PARALLAX OF GALACTIC CEPHEIDS FROM SPATIALLY SCANNING THE WIDE FIELD CAMERA 3 ON THE HUBBLE SPACE TELESCOPE: THE CASE OF SS CANIS MAJORIS

For each Cepheid, we obtain HST observations in five to nine epochs at 6-month intervals, ensuring that the observations are always executed at orientations 180° apart to within the HST pointing precision (about 001). The reason is that our measurements are inherently one-dimensional; we obtain very accurate positions perpendicular to the scanning direction, and much less accurate (often less so than direct observations) in the direction along the scan. In order to optimally measure the variation in position of the Cepheid, we need to ensure that the direction of resolution is always nearly the same.

To the extent possible, we also need the scan direction to be fixed with respect to the detector frame and close to the detector Y direction. This minimizes the impact of low-level geometric distortion, for which only the X component is needed, and of the charge transfer efficiency (CTE) effects that are well documented with space-based charge-coupled devices (CCDs; see, e.g., Anderson & Bedin 2010). In particular, the CTE losses are much smaller in the X than in the Y direction (Anderson 2014a); consequently, it is desirable for the measurement direction to nearly coincide with the detector X direction, implying a scan along Y. (A small angular offset is introduced in order to vary the pixel phase along the scan.) The motion of the Cepheid in the resolution direction is the result of the combination of the appropriate component of the proper motion and of the parallax of the target, compared to that of the reference stars. Ideally, the date and orientation of each observation should be chosen to maximize the projection of the parallactic motion along the resolution direction. Because of the 180° change requirement, the date and allowed orientation range of each observation are constrained, typically resulting in a projection factor of 0.8–0.9. We allow for a slack of up to 1 week in the scheduling of each observation.

At each epoch, we obtain four or five scanned observations. The first four are straight scans in the sequence: Forward, deep; Backward, shallow; Forward, shallow; Backward, deep. This sequencing helps average out time variations between deep and shallow scans, which could otherwise lead to larger systematic differences between deep scans (for most reference stars) and shallow scans (for the Cepheid and the brighter reference stars).

If possible, a fifth scan is obtained in so-called "serpentine" mode, in which the scan speed is increased to a value sufficient to avoid saturation of the target Cepheid in the broadband filter F606W, typically to about 1''–4'' s⁻¹ (up to an order of magnitude faster than the straight scan). With such a high scan speed, the length of the scan in the standard 350 s exposure time exceeds the size of the detector. Thus, in order to fit the length of the scan, it is necessary to "fold" the scan itself: the telescope describes a series of parallel forward and backward scans, offset by a user-selectable amount in the X direction. We use a separation of 4'', about 100 pixels. These scans are more complex to analyze and are more affected by potential cross-contamination (overlap between scans pertaining to different stars) because of the higher density of scans, but they offer the potential for direct comparison of the Cepheid and many of the reference stars within the same filter, and they can improve the measurement precision since more pixels are covered. Discussion of the analysis of serpentine scans can be found in Section 2.6.1.

As described in Paper I, in addition to the scanned observations we also obtain short, pointed observations of the field in order to determine multiband photometry of the reference stars in several medium-band filters, including WFC3's analogs of the Strömgren filters. The photometry thus obtained, combined with infrared JHK photometry from the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006), space-based photometry at 3.6 and 4.5 μm from the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010), and ground-based medium-resolution spectra, is used to obtain spectrophotometric distance estimates of as many of the reference stars as possible, which is a critical step in converting the relative parallax measurement for the Cepheid target into an absolute measurement. Details on the spectrophotometric data and distance estimates are in Section 3; the estimate of absolute parallax is discussed in Section 4.

Here we present the results of the analysis for the first of these 18 targets, the fundamental-mode Cepheid SS Canis Majoris (SS CMa), with a period of 12.35 days and a mean magnitude $\langle V\rangle$ = 9.9 mag. SS CMa was identified as variable by Hoffmeister (1929), and a period was determined by Oosterhoff (1935). We have chosen to complete the analysis of SS CMa because it is one of the first few Cepheids for which five epochs of observations have been completed, and because we believe that it is representative of the possible accuracy of the parallax measurements for the rest of the targets of our program.

Figure 3 shows a mosaic of the region of sky around SS CMa in the filter F547M, as obtained from the very short observations included in our program. The area represents approximately two WFC3/UVIS fields of view, stacked vertically with an overlap of ∼20''. Reference stars and their designations are indicated. The Cepheid (Star 0) is saturated. The diagonal bands are caused by the gap between the two detectors in WFC3/UVIS; more observations in the future will help cover this gap. Figure 4 shows a normal scan (top) and a serpentine scan (bottom); serpentine scans are discussed further in Sections 2.4 and 2.6.1.

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The Cepheid SS CMa has been discussed in the literature as a potential binary. Evans & Udalski (1994) identify a nearby faint blue star (Star 29 in Figure 3; 13'' from the Cepheid, V ≈ 15.51–15.58 mag), which they argue is likely to be a physical companion, on the basis of its estimated distance modulus and of probabilistic arguments. If Star 29 were physically bound to SS CMa, it would imply a separation much larger than the typical upper limit for physical companions (∼4000 AU; Evans et al. 2016). We will show in Section 4 that astrometric and spectrophotometric evidence suggests that Star 29 is significantly closer than the Cepheid and therefore not a physical companion. Szabados (1996) reports an apparent difference of ∼15 km s⁻¹ between the radial velocities (RVs) measured by Joy (1937) and Coulson & Caldwell (1985). R. I. Anderson et al. (2016, in preparation) also report a long-term RV change. If interpreted as caused by binarity, such changes would indicate an unresolved stellar companion on a currently unconstrained orbit with period likely on the order of decades.

Binarity can potentially bias the astrometric parallax determination if the orbital motion of the Cepheid itself, sampled at the times of the astrometric observations, has a sufficient component to contribute to the parallax signature. This is most likely for orbits with period ∼1 yr; very long period binaries will produce primarily a proper-motion bias, and short period binaries have a small astrometric signature that will typically average out in the measurements.

For the case of SS CMa, we have obtained new RV measurements, discussed in detail in Section 5, which demonstrate that the contribution of binary motion compatible with the observations is most likely below a few microarseconds, thus significantly smaller than the uncertainty in our parallax measurement.

In Paper I we demonstrated that for sufficiently bright sources, scanning-mode HST observations, in which the telescope is slewed during the exposure and each star leaves a trail of light nearly along a pixel axis, can achieve positional measurements with a precision up to 0.5–1 millipixels (1 millipixel, or mpix, is about 40 μas) in the direction perpendicular to the scan, about a factor of 10 better than optimal measurements from pointed images. However, we also discovered that at the millipixel level, several systematic effects come into play and need to be properly calibrated in order to fully realize a corresponding accuracy of ≲40 μas in the parallax measurement.

