Theia: Faint objects in motion or the new astrometry frontier

Because they are DM-dominated (see Fig. 2.1, where the number of stars versus the mass-to-light ratio is displayed), dwarf Spheroidal galaxies (dSphs) are excellent laboratories to test the distribution of DM within the central part of small galaxies and disentangle the influence of complex baryonic processes from that of dark matter at these scales.

Simulations (e.g. Oñorbe et al. 2015; Read et al. 2016), show that the DM distribution (referred to as DM profile) in dSphs strongly depends on their star formation history. More specifically, these simulations find that CDM can be heated by bursty star formation inside the stellar half light radius $R_{1/2}$, if star formation proceeds for long enough. As a result, some dSphs like Fornax have formed stars for almost a Hubble time and so should have large central dark matter cores, while others, like Draco and Ursa Minor, had their star formation truncated after just $\sim 1-2$ Gyrs and should retain their steep central dark matter cusp.

Large DM cores could also be attributed however to strong self-interactions. Hence finding evidence for such cores in the faintest dSphs (which are even more DM dominated (Wolf et al., 2010) than the classical ones), would bring tremendous insights about the history of baryonic processes in these objects and could even dramatically change our understanding of the nature of dark matter. Indeed, self-interacting DM (Spergel & Steinhardt, 2000) is expected to scatter in the dense inner regions of dSphs, and thus leads to homogeneous cores. Finding such a core DM distribution in dSphs could then reveal a new type of particle forces in the dark matter sector and provide us with new directions to build extensions of Standard Model of particle physics. On the other hand, finding cuspy DM profiles in all dSphs (including the faintest ones) would confirm $\Lambda$CDM and place strong constraints on galaxy formation. As shown in Figs. 3.16 and 3.21, with its micro-arcsecond astrometric precision, Theia has the ability to determine whether the DM distribution in dSphs is cuspy or has a core and therefore bring possible very significant breakthrough regarding the nature of DM.

To determine the inner DM distribution in dSphs, one needs to remove the degeneracy between the radial DM profile and orbital anisotropy that quantifies whether stellar orbits are more radial or more tangential in the Jeans equation (Binney & Mamon, 1982). This can be done by adding the proper motions of stars in dSphs. Fig. 2.2 shows that for the Draco dSph (which was obtained using single-component spherical mock datasets from the Gaia Challenge Spherical and Triaxial Systems working group,¹¹1 http://astrowiki.ph.surrey.ac.uk/dokuwiki/doku.php?id=tests:sphtri and the number of stars expected to be observed by Theia), the inclusion of proper motions lifts the cusp/core degeneracy that line-of-sight-only kinematics cannot disentangle.

We remark in addition that Theia will be able to perform follow-ups of Gaia’s observations of dSphs streams of stars if needed. Not only will Theia provide the missing tangential velocities for stars with existing radial velocities, but it will also provide crucial membership information - and tangential velocities - for stars in the outer regions of the satellite galaxies that are tidally disrupted by the Milky Way.

For over two decades cosmological simulations have shown that Milky Way-like DM halos have triaxial shapes, with the degree of triaxiality varying with radius (e.g. Dubinski, 1994; Kazantzidis et al., 2004): halos are more round or oblate at the center, become triaxial at intermediate radii, and prolate at large radii (Zemp et al., 2012).

These departures from spherical symmetry can be tested by precise measurement of the velocity of Hyper Velocity Stars (HVS), entirely independently of any other technique attempted so far (such as the tidal streams). HVS were first discovered serendipitously (Brown et al., 2005; Hirsch et al., 2005; Edelmann et al., 2005), and later discovered in a targeted survey of blue main-sequence stars (Brown 2015 and references therein). They are located between 20 and 100 kpc from the Galactic Center and have radial velocities that significantly exceed the Galactic escape velocity.

Because these velocities exceed the plausible limit for a runaway star ejected from a binary, in which one component has undergone a supernova explosion, the primary mechanism for a star to obtain such an extreme velocity is assumed to be a three-body interaction and ejection from the deep potential well of the supermassive black hole at the Galactic center (Hills, 1988; Yu & Tremaine, 2003).

By measuring the three-dimensional velocity of these stars, we will reconstruct the triaxiality of the Galactic potential. In a spherical potential, unbound HVS ejected from the Galactic center should travel in nearly a straight line, as depicted in Fig.2.3. However, for triaxial halos, the present velocity vector should not point exactly from the Galactic Center because of the small curvature of the orbit caused by non-spherically symmetric part of the potential (Gnedin et al., 2005; Yu & Madau, 2007). While both the halo and stellar disc induce transverse motions, the effect is dominated by halo triaxiality at the typical distance of HVS. The deflection contributed by the disc peaks around 10 kpc but quickly declines at larger distances, while the deflection due to the triaxial halo continues to accumulate along the whole trajectory. Fig. 2.4 actually shows the spread of proper motion for one star, HVS5, for different halo shapes (different halo axis ratios and different orientations of the major axis).

Proper motions of several HVSs were measured with the Hubble Space Telescope (HST) by Brown et al. (2015), using an astrometric frame based on background galaxies (the FOV was too small to include any quasars). However, these measurements were not sufficiently accurate to constrain the halo shape or the origin of HVS. Theia has a sufficiently large FOV to include about 10 known quasars from the SDSS catalog around most HVSs. This will provide a much more stable and accurate astrometric frame, and will allow us to constrain the halo axis ratios to about 5$\%$.

Fig.2.5 shows indeed that with a precision of $4\,\mu$as/yr (see Sec. 3.2) we can constrain the orientation of the halo major axis and measure the axis ratios to an accuracy of $\delta(q_{Z}/q_{X})<0.05$ for the typical HVS distance of 50 kpc. For comparison, Gaia at the end of its mission would achieve only $40-150\,\mu$as/yr, which is highly insufficient to provide useful constraints on the axis ratios.

Finally, an accurate measurement of HVS velocities may lead to improved understanding of the black hole(s) at the Galactic center. Indeed, theoretical models show that HVSs will have a different spectrum of ejection velocities from a binary black hole versus a single massive black hole.

The orbits of DM particles in halos²²2For an analysis of orbital content of DM halos see Valluri et al. (2010, 2012); Bryan et al. (2012); Valluri et al. (2013). cannot be detected directly since DM particles interact only weakly with normal matter. However, in a triaxial potential such as described above, it is expected that a large fraction of the DM orbits do have any net angular momentum. Hence these particles should get arbitrarily close to the center of the cusp, regardless of how far from the center they were originally. This allows dark matter particles, which annihilate within the cusp to be replenished for a timescale $10^{4}$ longer than in a spherical halo (analogous to loss cone filling in the case of binary black holes Merritt & Poon, 2004).

Recent cosmological simulations show that the orbital distributions of halo stars are similar to those of DM particles (Valluri et al., 2013, see Fig 3.18). The orbits reflect both the accretion/formation history and the current shape of the potential because DM halos are dynamically young (i.e. they are still growing and have not attained a long term equilibrium configuration where all orbits are fully phase mixed). This opens up the very exciting possibility that one can infer the orbital properties of DM particles by assuming that they are represented by the orbits of halo stars.

By combining the high accuracy determination of the shape (see Sec.2.1.2), the radial scale length, and density normalization of the dark matter halo of the Milky Way with accurate positions and velocities for halo field stars (which are obtained for free in by targeting HVS), we estimate that it will be possible to derive the orbits of 5000-10,000 field stars and thereby to infer the orbital distribution of dark matter particles.

A central prediction of $\Lambda$CDM in contrast to many alternatives of DM (such as warm DM, e.g. Schaeffer & Silk 1984 or interacting DM, e.g. Boehm et al. 2014) is the existence of numerous $10^{6}$ to $10^{8}$ M_⊙ DM subhalos in the Milky Way halo. Their detection is extremely challenging, as they are very faint and lighter than dSphs. However, N-body simulations of the Galactic Disc show that such a DM halo passing through the Milky Way disc would warp the disc and produce a motion (bending mode), as shown in Fig. 2.6. This opens new avenues for detection as such perturbations of the disc would result in anomalous motions of the stars in the disc (e.g. Feldmann & Spolyar 2015 for recent analysis), that could give rise to an astrometric signal.

These anomalous bulk motions develop both in the solar vicinity (Widrow et al., 2012) and on larger scales (Feldmann & Spolyar, 2015), see Fig.2.7. Therefore, measuring very small proper motions of individual faint stars in different directions towards the Galactic disc could prove the existence of these subhalos and confirm the CDM scenario. Alternatively, in case they are not found, Theia’s observations would provide tantalizing evidence for alternative DM scenarios, the most popular today being a warmer form of DM particle, though these results could also indicate dark matter interactions (Boehm et al., 2014).

A field of view of $1^{\circ}\times 1^{\circ}$ in the direction of the Galactic disc has $\sim 10^{6}$ stars with an apparent magnitude of $R\leq 20$ (given by the confusion limit). Given Theia’s astrometric precision per field of view, Theia could detect up to 3 impacts on the disc from sub-halos as small as a few $10^{6}\,\mbox{M}_{\odot}$.

