The Geometry of the Sagittarius Stream from Pan-STARRS1 3π RR Lyrae

For fitting this model, the sample of RRab stars near the Sgr orbital plane is split into bins of ${\tilde{{\rm{\Lambda }}}}_{\odot }\pm \tfrac{{\rm{\Delta }}{\tilde{{\rm{\Lambda }}}}_{\odot }}{2}$, each ${\rm{\Delta }}{\tilde{{\rm{\Lambda }}}}_{\odot }=10^\circ$ wide; the data are not binned in D. In each bin, we fit (independently) the parameters of the stream, ${D}_{\mathrm{sgr}}$ and ${\sigma }_{\mathrm{sgr}}$, along with the halo model parameter n. Whereas it is obvious why the stream-related model parameters should be fitted individually for each ${\tilde{{\rm{\Lambda }}}}_{\odot }$ bin, the reason for also fitting the halo power-law index n individually is to account for incompleteness of the data. The flattening parameter q is kept fixed at 0.71, because fitting for q did not improve the results for the stream-related model parameters.

To constrain the geometry of the Sgr stream in a probabilistic manner, we calculate the joint posterior probability ${p}_{\mathrm{RRL}}({\boldsymbol{\theta }}| { \mathcal D })$ of the parameter set ${\boldsymbol{\theta }}=({f}_{\mathrm{sgr}},{D}_{\mathrm{sgr}},{\sigma }_{\mathrm{sgr}},n)$, given the data set ${ \mathcal D }=(D,\delta D,l,b)$. The marginal posterior probability of the parameter set ${\boldsymbol{\theta }}$, ${p}_{\mathrm{RRL}}({\boldsymbol{\theta }}| { \mathcal D })$ is related to the marginal likelihood ${p}_{\mathrm{RRL}}({ \mathcal D }| {\boldsymbol{\theta }})$ through

$$\begin{eqnarray}&&{p}_{\mathrm{RRL}}({\boldsymbol{\theta }}| { \mathcal D })\propto {p}_{\mathrm{RRL}}({ \mathcal D }| {\boldsymbol{\theta }})p({\boldsymbol{\theta }})\end{eqnarray} \tag{ 5 }$$

where $p({\boldsymbol{\theta }})$ is the prior probability of the parameter set.

We evaluate

$$\begin{eqnarray}&&\mathrm{ln}\ {p}_{\mathrm{RRL}}({\boldsymbol{\theta }}| { \mathcal D })=\displaystyle \sum _{i}\mathrm{ln}\ {p}_{\mathrm{RRL}}({{ \mathcal D }}_{i}| {\boldsymbol{\theta }})+\mathrm{ln}\ p({\boldsymbol{\theta }})\end{eqnarray} \tag{ 6 }$$

with ${p}_{\mathrm{RRL}}({{ \mathcal D }}_{i}| {\boldsymbol{\theta }})$ given by Equation (1), and i indexes the RRab stars.

We use the following prior probability for the model parameters, $p({\boldsymbol{\theta }})$: for ${\sigma }_{\mathrm{sgr}}$, we choose a prior that is uniform in $\mathrm{ln}$, whereas for the other parameters, we adopt uniform priors. Specifically, we adopt

$$\begin{eqnarray}\mathrm{ln}\ p({\boldsymbol{\theta }}) & = & -\mathrm{ln}({\sigma }_{\mathrm{sgr}})\\ & & +\mathrm{Uniform}(0.05\leqslant {f}_{\mathrm{sgr}}\lt 1)\\ & & +\mathrm{Uniform}(1.7\leqslant n\lt 5.0)\\ & & +p({D}_{\mathrm{sgr}}| {\tilde{{\rm{\Lambda }}}}_{\odot }).\end{eqnarray} \tag{ 7 }$$

The prior for ${D}_{\mathrm{sgr}}$ depends on ${\tilde{{\rm{\Lambda }}}}_{\odot }$, and is uniform within ${D}_{\mathrm{minprior}}({\tilde{{\rm{\Lambda }}}}_{\odot })$, ${D}_{\mathrm{maxprior}}({\tilde{{\rm{\Lambda }}}}_{\odot })$ as indicated in Figure 4 and listed in Tables 2 and 3. Whereas the prior is generally wide, a quite restrictive prior was chosen for $20^\circ \leqslant {\tilde{{\rm{\Lambda }}}}_{\odot }\lt 30^\circ$ and $30^\circ \leqslant {\tilde{{\rm{\Lambda }}}}_{\odot }\lt 40^\circ$, because the fit otherwise behaves poorly because of the background sources along these l.o.s.

