Probing the Dark Matter Halos of External Galaxies with Stellar Streams

For simplicity and computational efficiency, we approximate streams as orbits. Although streams do not exactly follow orbits (e.g., H. Lux et al. 2013; J. L. Sanders & J. Binney 2013), this approach is still frequently used for fitting streams in the MW, while taking into account the biases it may induce (e.g., S. E. Koposov et al. 2010; D. Hendel et al. 2017; K. Malhan & R. A. Ibata 2019). Because these studies assume that stream stars lie on the same orbit, many find their estimates of the MW DM halo mass to be inflated, as well as biases in the halo flattening parameters that are exacerbated by the form of potential assumed. However, these biases do not completely negate the ability of orbit fitting to recover the broad mass and shape of the DM halo. Furthermore, given the limited dimensionality of the observational data we expect to have in the near future, this simplistic approach is well suited to rapidly explore the quality of information in external streams as well as a large range of setups.

We also acknowledge the loss of information about the stream progenitor, i.e., its mass and age, with this assumption. However, we would like to note that the large-scale morphology of a stream is determined by the progenitor’s orbit, and the mass of the progenitor can be reasonably estimated via the stream’s width or the amount of light within the stream track (K. V. Johnston et al. 2001; D. Martínez-Delgado et al. 2015).

We model the orbits using a leap-frog integrator which is part of the stream modeling code developed in D. Erkal et al. (2019), although we emphasize that we only use orbits as representing a stream’s track and not streams themselves. Henceforth, we use orbits and mock streams interchangeably throughout the paper. Similar to D. Martínez-Delgado et al. (2021), we simulate the orbit in coordinates centered on the host galaxy and then make a mock observation of the stream, assuming that the observer is looking along the z-axis. In this work, we use the x- and y-coordinates of the stream as the stream observables, i.e., we effectively place the stream at infinity since we ignore the trigonometric effects associated with the stream being at a finite distance from the observer.

For our mock streams, the progenitor in each chosen gravitational potential is launched from apocenter (chosen as y = 0 for simplicity) and integrated backward in time from its starting position and velocity in steps of 0.1 Myr until it completes half of a wrap around the host galaxy, i.e., until its winding angle (which we call the phase angle θ) reaches −π. We then integrate an orbit forward in time until it reaches a phase angle of θ ≈ π, again to avoid any overlap and get a full wrap of a stream.

Each orbit is integrated in the three gravitational potentials described in Table 1. The Φ_DM potential is a DM-only model with a Navarro–Frenk–White (NFW) profile (J. F. Navarro et al. 1997) whose parameters are similar to the MWPotential2014 model in J. Bovy (2015). Next, Φ_Gal is a more realistic galactic potential that contains DM and baryonic components similar to the MW. It consists of a Miyamoto–Nagai disk (M. Miyamoto & R. Nagai 1975), a Hernquist bulge (L. Hernquist 1990), and a NFW halo, and is nearly identical to MWPotential2014 except we use a Hernquist bulge. Our final potential is a simplified power law of the form

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{PL}}}(r)=-\frac{{v}_{c}^{2}}{\gamma }{\left(\frac{{r}_{0}}{r}\right)}^{\gamma },\end{eqnarray} \tag{ 1 }$$

where r₀ is a scale radius, v_c is the circular velocity at the scale radius of the host, and γ determines the radial form of the potential. We emphasize our consideration of spherical halos as a first step toward evaluating the feasibility of recovering DM halo properties with external streams, but we note most real galaxy halos are often distorted and nonspherical (B. Allgood et al. 2006).