Perhaps the most problematic systematic effect is in the insufficiently characterized, and variable, geometric distortion solution for the WFC3/UVIS camera. The mapping between pixels and the sky is well established and calibrated for direct images, with a residual uncertainty currently estimated below 0.006 pixels (rms) over the field (Bellini et al. 2011)—fully adequate for the astrometric interpretation of direct, pointed images. However, this uncertainty is an order of magnitude higher than the precision achievable in scanning mode and the requirements of our program. Calibration observations of the field of M35, obtained 1 yr apart in 2012 and 2013, show that much of the residual geometric distortion at the sub-0.01 pixel level is static and smooth, in that the residuals vary slowly over the field of view (on scales of ∼100 pixels) and are highly correlated from year to year.

Figure 5 shows that the pattern of residual geometric distortion is repeatable over short (orbital) timescales. Each line shows the variation in the measured X position along the scan for the same bright star, located at the same detector position, in several consecutive scans over a two-orbit period, after subtracting the jitter pattern for that observation. To reduce pixel-to-pixel noise, the lines have been smoothed with a 20-pixel length. Without residuals in the geometric distortion or other disturbance factors, these lines should all be consistent with zero (i.e., a constant X position along the scan). Instead, there is a definite pattern of deviations, and this pattern is consistent from scan to scan. The horizontal dashed lines indicate the nominal precision of 10 mpix for pointed observations, which is also the expected accuracy of the standard geometric distortion solution. This distortion pattern repeats closely even a year later: Figure 6 shows the median pattern for the same star in the same sets of observations taken 1 yr apart. (Consistent with our treatment of all scans, an overall tilt of the lines has been solved for and subtracted.) Again, the patterns are very similar, strongly suggesting that the residual geometric distortion is stable over time. The inset in Figure 6 shows the measured differential geometric distortion in each cell in Year 1 (abscissa) versus Year 2 (ordinate); the two quantities are highly correlated (r ≈ 0.70), showing that a significant part of the distortion remains the same from year to year over the whole field of view.

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However, the differences between scans in the same year, or between years, are larger than nominal statistical errors, which are below 1 mpix per cell. Time-dependent geometric distortion, identified and discussed in Paper I for both calibration and SY Aur observations, contributes to these differences. We thus attempt to characterize and correct for both a static and a time-dependent component of the geometric distortion correction, in different ways.

2.3.1. Static Correction to the Geometric Distortion Solution

Even neglecting the time dependence of the geometric distortion solution, any residual static term will affect our solution. The reason is that parallax observations need to take place approximately at 6-month intervals, and orbital geometry mandates that observations 6 months apart cannot be taken at the same orientation, although they can generally be taken at orientations 180° apart. Thus, for a typical target, there will be three epochs taken at one orientation, and two taken at an orientation different by 180°. Changing the telescope roll by 180° ensures that the resolution direction, perpendicular to the scan direction and thus typically along the detector X direction, remains the same on the sky, thus greatly simplifying the analysis and improving the accuracy of the results.

Target and reference stars can be placed essentially at the same detector location for each orientation; as we are interested in relative variations in the stars' position, small errors in the geometric distortion solution, which typically behave smoothly over the detector, will cancel out. However, this is not the case across orientations, when each star's location moves to a completely different place in the detector, and thus the accuracy in the geometric distortion solution comes in fully.

In order to improve the static geometric distortion solution, we have analyzed calibration observations taken of two open clusters, M48 and M67, which offer a wealth of bright stars and thus allow a dense sampling of the detector in as few as 10 dithers. Such observations have been obtained as part of the Cycle 22 WFC3 calibration program and have demonstrated that a static term, sampled on a grid of 100 × 100 pixel cells, can account for about half of the deviation from an accurate solution. (The differences remain larger than the statistical measurement errors, which are below 1 mpix, in part because of the time-dependent correction discussed in Section 2.3.2.) We therefore employ this solution as a correction to the default geometric distortion solution obtained by Bellini et al. (2011). The pattern of static geometric distortion thus obtained is shown in Figure 7; the top panel is for F606W, and the bottom panel is for F621M. The two patterns look remarkably similar; Figure 8 shows the strong correlation (r ≈ 0.7) between the geometric distortion corrections thus obtained, despite the fact that the observations targeted different star fields, in different filters, and were taken more than 1 month apart.

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2.3.2. Time-dependent Correction

Based on our experience observing the field of SY Aur (Paper I), we developed simulation tools to optimize the observations of other Cepheids, and we applied these to the field of SS CMa. As in SY Aur, we selected positions, scan speeds, scan lengths, and filters to allow the highest-quality parallax measurements from the field (see Paper I for these details). The field of SS CMa provided more than twice as many reference stars of intermediate brightness, used to register the deep and shallow scans, as the field of SY Aur.

In addition to the straight scans we discussed in Paper I—two each in broadband and narrowband filters at each epoch—we have also obtained and processed serpentine scans, in which the telescope moves through the field in a boustrophedonic pattern (down to up, shift right, up to down, shift right, etc.), as shown in the bottom panel of Figure 4. With this pattern scanned at a rate of 15 s⁻¹, the target Cepheid does not saturate even in the broadband filter. However, at this rate the telescope will traverse about 525'' over a 350 s exposure, almost four times the WFC3 field of view. Taking exposures shorter than 350 s is not desirable because of how memory is managed in WFC3; therefore, in order to keep the target and most of the reference stars on chip for the largest fraction of the time, and thus to collect as many photons as possible, the scan pattern is folded into multiple near-straight scan lines, all approximately along the Y direction, with small "crossbars" between them. In the case of SS CMa, the F606W serpentine scans had three legs, and the overall length was such that no crossbars are visible for the Cepheid; the turnaround occurred while the Cepheid was off-chip. Owing to planning priorities, no serpentine scan was obtained in the fifth and final epoch.

The advantage of the serpentine scanning pattern is that the Cepheid and most of the reference stars are observable in the same exposure; thus, it is possible to solve directly for relative and absolute parallaxes without the potential for inconsistencies in the geometric distortion solution across filters. The disadvantages are that (1) the telescope scans faster than in the narrowband frame with equivalent count rate, therefore providing less local contrast against the sky background for faint reference stars (this is partially compensated by the larger number of points per star); (2) the exposures are more crowded, with a greater chance that otherwise fine reference stars will be marred by an overlapping trace from a different star; and (3) the motion of the telescope is more complex, and it is more difficult to identify fixed points such as the start and end of each scan. For example, for SS CMa the scan length in the middle leg exceeds the size of the WFC3 instantaneous field of view, so that neither its start nor its end is visible. Thus, it is more challenging to determine the position of the reference points along each leg, other than the overall start and end. These difficulties notwithstanding, we have been able to process serpentine scans with methods similar to our other scans, and the results are incorporated in our astrometric solution (see Section 2.6.1).

