PARALLAX BEYOND A KILOPARSEC FROM SPATIALLY SCANNING THE WIDE FIELD CAMERA 3 ON THE HUBBLE SPACE TELESCOPE^*

Owing to its superior angular resolution and relative stability, HST is a promising platform for obtaining high-accuracy relative astrometry for sources within its field of view. The theoretical limiting precision for the measurement of a point source is approximately its root-mean square width divided by the signal-to-noise ratio (S/N) of the observation. The S/N in a normal exposure is limited by the number of photons that can be collected before saturation, typically ∼10⁵ for most imagers, thus suggesting a theoretical limit of about 0.1 mas per pointed observation. In practice, HST has so far been limited to a best-case single-measurement precision of ∼0.3–0.4 mas (∼0.01 WFC3-UVIS pixel; Bellini et al. 2011), both with the FGS and with its imagers.

There are several reasons for these limitations. With the most efficient imagers, light is discretely sampled in pixels with angular size ∼40 and 50 mas (WFC3 UVIS and ACS WFC, respectively). Measuring the position of a source to better than ∼1% of the pixel size has proved very difficult (Bellini et al. 2011); among the contributors to a noise floor may be zonal and temporal variations in the effective point-spread function (PSF), and small-amplitude irregularities in the geometric distortion which cannot reliably be calibrated with existing data. Any variations in the jitter over the image—produced, for example, by instantaneous rotations of the telescope—will introduce an additional field variation of the effective PSF. With the FGS, measurements are not limited by pixel size, but they are carried out one star at a time, so stability of the telescope and of the focal plane is paramount; in addition, the limited rate at which photons can be collected with the phototube detectors also limits the achievable S/N independent of the source brightness.

Many of the limitations of pointed observations can be overcome via a new observing mode with WFC3, spatial scanning under FGS control, developed for WFC3 in 2011 to obtain photometry during exoplanet transits. In this mode, the target field is observed while the telescope is slewing in a user-defined direction and rate (to a maximum of 78 s⁻¹). Up to 5'' s⁻¹, the telescope can maintain FGS guiding at all times; for faster scan rates, the telescope must be controlled by gyroscopes, leading to a smoother but less accurate motion. Each source thus describes a "trail" on the detector (see Figure 1). In the simplest mode, the motion is straight and uniform, resulting in a straight trail with constant brightness (counts per unit length) after accounting for geometric distortion in the detector. The trails for all sources are parallel in the distortion-corrected frame. More complex "serpentine" scans are also possible and can be preferable for sparse fields or exceptionally bright targets (see Section 4).

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Some of the advantages of this method are immediately obvious. The light from each source is spread over a much larger number of pixels, allowing a larger global S/N to be achieved for each source.⁴ Furthermore, each long trail provides thousands of separate position measurements in the cross-trail direction, one for each pixel traversed, thus averaging out the impact of single-pixel and local irregularities. Scanning at an angle relative to the detector also provides sub-pixel sampling of the undersampled WFC3 PSF. Because the measurements are time-resolved (e.g., 25 pixels per second at a scan rate of 1'' s⁻¹), the telescope jitter can be subtracted as a function of time, negating the impact of even large (∼1 pixel) jitter events which are not uncommon. With spatial scans, we expect to routinely achieve measurement precision of one-thousandth of a WFC3-UVIS pixel (1 millipixel, or mpix, corresponding to 40 μas) or better.

The disadvantage of this method is that precise measurements can be made in only one direction at a time, the direction perpendicular to the scanning motion, as positions in the direction along the motion are blurred by the motion itself. Thus, a precise two-dimensional measurement of relative positions requires in principle two observations. It is advantageous to choose scans to occur along the parallel readout direction (i.e., the Y-detector axis), as this limits the dominant direction of imperfect charge transfer and its attendant smearing to occur along the direction not being measured. The much smaller effect of the serial charge transfer efficiency is addressed in Section 2.5.2.

The disadvantage of obtaining positional changes in just one dimension is minimal for the measurement of parallaxes, as the motion of interest takes place in a predictable direction. By choosing the scan direction appropriately, the measurement can be made for the dominant parallax component.

The main characteristics of the planned observations relate to the brightness of the source, the availability of reference stars within the detector field of view, and the desired timing of the observation versus the allowed telescope roll angles. An unusually large degree of planning and simulating is needed to obtain useful observations in this mode.

As in all narrow-field astrometric observations, WFC3 spatial scan observations can only measure the relative parallax of the target—the difference between the parallax of the target and that of nearby reference stars. All stars in the field of view move along similar parallactic ellipses with the same shape, orientation, and phase because the apparent parallactic ellipse traced annually on the sky is simply the reflex of the motion of the Earth around the barycenter of the solar system. However, the amplitude of the motion of each star (e.g., its semimajor axis) scales inversely with its distance from the Sun, and represents the parallax. Since the absolute pointing of each observation is not known to better than a few tenths of an arcsecond, only the difference between parallaxes of stars in the field can be measured to useful precision.⁵

For this reason, it is critical for the field to contain a sufficient number of reference stars with independent distance estimates to correct relative parallaxes to absolute. Our typical observations in the plane of the Galaxy measure between 50 and 200 stars in a field of 5 × 27 defined by scanning WFC3-UVIS. It is important to note that distant stars, such as K giants several magnitudes fainter than the target Cepheid, are especially valuable in providing a correction to absolute parallax. For example, if a reference star is at 5 kpc (thus a parallax of 200 μas), even a relatively crude estimate of its absolute magnitude, perhaps with a 0.3 mag error, leads to a 30 μas uncertainty in its contribution to the correction to absolute parallax. With several such stars (a typical field will have 2–10 at this distance or beyond) and the use of a comprehensive set of UV, Strömgren, broad, and NIR bands to estimate stellar parameters, the uncertainty in this correction is expected to be well below our target measurement precision of 1 mpix (40 μas). Nonetheless, the correction to absolute parallax is a critical feature of our program, and the related uncertainties will be discussed in Section 2.6.