Additionally, we note that the launch velocity of a stream (orbit) varies across potentials in order to produce streams which have the same (1) on-sky track and (2) orbital properties in each. The launch velocity (at apocenter) for each potential is determined via

$$\begin{eqnarray}&&{v}_{\mathrm{apo}}^{2}=\displaystyle \frac{2[U({r}_{\mathrm{apo}})-U({r}_{\mathrm{peri}})]}{1-{\left(\tfrac{1+e}{1-e}\right)}^{2}},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray*}&&{r}_{{\rm{peri}}}={r}_{{\rm{apo}}}\frac{(1-e)}{(1+e)}.\end{eqnarray*}$$

U(r_apo) and U(r_peri) refer to the potential energy for a given potential at apocenter and pericenter, and e is the eccentricity of the orbit. For the stream setups discussed later in Section 2.3, this equation produces apocentric velocities ranging between ∼100 and 170 km s⁻¹.

Our first two potentials are used to evaluate how well the mass enclosed (within the apocenter of the stream) M_DM and scale radius r_s of the DM halo can be recovered with and without the presence of baryons, while Φ_PL is used to probe how well the overall radial profile of the potential can be recovered. Figure 1 illustrates the relationship between the radial profile of the potential γ and the precession of a stream’s apocenter, showing how we are able to use the precession of the stream apocenters (i.e., apsidal precession) to measure the potential’s radial form since streams precess differently in different gravitational potentials (K. V. Johnston et al. 2001; D. Hendel & K. V. Johnston 2015).

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To create our mock observations, we transform each mock stream into projected “on-sky” coordinates, i.e., the two-dimensional distance from the host (${r}_{{\rm{track}}}=\sqrt{{x}^{2}+{y}^{2}}$) versus the angle along the stream θ, which we visualize in Figure 2. We then take mock stream-track observations along the orbit in segments of θ = 0.3 rad (∼17°) and measure the projected radius at each of these angles by drawing from Gaussian distributions centered on the orbit. We assume an observational error of σ_track = 1 kpc. This value is motivated by D. Martínez-Delgado et al. (2021), who found similarly sized errors when measuring the track of a stream around M104, which is at a distance of ∼9.5 Mpc. We then repeat this procedure using an optimistic track uncertainty of σ_track = 0.2 kpc motivated by the requirement to resolve the widths of dwarf galaxy streams with future observational facilities (S. Laine et al. 2018). Additionally, we create a mock RV measurement at θ = 0 (i.e., where the stream crosses the positive x-axis) assuming an observational uncertainty of σ_rv = 10 km s⁻¹. This uncertainty is motivated by studies of external streams using current ground-based telescopes (e.g., M. A. Fardal et al. 2013; I. Escala et al. 2020; S. Pearson et al. 2022). Note that since the line-of-sight direction is along the z-axis, our RV is just v_z.

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With these mock observables, we construct a likelihood function to compare them with a model stream:

$$\begin{eqnarray}&&{\mathscr{L}}=\displaystyle \prod _{i=1}^{N}\frac{1}{\sqrt{2\pi {\sigma }_{{\rm{track}}}^{2}}}\exp \left(-\frac{{({r}_{i}^{{\rm{obs}}.}-{r}_{i}^{{\rm{model}}})}^{2}}{2{\sigma }_{{\rm{track}}}^{2}}\right),\end{eqnarray} \tag{ 3 }$$

where ${r}_{i}^{{\rm{obs.}}}$ is the “observed” on-sky distance of the mock stream to the host in the angular bin θ_i, ${r}_{i}^{{\rm{model}}}$ is the on-sky distance of the model to the host in the angular bin θ_i, and σ_track is the track uncertainty. We note that ${r}_{i}^{{\rm{model}}}$ is calculated by interpolating the model to where the “observed” data lie in θ. When including a mock RV observation in the fit, we combine Equation (3) with a likelihood of similar form for rv and σ_rv in the place of r and σ_track.

When generating orbits to compare with the mock streams, we fix one of the on-sky coordinates of the stream progenitor from which we launch orbits (as in D. Martínez-Delgado et al. 2021). This is done since orbits launched from locations along the stream will produce equivalent observables which would result in a degeneracy. For simplicity, we fix the y-coordinate to y = 0.

