) within 500 kpc and 1000 km s⁻¹ in the Sloan Digital Sky Survey, the error in the distance is dominated by the uncertainty of the Hubble constant and the peculiar velocity. The heliocentric velocity is 2237 km s⁻¹ after correcting for the Virgo, Great Attractor, and Shapley superclusters (Mould et al. 2000). We estimated the residual error to be 100 km s⁻¹, which corresponds to an infalling velocity toward an imaginary dark matter halo with 10¹¹ M_⊙ at a separation of 50 kpc. By adopting a local Hubble constant of 73 km s⁻¹ with 2.5% uncertainty (Riess et al. 2016), the distance has a 1σ uncertainty of 5%. We varied this distance by ±3σ to investigate its effect on the dark matter rotation curve. As listed in Table 4 and shown in Figure 12, the cusp-like NFW model is again advocated against the core-like ISO model for both distances.

(4) The scale height of the gas disk. For a thick disk, emissions from adjacent rings are projected to be in the same pixels, which causes beam smearing and affects the measurements of the inclination and position angles. By increasing and decreasing the scale height by a factor of 5 to 500 and 20 pc, respectively, the NFW model is still much better than the ISO model, as shown in Figure 12 and Table 4.

(5) The noncircular motion. As shown in Figure 6, the median amplitude of the residual velocity obtained by ^3D Barolo fitting is small, i.e., ∼2 km s⁻¹. During the fitting, the line-of-sight velocities are assumed to be entirely circular motions, while noncircular motions cause the real circular velocity to be underestimated. To quantify the amplitude of noncircular motions, we carried out harmonic decomposition with the GIPSY task RESWRI (Begeman 1989). As detailed in previous studies (Schoenmakers et al. 1997; Trachternach et al. 2008), the line-of-sight velocity of each ring can be decomposed into

$$\begin{eqnarray}&&{v}_{\mathrm{los}}(r)={v}_{\mathrm{sys}}(r)+{{\rm{\Sigma }}}_{m=1}^{N}({c}_{m}(r)\cos m\psi +{s}_{m}(r)\sin m\psi ),\end{eqnarray} \tag{ 11 }$$

where r is the radial distance of each ring from the dynamical center, v_sys(r) is the system velocity, ψ is the azimuthal angle in the plane of the disk, and ${v}_{\mathrm{los}}(r)={v}_{\mathrm{sys}}(r)+{c}_{1}(r)\cos \psi$ corresponds to a pure circular motion scenario. In this study, we expanded the velocity up to m = 3, as has been adopted in other studies (Schoenmakers et al. 1997; Trachternach et al. 2008).

To run RESWRI, we used the 2D velocity field produced by ^3D Barolo as the input, fixed the dynamical center and system velocity to those determined by ^3D Barolo, and set the rotation velocity, inclination angle, and position angle of each ring as free parameters. With the derived c_m and s_m , the amplitude of each noncircular harmonic component with m > 1 is defined as

$$\begin{eqnarray}&&{A}_{m}(r)=\sqrt{{c}_{m}^{2}(r)+{s}_{m}^{2}(r)},\end{eqnarray} \tag{ 12 }$$

and for m = 1, where c₁ is the circular motion,

$$\begin{eqnarray}&&{A}_{1}(r)=\sqrt{{s}_{1}^{2}(r)}.\end{eqnarray} \tag{ 13 }$$

The total amplitude of noncircular motion is given by

$$\begin{eqnarray}&&{A}_{\mathrm{tot}}(r)=\sqrt{{s}_{1}^{2}(r)+{c}_{2}^{2}(r)+{s}_{2}^{2}(r)+{c}_{3}^{2}(r)+{s}_{3}^{2}(r)}.\end{eqnarray} \tag{ 14 }$$

The measured harmonic component can be used to quantify the elongation of the potential _pot at each radius as follows:

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{pot}}\sin 2{\phi }_{2}=({s}_{3}-{s}_{1})\displaystyle \frac{1+2{q}^{2}+5{q}^{4}}{{c}_{1}(1-{q}^{4})},\end{eqnarray} \tag{ 15 }$$

where ϕ₂ is an unknown angle between the minor axis of the elongated ring and the observer in the plane of the ring, and q = cos i, with i being the inclination angle of the disk.