The optimal duty cycle for observing utilizes an exposure time of 350 s, just long enough to dump the buffer containing the previous observation during the next exposure, and a scan length of ∼144'', as long as possible while ensuring that both the start and end points of the scan are visible for the primary target within the 163'' field of view. This combination allows the largest number of scans per orbit, four or five depending on visibility, and near-continuous observations, and it yields a fiducial scan rate of ∼04 s⁻¹.

While any filter may be used for scanning, it is best to use a filter that has low wavefront errors and that passes light where CCD fringing is low. The use of longer wavelengths and redder filters would degrade resolution and increase fringing. Working too far to the blue can severely undersample the PSF (by a factor of two at 4400 Å) and diminish the S/N of red giants, which provide the best calibration to absolute parallax. The throughput of the filter is an important consideration as it determines the S/N of the target within the limited, useful dynamic range (6 × 10⁴ e⁻ pixel⁻¹ to 3 × 10² e⁻ pixel⁻¹, about 5.5 mag). It is also advantageous to choose a filter with a well-calibrated geometric distortion field (Bellini et al. 2011), as this provides the source of initial transformation from detector coordinates to the sky. The F606W filter (Δλ = 2300 Å, λ₀ = 5907 Å) is an attractive choice which makes useful measurements at the fiducial scan rate for stars at 10.6 < V < 16 mag. Unreddened stars at D = 2 kpc in this brightness range will have −1 < M_V < 4.5 mag. Extinction along lines of sight typical for long period Cepheids in the disk, 1.5 to 3 mag, will reduce the visible luminosity range accordingly. Red giants in this brightness range will be at 5 < D < 10 kpc in the absence of reddening (all magnitudes are quoted in the Vega system), with the lower end of the range providing a better estimate for sight lines through the disk.

Long-period Cepheids (P > 10 days, −4 > M_V > −7) within 5 kpc are more challenging parallax targets because they are very bright (6 < V < 10 mag) and thus would saturate in a broad-band exposure at the fiducial scan rate. In order to avoid saturation, either a faster scan or a narrower filter is needed, with the reduction in counts proportional to the inverse of the increased speed or the decreased filter width. Scans faster than 5'' s⁻¹ rely on less precise gyro guiding, and given the importance of maintaining a fixed scan direction, we prefer to avoid their use. In most cases, scans at a similar rate but employing a medium- or narrow-band filter—e.g., F621M (Δλ = 631 Å, saturation limit V = 9.2 mag) or F673N (Δλ = 100 Å, saturation limit V = 7.2 mag)—are sufficient to observe the Cepheids without saturation.⁶

However, in such observations there are typically too few (5–10) sufficiently bright stars that can be measured with enough precision to provide the desired target precision in the reduction to absolute parallax (based on the shot noise of the source, 20 μas precision requires ∼1000 counts per row and a trail length of a few thousand pixels). Thus, we rely on a hybrid approach: shallow, narrow-band scans to measure the Cepheid and a few reference stars, followed by deep, broad-band scans to measure 50–200 distant reference stars. The stars bright enough to be measured in the shallow scan and yet faint enough to avoid saturation in the deep scan serve as a bridge between the target Cepheid and the bulk of the reference stars; for this purpose, we do not require any knowledge of their distance or motion. Since the broad and shallow scans are taken concurrently, there is no significant astrophysical motion of the reference stars between scans, and the uncertainties associated with long-term motions (e.g., parallax) are negligible in this step. However, linking together shallow and deep scans can be a significant component in the final parallax error budget, as discussed in Section 4.

Figure 1 shows a typical scan image for the SY Aur field. (See Table 1 for a listing of scanning observations of SY Aur.) The nearly vertical "trails" are the images that each star leaves as the telescope scans over 144'', 88% of the length of the field of view. The area covered by 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 have trails that enter or leave the frame during the scan. Cosmic rays are the only compact sources in the frame and are readily identified by their lack of vertical extent, allowing us to disregard impacted pixels.