To explore the likelihood volume in parameter space, we implement an MCMC using the zeus (M. Karamanis et al. 2021) Python package to find the progenitor’s parameters (x, z, v_x, v_y, v_z) and the host potential parameters: M_NFW (M₂₀₀) and r_s in Φ_DM and Φ_Galaxy, v_c and γ in Φ_PL, which create an orbit model that best fits the observations. We define the priors on our varied parameters in Table 2, most of which are uniform distributions (for simplicity, and assigning equal probability to all values in a given range) with the exception of normal distributions for the velocities. This is due to our expectations for extragalactic stream stars to move (1) at speeds closer to those of halo stars (typically ∼200 km s⁻¹ for the MW), and (2) in any direction with respect to their host galaxy. We also employ the following restrictions: the total velocity of the progenitor must be less than 1000 km s⁻¹, 0 < v_c < 1000 km s⁻¹ in Φ_PL, and the maximum orbit integration time must be <14 Gyr to ensure a realistic model is created.

In accordance with zeus’s documentation, each MCMC run is initialized with the number of walkers equal to 2× the number of dimensions to be explored, i.e., 14 walkers (see Figure 5 in M. Karamanis et al. 2021). We cap each run at 10,000 total steps to ensure enough time for the sampler to properly explore the posterior with 5000 burn-in steps to discard the part of the chains from before convergence.

Next, we assess how well the DM halo properties of each stream–potential pair are recovered within the limit of 2D or 3D stream data (i.e., without or with a RV) with current and expected future observational power by performing our procedure described in Sections 2.1 and 2.2.

We do this for several different data availability scenarios:

1. 1.  
   Stream track information only with σ_track = 1 kpc.
2. 2.  
   Stream track (same σ_track) + one RV with σ_rv = 10 km s⁻¹.

Then for the streams run in the Φ_Gal potential, we also make and fit mock observations with a higher-precision track uncertainty:

1. 1.  
   Stream track information only with σ_track = 0.2 kpc.
2. 2.  
   Stream track (same σ_track) + one RV with σ_rv = 10 km s⁻¹.

We note that each RV measurement is taken at the location where the stream is launched, i.e., at y = 0.

Finally, we determine which stream properties provide the most information on their host potentials by repeating our approach for streams with varied lengths, inclinations, apocenters, and eccentricities as described by Table 3 in the different data availability scenarios. Every variation is made with respect to the fiducial case, which we define as the stream with the following properties: L = 1 wrap around the host, inclined at ϕ = 45° along the line of sight, an apocenter of r_apo = 30 kpc, and an eccentricity of e = 0.4 (repeated values in Table 3). More specifically, we vary only one stream parameter at a time for each of our tests to explore its impact on our ability to constrain the DM halo parameters. For example, we vary the stream length while keeping the inclination angle, apocenter, and eccentricity the same as the fiducial case, then vary the inclination angle while keeping the length, apocenter, and eccentricity the same as the fiducial, and so on.

For our fit to the fiducial stream (L = 1 wrap, ϕ = 45 deg, r_apo = 30 kpc, e = 0.4) in Φ_DM, neither the scale radius r_s nor the DM mass enclosed within the projected distance to the stream M_DM[R = 30 kpc] can be recovered when fitting models using only the stream track with σ_track = 1 kpc, as shown in the bottom-right panels of Figure 3. This is largely due to the degeneracies encountered between halo and stream properties when working with stream data of limited dimensionality. The most notable of these degeneracies include those between halo mass and stream velocity, and halo mass and scale radius. We will explore and discuss how they affect our results further in Section 4.1.

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Once a single RV measurement is added to the fit, we can constrain the scale radius to be ${r}_{s}=14.{3}_{-4.3}^{+5.0}$ kpc, where the true value lies within ∼1σ of the median as shown in Figure 4. In this case, the RV measurement provides a large boost in constraining power, leading to an accurate and precise radial profile measurement. Additionally, a RV measurement improves the ability to recover the enclosed DM mass M_DM with much greater precision. However, we find our measured value of M_DM is ∼2 times higher than the true value, which lies just within ∼2σ of the median (middle points, top-left panels of the summary figures in the following sections). Some further investigation reveals this inflated mass measurement is due to a bias caused by how well the scale radius is constrained. This bias affects a number of our results for the enclosed DM mass, which we will note throughout our findings as well as discuss further in Section 4.1.

