1.1.

Expected Astrometric Performance of the WFIRST Wide-Field Imager

The astrometric performance of the Wide-Field Channel (WFC) in the WFI on WFIRST will depend on numerous elements, including hardware characteristics, the stability of the platform, characterization of the optics and of the detector (down to the individual pixel), the ability to design observations with the required properties for reference stars, and calibration of both the point-spread function (PSF) and of the geometric distortion (GD) of the focal plane. Here, we summarize the assumptions used in this work; Table 1 collects some performance estimates for different types of observations.

Table 1

Approximate expected astrometric performance of the WFIRST WFI for different types of observations. All estimates are for well-exposed point sources (refer to Ref. 2 and Table 4 for depths of the core survey programs). See the referred sections for details about the assumptions leading to each number. These estimates are order-of-magnitude only.

Reference 6 discusses astrometric science in the WFIRST exoplanet microlensing (EML) survey field but makes very optimistic sensitivity estimates and ignores potential sources of systematic errors. For the purpose of this document, we assume that the single-exposure precision for well-exposed point sources is 0.01 pixel or about 1.1 mas; this is consistent with current experience on space-based platforms such as HST, as long as a comparable level of calibration activities are carried out.⁷ Depending on the platform stability and the quality and frequency of calibrations, this precision can be substantially improved by repeated, dithered observations as

Improvements in astrometric measurements can also be obtained by special techniques, such as spatial scanning and centering on diffraction spikes, described in Sec. 2.4, and by improvements in the pixel-level calibration, as discussed in Sec. 4.2.

Thanks to its large FoV and the availability of accurate reference sources from Gaia, each WFC exposure can achieve an absolute positional accuracy of 0.1 mas or better (see Sec. 3 for details). Although WFIRST can directly measure only relative parallaxes and PMs within its FoV, the ability to use reference stars in common with Gaia will allow parallaxes and PMs to be converted to an absolute reference frame to a worst-case accuracy of $10\mu \mathrm{as}$ in parallax and $10\mu \mathrm{as}{\mathrm{yr}}^{-1}$ in PM.

2.1.

Motions of Local Group Galaxies

The range and reach of WFIRST astrometry complement and extend Gaia and Large Synoptic Survey Telescope (LSST) astrometry. Figure 1 compares the reach of current and planned PM surveys to the PMs corresponding to known velocities and distances of LG objects. Since the known orbital and internal velocities refer in almost every case to the radial component, these are intended only to represent the order of magnitude one might expect for the PMs (indeed, as in the case of M31, the orbital PMs may be significantly smaller than that inferred by radial velocity measurements alone). From this figure, it is clear that to measure PMs of satellites beyond the Milky Way’s (MW) virial radius will require better precision than LSST can achieve, at larger distances (and thus fainter magnitudes). This is the window of opportunity for WFIRST.

Fig. 1

PMs accessible to various current and planned surveys and measurements, compared to the PMs corresponding to characteristic velocities and distances for LG objects. Shaded regions show the distances and PMs for single stars accessible to the Gaia (magenta) and LSST (purple) surveys, compared to the approximate reach of the WFIRST HLS field for bright K giants assuming 15 exposures over 5 years (yellow) and the additional reach for cross-matches with HST imaging (orange). The diagonal lines show the PM associated with several characteristic transverse velocities as a function of distance: the typical range of orbital velocities in the Galactic halo (magenta and cyan) and the typical internal velocity dispersion of a dSph galaxy (green). Thick vertical lines mark heliocentric distances to: the large (blue dashed) and small (blue dot-dashed) Magellanic Clouds, the edge of the MW halo (gray), M31 (red), and the approximate edge of the LG (green). Gray dotted vertical lines mark heliocentric distances to other dwarf galaxies in the LG, including satellites of the MW and M31.⁸ Current PM measurements by HST^9–12 and Gaia¹³ for MW GCs and satellite galaxies and for individual stars in the Sgr tidal stream¹⁴ are plotted as black points/symbols.

WFIRST astrometry is a crucial component of constraints on the nature of dark matter (DM) from the orbital and internal PMs of the satellite galaxies of the MW. The orbits of dwarf satellites can be used to help map the MW’s own DM halo out to its virial radius and beyond, placing our galaxy into a cosmological context. A complete knowledge of the MW halo’s properties enables tests of DM models through comparisons with predictions from simulations for its mass and shape, accretion history, and the mass and orbit distributions of its satellite galaxies. HST PMs of dwarf satellite galaxies, and Gaia astrometry in the inner Galaxy, will both make great strides toward this goal; however, Gaia has insufficient depth and HST insufficient FoV to reach to the edge of the MW halo (where the total mass is uncertain to a factor of 4; see Ref. 15) or to obtain internal PM measurements for many dwarf galaxies, which are crucial to break velocity degeneracies and understand the small-scale distribution of DM (see also work by the Gaia Challenge group,¹⁶ summarized in Fig 2.2 of Ref. 17).

The HST proper motion (HSTPROMO) campaign has used the HST, which has image quality similar to that expected for WFIRST, to measure PMs of both bound objects and stream stars in the MW (e.g., Refs. 10 and 18). In the coming years, HST will set a PM baseline for many more distant satellites:¹⁹ by the time WFIRST is ready, this baseline will be roughly 8 to 10 years for these satellites and far longer for dwarf galaxies with earlier observations. With its larger FoV and more sensitive detectors, WFIRST should be able to expand these measurements to more distant galaxies and achieve better accuracy due to the larger number of calibration objects available (and to the establishment of Gaia’s astrometric frame, see Sec. 3). Figure 2 shows the estimated number of stars in each of the MW’s satellites that are accessible to WFIRST at the depth of the planned high-latitude survey (HLS, $J<26.7$, blue) core program, as well as for a possible spatial-scanning mode on WFIRST ($H<16$, red). The upper panel of the figure shows the expected tangential velocity error for each dwarf assuming PM precisions of $25\mu \mathrm{as}{\mathrm{yr}}^{-1}$. Internal PM dispersions, as was recently done for the Large Magellanic Cloud with HST,²⁰ are reachable with WFIRST for the galaxies shown in cyan. These include three ultrafaint galaxies (Segue I, Draco, and Ursa Minor) that are sufficiently DM-dominated to distinguish between cold DM models (which predict a cuspy inner mass profile) and warm or “fuzzy” DM models (which predict cored inner profiles). Draco and Ursa Minor each have $\sim 80$ stars at $H<16$ and their internal velocity dispersions are marginally resolved even at $25\mu \mathrm{as}{\mathrm{yr}}^{-1}$, making them particularly good cases for spatial scanning to improve the internal PM accuracies by an order of magnitude.