2.4.1. The M35 Calibration Program

2.4.2. Matching Trails to Stars

2.4.3. Minirow Fits

The next step in the analysis process is measuring the cross-scan position of each star as a function of position along the trail. For each trail, we extract the pixel values and quality flags at each integer pixel step along the trail within ±10 pixels (±4 × the full width at half-maximum intensity) of the nominal trail center. We call this individual block of information a "minirow," as it is a small part of the detector row where signal from the star of interest is located (see Figure 2). Individual trails may be composed of a few hundred to several thousand minirows.⁷ Each minirow thus consists of a short (typically 21 pixel) array of data values versus X pixel location at a fixed Y location. Each pixel is also assigned a weight based on the detector noise model including zero weight for bad pixels; pixels too close to a spoiler are given zero weight, and if pixels within ±2 pixels of the peak are rejected, the minirow is rejected as being invalid. The fit involves three parameters: the amplitude, or scaling factor for the line-spread function (LSF), which sums to unity over its full length; the center position, or amount by which the LSF must be offset and spline-interpolated along the detector X direction for an optimal match; and the background, or constant level that must be added to the line profile to match the data. The LSF, oversampled by a factor of four, is previously derived as a function of X and Y detector position from the empirical PSF in images of star clusters (Bellini et al. 2011). Positional uncertainties are determined from the χ² of the minirow fit with a minimum floor of 0.01 pixel imposed at 10⁴ e⁻ pixel⁻¹ to reflect the finite precision of the geometric distortion field obtained from Bellini et al. (2011).

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2.4.4. Position in Rectified Coordinates

2.4.5. Jitter and the Reference Scan Line

A feature that is immediately apparent from the values of the fitted X position versus position along the scan (see Figure 2) is that the values vary locally much more than the estimated error in each measurement, with variations of ∼0.1 pixels and occasional jumps of up to 1 pixel. Comparing different trails at equivalent positions along the scan (e.g., at the same distance from the start of the scan), as shown in Figure 3, reveals that these variations are highly correlated from star to star, and are in fact caused by irregularities in the telescope motion—equivalent to the so-called "jitter" in pointed observations. Unlike pointed observations, for which the jitter produces a modest blurring of the PSF, scanned observations can be used to measure and correct for the telescope jitter perpendicular to the direction of motion (our preferred measurement axis). In addition, the presence of jitter allows a more accurate measurement of the relative position of each star along the scan direction, from a template fit or cross-correlation of the jitter features. Each star will be affected by different jitter features, depending on where it is located in the field of view and thus for what fraction of the exposure it is on the detector; to avoid systematic offsets due to asymmetric jitter, it is thus necessary to correct for the jitter at the pixel-by-pixel level.

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For this purpose, we define a reference scan line from the weighted average of all stars that yield good fits. The reference scan line represents the instantaneous offset from the ideal rectilinear motion of the telescope as a function of position along the scan (which is a proxy for time). A proper definition of the reference scan line requires accurate (<0.25 pixel) knowledge of each star's position along the scan direction, to avoid blurring of jitter features. We carry out this process iteratively. We obtain a first approximation to the reference scan line from the along-scan position determined either from a star catalog or from the apparent start/end position of the trail; these are typically accurate to ∼1 pixel. Then we fit this approximate reference scan line to each trail, thus obtaining a better position along the scan, and produce a new reference scan line from the updated positions in rectified pixel space. The final reference scan line is oversampled by a factor of four; for the best-measured trails, the estimated uncertainty in the along-scan position is a few percent of a pixel, sufficient to resolve the jitter frequency. Figure 3 shows the comparison of bright star trails aligned in scan time and the residuals after subtraction of one from another.

2.4.6. Trail-to-trail Separation along the Scan: Variable Rotation

One fundamental assumption of the scan method is that trails for different stars are essentially parallel, so their separation remains constant (on the sky, after correcting for geometric distortion) throughout the observation. On short time scales (<1 s), the parallelism of trails is demonstrated by the correspondence of their jitter features. However, on longer time scales (≳ 1 minute, corresponding to several hundred pixels traversed), the separation of trails occasionally appears to vary by several mpix—difficult to measure individually, but quite apparent in a statistical sense. Initially we investigated the possibility of deviations from the nominal geometric distortion; however, this interpretation would require that the deviations be repeatable across observations of different fields, which turns out to be inconsistent with the properties of the calibration data obtained for M35. Instead, the nonparallelism of trails exhibits a pattern consistent with slow rotations of the telescope's field of view (equivalently, its roll angle) throughout the observation. Under this interpretation, a single parameter—the instantaneous differential roll angle—causes small changes in the separation of each pair of trails in a pattern strictly dependent on their relative separation in the scan direction. Figure 4 shows the scan patterns formed by fixed or variable field rotation. Indeed, solving for a slow variation of the telescope roll angle (typically fitted as a fifth-degree polynomial in position along the scan, a proxy for time) removes the variation in separation between trails below statistical significance. Empirically, the instantaneous roll angle changes by 0001 to 0003 over an observation.

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The most likely reason for a variation in the instantaneous roll angle is in imperfections of the geometric solution in the FGS. Under FGS control, the telescope tracks the position of the guide stars throughout the observation, and makes continuous roll adjustments on the basis of their position in the focal plane to ensure that the roll angle remains constant. However, this requires very good knowledge of the geometric distortion solution in both FGSs over the more than 2' that each guide star traverses during a scan. A local error of ∼0.5 mas over 30'' would suffice to explain the observed changes in roll angle. Crucially, we observe that the variation in roll angle is consistent across observations of the M35 field, regardless of filter, direction, and time of the scan; however, different fields, or observations of the same field at different roll angles (which changes the position of the guide stars within the FGSs), have essentially a different but again internally consistent set of variations of roll angle. Both of these properties are consistent with the FGS geometric distortion interpretation (Nelan 2012).