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Similar to the case in Φ_DM, fitting only with the fiducial stream track in Φ_Gal does not recover the scale radius of the DM halo nor the enclosed DM mass when the stream-track uncertainty is 1 kpc. This changes when a RV measurement is added to the fit, as shown in Figure 5, allowing for a moderate scale radius constraint ${r}_{s}=19.{7}_{-7.7}^{+12.8}$ kpc and a relatively accurate and precise constraint on the enclosed DM mass ${M}_{{\rm{DM}}}[R=30{\rm{kpc}}]=22.{3}_{-10.0}^{+13.6}\times 1{0}^{10}\,{M}_{\odot }$ (middle points, top-right panels of later summary figures). As our constraint on r_s in this case is not incredibly precise, we believe the M_DM value recovered is reflective of a measurement of the enclosed mass with very little to no influence from the r_s constraint, which we explain further in Section 4.3.

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Reducing the stream track to σ_track = 0.2 kpc in Φ_Gal allows the scale radius to be recovered without any RV information, ${r}_{s}=12.{7}_{-6.3}^{+11.7}$ kpc (middle points, bottom-right panels of later summary figures). However, the corresponding enclosed mass measurement is biased to be ∼2.3 times more massive than the true M_DM, which lies just outside of 1σ. Figure 6 shows that with a RV, the scale radius constraint strengthens to ${r}_{s}=16.{7}_{-4.0}^{+4.9}$ kpc, while the enclosed mass measurement becomes less inflated (∼1.8 times greater than the true value) and much more precise.

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Finally, Figure 7 shows how when fitting the fiducial stream in Φ_PL, γ remains unconstrained whether using only observations of the stream track with σ_track = 1 kpc or when incorporating a RV measurement. In both cases, the γ posterior is skewed toward negative values due to a large volume in parameter space allowed by our priors, as well as a degeneracy between γ and inclination angle. We discuss both of these in more detail in Section 4.1.

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To assess the information stored in the length of an extragalactic stream, we vary the number of times a stream wraps around its host (first column of Table 3). Figure 8 shows the results of our fits to M_DM (top panels) and scale radius r_s (bottom panels) of the DM halo for varying stream lengths in Φ_DM (left side) and Φ_Gal (right side), color-coded by data availability.

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In Φ_DM, only a stream with two wraps around the host can provide a strong constraint on r_s without RV information (${r}_{s}=15.{9}_{-2.5}^{+3.3}$ kpc). The measurement of M_DM for this case is also precise, but biased to be ∼1.8× higher than the truth. When a RV is added to the fit, r_s is constrained for all explored stream lengths with increasing precision as the length of the stream increases. In particular, adding a RV to the stream track for the two-loop case slightly tightens the r_s constraint, and lowers the M_DM measurement to be ∼1.6× greater than the truth with much smaller uncertainties. We argue a constraint on the range of mass enclosed can still be made in this case as both distributions are tight and precise with the true value just outside of 1σ.

In Φ_Gal, no constraints on the enclosed DM mass or scale radius are made for stream tracks observed with a σ_track = 1 kpc uncertainty (circular points) without a RV measurement included in the fit. Adding a RV allows for a tighter constraint on r_s for the L = 0.5 wrap stream case. Meanwhile, the scale radius measurement for a multiloop stream is extremely precise but biased to lower values. This is, again, due to the degeneracy between mass and velocity (as we will explain further in Section 4.1), resulting in biased enclosed mass measurements. However, a lower bound on the DM mass can tentatively be placed without a RV for this case, M_DM[R = 30 kpc] ≥ 8.5 × 10¹⁰ M_⊙ (discussed further in Section 4.2), and with a RV the bound becomes M_DM[R = 30 kpc] ≥ 9.1 × 10¹⁰ M_⊙.