The analysis of the scan data for SS CMa largely follows the pattern we described in Paper I for SY Aur. The key steps are (1) identifying the pixels associated with each trail (star); (2) defining a minirow-by-minirow detector X position at each location along the trail by a one-dimensional fit of the observed signal along each minirow with a spatially variable line-spread function (LSF); (3) converting the position into rectified coordinates using the distortion map; (4) removing the effects of jitter and variable rotation; (5) determining the relative rectified X position for each star in each image; (6) combining the measurements from multiple scans within each epoch, including both deep and shallow frames; and, finally, (7) estimating the parallax of each target on the basis of the combined astrometric and spectrophotometric information. Key differences in the processing for SS CMa are (1) the availability of an improved geometric distortion solution via the static correction discussed in Section 2.3.1; (2) the use of serpentine scans, which are combined with deep and shallow frames for each epoch; and (3) the introduction of an empirical correction for the X-direction CTE loss (X-CTE). Thanks to the number and quality of reference stars, the nominal error of the parallax for SS CMa is significantly smaller than for SY Aur. We will now discuss in detail each step, highlighting the changes with respect to Paper I.

Our astrometric measurements are based on the spatial scan exposures listed in Table 5; Figure 4 shows typical examples of straight and serpentine scans.

The nearly vertical "trails" are the images that each star leaves as the telescope scans over the field. The length of the straight scans is ≈144'', 88% of the length of the field of view of WFC3/UVIS; thus, the part of the sky covered during the scan is almost twice the normal field of view of the camera. Stars near the center of the region spanned in the detector Y direction will have trails that start and end within the frame, while stars farther from the center along Y have trails that enter or leave the frame during the scan. For serpentine scans, the telescope motion is more complex; the vertical portions are scanned at 14 s⁻¹, about 35 pixels s⁻¹, a speed chosen to avoid saturation of SS CMa in filter F606W. However, at that scan speed, and given the desired exposure time of 350 s, the total length of the scan is ≈525'', over three times the field of view of the detector. Therefore, the scan is folded, consisting of three vertical legs separated by 4'' in the detector X direction. The turnarounds for the Cepheid occur just outside the detector field of view; other stars, at lower (higher) Y position, can have their first (second) turnaround within the field of view.

Note that all celestial sources in the field are extended because of the motion of the telescope. Cosmic rays are the only compact sources in the frame and are readily identified by their lack of spatial extent, allowing us to identify and disregard impacted pixels.

We used a master catalog of stars to first simulate and then match the observed trails to these stars. The fidelity of the simulations is a few pixels, and in many cases minor adjustments are needed to ensure that the full 15-pixel window around each trail pixel is fitted. We also use the simulations to identify the regions within each star's trail that are affected by nearby star trails, and we disregard the impacted pixels if the simulation indicates a bias of greater than 1.5 mpix for a given row.

The case of serpentine scans is more complex. For serpentine scans, each star's trail is marked by multiple reference points: the overall start and end, which are the start of the first leg and the end of the last leg, and whose location is determined by the shutter opening and closing; and the turnarounds between legs, whose location is determined by the motion of the telescope. For any given star, only a subset of these points is visible in the image. Furthermore, we have found that the relative location of these reference points cannot be predicted with sufficient precision from image to image, and therefore it has to be determined empirically from the data themselves. Locating individual trails in a serpentine image is thus a two-step process: first, locate the start, end, and turnaround points for a subset of well-exposed stars at various locations in the field of view; second, determine the geometry of the scan (with an accuracy of ∼3 pixels) and apply this geometry to predict the serpentine trails of all the stars visible in the scan using our star catalog. The latter include stars too faint to be profitably measured, but that are still bright enough to "spoil" the scans of brighter stars, as described above. The trail map must then be inspected and reference points tweaked to improve the match with the data; at the end of this labor-intensive process, we were generally able to identify and locate the trails of individual stars to within ∼1 pixel.

For the purpose of subsequent analysis, trails in serpentine scans are split into individual passes, each including the trails of all relevant stars in that pass. This allows us to associate together photons collected at the same time and consistently solve for time-dependent effects, such as the variable field rotation discussed below. It also allows for a more careful identification of spoilers, since different legs can have different spoiler impact. In the case of SS CMa, each three-leg serpentine scan results in three separate position measurements for all the stars, thereby associating together measurements taken at the same time. Note that the serpentine scan was not obtained during the fifth epoch; thus, Epochs 1 through 4 each have seven sets of measured positions (five for the Cepheid), while Epoch 5 has four sets of positions (two for the Cepheid).

As in Paper I, we independently fit each 15-pixel minirow along the trail to determine the X position of the star at that value of Y. The fit uses an empirical LSF appropriate to the filter and detector position, obtained by integrating the empirical point-spread functions (PSFs) from Bellini et al. (2011) and, like the latter, oversampled by a factor of 4. Data quality flags from the detector characterization, as well as flags from source-contaminated pixels, are used to avoid fitting bad pixels. The end result is an array of detector X positions and uncertainties as a function of the Y-axis position (equivalent to time) along the scan.

Also following the same procedure as in Paper I, we start with the geometric distortion solution for F606W from Bellini et al. (2011), which uses a definition of the PSF position that is consistent with the empirical determination of the PSF itself. This geometric distortion map is used to transform the detector X and Y positions to sky coordinates. We obtain a similar solution for F621M from calibration observations of ω Cen. In addition, in this paper we also correct for the residual geometric distortion obtained from calibration observations of M67 and M48 (Section 2.3.1). The original solution from Bellini et al. (2011) is expected to have an accuracy of ∼0.01 pixels on scales of ∼40 pixels; this accuracy is sufficient to reach position precision of 1 mpix (40 μas) for full-length scans, which would be a significant contribution to our overall error budget. By applying our new correction, we expect that the local residuals will be reduced to ∼3 mpix on a scale of 100 pixels, with a projected contribution to the final error budget of ∼20 μas. Again as in Paper I, we use the time-dependent velocity aberration values provided by the STScI pipeline, interpolated to account for its variation during the observation, to correct for the corresponding plate-scale changes along the scans. Later, we account for perturbations in the geometric distortion of the field caused by the day-night thermal cycle of HST when registering different scans.

