THE NATURE AND ORBIT OF THE OPHIUCHUS STREAM

The PS1 survey has observed the entire sky north of declination −30° in five filters covering 400–1000 nm (Stubbs et al. 2010; Tonry et al. 2012). The 1.8 m PS1 telescope has a 7 deg² field of view outfitted with a billion-pixel camera (Hodapp et al. 2004; Onaka et al. 2008; Tonry & Onaka 2009). In single-epoch images, the telescope can detect point sources at a signal-to-noise ratio (S/N) of 5 at 22.0, 22.0, 21.9, 21.0, and 19.8 mag in PS1 grizy_P1 bands, respectively. The survey pipeline automatically processes images and performs photometry and astrometry on detected sources (Magnier 2006, 2007). The uncertainty in photometric calibration of the survey is ≲0.01 mag (Schlafly et al. 2012), and the astrometric precision of single-epoch detections is 10 milliarcsec (Magnier et al. 2008, hereafter mas).

Based on the findings of Bernard et al. (2014b), we have used the dereddened fiducial of the old globular cluster NGC 5904 (from Bernard et al. 2014a, and shifted to the distance modulus of 14.9 mag) to select ∼170 candidate Ophiuchus stream members from the PS1 photometric catalog. The candidates were selected if their position was within 45 of the best-fitting great circle containing the stream (see Figure 1 of Bernard et al. 2014b), and if their dereddened ${g}_{P1}-{i}_{P1}$ color and i_P1-band magnitude were within 0.1 and 0.5 mag of the fiducial isochrone, respectively. We observed the selected candidates using the DEIMOS spectrograph on Keck II (Faber et al. 2003) and using the Hectochelle fiber spectrograph on MMT (Szentgyorgyi et al. 2011) over a course of two nights.

Seven candidate blue horizontal branch stars were observed with DEIMOS on 2014 May 29th (project ID 2014A-C171D, PI: J. Cohen). The observations were made using the 08 slit and the high resolution (1200 G) grating, delivering a resolution of 1.2 Å in the 6250–8900 Å range. The spectra were extracted and calibrated using standard IRAF¹² tasks. The uncertainty in the zero-point of wavelength calibration (measured using sky lines) was ≲0.04 Å (≲2 km s⁻¹ at 6563 Å).

The line-of-sight velocities of stars observed by DEIMOS were measured by fitting observed spectra with synthetic template spectra selected from the Munari et al. (2005) spectral library.¹³ Prior to fitting, the synthetic spectra were resampled to the same Å per pixel scale as the observed spectrum and convolved with an appropriate Line Spread Function. The velocity obtained from the best-fit template was corrected to the barycentric system and adopted as the line-of-sight velocity, v_los. We added in quadrature the uncertainty in the zero-point of wavelength calibration (2 km s⁻¹ at 6563 Å) to the velocity error from fitting.

The remaining 163 Ophiuchus stream candidates were observed with Hectochelle on 2014 June 6th (proposal ID 2014B-SAO-4, PI: C. Johnson). Observations were made using the RV31 radial velocity filter, which includes Mg I/Mgb features in the 5150–5300 Å range. To improve the S/N of faint targets, we binned the detector by 3 pixels in the spectral direction, resulting in an effective resolution of R ∼ 38,000.

Hectochelle spectra were extracted and calibrated following Caldwell et al. (2009). To account for variations in the fiber throughput, the spectra were normalized before sky subtraction. The normalization factor was estimated using the strength of several night sky emission lines in the appropriate order. Sky subtraction was performed using the average of 20–30 sky fibers, using the method devised by Koposov et al. (2011). A comparison of observed and laboratory positions of sky emission lines did not reveal any significant offsets in wavelength calibration (i.e., no offsets greater than 0.5 km s⁻¹ at 5225 Å).

The line-of-sight velocities of stars observed by Hectochelle were measured using the RVSAO package (Kurtz & Mink 1998), by cross-correlating observed spectra with a synthetic spectrum of an A-type and a G-type giant star (constructed by Latham et al. 2002). The velocity obtained from the best fitting template was adopted. To the uncertainty in v_los, measured by RVSAO, we added (in quadrature) the uncertainty in the zero-point of wavelength calibration, which we measured using sky emission lines to be 0.5 km s⁻¹. Finally, the measured velocities were corrected to the barycentric system using the BCVCORR task.

A comparison of velocities measured from DEIMOS and Hectochelle spectra for star "bhb6" (Table 1), shows that the two velocity sets are consistent within stated uncertainties.

Even though the primary goal of spectroscopic observations was to obtain precise radial velocities, the wavelength range and the resolution of Hectochelle spectra are sufficient to allow estimates of element abundances.

We determined stellar parameters from the continuum-normalized, radial velocity-corrected spectra using the SMH code of Casey (2014), which is built on the MOOG code of Sneden (1973). Kurucz α-enhanced (0.4 dex) model atmospheres (Castelli & Kurucz 2004) and a line list compiled from Frebel et al. (2010) by Casey (2014) were used (see Table 2 in the electronic version of the Journal). First, effective temperatures were calculated from 2MASS photometry and color–temperature calibrations of González Hernández & Bonifacio (2009) and spectroscopic temperatures were optimized around this value using the SMH code, by removing abundance trends with line excitation potential. Parameters of $\mathrm{log}\;g$ and [Fe/H] were determined using 14 Fe i and 2–3 Fe ii lines to achieve ionization balance, and microturbulence was calculated by removing trends in abundances as a function of the reduced equivalent width of the lines. Estimates of the α-enhancement were obtained using only the few available clean Mg, Ca, and Ti (I and II) lines, which comprised a total of 6–8 lines per star. The abundances were measured from equivalent widths and the lines we used are listed in Table 2. Because Mg lines are strong and may be saturated, the values of [Mg/Fe] are significantly different than values of [Ca/Fe] and [Ti/Fe].

Proper motions are crucial constraints for determining the orbit of a stream (e.g., Koposov et al. 2010). To measure the proper motion of stars in the vicinity of the Ophiuchus stream, we combine the astrometry provided by USNO-B (Monet et al. 2003) and 2MASS (Skrutskie et al. 2006) catalogs with the PS1 catalog. The USNO-B catalog lists photometry and astrometry measured from photographic plates in five different band-passes (O, E, J, F, and N). The USNO-B plates were exposed at different epochs, and thus each object in the catalog can have a maximum of five recorded positions. The 2MASS catalog provides only one position entry per object.

To reduce the systematic offsets in astrometry between different catalogs, we first calibrate USNO-B and 2MASS positions to a reference frame that is defined by positions of galaxies observed in PS1 (which are on the ICRS coordinate system). We define galaxies as objects that have the difference between point-spread function (PSF) and aperture magnitudes in PS1 r_P1 and i_P1 bands between 0.3 and 1.0 mag.

The astrometric reference catalog is created by averaging out repeatedly observed positions of PS1 objects. Between 2012 May and June, the region in the vicinity of the Ophiuchus stream was observed four times in PS1 g_P1, r_P1, and i_P1 bands. To minimize the uncertainty in astrometry due to wavelength-dependent effects, such as the differential chromatic refraction, we only average out positions observed through the r_P1-band filter. Since the astrometric precision of single-epoch detections is 10 mas (Magnier et al. 2008), the precision of the average position is ∼5 mas or better.

The USNO-B astrometry is calibrated following Munn et al. (2004, see their Section 2.1). First, we calculate the positions of objects at each of the five USNO-B epochs, using software kindly provided by J. Munn. Then, for each USNO-B object we find the 100 nearest galaxies, calculate the median offsets in right ascension and declination between the reference PS1 position and the USNO-B position for these galaxies, and add the offsets to the USNO-B position in question. This is done separately for each of the five USNO-B epochs. The single-epoch 2MASS positions are calibrated using the same procedure.