Fig. 2

(a) Estimated observational error in tangential velocity assuming PM precision of $25\mu \mathrm{as}{\mathrm{yr}}^{-1}$. Galaxies in cyan have estimated velocity errors (${\delta }_{\mu }$) comparable to or less than their intrinsic velocity dispersions (${\sigma }_v$).(b) Number of stars in LG dwarf satellite galaxies brighter than the limiting apparent magnitude of the WFIRST HLS ($J<26.7$, blue) and the limit for spatial scanning ($H<16$, red). Plot courtesy Matthew Walker.

2.2.

Motions of Stars in the Distant MW Halo

Besides dwarf galaxies, the MW’s halo contains the tidally disrupted remains of previously accreted galaxies, known as tidal streams. We expect that tidal debris should extend to at least the virial radius of the MW,²¹ but currently the most distant MW halo star known is an M giant at around 250 kpc.²² The most distant known populations with statistical samples of stars (BHB, RR Lyr, and M giant stars) extend to around 150 kpc (half the virial radius) at the magnitude limit of current surveys,^23–25 but the WFIRST HLS fiducial depth will reach to the MW’s virial radius down to the main-sequence turnoff. The orbits of distant stars probe the extent and total mass of the MW dark halo; they also represent a unique population of recently accreted small galaxies. With the HLS’s projected depth, the transition between the MW’s and M31’s spheres of influence, and perhaps the splashback radius of the MW (e.g., Ref. 26), could also be detected. PMs from WFIRST are crucial to these endeavors since complete phase-space information for these stars is the best way to confirm that stars associated in position at large distances are from the same progenitor and to connect groups on opposite sides of the galaxy through their orbits, leading to constraints on the mass profile and flattening of the Galactic dark halo at large distances (e.g., Ref. 27).

High-velocity stars are another interesting target, whether for GO observations or as serendipitous objects in repeated survey fields. These stars’ orbits, which have an extremely wide radial range, can potentially also be used to constrain the overall shape and mass of the MW DM halo.²⁸

At distances of 100 to 300 kpc, preliminary work shows that PM precision of $25\mu \mathrm{as}{\mathrm{yr}}^{-1}$ or better is required to eliminate outliers in groups and connect structures on opposite sides of the galaxy using their phase-space positions (Fig. 3; Secunda et al. in preparation). A similar precision would be needed to identify high-velocity stars at or near the Galactic escape velocity.

Fig. 3

Identifying stellar structures in the distant Galactic halo. (a) Groups of stars identified in a mock stellar halo²⁹ in the range 100 to 300 kpc, using sky positions (shown) and distances only. (b) Same stars colored by progenitor galaxy. Green arrows highlight the contribution of interlopers to group 5 (dark orange) in (a). (c)–(e) View of the same groups in energy-angular momentum projection, which requires six-dimensional phase space information including PMs. (d) Some outliers are already identifiable at $25\mu \mathrm{as}{\mathrm{yr}}^{-1}$ and (e) structures are clearly distinguishable at $5\mu \mathrm{as}{\mathrm{yr}}^{-1}$ (green arrows). Groups 2 and 4 (dark and light blue, respectively) in (a) are from the same tidally disrupted progenitor galaxy but are found on opposite sides of the sky; with $\le 25\mu \mathrm{as}{\mathrm{yr}}^{-1}$ precision they can be associated through orbit integration, reflected in (d) and (e) by their similar values of angular momentum ($j_z$).

2.3.

Constraining the Low-Mass End of the Subhalo Mass Function

Astrometry from pointed guest observer (GO) projects could provide several routes to understanding the distribution of low-mass substructure in the MW. The abundance (or lack) of low-mass structure is a key to differentiating between cold and warm DM models since cold DM predicts abundant substructure at small scales while in warm DM the mass function is cut off at a characteristic scale related to the intrinsic temperature of the DM particle (and hence to its mass in the case of a thermal relic).

One route is to search for perturbations to tidal tails from distant globular clusters (GCs) or dwarf galaxies. Current searches are focused on quantifying substructure in the spatial distribution of tidal debris but are limited by knowledge of the MW background/foreground in the region of the stream as well as by Poisson fluctuations in the star counts (e.g., Ref. 30) and by the uncertain dynamical ages of streams, which depend on modeling the orbit (e.g., Ref. 31). WFIRST’s capability to reach deep into the stellar main sequence (MS) at these distances will help mitigate the shot-noise issue. More importantly, obtaining astrometry of fields including tidal streams would allow superior selection of stream stars relative to the background/foreground, improving the sensitivity to density fluctuations, and allowing better constraints on the time when material was first tidally stripped. Streams commonly stretch tens of degrees over the sky, so WFIRST’s large FoV is uniquely well suited to this application. A thin stream usually has a velocity dispersion of 1 to $10\mathrm{km}{\mathrm{s}}^{-1}$, so to provide a useful PM selection for a stream at 50 kpc would require relative PMs to a precision of about $20\mu \mathrm{as}{\mathrm{yr}}^{-1}$.

Another possibility is to search for deviations in the apparent positions of quasars due to strong lensing by dark substructures. For a distant quasar lensed by a ${10}^8{M}_{\odot }$ subhalo at 50 kpc, the Einstein radius of the lens is roughly $20\mu \mathrm{as}$ (presuming a singular isothermal sphere). One route would be to look for the time-dependence of the lensing around a single quasar: for a subhalo moving at $200\mathrm{km}{\mathrm{s}}^{-1}$, the time to cross the Einstein radius is about 10 days. Alternatively, a wide field containing many quasars could be examined for statistical deviations between exposures taken at different times (separated by longer than 10 days). Either approach would require absolute astrometry (i.e., consistent between exposures) accurate to $20\mu \mathrm{as}$.