As for the analysis of SY Aur, we define the one-dimensional position measurements for each scan line relative to the mean line of the sample. This mean line or reference line is determined by aligning all scan lines in time and taking their weighted average. The reference line thus contains the jitter history in the direction perpendicular to the scan, which is removed from all lines in their difference with the reference. Figure 9 shows the comparison of two bright star trails aligned in scan time and the residuals after subtraction of one from the other. We use the requirement that scan lines, relative to the reference, are parallel on the sky to measure the time dependence of the scan roll angle. The variable rotation history of the scans can be measured well for the two deep scans obtained at each epoch in F606W; Figure 10 shows that the rotation angle is very similar in both scans obtained at the same epoch. The variable rotation cannot be measured as accurately in the shallow scans, because of the lower signal-to-noise ratio for all stars except for the Cepheid, which by itself cannot constrain the rotation angle. As the variable rotation term appears to be constant within each epoch, we determine the correction by averaging the measured rotation in the two deep scans at each epoch, and we apply the resulting correction to all deep and shallow scans for that epoch. Figure 10 also shows that the variable rotation is markedly smaller in Epochs 3 through 5, most likely because of the improved FGS geometric solution adopted on 2013 July 22, between our Epochs 2 and 3 (E. Nelan & M. Lallo 2016, private communication).

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The fundamental measurements at each epoch of observation consist of the relative X position of all stars, obtained from the combination of all coeval scans, deep, shallow, and (if available) serpentine. In order to combine these scans, they must be astrometrically registered, which requires correcting for any differential geometric distortion. As in Paper I, we include in the distortion model a low-order polynomial correction with free coefficients for each observation after the first. We adopt a polynomial correction to the X coordinate when aligning two frames that depends on the pixel position of each star trail in the detector; the assumption is that any variation in the transformation from true to measured position is tied to the telescope and detector and therefore is best described in measured rather than true coordinates. Note that a generic first-degree polynomial includes by definition an X-scale term (the first-order correction in X), as well as a detector rotation, which is slightly different from but closely related to the field rotation previously considered. As for M35 and SY Aur, we find that a second-degree polynomial as a function of X and Y coordinates is adequate to describe the X-coordinate transformation between two scans. (Only terms of total degree up to the polynomial degree are included, so our second-degree polynomial contains terms in X × Y, but not in X² × Y or X × Y².) A second-degree polynomial in X and Y has five coefficients (plus a constant term), two of which describe an offset and a rotation, respectively.

For SY Aur we found that a dearth of intermediate-brightness stars, combined with the need to allow for second-order polynomial corrections between the frames, led to significant uncertainties in the transformation of the Cepheid and of the reference stars to a common frame, resulting in a dominant contribution to the final uncertainty.

In order to ameliorate this problem, we modified our strategy in two ways. First, the number of available intermediate-brightness reference stars was a primary consideration in the selection of our targets. Second, we have obtained and processed serpentine scans in F606W; these scans provide additional constraints between the Cepheid, which is not saturated, and a larger number of intermediate-brightness reference stars. (Similar data were collected in some of the SY Aur epochs, but we were unable to process them properly for Paper I.) Position measurements for intermediate-brightness and faint stars in serpentine scans are not quite as accurate as those in the straight F606W scans, for a number of reasons: the same total counts are spread over a larger area, thus resulting in additional background noise; the higher density of trails results in more spoilers; and the more complex telescope motion results in additional parameters to be fitted. Nonetheless, we find that serpentine scans, while more complex to analyze, add significantly to the precision and reliability of the final measurement, and we expect that the analysis of future targets will use the serpentine scans as well.

To account for a shift in detector X position due to imperfect X-CTE, we have obtained calibration observations for WFC3/UVIS in spatial scan mode, in which a bright star has been moved across the amplifier boundary at X = 2048 in consecutive exposures. To the extent that the X-CTE effect is linear in the distance to the relevant amplifier, the mean relative position of stars in the field does not change as the field of view is dithered in the X direction—if stars are moved to the right (+X direction), those to the left of the boundary (X < 2048) will experience an increase in their apparent displacement to the right, and those to the right of the boundary (X > 2048) will experience a decrease of their apparent displacement to the left, for a null net effect. (Individual stars will move slightly with respect to one another, as fainter stars will be affected more than brighter stars.) However, if a star is moved from left to right of the boundary, its apparent displacement due to X-CTE will reverse sign, and thus it will experience a large net motion. An analysis of calibration observations obtained in Fall 2014 shows that the net effect is 8 mpix for a star near saturation, thus implying that a bright star near the amplifier boundary was shifted by 4 mpix at that time owing to X-CTE.

Experience with WFPC2, ACS, and (more limited) WFC3/UVIS strongly suggests that CTE effects grow linearly with time in orbit and with distance to the relevant amplifier. For WFC3/UVIS, we assume that the CTE loss was zero at launch. If this is correct, and if the effect is furthermore antisymmetric with respect to the amplifier boundary, the impact of X-CTE on parallax determinations vanishes as long as the observations are obtained with the same center and rotated field of view. The reason is that a growing X-CTE will result in an apparent motion for each star that is indistinguishable from a proper motion; thus, each star will have a spurious term in its estimated proper motion, but the parallax estimate is unaffected. Even if the field of view is shifted, the impact on each star is minimal, as discussed above—as long as the star does not switch amplifiers.

However, second-epoch observations placed the Cepheid at the same X detector location (X ≈ 2000, left of the amplifier gap) as the odd epochs, thus shifting the field center by about 100 pixels between the two orientations. The aim was to minimize the impact of uncertainties in the geometric distortion for the Cepheid by placing it in approximately the same detector location. The magnitude of the X-CTE effect was not fully understood at that time. On the basis of current information, minimizing the impact of X-CTE effects is deemed more important, and starting from Epoch 3, observations were obtained with the same field center for all epochs.

In order to correct for the residual X-CTE effect—which generally only affects the Cepheid target, as other stars within 50 pixels of the amplifier boundary are swamped by the light of the target—we simply correct the position measured for the Cepheid in Epoch 2 by twice the estimated offset at that time, about 2.9 mpix. This correction mimics the effect of placing the Cepheid in the symmetrical position (X ≈ 2100) and thus nullifies the effect of X-CTE on the measured parallax. As discussed above, relative proper motions will be affected by X-CTE and thus can only be determined accurately if a good overall calibration for X-CTE is obtained.

2.6.1. Serpentine Scans

For each epoch of observation, our goal is to obtain a measurement of the relative positions of all stars, the Cepheid as well as all the reference stars, along the resolution direction (the distortion-corrected detector X axis projected onto the sky). This measurement must be as accurate and free of systematics as possible.