Having tied the positions for each object at one 2MASS and five USNO-B epochs to the PS1 astrometric reference frame, we can now check for any additional systematic uncertainties in the calibrated astrometry. We do so using the leave-one-out cross-validation. One of the six calibrated positions is withheld, and a straight line is fitted to the remaining five positions and the PS1 position. The straight line fit (i.e., essentially a proper motion fit, neglecting the parallax) is then used to predict the position of an object at the withheld epoch. The difference between the withheld position and the predicted position is labeled as ${\rm{\Delta }}{\rm{R.A.}}$ or ${\rm{\Delta }}\mathrm{decl}.$

Inspection of ${\rm{\Delta }}{\rm{R.A.}}$ or ${\rm{\Delta }}\mathrm{decl}.$ values has revealed that the positions of USNO-B objects depend on magnitude for some epochs (Figure 1). We have examined ${\rm{\Delta }}{\rm{R.A.}}$ and ${\rm{\Delta }}\mathrm{decl}.$ values in different regions of the sky, and have concluded that these astrometric issues affect individual photographic plates, and are not specific to a particular photographic bandpass. To remove this dependence, we subtract plate-specific and magnitude-dependent offsets (shown by yellow circles in Figure 1) from the original USNO-B positions before we calibrate the positions using PS1 galaxies. Since USNO-B does not provide uncertainty in positions, we adopt the rms scatter of ${\rm{\Delta }}{\rm{R.A.}}$ and ${\rm{\Delta }}\mathrm{decl}.$ values (shown by red solid circles in Figure 1) as an estimate of the uncertainty in position at a given magnitude and epoch.

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We have also examined whether ${\rm{\Delta }}{\rm{R.A.}}$ and ${\rm{\Delta }}\mathrm{decl}.$ astrometric residuals depend on the ${g}_{P1}-{i}_{P1}$ color. We find that the residuals do depend on the color, and that they can be as high as 100 mas. The most likely explanation for this dependence is the differential chromatic refracation. We correct for this dependence using a similar approach as above. For each photographic plate we bin ${\rm{\Delta }}{\rm{R.A.}}$ and ${\rm{\Delta }}\mathrm{decl}.$ residuals as a function of color, calculate the median for each color bin, and subtract that value from the observed positions of stars in that color bin, before we calibrate the positions using PS1 galaxies.

Finally, to measure the absolute proper motion of an object we fit a straight line to all available positions (min. 3, max. 7) as a function of time (the epoch baseline is 58 years). The proper motion of confirmed Ophiuchus stream members is listed in Table 1.

For verification, we have also measured the proper motion of 700 candidate QSOs, selected using WISE (Wright et al. 2010) $W1-W2\gt 0.8$ color criterion (see Section 2.1 by Nikutta et al. 2014). The median proper motion of candidate QSOs is 0.3 mas yr⁻¹ and the uncertainty of the median is 0.2 mas yr⁻¹, showing that there is no statistically significant offset in measured proper motions.

To measure the systematic uncertainty in proper motions, one should ideally calculate the rms scatter of proper motions of fairly bright and static sources. Bright QSOs would be an ideal choice for this measurement, because they are extragalactic point sources and because they are not used in the calibration process. Unfortunately, the candidate QSOs described above are too faint (r > 18 mag) to be used for this purpose (i.e., the uncertainty in their proper motions is already dominated by Poisson noise).

Instead, we measure the systematic uncertainty by calculating the median uncertainty in proper motion of bright stars. For stars brighter than r = 17 mag, the uncertainty in proper motion is ∼1.5 mas yr⁻¹ (and constant with magnitude), and we adopt this value as the systematic uncertainty. To determine whether this uncertainty is well-measured, we examined the distribution of χ² per-degrees of freedom values calculated from proper motion fits of bright stars (r < 17). The mode of this distribution is centered at 1, indicating that on average, the uncertainties in proper motion are well-measured (i.e., not overestimated or underestimated). However, the observed distribution has longer tails than expected (toward high χ² values), indicating that for some stars the uncertainties in proper motion are understimated (e.g., due to blending of sources in photographic plates).

The v_los distribution of stars observed by DEIMOS and Hectochelle is shown in Figure 2. In this figure, a group of 14 stars with 285 < v_los/km s⁻¹ < 292 clearly stands out. This group, which we identify as the Ophiuchus stream, is well-separated from the majority of stars which have $| {v}_{\mathrm{los}}| \lt 200$ km s⁻¹. The positions, velocities, and PS1 photometry of stars in this group are listed in Table 1.

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A closer look at v_los of stars in the Ophiuchus stream (Figure 3) suggests that their velocities are changing as a function of galactic longitude. To fit this possible velocity gradient, we use an approach similar to the one taken by Martin & Jin (2010, see their Section 2.1.1).

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We wish to find the most likely set of parameters θ for which the observations of stars listed in Table 1, ${\mathcal{D}}=\{{{\boldsymbol{d}}}_{{\boldsymbol{k}}}\}{}_{1\leqslant k\leqslant 14}$, match the model described below. In the current problem, each data point ${{\boldsymbol{d}}}_{{\boldsymbol{k}}}$ is defined by its line-of-sight velocity v_los,k at galactic longitude ℓ_k, ${d}_{k}=\{{v}_{\mathrm{los},k},{{\ell }}_{k}\}$. The velocity has an uncertainty of ${\sigma }_{{v}_{\mathrm{los},k}}$. The star "bhb6" has been observed twice, so for this star we adopt the weighted average (and its associated uncertainty) of the two line-of-sight velocity observations. The uncertainty in longitude is not considered because it is much smaller compared to the uncertainty in velocity. The data points are also considered to be independent. Therefore, the likelihood of these data points with the model defined by the set of parameters θ, is

$$\begin{eqnarray}&&p({\mathcal{D}}| \theta )=\displaystyle \prod _{k}p({{\boldsymbol{d}}}_{{\boldsymbol{k}}}| \theta ,{{\ell }}_{{\boldsymbol{k}}}),\end{eqnarray} \tag{ 1 }$$

where $p({{\boldsymbol{d}}}_{{\boldsymbol{k}}}| \theta ,{{\ell }}_{{\boldsymbol{k}}})$ is the likelihood of data point k to be generated from the model. Using Bayes theorem, the probability of a model given the data, $p(\theta | {\mathcal{D}})$, is

$$\begin{eqnarray}&&p(\theta | {\mathcal{D}})\propto p({\mathcal{D}}| \theta )p(\theta ),\end{eqnarray} \tag{ 2 }$$

where p(θ) represents our prior knowledge on the model.

We explicitly define the likelihood $p({{\boldsymbol{d}}}_{{\boldsymbol{k}}}| \theta ,{{\ell }}_{{\boldsymbol{k}}})$ as

$$\begin{eqnarray}&&p({v}_{\mathrm{los},k}| \theta ,{\sigma }_{{v}_{\mathrm{los},k}},{{\ell }}_{k})={\mathcal{N}}({v}_{\mathrm{los},k}| v({{\ell }}_{k}),{\sigma }_{k}^{\prime }),\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&{\mathcal{N}}(x| \mu ,\sigma )=(1/\sqrt{2\pi {\sigma }^{2}})\mathrm{exp}(-0.5({(x-\mu )}^{2}){/\sigma }^{2})\end{eqnarray} \tag{ 4 }$$

is a normal distribution, and $\theta =\{\frac{{{dv}}_{\mathrm{los}}}{d{\ell }},\bar{{v}_{\mathrm{los}}},s\}$ are parameters of our model:

• 1.  
  $\frac{{{dv}}_{\mathrm{los}}}{d{\ell }}$ is the velocity gradient along the galactic longitude direction,
• 2.  
  $\bar{{v}_{\mathrm{los}}}$ is the velocity of the stream at the reference galactic longitude ℓ₀ = 5°, and
• 3.  
  s accounts for the additional scatter in velocities (e.g., due to the intrinsic velocity dispersion of the stream).

