#  Relative astrometry in an annular field

M. Gai  INAF - Osservatorio Astrofisico di Torino – I-10025 Pino Torinese
(TO), Italy  A. Vecchiato  INAF - Osservatorio Astrofisico di Torino – I-10025
Pino Torinese (TO), Italy  A. Riva  INAF - Osservatorio Astrofisico di Torino
– I-10025 Pino Torinese (TO), Italy  A. G. Butkevich  INAF - Osservatorio
Astrofisico di Torino – I-10025 Pino Torinese (TO), Italy  Pulkovo
Observatory, Russian Academy of Sciences – Saint Petersburg 196140, Russia  D.
Busonero  INAF - Osservatorio Astrofisico di Torino – I-10025 Pino Torinese
(TO), Italy  Z. Qi  Shanghai Astronomical Observatory, Chinese Academy of
Sciences – Shanghai 200030, China  M. G. Lattanzi  INAF - Osservatorio
Astrofisico di Torino – I-10025 Pino Torinese (TO), Italy

(October 2021)

###### Abstract

Background. Relative astrometry at or below the micro-arcsec level with a
$1\,m$ class space telescope has been repeatedly proposed as a tool for exo-
planet detection and characterization, as well as for several topics at the
forefront of Astrophysics and Fundamental Physics. Aim. This paper
investigates the potential benefits of an instrument concept based on an
annular field of view, as compared to a traditional focal plane imaging a
contiguous area close to the telescope optical axis. Method. Basic aspects of
relative astrometry are reviewed as a function of the distribution on the sky
of reference stars brighter than $G=12\,mag$ (from Gaia EDR3). Statistics of
field stars for targets down to $G=8\,mag$ is evaluated by analysis and
simulation. Results. Observation efficiency benefits from prior knowledge on
individual targets, since source model is improved with few measurements.
Dedicated observations (10-20 hours) can constrain the orbital inclination of
exoplanets to a few degrees. Observing strategy can be tailored to include a
sample of stars, materialising the reference frame, sufficiently large to
average down the residual catalogue errors to the desired micro-arcsec level.
For most targets, the annular field provides typically more reference stars,
by a factor four to seven in our case, than the conventional field. The
brightest reference stars for each target are up to $2\,mag$ brighter.
Conclusions. The proposed annular field telescope concept improves on
observation flexibility and/or astrometric performance with respect to
conventional designs. It appears therefore as an appealing contribution to
optimization of future relative astrometry missions.

Astronomical instrumentation (799) — Space astrometry (1541) — Astrometric
exoplanet detection (2130) — Space vehicles instruments (1548)

##  1  Introduction

Astrometry from space bears the promise of achieving substantially photon- and
diffraction-limited precision to measurement of celestial object positions
(and related quantities: motion, parallax, separation, …). Moreover, such
precision may be achieved on either narrow or large angular separation among
targets, providing the tools to achieve full-sky (global or absolute)
measurements. The most advanced large scale implementation of such ideas is
currently Gaia (Prusti et al., 2016), which has recently delivered its Early
Data Release 3 (EDR3) (Gaia Collaboration et al., 2021), with remarkable
astrometric quality (Lindegren et al., 2021), expected to further improve in
future releases.

However, as any scientific progress is a stepping stone toward the next
challenge, a number of other space missions have been, and will be, proposed
to further advance our capability of understanding the Universe by better
measurements. Some of them are briefly recalled below.

Photon-limited precision at the micro-arcsec (hereafter, $\mu as$) level, or
better, translates into comparable accuracy only at the expense of significant
efforts in terms of instrument design and calibration efforts, which also
affects operation. Calibration of the optical response is obviously alleviated
in case of a nearly ideal instrument, with small deviation from the
diffraction limited Point Spread Function (PSF). Advanced metrology concepts,
improving on the knowledge of instrument response, have also been investigated
(Zhai et al., 2011).

The science case for high precision, narrow angle astrometry has been
presented e.g. in the context of the proposed space mission Theia (Malbet,
2021). Exo-planetary science missions based on astrometry have also been
investigated in other approaches (Shao et al., 2019; Bendek et al., 2018;
Tuthill et al., 2018). Also observatory class missions like the Roman Space
Telescope (Croft, 2021) bear the potential for impressive results, in that
case with lower angular precision, but on much fainter extra-galactic objects.
We retain as reference most of the goals of the above concepts, but we
introduce a significant modification in the instrument and operation
implementation concept, which we expect to be beneficial with respect to
calibration and control of systematic errors related to variation of the
telescope optical response.

The recently proposed idea (Riva et al., 2020) of a Ring Astrometric Field
Telescope for Exoplanets and Relativity (RAFTER), reviewed in Sec. 3, is
characterised by a field of view, and related detection system, deployed over
a circular strip centered around the projection of the optical axis. Also,
circular symmetry is preserved throughout the optical system, thus ensuring
circular symmetry of the instrument response, as described in Sec. 3.1.

The annular field of RAFTER (radius $\theta$, width $\delta\theta$) allows
simultaneous observation of source pairs at a significantly larger angular
separation (up to $2\theta$) than in case of a conventional round geometry
detector, which concentrates the same area $2\pi\theta\,\delta\theta$ within
radius $\sim\sqrt{2\theta\,\delta\theta}$. The principle is depicted in Fig.
2, with $\theta=1^{\circ}$ and $\delta\theta<<\theta$: the target star $T$ is
located at the centre, and four reference stars ($R1$, $R2$, $R3$ and $R4$)
are placed at extreme positions on either axes. The target and any object in
the whole shaded area (dashed circle) can be observed simultaneously, by
setting the telescope optical axis in suitable points on the dotted circle,
i.e. pivoting the field of view around the target. For example, a set of three
targets, labelled $A$, $B$ and $C$, as shown in Fig. 2, can be observed
pairwise by application of small pointing offsets among subsequent positions
$C_{A}$, $C_{B}$ and $C_{C}$ of the optical axis.

Such instrument concept allows selection of the actual field of view close to
a given target with some liberty, according to performance optimisation
criteria. In particular, we may select a region around the science target
including the brightest accessible reference star; other ad hoc criteria may
be adopted when convenient, e.g. maximising the number of field stars. Some
such options are investigated in our study and detailed in Sec. 5.1.

This framework appears to be efficient toward implementation of robust narrow
angle astrometry, exploiting available references (the Gaia catalogue) and/or
strengthening it with additional observations to improve on individual source
parameters.

Figure 1: Field ($2^{\circ}$ radius, shaded) accessible to the target star $T$
(center) by pointing the telescope along the dotted circle ($1^{\circ}$
radius): reference stars $R1$, $R2$, $R3$ and $R4$ are shown at $\pm
2^{\circ}$ on either coordinate.

Figure 2: Example of pairwise observation of three targets $A$, $B$ and $C$,
using three pointing offset positions $C_{A}$, $C_{B}$ and $C_{C}$ of the
optical axis.

In Sec. 2 we briefly review some of the compelling science topics supporting a
high precision relative astrometry mission; in Sec. 3 the RAFTER instrument
concept is recalled and some relevant aspects of operation with an annular
field are outlined. In Sec. 4 the practical case of relative astrometry on a
specific exoplanetary system is expounded in some detail. In Sec. 5 we derive
the main statistical implications of annular field observation around the
brightest stars in the sky. Finally, in Sec. 6, we draw our conclusions.

##  2 Overview of the science case

As shown in Sect. 3, RAFTER is a suggested technical implementation of a
relative astrometry mission similar e.g. to Theia. The science cases of these
two missions, thus, share a lot of similarities, differences emerging
especially in consideration of the different observing strategies allowed by
the annular field of view of RAFTER vs. the concentrated one of Theia.

Limiting ourselves to a short review of the possible scientific applications
of this mission concept, RAFTER has the possibility to contribute to the
detection and orbit characterization of Earth- to Superearth-sized exoplanets,
as well as to several General Relativity and Cosmology related issues. For
example, high precision astrometry on selected stellar samples will be able to
probe the distribution (shape, radial profile, lumpiness) of the Dark Matter
(DM) halo of the Milky Way, and possibly M31, by determination of the dynamics
of dwarf spheroidal galaxies (dSph), globular clusters (GC) and halo streams
within the Local Group as mapped by a significant fraction of their brightest
stars, and precise masses, distances and proper motions to binary stars with
black hole (BH) and neutron star (NS) companions will also be obtained from
$\mu as$ level measurement of their orbits. This includes the prototype of
this class, Cygnus X-1 (Miller-Jones et al., 2021).

Another potential contribution is in the field of Solar System dynamics. In
the Solar System there exist $\sim 100$ objects whose maximum visual magnitude
is $V\leq 10$ or brighter, for which RAFTER can provide Micro- and sub-micro-
arcsecond astrometry. Most of them are asteroids of the main belt, whose
ephemerides can be improved by these measurements, at least for the vast
majority of them that was not visited by a dedicated space probe, with
potential implications in the field of Solar System dynamics. Moreover, high-
precision determination of the orbits of the Galilean satellites can be
combined with data from the Juno or the planned ESA JUICE mission to obtain
improved constraints on the internal structure of Jupiter.