Smaller observational uncertainties (σ_track = 0.2 kpc; star symbols in Figure 8) allow for a moderate constraint on r_s and a weak constraint on the enclosed mass when fitting the fiducial stream without a RV measurement. For a stream with two loops, the r_s measurement and corresponding mass measurement is more precise than that measured with the larger stream-track uncertainty, but not as biased. When a RV is added, r_s can be constrained for all stream-length cases with the exception of the stream with two wraps, with precision increasing as the stream gets longer. The corresponding enclosed DM mass measurements are precise for every case when a RV is included, but become inflated as the stream length gets longer and the r_s constraints become more precise.

Figure 9 displays the results of our fits to γ in Φ_PL for each stream length, color-coded by data availability. As mentioned in Section 3.1, we often find our γ measurements dominated by degeneracies. Fortunately, we find the constraints on γ improve as the streams get longer. Once the stream length reaches at least two wraps around the host, the degeneracies are completely overcome and we recover γ = 0.1 ± 0.06 with and without a RV measurement with nearly equal accuracy and precision.

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We assess the information stored in the inclination angle of a stream by varying and fitting the fiducial orbit from face-on to nearly edge-on (second column of Table 3) viewing angles. Figure 10 displays our results for enclosed DM mass and scale radius measurements as a function of inclination for both Φ_DM and Φ_Gal, similar to Figure 8.

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In Φ_DM, no constraints can be made on either M_DM or r_s without RV information, except for the face-on stream case in which r_s is well constrained. Once a RV is added, however, the constraint on the scale radius for this stream becomes much weaker while it strengthens for the more inclined streams. In addition, M_DM is weakly constrained for the face-on stream with a RV but is measured more precisely as the inclination angle increases. The aforementioned bias between the r_s constraint and M_DM does not seem to affect the face-on or most inclined orbit as much as it does the fiducial case since the r_s measurements are less precise. In particular, the measured enclosed DM mass for the ϕ = 70° stream is ${M}_{\mathrm{DM}}[R=30\,\mathrm{kpc}]=21.{3}_{-6.7}^{+9.2}\times 1{0}^{10}\,{M}_{\odot }$ (Figure 10, upper-left panel), having only a slight effect from the bias.

In Φ_Gal, there is a much more visible trend between r_s and stream inclination, as shown in the right side of Figure 10. For the fits made with σ_track = 1 kpc, the scale radius can be recovered without any RV information for the stream with ϕ = 70°, but the mass cannot. The addition of a RV to this case leads to a tighter r_s constraint with a moderately biased M_DM measurement, and a lower bound can be placed at M_DM[R = 30 kpc] ≥ 8.5 × 10¹⁰ M_⊙.

With smaller stream-track uncertainties, weak constraints on r_s can be made for both ϕ = 0°, 70° without radial velocities. For the face-on stream, we can also make a strong constraint on the enclosed DM mass ${M}_{\mathrm{DM}}[R=30\,\mathrm{kpc}]=18.{5}_{-9.1}^{+12.3}\times 1{0}^{10}\,{M}_{\odot }$ (Figure 10, upper-right panel). Adding a RV strengthens the r_s constraint for the more inclined streams, and enables enclosed DM mass constraints for these cases with increasing precision as inclination increases (Figure 10, top-right panel). To a lesser extent than the fiducial case, the M_DM measurement for the ϕ = 70° stream case is only slightly affected by the M_DM and r_s bias as evinced by the tighter scale radius constraint, whereas the M_DM for the face-on inclination is affected by it very little, if at all.

Figure 11 shows the results of our fits to constrain the radial profile γ in Φ_PL for these streams. There is a clear trend between stream inclination angle and γ in which γ is measured more accurately for streams at lower inclination. The measurements are generally dominated by the previously stated degeneracy between inclination angle and γ until a RV measurement is added to the orbit with ϕ = 0°, in which case γ is measured to be γ = 0 ± 0.2.