The v(ℓ_k) is the predicted velocity of the stream at position ℓ_k

$$\begin{eqnarray}&&v({{\ell }}_{k})=\displaystyle \frac{{{dv}}_{\mathrm{los}}}{d{\ell }}({{\ell }}_{k}-{{\ell }}_{0})+\bar{{v}_{\mathrm{los}}}\end{eqnarray} \tag{ 5 }$$

and ${\sigma }_{k}^{\prime }=\sqrt{{s}^{2}+{\sigma }_{{v}_{\mathrm{los},k}}^{2}}$ is the quadratic sum of the additional scatter s in velocity and the uncertainty in line-of-sight velocity of data point k. The likelihood of all data points can be calculated using Equation (1).

Before we can sample from the posterior probability distribution over our model parameters, we need to define the prior probabilities of model parameters. As prior probabilities, we adopt priors that are uniform in these ranges: $270\lt \bar{{v}_{\mathrm{los}}}/\mathrm{km}\;{{\rm{s}}}^{-1}\lt 320$, $-8\lt \frac{{{dv}}_{\mathrm{los}}}{d{\ell }}/\mathrm{km}\;{{\rm{s}}}^{-1}\;{\mathrm{deg}}^{-1}\lt 8$, $0\;\leqslant s/\mathrm{km}\;{{\rm{s}}}^{-1}\lt 3$.

To efficiently explore the parameter space, we use the Goodman & Weare (2010) Affine Invariant Markov chain Monte Carlo Ensemble sampler as implemented in the emcee package¹⁴ (v2.1, Foreman-Mackey et al. 2013). We use 200 walkers and obtain convergence¹⁵ after a short burn-in phase of 100 steps per walker. The chains are then restarted around the best-fit value and evolved for another 2000 steps. To enable easy reconstruction of the posterior distribution, we provide chains in a Zenodo data repository (Sesar et al. 2015) for all of the data modeling in Section 3 of this paper.

We characterize the most probable set of parameters for our model using the maximum a posteriori values. We also report the median and equivalent 1σ confidence intervals using the 50th, 16th, and 84th percentiles, respectively (see Table 3).

Table 1.  Ophiuchus Stream Member Stars

Name	R.A.	Decl.	gP1	rP1	iP1	zP1	yP1	vlos	DM	μl	μb
	(deg)	(deg)	(mag)	(mag)	(mag)	(mag)	(mag)	(km s−1)	(mag)	(mas yr−1)	(mas yr−1)
bhb1	241.52271	−7.01555	16.05 ± 0.02	16.11 ± 0.02	16.22 ± 0.01	16.25 ± 0.02	16.25 ± 0.02	286.7 ± 1.8	${14.72}_{-0.06}^{+0.06}$	−4.1 ± 2.1	2.6 ± 2.1
bhb2	241.49994	−7.03409	16.02 ± 0.02	16.09 ± 0.02	16.21 ± 0.02	16.25 ± 0.02	16.23 ± 0.02	285.3 ± 1.9	${14.73}_{-0.06}^{+0.06}$	−4.4 ± 2.2	2.1 ± 2.2
bhb3	242.13551	−6.87785	15.96 ± 0.02	15.99 ± 0.01	16.11 ± 0.02	16.16 ± 0.02	16.13 ± 0.02	290.0 ± 1.8	${14.61}_{-0.05}^{+0.05}$	−5.2 ± 2.0	2.4 ± 2.0
bhb4	241.94714	−6.88995	16.22 ± 0.02	16.32 ± 0.01	16.48 ± 0.02	16.54 ± 0.02	16.56 ± 0.02	291.3 ± 2.2	${14.64}_{-0.06}^{+0.05}$	−5.1 ± 2.2	3.3 ± 2.1
bhb6	242.33018	−6.84405	15.67 ± 0.01	15.64 ± 0.01	15.59 ± 0.01	15.60 ± 0.02	15.54 ± 0.02	290.8 ± 1.5	${14.58}_{-0.05}^{+0.05}$	−5.6 ± 1.3	5.2 ± 1.4
bhb7	242.91469	−6.69329	15.66 ± 0.01	15.56 ± 0.02	15.57 ± 0.01	15.54 ± 0.02	15.50 ± 0.02	289.8 ± 1.5	${14.47}_{-0.06}^{+0.05}$	−7.4 ± 1.5	7.1 ± 1.7
bhb6	⋯	⋯	⋯	⋯	⋯	⋯	⋯	289.8 ± 0.8	⋯	⋯
rgb1	241.51689	−6.98511	17.71 ± 0.02	17.18 ± 0.02	16.91 ± 0.02	16.78 ± 0.02	16.71 ± 0.02	286.0 ± 0.8	${14.72}_{-0.06}^{+0.06}$	−6.2 ± 1.5	0.7 ± 1.4
rgb2	241.94649	−6.86113	17.34 ± 0.02	16.74 ± 0.02	16.46 ± 0.02	16.32 ± 0.02	16.25 ± 0.02	286.7 ± 0.6	${14.64}_{-0.05}^{+0.05}$	−4.5 ± 1.4	1.1 ± 1.4
rgb3	241.96089	−6.89873	17.64 ± 0.02	17.08 ± 0.02	16.83 ± 0.02	16.68 ± 0.02	16.62 ± 0.02	287.5 ± 0.7	${14.64}_{-0.06}^{+0.05}$	−6.5 ± 1.5	3.0 ± 1.4
rgb4	242.26139	−6.90190	17.05 ± 0.02	16.45 ± 0.02	16.17 ± 0.01	16.02 ± 0.02	15.93 ± 0.02	288.8 ± 0.5	${14.60}_{-0.05}^{+0.05}$	−23.7 ± 1.4	−14.8 ± 1.5
rgb5	242.14832	−6.79765	17.79 ± 0.02	17.23 ± 0.02	16.96 ± 0.02	16.82 ± 0.02	16.74 ± 0.02	288.0 ± 0.9	${14.60}_{-0.05}^{+0.05}$	−6.0 ± 1.3	2.4 ± 1.4
sgb1	241.95962	−6.86881	18.90 ± 0.02	18.53 ± 0.02	18.38 ± 0.02	18.31 ± 0.02	18.28 ± 0.02	289.4 ± 2.2	${14.63}_{-0.05}^{+0.05}$	−5.8 ± 2.0	1.2 ± 2.1
msto1	242.02040	−6.84122	19.22 ± 0.02	18.89 ± 0.02	18.74 ± 0.02	18.69 ± 0.02	18.62 ± 0.03	291.8 ± 2.2	${14.62}_{-0.05}^{+0.05}$	−6.9 ± 2.4	4.4 ± 2.4
msto2	242.18360	−6.84056	19.10 ± 0.02	18.70 ± 0.02	18.54 ± 0.02	18.45 ± 0.02	18.42 ± 0.03	286.4 ± 2.6	${14.60}_{-0.05}^{+0.05}$	−3.4 ± 2.3	2.3 ± 2.2

Note. The horizontal line separates stars observed by DEIMOS and Hectochelle. The name indicates the likely evolutionary stage inferred from isochrone fitting. The PS1 photometry is not corrected for extinction. The uncertainty in v_los includes the uncertainty from cross-correlation/fitting and the uncertainty in the zero-point of wavelength calibration. The DM indicates the average DM of the stream at the position of the star, and the uncertainties are 68% confidence intervals. The proper motion in the galactic longitude direction, μ_l, includes the $\mathrm{cos}b$ term.