In order to obtain a measurement of the relative parallax and proper motion (in the resolution direction) of the stars on the field, data from all available epochs must be combined. In addition, information on the distance of the reference stars is required in order to obtain an absolute parallax for the target. In principle, the process requires only a linear combination of the positions as measured at each epoch to determine relative parallaxes and proper motions; if the mean parallax of the reference stars can be estimated, determining the parallax of the target is then straightforward.

In practice, solving for the parallax of the target is much more complex. Small rotations (at the level of a few hundredths of a degree) between epochs, as well as small changes in plate scale and low-order geometric distortion that are known to occur, can substantially affect the projected positions of each star in each epoch and thus impact significantly its estimates of parallax and proper motion. Occasionally, reference stars can have anomalous data, either because of measurement problems (e.g., undetected faint stars close enough to affect the fit) or for astrophysical reasons (e.g., binary companions or other sources of photocenter motion).

The approach we have adopted solves simultaneously for the astrometric parameters of all the stars in the field and for the geometric registration of all the epochs, using the spectrophotometric parallax estimates as priors for their astrometric parallax. With this approach, the spectrophotometric parallax estimates help constrain the registration between epochs, including the relative low-order geometric distortions. It is thus critical that the parallax estimates be as accurate and robust as possible.

In the next section, we present the combination of spectroscopic and photometric data and the analysis that leads to spectrophotometric distance estimates for as many of the reference stars as possible.

In Section 4 we return to the determination of the Cepheid parallax using the spectrophotometric distance estimates obtained in Section 3.

Narrow-angle astrometry, such as what we can obtain with HST, is fundamentally differential in nature, and therefore it can only constrain the difference between the parallax of stars within the field of interest. In order to convert this relative parallax estimate into an absolute measurement, the parallax of other stars in the field must be estimated, and careful consideration must be given to possible systematics and random uncertainties in these estimates. Following the approach used in previous studies (e.g., Harrison et al. 1999; Benedict et al. 2007), we estimate the individual distances of several reference stars with respect to which the relative parallax of our target is measured in order to convert its value to an absolute parallax. High-quality distance estimates, based on both spectroscopy and multiband photometry, and careful consideration of the uncertainties involved are paramount in obtaining a reliable, accurate conversion to absolute parallax. For this purpose, we have obtained medium-resolution, classification-quality spectroscopy and multiband photometry ranging from UV to mid-infrared, including a number of medium-band Strömgren-like filters, for all the reference stars in the field of SS CMa. A combination of stellar model fitting and spectrophotometric classification, together with an understanding of the distribution of stars along the line of sight, has been used to estimate the distance to each reference star and its likely uncertainty. We also used prior estimates of the reddening along the line of sight, based on measurements of stars in 2MASS and Pan-STARRS, in order to constrain the range of possible reddening; however, the final reddening–distance law was also fitted for in our analysis.

With this information, the relative parallaxes of the reference stars and of the Cepheid can be placed into an absolute frame. In addition, estimates of the parallaxes of other stars in the field can confirm the quality of the astrometric measurements, identify outliers, and help constrain some of the low-order geometric distortion variations discussed earlier.

3.1.1. Photometry for Reference Stars

3.1.2. Spectroscopy of Reference Stars

3.1.3. Estimating Spectrophotometric Parallaxes

Spectroscopic parallaxes of stars in the field were determined, as in Paper I, by matching up to 14 bands of photometry to stellar isochrones, comparing medium-resolution spectroscopy to stellar spectra for classification standards, and using the Besançon Galaxy Model (Robin & Crézé 1986; Robin et al. 2003, and references therein) as a likelihood prior for stellar parameters. We used a version of the model with an updated thick disk that better fits Sloan Digital Sky Survey (SDSS) and 2MASS data (A. Robin 2013, private communication).

Our procedure for measuring the spectroscopic parallaxes of the astrometric reference stars in the field has been somewhat refined and improved since the procedure used for the field of SY Aurigae described in Paper I. We have now added photometry of the stars from two bands of spatial scanning, a broad (F606W) and another Strömgren (F621M) filter, one broad band from HST in the near-infrared (F160W), and two bands of medium-infrared data from the all-sky WISE mission. We add an uncertainty of 0.05 mag in quadrature to all photometric uncertainties to account for possible differences in photometric systems between models and observations. The stellar classification of star temperature and luminosity class is now done using the MKCLASS version 1.7 automated Morgan–Keenan classifier (Gray & Corbally 2014). Finally, we have improved our prior knowledge of the extinction along the line of sight as a function of distance using the 2MASS determinations from Marshall et al. (2005), who provided extinction versus distance estimates for the line of sight in the direction of SS CMA (private communication) with an uncertainty of 0.3 mag at a given distance. An example of the quality of the results is shown in Figure 11 for Star 18; the observed photometry (diamonds) is matched to a reddened model (dashed line), with the residuals shown on a larger scale in the bottom panel. The inset shows the observed spectrum for Star 18 (black) overlaid with the best-fitting model according to MKCLASS (red).

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Because it can be difficult to estimate extinction along a line of sight at very low Galactic latitude, and because our up to 14-band stellar photometry spanning 0.275–4.5 μm can aid the determination, we started with a weaker prior having a Gaussian width of 0.5 mag, and then, on the basis of the a posteriori extinction estimates, we applied a global correction to the 2MASS estimates before reverting to the 0.3 mag uncertainty for a final estimate. The maximum extinction we allowed in the fits was 1.2 times the total extinction to infinity along this line of sight estimated by Schlafly & Finkbeiner (2011), which in turn was based on a rescaling of the IRAS-based estimate of Finkbeiner et al. (1998). The scatter of the a posteriori extinction estimates for each star around the extinction prior at its (spectrophotometrically) estimated distance was 0.21 mag, with extinction values ranging from A_V ≈ 0 at distance modulus μ = 9 mag to A_V = 3 mag at μ = 13 mag. Figure 12 shows the resulting final law for A_V versus μ (red points and red line), together with the individual values for each of the reference stars in the field (blue squares). The relation thus obtained between extinction and distance, for stars both closer by and farther away than the Cepheid, will also serve to constrain the reddening estimated for the Cepheid. This will in turn provide information on the intrinsic colors of the Cepheids and improve the robustness of the global P–L calibration we expect to obtain.

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The final step in the astrometric solution consists of combining the multiple measurement epochs taken over the course of 2 yr at intervals of 6 months to fit to our standard astrometric model, which involves three parameters for each star: position, parallax, and proper motion along the measurement direction. The results we present here for SS CMa are based on five epochs of observation; four more epochs are being obtained as part of a recently approved program extension (Program GO-14206). The fifth epoch does not include the serpentine scan. The exposures used are listed in Table 5.