In practice, we determine the change in roll angle as a function of position along a scan field using an average obtained from deepest scans of that field. The mean is then applied to all scans of this field. This is especially helpful for shallow scans whose trails lack the precision necessary to effectively quantify the variations in roll during the scan.

The result of the analysis of each scan image treated individually is a set of position measurements for each star in the nominal X direction. These measurements are with respect to the reference scan line defined after the correction for variable roll. The position of the reference scan line is arbitrary; there is information only in the relative position of the stars with respect to one another.⁸ Each measurement is also associated with its estimated error, and several ancillary quantities are recorded for each trail, including the number of valid minirows, typical amplitude and χ², and the position of the trail on the detector.

By measuring stellar positions at three or more epochs at six month intervals, we may now determine the proper motion and parallax of the measured stars.

For some applications of astrometric measurements it may be beneficial or even necessary to first combine the measurements of multiple scans obtained at a common epoch. Applications include combining multiple scans to reduce sources of error, combining scans obtained under different conditions to calibrate the effect of the conditions on measured positions (e.g., thermal state), or combining scans of differing depth to improve the dynamic range of the measurements. In applications for which the target and reference stars can be well measured at a single exposure depth (i.e., a range of ∼5 mag), one can proceed directly to measuring parallax in Section 2.6. This is not the case for the measurement of SY Aur (or any of the 18 new Cepheid measurements now in progress).

2.5.1. Bridging the Dynamic Range Gap for Cepheids

2.5.2. Single-epoch Position Measurements

The goal of the aggregation step is to obtain for each star a single measurement of position perpendicular to the scan direction that optimally uses all the measurements obtained at the same epoch.

The aggregation step uses a model relating the "true" position of each star in the measurement direction to its measured position in each scan. Each star has one free parameter, its nominal position in the measurement direction; in total, there is one fewer parameter than stars, as our position measurements are relative, and thus insensitive to a bulk shift of all positions in the measurement direction. As a convention, we take the mean of the true positions of all stars observed in the field to be zero to set the arbitrary reference position. The true positions of N stars are thus described by N − 1 positional parameters.

In addition to the parameters describing the stars' true positions, the transformation from true to observed position involves several additional effects that need to be quantified and modeled. The most obvious ones are scan-to-scan offset and field rotation as discussed in Sections 2.4.5 and 2.4.6. The offset is a free parameter for each scan, as the absolute telescope pointing is not stable enough to constrain its scan-to-scan position to the required accuracy of tens of  μas. The roll angle of the telescope is not identical for scans with the same requested roll angle, and small variations (<0001) need to be taken into account. Both offset and rotation are defined with respect to the first scan by convention; thus, aggregating M scans requires M − 1 offsets and M − 1 rotations.

Analysis of the M35 data, seen in Figure 5, shows clearly that these two parameters are not sufficient. Residuals between model and measurements routinely reach 5–10 mpix, even for scans that are obtained in the same filter and orbit in succession, well in excess of the expected statistical measurement errors. Furthermore, such residuals, as shown in Figure 5, have clear spatial correlations that indicate that there is variable low-order distortion at the few mpix level. The residuals are somewhat greater between different filters, largely due to a static scale difference, and between observations obtained at different orientations, as these include both the previous variable term as well as errors in the static geometric distortion field. These residuals do not show any correlation with the sources' brightness or color, and thus are most likely related to variations in the geometric transformation between true and measured positions.

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For the M35 calibration data, a successful approach consists of including in the model a low-order polynomial correction with free coefficients for each observation (see Figure 5). We adopt a polynomial correction to the X coordinate when aligning two frames which 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 but closely related to the field rotation previously considered.

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 and reduce residuals of well-measured stars to ≲ 1 mpix, as shown in Figure 5. (Only terms of total degree up to the polynomial degree are included.) Figure 5 shows the residuals after these polynomial corrections are included in the model. A second-degree polynomial in X and Y has five coefficients and the transformation has a total of seven parameters, including rotation on the sky and a constant term. The leading polynomial term of the cross-filter match has degree (1, 0), corresponding to an overall scale factor, and indicates that the geometric solution we initially adopt has a scale in F673N that is about 1.5 × 10⁻⁵ different from the F606W scale, or up to 40 mpix near the edge. Several other terms also have significant power but generally vary in sign, with contributions of up to 50 mpix at the edge of the field. Not surprisingly, the coefficients of these terms are smallest when a pair of frames have a similar predicted time-dependent focus position of the telescope based on the HST orbital thermal model as shown in Figure 5. Thus, it appears that much of the role of the polynomial correction is to account for PSF and field distortion caused by the changing thermal state of HST.

To quantify the size of a color dependence of the geometric distortion (i.e., a color wedge) over the broad range of the F606W filter, we used the M35 data to compare the relative source positions in F673N and F606W versus the B − V color after accounting for variable distortion. The full color range of the stars was −0.15 < B − V < 1.4 mag, with a mean color of 0.57 mag. We assume that any color effect in F673N is negligible owing to its very narrow wavelength range (Δλ = 100 Å versus Δλ = 2300 Å), so the dependence on color is a measure of a chromatic term in F606W. We find a very small color shift from the M35 data of 0.6 ± 0.2 mpix for a 0.5 mag difference in color from the mean. Eliminating the two bluest stars with B − V < 0 mag reduces the effect further to −0.1 ± 0.2 mpix and is more appropriate for the colors encountered for field stars and Cepheids. Given the small empirical size of a chromatic shift and the fact that Cepheids lie near the middle of the color range of stars, we conclude that a chromatic effect in F606W is negligible, and we will explore additional chromatic effects with an expanded data set in the future.