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We gauge the quality of information contained in the distance of a stream to its host by varying the stream’s apocenter from the inner to outer halo, with the fiducial apocenter as the intermediate case (third column of Table 3). Figure 12 displays our M_DM and r_s results for streams with varying apocenters. Contrary to the previous figures, the top panels now show the enclosed DM mass as a function of distance from the host galaxy.

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Without a RV, only the scale radius for the stream with r_apo = 15 kpc can be somewhat constrained in the Φ_DM potential (Figure 12, bottom-left panel). With a RV, r_s can be recovered precisely at all explored apocenters with improved accuracy for streams with apocenters outside of the true scale radius. Additionally, a RV enables M_DM to be measured at apocenters beyond the true scale radius, but these measurements are each moderately biased by the high precision of the r_s constraint.

For the same streams in Φ_Gal (right side of Figure 12), no constraints can be made on r_s or M_DM without a RV measurement included in the fit for data with σ_track = 1 kpc. The scale radius can be constrained with a RV for the stream with r_apo = 60 kpc (Figure 12, bottom-right panel), though not as strongly as for the fiducial case. Despite the broadness of the r_s posterior for the furthest stream, the enclosed mass measurement ${M}_{\mathrm{DM}}[R=60\,\mathrm{kpc}]=31.{5}_{-10.7}^{+11.2}\times 1{0}^{10}\,{M}_{\odot }$ (Figure 12, upper-right panel) is quite accurate and precise. Much like for the fiducial case, it also does not seem to be affected by the bias induced by a r_s constraint.

When the stream-track uncertainties are smaller, r_s can be well recovered at every explored stream apocentric distance, with the constraint improving in accuracy and precision as the apocenter increases. Consequently, the more precise r_s constraints from streams with r_apo > 16 kpc, i.e., the true halo scale radius, produce strongly biased M_DM posteriors. Adding a RV slightly improves the precision of the r_s measurements, but does not significantly improve the accuracy for any scale radius or DM mass measurements, aside from the fiducial case.

Lastly, we find no correlation between the separation of the stream and host galaxy for constraining γ in Φ_PL, as shown in Figure 13. No constraints can be made as every case is dominated by degeneracies. However, we find the precision of the γ measurement improves as the stream’s apocenter gets farther away from the host when a RV is added to the fit.

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We gauge the amount of information stored in a stream’s eccentricity by varying the fiducial case to be more circular and more eccentric (last column of Table 3). Figure 14 displays our findings for the M_DM and r_s of a NFW halo for streams with varying eccentricities. In Φ_DM, the only constraint to be made without a RV is a weak recovery of r_s for the e = 0.7 stream case (bottom-left panel). However, when a RV is added, the scale radius is more strongly constrained for the fiducial and less eccentric streams. In addition, M_DM is strongly recovered for the e = 0.2 stream with only a slight offset from the truth from the r_s constraint (top-left panel).

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In Φ_Gal, no constraints can be made for stream data with σ_track = 1 kpc until a RV is added, in which case r_s is then strongly recovered for the e = 0.2 stream (bottom-right panel). Correspondingly, the enclosed DM mass measurement for this case, ${M}_{{\rm{DM}}}[R=30\,{\rm{kpc}}]=20.{2}_{-6.9}^{+8.8}\times 1{0}^{10}\,{M}_{\odot }$, is incredibly accurate and precise (top-right panel). We find it is also the strongest mass constraint with little bias from the r_s constraint, which we will explain further in Section 4.

Other than the fiducial case, improved track uncertainties only weakly constrain r_s for the e = 0.2 stream without a RV measurement (bottom-right panel). Adding a RV slightly tightens the previous r_s constraints, but it is not enough to help constrain r_s for the most eccentric case. It also allows the enclosed DM mass to be strongly constrained for the least eccentric case, again with little bias from the r_s constraint (top-right panel).

We display the results for fits to γ for these streams in Φ_PL in Figure 15, in which we find no constraints can be made except for the most eccentric case (e = 0.7) when a RV is included in the fit. The constraint itself is weak, but the general trend of γ as a function of eccentricity implies higher-eccentricity orbits enable more accurate measurements of the radial profile of the potential.