Together with the astrometric parameters of each star, the model includes up to second-order geometric parameters used to align each epoch with one another (offset and rotation), as well as any residual large-scale adjustment to the geometric distortion required to reduce the model residuals. This last part is identical to the single-epoch aggregation step, but it now substitutes the stationary-star assumption with the astrometric model for each star.

The full model can be formally described by the expression

$$\begin{eqnarray}{X}_{{ij}} & = & {X}_{i0}-{X}_{{\rm{ref}},j}+{{PMx}}_{i}\;({t}_{j}-{t}_{0})+{\pi }_{i}\;{f}_{j}\\ & & +{R}_{j}\;{Y}_{i0}+\langle {P}_{j}({X}_{{\rm{det}}},{Y}_{{\rm{det}}}){\rangle }_{{{\rm{trail}}}_{i}},\end{eqnarray} \tag{ 1 }$$

where the basic measurements are the positions X_ij—that is, the X position of the trail of star i in image j (relative to the reference scan line)—measured after correction for variable rotation, scale-corrected for velocity aberration and variable distortion, and projected onto a constant sky frame. The X coordinate is aligned with detector X and, by design, aligned with the bulk of the parallactic motion. The quantity X_i0 is the reference position of star i at time t₀, and X_{ref, j} is the offset of image j in the X direction—in essence, the position of the reference scan line for image j on the sky. The astrometric motion of star i in the X direction is described by the X component of the proper motion, PMx_i, and the parallax π_i, applied with the epoch-dependent parallax factor f_j. The term f_j is the projection (for unit parallax) of the parallactic motion in the X direction at the time of the observations, computed using the formulae on pp. B28 and C5 of The Astronomical Almanac (2013) and the orientation of the detector axes from the image headers.

As we did for SY Aur, the model position must be corrected for the relative rotation and geometric distortion of image j with respect to the reference image. The rotation term on the sky is R_j Y_i0, where R_j is the rotation of image j and Y_i0 is the static relative position of star i in rectified coordinates along the Y direction with respect to the center of the field. We find our rotations to be of order 10⁻⁵, so even a coarse measurement of Y_i0 with a precision of ∼1 pixel will suffice. The polynomial term is determined simultaneously with the astrometric parameters during the model-fitting procedure, as a second-degree P_j (X_det, Y_det), where X_det and Y_det are detector coordinates; the total correction is determined by evaluating the polynomial for image j at every location along the trail of star i in that image and averaging the result. The constant term is omitted from the polynomial because it is degenerate with the image offset X_ref,j.

Our proper-motion term is relative to the set of stars in the field and contains a contribution from the estimated X-CTE term per year at that star's location.

Note also that the model is formulated to be linear in the astrometric parameters, which in turn are linearly related to most measured quantities, i.e., positions on the detector. As a consequence, the errors in the derived astrometric parameters for the Cepheid are likely to be very nearly Gaussian (the same does not necessarily apply to distant stars for which the spectrophotometric constraints dominate the error distribution). As long as the parallax and its error distribution are used directly, there is no need to apply nonlinear corrections such as those suggested by Lutz & Kelker (1973). More generally, proper consideration of all prior information used in selecting and characterizing the population of our target Cepheids will be required in determining the optimal calibration for the P–L relation on the basis of our measurements (Hanson 1979; Francis 2013; see also the discussion in Benedict et al. 2007). For SS CMa, the distance modulus bias is most likely smaller than for most Cepheids in the Benedict et al. (2007) sample, because of the different geometry; we currently estimate that the correction is −0.03 ± 0.02 mag, based on its Galactic latitude and distance. Therefore, we do not include a Lutz–Kelker-type correction for the parallax of SS CMa at this point; full consideration of the distance probability distribution of for each Cepheid, taking into account both selection and observational biases, will be included in the analysis of the full sample of Cepheids.

As far as the astrometric model is concerned, parallaxes are also relative; however, the degeneracy in the conversion to absolute parallaxes can be broken by using the spectrophotometric distance estimates for the stars in the field discussed in Section 3. The distance estimate of the target star will be insensitive to uncertainties in the distance of the reference stars so long as the set contains objects that are bright and distant (e.g., red giants). Note also that in addition to providing a conversion to absolute parallax, individual spectrophotometric parallaxes are also helpful in constraining some epoch-to-epoch geometric transformations. We will discuss in detail in Section 4.2 the impact of the parallax constraints for reference stars on the multi-epoch solution and investigate the consequences of uncertainties, outliers, and other possible issues.

Each epoch after the first is allowed a net offset and a second-degree polynomial adjustment to match the first epoch; since there are about 20 stars useful for measurement at each epoch, these additional six parameters per epoch over which we marginalize do not place an undue burden on the solution.

Formally, the a priori distance estimates based on spectrophotometric parallaxes serve as Bayesian priors for the parallax of the stars in the field. A prior is not used for the Cepheid, so that its distance estimate is determined directly and only from its observed parallax.

The best values of the model parameters are determined by minimizing the total model χ² to achieve the most likely parameters. Among these parameters is the absolute parallax of SS CMa, which results directly from the model optimization. A modest fraction of the reference stars in the field are expected to be part of binaries with parameters that would cause a significant deviation from our simple astrometric model. This fraction depends on distance and spectral class, but is ∼10%–20% for F and G stars at 1 kpc on the basis of the distribution of binary properties in Duquennoy & Mayor (1991; see also discussion in Paper I). We run the global model iteratively after rejecting one outlier (Star n2) on the basis of its disproportionate contribution to the total χ². We also exclude Star n10 because its astrometric parallax is suspect, resulting in a very large estimated distance, which is inconsistent with its spectrophotometric information.

4.1.1. Multi-parametric Model for SS CMa: Primary Solution

Figure 13 shows the best estimate of the parallax for the stars in the SS CMa field. For reference stars, the reported parallax combines both astrometric and spectrophotometric information; no spectrophotometric information is used for the Cepheid SS CMa (the star labeled 0).

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For each star, the top panel shows the astrometric measurements at each epoch (dots with error bars) and the best-fitting parallax model (red line), both in milliarcseconds; the fitted proper motion is subtracted from both the measurements and the models for ease of display. The gray band shows the spectrophotometric parallax estimate with a 2σ uncertainty, when available. The bottom panel shows the astrometric residuals from the best model, also in milliarcseconds.