Figure 13 shows the result of the harmonic decomposition. As shown in Figure 13(a), the radial c_m fluctuates around 0 km s⁻¹ with amplitudes ≲2 km s⁻¹. The radial s_m shows similar behaviors with amplitudes ≲2 km s⁻¹. As a result, the A_m of each harmonic component is in general ≲2 km s⁻¹. Figure 13(d) shows that the total amplitude A_tot is only about 2 km s⁻¹. As shown in Figure 13(c), all of the amplitudes are small fractions of the circular velocities at the corresponding radii that are <15%.

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Compared to other galaxies with similar absolute magnitudes (Trachternach et al. 2008), AGC 242019 is indeed a galaxy without stronger noncircular motions. Compared to simulated galaxies where noncircular motions result in noticeable underestimations of circular velocities (Oman et al. 2019), the noncircular amplitude of ∼2 km s⁻¹ in AGC 242019 is much less than the simulated values of 20–30 km s⁻¹. Therefore, we expect that the noncircular motion of AGC 242019 should not affect the derived rotation curve of dark matter. Finally, the derived ${\epsilon }_{\mathrm{pot}}\sin 2{\phi }_{2}$, as shown in Figure 13(f), suggests a spherical gravitational potential.

(6) The triaxiality of a dark matter halo. In this study, a spherical dark matter halo is assumed, while a halo has been found to be moderately triaxial in numerical simulations (Jing & Suto 2002; Bailin & Steinmetz 2005). However, a typical triaxial mass distribution results in only a small deviation in the density from the spherical assumption. Within the scale radius of the halo, the difference is only 10%–20%, which is much smaller than the required variation of a factor of 3 to decrease the inner slope by 0.5 (Knebe & Wießner 2006). Numerical modeling of the rotation curve further suggests that the halo triaxiality cannot significantly change the shape of the curve to make an intrinsic cusp be a core (or vice versa) in the observed data (Kuzio de Naray et al. 2009; Kuzio de Naray & Kaufmann 2011).

(7) Beam smearing. The tool ^3D Barolo takes the beam smearing into account in the tilted-ring fitting (Di Teodoro & Fraternali 2015). It first builds a gas disk in three spatial dimensions and three velocity dimensions, and then it convolves this artificial disk to the observed spatial resolution for comparison with the observed 3D data cube to derive the best-fitting parameters. It has been shown that ^3D Barolo is able to recover the rotation curve even at a low spatial resolution, i.e., two resolution elements over the semimajor axis (Di Teodoro & Fraternali 2015). The semimajor axis of AGC 242019 is resolved into ∼seven beams in the H i map. In addition, the Hα map has a rebinned spatial resolution (4'' in diameter) that is twice as high as that of the H i map. Although our Hα clumps are mainly distributed along the major axis with a narrower spatial extent, the overall shape of the Hα-derived curve is consistent with the H i curve, demonstrating that the beam-smearing effect on the H i rotation curve has been largely removed by ^3D Barolo (Di Teodoro & Fraternali 2015).

(8) The contributions from molecular and ionized gas. The very low surface density of AGC 242019 results in a low midplane pressure with P_ext/k ≲ 10³ cm⁻³ K, which gives a molecular-to-atomic gas ratio of ≲5% (see their Figure 3 and Equation (12) in studies of nearby galaxies; Blitz & Rosolowsky 2006; Leroy et al. 2008). We also checked that the atomic gas alone is sufficient to place the galaxy in the extended Schmidt law (Shi et al. 2011, 2018), consistent with a negligible fraction of molecular gas.

The ionized gas mass can be estimated from the Hα luminosity by ${M}_{{\rm{H}}{\rm\small{II}}}={m}_{{\rm{p}}}\tfrac{{L}_{\mathrm{Ha}}}{3.56\times {10}^{-25}{N}_{e}}$ (Osterbrock & Ferland 2006), where m_p is the proton mass, L_Ha is the Hα luminosity in erg s⁻¹, and N_e is the electron volume density in cm⁻³. By assuming a low N_e of 100 cm⁻³, we found that, even at the peak spaxel as shown in Figure 4(c), an ionized gas mass surface density of 0.025 M_⊙ pc⁻² is too small to affect the derived rotation curve of dark matter.