In the $\Lambda$CDM model, galaxies and other large-scale structures formed from tiny fluctuations in the distribution of matter in the early Universe. Inflation predicts a spectrum of primordial fluctuations in the curvature of spacetime, which directly leads to the power spectrum of initial density fluctuations. This spectrum is observed on large scales in the cosmic microwave background and the large scale structure of galaxies, but is very poorly constrained on scales smaller than 2 Mpc. This severely restricts our ability to probe the physics of the early Universe. Theia can provide a new window on these small scales by searching for astrometric microlensing events caused by ultra-compact minihalos (UCMHs) of DM.

UCMHs form shortly after matter domination (at $z\sim 1000$), in regions that are initially overdense ($\delta\rho/\rho>0.001$; Ricotti & Gould 2009). UCMHs only form from fluctuations about a factor of 100 larger than their regular cosmological counterparts, so their discovery would indicate that the primordial power spectrum is not scale invariant. This would rule out the single-field models of inflation that have dominated the theoretical landscape for the past thirty years. Conversely, the absence of UCMHs can be used to establish upper bounds on the amplitude of the primordial power spectrum on small scales (Bringmann et al., 2012), which would rule out inflationary models that predict enhanced small-scale structure (Aslanyan et al., 2016).

Like standard DM halos, UCMHs are too diffuse to be detected by regular photometric microlensing searches for MAssive Compact Halo Objects (MACHOs). Because they are far more compact than standard dark matter halos, they however produce much stronger astrometric microlensing signatures (Li et al., 2012). By searching for microlensing events due to UCMHs in the Milky Way, Theia will provide a new probe of inflation. A search for astrometric signatures of UCMHs in the Gaia dataset could constrain the amplitude of the primordial power spectrum to be less than about $10^{-5}$ on scales around 2 kpc (Li et al., 2012). Fig. 2.8 shows that with its higher astrometric precision, Theia would provide more than an order of magnitude higher sensitivity to UCMHs, and around four orders of magnitude greater mass coverage than Gaia. These projections are based on 8000 hr of observations of 10 fields in the Milky Way disc, observed three times a year, assuming that the first year of data is reserved for calibrating stellar proper motions against which to look for lensing perturbations. Fig. 2.9 shows that Theia would test the primordial spectrum of perturbations down to scales as small as 700 pc, and improve on the expected limits from Gaia by over an order of magnitude at larger scales.

The results will be independent of the DM nature, as astrometric microlensing depends on gravity only, unlike other constraints at similar scales based on dark matter annihilation, from the Fermi Gamma Ray Space Telescope (Bringmann et al., 2012). Theia’s sensitivity will be four orders of magnitude stronger than constraints from the absence of primordial black holes (PBHs), and more than an order of magnitude better than CMB spectral distortions (Chluba et al., 2012), which give the current best model-independent limit on the primordial power spectrum at similar scales.

The ultimate exoplanetary science goal is to answer the enigmatic and ancient question, “Are we alone?” via unambiguous detection of biogenic gases and molecules in the atmosphere of an Earth twin around a Sun-like star (Schwieterman et al., 2016). Directly addressing the age-old questions related to the uniqueness of the Earth as a habitat for complex biology constitutes today the vanguard of the field, and it is clearly recognized as one unprecedented, cross-technique, interdisciplinary endeavor.

Since the discovery of the first Jupiter-mass companion to a solar-type star (Mayor & Queloz, 1995), tremendous progress has been made in the field of exoplanets. Our knowledge is expanding ever so quickly due to the discovery of thousands of planets, and the skillful combination of high-sensitivity space-borne and ground-based programs that have unveiled the variety of planetary systems architectures that exist in the Galaxy (e.g., Howard 2013; Mayor et al. 2011). Preliminary estimates (e.g., Winn & Fabrycky 2015) are now also available for the occurrence rate $\eta_{\uplus}$ of terrestrial-type planets in the Habitable Zone (HZ) of stars more like the Sun ($\eta_{\uplus}\sim 10\%$) and low-mass M dwarfs ($\eta_{\uplus}\sim 50\%$).

However, transiting or Doppler-detected HZ terrestrial planet candidates (including the recent discovery of the $m_{\rm p}\sin i=1.3$ $M_{\oplus}$ HZ-planet orbiting Proxima Centauri) lack determinations of their bulk densities $\varrho_{\rm p}$. Thus, the HZ terrestrial planets known to-date are not amenable to make clear statements on their habitability. The K2, TESS, and PLATO missions are bound to provide tens of HZ Earths and Super Earths around bright M dwarfs and solar-type stars for which $\varrho_{\rm p}$ estimates might be obtained in principle, but atmospheric characterization for the latter sample might be beyond the capabilities of JWST and the Extremely Large Telescopes (ELTs). The nearest stars to the Sun are thus the most natural reservoir for the identification of potentially habitable rocky planets that might be characterized via a combination of high-dispersion spectroscopy and high-contrast imaging with the ELTs (Snellen et al., 2015) or via coronagraphic or interferometric observations in space (Leger, 2015).

Unlike the Doppler and transit methods, astrometry alone can determine reliably and precisely the true mass and three-dimensional orbital geometry of an exoplanet, which are fundamental inputs to models of planetary evolution, biosignature identification, and habitability. By determining the times, angular separation and position angle at periastron and apoastron passage, Theia’s exquisitely precise position measurements will allow the prediction of where and when a planet will be at its brightest (and even the likelihood of a transit event), thus (a) crucially helping in the optimization of direct imaging observations and (b) relaxing important model degeneracies in predictions of the planetary phase function in terms of orbit geometry, companion mass, system age, orbital phase, cloud cover, scattering mechanisms and degree of polarization (e.g., Madhusudhan & Burrows 2012). Only Theia observations have the potential to 1) discover most of the potentially habitable planets around the nearest stars to the Sun, 2) directly measure their masses and system architectures, and 3) provide the most complete target list and vastly improve the efficiency of detection of potential habitats for complex exo-life with the next generation of space telescopes and ELTs.

Our core program is focused on the use of Theia’s surgical single-measurement positional precision in pointed, differential astrometric mode ($<1\mu$as), in order to exploit the mission’s unique capability to search for the nearest Earth-like planets to the Sun. The amplitude $\alpha$ of the astrometric motion of a star due to an orbiting planet is (in micro-arcseconds):

The sample selected for our core program is comprised of 63 of the nearest A, F, G, K, and M stars (Fig. 2.10). Many of them are found in binary and multiple systems. Binary stars are compelling for Theia for a number of reasons. First, they are easier targets than single stars. For close Sun-like binaries, the magnitude of both components is lower (and sometimes much lower) than $V=9$ mag, which is the equivalent magnitude of a typical reference star field composed of 6 $V=11$ mag stars.

Furthermore, as the photon noise from the references is the dominant factor of the error budget, the accuracy for binaries increases faster with telescope staring time than around single stars. For binaries, the references only need to provide the plate scale and the reference direction of the local frame, the origin point coordinates are constrained by the secondary/primary component of the binary. Finally, when observing a binary, the astrometry on both components is obtained simultaneously: the staring time is only spent once as both components are within the same FoV. These two effects combined cause the observation of stars in binary systems to be much more efficient (as measured in $\mu$as$\times h^{-1/2}$) than that of single stars.

The binary star sample has projected separations $\theta_{\mathrm{AB}}$ in the range 0.6-100 arcsec. We have evaluated the limiting values of $\vartheta_{\mathrm{AB}}$ for which the companion does not constitute a problem for the direct detection by e.g. a 10-m space coronagraph in the visible, a near-infrared space interferometer with four arms equipped with 1-m mirrors, and an extremely large telescope observing at 2.2 $\mu$m. We find that for all configurations $\vartheta_{\mathrm{AB}}\geq 8.0\times(F_{B}/F_{A})^{1/3}$ arcsec, where $F_{A}$, $F_{B}$ are the visible fluxes of the primary (A) and secondary (B), respectively. In summary (see caption to Fig.  2.10), $62\%$ of the binary sample can have both components observed as part of the core program. For the 6 systems not fulfilling the requirement, we include the targets in one of the components of our manifold secondary program (see below).

We further stress that the complete census of small and nearby planets around solar-type stars is unique to high-precision astrometry. On the one hand, Sun-like stars have typical activity levels producing Doppler noise of $\sim 1$ m/s (or larger), which is still 10 times the signal expected from an Earth-analog (Lovis et al. 2011). We have run detailed simulations of detectability of the 9 cm/s RV semi-amplitude of the planetary signal induced by an Earth-twin around a solar analog, using 10 cm/s per-measurement precision appropriate for an instrument such as ESPRESSO, and in the presence of representative values ($1-2$ m/s) of uncorrelated and correlated RV jitter of stellar origin. We simulated 5- through 10-years observing campaigns with very intensive monitoring ($\geq 100$ observations per season), similar to the time sampling adopted by Anglada-Escudé et al. (2016) to detect the HZ terrestrial planet candidate to Proxima Centauri. We then ran different periodogram analysis algorithms (GLS, BGLS, FREDEC), and estimated the recovery rate of the signals. With a bootstrap false alarm probability $<1\%$, we found that there was $<40\%$ chance of a clear detection. On the other hand, Theia astrometry will be almost insensitive to the disturbances (spots, plages) due to stellar activity, having typical activity-induced astrometric signals with amplitude below 0.1 $\mu$as (Lagrange et al. 2011).