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4.1.1. Halo Mass and Progenitor Velocity Degeneracy

4.1.2. Halo Mass and Scale Radius Bias

4.1.3. Radial Profile in Φ_PL

Many of our γ results in Φ_PL are affected by the tendency of the posterior samples to pile up at γ = −1. Initially, we believed this to be due to a degeneracy between a stream’s inclination angle along the line of sight and the potential (visualized in Figure 16, left panel) in which an orbit with a certain inclination in one potential will look the same as an orbit with a different inclination angle in another potential when observed in projection. We believed allowing the orbit to loop around multiple times allowed the MCMC to hone in on the correct solution in ϕ–γ space (Figure 16, right panel). However, we realize this does not completely explain why the MCMC prefers a γ value of −1. While the correlation between γ and inclination does contribute to the runaway to some extent, we believe it is mainly due to our priors. When γ = −1, a wide range of velocities can create orbits within our uncertainties which look the same when observed in projection, thus leading to a large prior volume that dominates the posterior. As we will show in Section 4.3, including RV measurements can help remedy this tendency.

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Despite the numerous degeneracies encountered with stream data limited only to “on-sky” positions, we have shown that it is still possible to infer properties of an extragalactic DM halo. With current stream-track uncertainties, the overall radial profile can be constrained, as shown in Figure 17, and a lower bound can tentatively be placed on the enclosed DM mass when the stream has multiple well-measured wraps around its host galaxy. While we acknowledge the biased nature of our measurement of M_DM in this case (discussed in Sections 4.1.1 and 4.1.2), we would like to note that the true enclosed mass is still within our 2σ uncertainties. Therefore, while it may not be a true measurement of the halo mass itself, it most certainly restricts the range of possible masses. We show the enclosed mass distribution within 15 kpc for this case in both Φ_DM and Φ_Gal in the Appendix (Figure 21), in which it has a clear boundary due to the baryonic component (∼7 × 10¹⁰ M_⊙), ruling out lower masses in Φ_Gal.

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In addition, the radial profile can be recovered for many stream cases with the smaller stream-track uncertainties (σ_track = 0.2 kpc) expected from future observations, i.e., the fiducial, ϕ = 0°, 70°, r_apo = 15, 60 kpc cases. They also enable enclosed DM mass measurements for a few cases, i.e., ϕ = 0°, e = 0, however these should be taken lightly as a mass constraint is more trustworthy with a RV.

We note that previous studies have also shown that the flattening of a DM halo can also be inferred with stellar streams (J. Nibauer et al. 2023). In that work, the radial profile of the external galaxy was fixed and only the flattening was varied. In contrast, we have fixed the DM halos to be spherical and only varied the radial profile in this work. In the future, we will explore how well the radial profile and flattening can be simultaneously inferred using only the stream track.

4.3.1. What Can Be Done with a Single Radial Velocity Measurement?

4.3.2. What Happens as More Radial Velocity Measurements Are Added?

In addition to the data availability scenarios we test for our set of extragalactic streams, we also explore the impact of including more RV measurements on our fits to the DM halo. We do so by fitting the fiducial stream in each potential once again with σ_track = 1 kpc, and including two, then three RV measurements. Each additional RV is taken at $\theta =\pm \frac{\pi }{2}$, respectively, and incorporated into each fit as an additional term to the likelihood described in Section 2.2.