2.4.

Detection and Characterization of Exoplanets

Very accurate PM estimates will make it feasible to search for the astrometric signature of exoplanets around nearby stars. For competitive constraints on exoplanet masses and orbital parameters, an instantaneous precision of better than $10\mu \mathrm{as}$ is required. The best constraints can be achieved for the most nearby stars ($d\lesssim 10\mathrm{pc}$). A dedicated GO program that specifically observes those most promising targets with a flexible schedule is therefore complementary to the EML.

Because of their close proximity, the target stars are generally very bright. This makes high-precision astrometry possible by using one of two different methods: spatial scanning and diffraction spike modeling.

2.5.

Detailed Structure of the Inner Milky Way

Gaia will revolutionize our understanding of MW structure in the outer parts of the MW, including the halo. However, Gaia has a very limited view of the inner MW due to the significant extinction in the Galactic plane at optical wavelengths (Fig. 4 shows an illustrative example using a simulated galaxy) as well as crowding (not accounted for in Fig. 4). WFIRST will probe significantly deeper into the inner MW and be less seriously affected by crowding than Gaia, allowing us to map the structure and kinematics of this region and complement the Gaia view. As an example, the EML survey will obtain precise parallaxes and ultraprecise PMs for over 50 million stars in a small area of the Galactic bulge, enabling a detailed analysis of their kinematics and density distribution. Currently, studies of bulge stellar populations are limited by the quality of the PM and the need to remove foreground disk stars, typically achieved via kinematic or photometric filters (see, e.g., Ref. 43). Both are statistical in nature and do not provide a direct determination of the distance to individual stars. According to the current requirements, a mission-long astrometric accuracy of $10\mu \mathrm{as}$ or better (with a stretch goal of $3\mu \mathrm{as}$) should be achieved at $H_{\mathrm{AB}}=21.6$ (EML 20; see Sec. 5.2).

Fig. 4

Simulated completeness of the distribution of red clump stars with distances $|z|<500\mathrm{pc}$ from the Galactic plane, based on the MW-like simulated galaxy in Ref. 36. (a) Stars that Gaia would detect in the optical with $G<20$; (b) those seen by WFIRST in the IR with $K<26$. Gray contours in both panels show the density of the complete distribution on a logarithmic scale; the black cross marks the location of the Sun. The synthetic red-clump catalog was constructed by drawing stars in the range $-0.48<M_I<-0.08$, $0.8<V-I<1.4$³⁷ from the model isochrones in Ref. 38, distributed according to the age and stellar mass density of the simulated star particles.³⁹ The three-dimensional extinction map in Ref. 40 was interpolated to determine apparent magnitudes and reddening, and Gaia $G$ magnitudes were calculated using pygaia.⁴¹ This simulated view ignores the effects of crowding (which significantly affect Gaia in the plane but are anticipated to be relevant for WFIRST only within $\sim 0.5\mathrm{deg}$ of the Galactic Center; see Sec. 2.6) and does not include a prominent Galactic bar (see Ref. 42). Gaia will largely be limited to heliocentric distances $<4\mathrm{kpc}$ in the plane, and WFIRST can measure parallaxes and velocities of stars well beyond the Galactic Center.

At comparable accuracy in relative parallaxes, distances to individual stars can be measured to 9% at the bulge (3% if the stretch goal is achieved), and useful distance discrimination should be obtained to significantly fainter magnitudes. The two tangential components of the space velocity can be recovered to the same accuracy (in this regime, the distance error is dominant over the PM error in deriving the space velocity). This information will enable a much cleaner determination of the kinematics of the bulk of bulge stars in the EML survey field and readily identify subgroups of stars—disk or bulge—with anomalous kinematics. If depth effects can be accounted for, the end-of-mission PM accuracy translates to a velocity precision of $\sim 1\mathrm{km}{\mathrm{s}}^{-1}$; together with the very large number of stars measured, this will permit a clear component separation of the spatially overlapping bulge and halo populations (see, e.g., Ref. 44) and potentially identifying complex structures such as the anomalous motions found in the X-shaped regions of the bulge (see, e.g., Ref. 45). In principle, Gaia will achieve comparable precision over all of the bulge but only for the bright red giants at $G\sim 15$ or brighter; the uncertainties will be considerably larger ($\sim 2$ orders of magnitude) at Gaia’s faint limit.

Regions within 0.5 deg of the Galactic Center will likely suffer from crowding at $H<21$, as has been seen with HST-WFC3IR studies of this region,⁴⁶ which would limit the astrometric precision to $>0.5\mathrm{mas}{\mathrm{yr}}^{-1}$ for stars fainter than this. Beyond this region, the stellar density is not typically high enough to impact astrometric precision down to $H<24$. However, the exact determination of how the WFIRST astrometric precision will scale with stellar density, SNR, PSF knowledge, and survey depth will require image-level simulations in the future.

2.6.

Star Formation in the Milky Way

With the advent of large IR surveys of the Galactic Plane, many new young star clusters have been identified. The most massive of these young clusters are ideal laboratories for studies of star and cluster formation, stellar evolution, and cluster dynamics, but detailed studies of these regions are hampered by high and spatially variable extinction, high stellar densities, and confusion with foreground and background stars. Many of these limitations can be overcome with the addition of PMs observed in the IR, to separate out the comoving cluster members from the contaminating field population.^7,47 Furthermore, measurements of the internal velocity structure of star clusters provide constraints on the unseen stellar population from dynamical mass measurements, thereby informing cluster evolution models.⁴⁸

WFIRST is ideally suited for studies of massive young clusters and star forming regions in the MW, given its wide FoV at IR wavelengths, high spatial resolution, and potential for precise photometry and astrometry. For rapidly moving populations in the center of the Galaxy, a PM precision of $\sim 0.5\mathrm{mas}{\mathrm{yr}}^{-1}$ per star is needed to separate cluster members from field stars. To obtain internal velocities or separate clusters in the disk, a PM precision of $0.05\mathrm{mas}{\mathrm{yr}}^{-1}$ or better is desired. Even higher astrometric precisions would enable searches for binaries and higher-order multiples.