For the full sample of the nearest stars considered in Fig.  2.10 we achieve sensitivity (at the $6-\sigma$ level) to planets with $M_{p}\leq 3$ $M_{\oplus}$ (See section 3.6). If we consider $\eta_{\mathrm{Earth}}\sim 10\%$, for the sample of 63 stars closest to our Solar System we thus expect to detect $\sim 6$ HZ terrestrial planets. Of these, 5 would be amenable for further spectroscopic characterization of their atmospheres. Theia can perform the astrometry of the relevant stars and make a thorough census (95% completeness) of these planets by using less than 10% of a four years mission. As indicated above, this program will also be valuable for understanding planetary diversity, the architecture of planetary systems (2-d information plus Kepler’s laws, results in 3-d knowledge) including the mutual inclination of the orbits, a piece of information that is often missing in our exploration of planetary systems.

We envision a fourfold secondary program to exploit the Theia potential to elucidate other important questions in exoplanetary science. It will make use of an additional $\sim 7\%$ of mission time.

The unexpected discovery of numerous giant planets in binaries has indeed sparked a string of theoretical studies, aimed at understanding how planets can form and evolve in these environments (see Thebault & Haghighipour 2014). Of particular interest is the subsample of close binaries with separation 10-40AU, for which 5 exoplanets have been detected relatively close to the habitable zone around one stellar component (S-type orbits), but on orbits which are also close to the dynamical stability limit imposed by the companion’s perturbations (Haghighipour 2004, Thebault 2011, Satyal & Musielak 2016). In such a highly perturbed environment, these exoplanets are very difficult to form following the standard planet-formation scenario, and their very existence presents a challenge to theoretical studies (Thebault & Haghighipour 2014). Several scenarios have been proposed in recent theoretical studies to solve this apparent paradox, such as the outward migration of growing embryos (Payne et al. 2009), the accretion by sweeping of small debris (Xie et al. 2010), early orbital evolution of the binary (Thébault et al. 2009), and even more radical solutions such as a binary-specific channel for planet formation (Duchêne 2010). The contribution of Theia could be of great value for these ongoing studies, because it will survey at least 6 systems in this crucial 10-40AU separation range (11 in the 5-100AU range, see Table 2.1), with a sensitivity down to terrestrial planets in the habitable zone, which is out of reach for present observation facilities. The expected additional constraints on the occurrence rate of planets in tight binaries could prove decisive in helping to discriminate between the different planet-formation-in-binaries scenarios that have been proposed.

Another unique use of Theia is the study of non-transiting, low-mass multiple-planet systems that have already been detected with RVs. Theia astrometry will confirm or refute controversial detections, remove the $\sin i$ ambiguity and measure actual planetary masses. Furthermore, it will directly determine mutual inclination angles, which are critical to study a) the habitability of exoplanets in multiple systems, since they modify the orientation of the spin axes and hence the way the climates change across time (e.g. Laskar & Robutel 1993; Armstrong et al. 2014)) the dynamical evolution history of multiple systems, as e.g. coplanar orbits are indicative of smooth evolution, while large mutual inclinations and eccentricities point toward episodes of strong interactions, such as planet-planet scattering. Fig. 2.11 illustrates a case where degeneracy in RV can be removed by astrometry.

Gaia has the potential to detect thousands of giant planetary companions around stars of all ages (including pre- and post-main-sequence), spectral type, chemical abundance, and multiplicity (Casertano et al. 2008; Sozzetti et al. 2014; Perryman et al. 2014; Sahlmann et al. 2015). Theia will cherry-pick on Gaia discoveries and identify systems amenable to follow-up to search for additional low-mass components in such systems, particularly in the regime of stellar parameters difficult for radial velocity work (e.g., early spectral types, young ages, very low metallicity, white dwarfs). Some of the systems selected might also contain transiting companions identified by TESS and PLATO (and possibly even Gaia itself), or planets directly imaged by SPHERE or E-ELT.

To-date, among the few planetary mass objects that have been associated to brown dwarf (BD) hosts using direct imaging and microlensing techniques, only one is likely to be a low-mass planet (Udalski et al. 2015, and references therein). However, there are both observational (Scholz et al., 2008; Ricci et al., 2012, 2014) as well as theoretical (Payne & Lodato, 2007; Meru et al., 2013) reasons to believe that such systems could also be frequent around BDs. The recent identification of a trio of short-period Earth-sized planets transiting a nearby star with a mass only $\sim 10\%$ more massive than the Hydrogen-burning limit (Gillon et al., 2016) is a tantalizing element in this direction.

In its all-sky survey, Gaia will observe thousands of ultra-cool dwarfs in the backyard of the Sun with sufficient astrometric precision to reveal any orbiting companions with masses as low as that of Jupiter (Sozzetti 2014).

Theia will push detection limits of companions down to terrestrial mass. If the occurrence rate of $P\leq 1.3$ d, Earth-sized planets around BDs is $\eta=27\%$ as suggested by He et al. (2016) based on extrapolations from transit detections around late M dwarfs, the Theia measurements, probing for the first time a much larger range of separations with respect to transit surveys with sensitivity to low-mass planets, will unveil a potentially large number of such companions, and place the very first upper limits on their occurrence rates in case of null detection.

The brightest Galactic X-ray sources are accreting compact objects in binary systems. Precise optical astrometry of these X-ray binaries provides a unique opportunity to obtain quantities which are very difficult to obtain otherwise. In particular, it is possible to determine the distances to the systems via parallax measurements and the masses of the compact objects by detecting orbital motion to measure the binary inclination and the mass function. With Theia, distance measurements are feasible for $>$50 X-ray binaries (in 2000h), and orbital measurements will be obtained for dozens of systems. This will revolutionize the studies of X-ray binaries in several ways, and here, we discuss goals for neutron stars (NSs), including constraining their equation of state (EoS), and for black holes (BHs).

Matter in the NS interior is compressed to densities exceeding those in the center of atomic nuclei, opening the possibility to probe the nature of the strong interaction under conditions dramatically different from those in terrestrial experiments and to determine the NS composition. NSs might be composed of nucleons only, of strange baryons (hyperons) or mesons in the core with nucleons outside (a hybrid star), or of pure strange quark matter (a quark star). A sketch of the different possibilities is given in Fig. 2.12. Via the equation of state (EoS), matter properties determine the star’s radius for a given mass. In particular, since general relativity limits the mass for a given EoS, the observation of a massive NS can exclude EoS models. Presently, the main constraint stems from the measurements of two very massive NSs in radio pulsar/white dwarf systems which have been reported with high precision (Demorest et al., 2010; Antoniadis et al., 2013; Fonseca et al., 2016).

The key to constraining the NS EoS is to measure the masses and radii of NSs. While masses have been measured for a number of X-ray binary and radio pulsar binary systems (e.g., Lattimer, 2012; Özel & Freire, 2016), the errors on the mass measurements for most X-ray binaries are large (see Fig. 2.13, left). The ultimate constraint on the EoS would be a determination of radius and mass of the same object, and a small number of such objects might be sufficient to pin down the entire EoS (e.g., Özel & Psaltis, 2009), see Fig. 2.13 (right), where several $M$-$R$ relations for different EoSs are shown. Current techniques to determine radii rely on spectroscopic measurements of accreting neutron stars, either in quiescence (Heinke et al., 2014) or during thermonuclear (type I) X-ray bursts (Özel & Freire, 2016), and also timing observations of surface inhomogeneities of rotating NSs (Miller & Lamb, 2016; Haensel et al., 2016).

Theia will contribute by obtaining precise mass constraints with orbital measurements (Tomsick & Muterspaugh, 2010) and by improving distance measurements. Distances must be known accurately to determine the NS radii. For that purpose, Theia data can be combined with existing and future X-ray data, e.g., from Athena, which is scheduled as an ESA L2 mission. The Athena Science Working Group on the endpoints of stellar evolution has observations of quiescent neutron star X-ray binaries to determine the NS EoS as its first science goal; however, their target list is restricted to systems that are in globular clusters. Theia will enable distance measurements for many more NS X-ray binaries, allowing Athena to expand their target list.

Other techniques for constraining the NS EoS might also be possible in the future: detecting redshifted absorption lines; determining the moment of inertia of the double pulsar J0737$-$3039; and the detection of gravitational wave emission from the inspiral of a NS-NS merger. However, the mass and distance measurements that Theia would obtain use techniques that are already well-established, providing the most certain opportunity for greatly increasing the numbers of NSs with mass or radius determinations.

In addition to the goal of constraining the NS EoS, NS masses are also relevant to NS formation and binary evolution. Current evolutionary scenarios predict that the amount of matter accreted, even during long-lived X-ray binary phases, is small compared to the NS mass. This means that the NS mass distribution is mainly determined by birth masses. Determining the masses of NSs in X-ray binaries, therefore, also provides a test of current accretion models and evolutionary scenarios, including the creation of the NSs in supernovae.

BHs are, according to the theory of general relativity, remarkably simple objects. They are fully described by just two parameters, their mass and their spin. Precise masses are available for very few BHs and the recent detection of gravitational waves (Abbott et al. 2016) has demonstrated that BHs can have considerably higher masses than expected based on our understanding of stellar evolution and the fate of massive stars. Although BHs leave few clues about their origin, one more parameter that can be determined is the proper motion of BHs in X-ray binaries. Measurements of proper motions provides information about their birthplaces and formation, including whether they were produced in a supernova (or hypernova) or whether it is possible for massive stars to collapse directly to BHs. A few BH X-ray binaries have proper motion measurements (e.g., Mirabel et al., 2001), but this number will rise dramatically with the astrometry measurements that Theia will provide.