Figure 18 shows our constraints on the enclosed mass and scale radius in Φ_DM and Φ_Gal for the fiducial stream with multiple RVs compared to the cases with just one or no radial velocities. In Φ_DM, we find a second RV only slightly improves upon the constraints made for M_DM and r_s with one RV, while the fit with a third RV significantly improves the enclosed DM mass measurement in both accuracy and precision (${M}_{{\rm{DM}}}[R=30\,{\rm{kpc}}]=23.{3}_{-6.0}^{+6.8}\times 1{0}^{10}\,{M}_{\odot }$,) thus lessening the bias between M_DM and r_s. Similarly, there is not much of a difference between the constraints made with one or two radial velocities in Φ_Gal, other than a slight improvement in the enclosed DM mass measurement with two RVs. With a third RV, there is a slight improvement in the precision of the r_s measurement and an even slightly more improved mass measurement of ${M}_{{\rm{DM}}}[R=30\,{\rm{kpc}}]\,=17.{8}_{-6.6}^{+8.0}\times 1{0}^{10}\,{M}_{\odot }$. As was the case in Φ_DM, the additional radial velocities aid in overcoming the mass and velocity degeneracy as well as the bias induced by a r_s constraint.

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Figure 19 displays our results for these fits in Φ_PL. We find that a second RV boosts the accuracy of the radial profile measurement in Φ_PL enough to constrain it to $\gamma =-0.{2}_{-0.2}^{+0.4}$ with a similar precision as the fit with one RV. However, we also find the inclusion of a third RV measurement provides about the same amount of constraining power on γ as the fit with two RVs. This indicates a nonlinear relationship between the improvement of a constraint and the number of radial velocities included in the fit.

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Overall, the inclusion of multiple RV measurements in the fits serves to boost the strength of the constraints as well as helping to overcome degeneracies and biases. However, the amount by which an additional RV improves our measurements varies, possibly due to the potential. We find that in Φ_PL, an additional RV significantly improves the accuracy of the constraint on γ. In contrast, we find additional radial velocities provide only slight improvements upon the constraints made with one RV in both Φ_DM and Φ_Gal. We would also like to note that the amount of help given by additional RVs could also vary based on the DM halo information gleaned from the intrinsic properties of the stream.

Currently, we have to rely on the observability of a visible tracer, such as a planetary nebula or globular cluster, within an extragalactic stream to obtain even one RV measurement. Nonetheless, the upcoming spectrograph MOSAIC on the Extremely Large Telescope is expected to be powerful enough to get spectra of individual extragalactic stream stars. Its high-multiplex mode will be sensitive to the Ca II infrared triplet lines for stream stars around at least 10 galaxies within 10 Mpc (D. Martínez-Delgado et al. 2025, in preparation). Therefore, it is likely that we will be able to obtain more than one RV measurement for an extragalactic stream observed with MOSAIC.

In general, we find that longer streams, i.e., streams with more than one wrap around its host galaxy, provide the most accurate and precise constraints on the overall radial profile both with and without a RV in the fit. When varying the inclination angle of the observed stream, we find streams with lower inclinations better constrained γ when modeled in Φ_PL, while streams with higher inclinations yield better r_s constraints in Φ_DM and Φ_Gal. When varying stream eccentricity, we find that streams on more eccentric orbits (e > 0.4) provide better constraints on the radial profile when modeled in Φ_PL, while streams on moderate to less eccentric orbits provide better constraints on M_DM and r_s in Φ_DM and Φ_Gal. Finally, we find that streams with apocenters beyond the scale radius of the DM halo are better for radial profile and enclosed DM mass constraints.

The DM halo properties measured by external stellar streams can be compared with other independent inference methods for consistency. For example, a handful of the galaxies hosting external streams mentioned throughout this paper have corresponding rotation curve measurements (e.g., R. H. Sanders 1996; M. Valdez-Gutiérrez et al. 2002; T. Kolcu et al. 2023) which are fundamental for inferring the innermost parts of galactic DM halos. Other methods used to probe galactic DM include globular cluster kinematics (e.g., K. A. Woodley et al. 2010; M. Reina-Campos et al. 2023), gravitational lensing (e.g., R. Massey et al. 2010; R. Mandelbaum 2015), and Schwarzschild modeling (e.g., R. C. E. van den Bosch et al. 2008; G. Santucci et al. 2022), each of which bring their own insight into DM. While stellar streams provide a unique perspective into both the total mass and structure of DM halos, it is crucial to cross-correlate independent measurements to build a better understanding of galactic DM halo properties in general.