In principle, imperfect charge-transfer efficiency (CTE) can smear charge along the readout axes and shift astrometry. The effect is 1–2 orders of magnitude smaller along the X axis for HST CCDs owing to the greatly reduced pixel transfer (and trapping or detrapping) time than for the Y axis. For this reason we chose to scan observations along the Y-axis and measure astrometry along the X-axis. However, a small amount of charge is apparent trailing hot pixels in the X axis opposite the read direction indicating the presence of short timescale charge traps and imperfect X-CTE. This deferred charge is about 0.08% for a pixel with 10,000 electrons and becomes a smaller fraction for brighter pixels. The resulting astrometric shift is about twice as large for a given charge total in a two-dimensional PSF than the one-dimensional minirow of a scan due to the larger shift affecting the fainter PSF wings. Using these calibrations, we estimate the shift of SY Aur in the shallow scan from fast trapping would be ∼0.4 mpix and even smaller, about <0.2 mpix, for a Cepheid with peak counts of >30, 000 electrons. Depending on the relative position of stars and amplifiers across epochs, the effect on a star's measured parallax may fully or partially cancel. For SY Aur we conclude that the effect (<3% at 2 kpc) can be neglected. In the future we will try to directly measure the change in scan line astrometry in the X-direction by dithering a source back and forth across the line dividing the read-out X-direction for use in a larger sample analysis. We also note that no evidence of temporal changes in the distortion of WFC3 (i.e., of the skew) has been found but this will be investigated with future observations of M35.

For the SY Aur field we observed the Cepheid at 8.7 < V < 9.4 mag in F673N at the fiducial scan rate of 04 s⁻¹, resulting in peak counts for the Cepheid of 10,000–15,000 electrons, thus providing unsaturated measurements to complement the contemporaneous F606W scans (where the Cepheid is oversaturated by a factor of 5–6 and F621M would saturate by a factor of 1–1.5 if used). Unfortunately, the anti-galactocentric direction of SY Aur (Galactic longitude ℓ = 16475) results in a rather sparse field with only three stars that are both well-measured in F673N (7 < V < 13 mag) and not saturated in F606W (V > 11 mag). Because the calibration requirements to reach millipixel precision from spatial scanning were not known at the start of this pilot project, the selected field of SY Aur is less than optimal.

A few more stars exist at 13 < V < 15 mag, providing a little additional constraining power. The paucity of stars available to define the transformation between the shallow and deep frame results in a relatively noisy mapping which is not very robust. While the mean precision of each Cepheid position measurement in F673N is 0.4 mpix (see Figure 6), transforming the unsaturated Cepheid position in F673N to F606W with so few stars degrades the Cepheid position to about 2 mpix precision. As shown in Figure 6, for other Cepheid fields selected in directions richer with reference stars, we find that we can constrain the Cepheid position in the deeper scan to 1 mpix.

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The final step is to utilize the multiple measurement epochs taken over the course of two years at intervals of six months, in order to estimate the parallax and proper motion of each star in the field. This model involves several considerations. First, the position of the stars in the field, relative to one another, naturally changes over time as a consequence of their astrometric motion in the measurement direction. This motion is modeled as the combination of a linear term, which is the projection of the proper motion along the measurement direction, and a periodic term, which is the projection of the parallactic motion. Note that the shape and phase of the parallactic motion as well as the component in the measured direction are fully known from the position of each star, the motion of Earth with respect to the barycenter of the solar system, and the orientation of WFC3; the only free parameter is its amplitude, which scales directly with the parallax of the target. Thus, the standard astrometric model involves three parameters for each star: position, parallax, and proper motion along the measurement direction.

Observations spaced every half year, at the time of the maximum and minimum parallactic excursion, give optimal discrimination between parallactic and proper motion. In practice, HST observations of SY Aur could be scheduled at the optimal time (mid September and early March) at an orientation no closer than 35° (or 180° + 35°) from the optimal, reducing the apparent parallax component to 80% of its maximum value. The available roll angle at a given time depends on HST's orientation with respect to the Sun (in relation to the telescope aperture and solar panels) and the availability of guide stars. Although the observation orientation may not be optimal, symmetry ensures that the same orientation flipped by 180° will be available six months later.

At least three epochs are necessary to be able to disentangle parallax and proper motion for each star. In practice, three epochs will not suffice; additional free parameters are involved in registering the observations, and the errors in the derived parameters would be much larger than the measurement errors. Four epochs are in general sufficient to obtain good constraints on the parallax and proper motion separately; when available, five epochs help reduce the covariance between derived parallaxes and proper motions, and improve the precision of the final measurement beyond the obvious factor $\sqrt{5/4}$. For SY Aur we have in fact five epochs, although two lack repetition of the shallow or deep scan.

In addition to the astrometric parameters of each star, the model includes 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 in Section 2.5.2, but it now substitutes the stationary star assumption with the astrometric model for each star. We also now utilize epochs obtained with orientation differences of 180°.