Currently, the cutting edge of research in BH X-ray binaries involves constraining BH spins, including the rate of spin and the orientation of the spin axis. Techniques for determining the rate of spin include measuring of the relativistic broadening of the fluorescent iron $K_{\alpha}$ line in the X-ray emission and the study of the thermal continuum X-ray spectra (Remillard & McClintock, 2006; Miller, 2007). Concerning the direction of their spin axes, there is evidence that the standard assumption of alignment between the BH spin and orbital angular momentum axes is incorrect in some, if not many, cases (Maccarone, 2002; Tomsick et al., 2014; Walton et al., 2016), likely requiring a warped accretion disc. Theoretical studies show that such misalignments should be common (King & Nixon, 2016). However, binary inclination measurements rely on modeling the ellipsoidal modulations seen in the optical light curves (Orosz et al., 2011), which is subject to systematic uncertainties, and Theia will be able to provide direct measurements of orbital inclination for many of the BH X-ray binaries that show evidence for misalignments and warped discs (see Sec.3.7 for targets).

About thirty years ago Bohdan Paczyński (Paczynski 1986) proposed a new method for finding compact dark objects, via photometric gravitational microlensing. This technique relies on continuous monitoring of millions of stars in order to spot its temporal brightening due to space-time curvature caused by a presence and motion of a dark massive object. Microlensing reveals itself also in astrometry, since the centre of light of both unresolved images (separated by $\sim$1 mas) changes its position while the relative brightness of the images changes in the course of the event. Astrometric time-series at sub-mas precision over course of couple of years would provide measurement of the size of the Einstein Ring, which combined with photometric light curve, would directly yield the lens distance and mass. Most microlensing events are detected by large-scale surveys, e.g., OGLE and, in future possibly also the LSST. At typical brightness of V=19-20mag only Theia would be capable at providing good-enough astrometric follow-up of photometrically detected microlensing events. Among 2000 events found every year, at least a couple should have a black hole as the lens, for which the mass measurement via astrometric microlensing would be possible with Theia.

Detection of isolated black holes and a complete census of masses of stellar remnants will for the first time allow for a robust verification of theoretical predictions of stellar evolution. Additionally, it would yield a mass distribution of lensing stars as well as hosts of planets detected via microlensing.

Theia will determine a complete census of potentially habitable terrestrial-type planets around a selected sample of the nearest stars achieving sensitivity to masses in the rocky Super-Earth regime ($1<M_{p}<5$ $M_{\oplus}$). These objects are identified as most amenable to spectroscopic follow-up with next-generation direct-imaging devices (both from the ground and in space) for identification of atmospheric bio-signatures indicative of a complex biology on the surface (Kopparapu et al. 2014; Rogers 2014). In collaboration with several groups across Europe and USA, we performed a Double Blind Test and reached the conclusion that a minimum condition to obtain detections of small planets (even in multiple planet systems), is a signal-to-noise ratio $S/N=6$. The signal $S$ is computed using the equation of the astrometric signal (see Section 2.2.2). The noise $N$ is given by the end of mission accuracy derived as $\sigma=\sigma_{0}\times[(t_{\mathrm{vis}}/1h)^{-1/2}\times(N_{\mathrm{vis}})^{-1/2}+\sigma_{\mathrm{sys}}^{2}]^{1/2}$, where $\sigma_{0}=0.98$ $\mu$as is characteristic of the instrument (with a 0.80 m primary mirror. See Sect. 6.2 and Sect. 6.4.5), $N_{\mathrm{vis}}$ is the number of visits per star, $t_{\mathrm{vis}}$ is the duration of each visit, and $\sigma_{\mathrm{sys}}=0.125$ $\mu$as is a systematic noise floor term, which includes possible unmodeled jitter from spots with rms amplitude of 0.07 $\mu$as (Lagrange et al. 2011).

Taking into account this requirement, the following program will be executed. Observations of the most suitable nearby 63 A-F-G-K-M stars in 50 individual target fields (including binary systems as described in Section 2.2.2) will be obtained with $N_{\mathrm{vis}}=50$; $t_{\mathrm{vis}}=0.8$ h, after deduction of the 30% overhead for slews between targets ($\sim 10$ deg). The total duration for such a program is $\sim$0.3 yr ($50\times 50\times 0.8=2000$ h), or 10% of the 3.5 year observing mission time minus slew overhead. Table 3.2 gives an extract of the target list where stars are ranked by increasing detectable planetary mass.

a) Planetary systems in S-type binary systems We will use Theia to survey the 16 most suitable stellar systems, with separations in the overall range 5-100 AU, with sensitivity to terrestrial planets in the HZ of each component, which is out of reach for present observational facilities. Note that this secondary program comes at no cost in terms of observing time, but it constitutes a natural and valuable byproduct of the core program described above. The sample size is large enough to investigate the impact of close-in stellar companions in the formation and evolutionary history of such systems.

b) Follow-up of known Doppler systems

A sample of $\sim 20$ bright stars ($R\leq 10$ mag) hosting Doppler-detected systems with low-mass planets will be observed at 50 epochs to determine actual masses and mutual inclination angles. We will invest 1/2 h of integration time per visit, excluding overheads ($20\times 50\times 0.5=500$ h). The sample size is large enough to allow for in-depth studies of the dynamical evolution history as well as possible habitability of such systems.

c) Planetary systems on an off the main sequence We envisage some $\sim 500$ hrs of observing time (excluding overheads) devoted to this program among 20-30 highly valuable systems with Gaia detections of intermediate-separation giant planets (transiting and not) around relatively bright ($R\leq 13$ mag) young stars, very metal-poor stars, and early-type dwarfs. For the specific case of young systems hosting large circumstellar dust or ’older’ debris discs, artifacts that might mimic planetary signals will be removed by means of follow-up measurements such as direct imaging or astrometry at radio frequencies with SKA (Kral et al. 2016).

d) Terrestrial planets around Brown Dwarfs Approximately 500 hrs of mission time (excluding overheads) will be allocated to the program of low-mass planet astrometric detection with Theia around brown dwarfs. The target sample will be constituted of $\sim 20$ benchmark early L-type dwarfs ($R\leq 18$ mag) with and without detected companions by Gaia. In absence of detection, the sample size will allow to determine a lower frequency $f_{p}$ of terrestrial-mass companions with respect to the He et al. (2016) estimates at the $\sim 5\,\sigma$ level.

Required by: Dark matter studies, compact objects, and general astrophysics observations (open time).

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  R1A Differential centroids must be precise to 10 $\mu\mbox{as}$ per epoch.

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  R1B Field of views must encompass a diameter of $0.5^{\circ}$ for reference system materialization.

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  R1C The mode must provide sensitivity to faint objects ($R>20$).

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  R1D The mode must allow measurements of up to $10^{5}$ objects per FoV.

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  R1E The mode must provide good quantum efficiency in visible wavelengths (400-900 nm).

Required by: Exoplanets and general astrophysics observations (open time).

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  R2A Differential centroids must be precise to 1 $\mu\mbox{as}$ per epoch.

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  R2B Field of views must encompass a diameter of $0.5^{\circ}$ to provide enough reference objects.

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  R2C The mode must provide sensitivity to bright objects ($R<10$).

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  R2D The mode must allow measurements of up to $10^{2}$ objects per FoV.

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  R2E The mode must provide good quantum efficiency in visible wavelengths (400-900 nm).

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  R3A The spacecraft must allow up to 20 000 re-pointings over the mission life-time depending on the final target list.

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  R3B Calibrations must allow the control of systematics to a level better than 150 nanoarcseconds per epoch for the Exoplanet observations.

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  R3C Calibrations must allow the control of systematics to a level better than 10 $\mu$as per epoch for the Dark Matter observations.

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  R5A The spacecraft and mission profile must be compatible with an Ariane 6.2 launch.

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  R5B Adopted technologies must be at $\mbox{TRL}\geq 5$ or reach that level before 2021.

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  R5C The mission must be launch-ready by 2029.

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  R5D The ESA CaC must be $\leq$€550M.

Achieving micro-arcsecond differential astrometric precision requires the control of all effects that can impact the determination of the relative positions of the point spread function. The typical apparent size of an unresolved star corresponds to 0.2 arcseconds for a 0.8 m telescope operating in visible wavelengths.

The challenge is therefore to control systematics effects to the level of 1 part per 200 000. The precision of relative position determination in the Focal Plane Array (FPA) depends on i) the photon noise, which can be either dominated by the target or by the reference stars; ii) the geometrical stability of the focal plane array, iii) the stability of the optical aberrations, iv) the variation of the detector quantum efficiency between pixels. These effects impair other missions that could perform differential astrometry measurements, like HST, Kepler, WFIRSTor Euclid, posing fundamental limits to their astrometric accuracy. All these effects are taken into account in the Theia concept.

To address the challenges outlined in section 4.1.1 and fulfill the requirements from section 3, two different possible concepts can be adopted. A NEAT-like mission consisting on a formation flight configuration (Malbet et al., 2012) or an Euclid-like mission,³³3Euclid red book: http://sci.esa.int/euclid/48983-euclid-definition-study-report-esa-sre-2011-12/ but with a single focal plane and instrument metrology subsystems. Both concepts consist in adopting a long focal length, diffraction limited, telescope and additional metrological control of the focal plane array. The proposed Theia/M5 mission concept is the result of a trade-off analysis between both concepts.