${D}_{\mathrm{minprior}}$, ${D}_{\mathrm{maxprior}}$ are basically constrained by the minimum and maximum distances in the ${\tilde{{\rm{\Lambda }}}}_{\odot }$ slice in particular case, but are also defined in order to mask dense regions at short heliocentric distances as well as to separate the leading and trailing arms where both are present at the same l.o.s.

The most probable model given the data is explored using the Affine Invariant Markov Chain Monte Carlo (MCMC) ensemble sampler (Goodman & Weare 2010) as implemented in the emcee package (Foreman-Mackey et al. 2013).

The approach was verified with mock data, using a halo component that was sampled from the underlying halo model, superimposed by a mock stream that was inserted as a stellar density sheet; its number density is uniform perpendicular to the l.o.s. and Gaussian along it. The fraction of stream stars with respect to the halo stars, described by ${f}_{\mathrm{sgr}}$, was then successively lowered; i.e., the fit was carried out in the limits of many and few stars in each ${\tilde{{\rm{\Lambda }}}}_{\odot }$ slice to make sure that reasonable fits can be obtained for densities such as that present for the PS1 3π RR Lyrae candidates, which is ∼0.5–1 deg⁻² for most parts of the sky.

We now illustrate which practical issues are entailed in fitting the model to the data in a ${\tilde{{\rm{\Lambda }}}}_{\odot }$ bin. Each distance and depth estimate $({D}_{\mathrm{sgr}},{\sigma }_{\mathrm{sgr}})$ is obtained by optimizing Equation (6) using an MCMC procedure (Foreman-Mackey et al. 2013). Figures 2 and 3 show fits to individual slices in ${\tilde{{\rm{\Lambda }}}}_{\odot }$. Figure 2 gives the fit for a $10^\circ$ wide slice centered on ${\tilde{{\rm{\Lambda }}}}_{\odot }=50^\circ$. In this direction, only the leading arm is present. The plot indicates the prior on ${D}_{\mathrm{sgr}}$, in this case, set only by the minimum and maximum distances available from sources in the ${\tilde{{\rm{\Lambda }}}}_{\odot }$ slice in the particular case. The distribution of the sources is shown, overplotted with the model from the best-fit parameters given as a solid blue line. The transparent blue lines represent samples drawn from the parameter probability density function, illustrating the spread of models; the downturn of the models at small distances below 40 kpc is also a reflection of our sample incompleteness (here at the bright end). In the case in Figure 2, showing ${\tilde{{\rm{\Lambda }}}}_{\odot }=55^\circ$, a halo profile much steeper than expected from n = 2.62 given in the model of Sesar et al. (2013) is obvious; local variations in n presumably reflect simply the halo substructure. The estimate of ${D}_{\mathrm{sgr}}$ and ${\sigma }_{\mathrm{sgr}}$ is clearly seen as being sensible in Figure 2. Here, the variance on the estimated parameters is very small, and the parameters fit well to what one would guess by visual inspection. Even for ${\tilde{{\rm{\Lambda }}}}_{\odot }$ slices where the fit is poorer (both by visual inspection and by the variance of the distance estimate) a sensible distance estimate, not driven by the priors, is found as we show below.

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Figure 3 gives the fit for ${\tilde{{\rm{\Lambda }}}}_{\odot }=155^\circ$, where both leading and trailing arms are along the l.o.s. Using distinct priors on ${D}_{\mathrm{sgr}}$ separates the two debris streams and gives precise estimates on distance and depth of both leading and trailing arms (see also Figure 4 around ${\tilde{{\rm{\Lambda }}}}_{\odot }=155^\circ$). This illustrates the importance of carefully set priors.