The full model can be formally described by the expression

$$\begin{eqnarray} X_{ij} &=& X_{i0} - X_{{\rm ref},j} + pmx_i \, (t_j-t_0) + \pi _i \, f_j + R_j\, Y_{i0}\nonumber\\ \ms && + \langle P(X_{\rm det}, Y_{\rm det}) \rangle _{{\rm trail},ij}. \end{eqnarray} \tag{ 1 }$$

Here 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. Note that the proper motion can only be relative, since any change of all proper motions by the same amount can be subsumed into a change of X_{ref, j} for each epoch. 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 spectrophotometric distance estimates for the stars in the field. Finally, 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. (Since typical rotations are of order 10⁻⁵, a measurement of Y_i0 with a precision of ∼1 pixel will suffice.) The polynomial term is determined as part of the model-fitting procedure, typically as a second- or third-degree polynomial P(X_det, Y_det), where X_det and Y_det are detector coordinates; the total correction is determined by evaluating the polynomial at every location along the trail of star i in image j and averaging the result. By convention, the rotation and polynomial term apply to each frame in relation to the first in a set.

Again, the M35 calibration data provide good guidance, as they were obtained at two orientations which differed by 180°. These data are useful for defining a family of polynomials with the smallest number of terms needed to adequately account for variable distortion (at the same orientation) or static and variable distortion (at flipped orientation).

In the final step we need to convert the relative parallaxes into absolute, thus yielding a geometric distance estimate. This is accomplished by estimating the reference star parallaxes from the spectrophotometric absolute magnitude estimates that come from multiband photometry and medium-resolution spectroscopy. The distance estimate of the target star will be insensitive to uncertainties in the distance of the reference stars as long as the set contains objects which are bright and distant (e.g., red giants; see Section 2.3).

For the SY Aur field (and for other Cepheid fields in progress), we obtained direct imaging with HST during the scanning observations and measured photometry of all reference stars in the UV (F275W, F336W), Strömgren (F410M, F467M, F547M), and broad-band (F850LP) systems. To this photometry we added J, H, and K-band photometry from the Two Micron All Sky Survey (2MASS) to provide a set of up to 9 bands of photometry from 0.2 to 2.2 μm. All the photometry was of high S/N, with the exception of F275W where only a third of the stars yielded a measurement (${\hbox{\sl F275W}} < 22.8$ mag). Missing or excluded photometry was recorded for stars which suffered cosmic ray hits, suffered blending in the 2MASS data (as identified with HST F850LP imaging), and for half the field not covered by F410M imaging. In practice, the average number of bands with useful measurements per star was between six and seven.

To estimate the spectroscopic parallax for each reference star, we generated a sample of 29,000 mock stars in the direction of SY Aur using the Besançon galaxy model (Robin & Crézé 1986; Robin et al. 2003, and references therein). The thick disk of the model has been updated to better fit Sloan Digital Sky Survey and 2MASS data (A. Robin 2013, private communication).

This sample of mock stars has a distribution of the four parameters (log g, initial mass, metallicity, and extinction) as expected from the Besançon model along the sight line to SY Aur (see Table 2). The extinction to the edge of the Galaxy in this direction, the upper limit for stars along the line of sight, is given from Schlafly & Finkbeiner (2011). For each mock star we select the stellar model from the Padova isochrone tables (Bressan et al. 2012) whose parameters best match the mock parameters.⁹ Each stellar model includes a determination of the absolute magnitude (including the mock extinction appropriate for each band) of that model star in each of the measured bands, thus producing 29,000 mock stellar models. The comparison of each mock model to the measurements of a reference star produces a distance estimate (and an estimate of the four nuisance parameters) and a likelihood that the model is good based on the size of the χ² statistic between model and data. The net effect of the Besançon model as a prior, as opposed to no prior on the likelihood of combinations of stellar parameters, is a small increase in the median distance estimate of 8% or a reduction in the median reference parallax of 50 μas.

We independently determined the temperature and luminosity class of the majority of the reference stars via medium-resolution optical spectra compared to template spectra. The spectra were obtained with the Kast double spectrograph (Miller & Stone 1993) on the 3 m Shane reflector at Lick Observatory and DIS on the APO 3.5 m. Standard procedures were used for the data reduction.

In order to combine the photometric and spectroscopic results, we treated the spectroscopic term as an additional contribution to the χ² statistic on the basis of the agreement between the temperature and the log g term given for the best-matching spectral template, including both spectral and luminosity class. The spectral templates used were from Pickles (1998) and the ELODIE database (Prugniel & Soubiran 2001). This is especially important for discriminating dwarfs from giants, for which the spectroscopic contribution is often more powerful at discrimination than photometry. Each star's distance (and expected parallax) was determined from the normalized, summed product of mock distances and likelihoods (i.e., the Bayesian mean), with uncertainties fixed at 0.3 mag as derived from Monte Carlo simulations of the models. The parallax uncertainty (i.e., systematic uncertainty in frame parallax) of the set of 34 fitted reference stars is 12 μas, well below our target uncertainty. As expected, most of this precision comes from the most distant stars which are primarily red giants. A single red giant at a distance of 5 kpc would give an uncertainty of 30 μas. Indeed, the five most distant stars alone give an uncertainty in the reduction to absolute parallax of 18 μas, equivalent to three distant red giants. Assuming a larger per-star uncertainty of 0.5 mag increases the uncertainty in the constant parallax term to 21 μas, still well below our overall accuracy for this field. These estimates allow us to break the degeneracy between relative parallaxes and obtain an actual parallax estimate for each star, including the Cepheid.