The Theia PLM concept consists on a single Three Mirror Anastigmatic (TMA) telescope with a single focal plane (see Fig. 4.23) covering a $0.5^{\circ}$ field-of-view with a mosaic of detectors. To monitor the mosaic geometry and its quantum efficiency, the PLM includes a focal plane metrology subsystem. While to monitor the telescope geometry, a dedicated telescope metrology subsystem is used.

To reach sub-microarcsecond differential astrometry a diffraction limited telescope, with all aberrations controlled, is necessary. Controlling the optical aberrations up to third order in the large Theia Field-of-View required a large exploration of different optical design concepts. A trade-off analysis was performed between different optical designs, which resulted in two optical concepts that can fulfill requirements. Both are based on a Korsch Three Mirror Anastigmatic telescope; one is an on-axis solution (Fig. 4.24) while the second is an off-axis telescope (Fig. 4.25). In both cases only three of the mirrors are powered mirrors. While the on-axis solution adopts a single folding mirror, the off-axis solution adopts two folding mirrors. For reasons discussed in Section 4.2.1, we choose the on-axis design as our baseline.

To achieve the precision by centroiding as many stars as possible, a mosaic of CCD or CMOS detectors will be assembled on the focal plane. The detectors must feature small pixels ($\sim 10\mu$m) and well controlled systematic errors. Detailed in orbit calibration of their geometry and response will be monitored via a dedicated laser metrology system.

Theia/M5 most stringent science requirement results in a centroid error calibration of $10^{-5}$ pixel. Even in the absence of optical system errors, systematics greater than $\mu$pixel are caused by the non-perfect detectors. These are caused by the non-uniform quantum efficiency and by pixel offsets. The pixel layout is not perfectly regular, and differences exists between the exact positions and a perfect regular grid structure. With CCD and CMOS detectors, non-uniform QE mitigation strategies are well known and are calibrated by flat fielding. But to reach a $\mu$pixel differential accuracy the pixel offsets and intra-pixel non-uniformity also have to be calibrated.

To monitor such distortions of the focal plane array, and to allow the associated systematic errors to be corrected, Theia/M5 relies on metrology laser feed optical fibers placed at the back of the nearest mirror to the detectors. The fibers illuminate the focal plane and form Young’s fringes that are observed simultaneously by all FPA detectors (Fig. 4.27). These fringes allow to solve the XY position of each detector and pixel. To measure the QE (inter-pixel, and intra-pixel), the light beams have their phase modulated by optical modulators. The arrays are read at 50 Hz providing many frames yielding high accuracy. By measuring the fringes at the sub-nanometer level using the information from all the pixels, it is possible to determine the QE map and solve the position of reference stars compared to the central target with a differential accuracy of $\sim$1 $\mu\mbox{as}$ or better per hour.

In addition to measuring the FPA physical shape, the rest of the telescope needs monitoring to control time-variable aberrations at sub $\mu$as level. Even at very stable environments such as L2 the telescope geometry is expected to vary for different reasons: structural lattice reorganization (as the micro-clanks observed in ESA/Gaia), outgassing and most importantly, thermo-elastic effects due to the necessary variation of the Solar Aspect Angle during the mission for pointings to the different science targets. The telescope metrology subsystem is based on a concept of linear displacement interferometers installed between the telescope mirrors, with the role to monitor perturbations to the telescope geometry.

The demanding image requirements for the point spread function and the relative centroid motion due to temporal variations on the optics positions and shapes are the driving design parameters. An extensive trade-off analysis was performed with a series of optical designs, resulting in two concepts. As discussed earlier, the best solutions adopt a three powered mirrors, one configuration on-axis, and another off-axis.

The on-axis design (Fig. 4.24) is chosen as the baseline for the Theia/M5 proposal. The reasons are that the off-axis configuration does not achieve the same aberration control and requires additional folding mirrors with respect to the on-axis, thus adding complexity to the mechanics and the metrological controls.

A detailed study of the astrometric sensitivity of the optical design to thermo-elastic effects is under way, and it is considering both designs. Results for the on-axis design show that it is possible to model and correct astrometric displacements on the entire FoV and keep the median residuals under $\sim 125$ nas, defining the level of systematics control.

The baseline solution (Fig. 4.24) adopts mirrors operating on-axis and a fold mirror (M3). The fold mirror has a central hole to allow the on-axis light go through. The fold mirror is located at the exit pupil near the Cassegrain focus at the same time. The pupil is re-imaged on the same fold mirror by the M4 mirror, and it goes through it again to be imaged on the focal plane. Due to this hole, the centre of the FoV is lost. Theia scientific cases were verified and they do not required the central $0.06^{\circ}$ of the FoV. A further straylight impact assessment is required for future studies.

Static figure errors of the primary 0.8m mirror will produce centroid offsets that are mostly common-mode across the entire field of view. Differential centroid offsets are significantly smaller than the field dependent coma and are negligible. Similarly, changes in the primary mirror surface error produce mostly common-mode centroid shifts and negligible differential centroid offsets. Static figure errors of the other mirrors will produce static biases on the centroids. However, temporal variation of the shape of the mirrors after the primary would produce non-common centroid shifts. As $\lambda/200$ temporal variations of the figure would produce sub-$\mu$as shifts, a monitoring of wavefront variations better than $\lambda/1000$ is required to ensure control of errors caused by secular mirror-deformations and optimal focusing of the telescope.

To ensure optimal focusing of the optics, the Theia concept adopts a 5 degrees-of-freedom, Gaia-like, mechanism at the secondary mirror (Urgoiti et al., 2005; Compostizo et al., 2011) to enable sub-micrometer repositioning after launch. And an active control of the temperature of the telescope is proposed (Sect. 5.2.2) by Thales Alenia Space.

The mirrors can adopt low temperature optimized Zerodur (Jedamzik & Westerhoff, 2014) or ULE, using light-weighting on M1 (Krödel & Devilliers, 2009; Krödel et al., 2010, 2014). To minimize thermo-elastic impacts in the position of the mirrors, SiC, Si₃N₄ (CTE at $10^{-6}$K^-1) or CFRP based materials ($10^{-8}$K^-1 was reached for Ti+CFRP for LISA, e.g. Verlaan et al., 2012) could be adopted for the telescope structure. Aluminum coatings can be adopted. The estimated total telescope mass considering Zerodur and SiC is 226.7 kg (272.0 kg with 20% margins).

The Theia science cases require a FoV of $\sim 0.5^{\circ}$. Our concept is based on filling the focal plane array with 24 detectors arranged in a circular geometry (Fig. 4.26). Each detector comprises at least $4096$(H)$\times 4096$(V) pixels of $\sim 10$ $\mu\mbox{m}$. At the border of the FPA, four Shack-Hartmann wave front sensors sensitive to optical path differences of $\lambda/1000$ are placed – performances are similar to Gaia WFS (Vosteen et al., 2009). They can operate as a trigger by verifying that there is no variation of the shape of the optical surfaces before an observation starts, as an additional source of calibration information, and they enable a fully-deterministic and optimal focusing of the Theia telescope along the mission. To minimize straylight by bright objects, the Theia WFS telescopes could be treated with highly absorbing coatings such as NPL Super Black (Brown et al., 2002) or Surrey Nanosystems Vantablack (Theocharous et al., 2014).

Two detector technologies can be adopted: CCDs or CMOSes. CMOS detectors present a high QE over a larger visible spectral band, that can also reach infrared wavelengths depending on the sensitive layer, and programmable readout modes, faster readout, lower power, better radiation hardness, and the ability to put specialized processing within each pixel. On the other hand, there are many known CMOS detector systematics, even for advanced detectors as the Teledyne H4RG10. The most challenging effects are: fluence-dependent PSF, correlated read noise, inhomogeneity in electric field lines and persistence effects (e.g. Simms, 2009). For the Theia/M5 proposal we select CCDs due to their more mature status, including in astrometric missions such as ESA/Gaia. CCDs also have systematic effects but they are much better understood and have mitigation strategies in place (eg. charge transfer inefficiency, (e.g. Prod’homme et al., 2012; Short et al., 2013; Massey et al., 2014).

In our baseline design we select CCDs, like the e2v CCD273-84 detectors originally developed for the Euclid/VIS instrument (Short et al., 2014), with a minor modification to its pixel size, from 12 $\mu\mbox{m}$ to 10 $\mu\mbox{m}$. Nevertheless, depending on the level of characterization of the CMOS detectors that are being developed for Euclid, JWSTand WFIRST, and on results from ESA TRP activities with European companies (as e2v, Leonardo and Selex) to further develop and characterize this type of technology, Theia’s FPA could employ CMOSes.

The typical characteristics of detectors required for the Theia/M5 Camera can be seen in Tab.4.4 indicating that complete feasibility of the focal plane data can be achieved with very high TRL device as the e2v CCD273-84.

The main Theia science cases have a wide dynamic range. As CCDs would not be able to readout the exoplanet target stars at a high enough rate, there are two possible solutions for a CCD-based FPA. Either two of the detectors in the focal plane array can be replaced by CMOS detectors, and thus individual windows could be read at high enough rates, or two of the CCD detectors of the focal plane could be covered with a filter to prevent saturation of the brightest target stars. We consider two detectors always to provide dual redundancy. During the exoplanet observation, the target star would always need to be located at one of such detectors. Also, to read out the CCD detectors while observing faint targets for DM related science cases, a shutter mechanism using a slow leaf like Euclid ’s design can be adopted (Glauser et al., 2010). We note that these solutions are only necessary if CCD detectors are chosen over CMOS detectors during the Phase-A studies.