More details on a few selected aspects of the scientific case are given in the
following subsections.

###  2.1  Exoplanets: State of the Art

The current exoplanet count is at level 5000. Specifically, as of 2021
November 1st, NASA Exoplanet Archive (exoplanetarchive.ipac.caltech.edu) lists
4566 confirmed planets (along with 4663 TESS transit candidates), while the
less restrictive Extrasolar Planets Encyclopaedia (exoplanet.eu) lists 4868
confirmed in 3597 systems. Most such planets have been detected through
photometric transit and radial velocity measurements ($\sim 71$% and 20%,
respectively), while the others have been discovered using astrometry,
transits timing, microlensing and other methods. Perryman (2018) offers
comprehensive discussion of various techniques employed to detect exoplanets.

The systems with confirmed planets evidence a rich diversity of objects. For
example, planetary masses range over six orders of magnitude, extending from a
70 $M_{\mathrm{J}}$ brown dwarf (Casewell et al., 2020) down to a 0.02
$M_{\oplus}$ super Mercury (Rappaport et al., 2012). A remark concerning
terminology is due: although brown dwarfs are classified as substellar
objects, we do not differentiate between them and planets for brevity. This
should raise no ambiguity in the text.

The current exoplanetary census contains 43 confirmed Earth-like planets, i.e.
with masses ranging from 0.6 to 2.3 $M_{\oplus}$. Most of them (36) have been
detected from primary transit searches, while microlensing observations and
radial velocity surveys revealed five and one planet, respectively. The
discovery of the last planet in the list, KOI-55 c, is remarkable: its
existence was inferred from the brightness pulsation of its host star
(Charpinet et al., 2011).

The large number of detected exoplanets encourage the belief that many, if not
all, stars host planetary systems. The major detection techniques, i.e.
photometric transit and radial velocity, however, suffer from a strong
selection effect. Transits are observable only in case of edge-on systems,
when our line-of-sight is almost parallel to the orbital plane. The
probability that orientation of a planet orbit is favorable for transit
detection is determined by the ratio of the stellar radius $R_{\star}$ to the
size of planet orbit $a_{\mathrm{p}}$ (Borucki & Summers, 1984; Perryman,
2018):

$p\simeq
0.005\left(\frac{R_{\star}}{R_{\oplus}}\right)\left(\frac{a_{\mathrm{p}}}{1\,\mathrm{AU}}\right)^{-1}\,.$
(1)

The geometric probability depends on neither star distance nor planet size.
For example, for a Sun twin ($R_{\star}=R_{\odot}$), the probability is 0.5%
for the Earth and 0.1% for Jupiter. For a solar type star, the probability is
above 1% if $a_{\mathrm{p}}\lesssim 0.5\,\mathrm{AU}$, while for a A0V host
star with $2.5R_{\odot}$ a 1%-probability zone extends to 0.8 AU.

Although conditions are more relaxed for radial velocity measurements,
discovery efficiency is still maximal for edge-on orbit orientation. The
variation of star velocity due to orbital motion becomes less detectable as
orbit inclination and/or companion mass decrease ($m\sin i$ effect).

Thus, only a small fraction of exoplanets can be addressed by the two most
widely used detection techniques.  Face-on orbits, when orbital plane is
nearly perpendicular to the line of sight, are mostly undetectable by either
transit or radial velocity methods. In contrast, astrometric measurements are
not subject to the above mentioned geometric limitations and can detect reflex
motion of the star in any orbit configuration.

We address astrometric exoplanet detection in more detail in Sect. 4. Like
radial velocities, astrometric measurements are more sensitive to massive
components. There is a marked difference in the way that observed effects
depend on star distance. Both eclipse depth and radial velocity amplitude are
distance-independent, though the related photometric accuracy, of course, is
affected. As a geometric effect, however, the astrometric signature scales
inversely with distance (see Eq. (6) below). This naturally favors usage of
astrometry to search for planets orbiting stars in the solar vicinity.

It is worth mentioning that each detection technique has its own advantages
and disadvantages; simultaneous exploitation of different techniques offers
the best chances for determination of various parameters of components and
host stars. The multi-planet system $\pi$ Mensæ, for which combination of
spectroscopic observations carried out at the ESO’s Very Large Telescope,
photometric transits observed by TESS, and Gaia astrometric data provided the
3D architecture of its planetary orbits (Damasso et al., 2020), gives a good
examples of such synergy.

Moreover, the combination of new and previous astrometric measurements can
provide new insight and better parameter determination on several cases, e.g.
resolving binaries with stellar or substellar companions by proper motion
anomaly (Kervella, P. et al., 2019).

###  2.2 General relativity and Cosmology-related experiments

One interesting possibility is that of searching for possible deviations from
Newtonian dynamics in near Wide Binaries (WB). This opportunity was explored
in a recent paper by Banik & Kroupa (2019) that simulated the feasibility of
detecting MOND-like astrometric signals in the orbit of Proxima Centauri
around $\alpha$ Centauri A and B. This work concluded that a successful
detection requires about $T=10$ years of observations with $f=3$ observations
per year at an accuracy of $\sigma\leq 0.33~{}\mu\mathrm{as}$, which is within
the reach of the proposed Theia mission (The Theia Collaboration et al.,
2017), and thus of the RAFTER mission as it will be explained below. This
duration can be reduced by increasing the cadence of the observations, as a
result of a $\mathrm{SNR}\propto T^{5/2}f^{1/2}$. Moreover, the annular field
of view gives the RAFTER concept a better opportunity with respect to Theia
for this specific science case. As mentioned in the cited paper, in fact,
basically this experiment aims at measuring the relative acceleration between
Proxima and $\alpha$ Centauri. In the case of Theia, whose field of view of
$0.5^{\circ}$ is smaller than the angular separation of $2.18^{\circ}$ between
these two stars, this goal can be reached only by building an accurate
reference frame with a sufficient number of stable reference stars, whereas
the $\sim 1^{\circ}$ radius of the annular field of RAFTER allows, in
principle, to measure directly such a relative acceleration.

The actual number of existing stellar-mass black holes is an unsolved question
that can potentially affect our understanding of galactic evolution in the
early universe (Wheeler & Johnson, 2011). Moreover, the hypothesis that Dark
Matter may be constituted by Primordial Black Holes (PBH) in the solar-mass
range had been thoroughly investigated in the past and rejected by several
observational constraints (see e.g. Jedamzik (2020) for a list of relevant
references). In the same paper, however, the author casts doubts on these
conclusions, arguing that these limits rely on the correctness of the PBH
binary semi-major and eccentricity distribution, and that it can evolve to be
made consistent with present data. One technique that can be used to detect
these PBH is the astrometric microlensing, and RAFTER can open up new
possibilities in this respect.

Microlensing happens when a massive object passes in between an observer and a
background source of light, aligned exactly or almost exactly with them. In
such a case, the light of the background source is deflected by the massive
objects, which behaves like a lens whose optical effects, unlike normal
lenses, are produced by its gravity pull. The background object can appear
magnified and distorted at the observer’s position.

This phenomenon stems from the well-known effect of light deflection, stating
that a mass $M$ shifts the observed position of a light source from its
nominal position on an amount given by $\alpha=4GM/(c^{2}b)$, where $b$ is the
impact parameter of the null geodesic connecting the source and the observer.

In the microlensing phenomenon, the original source is seen very close to the
deflecting object by the observer, and as a consequence it is split into two
distinct images, each presenting two main effects (Boden et al., 1998): a
_photometric_ effect in which the intensities of the two images are magnified,
and an _astrometric_ effect in which the background source is displaced from
its nominal position by a small angle.

Quantitatively, the appearance of microlensing is governed by the lens mass
and by the three distances between the source (s), the lens (l) and the
observer (o). The impact parameter can be written as function of these
quantities as $b=d_{\mathrm{so}}/(d_{\mathrm{sl}}d_{\mathrm{lo}})$, and in the
literature the angular displacement is often defined in units of the so-called
_angular Einstein radius_

$\theta_{E}=\sqrt{\frac{4GM}{c^{2}}\frac{d_{\mathrm{sl}}}{d_{\mathrm{so}}d_{\mathrm{lo}}}}=\sqrt{\frac{4GM}{c^{2}}\left(d_{\mathrm{lo}}^{-1}-d_{\mathrm{so}}^{-1}\right)},$
(2)

which represents the displacement when the three objects are perfectly
aligned.