An important factor to consider for this science case is that cluster members span a large range in brightness. The brightest and most massive cluster members in clusters beyond 4 kpc often have $J=9$ or brighter. Careful calibration of persistence, shorter integration times, or possibly narrow-band filters will be needed to reach both bright, massive members and faint, low-mass members of clusters.

2.7.

Isolated Black Holes and Neutron Stars

Our Galaxy likely contains ${10}^7⏧{10}^8$ stellar mass black holes and orders of magnitude more neutron stars.⁴⁹ Measuring the number and mass statistics of these stellar remnants will provide important constraints on the initial stellar mass function, the fate of massive stars and the initial-final mass relation, the star formation history of our Galaxy, and the fundamental physics of compact objects. WFIRST has the ability to find such objects in large numbers through gravitational microlensing when a background star passes behind the compact object and is magnified photometrically. However, only the addition of WFIRST astrometry will enable us to measure the precise masses of these objects through astrometric microlensing. The apparent astrometric shift of the background star due to microlensing, which is proportional to $M^{1/2}$, is $\sim 1$ milliarcsecond for a $10M_{\odot }$ black hole at 4 kpc lensing a background star at 8 kpc (Fig. 5). Thus, the necessary astrometric precision to detect isolated black holes is $<150\mu \mathrm{as}$; a factor of 2 to 3 better precision would also allow the detection of neutron stars.

Fig. 5

Astrometric shift of a background bulge star (source, $d=8\mathrm{kpc}$) lensed by a foreground compact object such as a black hole, neutron star, or white dwarf (lens, $d=4\mathrm{kpc}$). The astrometric shift changes as a function of the projected source-lens separation on the sky, $u$, in units of the Einstein radius. For the $10M_{\odot }$ case, the Einstein radius is $\sim 4$ mas and the time for the source to cross the Einstein ring is typically $>100\mathrm{days}$.

While Gaia or ground-based adaptive optics systems may detect one or a few isolated black holes,⁵⁰ WFIRST’s IR capabilities and monitoring of the Galactic Center and bulge fields will yield the much larger samples needed to precisely measure the black hole and neutron star mass function and multiplicity. Microlensing by massive objects typically has long timescales, with Einstein crossing times $>100\mathrm{days}$ for black holes, so WFIRST astrometry should be stable on these timescales; i.e., routinely calibrated on sky if possible.

2.8.

Globular Clusters

In the last decade, a wealth of revolutionary studies has dramatically changed the traditional view of GCs as the best examples of “simple stellar populations:” stars with the same age and chemical composition. The presence of multiple stellar populations (MPs) in GCs has been widely established along all the stellar evolutionary phases (e.g., Ref. 51, 52, and references therein): spectroscopic studies have found significant star-to-star variation in light elements (e.g., Ref. 53 and references therein), whereas high-precision photometry, mostly from HST data, has clearly revealed the presence of distinct sequences in color-magnitude diagrams (CMDs) at all wavelengths (e.g., Refs. 52 and 56; see also Fig. 6 of Ref. 57). Several GCs have also shown the presence of significantly He-enhanced subpopulations (e.g., Refs. 58 and 59) and even subpopulations with distinct iron content in a few cases such as $\omega$ Cen, M22, Terzan 5, M54, NGC 5824, and M2.^60–63 These observational findings present formidable challenges for theories of the formation and evolution of GCs and have inaugurated a new era in GC research in which understanding how multiple stellar systems form and evolve is not just the curious study of an anomaly but a fundamental key to understanding star formation.

Fig. 6

(a) The $m_{\mathrm{F}275\mathrm{W}}$ versus $m_{\mathrm{F}275\mathrm{W}}-m_{\mathrm{F}336\mathrm{W}}$ CMD of $\omega$ Cen, showing several subpopulations of stars in all evolutionary sequences (from Ref. 54). MPs reveal themselves in high-precision photometry from the UV to the near-IR. (b) $m_{\mathrm{F}160\mathrm{W}}$ versus $m_{\mathrm{F}110\mathrm{W}}-m_{\mathrm{F}160\mathrm{W}}$ CMD of an outer field of $\omega$ Cen, corrected for differential reddening. Field stars (orange dots) are identified using PMs. From Ref. 55.

Measuring the PMs of stars in GCs is the most effective way to constrain the structure, formation, and dynamical evolution of these ancient stellar systems and, in turn, that of the MW itself. High-precision HST astrometry of GCs is now becoming available for a large number of objects (e.g., Ref. 64), but current PM catalogs are limited by the small FoV of HST, either to the innermost few arcminutes or to pencil-beam locations in the outskirts. While most dynamical interactions do happen in the center of GCs, answering many outstanding questions will require high-precision PMs of faint cluster stars over wide fields, for which WFIRST is by far the best tool. Here, we discuss a few examples of such investigations.

2.8.1.

Multiple-population internal kinematics

The PM-based kinematic properties of MPs have so far been characterized for only three GCs: 47 Tuc,⁶⁵ $\omega$ Cen,⁶⁶ and NGC 2808.⁶⁷ The short two-body relaxation timescale in the inner regions of these clusters, where most observations have so far been focused, implies that any initial differences in the kinematic properties of different stellar populations have likely been erased. The cluster outskirts, however, have much longer relaxation timescales and could still retain fossil kinematic information about the early stages of cluster evolution (e.g., Ref. 68). The outer regions can thus provide a wealth of information and constraints on the formation and early dynamics of MP clusters, on the subsequent long-term dynamical evolution driven by two-body relaxation, and on the role of the Galactic tidal field in the outskirts of clusters. For instance, it has been shown that second-generation stars in 47 Tuc,⁶⁵ NGC 2808,⁶⁷ and $\omega$ Cen⁶⁹ are characterized by an increasing radial anisotropy in the outer regions with respect to first-generation stars (see Fig. 7). Even further out, at distances approaching the tidal radius, the effects of the external tidal field are expected to lead to a more isotropic velocity distribution.