The focal-plane metrology is used to monitor the position of the detectors and pixels relative to each other to conduct the astrometric measurement. It is also used to calibrate the inter- and intra-pixel response of the detector during periodic focal plane-calibration.

Three testbeds have been set up to demonstrate that this metrology concept can reach $10^{-5}$ pixel levels. Two were built in USA, at the NASA/JPL: the MCT (Micro-pixel Centroid Testbed) and the VESTA (Validation Experiment for Solar-system STaring Astrometry) (Nemati et al., 2011). The best results obtained at the JPL testbeds were $10^{-4}$ pixels after using flat field and pixel offsets corrections, measured respectively with a supercontinuum source (incoherent broadband) and a single mode stabilized laser. By averaging relative star positions over groups of detector positions (using 100 independent positions, i.e. a space of $\sim 40\times 40$ pixels for Nyquist-sampled centroids), the final error went down to $5\times 10^{-5}$ pixels.

A metrology testbed was built in Europe, in France, at IPAG/Grenoble: the NEAT-demo(Crouzier et al., 2014). A schematic of this testbed setup is shown in Fig. 4.33. It has been determined that the level of stray light inside the vacuum chamber of the testbed was too high to obtain a useful measure of pixel offsets due to the difficulty to attenuate properly coherent stray light. Nevertheless, even without a stringent control of stray light the NEAT-demo testbed reached a calibration of $6\times 10^{-5}$ the pixel size (Crouzier et al., 2016a, b).

The FPA metrology block diagram can be seen in Fig. 4.32. The system consists of the metrology source, the metrology fiber launchers and the focal plane detectors. The focal plane detectors are the detectors of the Theia/M5 camera itself, described in Sect. 4.2.2; the detectors alternatively measure the stellar signal (54 s observations) and the metrology (1 s per axis every minute). The metrology fiber launchers consist of a set of optical fibers (at least four) attached to a fiber launcher at the back of the M3 mirror.

The fiber tips are at a distance of $\sim 40$ cm from the FPA. Two single mode fibers with a numerical aperture of 0.4 and an optical power of 1 mW at the tips yield about $10^{8}\ e^{-}$/s per pixel on the detector. Because of the proximity between the fiber tips and the focal plane, the large numerical aperture of 0.4 is required. Even with this aperture, the flux will be lower at the corners of the FPA, thus if full calibration accuracy is required for the extreme positions of the FPA, more power per fiber will be required. A power of 1 mW per fiber tip allows a characterization of the center of the focal plane at the required level in a relatively short period. To reach the full calibration accuracy $10^{10}$ photons are needed, this total is obtained every 100 minutes of operations (accounting for the 1 second of metrology per axis available every minute). Using two 15 mW lasers will ensure redundancy and a flux of at least 1 mW at the fiber tips, accounting for the losses in the metrology system. With this setup, an estimate of the power required to run the Focal Plane Array metrology subsystem is 29 W (35 W with 20% margin). The total mass of the system is estimated to be 7.9 kg (9.4 kg with 20% of margin).

To monitor the distortions of the telescope geometry and to allow the associated systematic errors to be corrected up to sub-microarcsecond levels, Theia/M5 relies on a linear metrology concept. All pairs of mirrors of the telescope are virtually connected using a set of six independent interferometric baselines per mirror pair, each baseline measuring a distance variation. These baselines are organized into virtual hexapod structures (see Fig. 4.36). Accordingly, based on the independent distance determinations obtained by each baseline, the relative positions and angles between each pair of mirrors can be fully and rigidly determined, as any modification of the mirror-pair geometry impacts into all six interferometric baselines. This provides a small redundancy, considering that the mirrors are axis-symmetric in the ideal case.

The relative position variations of the mirrors must be known with a precision at the hundreds of picometers level to reach sub-microarcsecond level differential astrometric corrections in the whole Theia focal plane due to telescope geometry variations. Based on detailed analysis of Zemax simulations of perfect and thermally distorted SiC structure telescopes, to fulfill the most stringent Theia/M5 science case, and attain a soil of controlled systematics up to 125 nano-arcseconds over the FPA, the individual interferometer baselines between the pairs of mirrors must be capable of differential measurements of 50 picometers over the observation time (that can reach several minutes).

Existing space based interferometers from TNO, as the ESA/Gaia Basic Angle Monitor are already capable of reaching more precise measurements than those required by Theia/M5 – BAM can perform $\sim 1.5$ pm optical path difference measurements (Gielesen et al., 2013). A Thales telemeter developed for CNES can reach $\sim 100$ pm, and the Thales interferometer produced for the MTG (Meteosat Third Generation) satellite can reach 1 nm per measurement (Scheidel, 2011) – higher precisions can be reached by averaging over many measurements. These already existing instruments shows that Theia/M5 requirement for the telescope metrology baselines can be fulfilled.

The proposed Theia telescope metrology subsystem has three main components: the laser source, the micro-interferometers and the associated electronics. The laser source can be derived from the MTG instrument 15 mW laser, with a power increase and improved long-term stability performances. Alternatively a TRL9 TESAT YAG laser like the LISA Pathfinder source can be adopted. This laser provides 45 mW and can be locked to a cavity to provide excellent mid-term stability. For multi-year stabilization it can be locked to an Iodine gas cell, providing absolute frequency precision to better than 5 kHz. Theia PLM will have redundancy in the laser sources, and they are estimated to weigh $\sim 2$ kg each, or a total of 4 kg.

The laser sources will inject the beam into an optical switcher connected to mono-mode fibers that drive the beam to each micro-interferometer baseline. Each micro-interferometer baseline consist in the micro-interferometer optical bench and an associated retro-reflector. The retro-reflector can be a classical corner cube produced from Zerodur (e.g. Jedamzik & Westerhoff, 2014). The optical benches consist on a Zerodur bench and associated optics (e.g. Fig. 4.34). They fill a volume of $3.5\times 3.5\times(8-10)$ cm each, with a first estimated mass of 200 g. Molecular adhesion can be used to fix the optics and it could also be used to fix the bench and the retroreflectors directly to the borders of the Theia/M5 telescope mirrors. The dimensions and materials of the design can be further optimized during Phase-A studies.

There are two options for the associated detection and readout electronics: it could be shared with the FPA electronics or be developed specifically for the telescope metrology subsystem. Although implementation could be more complex, the first option provides simpler qualification and procurement processes and results in higher homogeneity and stability. Thus it is considered as the baseline. Additional FPGA based electronics (e.g. Actel RTG4) for the on-board processing of the metrology signal would fit into a mechanical box with $30\times 22\times 3.5$ cm, and has a mass estimate of $\sim 1$ kg. The electronics can be shared between the baselines.

The telescope metrology subsystem can be composed by 18 or 24 individual baselines. If only the telescope geometry is monitored, which is the minimum requirement for Theia/M5, three laser hexapods, each with six baselines, are required (see Fig. 4.36). Optionally the telescope to FPA geometry can also be monitored with an additional hexapod. A conceptual block diagram of this subsystem can be seen in Fig. 4.35, and a possible implementation concept in Fig. 4.36. Further optimization of this concept will take place during Phase 0/A studies. A small reduction of the number of baselines can be foreseen, for instance considering assurances of the TAS active thermal control and on detailed thermo-mechanical analysis of the spacecraft and payload structures.

An estimate of the power required to run all the telescope metrology subsystem lasers and electronics is $58.7$ W ($\sim 70.5$ W including 20% margins). The total mass reaches $12.4$ kg ($\sim 14.9$ kg with 20% margin).

At the heart of the Theia mission lies a careful and accurate calibration of the observations, and the use of optimized methods to extract as much science out of the data as possible. In order to achieve this goal, our team consists of experts from various astrometric and photometric (proposed) missions like Gaia, HST, SIM, and NEAT. The work is formally split up in several development units which are described below following processing order when applicable.

This unit will take care of receiving, decompressing and combining the raw measurement and ancillary data packets from the focal plane and instrumentation in general into self-contained data records, so that each of it can be easily processed afterwards. A record of all on-board events (e.g. anomalous events) and telemetry (e.g. temperatures) is constructed. Raw on-board attitude (pointing) from the satellite will also be processed here, although without any refinement yet, in order to tag the output records with not only their measurement time but also with their associated sky coordinates. The satellite position and velocity, as determined from ground-tracking data will be compiled for downstream processing.

The raw outputs from the previous stage include tags with the spacecraft pointing, position, velocity, temperature and time of observation data. Raw frames will be processed, which include correction for known detector systematics: nonlinearity, charge transfer ineficiency, in case of CCDs or nonlinearity, inter-pixel capacitance (IPC) which can include small anisotropies, afterimage (persistence) and reciprocity failure (flux-dependent nonlinearity) in case of CMOSes. Correlated read-noise would then be corrected for in the next step. The refined centroid and flux of the stars will be determined by iteratively fitting a model of the point spread function to the local images of each star. Initial broad-band photometry of the stars will also be determined during this process, although it will be refined (see Sect. 4.5.7) after metrological pixel response calibration.

After all calibrations have converged, relative astrometric parameters can be determined with the precision presented in the next section (4.5.5).