By defining the angular separation $u$ between the lens and the source in
units of $\theta_{E}$, the intensity of each image writes

$A_{1,2}=\frac{u^{2}+2}{2u\sqrt{u^{2}+4}}\pm\frac{1}{2};$ (3)

since typically $\theta_{E}\sim 1~{}\mathrm{mas}$, for on-ground observations
the image separation is below the resolution of the observing instrument, and
the lensed source appears as a single object with intensity

$A=A_{1}+A_{2}=\frac{u^{2}+2}{u\sqrt{u^{2}+4}}.$ (4)

Similarly, for what regards the _astrometric_ effect, the background source is
displaced from its nominal position by an angle

$\delta\theta=\frac{u}{u^{2}+2}\theta_{E}.$ (5)

While microlensing events are relatively easy to detect with photometric
measurements, its astrometric displacement is very difficult to measure, at
least from the ground. On the other side, estimating the mass of the lens
exploiting photometric measurements only is extremely difficult when the lens
is a single objects, as in the case of a PBH.

Without entering into much detail, which the interested reader can find in Lu
et al. (2016), it is sufficient to note that Eq. (2) shows how the mass of the
lens can be inferred by the knowledge of $\theta_{E}$. This quantity is
directly related to the astrometric displacement, while, for single-object
lenses, its estimation with photometric measurements only requires a nearly
direct passage of the source over the lens.

The potential of microlensing for determining trigonometric parallaxes is also
worth mentioning. As recently demonstrated by Rybicki et al. (2022), an
intensive photometric follow-up of the microlensing event Gaia19bld allowed
not only to compute the lens mass but also to derive its distance with a 10%
accuracy. Operating in continuous observing mode, RAFTER can provide dense
coverage of brightness variation during a microlensing event to find lens
properties from photometric data, in addition to the astrometric effect
described by Eqs. (2) to (5). Thus, RAFTER is potentially capable of using
both photometric and astrometric distance determination methods, giving
another good example of synergy between different observational techniques.

RAFTER can help to shed light on the problem of the existence and composition
of Dark Matter also by helping to improve the sample of Hypervelocity stars
(HVS). These objects are considered a promising powerful probe to infer
information about the gravitational potential of the Milky Way in the galactic
halo, namely the Dark-Matter-dominated region of our galaxy according to the
concordance $\Lambda$CDM model.

A recently published paper (Contigiani et al., 2019) proposes a new technique
able to put stringent constraints on the gravitational potential of the Milky
Way using the mass, position, and velocity distributions of the HVSs. In this
paper the authors put their method to test under an ideal but realistic
scenario based on the predicted results of the ESA _Gaia_ astrometric mission,
showing that 200 HVSs are able to provide estimations of the Navarro-Frenk-
White (NFW) potential parameters with sub-percent uncertainties.

The accuracy of these constraints depends on the size of the HVSs sample, and
previous works of the same authors (Marchetti et al., 2017, 2018) showed that
suitable HVS candidates can be identified with neural network techniques that
depend solely on the astrometric parameters. The reliability of the
identification, however, depends on the uncertainties on parallaxes and proper
motions, a requirement that conveniently matches the RAFTER purpose of
prolonging and improving the astrometric accuracy of the _Gaia_ catalog.

In addition to the improvement on the HVSs sample, RAFTER can contribute to
the Dark Matter investigation also by observations of halo Wide Binaries.
Their small binding energy, in fact, makes them very susceptible to external
perturbations (passing stars, molecular clouds, spiral arms, large-scale
tides, and massive objects in general), which allows them to be used to study
the medium in which they are immersed. Therefore, halo wide-binaries can be
used to study fine details of the gravitational potential of that Galactic
component, placing relevant constraints on the nature of Dark matter (Yoo et
al., 2004; Quinn et al., 2010). Given their extremely long periods (and large
semi-major axis) their confirmation as a true gravitational pair requires very
high-precision relative astrometry, currently available for only relatively
small samples (see e.g., Gould & Chanamé (2004); Coronado et al. (2018); Tian
et al. (2020)). Gaia DR3 will surely provide many new candidates (usually
through common proper-motion, supplemented with high-precision ground-based
radial velocities), but an instrument like RAFTER could be used to follow-up
on them, and confirm/discard their true binary nature.

##  3  The RAFTER concept

The telescope, recently described in the literature (Riva et al., 2020), is
designed with the goal of achieving good imaging quality, and above all good
astrometric performance over a comparably large field of view. The latter
requirement is implemented through the prescription of preserving circular
symmetry at each stage of the optical system, down to and including the focal
plane. The annular field thus provide invariance of the image characteristics
along the azimuthal coordinate (at fixed angular radius from the optical
axis), and smooth variation in the radial direction. This ensures that
systematic errors related to instrument optical response are minimised, at
least along one coordinate. The Roman Space Telescope (RST) also uses part of
the annular corrected field of its optical design (Pasquale et al., 2014) in
its Wide Field Imager, whose $3\times 6$ chips are arranged in a semi-lunar
(“smile”) layout.

The proposed design achieves sub-$\mu as$ systematic error on azimuth, even
with moderate perturbations to the nominal configuration, and $<2\,mas$
distortion over the radial range $[0^{\circ}.9,1^{\circ}.1]$, easily modelled.
The diffraction limited field ($>1.25$ square deg) is partially populated by
66 $4k$ detectors on a $4^{\prime}$ wide ring ($0.26$ square deg).

###  3.1  Optical system main characteristics

The RAFTER design is derived from a classical Three Mirror Anastigmat (TMA)
(Korsch, 1977), optimised with the explicit goal of exploiting a full ring
within the annular corrected field. The telescope layout is shown in Fig. 3,
in its CCD-compatible version with effective focal length $EFL=30\,m$. The
peculiarity is that the secondary (M2) and tertiary (M3) mirrors are placed
within each other, as well as the primary (M1) and a flat folding mirror (FM).
The input pupil and all mirrors are annular, i.e. they use only a circular
region. The flat FM has the double function of feeding the optical path from
M2 to M3, and from M3 to the focal plane (FP), a ring of CCD detectors around
the telescope main tube. The system is therefore highly symmetric and very
compact: $1.7\ m$ length, $1.2\ m$ diameter. Optical design characteristics,
including indications on robustness against perturbations, have been published
(Riva et al., 2020).

Figure 3: The RAFTER telescope layout. The focal plane (FP) is deployed on a
ring surrounding the input beam (baffling not shown).

###  3.2  A strawman mission: key elements

The actual design of payload and satellite is beyond the scope of the current
study, mostly focused on the instrument concept and related operation options.
A few aspects are proposed, in order to define the framework for the
subsequent discussion.

RAFTER is assumed to be operated on a pointed satellite, aimed at comparably
long observations of selected targets to achieve high precision, rather than a
scanning mission covering the whole celestial sphere (or large fractions
thereof), and necessarily devoting little time to individual sources.

The $1\,m$ class telescope is consistent with the main payload of a medium
class mission, driving satellite and operation design. Additional payloads are
not considered herein, but such option will be an obvious concern of the
supporting agencies throughout early mission definition phases. The operating
orbit can be selected with some freedom after finalisation of the science case
priorities; GEO and L2 are obvious possibilities suggested by previous
missions. The mission lifetime is assumed to be of order of five years. Longer
duration, obviously impacting costs, is beneficial to an astrometric mission,
allowing observation of more targets and/or improved determination of their
dynamical parameters, above all for long period orbits.

At any given time, the sky area accessible will be limited by attitude
consideration, including the need to avoid the Sun on the payload, and to feed
the solar array providing the power supply. Besides, this results in a
geometric constraint on observations, so that most targets may be observed a
few times a year, e.g. in epochs grouped over about a semester.

We assume a typical observation time of order of one hour. If the fractional
time required to switch between subsequent targets is $\sim 10\%$, it may be
possible to operate order of 20 observations per day, for a total of $\sim
7,000$ observations per year. Since several sources will require two or more
observations per year, an acceptable number of individual science targets is
on the order of few thousand.

Careful definition of the science case will therefore be required to maximise
the mission output, trading off the competing requirements of large samples
and fine sampling over the mission lifetime. In practice, a few targets,
requiring higher precision and/or time resolution, may be observed several
times, and others less frequently.

###  3.3  High cadence observations

Broadband observation of bright sources with a sizeable telescope results in
fast detector saturation. A long integration, required to achieve high
precision thanks to a correspondingly high photon budget, must therefore be
split into many shorter elementary exposures, with the data co-added either on
board or on ground. This also provides a valuable high frequency information
on pointing stability, which may be taken advantage of by the on-board
attitude and orbit control system (AOCS). Multiple exposure astrometry on
bright stars is also discussed in a separate paper (Gai et al., submitted to
PASP).

Full detector readout on a short timescale would be impractical, but, since
bright sources are few and usually wide apart, we can set windows around them
for selective readout, discarding less interesting regions. On-board
detection, required for proper management of readout windows, is routinely
used on Gaia.

The detector readout is assumed to be performed on a pixel period of $5\,\mu
s$, not very challenging with respect to modern science instrumentation. Full
frame readout on single-output $4k\times 4k$ devices requires about 80 s, thus
window readout is required for all chips imaging sources brighter than
$G\simeq 15.5\,mag$. Several large windows (e.g. $44\times 44$ pixels) can be
read with source magnitude $G\gtrsim 8\,mag$; then, size and number of windows
must be reduced. A single, small window ($10\times 10$ pixels) per chip may
allow imaging for stars as bright as $G\simeq 2.5\,mag$.