The best estimate of the parallax of SS CMa is 0.348 ± 0.038 mas, corresponding to a distance estimate of 2.87 ± 0.33 kpc. However, note that, as commented in Section 4.1, the error distribution is likely Gaussian only in parallax. A nonlinear conversion, e.g., to distance, will have a nonsymmetric error distribution, which must be taken into account in further processing. It is also necessary to consider any prior information used in selecting and characterizing the sample (Hanson 1979; Francis 2013; see also the analysis in Benedict et al. 2007).

The uncertainty in the conversion to absolute parallax (i.e., the systematic uncertainty in the frame parallax) of the set of 20 fitted reference stars is 7 μas, well below our target uncertainty. In the field of SS CMa, the precision of the conversion to absolute parallax benefits from the presence of some very distant stars, and especially Star 43, whose spectrophotometry indicates that it is a K giant at around 30 kpc (Section 4.3). However, even excluding Star 43, the rest of the reference stars indicate an uncertainty in the frame parallax of about 11 μas, still much smaller than our target uncertainty. This is not surprising; Figure 14 shows the typical precision of the correction to absolute parallax in random fields generated from the Besançon model for the direction of SS CMa, assuming a 15% typical uncertainty in spectrophotometric distance estimates and that only half of the stars in the field will be available for the conversion. A typical field would have ∼20 available reference stars and an uncertainty of 9 μas in the conversion to absolute parallax for an assumed rms uncertainty of 0.3 mag in the estimated distance moduli, comparable to the values for the actual data. (The uncertainty in the conversion to absolute parallax will scale roughly linearly with the assumed uncertainty in the distance moduli.) However, Star 43 is unusual in its own right, and it will be further discussed in Section 4.2.

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In the primary solution, we assumed that the spectrophotometric and astrometric distance estimates for each star are separate but valid measurements of the same quantity, the physical distance of each star. In practice, this assumption may not always be true, e.g., because of the possible binarity discussed earlier. On the other hand, the assumptions underlying our spectrophotometric distance estimates may not be valid for all stars, and there could be outliers—e.g., due to anomalous extinction, or a history of mass exchange—which could occasionally lead to a faulty estimate of the distance modulus.

A careful comparison of astrometric and spectrophotometric parallaxes for the reference stars can provide a powerful check of our procedures and our final accuracy. Spectrophotometric parallaxes, derived from a combination of spectra and multiband photometry in conjunction with stellar model tracks, a model of the density distribution of stars along the line of sight, and an extinction model, typically have fractional accuracy that varies little as a function of distance; thus, their absolute error is much smaller for distant stars than for nearby ones. On the other hand, astrometric parallaxes have absolute errors that are similar in magnitude in terms of parallax angle, and hence nearly independent of distance, although they do scale with apparent brightness; thus, for nearby stars, astrometric parallaxes are more accurate than spectrophotometric ones, and vice versa for distant stars. For typical stars in our analysis, the two accuracies are comparable at about 1 kpc. Consequently, stars beyond 1 kpc provide a solid check of astrometric measurements, while closer stars provide a verification of the spectrophotometric estimates.

However, the parallax estimated for each star in the full (primary) solution shown in Figure 13 is affected by its spectrophotometric prior and hence cannot be used directly for an independent check of the astrometric parallax thus obtained. On the other hand, dropping all of the spectrophotometric priors is not a viable option. First, all narrow-field parallax measurements are, by necessity, only relative; thus, without some spectrophotometric measurements, no absolute parallax can be derived. Second, as discussed in Section 4.1, without a spectrophotometric prior for the majority of the stars, we lack the ability to properly constrain the relative alignment and polynomial distortion for each epoch of observation, thus worsening the quality of the measurements and introducing substantial degeneracies in the solution process.

In order to carry out a meaningful test of the quality of our astrometric parallaxes, we repeat the multi-epoch fit by discarding the spectrophotometric prior for each star in turn, and we define the parallax obtained for that star as its pure astrometric parallax. For example, when measuring the pure astrometric parallax for Star 29, we discard the spectrophotometric prior only for Star 29, but retain the prior for all the other stars in the fields that have one. This allows the solution to converge with only a minor decrease in overall precision, resulting in a trigonometric parallax estimate to Star 29 that is completely independent of any photometric or spectroscopic information for that star. The pure astrometric parallax is listed as π_astro in Table 4, where the parallax resulting from the full solution is labeled π_full. We then repeat this process for all other stars for which a spectrophotometric prior is available; for stars without a spectrophotometric prior, including the Cepheid, the pure astrometric parallax is of course identical to the parallax from the full solution. This procedure not only allows us to assess the quality of our astrometric measurements with the spectrophotometric distance estimates but also mimics the handling of the Cepheid itself, for which no spectrophotometric prior is ever used, and thus provides a useful test of the validity of its parallax measurement.

Figure 15 shows a comparison of the pure astrometric and spectrophotometric parallaxes for the reference stars in the field of SS CMa. The Cepheid is not included, as no spectrophotometric prior is used for it. For the majority of the stars, there is a very reasonable agreement between them, with 17 out of 20 within nominal 2σ; only Stars 3, 38, and n4 are outside this range, and Star 10 has a very small value for the pure astrometric parallax, with a very large uncertainty.

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Excluded from Figure 15 is Star n10, for which the pure astrometric parallax is negative. In our solution, "negative" parallaxes are not necessarily disallowed; they can occur, for example, if the correction to absolute parallax is underestimated, and all stars are in reality closer than the astrometric parallax indicates. In that case, the true parallax would simply be larger than the resulting value, because of the larger correction to absolute parallax. However, in this case the solid agreement between astrometric and spectrophotometric parallaxes for most stars argues strongly against a large systematic error in the reduction to absolute parallax. Although we could limit the solution to require positive parallaxes, this step would be somewhat arbitrary, given that the actual value of the reference parallax is part of the optimization process, and could give excessive weight to stars with very low parallaxes in the solution. Therefore, we treat Star n10 as an astrometric outlier and exclude it from our solution. Based on the discussion in Paper I, we expect ∼10% of the reference stars to be astrometric outliers due to binarity, so the presence of an outlier in this sample is not surprising.