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Figure 5 shows the estimated ${\sigma }_{\mathrm{sgr}}$ of the stream (being half the l.o.s. depth) versus ${\tilde{{\rm{\Lambda }}}}_{\odot }$ for both the leading and trailing arms along with its uncertainty. Figure 5 quantifies what was qualitatively apparent from Figure 4(a): the stream tends to broaden along its orbit from ∼1.75 to 6 kpc for the leading arm, reaching even ∼10 kpc for the trailing arm. As expected, ${\sigma }_{\mathrm{sgr}}$ and thus the l.o.s. depth are largest close to the apocenters. This is the first systematic determination of the l.o.s. depth, although the uncertainties are still quite large for some parts of the stream. The leading arm's l.o.s. depth rises (and falls) toward (and away) from the apocenter. In contrast, ${\sigma }_{\mathrm{sgr}}$ for the trailing arm remains larger in the range $200^\circ \lt {\tilde{{\rm{\Lambda }}}}_{\odot }\lt 300^\circ$. At least in part, this is presumably because our l.o.s. direction forms a shallower angle with the stream direction than the leading arm. Except toward the apocenters, ${\sigma }_{\mathrm{sgr}}$ also rises toward the "end" (the largest ${\tilde{{\rm{\Lambda }}}}_{\odot }$) of the respective trailing or leading arm.

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In addition to the l.o.s. depth of the Sgr stream, its actual depth would be of great interest. As we know the angle between the normal to the stream and the l.o.s., we could deproject the l.o.s. depth ${\sigma }_{\mathrm{sgr}}$ to get the actual width of the stream.

First, we convert the polar coordinates of the projected $({\tilde{{\rm{\Lambda }}}}_{\odot },{\tilde{B}}_{\odot })$, as shown in Figure 4, into their Cartesian counterparts $({x}_{\mathrm{sgr}},{y}_{\mathrm{sgr}})$. We then calculate the deprojected depth ${\tilde{\sigma }}_{\mathrm{sgr}}$ for each bin i in ${\tilde{{\rm{\Lambda }}}}_{\odot }$ as

$$\begin{eqnarray}&&{\tilde{\sigma }}_{\mathrm{sgr},i}={\sigma }_{\mathrm{sgr},i}\cos ({\tilde{{\rm{\Lambda }}}}_{\odot ,i}-{\alpha }_{i})\end{eqnarray} \tag{ 8 }$$

with

$$\begin{eqnarray}&&{\alpha }_{i}=\tan \left(\displaystyle \frac{{y}_{\mathrm{sgr},i+1}-{y}_{\mathrm{sgr},i-1}}{{x}_{\mathrm{sgr},i+1}-{x}_{\mathrm{sgr},i-1}}\right).\end{eqnarray} \tag{ 9 }$$

Equation (9) approximates the tangent in $({x}_{\mathrm{sgr},i},{y}_{\mathrm{sgr},i})$ with a line through $({x}_{\mathrm{sgr},i-1},{y}_{\mathrm{sgr},i-1})$ and $({x}_{\mathrm{sgr},i+1},{y}_{\mathrm{sgr},i+1})$, thus the first and last ${\sigma }_{\mathrm{sgr}}$ of the leading and trailing arms are not deprojected.

The deprojected depths along with their uncertainties are given in Tables 6 and 7 in the Appendix. Figure 6 shows how the l.o.s. and deprojected depth of the Sgr stream vary strongly during the orbital period. The ${\tilde{\sigma }}_{\mathrm{sgr}}$ profile is flatter than the ${\sigma }_{\mathrm{sgr}}$ profile, and as expected, the trailing arm's deprojected depth ${\tilde{\sigma }}_{\mathrm{sgr}}$ is not noticeably boosted, in contrast to ${\sigma }_{\mathrm{sgr}}$, which is. But a variation during the orbital period is still present. Comparing the two depths emphasizes that the greater depth at the apocenters is a combination of the projection effect and the true broadening when the velocities become small near the apocenters.

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We can also quantify the amplitude A of the stream fit from Section 3, defined as the number of RRab stars in the stream per degree as a function of ${\tilde{{\rm{\Lambda }}}}_{\odot }$, i.e.,

$$\begin{eqnarray}&&A=\left(\#\mathrm{sources}\ \mathrm{within}\ {\tilde{{\rm{\Lambda }}}}_{\odot }\pm \displaystyle \frac{{\rm{\Delta }}{\tilde{{\rm{\Lambda }}}}_{\odot }}{2}\right)\times {f}_{\mathrm{sgr}}/({\rm{\Delta }}{\tilde{{\rm{\Lambda }}}}_{\odot }\times {\sigma }_{\mathrm{sgr}}).\end{eqnarray} \tag{ 10 }$$

The amplitudes for both the leading and trailing arms are given in Tables 8 and 9 in the Appendix.