2.6.1. Setting Up the Model

2.6.2. Multiparametric Model for SY Aurigae: Results

The resulting fits for the SY Aur field are shown in Figure 7. For the remaining 26 reference stars, the spectroscopic parallax constraints are shown in Figure 8 compared to the a posteriori parallax with parameter values given in Table 2. The proper-motion estimates are subtracted from both data and model for ease of examination. For bright stars with long scans, the parallax uncertainty is just under 40 μas, sufficient for a 16% measurement of parallax for two red giants, stars 4 and 10 at D = 3.4 and 4.1 kpc, respectively. The distance of the Cepheid SY Aur from its parallax is D = 2.3 kpc, making this star and others in the field the most distant stars with well-determined parallax. This distance is in good agreement with the expectation of D ≈ 2.4 kpc based on the Milky Way P–L relation (Benedict et al. 2007) as detailed in the next section.

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The precision of the Cepheid parallax at this distance should be about 4% based on optimal photon statistics, a designed 3600-pixel-long scan length, and position in the center of the field away from larger and more uncertain distortions at the edge of the field. Because this particular Cepheid at 8.8 < V < 9.4 mag is too bright to avoid saturation in the deep scan when V < 10.6 mag, its measurement precision would be lower, about 6%, owing to the reduced photon statistics in the shallow scan. However, the lack of bright stars in this field (only 3 with astrometric errors of <100 μas in the shallow scan; see Figure 6) degrades the precision of the transformation between the shallow and deep scan for the Cepheid to between 50 and 100 μas and thus the precision of this parallax to 12%. Through better understanding of the parameters which affect the parallax precision, we selected better Cepheid fields for new observations with more reference stars, as well as filters and scan speeds which maximize photon statistics, to get much closer to the attainable precision (see Figure 6). In addition, the precision of the SY Aur parallax from these data should improve through better empirical knowledge of the spatial transformation between WFC3-UVIS filters garnered from ongoing spatial scans of other Cepheid fields, so we consider this measurement to be preliminary until collection of the full-sample of 19 Cepheids.

While ground-based observatories have imaged these bright MW Cepheids in the NIR, systematic uncertainties in the flux scale between the ground and HST photometric systems would limit the precision of the Hubble constant independent of the measurement of their parallaxes. The long-term internal stability of the nonstandard HST photometric system in the NIR has been established to ∼1% (Kalirai et al. 2011). However, the NIR HST WFC3-IR bandpasses of F160W and F125W used to observe Cepheids are not well matched to ground-based bandpasses, which are set by natural breaks in atmospheric OH emission and water transmission. The difference between a typical ground-based H-band filter and its WFC3-IR equivalent, F160W, is large, with color terms demonstrating a 20% difference between the measured brightness for stars with J − H differing by 1 mag (Riess 2011a). In addition, ground-based systems suffer from photometric instabilities at the few percent level owing to nightly and hourly variations in the amount of precipitable water vapor and aerosols in the atmosphere. The best-understood NIR ground-based system, 2MASS, is calibrated to a precision of 0.02–0.03 mag (Skrutskie et al. 2006). Not surprisingly, systematic differences of 0.02 mag exist between the mean NIR magnitudes measured for the same Cepheids observed at different ground-based observatories, even after accounting for the known differences in their photometric systems (Monson & Pierce 2011). These remaining differences likely reflect the limitations with which the throughput of any ground-based system is known in the NIR and present a critical systematic uncertainty in relating ground-based NIR magnitudes of MW Cepheids to those in distant galaxies observed with HST.

While optical ground-based and HST systems are easier to cross-calibrate than those in the NIR, even the uncertainty between these systems resulted in a reported 5% systematic uncertainty in the determination of H₀ by Freedman et al. (2001).

The only way to ensure that future highest-quality Cepheid parallaxes from HST or Gaia are fully leveraged is to observe the nearest MW Cepheids with the same photometric system used to observe their distant counterparts. However, it is challenging to accurately measure the brightness of nearby, long-period Cepheids with HST. The 33 known MW Cepheids with P > 10 days and D < 5 kpc have 3.5 < H < 7.5 mag and 6 < V < 10 mag, and any would saturate in the shortest exposures possible with WFC3-IR in the F125W and F160W filters. In addition, the shortest possible exposure, 0.1 s with the smallest subarrays (unsaturated for H > 8 mag), limits the aperture radius to 32 pixels (∼4''), which is not ideal for distinguishing the sky level from the wings of the PSF. While it is possible to accurately measure saturated sources by fitting to the unsaturated wings of the light profile, it is important to minimize the degree of saturation, as exclusive reliance on pixels progressively farther out in the wings increases systematic uncertainties which propagate from uncertain ratios between the peak and wing fluxes, including their color dependence in broad bands.