5.1.1. Fuzzy Cold Dark Matter

Through numerical simulations, a halo of fuzzy cold dark matter is found to be composed of a soliton core superposed on an extended halo (Hu et al. 2000; Schive et al. 2014). The latter can be represented by the NFW model, while the former can be approximately described by

$$\begin{eqnarray}&&{\rho }_{c}(r)\approx \displaystyle \frac{1.9{\left({m}_{\psi }/{10}^{-23}\mathrm{eV}\right)}^{-2}{\left({R}_{c}/\mathrm{kpc}\right)}^{-4}}{{\left[1+9.1\times {10}^{-2}{\left(r/{R}_{c}\right)}^{2}\right]}^{8}}{M}_{\odot }\,{\mathrm{pc}}^{-3},\end{eqnarray} \tag{ 16 }$$

where m_ψ is the particle mass and R_c is the core radius. The soliton core is linked to the total halo through the core–halo relationship (Schive et al. 2014), which gives

$$\begin{eqnarray}&&{R}_{c}=1.6{\left({m}_{\psi }/{10}^{-22}\mathrm{eV}\right)}^{-1}{\left({M}_{{\rm{h}}}/{10}^{9}{M}_{\odot }\right)}^{-1/3}\,\mathrm{kpc}.\end{eqnarray} \tag{ 17 }$$

As shown in Figure 9(b), an NFW model fits the density profile of AGC 242019 very well, which shows negligible residual density for the soliton core to account for. As a result, the possible contribution to the dark matter density from the soliton core should not exceed the observed error at all radii, which can be used to constrain the m_ψ . For each radius, we estimated the density of the soliton core as a function of the particle mass m_ψ with the best-fit M_h = (3.5 ± 1.2) × 10¹⁰ M_⊙ through the above two equations. It is found that the measurements at the innermost two radii give the strongest constraints on m_ψ . The 3σ observed errors at two radii constrain the m_ψ range to be <0.06 × 10⁻²² or >2.7 × 10⁻²² eV. Compared to the constraint from the Lyα forest, in which m_ψ < 1.0 × 10⁻²² or >23 × 10⁻²² eV, the dynamics of AGC 242019 gives a factor of about 20 times smaller constraint on the upper bound of the lower range. As shown in Figure 14(a), if adopting the 3% HPD halo mass of 1.5 × 10¹⁰ M_⊙, m_ψ <0.11 × 10⁻²² or >3.3 × 10⁻²² eV, whose upper bound of the lower range is still 10 times lower than the constraint by the Lyα forest. If somehow our error is underestimated by a factor of 2, the upper bound of the lower range only increases by a factor of ∼1.6, and the lower bound of the upper range decreases by a factor of ∼1.2. Therefore, the constraint by AGC 242019, along with the one from the Lyα forest, is inconsistent with the typical m_ψ of ∼10⁻²² eV that is required to explain the dynamics of other galaxies with cored dark matter halos (Hu et al. 2000; Schive et al. 2014). It is thus found that there is no m_ψ value that can reconcile all of the observational facts.

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5.1.2. Self-interacting Dark Matter

Self-interacting dark matter transmits kinetic energy from the outer part inward to form a constant density core. For the interaction to be efficient, the scattering rate per particle should be important, that is, at least once over the galaxy age (Spergel & Steinhardt 2000; Rocha et al. 2013; Tulin & Yu 2018),

$$\begin{eqnarray}&&{\rm{\Gamma }}(r){t}_{\mathrm{age}}\approx \rho (r)(\sigma /m){v}_{\mathrm{rms}}(r){t}_{\mathrm{age}}\sim 1,\end{eqnarray} \tag{ 18 }$$

where is Γ(r) is the scattering rate per particle, ρ(r) is the dark matter density at a radius of r, σ/m is the cross section per particle mass, and v_rms is the relative velocity of dark matter particles. The above equation can be rewritten as

$$\begin{eqnarray}&&\displaystyle \frac{\sigma /m}{1\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}}\approx \left(\displaystyle \frac{9.2\,\mathrm{Gyr}}{{t}_{\mathrm{age}}}\right)\left(\displaystyle \frac{0.01\,{M}_{\odot }\,{\mathrm{pc}}^{-3}}{\rho (r)}\right)\left(\displaystyle \frac{50\,\mathrm{km}\,{{\rm{s}}}^{-1}}{{v}_{\mathrm{rms}}}\right).\end{eqnarray} \tag{ 19 }$$