Fig. 7

(a) Radial (blue) and tangential (green) PM dispersions as a function of color for the bluest (left) and reddest (right) parts of the MS of 47 Tuc (top) and of the Small Magellanic Cloud (bottom) (from Ref. 65). (b) Deviation from tangential-to-radial isotropy (horizontal line) for the five subpopulations in NGC 2808. Vertical lines mark the locations of $r_{\mathrm{h}}$, $1.5\times r_{\mathrm{h}}$, and $2\times r_{\mathrm{h}}$ (from Ref. 67).

Both WFIRST’s wide FoV and its improved sensitivity will make revolutionary steps forward in understanding the initial differences in MPs if sufficient PM accuracy can be achieved. Due to mass segregation, the most abundant stars in the outskirts of GCs are low-mass, faint main-sequence objects. Gaia can only measure stars as faint as the turn-off region in most clusters and, therefore, will not be able to provide enough statistics to properly characterize the kinematics of the outer cluster regions. The expected internal velocity dispersion of cluster stars near the tidal radius is of the order of $\lesssim 1$ to $3\mathrm{km}{\mathrm{s}}^{-1}$, even for the most massive clusters. The PM error adds in quadrature, so it should be less than half the intrinsic velocity dispersion, i.e., $\lesssim 1\mathrm{km}{\mathrm{s}}^{-1}$, in order to measure dispersions in cluster outskirts. At the typical distance of Galactic GCs, $\sim 10\mathrm{kpc}$, this translates into PM errors of the order of $\lesssim 20\mu \mathrm{as}{\mathrm{yr}}^{-1}$.

2.8.2.

Energy equipartition

It is widely assumed that GCs evolve toward a state of energy equipartition over many two-body relaxation times so that the velocity dispersion of an evolved cluster should scale with stellar mass as $\sigma \propto m^{-\eta }$, with $\eta =0.5$. Recently,⁷⁰ used direct $N$-body simulations with a variety of realistic initial mass functions and initial conditions to show that this scenario is not correct (see also Ref. 71). None of these simulated systems reaches a state close to equipartition: instead, over sufficiently long timescales the mass-velocity dispersion relation converges to the value ${\eta }_{\infty }\sim 0.08$ as a consequence of the Spritzer instability (see Fig. 8).

Fig. 8

Time evolution of the energy equipartition indicator $\eta$ for single main-sequence stars in N-body simulations, from Ref. 70. The time along the abscissa is expressed in units of the initial half-mass relaxation time $t_{\mathrm{rh}}$ (0). Complete energy equipartition ($\eta =0.5$; dotted line) is never attained, confirming previous investigations based on stability analysis.

These intriguing results have just started to be observationally tested (e.g., Refs. 69 and 72). To measure $\eta$, a wide range of stellar masses must be probed. Again, this task is out of reach for Gaia because of its relatively bright magnitude limit, but WFIRST will easily measure high-precision PMs down to the hydrogen-burning limit (HBL; $\sim 0.08M_{\odot }$) and out to the tidal radius, thus constraining both the current state of energy equipartition in a cluster and its past dynamical evolution. As in the previous case, PM errors of the order of $\lesssim 20\mu \mathrm{as}{\mathrm{yr}}^{-1}$ are needed.

2.8.3.

Hydrogen-burning limit and the brown-dwarf regime

WFIRST will also make it possible to study the luminosity functions of GCs beyond the HBL and into the brown-dwarf regime. Close to the HBL, old stars show a huge difference in luminosity for a small difference in mass, resulting in a plunge of the luminosity function toward zero for stars with masses just above this limit. Stars in GCs are homogeneous in age, distance, and chemical composition (within the same subpopulation), so at the typical GC age of $>10\mathrm{Gyr}$, stars with masses below the HBL will have faded by several magnitudes relative to those above it, creating a virtual cutoff in the luminosity function (e.g., Refs. 73 and 74; see also Fig. 9).

Fig. 9

Deep near-IR CMD of the GC M4, from Ref. 74. The white-dwarf and brown-dwarf regions are labeled, and low-mass stellar models are over-plotted in green and red. The expected end of the H-burning sequence is marked with red dashed lines and a shaded area.

The best place to observe the properties of stars approaching the HBL is once again outside a cluster’s core region, where contamination by light from much brighter red-giant-branch stars is negligible. The brown-dwarf regime in GCs is unexplored ground, so many new and intriguing discoveries may be waiting for WFIRST. Due to the relatively low number density of low-mass MS and brown-dwarf stars in the cluster outskirts, WFIRST is the perfect astronomical tool for these investigations as well. Proper-motion-based cluster-field separation is needed to create clean samples of cluster members; PM errors of the order of a few tenths of $\mathrm{mas}{\mathrm{yr}}^{-1}$ might be sufficient to separate cluster stars from field stars, whereas errors one order of magnitude smaller would also enable studies of the internal kinematics of cluster stars in these lowest-mass regimes.

4.1.

Geometric Distortion

GD is the most significant systematic contributor for astrometry that is not currently covered by explicit requirements for either the HLS or EML survey. A dedicated set of observations to autocalibrate the GD of the WFI is currently being considered. There are two main ways to solve for the GD: using previous knowledge of the stellar positions in the field from existing astrometric catalogs (the “catalog” method), or via autocalibration, in which stellar positions themselves are iteratively solved for together with the GD. Each of these approaches has different advantages and disadvantages, but both depend strongly on the precision with which the position of stars can be measured using appropriate PSFs (e.g., Refs. 8182.83.84.85.86.87.88.89.90.–91).