###  3.4  Calibration hints

The capability of an annular field with radius $\theta$ to observe at the same
time a selected target and any source within angular distance $2\theta$ was
anticipated in the Introduction and illustrated in Fig. 2.

By rotation around the optical axis, the same pair of objects can be observed
repeatedly in different detector positions, thus contributing to both
measurement statistics and calibration. Subsequent estimates of the pair
separation can be expected to differ by amounts related to the instrument
response variation over the field, assuming that the observation is short with
respect to the natural time scale of variation of both payload and
astrophysical sources. Repeated observations on longer time scales will remove
the degeneration between correlated instrument variations, and uncorrelated
evolution of different sources.

The goal precision of RAFTER is below the error level expected on individual
sources in the final Gaia catalogue, but comparable to the intrinsic precision
of the underlying Gaia reference frame. Therefore, statistics on a
sufficiently large sample will average down the resulting collective error to
adequate levels.

Pairwise observation of several nearby objects, not simultaneously fitting
within the annular field of view, allows measurement of the target against
different reference sources and with different sampling geometry with respect
to the instrument radial and azimuthal coordinates, as in Fig. 2.

###  3.5  Relative astrometry: the local reference frame

Each target is located with respect to the available population of field
stars, materialising a local representation of the global Gaia reference
frame. Since the goal measurement precision is below the $1\,\mu as$ range, we
may wonder whether the Gaia precision is adequate to the task.

Although a complete answer for each science case should be substantiated by a
thorough, dedicated investigation, we address briefly some of the main aspects
involved. Simple conceptual considerations and simulations seem to provide
encouraging hints with respect to the feasibility of our ambitious precision
goal.

The catalogue uncertainty on reference objects affects the evaluation of any
target’s dynamics in different ways:
\- individual position errors induce an offset on the photocenter, constant
over the mission lifetime.
\- individual proper motion errors induce a continuous photocenter drift,
appearing as a contribution to the linear motion of the target.
\- individual parallax errors generate a reflex parallax term on the target.

The first term is inessential for most purposes, and the second term does not
affect the measurement of the target’s astrometric wobbling. The third
contribution is potentially more critical, since it induces an apparent
oscillation on the target’s linear motion, which may be erroneously
interpreted as the astrometric effect of an orbiting companion.

Of course, astrometric measurements are intrinsically affected by troubles in
the detection of orbital periods close to one year, because it is difficult to
disentangle them from the modulation applied by the Earth’s motion around the
Sun. The difficulty is somewhat mitigated by geometry, since the target’s
orbit will not usually be similar to the Earth’s orbit projection involved in
the parallax method; however, the superposition might make the estimate more
prone to errors. The degeneration may be reduced by a long sequence of
astrometric observations, and sometimes broken by introduction of additional
astrophysical information.

###  3.6  Cosmic noise

A concern for high precision astrometry is the intrinsic astrometric
variability of reference stars around a selected target, induced by a number
of reasons: photospheric activity, starspots, undetected companions, and so
on. While the issue is quite relevant and deserves ad hoc studies for any
specific application and target, we can remark that the above assessment on
the number of field stars provides some encouraging element.

Each source may have its own amplitude and time scale of variation; however,
they can hardly be expected to have the same direction and phase. The
mitigation of catalogue errors by averaging them over a sufficiently large
sample of field stars (as in Sec. 5.5) is a sort of “brute force” statistical
approach. This may be expected to work as well in levelling out the individual
astrometric fluctuations from the photo-center estimate used as a reference to
evaluate the target motion.

##  4  Astrometry optimization on Exoplanets

Astrometric information is used in exoplanetary studies to solve two tasks:
discovery of new objects and characterization of known systems. Both these
tasks are based on examining effect of exoplanets, or, more generally, unseen
companions, on their host stars. Detection of planets relies on analysis of
deviation of apparent path of the host from the single stars. Size of this
effect is conveniently described by the so-called astrometric signature
(Butkevich, 2018; Perryman, 2018; Ranalli et al., 2018; Sozzetti et al., 2014)

$\upsilon=3\
\mu\mathrm{as}\times\left(\frac{M_{\mathrm{p}}}{M_{\oplus}}\right)\left(\frac{M_{\star}}{M_{\odot}}\right)^{-1}\left(\frac{a_{\mathrm{p}}}{1\
\mathrm{AU}}\right)\left(\frac{d}{1\ \mathrm{pc}}\right)^{-1}\,,$ (6)

where $d$ is the distance,
$a_{\mathrm{p}}=\left(M_{\star}/M_{\mathrm{p}}\right)a_{\star}$ is the semi-
major axis of the planet orbit, $M_{\star}$ and $M_{\mathrm{p}}$ the host star
and planet mass, respectively; $a_{\star}$ is the size of the stellar orbit
around the system’s barycenter.

For Earth-type planets, astrometric signatures are at the sub-$\mu$as level.
For example, if a $1\,M_{\oplus}$ planet orbits a Solar-mass star at
$a_{\mathrm{p}}=1\ \mathrm{AU}$, the signature ranges from 0.3 to 0.03 $\mu$as
for stars in the volume between $d=10$ and 100 pc. In contrast, massive
planets have more prominent effect on their host stars. For instance, the
impact of a Jupiter ($a_{\mathrm{p}}=5\ \mathrm{AU}$) amounts to $\simeq 1$
mas for a $0.5\,M_{\odot}$ M dwarf at 10 AU. However, planets at large
$a_{\mathrm{p}}$ need relatively long measurements for reliable detection.

###  4.1  Mission related constraints

Simulations showed that astrometry is most efficient in discovering exoplanets
with orbital period not exceeding the duration of observations (Perryman et
al., 2014; Sozzetti et al., 2014). In view of the anticipated mission lifetime
of 5 yr, therefore, we restrict our analysis to planets with orbital period
$P\lesssim 5$ yr. Making use the Kepler’s third law, we obtain the
corresponding constraint on the detectable orbit size

$a_{\mathrm{p}}\lesssim a_{\mathrm{max}}\simeq 3\
\mathrm{AU}\times\left(\frac{M_{\star}}{M_{\odot}}\right)^{1/2}\,.$ (7)

This condition means that for upper main sequence host stars ($M_{\star}\simeq
10\,M\odot$) the semi-major axis of planet orbit is limited to $\lesssim 10$
AU, while it should be below $\simeq 1$ AU for lower main sequence stars
($M_{\star}\simeq 0.1\,M\odot$).

Astrometric discovery of exoplanets crucially depends on the signal-to-noise
ratio $\upsilon/\sigma$, with $\sigma$ being the astrometric accuracy. We
adopt the ordinary three-sigma rule, that is, we assume $\upsilon\geq 3\sigma$
in the following. This criterion, together with the above condition for the
maximum orbit size, enables us to establish a lower limit to the planet mass.
Rearranging Eq. (6) and applying Eq. (7), we find that

$M_{\mathrm{p}}\gtrsim M_{\mathrm{min}}\simeq M_{\oplus}\left(\frac{\sigma}{3\
\mu\mathrm{as}}\right)\left(\frac{M_{\star}}{M_{\odot}}\right)^{1/2}\left(\frac{d}{1\
\mathrm{pc}}\right)\,.$ (8)

This equation is convenient for estimation of the minimum mass of
astrometrically detectable planet in terms of accuracy, host star mass and
distance.

Our simulations (Sec. 5.4) show that a precision level of few $\sigma=0.1\
\mu$as is achievable with the RAFTER design on one hour exposures on very
bright stars ($G\lesssim 4\ mag$); the value $\sigma=0.2\ \mu$as is used
below, in our assessment of planets that can be discovered. Table 4 gives
predictions for the accuracy, including the precision level expected on nearby
reference stars materialising the reference frame. However, these estimations
are affected by operations and other error sources, which are preliminarily
addressed in Sec. 5.4 to 5.6. Therefore, the value of 0.2 $\mu$as is to be
taken as a representative estimate of foreseen astrometric performance.

Fig. 5 illustrates the minimum detectable planet mass versus distance for
three different host star masses: solar ($1\,M_{\odot}$), low-
($0.1\,M_{\odot}$) and upper- ($10\,M_{\odot}$) main sequence. This plot shows
that an Earth-mass planet can be detected up to $d=15$ pc for solar-mass
stars, while for $0.1\,M_{\odot}$ stars , e.g. M6 dwarfs, the Earth discovery
limit is at $\simeq 47$ pc. For a $1\,M_{\odot}$ host star, the lower
detectable planet mass runs from 0.7 to $7\,M_{\oplus}$ as distance goes from
10 to 100 pc. For low-mass stars, the corresponding lower limit ranges from
0.2 to $2\,M_{\oplus}$.