For AGC 242019, the NFW model fits the rotation curve well down to the innermost radius. This sets the upper limit to the radius of a possible density core and the lower limit to the above ρ(r) if dark matter is self-interacting in AGC 242019. If the halo forms around z = 2, we get t_age = 10 Gyr. The v_rms is set to be the virial velocity at the virial radius, which is 47 km s⁻¹. We have σ/m <1.63 cm² g⁻¹ for AGC 242019.

As shown in Figure 14(b), existing studies prefer somewhat larger σ/m on galaxy scales (Zavala et al. 2013; Elbert et al. 2015; Kaplinghat et al. 2016) and smaller values on cluster scales (Randall et al. 2008; Harvey et al. 2015; Kahlhoefer et al. 2015; Kaplinghat et al. 2016). Such a velocity dependence of the cross section seems to reconcile the results over different scales (Kaplinghat et al. 2016). However, the cuspy dark matter halo of AGC 242019 may challenge this simple picture, whose upper limit on σ/m is somewhat in tension with the lower bound of the σ/m range as required by cored halos of other dwarf galaxies.

5.1.3. Warm Dark Matter

In warm dark matter, the density core has a size that can be approximately by Hogan & Dalcanton (2000) and Macciò et al. (2012),

$$\begin{eqnarray}&&{r}_{\mathrm{core}}^{2}=\displaystyle \frac{\sqrt{3}}{4\pi {{GQ}}_{\max }}\displaystyle \frac{1}{\langle {v}_{\mathrm{rms}}^{2}{\rangle }^{0.5}},\end{eqnarray} \tag{ 20 }$$

where v_rms is the velocity dispersion (i.e., the mass) of the halo. Here ${Q}_{\max }$ is the maximum phase density as given by

$$\begin{eqnarray}&&{Q}_{\max }=1.64\times {10}^{-3}\left(\displaystyle \frac{{\rho }_{{\rm{L}}}}{{\rho }_{\mathrm{cr}}}\right){\left(\displaystyle \frac{m}{\mathrm{keV}}\right)}^{4}\displaystyle \frac{{M}_{\odot }\,{\mathrm{pc}}^{-3}}{{\left(\mathrm{km}{{\rm{s}}}^{-1}\right)}^{3}},\end{eqnarray} \tag{ 21 }$$

where m is the mass of warm dark matter, and $\tfrac{{\rho }_{{\rm{L}}}}{{\rho }_{\mathrm{cr}}}$ is the local density relative to the critical density:

$$\begin{eqnarray}&&\displaystyle \frac{m}{\mathrm{keV}}=0.37\displaystyle \frac{1}{{\left({\rho }_{{\rm{L}}}/{\rho }_{\mathrm{cr}}\right)}^{0.25}}\ast {\left(\displaystyle \frac{\mathrm{kpc}}{{r}_{\mathrm{core}}}\right)}^{0.5}{\left(\displaystyle \frac{\mathrm{km}{{\rm{s}}}^{-1}}{\langle {v}_{\mathrm{rms}}^{2}{\rangle }^{0.5}}\right)}^{0.25}.\end{eqnarray} \tag{ 22 }$$

As shown in Figure 14(c), the innermost radius of the cuspy halo of AGC 242019 leads to m > 0.23 keV. The figure also includes the constraints from the LITTLE THINGS galaxies (Oh et al. 2015) that are derived from the core radius and the velocity at the outermost measurable radius listed in their Table 2. It seems that there is no consistent mass range that can explain all observational facts. Warm dark matter is known to have a "catch-22" problem: the requirement for the particle mass to solve the cusp–core problem will result in no formation of small galaxies in the first place. Our discovery of a cuspy dark matter halo in AGC 242019 will further challenge the warm dark matter scenario such that, even on kiloparsec scales, the required particle mass cannot reconcile with the constraint that accounts for cored halos in some other dwarf galaxies.