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We find $\frac{{{dv}}_{\mathrm{los}}}{d{\ell }}=4.0\pm 1.2$ km s⁻¹ deg⁻¹, $\bar{{v}_{\mathrm{los}}}=289.1\pm$0.4 km s⁻¹, and a very small velocity dispersion of $s={0.4}_{-0.4}^{+0.5}$ km s⁻¹. Thus, we detect a gradient in the line-of-sight velocity at a 4σ level.

The preliminary element abundances of five red giant branch (RGB) stars in the Ophiuchus stream (that is, the v_los ∼ 290 km s⁻¹ group), and observed by Hectochelle, are listed in Table 4. The uncertainties of the determined parameters are listed in the notes of Table 4.

We find the stars in the Ophiuchus stream to be poor in Fe ($[\mathrm{Fe}/{\rm{H}}]\sim -2.0$ dex) and enhanced in α-elements ([α/Fe] = 0.4 ± 0.1 dex). Their [Fe/H] are consistent within 0.05 dex (rms scatter), despite fairly large estimated uncertainties in individual measurements (≲0.2 dex). The small scatter in [Fe/H] suggests that these stars come from the same single stellar population.

Based on their position (within 5' of the Ophiuchus stream, as traced by Bernard et al. 2014b), kinematic and chemical properties, we conclude that all stars listed in Table 1 are high-probability members of the Ophiuchus stream.

The sample of Ophiuchus stream members, which we have identified above using velocities and metallicities, now gives us an opportunity to further constrain the distance and the CMD of the stream.

3.3.1. Model

To model the CMD of the stream, we use a probabilistic approach analogous to the one described in Section 3.1. In our data set, ${\mathcal{D}}$, each data point ${{\boldsymbol{d}}}_{{\boldsymbol{k}}}$ is now defined by its galactic longitude and latitude, and by its PS1 grizy_P1 magnitudes, ${{\boldsymbol{d}}}_{{\boldsymbol{k}}}=\{{{\ell }}_{k},{b}_{k},{g}_{k},{r}_{k},{i}_{k},{z}_{k},{y}_{k}\}$.

Our data set contains only the Ophiuchus stream stars that were identified based on spectroscopic data (i.e., velocity and metallicity). Thus, the set is uncontaminated but very sparse and has a complicated spatial selection function. Because of this, and because we are primarily interested in constraining the distance of the stream, we focus on finding the isochrone(s) that match the confirmed members in the color–magnitude space, and do not to model the projected shape of the stream on the sky (for now, but see Section 3.4).

To model the stream in color–magnitude space, we use a grid of theoretical PARSEC isochrones¹⁶ (release v1.2S; Bressan et al. 2012; Chen et al. 2014). Each isochrone ${\mathcal{I}}$ provides PS1 magnitudes ${m}^{\prime }={g}_{P1}^{\prime },{r}_{P1}^{\prime },{i}_{P1}^{\prime },{z}_{P1}^{\prime },{y}_{P1}^{\prime }$ for a star of initial mass M_init in a single stellar population of age t, metal content Z, and parametrized for the mass-loss on the RGB using the variable values of the Reimers law parameter η (Reimers 1975, 1977). At the reference galactic longitude ℓ₀ = 5°, the distance modulus of the stellar population is defined with parameter $\bar{\mathrm{DM}}$, and a gradient in distance modulus with galactic longitude is modeled with parameter $\frac{d\mathrm{DM}}{d{\ell }}$. We model the extinction in a PS1 band by adding the

$$\begin{eqnarray}&&{C}_{\mathrm{ext}}\cdot \left[{E}_{\mathrm{SFD}}(B-V| {{\ell }}_{k},{b}_{k})+E{(B-V)}_{\mathrm{off}}\right]\end{eqnarray} \tag{ 6 }$$

term to isochrone magnitudes, where ${E}_{\mathrm{SFD}}(B-V| {{\ell }}_{k},{b}_{k})$ is the reddening at position $({{\ell }}_{k},{b}_{k})$ in the Schlegel et al. (1998) dust map, and $E{(B-V)}_{\mathrm{off}}$ accounts for a possible zero-point offset. The extinction coefficients ${C}_{\mathrm{ext}}=\{3.172,2.271,1.682,1.322,1.087\}$ for PS1 $\{g,r,i,z,y\}$ bands are taken from Table 6 of Schlafly & Finkbeiner (2011). To account for the fact that stellar evolution models are not perfect, we introduce five ${\sigma }_{m,\mathrm{iso}}$ parameters, where $m=g,r,i,z,y$, that model the uncertainty in each PS1 grizy magnitude provided by PARSEC isochrones.

Given the above model of the stream, the likelihood of a star k with this model is defined by the comparison of the spectral energy distribution ${\{g,r,i,z,y\}}_{P1}$ with the prediction ${\{{g}^{\prime },{r}^{\prime },{i}^{\prime },{z}^{\prime },{y}^{\prime }\}}_{P1}$ of an isochrone ${{\mathcal{I}}}_{k}$ for a star with the initial mass M_init. Therefore,

$$\begin{eqnarray}&&p({{\boldsymbol{d}}}_{{\boldsymbol{k}}}| {{\mathcal{I}}}_{k})=\int \left[\displaystyle \prod _{m=g,r,i,z,y}{\mathcal{N}}({m}_{k}| {m}^{\prime }({M}_{\mathrm{init}}),{\sigma }_{{m}_{k}}^{\prime })\right]{{dM}}_{\mathrm{init}},\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}&&{{\mathcal{I}}}_{k}={\mathcal{I}}({{\ell }}_{k},{b}_{k},t,Z,\eta ,E{(B-V)}_{\mathrm{off}},\bar{\mathrm{DM}},\displaystyle \frac{d\mathrm{DM}}{d{\ell }},{{\boldsymbol{\sigma }}}_{\mathrm{iso}})\end{eqnarray} \tag{ 8 }$$

is the isochrone at the galactic position of star k, ${{\boldsymbol{\sigma }}}_{\mathrm{iso}}$ = $\{{\sigma }_{g,\mathrm{iso}},{\sigma }_{r,\mathrm{iso}},{\sigma }_{i,\mathrm{iso}},{\sigma }_{z,\mathrm{iso}},{\sigma }_{y,\mathrm{iso}}\}$, ${\mathcal{N}}(x| \mu ,\sigma )$ is a normal distribution, and

$$\begin{eqnarray}&&{\sigma }_{{m}_{k}}^{\prime }=\sqrt{{\sigma }_{{m}_{k}}^{2}+{\sigma }_{m,\mathrm{iso}}^{2}+{\left[0.1{C}_{\mathrm{ext}}{E}_{\mathrm{SFD}}(B-V{| {\ell }}_{k},{b}_{k})\right]}^{2}}\end{eqnarray} \tag{ 9 }$$

is the sum of the uncertainty in the isochrone magnitude $({\sigma }_{m,\mathrm{iso}}),$ observed magnitude of data point k (${\sigma }_{{m}_{k}}$), and extinction (10% fractional uncertainty in ${E}_{\mathrm{SFD}}(B-V)$; Schlegel et al. 1998). The likelihood of all data points can then be calculated by combining Equations (1) and (7).