The catalog method is less demanding of telescope time since it requires fewer images to calibrate the GD and monitor temporal variations, but it strongly depends on the quality of the astrometric catalog used as a reference since the residuals and the systematic errors present in the reference catalog can propagate in the GD solution. In addition, unless the reference catalog is based itself on images taken very close in time to the WFIRST calibration images, PMs (and their errors) can introduce significant residuals in the GD solution. A recent technical report on the catalog method (Ref. 92) also highlights the importance of using accurate PSFs that take into account for jitter and interpixel capacitance effects.

The autocalibration approach requires more images, and therefore more telescope time, but offers a self-consistent calibration solution and can be designed to be formally insensitive to proper-motion-related errors.

On-sky GD calibration has historically been performed using large dithered exposures (as wide as the FoV in some cases) of a homogeneously distributed, moderately dense stellar field. Stars in the EML survey fields are homogeneously distributed, and while their overall stellar density can be too high, this can be mitigated by using only the brightest stars in each field to calibrate the GD. If the exposure time has been carefully chosen, the bright, unsaturated stars will still be reasonably far apart from each other, and their surrounding neighbors typically a few magnitudes fainter, so that the bright stars can still be considered fairly isolated. The Baade window was successfully used by Refs. 83and 88 to calibrate the GD of two different ground-based, wide-field IR detectors (ESO WFI@MPG and HAWK-I@VLT, respectively). The GC $\omega$ Cen, another possible target field for WFIRST calibration, was used by Ref. 89 to calibrate the GD of the IR WFI VISTA InfraRed CAMera (VIRCAM) at the Visible and Infrared Survey Telescope for Astronomy (VISTA). The same technique of using only the brightest stars could be applied to the crowded regions in the core of $\omega$ Cen. On the other hand, the stellar density near the tidal radius ($\sim {48}^{\prime }$, Ref. 93, 2010 edition) may be too low. Other GCs could also be used for calibration, but because of its overall high number of members and its large tidal radius, $\omega$ Cen is the best target.

4.1.1.

Example of an autocalibration strategy

A possible autocalibration strategy for the WFIRST WFI could be modeled on the calibration described in Ref. 89 for the VIRCAM@VISTA detectors. The VIRCAM WFI comprises 16 $2\mathrm{k}\times 2\mathrm{k}$ VIRGO detectors, for a total FoV of about $1.3\times 1.0$ sq. deg, but the very large gaps between the chips bring down the effective FoV to 0.59 sq. deg. The calibration program (ESO proposal 488.L-0500(A), PI: Bellini) used a combination of small and large $5\times 5$ dithers. Large dithers were used to cover the gaps between chips, monitor low-frequency distortions, and construct a single common reference system for all observations; the small dithers were included to enable independent modeling of the high-frequency residuals of the GD within each chip (more in Ref. 89). The choice of a $5\times 5$ dither pattern was a compromise to obtain a sufficiently high-precision GD correction in a reasonable amount of telescope time. Extremely small dithers (from a few subpixels to a few pixels apart) are not strictly needed to characterize the PSF in well-populated star fields since nature distributes stars randomly with respect to the pixel boundaries (see also Ref. 94).

Figure 10 shows an example of a possible dither strategy for WFIRST following this plan. Figure 10(a) is a plan for a small $5\times 5$ dither pattern that covers each WFI detector from corner to corner. In the top left panel, the black dots mark the center of each of the 25 dithers, with the detector layout of the central dither shown in red and other dithers in gray. The bottom left panel shows the resulting depth-of-coverage map on a logarithmic scale, with a maximum of 25 different images covering the same patch of sky. Most of the map is covered by at least 22 images, but only 12 to 15 of these come from the same chip so that the same star is typically imaged in 12 to 15 different chip locations. The $5\times 5$ pattern never repeats the same shift along the $X$ or $Y$ direction, thus guaranteeing that the same stars will never fall on the same column or row, to minimize the impact of possible detector defects or degeneracies in the distortion solution.

Fig. 10

(a) Example of a small, $5\times 5$ dither pattern that covers each WFI detector from corner to corner. (b) Example of a large, $9\times 5$ dither pattern that covers the entire FoV from corner to corner. Top row: Dither pattern layout on-sky, with the center of each dither marked by a black dot, the WFI outline of the central dither shown in red, and outlines of the other dithers in gaey). Bottom row: Depth-of-coverage map (number of repeat observations as a function of position) for the assumed dither strategies, on a logarithmic scale. See Sec. 4.1 for details.

Figure 10(b) shows an example plan for a $9\times 5$ pattern of large dithers that covers the entire WFI FoV from corner to corner. Because of the rectangular shape of the WFI FoV, nine dithers on the $X$ axis are needed to cover the FoV with similar spacing to the five dithers along the $Y$ axis. As for the small dithers, the layouts and centers of each pointing are shown on top and the resulting depth-of-coverage map is shown on the bottom. In this case, the layout results in a maximum of 39 different images at the center of the pattern.

The dither patterns shown in Fig. 10 all have the same telescope rotation angle, but in order to properly calibrate the skew terms of the distortion, a few observations of the same field at different roll angles would be highly beneficial. It is not obvious to suggest exactly how many of these rotated exposures should be taken, but sampling the full circle every 45 deg 60 deg should suffice. The total FoV covered by the large dither pattern in Fig. 10 allows for the central pointing to be rotated by any angle and still be fully within the covered region.

The proposed dither strategy makes use of 25 small dithers and 45 large dithers for a given filter, plus six or eight additional pointings (60 deg or 45 deg sampling, respectively) to constrain the skew terms, for a total of 76 to 78 distinct pointings. Experience calibrating the HST GD shows that convergence in the GD solution is achieved when stellar positions transformed from one image to another taken with a different pointing have rms residuals comparable to the stellar centroiding errors. Simulations to assess the precision of the GD correction as a function of the adopted dither strategy are ongoing to determine, among other things, whether fewer pointings than the example shown here could be sufficient.

An additional complication introduced by WFIRST’s large FoV is due to the use of tangent-plane projections: adopting the same projection point for images taken more than a few arcminutes apart results in significant positional transformation residuals. The GD calibration therefore has to be carried out on the celestial sphere rather than on any given tangent plane, adding an extralayer of complexity (see also Ref. 89).