Figure 7 shows the amplitudes plotted versus the ${\tilde{{\rm{\Lambda }}}}_{\odot }$ bins. The value of A increases near the apocenter of the leading arm to about twice as much as away from its apocenter. Also near the apocenter of the trailing arm, A rises with respect to the value it has away from the apocenter, but not as strikingly as found for the leading arm. As the angular velocity decreases near the apocenter, we had expected to find an increased source density, and thus larger A, near the apocenters than in sections of the stream away from apocenters. In addition to this general statement, we find that the source density is about six times larger at the leading arm's apocenter than at the trailing arm's (compare also Figure 7 to Figure 4). We checked whether this can be partially explained by a selection effect, because the leading arm's apocenter has a smaller heliocentric distance than the trailing arm's. Using the selection function from Sesar et al. (2017c), we find that this falls far short of accounting for the difference in apocenter source densities between the leading and trailing arms, so incompleteness is not an issue here. Additionally, simulations like that of Dierickx & Loeb (2017) also show a similar behavior, see, e.g., their Figure 9, which shows a higher source density at the leading arm's apocenter.

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Sources orbiting in a potential show a precession of their orbits, which means that they do not follow an identical orbit each time, but actually trace out a shape made up of rotated orbits. This is because the major axis of each orbit is rotating gradually within the orbital plane.

Orbits in the outer regions of galaxies with a spherically symmetric gravitational potential are expected to have a precession between $0^\circ$ and $120^\circ$ (Belokurov et al. 2014). Assuming a spherically symmetric potential, the precession depends primarily on the shape of the potential and thus on the radial mass distribution (Belokurov et al. 2014). Additionally it is also a function of the orbital energy and angular momentum distribution (Binney & Tremaine 2008).

The estimates of angular mean distance ${D}_{\mathrm{sgr}}$ of the Sgr stream that were obtained during this work enable us to make statements about the precession of the orbit. To do so, the angle between the leading and trailing apocenters is measured.

We calculate this angle by fitting a model to the distance data in both the leading and trailing arms, namely fitting a Gaussian and a (shifted and scaled) log-normal. A comparable fit was carried out by Belokurov et al. (2014). The models used here are unphysical, but can be applied here because they describe the angular distance distribution ${D}_{\mathrm{sgr}}({\tilde{{\rm{\Lambda }}}}_{\odot })$ adequately in order to find the apocenters along with their uncertainties. As the angular distance distribution ${D}_{\mathrm{sgr}}({\tilde{{\rm{\Lambda }}}}_{\odot })$ of the leading arm appears to be symmetrical with respect to the assumed apocenters, and also appears to be Gaussian-like, a Gaussian model is fitted to the ${D}_{\mathrm{sgr}}({\tilde{{\rm{\Lambda }}}}_{\odot })$ of the leading arm. In contrast, the trailing arm shows a clear asymmetry. For this reason, we fit the trailing arm's distance distribution using a (shifted and scaled) log-normal, fitted for the range $105^\circ \leqslant {\tilde{{\rm{\Lambda }}}}_{\odot }\leqslant 265^\circ$. With comparable results, a parabola can be fitted to the data.

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The best-fit Gaussian model for the leading and trailing apocenters is shown in Figure 8. In this figure, blue and red lines show the best-fit Gaussian model for both the leading and trailing arms. The positions of the apocenters are each denoted by a circle symbol. Dashed lines mark the corresponding ${\tilde{{\rm{\Lambda }}}}_{\odot }$ of each apocenter.

We find the leading apocenter at ${\tilde{{\rm{\Lambda }}}}_{\odot }^{L}=63\buildrel{\circ}\over{.} 2\pm 1\buildrel{\circ}\over{.} 2$, reaching ${D}_{\mathrm{sgr}}^{L}=50.88\pm 0.45\,\mathrm{kpc}$, and the trailing apocenter at ${\tilde{{\rm{\Lambda }}}}_{\odot }^{T}=167\buildrel{\circ}\over{.} 6\pm 0\buildrel{\circ}\over{.} 44$, reaching ${D}_{\mathrm{sgr}}^{T}=91.12\pm 0.09\,\mathrm{kpc}$.