3.3.2. Priors on the CMD Model

Before we can calculate the probability of a model, we need to define the prior probabilities of model parameters. Below, we list our priors and describe the justification for each one. A summary of priors is given in Table 5.

Table 3.  Ophiuchus Stream Parameters

Parameter	MAPa	Median and Central 68% C.I.b
$\bar{{v}_{\mathrm{los}}}$	289.1	${289.1}_{-0.4}^{+0.4}$ km s−1
$\frac{{{dv}}_{\mathrm{los}}}{d{\ell }}$	4.0	${4.0}_{-1.2}^{+1.2}$ km s−1 deg−1
s	0.0	${0.4}_{-0.4}^{+0.5}$ km s−1
[Fe/H]	⋯	$-{1.95}_{-0.5}^{+0.5}$ dex
[α/Fe]	⋯	${0.4}_{-0.1}^{+0.1}$ dex
Age	11.7	${11.7}_{-0.3}^{+0.6}$ Gyr
Mass-loss parameter η	0.49	${0.48}_{-0.04}^{+0.02}$
Metallicity Z	2.3 × 10−4	$({2.3}_{-0.3}^{+0.4})\times {10}^{-4}$
$E{(B-V)}_{\mathrm{off}}^{c}$	0.011	${0.008}_{-0.009}^{+0.009}$ mag
$\bar{\mathrm{DM}}$	14.57	${14.58}_{-0.05}^{+0.05}$ mag
$\frac{d\mathrm{DM}}{d{\ell }}$	−0.20	$-{0.20}_{-0.03}^{+0.03}$ mag deg−1
σg,iso	0.0003	${0.012}_{-0.008}^{+0.016}$ mag
σr,iso	0.0003	${0.009}_{-0.006}^{+0.012}$ mag
σi,iso	0.0006	${0.007}_{-0.005}^{+0.009}$ mag
σz,iso	0.0003	${0.006}_{-0.005}^{+0.008}$ mag
σy,iso	0.0003	${0.007}_{-0.005}^{+0.009}$ mag
ℓmin	3.81	${3.84}_{-0.03}^{+0.03}$ deg
ℓmax	5.85	${5.86}_{-0.03}^{+0.03}$ deg
A	31.37	${31.38}_{-0.02}^{+0.02}$ deg
B	−0.80	$-{0.80}_{-0.03}^{+0.03}$
C	−0.15	$-{0.16}_{-0.04}^{+0.04}$ deg−1
Deprojected length	1.6	${1.5}_{-0.3}^{+0.3}$ kpc
σb	6.0	${6.9}_{-0.6}^{+0.7}$ arcmin
$\bar{{\mu }_{{\ell }}}$	−5.5	$-{5.6}_{-0.3}^{+0.3}$ mas yr−1
$\frac{d{\mu }_{{\ell }}}{{dl}}$	−1.5	$-{1.6}_{-0.6}^{+0.5}$ mas yr−1 deg−1
$\bar{{\mu }_{b}}$	2.4	${2.3}_{-0.3}^{+0.3}$ mas yr−1
$\frac{d{\mu }_{b}}{d{\ell }}$	2.0	${2.3}_{-0.4}^{+0.5}$ mas yr−1 deg−1
Pericenter	3.55	${3.57}_{-0.06}^{+0.05}\left({}_{-0.05}^{+0.35}\right)$ d kpc
Apocenter	17.0	${16.8}_{-0.4}^{+0.6}\left({}_{-2.9}^{+0.0}\right)$ kpc
Eccentricity	0.66	${0.65}_{-0.01}^{+0.01}\left({}_{-0.08}^{+0.0}\right)$
Orbital period	351	${346}_{-7}^{+11}\left({}_{-73}^{+2}\right)$ Myr
Radial period	239	${237}_{-5}^{+7}\left({}_{-50}^{+2}\right)$ Myr
Vertical period	346	${342}_{-7}^{+11}\left({}_{-75}^{+2}\right)$ Myr
Mass of the progenitor	⋯	∼2 × 104 M⊙

Notes.

^aMaximum a posterior value, where available. ^bThe median and the central 68% confidence intervals are measured from marginal posterior distributions. The intervals are calculated from the difference of the 16th and 50th, and 84th and 50th percentile. ^cRecall that ${(B-V)}_{\mathrm{off}}$ is the offset with respect to the reddening provided by the Schlegel et al. (1998) dust map, and is not reddening by itself. ^dThe numbers in parenthesis illustrate the range of values (with respect to the median) obtained when varying the distance of the Sun from the Galactic center and the circular velocity at solar radius (see Section 4).

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Based on spectroscopic data, the Ophiuchus stream is metal-poor ([Fe/H] = −1.95 ± 0.05 dex) and α-enhanced ([α/Fe] = 0.4 ± 0.1 dex). As shown by Salaris et al. (1993), the α-enhanced stellar population models are equivalent to scaled-solar ones with the same global metal content [M/H], where [M/H] for α-enhanced models can be calculated using their Equation (3)

$$\begin{eqnarray}&&[{\rm{M}}/{\rm{H}}]\approxeq [\mathrm{Fe}/{\rm{H}}]+{\mathrm{log}}_{10}(0.638\times {10}^{[\alpha /\mathrm{Fe}]}+0.362).\end{eqnarray} \tag{ 10 }$$

For the element abundance of the Ophiuchus stream, we estimate [M/H] ∼ −1.7 ± 0.2 dex. This means that we should adopt the normal distribution ${\mathcal{N}}({\mathrm{log}}_{10}(Z/{Z}_{\odot })| -1.7,0.2)$ as the prior probability of metallicity Z (where Z_⊙ = 0.0152 is the solar metal content used by this particular set of PARSEC isochrones). However, in the context of cross-validating our analysis, we decided to replace the above metallicity prior in favor of a (less informative) prior that is uniform in the 0.0001 < Z < 0.0004 range. Even though a less informative prior was adopted, at the end of Section 3.3.3 we find a very impressive consistency between the posterior distribution of metallicity Z (obtained using CMD fitting) and the spectroscopic estimate of Z (see bottom panel of Figure 4). In the end, it is important to note that our results would not have changed significantly if we used the more informative prior for metallicity content Z.

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Table 4.  Element Abundances of Ophiuchus Stream Stars

Name	Teff	log g	[Fe/H]	[Mg/Fe]	[Ca/Fe]	[Ti/Fe]
	(K)	(dex)	(dex)	(dex)	(dex)	(dex)
rgb1	5680	3.2	−2.00	−0.12	0.63	0.69
rgb2	5430	3.3	−1.95	0.06	0.62	0.47
rgb3	5700	3.1	−1.95	0.23	0.27	0.46
rgb4	5400	2.8	−1.90	0.24	0.57	0.42
rgb5	5720	2.9	−1.90	−0.14	0.83	...

Note. The uncertainty in T_eff is <200 K, <0.45 dex for $\mathrm{log}\;g$, ≲0.2 dex for [Fe/H], and ∼0.35 dex for abundances of α-elements.

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Table 5.  Prior Probabilities of CMD Parameters

Parameter	Prior Type	Range
Age t	uniform	8–13.5 Gyr
Mass-loss parameter η	uniform	0.2–0.5
Metallicity Z	uniform	0.0001–0.0004
$E{(B-V)}_{\mathrm{off}}$	uniform	−0.1–0.1 mag
$\bar{\mathrm{DM}}$	uniform	14.2–15.2 mag
$\frac{d\mathrm{DM}}{d{\ell }}$	uniform	−0.5–0.5 mag deg−1
σm,iso (m = g, r, i, z, y)	uniform	0–0.1 mag

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The presence of BHB stars, the [Fe/H] and the α-enhancement of the stream point to an old stellar population. Thus, for age we adopt a uniform prior in the 8 < t/Gyr < 13.5 range.