4.2.

Pixel-Level Effects

4.2.1.

Quantum efficiency variations

Variations in the quantum efficiency (QE) within a single pixel can affect the accuracy of localization and therefore the astrometric precision. Reference 95 measured the intrapixel response function for the H2RG detectors to be flown on JWST, which are direct precursors of those planned for WFIRST. They found that the variations in the response per pixel [shown in Fig. 11(b)] appear to be mainly caused by redistribution of charges from pixel-to-pixel rather than by variations in pixel sensitivity. The most important effect in redistributing charge between pixels was the diffusion of charge to neighboring pixels, followed by interpixel capacitance (measured between 0% and 4%). They also find that the type of small defect visible in Fig. 11 occurs in roughly 10% of pixels. Additional testing of next-generation detectors more closely resembling those to be used for WFIRST is ongoing, but we expect that they will exhibit lower levels of variation.

Fig. 11

(a) Figure 4 of Ref. 95, showing the pixel response at 650 nm for an 8-pixel-square region of an H2RG detector. (b) Pixel offsets in a $128\times 128$ region of an H2RG detector. (Figure courtesy Michael Shao.)

4.3.

Filters and Color-Dependent Systematics

Color-dependent systematic errors in the GD may have a significant impact on WFIRST. The bluest filters of the WFC3/UVIS detector on HST have color-dependent residuals of a few hundredths of a pixel (see Fig. 6 of Ref. 87). In that case, it is likely that the problem is due to a chromatic effect induced by fused-silica CCD windows within the optical system, which refract blue and red photons differently and have a sharp increase of the refractive index below 4000 Å. As a consequence, the F225W, F275W, and F336W filters are the most affected. WFIRST detectors are not based on silica CCDs, but the possibility of similar color-dependent systematic effects should be taken into account.

In addition, especially for wide-band filters, the GD affecting redder and bluer photons is likely to be slightly different due to diffraction and optical distortion. A test using WFC3/UVIS observations in the F606W filter of blue-horizontal-branch and red-giant-branch stars in $\omega$ Cen showed that the measured positions of blue and red stars are off by $\sim 0.002$ UVIS pixels on average, with respect to their true positions. Filter-dependent residuals could therefore introduce small but still significant color-dependent systematic effects in wide-band WFIRST filters, including global (position-independent) effects. These filter-dependent systematics are expected to be stable over time, so their calibration should be straightforward.

4.4.

Hysteresis in Readout Electronics

WFIRST should be aware that, on at least one existing instrument with HxRG detectors, the readout electronics are affected by significant hysteresis.⁸⁸ The effect was first observed by J. Anderson in data taken by the High Acuity Wide field K-band Imager (HAWK-I) on the Very Large Telescope (VLT) while observing a standard field to help calibrate JWST and the HAWK-I detector itself, and later by Ref. 88 in all the fields observed with HAWK-I during the commissioning of the detector (see their Table 2). Figure 12 shows the results of Anderson’s investigation on the top and of Ref. 88 on the bottom. Plotted are the positional residuals of stars in the “distortion-free” master frame transformed into the raw reference system of the detector and the raw positions of the same stars in the raw reference system of the detector. The master frame was constructed iteratively using the autocalibration method to solve for the GD. In both cases, a periodic signal in the astrometric residuals is observed in the calibration data, believed to be caused by the alternating readout directions for the different amplifiers. A detailed description of the effects of hysteresis on astrometry and how to minimize them is presented in Sec. 5 of Ref. 88. For the H2RG chips of the HAWK-I detectors, the hysteresis effect produces $\pm 0.035\text{pixels}$ (or $\pm 3.7\mathrm{mas}$) of systematic error, which can be easily modeled and corrected for. The WFIRST detectors will have 32 amplifiers, and scientific full-frame images will make use of all of them. If similar hysteresis effects to those of the H2RG of the HAWK-I camera are present as well in the WFIRST detectors, they will likely be easily minimized as was done for HAWK-I.

Fig. 12

Hysteresis in the HAWK-I detectors on the VLT, precursors to those planned for WFIRST. (a) Results from an investigation by Anderson. Four horizontal strips of the detector were analyzed, as shown on the far left. The central columns show the distortion as a function of $x$ position in each strip; the vertical scale is in pixels (1 pix = 100 mas). The raw distortion is shown to the left, whereas to the right are shown the residuals after subtraction of a smooth global polynomial, revealing a high-frequency periodic signal. On the far right are close-up views of the astrometric effect. The top row shows one polynomial-subtracted stripe; while the central row shows the residuals phase-wrapped by 128 columns. The resulting step function (left center row) has a half-amplitude of 0.035 pixel; the right center row shows residuals after subtraction. The unwrapped, corrected residuals are shown in the bottom row. (b) Results from Ref. 88. The left-hand panel shows $\delta x$ residuals for each of the four HAWK-I detectors after the polynomial correction is applied. The right-hand panel shows a periodogram, with a period of 128 pixels, containing all the points plotted on the left, with the median shown as a solid red line. The dashed red lines are at 0 and $\pm 0.05$ HAWK-I pixels (about $\pm 5.3\mathrm{mas}$).

4.5.

Scheduling

A small amount of extraattention to scheduling can give great payout for astrometric measurements. As suggested by Eq. (2), the ideal scheme for measuring PMs is to space revisits to the same field as evenly as possible over the longest possible time baseline, in the interest of minimizing both random and systematic errors. Maximizing the time baseline reduces the random error while increasing the number of epochs protects against systematics, which are often time-dependent with unknown correlations. Here, we discuss a few considerations for the core science and guest observing components of the WFIRST mission.

4.6.

Jitter

The current WFIRST requirements impose a maximum jitter rms of $\le 14\mathrm{mas}$, far larger than HST’s jitter rms of 2– to 5 mas. Jitter of the size allowed by WFIRST’s requirements could have a significant impact on the shape of the PSF. Tightening this limit would clearly result in better astrometry, but would also translate into a higher cost for some of the telescope components and is thus beyond the mandate of our working group. We have thus considered how to mitigate the effect of jitter at the level set by existing requirements.