Since Theia is aimed at the sub-microarcsecond accuracy for its differential observations over a field of about 0.5 degrees, a serious fully-relativistic model is indispensable. The fact that the accuracy is higher than for Gaia while the nature of the observations is differential largely compensate each other so that the relativistic model for Theia can have the same physical content as the model used for Gaia (Klioner, 2003, 2004). The latter has an accuracy of 0.1 microarcseconds for global astrometry. Nevertheless, the details of the Theia model and its optimal formulation still has to be investigated. The model of Theia should be based on the system of the hierarchy of the relativistic reference systems – BCRS, GCRS, etc. – recommended by the IAU and used for high-accuracy relativistic modeling over last 20 years (Soffel et al., 2003). The main components of the models are as follows.

First, Theia needs a relativistic treatment of its barycentric orbit as well as its orientation. To calculate the differential aberration in the Theia’s FoV, its velocity should be known with an accuracy of about 20 mm/s — a rather relaxed requirement. To account for the parallax (or planetary aberration) when observing Near-Earth Objects, it is desirable to know its position with an accuracy of a few hundred meters. This should be better quantified at a later stage of the mission.

Second, the propagation of light from the source to the satellite has to be accurately modeled. This includes effects due to several types of gravitation field - monopole, quadrupole and multipolar, gravitomagnetic due to translational and rotational motion of the gravitating bodies. The differential nature of observations relaxes the accuracy requirements in some cases, but in other cases (e.g. observations close to Jupiter or other giant planets) the effects should be computed with an accuracy that meets the observational accuracy. The physical content of these effects is well-known (e.g. Klioner & Zschocke, 2010; Teyssandier, 2012; Hees et al., 2014; Kopeikin et al., 2011). Detailed optimal formulation still should be found.

Finally, Theia requires a high-accuracy definition of the motion of the observed sources with respect to the barycentre of the solar system. Here one can expect that most of the sources can be described either by the dynamical equations of motion (e.g. for solar system objects or for components of non-single stellar systems in their motion relative to the common barycentre) or by the model assuming that the source moves with a constant velocity (the model used both for Hipparcos and Gaia for single stars). In some special cases the latter model may require an update taking into account non-linearity of the relative motion as well as light-travel effects. Again, the physical content of such an update is well known and understood Kopeikin et al. (2011).

The Optical Field Angle Calibration (OFAD) can only be calibrated in space, with observations of a well known star field, with multiple observations at different pointings over a short period of time. The observations are put into a simultaneous solution of the chosen mathematical equation for the mirror form and the terms representing the metrology components of the mirror.

Thermo-mechanical deformations, due to the variation of the Solar Aspect Angle (SAA) in different repointings of the instrument and the non-zero Coefficient of Thermal Expansion (CTE) of the material chosen for the telescope and Focal Plane Array structures, will produce relative astrometric shifts along the lifetime of the mission. To monitor these deformations the Science Payload includes three different subsystems: wavefront sensors, a focal plane metrology subsystem and a linear metrology subsystem. Additionally, continued monitoring of the calibration field throughout the mission allows for redundant monitoring of temporal changes in the mirror and instruments that would require further calibration to maintain precision and stability for astrometry.

The Shack-Hartmann wavefront sensors are responsible for a continuous monitoring of the focus of the telescope and for a continuous monitoring of the quality of the wavefront arriving at Theia detectors due to deformations of the telescope mirrors from thermal variations.

A Gaia-like WFS solution is able to sense $\lambda/1000$ variations of the wavefront, and Theia FPA concept includes four of such WFS at each corner of the FoV to allow sub-$\mu$as monitoring.

The Focal Plane Array Metrology subsystem includes an interferometer. It is responsible for frequent monitoring and calibration of the deformations of the positions of each pixel of the detectors of the FPA at a level of $<10^{-5}$ (Crouzier et al., 2016b). This subsystem measures the geometrical parameters of the FPA with respect to M3, of each detector with respect to the center of the FPA and of each pixel with respect to the center of the detector.

The Telescope Metrology subsystem contains 18 linear interferometers that continuously measure the relative positions and angles of the mirrors of the telescope. Between each pair of mirrors, a "Virtual Laser Hexapod" is created using retro-reflectors, monitoring any thermo-mechanical effects.

Detailed Zemax simulations of the Theia telescope deformed under a set of worse case scenarios (dT = 100 mK, for different SAAs) and analysis of the astrometric displacements caused by such thermo-mechanical deformations show that the OFAD can be modelled in terms of an 8th order Chebyshev polynomial.

The existence of this low order expansion shows that it is possible to calibrate at sub-micro arcsecond level the astrometric displacements in the FPA caused by variations of the telescope structure by using the Telescope Metrology subsystem measurements at the $50pm$ level.

After the OFAD calibration is completed, additional calibrations are made for color response using stars of different colors in the same field across the field of view of the instrument.

For each hour of observation the final relative positions precision (RMS) is estimated to be 1, 14, 94, 1100 $\mu\mbox{as}$ at R-band magnitudes 7, 15, 19, 24 respectively. The resulting astrometric parameters for a 40h and 1000h science case during a 4 year mission is shown in Table 4.6 and Figs. 4.38 and  3.21. A position calibration noise floor of 0.125 $\mu\mbox{as}$ has been used (see Sect. 4.2.4).

Theia is a small-field relative astrometry mission, meaning that the derived astrometric parameters for the target stars in a field will have position, parallax and proper motion relative to a local reference frame tied to a global one. At the time of the Theia mission, the most accurate and complete optical reference frame will be that of the Gaia catalog⁵⁵5Note that for the hyper velocity stars science case the current SDSS quasars will already suffice to perform measurements (see Sect. 2.1.2).. Typically hundreds to thousands of Gaia sources will be visible in a single Theia frame. By using Gaia global astrometry parameters as priors, the astrometric solution of all the stars observed by Theia will be automatically tied to the Gaia frame, without the need of forcing physical priors on sources such as quasars or remote giant stars. Let us note that since most science cases require parallaxes and proper motions, it is possible to construct a kinematic reference frame for Theia at almost Gaia accuracy. While Gaia positions will degrade linearly over time, the accuracy of proper motions and parallaxes will remain almost the same as for the Gaia epoch⁶⁶6A positional reference frame derived with Gaia stars will be degraded by a factor of $5-6$ at the Theia epoch, while proper motions and parallaxes themselves degrade to a much lesser degree, See Sect. 1.5.4 and 1.5.5 of the Hipparcos and Tycho catalogues (Perryman et al., 1997)..

Zonal systematic errors in the Gaia absolute reference frame will set the uncertainty floor on each astrometric field. In what follows, we discuss the possible impact of these correlations into Theia’s absolute proper motion and parallax measurements.

For Gaia, the correlations of astrometric parameters resulting from the determination of attitude parameters have been studied before launch in detail (see Holl & Lindegren, 2012; Holl et al., 2012) and are expected to induce correlations at angular separations $<0.7$ deg, which could be a serious concern for Theia as this corresponds to its proposed FOV. However, these correlations were estimated to be (much) below $r=0.5$% (Holl et al., 2009). Bright stars ($V<13$) and low star-density regions will have the highest correlations. Table 4.7 gives a general overview of the limit to which one can average Gaia parallax and proper motions for various possible correlation coefficients (precise values of correlations will not be known until a few years into the Gaia mission), and hence the ultimate accuracy by which parallax and proper motion measurement of Theia can be expressed in the Gaia reference frame. The limiting accuracy for the correlation coefficients between $5\times 10^{-3}$ and $5\times 10^{-5}$ is plotted as function of magnitude in Fig. 3.21. In principle all Gaia reference stars observed in a Theia fields can be used to fix the reference frame, which will be of varying magnitude and number, therefore the reachable absolute astrometry parameter accuracy of Theia will differ from field to field.

In pre-launch studies the basic-angle metrology specification of 0.5 $\mu\mbox{as}$ over 5 min, the global parallax zero-point has been estimated to be around 0.1 $\mu\mbox{as}$. With regard to the proper motions, Gaia itself will be tied to the radio ICRF with an expected precision in proper motions of 0.2-0.3 $\mu\mbox{as}$/yr (Mignard & Klioner, 2012; Mignard, 2012). The Tycho-Gaia solution of $\textit{Gaia}^{\prime}s~$ DR1 release contains systematic errors that are not representative for the final (Gaia only!) solution due to incomplete calibration models, short Gaia data span, and mixing with Tycho data. Therefore it can currently not be reliably assessed if the calibration errors of the final Gaia data release (that will be available at the Theia launch) will cause any departure from the estimated pre-launch systematic errors.

In summary, anchoring Theia observations to the Gaia reference frame is a critical aspect to enable global astrometric capabilities of a differential astrometry mission.

Since specific targets and general astrophysics programs (e.g. very distant halo stars, objects in nearby galaxies) might need to push global astrometric precision to the limit, future mission development plans will have dedicated work-packages and a team specifically devoted to this task. Similarly, optimization of the pointing to capture good reference stars and/or distant sources will be done for all science cases and observations at a later stage.

Astrometry and photometry are closely linked. In astrometry, the most accurate way to determine the centroid of a source involves the fitting of a PSF to the image data. Photometry can also be derived using such PSF fitting. The same error sources in astrometry also affect photometric accuracy. Detailed calibration of the detector enable both more accurate astrometry and more accurate photometry. DICE experiment (Crouzier et al., 2016a) concludes that in order to reach an error below $5\times 10^{-6}$ pixel on the centroid, the pixel response non uniformity (or PRNU) must be know to better than $1.2\times 10^{-5}$. Such a precision can only be obtained if a precise photometric calibration is used.