Metal-poor and old populations have η ∼ 0.4 (Renzini & Fusi Pecci 1988). Therefore, for the mass-loss parameter η we adopt a uniform prior in the 0.2 < η < 0.5 range. For the uncertainty in isochrone magnitudes, we adopt a prior that is uniform in the 0 ≤ σ_m,iso < 0.1 mag range, where m = g, r, i, z, y.

According to Bernard et al. (2014b), the Ophiuchus stream is located about 9 ± 1 kpc from the Sun. Thus, for $\bar{\mathrm{DM}}$ we adopted a uniform prior in the $14.2\lt \bar{\mathrm{DM}}\lt 15.2$ mag range (corresponding to the 7–11 kpc range). For the gradient in distance modulus, a uniform prior in the $| \frac{d\mathrm{DM}}{d{\ell }}| \lt 0.5$ mag deg⁻¹ range is adopted. Since the reddening in the region of interest is greater than 0.1 mag, we assume that the possible systematic offset in $E(B-V)$ values provided by the Schlegel et al. (1998) dust map is less than 0.1 mag (i.e., $| E{(B-V)}_{\mathrm{off}}| \lt 0.1$ mag).

3.3.3. Posterior Distributions of CMD Parameters

To efficiently explore the parameter space, we again use the emcee package. We use 1000 walkers and obtain convergence after a short burn-in phase of 100 steps per walker. The chains are then restarted around the best-fit value and evolved for another 4000 steps. The maximum a posterior values, the median and the central 68% confidence intervals of model parameters are listed in Table 3.

We find the stream to be ∼12 Gyr old and to have a distance modulus of 14.58 ± 0.05 mag (i.e., a distance of 8.2 kpc) at the reference galactic longitude l₀ = 5° (top panel of Figure 4). Most importantly, we detect a gradient of −0.20 ± 0.03 mag deg⁻¹ in distance modulus. This gradient is inconsistent with zero (i.e., with the no gradient hypothesis) at a 7σ level, and confirms the suggestion by Bernard et al. (2014b) that the eastern part of the stream is closer to the Sun (Figure 5). The gradient in distance modulus is mostly constrained by BHB and MSTO/SGB stars. When these stars are not used to constrain the CMD of the stream, the marginal posterior distribution of the gradient in distance modulus becomes multimodal and poorly constrained.

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To verify whether the observed gradient in distance modulus is real, we compared derredened colors and magnitudes of BHB stars in the Ophiuchus stream. Stars bhb1 and bhb3 have identical dereddened ${g}_{P1}-{i}_{P1}$ color (${g}_{P1}-{i}_{P1}=-0.43$ mag), and thus should have identical absolute magnitudes.¹⁷ However, their dereddened i_P1-band magnitudes differ by 0.1 mag, and the brighter star in the pair is located 0.5 deg eastward (in the galactic longitude direction), in agreement with the observed distance modulus gradient of −0.2 mag deg². The stars bhb6 and bhb7 also show similar behavior.

In Figure 6, we compare CMDs of the Ophiuchus stream and field stars. The CMD of field stars (grayscale pixels) was obtained by binning the ${g}_{P1}-{i}_{P1}$ colors and i_P1-band magnitudes of stars located more than 18' from the equator¹⁸ of the Ophiuchus stream.

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In Section 3.3.2, we adopted a uniform prior for the metallicity content Z in order to test the predictive power of our data set. As shown in the bottom panel of Figure 4, the peak of the marginal posterior distribution of Z is consistent with the mean value of Z estimated from spectroscopic data (solid vertical line), and the distribution is even narrower than the distribution of Z estimated from spectroscopy (dashed vertical lines). This result demonstrates the predictive power of our data. It shows how a combination of good coverage of the CMD, PS1 photometry, and detailed modeling can provide an accurate and precise estimate of the metallicity of single stellar populations.

The longitude-dependent CMD model we have built in Section 3.3, and the luminosity functions associated with the model, allow us to assign a likelihood that a star is a member of the Ophiuchus stream, based on the star's galactic longitude ℓ, ${g}_{P1}-{i}_{P1}$ color, and i_P1-band magnitude. The distribution of field stars in the ${g}_{P1}-{i}_{P1}$ versus i_P1 CMD (grayscale pixels in Figure 6), on the other hand, enables us to estimate the likelihood that a star is associated with the field. As we show in this section, these two probability density functions (PDFs), when combined with positional and proper motion data, can be used to simultaneously trace the extent of the Ophiuchus stream and determine its proper motions across the sky.

In principle, we could measure the proper motion of the Ophiuchus stream using the proper motion of its confirmed members. However, since our sample of confirmed members contains only 14 stars, there is a possibility that one or two stars with incorrectly measured proper motions may bias the results. As an example, stream member "rgb4" is clearly an outlier in proper motion as it has μ_ℓ ∼ −24 mas yr⁻¹, while the remaining members have μ_ℓ ∼ −6 mas yr⁻¹. A visual inspection of digitized photographic plates has revealed that "rgb4" is blended with a neighbor of similar brightness, which affects the measured position of the star and its proper motion.

Fortunately, we do not need to rely only on confirmed members and can use a much larger sample of stars in the vicinity of the Ophiuchus stream to constrain its proper motion and extent. As we detail below, we use a probabilistic approach (see Sections 3.1 and 3.3) and model the distribution of stars simultaneously in coordinate, proper motion, and color–magnitude space as a mixture of stream and field (i.e., non-stream) stars. Even though we do not a priori know which star is a true member of the stream, we assume that as an ensemble, the stream stars have certain characteristics which make them distinguishable from field stars (e.g., common proper motion, distance, position on the sky and in the CMD), and that the scatter in these characteristics is sufficiently small to overcome the fact that there are a lot more field than stream stars. The narrow width of the stream in color–magnitude (Figure 6) and coordinate space (Figure 1 of Bernard et al. 2014b) support this assumption. After all, if the stream did not have these characteristics, it likely would not have been detected by Bernard et al. (2014b) in the first place.

Even though the probabilistic approach we describe below uses all of the stars in the vicinity of the Ophiuchus stream to constrain the extent and proper motion, we expect that most of the signal will come from main sequence turn-off (MSTO) stars associated with the stream. As shown in Figure 6, the stream's MSTO is bluer than the field population, which means that stars in this region of the CMD are much more likely to be associated with the stream than with the field population. Thus, the statistical weight of such stars will be greater than, for example, the weight of stream's RGB stars, which occupy the region of the CMD that is heavily dominated by field stars.

Assuming the stream extends between galactic longitudes ℓ_min and ℓ_max, the likelihood that a star with galactic longitude ℓ_k, latitude b_k, proper motions in galactic coordinates of μ_ℓ,k and μ_b,k, color ${(g-i)}_{k}$ and magnitude i_k is drawn from the mixture model, is equal to

$$\begin{eqnarray}p({{\boldsymbol{d}}}_{{\boldsymbol{k}}}| \theta ,{{\ell }}_{k}) & = & {{fp}}_{\mathrm{str}}({{\boldsymbol{d}}}_{{\boldsymbol{k}}}| {\theta }_{\mathrm{str}},{{\ell }}_{k})\\ & & +(1-f){p}_{\mathrm{fld}}({{\boldsymbol{d}}}_{{\boldsymbol{k}}}{| \theta }_{\mathrm{fld}},{{\ell }}_{k}),\end{eqnarray} \tag{ 11 }$$

where ${{\boldsymbol{d}}}_{{\boldsymbol{k}}}\equiv \{{{\ell }}_{k},{b}_{k},{\mu }_{{\ell },k},{\mu }_{b,k},{(g-i)}_{k},{i}_{k}\}$ contains measurements for data point (star) k, and $\theta \equiv \{{\theta }_{\mathrm{str}},{\theta }_{\mathrm{fld}}\}$ contains parameters that model the distribution of stream and field stars, respectively. The parameter f specifies the fraction of stars in the stream (out of all stars between ℓ_min and ℓ_max) and is 0 ≤ f ≤ 1 for all ℓ_k where ℓ_min < ℓ_k < ℓ_max, otherwise, it is f = 0.