For massive stars, the astrometric effect from an Earth orbiting a
$10\,M_{\odot}$ star remains undetectable by RAFTER even at 10 pc. This result
lends itself to a straightforward interpretation. For such a star, the minimum
detectable orbit size from Eq. (7) is $a_{\mathrm{p}}\lesssim 9.5$ AU. The
corresponding astrometric signature of an Earth-mass planet at 10 pc is
$\simeq 0.3\ \mu$as, i.e. a detection level of $1.5\sigma$, well below the
adopted $3\sigma$-threshold. The minimum detectable planet mass runs from 2 to
$20\,M_{\oplus}$, over distances ranging from 10 to 100 pc, for a
$M=10\,M_{\odot}$ host star. Eq. (8) suggests that, at 10 pc, the astrometric
motion caused by an Earth-mass planet is detectable for $M_{\star}\lesssim
2.2\,M_{\odot}$, i.e. for A2V and later main-sequence star.

It is worth stressing that the detection limits refer to a mission lifetime of
5 yr, scaling as a function of such parameter.

Figure 4:  Minimum mass of detectable planet as a function of distance. The
estimation assumes astrometric $SNR=3$. The solid, dashed and dash-dotted
lines correspond to a solar-mass, upper- and lower-main sequence host star,
respectively.

Figure 5:  Path on the sky of HD147513, over 5 yr. The coordinates are
reckoned from the starting position at J2026.0. The red dots give an example
of observations sampled at the rate of 6 observation sets per a year. The
shaded rectangle indicates the vicinity of a crossing point shown in Fig. 6.
The area size is greatly exaggerated in this drawing.

###  4.2  Pinpointing exo-planets

We now briefly discuss how high-precision relative astrometry can be used for
characterization of known exoplanetary systems. Many exoplanets have been
discovered from radial velocity measurements. This technique can determine
only the combination $a_{\star}\sin i$, where $i$ is the orbit inclination.
Accordingly, the planet mass is uncertain by the unknown factor $\sin i$.
In contrast, astrometry is capable of determining the inclination, together
with the other Keplerian elements, and, therefore, it can provide valuable
complementary information, allowing full characterization of planetary
systems.

As an example, we consider HD147513b, a Jupiter mass planet orbiting a G3/G5V
star with $M=0.92\ M_{\odot}$ (Mayor et al., 2004). This system belongs to the
Gaia primary sources, and its astrometric data have high accuracy, in
particular the parallax relative error is $\sigma_{\varpi}/\varpi=8.5\times
10^{-4}$. The system characteristics are summarised in Table 1, deriving the
astrometric parameters and radial velocity from the Gaia EDR3 (Gaia
Collaboration et al., 2020) and DR2 (Gaia Collaboration, 2018), respectively,
while the orbital elements and the host star data are from the Extrasolar
Planets Encyclopaedia.111http://exoplanet.eu/

With parallax $\varpi=77.565\ mas$, the distance to the system is 12.9 pc.
Because of the inclination uncertainty, only the lower limit to the planet
mass is known: $M_{p}\gtrsim 1.21\,M_{\mathrm{J}}=385\,M_{\oplus}$. This, in
turn, results in an astrometric displacement due to the orbital motion at the
level of at least 130 $\mu$as.

In terms of our reference accuracy (0.2 $\mu$as), this value is associated to
an astrometric signal-to-noise ratio $\mathrm{S}/\mathrm{N}\simeq 650$ (with
maximum value of the inclination, $i=90^{\circ}$), or better with smaller
inclination. A full astrometric solution, requiring several samples per orbit,
can constrain the inclination, planetary mass and other parameters.

Table 1: Parameters of HD147513b (Gaia source ID 6018047019138644480) Parameter | Value | Uncertainty | Unit
---|---|---|---
$\alpha$ | 246.0058026652 | 0.057 | deg; mas
$\delta$ | -39.1929655698 | 0.038 | deg; mas
$\varpi$ | 77.565 | 0.066 | mas
$\mu_{\alpha*}$ | 73.748 | 0.082 | mas yr-1
$\mu_{\delta}$ | 3.367 | 0.059 | mas yr-1
$v_{r}$ | 12.889 | 0.132 | km s-1
Apparent magnitude, $V$ | 5.37 |  | mag
Mass, $M_{\mathrm{p}}\sin i$ | 1.21 |  | $M_{\mathrm{J}}$
Semi-major axis, $a_{\mathrm{p}}$ | 1.32 |  | AU
Orbital period, $P$ | 528.4 | 6.3 | day
Eccentricity, $e$ | 0.26 | 0.05 |
Argument of pericenter, $\omega$ | 282 | 9 | deg
Pericenter passage epoch, $T_{\mathrm{p}}$ | 2451672.0 | 11 | JD

Fig. 5 illustrates the apparent path of HD147513 on the celestial sphere,
computed for the period from 2026 to 2031, using the data in Table 1. The
trajectory shows a looping motion, i.e. the path intercepts itself several
times at different times. It is clear from simple geometrical considerations
that this happens when the parallax is larger than the annual proper motion,
hence the feature is common to most nearby stars. The time distribution of the
crossing events depends on the relationship between parallax and absolute
value of proper motion $\mu=(\mu_{\alpha*}^{2}+\mu_{\delta}^{2})^{-1/2}$. For
HD147513, with $\varpi=77.565$ mas and $\mu=73.825$ mas yr-1, the crossings
are separated by 0.84, 1.29 and 1.62 yr.

The crossing points are favorable for planet discovery and orbit
characterization by relative astrometry. Indeed, when a star returns to one
such region, it fits in the same local frame, just slightly deformed due to
proper motion and parallax of the reference (field) stars. Thus, repeated
transits offer better opportunities to detect and measure the peculiarities of
the target’s motion, compared to other fields crossed just once by the star.

Therefore, the observation schedule can be optimised to take advantage of
crossing conditions. The red dots in Fig. 5 exemplify a distribution of
observations with a uniform 6 point/yr cadence, suited to conventional full
astrometric solution, still approximately hitting eight of the twelve crossing
points.

The fine structure of orbits close to a crossing point is further illustrated
in Fig. 6, where the stellar path is shown for three different values of
inclination; the two visits are separated by a time elapse of 0.84 yr.
Different inclinations imply different planet masses: the mass is minimum
(1.21 $M_{\mathrm{J}}$) for an edge-on orbit with $i=90^{\circ}$, while larger
values are related to smaller inclinations: 1.40 $M_{\mathrm{J}}$ for
$60^{\circ}$, and 2.42 $M_{\mathrm{J}}$ for $30^{\circ}$. The plot shows that,
for the first passage, the green line is separated from the blue and red lines
by about 1 and $3\ \mu$as, respectively, and by some 50% less in the second
transit. This difference is due to the changing orientation of the host star
orbital position relative to the line-of-sight.

Using the same set of reference stars, differential astrometry is expected to
localize the stellar path at sub-$\mu$as precision. This, in turn, should make
it possible to estimate the inclination to order of $6^{\circ}$ ($1\sigma$) in
one crossing, because inclination affects the apparent stellar path.

This example evidences the relative astrometry capability to efficiently break
the well-known degeneracy between planet mass and inclination, with few
measurements. However, a full astrometric solution can provide all seven
Keplerian elements, provided good enough observations (Perryman, 2018). With
its very high astrometric accuracy, therefore, RAFTER has a good potential for
characterization of known planetary systems, for constraining planet masses as
well as for improvement or independent determination of three-dimensional
orbits.

Figure 6: Zoomed view of the $14\times 10\ \mu$as area marked in Fig. 5. The
moments of observation when the star crosses the same area are separated by
0.84 yr. The red, blue and green lines show the path of the host star of the
HD145513 system on the sky for inclination $i=30$, 60 and 90 deg,
respectively. The first passage is drawn with the solid lines, while the
dashed lines represent the second passage. The arrows indicate the direction
of motion. The black dots illustrate positions of the star calculated at a
fixed moment of time during the first passage. The black squares exemplify the
same for the second passage.

Table 2: Reference stars for HD147513b within $0^{\circ}.5$, down to $G=10\,mag$ Distance | G | RA | Dec
---|---|---|---
(deg) | (mag) | (deg) | (deg)
0.373 | 9.623 | 246.359 | -38.941
0.390 | 8.728 | 246.321 | -38.889
0.421 | 9.092 | 246.047 | -38.773
0.439 | 8.812 | 246.539 | -39.344

Table 3: Reference stars for HD147513b within $2^{\circ}$, down to $G=7\,mag$.  Distance | G | RA | Dec
---|---|---|---
[deg] | [mag] | [deg] | [deg]
0.714 | 6.110 | 245.136 | -39.431
0.851 | 6.621 | 245.061 | -39.630
1.630 | 5.394 | 246.132 | -37.566
1.777 | 6.408 | 245.781 | -37.425
1.882 | 6.679 | 245.916 | -41.074

##  5  Observations with an annular field

With respect to a conventional contiguous circular or rectangular field, an
annular field observes at a given time several stars at some distance from a
target (located within the focal plane ring populated by detectors), and does
not “see” simultaneously many other nearby sources (inside the annulus). Such
limitation can be overcome by a set of partially overlapped observations, as
depicted in Fig. 2. Some of the peculiar differences of annular and
conventional contiguous field observations are evaluated below, by comparison
of three cases:

  1. 1.

a circular field with diameter $0^{\circ}.5$, comparable that proposed for
Theia ($\sim 0.2$ square deg);

  2. 2.

a thin annular field with diameter $2^{\circ}$ and width $2^{\prime}$, radius
$58^{\prime}\leq\rho\leq 1^{\circ}$, with approximately the same area;

  3. 3.

a thick annular field, width $4^{\prime}$ ($56^{\prime}\leq\rho\leq
1^{\circ})$, corresponding to RAFTER, with area $\sim 0.4$ sq. deg.