For a more detailed discussion of the apocenter substructure, reaching up to 120 kpc from the Sun, we refer to Sesar et al. (2017c), Section 3.

The differential orbital precession ${\omega }_{\odot }={\tilde{{\rm{\Lambda }}}}_{\odot }^{T}-{\tilde{{\rm{\Lambda }}}}_{\odot }^{L}$ is $104\buildrel{\circ}\over{.} 4\pm 1\buildrel{\circ}\over{.} 3$, corresponding to a difference in heliocentric apocenter distances of 40.24 ± 0.45 kpc.

The actual Galactocentric orbital precession is slightly lower than the difference between the heliocentric apocenters. The Galactocentric distances and angles of the leading and trailing apocenters are calculated by taking into account that the Galactocentric distance of the Sun is 8 kpc. Consequently, the opening angle between the positions of the two apocenters, as viewed from the Galactic center, is then ${\omega }_{\mathrm{GC}}=96\buildrel{\circ}\over{.} 8\pm 1\buildrel{\circ}\over{.} 3$. The Galactocentric distance of the leading apocenter is then 47.8 ± 0.5 kpc, and that of the trailing apocenter 98.95 ± 1.3 kpc, resulting in a difference in mean Galactocentric apocenter distances of 47.45 ± 1.4 kpc.

Aside from the apocenter precession of the stream (see Section 4.3), the orbital plane itself might show a precession. To test this we obtain the weighted latitude of the stream RRab, $\langle {\tilde{B}}_{\odot }\rangle$, as a function of ${\tilde{{\rm{\Lambda }}}}_{\odot }$. The weight of each star is the probability that the star is associated with the Sgr stream.

The fit as described in Section 3.1 was carried out for each bin i in ${\tilde{{\rm{\Lambda }}}}_{\odot }$, resulting in a parameter set ${{\boldsymbol{\theta }}}_{i}=({f}_{\mathrm{sgr},i},{D}_{\mathrm{sgr},i},{\sigma }_{\mathrm{sgr},i},{n}_{i})$ describing the stream and halo properties in the ${\tilde{{\rm{\Lambda }}}}_{\odot }$ bin in each case.

We now again make use of the model for the observed heliocentric distances, Equation (1) with the halo described by Equation (3) and the stream described by Equation (4). We calculate ${p}_{\mathrm{sgr}}({l}_{j},{b}_{j},{D}_{j}| {{\boldsymbol{\theta }}}_{i})$ as the fraction of the likelihood that a star j is associated with the Sgr stream divided by the sum of the likelihood that it is associated with the Sgr stream and the likelihood that it is associated with the halo:

$$\begin{eqnarray}&&{p}_{\mathrm{sgr},j}({l}_{j},{b}_{j},{D}_{j}| {{\boldsymbol{\theta }}}_{i})=\displaystyle \frac{{p}_{\mathrm{stream}}(D| {{\boldsymbol{\theta }}}_{i})}{({p}_{\mathrm{halo}}(D| {{\boldsymbol{\theta }}}_{i})+{p}_{\mathrm{stream}}(D| {{\boldsymbol{\theta }}}_{i}))}.\end{eqnarray} \tag{ 11 }$$

The weighted latitude $\langle {\tilde{B}}_{\odot }\rangle$ in a bin i is then calculated as

$$\begin{eqnarray}&&\langle {\tilde{B}}_{\odot }{\rangle }_{i}=\displaystyle \frac{{\displaystyle \sum }_{j}({\tilde{B}}_{\odot ,i}\times {p}_{\mathrm{sgr},j})}{{\displaystyle \sum }_{j}({p}_{\mathrm{sgr},j)}}.\end{eqnarray} \tag{ 12 }$$

We then use the difference in $\langle {\tilde{B}}_{\odot }\rangle$ for the leading and trailing arms to quantify the precession of the orbital plane.

The resulting $\langle {\tilde{B}}_{\odot }\rangle$ for both the leading and trailing arms are given in Tables 8 and 9 in the Appendix. Figure 9 shows $\langle {\tilde{B}}_{\odot }\rangle$ plotted versus the ${\tilde{{\rm{\Lambda }}}}_{\odot }$ bins.