It is important to note that the accuracy of the partially constrained solutions obtained by dropping the spectrophotometric prior for one of the reference stars can be potentially compromised, especially if that star is near a corner of the field. The reason is that the low-order polynomial distortion that we adopt to register the measurements across epochs may lack a critical constraint near that corner, while on the other hand its value is required at that location. Consequently, a quasi-degeneracy in the multi-epoch astrometric solution exists for that star. This is especially apparent for stars such as Star 3, which is the only bright star near the bottom left of the field (see Figure 3). The full solution (Figure 13) shows for Star 3 a normal parallax value and uncertainty of 0.418 ± 0.044 mas, and its residuals are fairly typical. On the other hand, the partially constrained solution in which its parallax prior is dropped is 0.167 ± 0.111 mas, with a very different value and a much larger uncertainty than the fully constrained solution. Inspection of the two solutions shows that they differ by about 40% in the Y² polynomial term, resulting in a differential offset of up to 10 mpix in Epoch 5. The case of Star 10 is even more extreme, with an increase of the nominal error from 0.096 mas to 0.260 mas for the pure astrometric parallax. For most other stars, the uncertainty increases by 10%–70% when the spectroscopic prior is dropped. (Another exception is Star 43, which has a comparatively poor fractional accuracy in the trigonometric parallax because of its very large distance.) Figure 16 shows the difference between pure astrometric and spectrophotometric parallax as a function of distance from the center of the field: two of the three stars with difference larger than 0.3 mas are more than 2000 pixels from the field center.

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In summary, we conclude that there is in general good consistency between the astrometric parallaxes we measure from this set of HST spatial scans and the distance estimates obtained from our spectroscopic and multiband photometric measurements. Moreover, as expected, the availability of such estimates for a large fraction of the reference stars is necessary to constrain the overall astrometric solution.

Among the reference stars in the field, Star 29 was suggested by Evans & Udalski (1994) as a possible binary companion to SS CMa, on the basis of its blue color, estimated distance, and likelihood of the relatively small separation of 13''. However, we classify Star 29 as G6 V on the basis of its spectrum (see Table 2), and its estimated spectrophotometric distance modulus is 10.2 ± 0.3 mag, which places it significantly closer than the Cepheid. As shown in Figure 16, the astrometric parallax for Star 29 is in agreement with its spectrophotometric distance, and we conclude that Star 29 is not physically associated with the Cepheid SS CMa, consistent with the upper limit on companion separation found by Evans et al. (2016).

Another interesting reference star is Star 43. Although faint in the visible (V₆₀₆ = 15.26 mag), this star is quite bright in the near-infrared (H = 10.76 mag), and it is classified from the spectrum as a K5 III, with spectrophotometric distance modulus 17.4 ± 0.5 mag. This places the star about 30 kpc from the Sun, well outside the disk and the spheroid of the Galaxy. (Note that this star by itself carries about half the weight of the conversion to absolute parallax for this field.) The trigonometric parallax without spectroscopic prior is 0.018 ± 0.065 mas, hence not significantly detected, but certainly indicative of the star being beyond 10 kpc. Such stars are likely rare and are often found through variability studies or special spectral features (see, e.g., the carbon-star selection in Huxor & Grebel 2015). The Besançon model (Robin & Crézé 1986; Robin et al. 2003, and references therein) indicates that only 1 in 50 fields at the Galactic coordinates of SS CMa would have a star of comparable brightness beyond 10 kpc, and about 1 in 150 beyond 30 kpc.

We observed SS CMa between 2013 April and 2015 November using three different echelle spectrographs: (1) the Hamilton spectrograph (Vogt 1987) at the Shane 3 m telescope located at Lick Observatory; (2) the Hermes spectrograph (Raskin et al. 2011) at the Flemish 1.2 m Mercator telescope located at the Roque de los Muchachos Observatory on the island of La Palma, Spain; and (3) the Coralie spectrograph (Queloz et al. 2001) at the Swiss 1.2 m Euler telescope located at La Silla Observatory, Chile. Data from the Hermes and Coralie spectrographs were reduced using dedicated pipelines. Hamilton spectra were reduced using standard IRAF routines. RVs were determined by cross-correlation using a numerical mask representative of a solar spectral type (Baranne et al. 1996; Pepe et al. 2002).

RVs from Hermes and Coralie were found to be compatible with each other to within 10–20 m s⁻¹, and no zero-point offset was applied. RVs from the Hamilton spectrograph were brought to the Coralie/Hermes zero-point via observations of stable standard stars (HR 4027 and one or more of the following: HD 26161, HR 124, HR 7373, or NSV 7543) using the velocities and zero-point offsets presented by Nidever et al. (2002). Given the uncertainties involved with this zero-point correction, and factoring in the intrinsic precision of the Hamilton RVs, we estimate an uncertainty of approximately 200 m s⁻¹ for these data. The individual pipeline-estimated RV uncertainties for Coralie and Hermes range between 20 and 80 m s⁻¹. We opt to not include the literature RV data from Joy (1937) and Coulson & Caldwell (1985) in this analysis, since zero-point offsets (1 ± 0.5 km s⁻¹ for Coulson & Caldwell 1985) and low precision (typical uncertainties larger than 1 km s⁻¹) dilute the precision of our new measurements, while not adding significant information for the timescales of interest (1–2 yr).

To investigate a possible astrometric signal caused by binarity, we constrain possible values of the projected semimajor axis, ${a}_{1}\mathrm{sin}i$, for SS CMa by modeling the RV data. Our model consists of a sum of a 9-harmonic Fourier series, representing the intrinsic velocity variation during the pulsation, and a circular orbital motion. We adopt circular orbital motion for simplicity, and since there is no evidence for an eccentric orbit in the available data. We adopt a constant pulsation period of P = 12.3535 days, which minimizes the scatter in the phased RV data set. We then convert the projected semimajor axis into an astrometric term by assuming a distance of 3 kpc. As usual with RV information, this represents a minimum orbital signature, corresponding to an edge-on orbit.

The time sequence of RV measurements indicates a nearly constant systemic RV, excluding even fairly low amplitude deviations (above ∼400 m s⁻¹), consistent with a null detection of orbital motion to within the uncertainty of our measurements. We estimate upper limits on astrometric signals caused by undetected companions, noting that small variations in the pulsation RV pattern already found in several Cepheids (Anderson 2014b) can mimic the effect of low-mass spectroscopic companions (Anderson et al. 2016). Hence, the mere detection of time-variable low-amplitude changes in systemic velocity is not necessarily a clear indication of spectroscopic binarity, as the available sampling of the RV data is insufficient to fully separate these two possibilities. If interpreted as orbital motion, the variation seen in the RV data would result in the astrometric signature shown in Figure 17; darker colors correspond to lower χ². The maximum impact is around a period of 1 yr with an orbital amplitude of ∼4 μas, while the lowest χ² occurs for an amplitude of 1.5 μas. In order to account for the uncertain contribution of the possible orbital motion, we conservatively add a term of 4 μas in quadrature to the nominal parallax error for SS CMa. Monitoring of this and other Cepheids in our program will continue, and the results will be presented separately in greater detail (R. I. Anderson et al. 2016, in preparation).

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