With respect to the source HD147513b, described in Sec. 4.2, we have reference
stars available in either case in significantly different magnitude ranges. In
case 1, within $0^{\circ}.5$ from the target, we have four stars, down to
magnitude $G=10\,mag$, as listed in Table 3. However, in cases 2 and 3, there
are five stars within $2^{\circ}$ distance brighter than $G=7\,mag$, listed in
Table 3. The annular field makes available brighter reference stars, with a
gain of about $2\,mag$ in this case. Such kind of evaluation is performed in
our analysis to all bright targets ($G\leq 8\,mag$) in the sky, building up
some relevant statistical results.

We select a sample of bright sources, and nearby reference stars, as described
in Sec. 5.1. The main aspects considered in our analysis, based on the general
considerations anticipated in Sec. 1, are:

  1. 1.

the brightest accessible reference star (Sec. 5.2);

  2. 2.

the number of available bright reference stars (Sec. 5.3);

  3. 3.

the photon limited uncertainty on the position of the photo-center of the set
of reference stars, against which the target is located (Sec. 5.4);

  4. 4.

the uncertainty on target motion due to the limited knowledge on reference
stars, i.e. the catalogue induced errors (Sec. 5.5);

  5. 5.

the potential improvement on the catalogue data of the observed sample of
reference stars achievable by exploitation of the new astrometric measurements
(Sec. 5.6);

  6. 6.

the benefits of a larger sample, achieved by setting a slightly fainter
limiting magnitude for the reference stars (Sec. 5.7).

In our assessment, all above aspects are considered independent of each other,
for simplicity. In a practical case, observations should more appropriately be
solved for both target and field stars, thus actually improving astrometry on
all objects. Calibration is an intrinsic part of the process. The issue of
such an “holistic” approach will be further investigated in future studies.

###  5.1  Sources from Gaia EDR3

The annular field performance is evaluated by simulation, using the Gaia EDR3
catalogue for the astrometric parameters and magnitude of our star sample. The
target stars are the 62,723 bright objects down to $G=8\,mag$, whereas field
stars are limited to $G=12\,mag$ (about three million objects) for most of the
current exercise.

For each target, the field center is selected according to the criterion of
including the brightest accessible partner star compatible with each case of
focal plane geometry. The initial search is focused on a circular region with
radius corresponding to the maximum detector size, as from Fig. 2. In case 1,
such radius is $0^{\circ}.5$, whereas in cases 2 and 3 it is $2^{\circ}$, thus
providing access to a sky area 16 times larger (the ring width is mostly
irrelevant).

The field of view is then placed according to the selected target-bright
partner (BP) pair. In the central field case, the centre is placed in the mid
point of the line joining the pair, whereas in the annular cases the optical
axis is displaced sufficiently to place both stars in the annulus, i.e. at
$1^{\circ}$ radius (see Fig. 2). Further optimization of the observing region
may depend on additional science requirements (e.g. including or avoiding
specific field stars), but the subject is not addressed in the current simple
framework. The total number of accessible field stars is listed in Table 4,
together with some of the results of our analysis (described in the
following).

Table 4: Statistics of the stellar sample; data from Gaia EDR3, $G\leq 12\,mag$ | Central | Annular 1 | Annular 2
---|---|---|---
Total number of accessible field stars | 1,310,469 | 1,380,708 | 2,683,834
Bright partner median magnitude [mag] | 7.390 | 5.011 | 5.011
Bright partner RMS magnitude [mag] | 1.235 | 1.027 | 1.027
Individual bright partners | 48,667 | 9,123 | 9,123
Median number of field stars, BP pointing | 16 | 17 | 32
RMS number of field stars, BP pointing | 17.2 | 16.6 | 32.4
Median number of field stars, scan | 59.0 | 233.5 | 467
RMS number of field stars, scan | 63.2 | 233.0 | 465.9
Field cumulative precision, median [$\mu as$] | 0.224 | 0.155 | 0.143
Field cumulative precision, RMS [$\mu as$] | 0.112 | 0.106 | 0.064

###  5.2  Brightest partner matching each target

The field of view can be selected to include the target and the brightest
accessible field star, hereafter Bright Partner (BP), considered as the most
precise available astrometric reference thanks to its favourable photon
budget.

For each target, the BP magnitude is brighter in the annular field cases (2
and 3) than for the central field by about two magnitudes, as shown in Fig. 7.
The number of BPs is larger than the number of stars at a given magnitude
(left panel in figure), because several nearby targets are paired to a common
reference bright star by the field selection strategy. Conversely, the central
field case must resort to a larger number of fainter BPs in the close
neighbourhood of the target (right panel). This results in a higher sampling
of bright stars over the set of measurements, used as BP to several fainter
targets. The median and RMS values of BP magnitude are listed in Table 4.

Due to the annular fields definition, the same bright targets and BPs are
selected in both cases 2 and 3.

Figure 7: Bright partner magnitude: collective (one per target, left) and
individual (duplicates removed, right).

###  5.3  Field scan for more reference stars

An alternative observing strategy is considered, in order to evidence the
annular field flexibility, consisting in selecting different fields for each
target (by pointing offset, see Fig. 2) throughout observations, thus piling
up larger sets of reference stars. The two observing strategies will be
referred to in the following as “BP pointing” and ”scan”, respectively.

Releasing the requirement of simultaneous observation of target and BP, it is
possible to point the instrument with more flexibility within the accessible
area, thus increasing the number of observed field stars. E.g., we may assume
three observations per year over five years (nominal mission lifetime),
totalling 15 visits uniformly distributed in time, placed with a pointing
offset such that the accessible region around each target is “scanned” in a
roughly uniform way. This is beneficial e.g. in averaging down the catalogue
errors affecting the determination of each target’s motion, as discussed in
Sec. 5.5.

The histogram of the number of field stars down to $G=12\,mag$, for BP
pointing, is shown in Fig. 8 (left). Both central field and thin annular field
cases provide comparable distributions (blue and red lines), according to
their equal sky area coverage, with a median value of 16 and 17 reference
stars, respectively, whereas the thick annular field case (green line)
features a significantly larger number of field stars ($>30$), thanks to its
larger area.

Using multiple pointing offsets, 15 in our case, it is possible to accumulate
observations over more field stars, within an accessible area which is larger
for the annular than for the central field case. The resulting population of
reference stars increases in any case, but the central field reaches a median
star count $\sim 60$, against $>230$ and $>460$ respectively for the thin and
thick annular field cases. The statistics is evidenced by the histograms in
Fig. 8 (right), and some of the resulting relevant figures are listed in Table
4.

Figure 8:  Histogram of field star population within $G=12\,mag$, BP pointing
(left) and field scan (right) observations.

###  5.4  Photon limited uncertainty on reference frame

The field stars represent a materialization of the Gaia reference frame,
providing a “grid” against which the target’s motion can be evaluated. The
impact of catalogue errors will be discussed in Sec. 5.5; here we just address
the photon limited precision associated to the whole set of field stars
selected for each target. For the current exercise, we assume BP pointing,
with one hour exposure; the results from field scan are obviously expected to
be even better, depending on implementation.

The target position with respect to the set of $N$ field stars can be defined
by its angular separation to each of them, whichever their actual current
positions. The location uncertainties $\sigma_{n},\,n=1,\ldots,$N associated
to each star’s magnitude is therefore combined by weighted average to provide
the collective photo-center uncertainty $\sigma_{C}$ (target excluded):

$\frac{1}{\sigma_{C}^{2}}=\sum_{n=1}^{N}\frac{1}{\sigma_{n}^{2}}\,.$ (9)

This can be considered as the measurement precision of the reference frame in
the current observation (single epoch).

For each target, the above collective uncertainty $\sigma_{C}$ is computed,
and its statistics is considered. The distribution of the results, shown in
Fig. 9 (left), is for most targets below $1\,\mu as$ for the central field
case (blue line), and it improves progressively for the thin (red line) and
thick (green line) annular field cases. The annular fields also evidence a
more compact distribution, dominated by the higher precision of the lower
magnitude BP.

Figure 9:  Cumulative field photocenter precision (left) and individual epoch
precision, including field photocenter error (right).