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This gives evidence for the leading arm staying in or close to the plane defined by ${\tilde{B}}_{\odot }=0^\circ$, whereas the trailing arm is found within within $-5^\circ$ to $5^\circ$ around the plane. From this, we find a separation of $\sim 10^\circ$, as also derived by Law et al. (2005).

The best previous estimates of the heliocentric distances for a large part of the Sgr stream come from Belokurov et al. (2014), who used blue horizontal branch (BHB) stars, subgiant branch (SGB) stars, and red giant branch (RGB) stars from the Sloan Digital Sky Survey Data Release 8 (SDSS DR8). In Figure 10 we compare our heliocentric distances, ${D}_{\mathrm{sgr}}\pm \delta {D}_{\mathrm{sgr}}$, to those from Belokurov et al. (2014) (Figure 6 therein). We show the $1\sigma$ uncertainties from Belokurov et al. (2014) where available, and assume the uncertainties to be 10% if not stated otherwise.

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Overall, the two estimates are in good agreement, attesting to the quality of the analysis by Belokurov et al. (2014). The distances from Belokurov et al. (2014) may be systematically slightly larger; the fact that the RRab distances we use are directly tied to parallaxes from the Hubble Space Telescope (HST) and Gaia DR1 (Sesar et al. 2017a) should lend confidence to the distance scale of this work. Our new estimates for the mean distance are three times more precise, and presumably also accurate. The typical mean distance uncertainty in Belokurov et al. (2014) is 1–2 kpc and up to 0.1 ${D}_{\mathrm{sgr}}$ for most parts of the stream, whereas our work shows comparable or smaller $\delta {D}_{\mathrm{sgr}}$ (see Tables 4 and 5).

As mentioned before, our new map of the Sgr stream also has considerably more extensive angular coverage.

The high individual distance precision to the RRab of 3% allows us to map the l.o.s. depth of the stream, which Belokurov et al. (2014) could not do, or at least did not. For these reasons, our work improves the knowledge of the geometry of the Sgr stream significantly.

However, care must be taken in parts of the Sgr stream where the number of sources is comparatively low. The trailing arm's distance estimate for the bin centered on ${\tilde{{\rm{\Lambda }}}}_{\odot }=125^\circ$ results from only 28 sources within the prior indicated by Figure 4(a), i.e., $D\gt 40\,\mathrm{kpc}$. In this bin, the estimated ${D}_{\mathrm{sgr}}$ is smaller than the ${D}_{\mathrm{sgr}}$ estimated for nearby bins, and the same applies for the estimated width of the stream, which appears to be too narrow.

In both analyses, the apocenters of the leading and trailing arms are derived. We find the leading apocenter at ${\tilde{{\rm{\Lambda }}}}_{\odot }^{L}=63\buildrel{\circ}\over{.} 2\pm 1\buildrel{\circ}\over{.} 2$, reaching ${D}_{\mathrm{sgr}}^{L}=50.88\pm 0.45\,\mathrm{kpc}$, and the trailing apocenter at ${\tilde{{\rm{\Lambda }}}}_{\odot }^{T}=167\buildrel{\circ}\over{.} 6\pm 0\buildrel{\circ}\over{.} 44$, reaching ${D}_{\mathrm{sgr}}^{T}=91.12\pm 0.09\,\mathrm{kpc}$. The differential orbital precession ${\omega }_{\odot }={\tilde{{\rm{\Lambda }}}}_{\odot }^{T}-{\tilde{{\rm{\Lambda }}}}_{\odot }^{L}$ is $104\buildrel{\circ}\over{.} 4\pm 1\buildrel{\circ}\over{.} 3$, with a difference in heliocentric apocenter distances of 40.24 ± 0.45 kpc. Taking into account that the Galactocentric distance of the Sun is 8 kpc, the corresponding Galactocentric angle from our analysis is ${\omega }_{\mathrm{GC}}=96\buildrel{\circ}\over{.} 8\pm 1\buildrel{\circ}\over{.} 3$. The Galactocentric distance of the leading apocenter is then 47.8 ± 0.5 kpc, and that of the trailing apocenter 98.95 ± 1.3 kpc, resulting in a difference in mean Galactocentric apocenter distances of 47.45 ± 1.4 kpc.