The likelihood ${p}_{\mathrm{str}}({{\boldsymbol{d}}}_{{\boldsymbol{k}}}| {\theta }_{\mathrm{str}},{{\ell }}_{k})$ is a product of spatial likelihood ${p}_{\mathrm{str}}^{\mathrm{sp}}$, proper motion likelihood p_str^pm, and the color–magnitude likelihood p_str^cm

$$\begin{eqnarray}&&{p}_{\mathrm{str}}({{\boldsymbol{d}}}_{{\boldsymbol{k}}}{| \theta }_{\mathrm{str}},{{\ell }}_{k})={p}_{\mathrm{str}}^{\mathrm{sp}}({b}_{k}{| \theta }_{\mathrm{str}}^{\mathrm{sp}},{{\ell }}_{k})\\ &&\quad \times {p}_{\mathrm{str}}^{\mathrm{pm}}({\mu }_{{\ell },k},{\mu }_{b,k}{| \theta }_{\mathrm{str}}^{\mathrm{pm}},{{\ell }}_{k})\\ &&\quad \times {p}_{\mathrm{str}}^{\mathrm{cm}}({(g-i)}_{k},{i}_{k}{| \theta }_{\mathrm{str}}^{\mathrm{cm}},{{\ell }}_{k},{b}_{k},{E}_{\mathrm{SFD}}(B-V{| {\ell }}_{k},{b}_{k})).\end{eqnarray} \tag{ 12 }$$

The likelihood for field stars, ${p}_{\mathrm{fld}}({{\boldsymbol{d}}}_{{\boldsymbol{k}}}| {\theta }_{\mathrm{fld}},{{\ell }}_{k})$, has the same decomposition.

In galactic coordinates, the distribution of stream stars in the latitude direction is modeled with a Gaussian of width σ_b, where the latitude position of the Gaussian changes as a quadratic function of the galactic longitude

$$\begin{eqnarray}&&{p}_{\mathrm{str}}^{\mathrm{sp}}({b}_{k}{| \theta }_{\mathrm{str}}^{\mathrm{sp}},{{\ell }}_{k})={\mathcal{N}}({b}_{k}| \nu ({{\ell }}_{k}),{\sigma }_{b}),\end{eqnarray} \tag{ 13 }$$

where $\nu ({{\ell }}_{k}| A,B,C)=A+B({{\ell }}_{k}-{{\ell }}_{0})+C{({{\ell }}_{k}-{{\ell }}_{0})}^{2}$ is the galactic latitude of the equator of the stream and ℓ₀ = 5°.

The spatial distribution of field stars is modeled in a similar fashion, except the Gaussian has a width ${\sigma }_{b}^{\prime }$, and its latitude position is a linear function of the galactic longitude

$$\begin{eqnarray}&&{p}_{\mathrm{fld}}^{\mathrm{sp}}({b}_{k}{| \theta }_{\mathrm{fld}}^{\mathrm{sp}},{{\ell }}_{k})={\mathcal{N}}({b}_{k}{| \nu }^{\prime }({{\ell }}_{k}),{\sigma }_{b}^{\prime }),\end{eqnarray} \tag{ 14 }$$

where ${\nu }^{\prime }({{\ell }}_{k}| {A}^{\prime },{B}^{\prime })={A}^{\prime }+{B}^{\prime }({{\ell }}_{k}-{{\ell }}_{0})$. We use the above model for p_fld^sp because it is easy to implement, and because for large ratios of ${\sigma }_{b}^{\prime }/{\sigma }_{b}$ the Gaussian that models the spatial distribution of field stars approximates to a plane with respect to the much narrower Gaussian that describes the spatial distribution of stream stars.

At the reference galactic longitude ℓ = 5°, the stream is assumed to have proper motion $\bar{{\mu }_{{\ell }}}$ and $\bar{{\mu }_{b}}$, with possible gradients in proper motion of $\frac{d{\mu }_{{\ell }}}{d{\ell }}$ and $\frac{d{\mu }_{b}}{d{\ell }}$ (i.e., gradients as a function of galactic longitude). The proper motion likelihood of stream stars is then

$$\begin{eqnarray}&&{p}_{\mathrm{str}}^{\mathrm{pm}}({\mu }_{{\ell },k},{\mu }_{b,k}{| \theta }_{\mathrm{str}}^{\mathrm{pm}},{{\ell }}_{k})\\ &&\quad ={\mathcal{N}}({\mu }_{{\ell },k}{| \mu }_{{\ell }}({{\ell }}_{k}),{\sigma }_{k}^{\prime })\\ &&\qquad \times {\mathcal{N}}({\mu }_{b,k}{| \mu }_{b}({{\ell }}_{k}),{\sigma }_{k}^{\prime }),\end{eqnarray} \tag{ 15 }$$

where ${\mu }_{{\ell }}({{\ell }}_{k})$ = $\frac{d{\mu }_{{\ell }}}{d{\ell }}\left({{\ell }}_{k}-{{\ell }}_{0}\right)+\bar{{\mu }_{{\ell }}}$ and ${\mu }_{b}({{\ell }}_{k})$ = $\frac{d{\mu }_{b}}{d{\ell }}\left({{\ell }}_{k}-{{\ell }}_{0}\right)+\bar{{\mu }_{b}}$ are the predicted proper motions of the stream at galactic longitude ℓ_k, and ${\sigma }_{k}^{\prime }=\sqrt{{\sigma }_{\mathrm{pm}}^{2}+{\sigma }_{\mu ,k}^{2}}$ is the quadratic sum of the intrinsic proper motion dispersion and the uncertainty in the corresponding proper motion of data point k. The purpose of parameter σ_pm is to account for any additional scatter in proper motions (e.g., due to unaccounted errors). The proper motion likelihood of field stars has the same form (but different parameters) as the proper motion likelihood of stream stars.

The likelihood that a star is drawn from the stream's CMD is defined as

$$\begin{eqnarray}&&{p}_{\mathrm{str}}^{\mathrm{cm}}({(g-i)}_{k},{i}_{k}{| \theta }_{\mathrm{str}}^{\mathrm{cm}},{{\ell }}_{k},{b}_{k},{E}_{\mathrm{SFD}}(B-V{| {\ell }}_{k},{b}_{k}))\\ &&\quad =\zeta \displaystyle \int \displaystyle \int {\mathcal{N}}({(g-i)}_{k}^{\prime }| g-i,{\sigma }_{{(g-i)}_{k}})\\ &&\qquad \times {\mathcal{N}}({i}_{k}^{\prime }| i,{\sigma }_{{i}_{k}})p(g-i,i| \mathrm{str})d(g-i){di},\end{eqnarray} \tag{ 16 }$$

where ${\sigma }_{{(g-i)}_{k}}$ and ${\sigma }_{{i}_{k}}$ are the uncertainty in color and magnitude of data point k, and ζ is a normalization constant calculated such that the integral of Equation (16) over the considered region of CM space is unity. To account for the extinction and the gradient in distance, we use color ${(g-i)}_{k}^{\prime }={(g-i)}_{k}-1.49{E}_{\mathrm{SFD}}(B-V| {{\ell }}_{k},{b}_{k})$ and magnitude ${i}_{k}^{\prime }={i}_{k}-1.682{E}_{\mathrm{SFD}}(B-V| {{\ell }}_{k},{b}_{k})-\frac{d\mathrm{DM}}{d{\ell }}({{\ell }}_{k}-5)$, where $\frac{d\mathrm{DM}}{d{\ell }}=-0.20$ mag deg⁻¹ is the most probable gradient in distance modulus (see Table 3).