The precision associated to individual observations is based on the photon
limited precision (Gai et al., 2017; Mendez et al., 2014) for unresolved
stars, diffraction limited images, and considering a $30\%$ degradation factor
from realistic instrument disturbances, for an exposure time of one hour per
epoch. This is composed with a calibration error related to the field
photocenter precision, choosing as a conservative value the median value of
the distribution (Fig. 9, left) $+3\sigma$, which includes $>98\%$ of the
cases. The resulting individual measurement uncertainty is shown in Fig. 9
(right); the solid blue line refers to the central field, the dashed red line
to the thin annular field, and the dotted green line to the thick annular
field.

Assuming other error sources are kept at bay, sub-$\mu as$ precision is
achieved for point-like sources brighter than $G\simeq 8\,mag$; the noise
floor is therefore dominated by the field star statistics, and can be improved
with observing strategy (e.g. field scan).

Figure 10:  Catalogue limited position errors, single pointing (left) and with
field scan strategy (right).

###  5.5  Catalogue residual errors

Repeated observations of our $G\leq 8\,mag$ targets, to determine their
dynamics, is affected by residual catalogue errors from the set of field stars
used as references, according to the preliminary remarks in Sec. 3.5. Such
residual errors depend therefore on observing strategy, as suggested in Sec.
5.3: using different field stars will reduce the random component. The
selected field star populations are those shown in Fig. 8, respectively for
fixed pointing (left), and for a set of 15 different pointing offsets (right).

Figure 11:  Catalogue limited proper motion errors, single pointing (left) and
with field scan strategy (right).

The statistics from our simulation is shown in Figs. 10, 11 and 12,
respectively for position, proper motion and parallax. In the left panel, the
results of repeated observation of the same field, i.e. using the same
reference stars for each target, are shown. In the right panel, the values are
related to the alternative field scan strategy, which selects different sets
of field stars within the accessible area around the target (15 different
pointing offsets throughout visits). In the latter case, the larger accessible
sky area provided by the annular field makes available many more stars than
the central field case, with a further improvement with the width of the
annular field. Since each star is supposed to be affected by a random
catalogue error, the net effect is a more effective reduction of the overall
uncertainty on the actual target position. Systematic catalogue errors are
supposed to be negligible.

Figure 12:  Catalogue limited parallax errors, single pointing (left) and with field scan strategy (right). Table 5: Residual catalogue error  | Central | Annular 1 | Annular 2
---|---|---|---
Median position, BP pointing [$\mu as$] | 6.25 | 6.06 | 4.41
RMS position, BP pointing [$\mu as$] | 2.59 | 2.57 | 1.76
Median position, scan [$\mu as$] | 3.25 | 1.64 | 1.16
RMS position, scan [$\mu as$] | 1.30 | 0.62 | 0.44
Median proper motion, BP pointing [$\mu as/yr$] | 1.25 | 1.21 | 0.88
RMS proper motion, BP pointing [$\mu as/yr$] | 0.52 | 0.51 | 0.35
Median proper motion, scan [$\mu as/yr$] | 0.65 | 0.33 | 0.23
RMS proper motion, scan [$\mu as/yr$] | 0.26 | 0.12 | 0.09
Median parallax, BP pointing [$\mu as$] | 6.25 | 6.07 | 4.41
RMS parallax, BP pointing [$\mu as$] | 2.58 | 2.57 | 1.75
Median parallax, scan [$\mu as$] | 2.60 | 1.31 | 0.93
RMS parallax, scan [$\mu as$] | 1.04 | 0.49 | 0.35

The results are in the range of a few $\mu as$ on positions and parallaxes,
and $\sim 1\mu as/yr$ on proper motion, just using a total observing time of
15 hours per target. This may be adequate to some science topic, but
insufficient for others; however, smart tuning of the observing strategy
appears able to provide impressive performance even with parsimonious usage of
limited time resources. Relevant values are summarised in Table 5.

###  5.6  Catalogue maintenance and improvement

The astrometric measurements from observations may be used to improve, up to a
point and in suitable conditions, the catalogue precision on field stars, or
at least to preserve it against natural degradation. Not only field star
errors can be averaged out effectively to get proper determination of any
target’s kynematic parameters, but in principle astrometry on field stars can
be improved as well, assuming they are observed repeatedly throughout the
mission lifetime. We again assume three observations per year over five years,
uniformly distributed. This simulation is performed on the whole sample of
Gaia objects down to $G=12\,mag$, each considered as a single star with a
simple linear motion described by the five astrometric parameters in the
catalogue. Each star is located against the reference system materialised by
the set of remaining objects, using the single epoch precision in Fig. 9
(right).

The full sample provides a realistic distribution of positions on the sky,
proper motion and parallax, suited to our immediate goal of evaluating the
potential improvement on astrometric parameters of field stars achievable with
the proposed observing strategy. In practice, depending on target and sampling
scheme selection from the main science case, the number of field star subject
to astrometric bootstrap may range, e.g. between few ten thousand to few
hundred thousand objects.

Figure 13:  Photon limited errors on reference star positions.

Figure 14:  Photon limited errors on reference star proper motions.

Figure 15:  Photon limited errors on reference star parallaxes.

The results of the simulation are shown in Figs. 15, 15 and 15, respectively
for positions, proper motions and parallaxes. The corresponding statistics
(median errors and their RMS spread) is reported in Table 6; median errors
improve from the central, through the thin annular, to the thick annular field
case. The RMS of experimental errors may be considered as an expectation on
the variability expected among different fields, depending mainly on source
density. The estimation does not include the residual catalogue (absolute)
errors, i.e. it is strictly relative astrometry.

Table 6: Astrometric catalogue bootstrap by observation of field star | Central | Annular 1 | Annular 2
---|---|---|---
Median position error [$\mu as$] | 3.25 | 1.64 | 1.16
RMS position error [$\mu as$] | 1.30 | 0.62 | 0.44
Median proper motion error [$\mu as/yr$] | 0.65 | 0.33 | 0.23
RMS proper motion error [$\mu as/yr$] | 0.26 | 0.12 | 0.09
Median parallax error [$\mu as$] | 2.60 | 1.31 | 0.93
RMS parallax error [$\mu as$] | 1.04 | 0.49 | 0.35

###  5.7  Pushing toward fainter magnitude

Most of the evaluations in previous sections have been performed on a slice of
the Gaia EDR3 catalogue limited to $G\leq 12\,mag$, for reasons of
practicality. The number of available field stars obviously increases by
setting a fainter limiting magnitude. It may be noted that fainter stars, e.g.
down to $G\leq 15\,mag$, are still quite bright for a $1\,m$ class telescope,
requiring elementary exposure time below $1\,min$, and providing an epoch
location uncertainty of $\lesssim 10\,\mu as$ in one hour. It may be expected
that, in general, random error components on the reference frame
materialization (implemented by the whole set of field stars) will improve by
setting a fainter limiting magnitude, i.e. using more stars, albeit each
affected by comparably large individual errors.

Table 7:  Number of field stars as a function of magnitude  G | Central | Thin Annular | Thick Annular
---|---|---|---
mag | Median | RMS | Median | RMS | Median | RMS
12 | 59.0 | 70.9 | 185.5 | 220.6 | 371.0 | 441.1
13 | 124.0 | 134.9 | 400.8 | 659.7 | 801.5 | 1,319.4
14 | 258.5 | 458.9 | 880.9 | 2,082,4 | 1,761.8 | 4,164.8
15 | 483.0 | 1,491.3 | 1,903.6 | 6,404.5 | 3,807.2 | 12,808.9

Figure 16:  Number of field stars as a function of the limiting magnitude, for
the central (left) and thick annular field (right).

The star density in the range $12\leq G\leq 15\,mag$ is more dependent on
Galactic latitude, compared to brighter sources, which are somewhat more
uniformly distributed. A simple assessment of average sky density has been
performed by selecting from the Gaia catalogue a sample of all sources over
the range $G\leq 12\,mag$, within a $2^{\circ}$ radius around 100 bright
objects in the range $4.2\lesssim G\lesssim 6\,mag$, distributed on rather
sparse positions on the sky.
The number of stars down to a given magnitude in this sample is shown in Fig.
16, respectively for the central (left) and thick annular (right) case. The
thin annular case, intermediate between the others, is omitted for the sake of
brevity. The typical number of reference stars increases with the limiting
magnitude, and it is significantly larger in the latter case, as expected.
Statistical values are listed in Table 7; the large spread evidenced by the
RMS is mainly due to the variation in star population with Galactic latitude.