If we assume that the trailing arm's apocenter is close to the maximum extent of the derived ${D}_{\mathrm{sgr}}$, as is done by fitting a Gaussian to the five closest points near the maximum extent, we find the trailing apocenter at ${\tilde{{\rm{\Lambda }}}}_{\odot }^{T}=173^\circ 4\pm 2\buildrel{\circ}\over{.} 0$, reaching ${D}_{\mathrm{sgr}}^{T}=92.7\pm 1.3\,\mathrm{kpc}$. The differential orbit precession ${\omega }_{\odot }\,={\tilde{{\rm{\Lambda }}}}_{\odot }^{T}-{\tilde{{\rm{\Lambda }}}}_{\odot }^{L}$ is then $108\buildrel{\circ}\over{.} 9\pm 2\buildrel{\circ}\over{.} 4$, with a difference in heliocentric apocenter distances of 41.82 ± 0.45 kpc. The Galactocentric angle is then slightly larger than for the log-normal fit, ${\omega }_{\mathrm{GC}}=101\buildrel{\circ}\over{.} 0\pm 2\buildrel{\circ}\over{.} 4$; the Galactocentric distance of the leading apocenter is 49.2 ± 0.5 kpc and that of the trailing apocenter is 100.7 ± 1.3 kpc, resulting in a difference in mean Galactocentric apocenter distances of 51.5 ± 1.4 kpc.

Belokurov et al. (2014) give the position of the leading apocenter as ${\tilde{{\rm{\Lambda }}}}_{\odot }^{L}=71\buildrel{\circ}\over{.} 3\pm 3\buildrel{\circ}\over{.} 5$ with a Galactocentric distance ${R}^{L}=47.8\pm 0.5\,\mathrm{kpc}$, and the position of the trailing apocenter as ${\tilde{{\rm{\Lambda }}}}_{\odot }^{L}=170\buildrel{\circ}\over{.} 5\pm 1^\circ$ with a Galactocentric distance ${R}^{L}\,=102.5\pm 2.5\,\mathrm{kpc}$. They state the derived Galactocentric orbital precession as $\omega =93\buildrel{\circ}\over{.} 2\pm 3\buildrel{\circ}\over{.} 5$.

To summarize the comparison:

• 1.  
  Our analysis is done from one single survey and type of star, whereas the work by Belokurov et al. (2014) relies on BHB, SGB, and RGB stars. The extent and depth of PS1 3π enables us to provide a more extensive angular coverage of sources. This has resulted in the first complete (i.e., spanning $0^\circ \lt {\tilde{{\rm{\Lambda }}}}_{\odot }\lt 360^\circ$) trace of the Sgr stream's heliocentric distance from a single type of star originating from a single survey.
• 2.  
  The heliocentric mean distances of the stream from Belokurov et al. (2014) may be systematically slightly larger; the fact that the RRab distances we use are directly tied to HST and Gaia DR1 parallaxes (Sesar et al. 2017c) should lend confidence to the distance scale of this work.
• 3.  
  Along with the extent of the Sgr stream, we can give its l.o.s. depth ${\sigma }_{\mathrm{sgr}}$, and deproject ${\sigma }_{\mathrm{sgr}}$ in order to get its true width.
• 4.  
  Our analysis shows a Galactocentric orbital precession about $4^\circ$ larger than measured by Belokurov et al. (2014), or $8^\circ$ larger if assuming that the trailing arm's apocenter is close to the maximum extent of the derived ${D}_{\mathrm{sgr}}$. This is within the error range given by Belokurov et al. (2014). Generally speaking, the higher the Galactocentric orbital precession, the smoother the dark matter density is as a function of the Galactocentric radius. Logarithmic haloes should show an orbital precession of about $120^\circ$ (Belokurov et al. 2014), whereas a smaller orbital precession angle indicates a profile with a sharper drop in the radial dark matter density (Belokurov et al. 2014). Finding this result, together with the result of Belokurov et al. (2014) as well as the simulation by Dierickx & Loeb (2017), is a strong indicator that a steeper profile than the logarithmic one should be considered for the dark matter halo of the Milky Way.