5.1.4. The Modified Newtonian Dynamics

It is found that the effect of baryonic feedback on the density profile of dark matter depends on the stellar-to-halo mass ratio (M_*/M_halo; e.g., Di Cintio et al. 2014; Tollet et al. 2016), as shown in Figure 15. In the low-M_*/M_halo regime (<0.01%), the feedback is not strong enough to expel baryons, and no dark matter core forms. In the high M_*/M_halo regime (>3%), the significant contribution to the potential from baryonic matter cancels out the feedback effect and even produces a cuspier profile. In between, the stellar feedback can efficiently alter the density profile, and the flattest density profile forms around M_*/M_halo = 0.6%. The above result is independent of model parameters such as star formation threshold, initial mass function, supernova energy, etc. However, AGC 242019 has a stellar mass of (1.37 ± 0.05) × 10⁸ M_⊙ and a halo mass of (3.5 ± 1.2) × 10¹⁰ M_⊙, leading to an M_*/M_halo of (0.39 ± 0.13)%, which is close to the ratio where the flattest slope should form in simulations. Thus, AGC 242019 is inconsistent with the above M_*/M_halo dependence of the inner dark matter profile.

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Some other studies emphasize the importance of star formation threshold on the effect of the stellar feedback (e.g., Governato et al. 2010; Benéz-Llambay et al. 2019). If the threshold for gas to form stars is high, a large amount of gas can accumulate in the center of a halo and dominate the potential before star formation takes place. The subsequent feedback-driven massive outflows or repeated multiple outflows alter the orbits of dark matter to produce a dark matter core. On the other hand, for a low star formation threshold, gas is expelled by feedback before it contributes significantly to the potential. A dark matter cuspy profile is thus preserved. This scenario at least partly explains the formation of dark matter cores in some simulations (Di Cintio et al. 2014; Fitts et al. 2017; Macciò et al. 2017) but not in others (Bose et al. 2019). The simulation by Benéz-Llambay et al. (2019) predicts an intact cuspy dark matter density profile independent of the M_*/M_halo if the star formation threshold is low. As shown in Figure 15, AGC 242019 is consistent with their prediction. As a UDG, AGC 242019 does have a low gas and stellar mass surface density with ongoing star formation as revealed by the GALEX far-UV image. This indicates that on subkiloparsec scales, star formation in AGC 242019 can proceed at a very low gas mass surface density that is on the order of 1 M_⊙ pc⁻² or 0.4 × (100 pc/h) cm⁻³, where h is the scale height of the gas disk. However, its star formation efficiency (SFR/gas = 0.03 Gyr⁻¹) is much lower than that in spiral galaxies (∼0.3 Gyr⁻¹; Leroy et al. 2008; Shi et al. 2011), inconsistent with that adopted in the simulations. Therefore, a low star formation threshold should not be the only physical cause of a cuspy profile in AGC 242019.

Besides the star formation threshold, the duration of star formation may also be important, as recognized in some simulations (Read et al. 2016a). As long as star formation proceeds long enough, e.g., ∼10 Gyr for a halo mass of 10⁹ M_⊙ and a longer timescale for a larger halo, a halo core always forms. The UDG AGC 242019 has a halo mass of (3.5 ± 1.2) × 10¹⁰ M_⊙, and its low concentration implies a late formation time (see next section), which may explain its cuspy profile given the above scenario.

The UDGs are low stellar mass dwarfs but with sizes typical of spiral galaxies (Abraham & van Dokkum 2014; Leisman et al. 2017). They are found in all environments, including galaxy clusters, galaxy groups, and the field. Many mechanisms have been proposed to understand their origin: they may be normal dwarf galaxies but experience star formation feedback that redistributes gas and stars to larger radii (Governato et al. 2010; Di Cintio et al. 2014; Jiang et al. 2019), they may live in a high-spin dark matter halo with extended gas distributions and low efficiencies in converting gas into stars (Amorisco & Loeb 2016), and some environmental effects, such as ram pressure stripping or tidal puffing, may be important in their formation (Yozin & Bekki 2015; Jiang et al. 2019).