One way to decrease the exposure time and the degree of saturation is through spatial scanning during integration. Spatial scanning of the telescope reduces the effective exposure time a pixel "sees" a source in proportion to the inverse of the scan speed. The highest scan speed available with HST (requiring gyroscope guiding) is 78 s⁻¹ which, for a pixel size of 013 for WFC3-IR, is 0.017 s, or 0.03 s including flux from the off-peak integration. At this speed saturation begins at H = 7 mag. This is just under half the minimum integration of the smallest subarray allowed. In the optical the advantage of scanning bright targets is even greater, where the 004 pixel of WFC3-UVIS and the PSF produce a minimum effective exposure time of 0.01 s, a factor of 50 shorter than the minimum allowed exposure, a saturation limit of V = 7 mag, and without uncertainties associated with a variable time of shutter flight. For increasingly brighter targets above the saturation limit, successive pairs of pixels adjacent to the peak saturate at a rate of approximately one new pixel pair per magnitude, and information is lost in blocks. For optical photometry with WFC3-UVIS, Gilliland et al. (2010) have shown that with a gain setting of 2 or higher, full-well saturation occurs before digital saturation, so that saturated star photometry is well measured by including the sum of the blooming charge in the total without loss in precision or accuracy.

Another advantage of scanning bright stars instead of taking pointed images with subarrays comes from improved sampling of the detector. Pixel-to- pixel variations in the flat fields or positional variation in quantum efficiency produces errors of about 0.01 mag with the WFC3-IR detector (Riess 2011b). In addition, errors which depend on pixel phase (e.g., resulting from imperfect knowledge of the PSF) are reduced by scanning at an angle which varies the pixel phase, yielding a result that is independent of pixel phase.

On JD 2,455,989 we obtained scanned observations of the MW Cepheid SY Aur with WFC3-IR F160W at a commanded scan rate of 75 s⁻¹.

We fit an empirical light profile (i.e., cross-section of a trailed PSF) to each minirow (i.e., a 21 pixel row centered on the source). The effective exposure time for each nondestructive sample of the HgCdTe detector is a fixed number (determined from ground-based testing, 0.853 s to read out the 512 × 512 pixel subarray utilized) plus an additional increment or decrement of exposure time to account for the altered position of the moving source relative to the detector. This additional time is the standard interval between reads multiplied by the fractional increase or decrease of the array that must be read out before encountering the approaching or receding scan line (the sign of which depends on the scan direction and whether the scan is above or below the midline of the instrument which determines the readout direction).¹⁰

The difference in integration between an approaching and receding scan is quite large, amounting to an integration interval of 0.7092 s over a length of 44 pixels between samples when scanning toward the readout amplifier or 1.065 s over 65 pixels when scanning away from the readout. For each sample time interval, the count rate is measured from the product of the mean minirow count in the sample and the length of the sample scan divided by the time interval of the sample. The length divided by the interval is an empirical measure of the scan rate, which was found to be about 4% higher than requested. After rejecting the central pixel of each minirow which is slightly saturated, the measured count rate is 15.75 million e⁻ s⁻¹ (F160W = 6.706 mag) scanning toward and 15.49 million e⁻ s⁻¹ (F160W = 6.725 mag) scanning away. This 2% difference arises from edge effects causing us to overcount or undercount a partially filled pixel. We take the average of 6.716 ± 0.005 mag to cancel the edge effects. We increase this by 0.01 mag to 6.706 ± 0.005 mag to account for a 1% systematic underestimate of bright sources in short exposures where trapping is seen to reduce the measured charge by 1%. The row-to-row scatter is 3.25% and appears to result from nonsmooth scanning. It is worth noting that a fractional error in the WFC3-IR sample time, if uniform over all sample patterns (i.e., from the WFC3 clock running slow or fast), would be negated in the measurement of relative distances between MW and extragalactic Cepheids (when observed using the same clock).

Most known MW Cepheids have had their NIR light curves measured by Laney & Stobie (1992) or Monson & Pierce (2011). Their periods and phases have typically been determined to the third or fourth decimal place from optical light curves, which assures that photometry from a single epoch can be easily transformed to the mean magnitude with less than 1% statistical uncertainty (mean magnitudes are the standard measure used for Cepheid distance determinations as shown in Figure 9). For SY Aur, the expected difference between the single epoch and the mean in F160W is 0.07 mag, for a final result of ${\hbox{\sl F160W}}=6.64 \pm 0.01$ mag. This is 0.04 mag fainter than the zeropoint from Monson & Pierce (2011) which is within the expected uncertainties of the relative zeropoints of the two system. The extinction to SY Aur is estimated to be A_V = 1.40 mag, A_F160 = 0.27 and A_K = 0.16 mag based on the B − V color excess of 0.45 mag in the compilation of Tammann et al. (2003). This is in good agreement with the mean reference star value of A_K = 0.14 mag (and the value of A_K = 0.17 mag interpolated from their values at the parallax of SY Aur) determined from the fits of the stellar models to the reference star photometry (the Cepheid is a bit farther than the mean reference star and so it is not surprising that its extinction is modestly higher). With a mean visual mag of 9.06 and a period of 10.1 days, the Milky Way P–L relation from (Benedict et al. 2007) gives a distance modulus of μ₀ = 11.72 ± 0.20 mag where the uncertainty is dominated by variance in the reddening law. Using instead the F160W-band zeropoint from the parallaxes of Benedict et al. (2007) gives 11.94 ± 0.10 mag. Either is in good agreement with the preliminary parallax distance modulus of μ₀ = 11.84 ± 0.12 mag measured here where the uncertainty is statistical only as discussed in the previous section.

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