In Equation (16), $p(g-i,i| {\theta }_{\mathrm{str}}^{\mathrm{cm}})$ is the PDF of the Ophiuchus stream in the ${g}_{P1}-{i}_{P1}$ versus i_P1 color–magnitude space at galactic longitude ℓ₀ = 5°. This PDF was constructed by sampling isochrones from the stream's CMD model (Section 3.3.3), multiplying them with their luminosity functions, and then summing them up in a binned ${g}_{P1}-{i}_{P1}$ versus i_P1 CMD.

The likelihood that a star is drawn from the field CMD is calculated as

$$\begin{eqnarray}&&{p}_{\mathrm{fld}}^{\mathrm{cm}}({(g-i)}_{k},{i}_{k}{| \theta }_{{bkg}}^{\mathrm{cm}},{{\ell }}_{k},{b}_{k})\\ &&\quad =\zeta \displaystyle \int \displaystyle \int {\mathcal{N}}({(g-i)}_{k}| g-i,{\sigma }_{{(g-i)}_{k}})\\ &&\qquad \times {\mathcal{N}}({i}_{k}| i,{\sigma }_{{i}_{k}})p(g-i,i{| \theta }_{{bkg}}^{\mathrm{cm}},{{\ell }}_{k},{b}_{k})d(g-i){di}.\end{eqnarray} \tag{ 17 }$$

In Equation (17), the observed color and magnitude are not corrected for extinction or any gradients.

The PDF of field stars (i.e., the CMD), $p(g-i,i| {\theta }_{{bkg}}^{\mathrm{cm}},{{\ell }}_{k},{b}_{k})$ in Equation (17), depends on the galactic position. We construct $p(g-i,i| {\theta }_{{bkg}}^{\mathrm{cm}},{{\ell }}_{k},{b}_{k})$ by dividing the Ophiuchus region into 1° × 1° spatial pixels that overlap by 05 in galactic longitude and latitude directions. For each spatial pixel, we bin ${g}_{P1}-{i}_{P1}$ colors and i_P1-band magnitudes of stars located more than 18' from the equator¹⁹ of the stream, and normalize the resulting CMD to unity area. An example CMD of field stars is shown in Figure 6 as grayscale pixels.

In total, our model contains 20 parameters. For all of the parameters, we have adopted uniform priors within reasonable bounds. The allowed ranges of model parameters were determined by examining positions and proper motions of confirmed members and other stars. In addition to adopted priors, we also require that the parameters satisfy the following constraints:

• 1.  
  the spatial width of the stream must be smaller or equal than the width of the spatial distribution of field stars: ${\sigma }_{b}\leqslant {\sigma }_{b}^{\prime }$,
• 2.  
  the additional scatter in proper motion of stream stars must be smaller than the scatter in proper motions of field stars: ${\sigma }_{\mathrm{pm}}\leqslant {\sigma }_{\mathrm{pm}}^{\prime }$, and
• 3.  
  the galactic latitudes of confirmed members (b^conf_k) must be within 3σ_b of the equator of the stream: $| \nu ({l}_{k}^{\mathrm{conf}})-{b}_{k}^{\mathrm{conf}}| \leqslant 3{\sigma }_{b}$, where $\nu ({l}_{k}^{\mathrm{conf}})$ is the galactic latitude of the stream at the position of confirmed members, and 1 ≤ k ≤ 14.

As our data set, we use stars brighter than i_P1 = 20 mag with measured proper motions, and located in a 4 × 4 deg² area centered on the Ophiuchus stream. To explore the parameter space, we use 200 emcee walkers and obtain convergence after a short burn-in phase of 100 steps. The chains are then restarted around the best-fit value and evolved for another 2000 steps. The maximum a posterior values, the median and the central 68% confidence intervals of model parameters are listed in Table 3.

We find the stream to be confined between galactic longitudes of 381 and 585 (Figure 7). When combined with the distance of the stream (Figure 5), this result implies that the deprojected length of the stream is 1.6 ± 0.3 kpc. Thus, the stream is very foreshortened in projection, by a ratio of 6:1. The galactic latitude of the equator of the stream is at ${b}_{\mathrm{stream}}({\ell })$ = 31.37–0.80$({\ell }-5)-0.15{({\ell }-5)}^{2}$ deg, and the width of the stream is σ_b ∼ 6 arcmin (in the galactic latitude direction). In direction perpendicular to the stream's equator, the stream is ∼35 wide, which is similar to width of ∼3' measured by Bernard et al. (2014b).

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Using the ${g}_{P1}-{i}_{P1}$ color and i_P1-band magnitude, and given the CMD model of the stream (Section 3.3.3), we can evaluate the probability that a star is a member of the Ophiuchus stream. We have calculated these probabilities for all of the stars in the vicinity of the Ophiuchus stream and have created a probability-weighted number density map of the stream, shown as grayscale pixels in Figure 7. An inspection of the number density map did not reveal a significant overdensity of stars along the stream that would indicate the presence of a progenitor.

Our data indicate that the proper motion of the stream changes as a function of galactic longitude (Figure 8). The gradients in proper motion are significant at ≳3σ level, and while their absolute values are similar (∼2 mas yr⁻¹ deg⁻¹), the gradients have opposite signs. For comparison, the gradients in proper motions of field stars are 10 times smaller, $\frac{d{\mu }_{{\ell }}^{\prime }}{d{\ell }}\sim 0.1$ mas yr⁻¹ deg⁻¹ and $\frac{d{\mu }_{b}^{\prime }}{d{\ell }}\sim 0.2$ mas yr⁻¹ deg⁻¹. Overall, the proper motions of field stars (${\mu }_{{\ell }}^{\prime }\sim -6$ mas yr⁻¹ and ${\mu }_{b}^{\prime }\sim 0.2$ mas yr⁻¹) are consistent with apparent motions of a population at ∼8 kpc (due to the motion of the Sun around the Galaxy). For comparison, the apparent motion of the compact radio source Sgr A^* at the Galactic center is ${\mu }_{{\ell }}^{{\mathrm{SgrA}}^{*}}=-6.38$ mas yr⁻¹ and ${\mu }_{b}^{{\mathrm{SgrA}}^{*}}=-0.20$ mas yr⁻¹ (Reid & Brunthaler 2004).

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The stream parameters we have obtained so far can be used to place a lower limit on the mass of the initial population of the Ophiuchus stream. The fraction of stars f in the stream between longitudes ℓ_min and ℓ_max, can be converted to the number of stars in the stream, N_stars. We find that there are N_stars = 300 ± 30 stars brighter than i_P1 = 20 mag in the Ophiuchus stream. If we adopt the luminosity function associated with the most probable CMD model of the stream and assume Kroupa (1998) initial mass function (not corrected for binarity), this number of stars implies that the initial population of the Ophiuchus stream had to have a mass of at least M_init = (7.0 ± 0.7) × 10³ M_⊙.