The dark matter halo of AGC 242019 has a mass of (3.5 ± 1.2) × 10¹⁰ M_⊙, which is typical of a halo hosting a dwarf. This suggests that AGC 242019 is not a failed massive galaxy, unlike other UDGs found in clusters (Beasley et al. 2016; van Dokkum et al. 2016). Although UDGs may have diverse origins, our measurements are more reliable. In the studies of Beasley et al. (2016) and van Dokkum et al. (2016), the halo mass was inferred from one to two velocity data points by assuming a halo concentration.

The cuspy halo of AGC 242019 also suggests that the feedback has not been strong enough over its history to expel a large amount of baryonic matter to large distances. Otherwise, a cored halo should have already formed, like in other dwarf galaxies, as suggested by feedback models (Navarro et al. 1996; Governato et al. 2010). Thus, AGC 242019 has experienced weak feedback over its history. This seems to be consistent with the deviation of UDGs from the Tully–Fisher relationship; gas and stars are not expelled out of the disk, so that a UDG contains more baryons at a given maximum circular velocity that roughly represents the halo mass (Mancera Piña et al. 2020). The UDG AGC 242019 has a maximum circular velocity of 47 km s⁻¹ and a baryonic mass of (9.88 ± 0.36) × 10⁸ M_⊙, placing it slightly above the Tully–Fisher relationship too. With the accurate measurement of the halo mass, AGC 242019 is found to be off both the M_* versus M_halo and M_baryon versus M_halo relationships (Santos-Santos et al. 2016). The low SFR of AGC 242019 also suggests weak ongoing stellar feedback as implied by the relationship between the SFR and the ionized gas velocity dispersion (e.g., Yu et al. 2019). Our regular velocity field with small noncircular motion, as shown in Section 4.2, is consistent with weak ongoing feedback too.

The halo of AGC 242019 has a small concentration of 2.0 ± 0.36, which is much smaller than the median concentration of 15 at the same halo mass in the local universe, as expected from simulations (Macciò et al. 2007). This difference is most likely because AGC 242019 is an isolated halo that originated from a tiny initial density peak in early times and collapsed recently. It is found that the halo concentration decreases with a later formation time (Zhao et al. 2003; Lu et al. 2006). A "young" halo thus suggests late formation of AGC 242019, which seems to be consistent with the findings of UDGs in cosmological simulations (Rong et al. 2017).

The specific angular momentum can be derived from the rotation curve combined with the stellar and gas mass profiles. The mass profiles were extended to 15 kpc following the same procedure that estimates the baryonic contribution to the rotation curve (see Sections 3.1.3 and 3.1.4). The rotation curve is also extended to 15 kpc by combining the best-fit NFW rotation curve plus the baryonic contribution. As shown in Figure 16(a), it is found that AGC 242019 has an angular momentum that is consistent with dwarf irregular galaxies at the same baryonic mass (Butler et al. 2017). However, as discussed before, AGC 242019 has a higher baryon/halo mass ratio as compared to galaxies in general. We derived the halo masses of dwarf irregulars following Butler et al. (2017) and compared them with AGC 242019. As shown in Figure 16(b), AGC 242019 still has a similar specific angular momentum at a given halo mass as compared to the average of dwarf irregulars. The result may be against the model that a UDG forms in a high-spin dark matter halo (Amorisco & Loeb 2016).

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As shown in Section 5.2, AGC 242019 has a low star formation efficiency, consistent with the model of Amorisco & Loeb (2016). If placing it on the extended star formation law (Shi et al. 2011, 2018), we found that AGC 242019 follows the law as shown in Figure 16(c). This suggests that its low star formation efficiency is regulated by its low stellar mass surface density.

In summary, AGC 242019 forms in a dwarf-sized halo with late formation time. It has weak feedback, low star formation efficiency, and a normal specific angular momentum of baryons at a given halo mass.