The instrumental effects to consider in the photometric calibration are bias, dark, flat-fielding (PRNU), fringing, background, saturation and non-linearity, contamination, pixel position and characterization of the read-out-noise. All these effects are calibrated through specially designed calibration images from pre-launch measurements and during mission operation from onboard interferometric fringe measurements and the use of the shutter.

The two common methods to derive photometry from CCD images are aperture photometry and PSF fitting. PSF fitting strategy is preferred for Theia, as it is used anyway for astrometry processing, as stated above. Estimates of the relative photometry precision are shown in Fig. 4.39.

Differential photometry approach can also be used for Theia instead of (or together with) absolute photometry. Constant sources in the field are used to settle the relative photometry of the sources of interest. Finally, in order to standardize the instrumental photometry a set of calibration sources with standard photometry in the field of view are needed. Gaia mission provides standard homogeneous good quality photometry of constant sources in all the sky.

The proper management of the development of the scientific processing software is crucial to ensure the timely production and quality of the Theia products. This management has to rely on professional software (SW) development procedures and standards and has to ensure an adequate coordination of the geographically disperse contributions to the effort. For this we propose the creation of a "core development" team that will have the responsibility of:

1. 1.

   Acting as coordinator of the SW development, defining and enforcing development standards in the Theia consortium. Specifically, the core team will be responsible for the QA of the development, including interface definition, version control, enforcement of ECSS standards.

2. 2.

   Receiving the contributions from the different sub tasks and integrating them into a common framework. We will maintain a common code repository for collection and control of the code and versioning.

3. 3.

   Enforcing the definition and implementation of unit tests for the SW. We will set up a continuous integration environment where the unit tests will be run at least daily to ensure repository code integrity and validity.

4. 4.

   Defining, implementing and running integration tests for the SW. At each SW release the full processing chain will be tested end-to-end to ensure its correctness and integrity. These tests will be defined to cover both technical and scientific validation of the code. Executing the pipeline. The code will be deployed in the processing system to produce the scientific data from the Theia telemetry. The core team will take responsibility for code deployment, execution control, management and delivery of science-ready products.

5. 5.

   The core team will be responsible for monitoring the performance of the instruments by producing, in addition to the final relative corrected position, files with information about the general performance of every data set — generating warnings when there are failures or the instrument has gone through significant changes.

Theia is an astrometry mission that needs to point to different directions of the sky. L2 is the selected option for the orbit, since it is very favorable for overall stability because of the absence of gravity gradients, the time available for observation, the environmental conditions characterized by low total ionizing doses. In addition, the thermal conditions over the orbit remains the same, thereby simplifying thermo-elastic design issues. The Theia spacecraft will be directly injected into a large Lissajous or Halo orbit at L2.

To avoid parasitic light from the Sun onto the telescope and the detector, Theia spacecraft have baffles that protect them from Sun light at angle larger than $\pm 45^{\circ}$ in the Sun direction.

Considering the M5 call programmatics, the targeted orbit and the Theia mass range, the baseline launcher is Ariane 6.2. The launch strategy would consist in a unique burn of upper stage injecting directly the S/C onto a L2-transfer trajectory, avoiding a coasting on a parking orbit. The separation of the satellite would occur after about 1/4 of rotation around the Earth, preventing the Sun illumination to enter into the instrument during ascent and up to the separation.

After the completion of its last burn, the Ariane upper stage would re-orient the S/C into a 3-axes stabilized attitude, with the Sun in the Sunshield normal direction. As for Herschel satellite, Theia spacecraft would require a reactive Safe Mode control to maintain roughly the Sun in this direction and ensure the instrument protection.

The use of Ariane 6.2 enables a large volume for the spacecraft and its payload. The Ariane fairing allows a maximum diameter of 4500 mm, which conditions: (i) The maximum primary mirror diameter; (ii) the maximum Service module size and thus its internal accommodation capacity, (iii) the maximum Sun shield and V-groove screens size, linked also with the telescope size and required sky accessibility requirements.

Although not currently specified, the Ariane 6.2 launcher will allow a maximal spacecraft wet mass well in excess of the 2145 kg allowed on Soyuz for a direct transfer to the L2 point. It shall be noticed that considering Ariane 6.2 performance and the current Theia mass budget, a dual launch is an attractive option that would optimize Theia launch costs if a co-passenger can be identified during the assessment.

The time baseline to properly investigate the science program of Theia is 4 years including some time devoted to orbit maintenance. A total of approximately 6 months has been estimated for the orbit transfer including the spacecraft and instrument commissioning. From the total of $\sim 35000$ h dedicated for the scientific program, about 15 min per slew will be dedicated to reconfiguration and station-keeping. The thermal stabilization time is in addition to the slew time.

The primary objective requires 20500 h of nominal time (first column of Table 3.3) and about 4000 h will be dedicated to open time observations. The Theia Collaboration is keeping large margins to consider the spacecraft thermal stabilization aspects that can only be correctly known after a detailed thermal modeling of the spacecraft and payload modules and mission scheduling. The mission timeline is flexible and can be optimized together with the target list and the main instrumental characteristics.

The proposed mission architecture relies on a Korsch three mirror telescope accommodated vertically on top of a platform including all support subsystems.

A high thermal stability of the telescope necessary to ensure its performances is obtained through the use of a Sun Shield on which is accommodated the Solar Array and a vertical V-groove screen.

The micro-arcsecond performance specification places stringent requirements on the pointing stability. This demands a relative pointing error (RPE) of $\sim~20$ mas ($1\sigma$) in spacecraft $x$ and $y$ direction, and a fraction of arcsec ($1\sigma$) in $z$ (roll) over the image accumulation time. This can be achieved using chemical (monopropellant hydrazine) propulsion for the transfer corrections, monthly station keeping manoeuvres, and large (180 deg) slew manoeuvres, and then cold gas micro-propulsion system (MPS) for fine acquisition. This can also be achieved by performing the pointings of the satellite using reaction wheels for large attitude motions between targets and then cold gas. At the approach of the correct pointing, the Satellite will use a MPS for fine relative motion acquisition. During the scientific observation the reaction wheels are stopped to avoid the generation of mechanical noise. Such a synergetic concept was developed for the Euclidscience mode AOCS (e.g. Bacchetta et al., 2015).

The satellite key features are presented in Fig. 5.41. The design of satellite is mainly based on the Euclidservice module with a downscaled size to better suit to specific Theia needs and minimize the mass to leave the door open to dual launch opportunities. It uses a central tube and standard 1194mm interface compatible with Ariane planned adapters and an irregular hexagonal shape structure contained inside a 3.5m diameter circle, providing large volumes for platform units and fluid tanks accommodation. Similarly to Euclid and Herschel satellites, Theia Korsch telescope is accommodated on top of the service module in a vertical position leading to a spacecraft height of about 5m. This concept allows to optimize the payload size, Ariane fairing allowing large volumes.

A key driver for the Theia mission is the payload structural stability. Particularly during long observations of 10h or more, the payload shall remain stable within 30 mK to avoid nanometer distance variations of the primary-secondary mirror. This stringent requirement demands a very stable payload structure with low thermo-elastic deformations. The preliminary analyses led to the selection of a telescope structure largely making use of SiC or Si₃N₄ ceramic materials. Such materials can be used in a large number of structure components (e.g. truss, brackets, plates).

The payload structural stability requirement imposes a very stable thermal concept for the spacecraft (Fig. 5.41). The preliminary design is making use of a Sun shield supporting also the Solar array. Here too, the concept derives from Euclid and Herschel. However, Theia will require additional V-groove vertical screen(s) aiming at minimizing the temperature variations from one target to the other due to different solar impingement on the satellite. In addition, an active control of the payload structure temperature will be added through the use of high precision thermistors and optimally spread heaters. Thermal stability levels more strict than Theia requirements have been demonstrated by Gaia’s . Nevertheless, there is still space for some optimization effort related to the V-grooves and the active thermal control system.

Strong heritage does exist on Theia spacecraft avionics and AOCS, particularly coming from Euclid mission having very similar needs in terms of pointing performances. The proposed AOCS configuration would use Star Trackers, FOG gyroscope and Fine Sun Sensors as main sensors, reaction wheels for fast repointing between targets and probably cold gas $\mu$-propulsion (or mini Radio-frequency Ion Thruster, mini-RIT), depending on Euclid final performances. Associated fluid tanks (Nitrogen or Xenon) would be accommodated around the central tube in a symmetrical configuration to keep a centered CoG. This AOCS concept is perfectly compatible with the preliminary pointing performances required by the Theia mission.

The electrical power subsystem would be built a fixed Solar Array installed on the Sun shield as on Euclid with a size compatible with Theia needs and with extreme foreseen Sun depointing angles. A battery has been added to power the S/C during the launch operations and up to separation.

The downlink data rate needs and operational constraints are equivalent to the ones of Euclid, profiting from its heritage. Similarly, Data Handling units would derive from Euclid ones, with a Control and Data Management Unit (CDMU) including the software and a Mass Memory Unit with several TB storage capacity.

The required $\Delta V$ for transfer to L2, attitude control, station-keeping maneuvers and end of life disposal will be performed by an hydrazine-based propulsion system operation in blow down. The associated tank will be accommodated inside the central tube.