Preliminary simulations of WFIRST’s geometric-distortion corrections based on EML-like Bulge images,⁹² which make use of time-constant, spatially variable library PSF models adapted from WebbPSF for WFIRST,¹⁰¹ show that there is significant degeneracy (at the 0.02 to 0.05 pixel level) between the achievable geometric-distortion correction and pixel-phase errors in stellar positions.

One way to break the degeneracy is to spatially perturb the library PSF in each individual exposure, so as to tailor the library PSF to the particular jitter status of each image. Because of the GD, jitter-induced PSF variations are expected to affect the PSF of different WFI detectors in a different way. Using this workaround to calibrate the GD would require images with roughly 20 to 40 thousand bright and isolated sources, homogeneously distributed across the WFI FoV, in order to map local PSF variations on scales of 500 pixels or so.

Another possibility would be to exploit the enormous number of images in the EML survey to map PSF variations at different jitter rms values, thereby creating a jitter-sampled set of spatially variable model PSFs. This technique will be limited by the use of only two filters for the EML survey since it is unclear whether the models generated by these two filters will apply equally well to the rest of the filterset.

4.7.

Data Management

Both pointed and spatially scanned astrometric observations will be constrained by the data downlink rate. For the WFIRST reference mission at L2, the downlink rate is estimated to be about 1.3 TB ${\mathrm{day}}^{-1}$, barring addition of extra ground stations.¹⁰² Particularly for pointed astrometric observations, this limits the number of reads per exposure that can be downloaded. The current plan is to allow configuration of the options for averaging and saving reads, similar to JWST. For astrometry, two options are particularly important:

4.8.

High-Level Data Products and Archive

The combination of a wide FoV, the plentiful availability of reference stars with extremely accurate astrometry from Gaia, and the resolution and stability of a space-based platform makes it possible for WFIRST to achieve extremely accurate absolute astrometry: better than $100\mu \mathrm{as}$ for essentially all WFC imaging products (see Sec. 3 and Ref. 75). Any field observed more than once by the WFI is thus a potential astrometric field, providing the community with a wealth of opportunities for high-precision astrometry studies in many different domains, including many of the science cases described in Sec. 2 of this report. However, for this potential to be realized for all users throughout the mission, we strongly recommend that the data processing pipeline and the archive incorporate from the outset the necessary elements to obtain, propagate, and maintain in practice the astrometric accuracy that the mission characteristics make possible in principle.

Many of the features that make WFIRST an excellent astrometric instrument, notably the requirements for excellent PSF modeling and thermal stability, are already addressed by the core science programs (the HLS and EML surveys). These requirements are summarized in Appendix A. Given their use of repeated visits to the same fields, both of these programs also offer an opportunity to produce excellent astrometry for all observed stars in their footprints with little extra analysis. We recommend that derived PMs should be provided as part of the object catalogs for both surveys once multiple epochs have been observed. In the HLS particularly, a further cross-match of stars to the LSST catalog would extend the LSST survey into the IR regime for the region covered by HLS, identify variable stars that can be used as standard candles (particularly for RR Lyrae stars, as the period-luminosity relation is much tighter in the IR) and allow for cross-validation of PMs.

Specifically, we recommend that the mission considers the possibility of achieving the following goals:

These recommendations are based on part on our experience with HST. The astrometric accuracy available for HST data early in the mission was originally limited by the quality of the GS positions; therefore, modest effort was placed into improving other components of the astrometric fidelity, such as the knowledge and time evolution of the relative positions of instruments and guiders. Now that substantially better positional accuracy is available for guide and reference stars, retrofitting the HST pipeline and archive to improve the final absolute astrometric accuracy of HST-processed data has proven complex and resource-intensive. We recognize that several of these recommendations go beyond the current science requirements and may exceed the baseline capabilities of the mission, as currently planned. However, incorporating these considerations, to the extent possible, into the design of the WFIRST data processing and archive systems will greatly improve the quality and accessibility of mission data for astrometry, improving science outcomes in this area and ultimately reducing total development costs when compared with adding similar capabilities at a later time.

5.1.

A.1 Astrometry with the High-Latitude Survey

The current requirements related to astrometry, as of July 2017, include the following:

The reference dither pattern for the HLS is a set of 3 to 4 dithers of a size intended to cover the chip gaps, repeated to tile the field and in each of four filters: Y, J, H, and F184. A second pass over each field follows 6 months later at a different roll angle.

The projected, approximate bright and faint point-source limits of the HLS are summarized in Table 4. The faint limits are as stated in the requirements above. The bright limits were estimated based on the statement in the reference mission that pixels in the HLS will be read nondestructively every 5.4 s, and assuming that any pixels that saturate before the fourth such read will be hard-saturated (see the discussion in Ref. 75). Using the GS ETC, assuming 25% of light in the central pixel for all filters and 65,000 electrons as the saturation level, the values listed in the table give the approximate bright limit (probably accurate to within 0.3 to 0.4 magnitudes).

Table 4

Bright and faint point-source limits for the WFIRST HLS (AB magnitudes), based on the requirements described in Sec. 5.1 A.1.

5.2.

A.2 Astrometry with the Exoplanet MicroLensing Survey

The current astrometry-related requirements being discussed (as of June 29, 2017) for the EML survey are:

Currently, both narrow (two-pixel wide) and large (10-in. wide) possibilities for dithering are being explored for this core project. From the perspective of astrometric calibration, large dithers are crucial to accurately measure skew and kurtosis in the wings of the PSF.

Parallaxes and PMs over the bulge field are part of the mission of this program to characterize the masses of the star-planet pairs that will be discovered. The survey is therefore requesting the first two (spring/fall) and last two bulge observing seasons over the full time-baseline of the mission. This is also optimal for general astrometry in the bulge fields but may pose problems for understanding long-term variations in the PSF (on timescales of a year or so) prior to the end of the mission.