Part of the Sgr stream's leading arm in the Galactic northern hemisphere is "bifurcated," or branched, in its projection on the sky (Belokurov et al. 2006). Starting at ${\rm{R}}.{\rm{A}}.\,\sim \,190^\circ$, the lower and upper declination branches of the stream, labeled A and B respectively (Belokurov et al. 2006), can be traced at least until ${\rm{R}}.{\rm{A}}.\,\sim \,140^\circ$. As stated by Fellhauer et al. (2006), the bifurcation likely arises from different stripping epochs, the young leading arm providing branch A and the old trailing arm branch B of the bifurcation. Belokurov et al. (2006) state that the SGB of branch B is significantly brighter and hence probably slightly closer than A, but the branch itself is reported to have much lower luminosity than A.

Their Figure 4 shows a noticeable, but small, difference in the distances estimated for branches A and B of 3–15 kpc, qualitatively consistent with the simulations by Fellhauer et al. (2006). However, Ruhland et al. (2011) found from an analysis of BHB stars in the stream that the branches differ by at most 2 kpc in distance. To follow up on this, we measured the RRab mean distances for small patches in both branches, as shown by the polygons in Figure 11, fitting a halo and stream model as described above in Section 3. This fitting led to the distance estimates as shown in Figure 11 and in Table 10 in the Appendix. Indeed a small difference in distance between the two branches can be found, branch B being closer than branch A, as in the simulation by Fellhauer et al. (2006). But the sparse sampling by the RR Lyrae makes this analysis inconclusive.

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Analogous to the bifurcation of the leading arm found by Belokurov et al. (2006), Koposov et al. (2012) found a similar bifurcation in the trailing arm of the Sgr stream, consisting of two branches that are separated on the sky by $\sim 10^\circ$.

This bifurcation was later confirmed and studied in greater detail by Slater et al. (2013), using MSTO and red clump (RC) stars from the Pan-STARRS1 survey, and by Navarrete et al. (2017), who have examined a large portion of approximately $65^\circ$ of the Sgr trailing arm available in the imaging data from the VLT Survey Telescope (VST) ATLAS survey, using BHB and SGB stars, as well as RR Lyrae from the Catalina Real-Time Transient Survey.

They found the trailing arm appearing to be split along the l.o.s., with the additional stream component following a distinct distance track, and a difference in heliocentric distances exists of $\sim 5\,\mathrm{kpc}$. The bulk of the "bright stream" (Slater et al. 2013) is below the Sgr orbital plane (thus ${\tilde{B}}_{\odot }\lt 0^\circ$), while the "faint stream" lies mostly above the plane (${\tilde{B}}_{\odot }\gt 0^\circ$).

We compare here our distance distributions to the findings of Slater et al. (2013) and Navarrete et al. (2017) for different regions in $({\tilde{{\rm{\Lambda }}}}_{\odot },{\tilde{B}}_{\odot })$.

Navarrete et al. (2017) report a bifurcation in the $({\tilde{{\rm{\Lambda }}}}_{\odot },{\tilde{B}}_{\odot })$ plane with a separation of $\sim 10^\circ$. Likely due to our relatively sparse source density, we cannot find an indicator for a bifurcation in the $({\tilde{{\rm{\Lambda }}}}_{\odot },{\tilde{B}}_{\odot })$ plane that would lead to a "bright stream" and a "faint stream."

We then checked whether we can identify l.o.s. substructures, and made histograms of the heliocentric distance distribution for several patches along the trailing arm of the Sgr stream.

In Figure 12, we give a histogram of our distance estimates in one of the regions probed by Navarrete et al. (2017) and Slater et al. (2013). This specific region was also probed using RR Lyrae by Navarrete et al. (2017) (see their Figure 9). We give our estimates of the heliocentric distance D and the distance modulus m−M (Sesar et al. 2017c). Blue markers represent substructures found by Navarrete et al. (2017). A similar shape of the distance distribution is found, and we also detect the substructures they call "SGB 1" and "SGB 2." We find "SGB 1" at a slightly larger distance than Navarrete et al. (2017). We find "SGB 2" split into two components.

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We were also able to identify similar substructures to those found by Navarrete et al. (2017) and Slater et al. (2013) within other patches of the trailing arm of the Sgr stream, and count them as tentative but marginally significant because of the relatively low density of our tracers.