##  6  Conclusions

The potential for relative astrometry of a $1\,m$ class space telescope
endowed with an annular field of view with $\sim 1^{\circ}$ radius is
evaluated by comparison with a conventional instrument concept with a compact
focal plane detector. Basic concepts of astrometry as a tool for crucial
parameter determination are recalled, in particular with respect to real mass
determination of exo-planetary systems. We evidence that smart observing
plans, exploiting prior knowledge on individual targets, result in efficient
measurement, e.g. pinpointing orbit inclinations, and therefore true planet
masses, with a minimum number of dedicated observations. The statistics on
reference field stars brighter than $G=12\,mag$, for targets down to
$G=8\,mag$, is evaluated on the Gaia EDR3 catalogue; the potential of
sufficiently wide sets of Gaia sources as local materialization of the
reference frame at the $\mu as$ level is discussed. The annular field provides
the capability of finding adequate reference objects to a larger distance from
the selected target than the central field, thus improving on the overall
astrometric measurement because more and/or brighter field stars are made
available. In particular, in our scenario, each target may find in the annular
field individual reference stars typically $2\,mag$ brighter than in the
central field, or many more stars (four to seven times as many in the example
described), resulting in a more reliable reference frame. Repeated observation
of selected sources (several ten thousand, to few hundred thousand) may
“refresh” their astrometric parameters, mitigating the natural catalogue
degradation with time.

##  Acknowledgments

The activity has been partially funded by a grant from the Italian Ministry of
Foreign Affairs and International Cooperation (ASTRA: Astrometric Science and
Technology Roadmap), and by the Italian Space Agency (ASI) under contract
2018-24-HH.0.

## References

  * Banik & Kroupa (2019) Banik, I., & Kroupa, P. 2019, Mon. Not. R. Astron. Soc., 487, 1653, doi: 10.1093/mnras/stz1379
  * Bendek et al. (2018) Bendek, E., Tuthill, P., Guyon, O., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10698, Proc. SPIE, 106980G, doi: 10.1117/12.2313919
  * Boden et al. (1998) Boden, A. F., Shao, M., & Van Buren, D. 1998, Astrophys. J., 502, 538, doi: 10.1086/305913
  * Borucki & Summers (1984) Borucki, W. J., & Summers, A. L. 1984, Icarus, 58, 121, doi: 10.1016/0019-1035(84)90102-7
  * Butkevich (2018) Butkevich, A. G. 2018, MNRAS, 476, 5658, doi: 10.1093/mnras/sty686
  * Casewell et al. (2020) Casewell, S. L., Belardi, C., Parsons, S. G., et al. 2020, MNRAS, 497, 3571, doi: 10.1093/mnras/staa1608
  * Charpinet et al. (2011) Charpinet, S., Fontaine, G., Brassard, P., et al. 2011, Nature, 480, 496, doi: 10.1038/nature10631
  * Contigiani et al. (2019) Contigiani, O., Rossi, E. M., & Marchetti, T. 2019, Mon. Not. R. Astron. Soc., 487, 4025, doi: 10.1093/mnras/stz1547
  * Coronado et al. (2018) Coronado, J., Rix, H.-W., & Trick, W. H. 2018, MNRAS, 481, 2970, doi: 10.1093/mnras/sty2468
  * Croft (2021) Croft, R. A. C. 2021, mnras, 501, 2688, doi: 10.1093/mnras/staa3769
  * Damasso et al. (2020) Damasso, M., Sozzetti, A., Lovis, C., & et al. 2020, Astron. Astrophys., 642, A31, doi: 10.1051/0004-6361/202038416
  * Gai et al. (2017) Gai, M., Busonero, D., & Cancelliere, R. 2017, Publ. Astron. Soc. Pac., 129, 054502, doi: 10.1088/1538-3873/aa5c9c
  * Gaia Collaboration (2018) Gaia Collaboration. 2018, VizieR Online Data Catalog, I/345
  * Gaia Collaboration et al. (2020) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2020, Astron. Astrophys., doi:10.1051/0004-6361/202039657
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., Prusti, T., & et al. 2021, aap, 649, A1, doi: 10.1051/0004-6361/202039657
  * Gould & Chanamé (2004) Gould, A., & Chanamé, J. 2004, ApJS, 150, 455, doi: 10.1086/381147
  * Jedamzik (2020) Jedamzik, K. 2020, J. Cosmology Astropart. Phys, 2020, 022, doi: 10.1088/1475-7516/2020/09/022
  * Kervella, P. et al. (2019) Kervella, P., Arenou, F., Mignard, F., & Thévenin, F. 2019, A&A, 623, A72, doi: 10.1051/0004-6361/201834371
  * Korsch (1977) Korsch, D. 1977, Appl. Opt., 16, 2074, doi: 10.1364/AO.16.002074
  * Lindegren et al. (2021) Lindegren, L., Klioner, S. A., Hernández, J., & et al. 2021, aap, 649, A2, doi: 10.1051/0004-6361/202039709
  * Lu et al. (2016) Lu, J. R., Sinukoff, E., Ofek, E. O., Udalski, A., & Kozlowski, S. 2016, Astrophys. J., 830, 41, doi: 10.3847/0004-637X/830/1/41
  * Malbet (2021) Malbet, F. 2021, Exp. Astron.
  * Marchetti et al. (2018) Marchetti, T., Contigiani, O., Rossi, E. M., et al. 2018, Mon. Not. R. Astron. Soc., 476, 4697, doi: 10.1093/mnras/sty579
  * Marchetti et al. (2017) Marchetti, T., Rossi, E. M., Kordopatis, G., et al. 2017, Mon. Not. R. Astron. Soc., 470, 1388, doi: 10.1093/mnras/stx1304
  * Mayor et al. (2004) Mayor, M., Udry, S., Naef, D., et al. 2004, Astron. Astrophys., 415, 391, doi: 10.1051/0004-6361:20034250
  * Mendez et al. (2014) Mendez, R. A., Silva, J. F., Orostica, R., & Lobos, R. 2014, Publ. Astron. Soc. Pac., 126, 798, doi: 10.1086/678041
  * Miller-Jones et al. (2021) Miller-Jones, J. C. A., Bahramian, A., Orosz, J. A., & et al. 2021, Science, 371, 1046, doi: 10.1126/science.abb3363
  * Pasquale et al. (2014) Pasquale, B., Content, D., Kruk, J., & et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9293, International Optical Design Conference 2014, ed. M. Figueiro, S. Lerner, J. Muschaweck, & J. Rogers, 929305, doi: 10.1117/12.2177847
  * Perryman (2018) Perryman, M. 2018, The Exoplanet Handbook (Cambridge Univ. Press)
  * Perryman et al. (2014) Perryman, M., Hartman, J., Bakos, G. Á., & Lindegren, L. 2014, ApJ, 797, 14
  * Prusti et al. (2016) Prusti, T., De Bruijne, J., Brown, A., et al. 2016, Astron. Astrophys., 595, A1
  * Quinn et al. (2010) Quinn, D. P., Wilkinson, M. I., Irwin, M. J., et al. 2010, in Astronomical Society of the Pacific Conference Series, Vol. 435, Binaries - Key to Comprehension of the Universe, ed. A. Prša & M. Zejda, 453
  * Ranalli et al. (2018) Ranalli, P., Hobbs, D., & Lindegren, L. 2018, Astron. Astrophys., 614, A30, doi: 10.1051/0004-6361/201730921
  * Rappaport et al. (2012) Rappaport, S., Levine, A., Chiang, E., & et al. 2012, ApJ, 752, 1, doi: 10.1088/0004-637X/752/1/1
  * Riva et al. (2020) Riva, A., Gai, M., Vecchiato, A., et al. 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11443, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 114430P, doi: 10.1117/12.2576806
  * Rybicki et al. (2022) Rybicki, K. A., Wyrzykowski, L., Bachelet, E., et al. 2022, A&A, 657, A18, doi: 10.1051/0004-6361/202039542
  * Shao et al. (2019) Shao, M., Turyshev, S. G., Zhai, C., et al. 2019, Bull. American Astron. Soc., 51, 74
  * Sozzetti et al. (2014) Sozzetti, A., Giacobbe, P., Lattanzi, M. G., et al. 2014, MNRAS, 437, 497
  * The Theia Collaboration et al. (2017) The Theia Collaboration, Boehm, C., Krone-Martins, A., et al. 2017, arXiv e-prints, arXiv:1707.01348.  https://arxiv.org/abs/1707.01348
  * Tian et al. (2020) Tian, H.-J., El-Badry, K., Rix, H.-W., & Gould, A. 2020, ApJS, 246, 4, doi: 10.3847/1538-4365/ab54c4
  * Tuthill et al. (2018) Tuthill, P., Bendek, E., Guyon, O., & et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10701, Optical and Infrared Interferometry and Imaging VI, ed. M. J. Creech-Eakman, P. G. Tuthill, & A. Mérand, 107011J, doi: 10.1117/12.2313269
  * Wheeler & Johnson (2011) Wheeler, J. C., & Johnson, V. 2011, Astrophys. J., 738, 163, doi: 10.1088/0004-637X/738/2/163
  * Yoo et al. (2004) Yoo, J., Chanamé, J., & Gould, A. 2004, ApJ, 601, 311, doi: 10.1086/380562
  * Zhai et al. (2011) Zhai, C., Shao, M., Goullioud, R., & Nemati, B. 2011, Proceedings of the Royal Society of London Series A, 467, 3550, doi: 10.1098/rspa.2011.0255