Eric Michel
email: eric.michel@obspm.fr

Kevin Belkacem

Benoît Mosser

Reza Samadi

Misha Haywood

David Katz

Benoit Famaey

Tiago L Campante

Mário J P F G Monteiro

Margarida S Cunha

-Eric Michel

Benoit Mosser

Reza Samadi -Lesia

Mario J P F G Monteiro

Juan Carlos Suarez -Ramon

Sébastien Deheuvels

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Chronos

Chronos

Take the pulse of our Galactic neighbourhood After Gaia: time domain information, masses and ages for stars 1) Introduction:

Understanding our Galaxy with Gaia and time domain constraints.

Galactic archaeology has entered a new era with the data provided by the Gaia mission, allowing to get new insights into fundamental questions about the build-up of our Galaxy and its different stellar components. For instance, is the actual backbone of our Galaxy a "disky" stellar component, or was there an early in-situ stellar halo? What is the role played by early mergers and accretions of satellite galaxies? What was (and still currently is) the respective role of external accretions and internal dynamical instabilities (bar, spiral arms), as well as their interplays, in shaping the star formation history and the dynamics of the different stellar components? Answering all these questions requires determining the ages of stars.

The Gaia mission (Gaia Collab. et al. 2018a) is providing us with an unprecedented insight of our Galaxy. Completed by the associated large spectroscopic surveys (such as APOGEE and WEAVE), Gaia is characterizing in detail and with a unique precision, positions (including distances), movements, effective temperatures, luminosities, chemical compositions for about a billion stars over a large portion of our Galaxy and nearby satellite galaxies (e.g., Gaia Collab et al. 2018b).

These precise data shed a new light on the above fundamental questions about how our Galaxy has formed and evolved. They reveal unexpected trends, suggesting for instance that a large fraction of the oldest stars in the Galaxy have "disky" orbits [START_REF] Sestito | Tracing the formation of the Milky Way through ultra metal-poor stars[END_REF], that the stellar halo is almost fully accreted (e.g., Di [START_REF] Matteo | The Milky Way has no in-situ halo but it has a thick disc[END_REF] and that these accreted components in turn played an important role in shaping the in-situ "disky" components (e.g., [START_REF] Helmi | The merger that led to the formation of the Milky Way's inner stellar halo and thick disk[END_REF], while it appears that more recent accretions are still influencing disk dynamics today [START_REF] Antoja | A dynamically young and perturbed Milky Way disk[END_REF][START_REF] Laporte | Footprints of the Sagittarius dwarf galaxy in the Gaia data set[END_REF]. The respective roles of such external perturbations and internal instabilities such as the Galactic bar, the structure of which also being constrained with Gaia (e.g., [START_REF] Monari | Signatures of the resonances of a large Galactic bar in local velocity space[END_REF] , and the interplay between these external and internal dynamical mechanisms, are however still under investigation. To reach a more precise view of the situation, we however absolutely need to enrich the very precise astrometric-kinematic-chemical view of the Galaxy provided by Gaia and large spectroscopic surveys with high precision ages which can be provided through seismic information (or more generally time domain information).

A large fraction of stars shows multi-periodic photometric variations due to stellar pulsations, but also rotational modulation, granulation... These phenomena and the measurement of their photometric signature carries valuable information of different nature than the astrometric-kinetic-chemical one, like mass, age and (unprojected) rotation estimates. With CoRoT [START_REF] Baglin | CoRoT: a high precision photometer for stellar ecolution and exoplanet finding[END_REF]) and Kepler [START_REF] Borucki | Kepler Planet-Detection Mission: Introduction and First Results[END_REF], we have learned to derive this information from photometric light curves of tens of thousands of stars and we can foresee an upgrade of this capacity in the number of stars but also in type of stars and in type of applications. This is why we are firmly convinced that it will be a top priority for the scientific community in 2035-50 to improve our view and understanding of our Galaxy, by adding a time domain dimension to the Gaia Survey. Chronos will complete the Gaia survey with high-temporal resolution for a million stars with 8≤mV≤11.

-in Sect.2 we discuss the nature of the information brought by the variability dimension and show why it is original and complementary to information on stars brought by Gaia and spectroscopic surveys and why a strong potential synergy exists. We also stress the specific constraints characterizing time domain measurements with a special emphasis on the project considered here.

-in Sect.3 we show how our understanding of the Galaxy will progress with the variability information. In order to be practical, we realise this study in the framework of a specific example of survey focussing on red giants for which an instrumental option is proposed in the next Section.

-in Sect.4 we describe a possible option for an instrument and a mission profile capable of providing such a variability survey. We compare this instrument with existing or planed missions and show how unique and well suited it is for our purpose.

-Section 5 addresses possible improvements for the scientific objective and technical challenges for the mission concept presented here.

-in Sect.6 we summarize and conclude in a broad scientific perspective.

2) Specific nature of time domain characterization and synergy with Gaia and large spectroscopic surveys

As already mentioned, photometric variability exists at different amplitude levels and different time scales in a large proportion of stars. This variability can be due to stellar pulsations, but also to other phenomena like rotation modulation associated with activity or binary tide effects, granulation…

2a) Specificity of time domain information

Considering stellar pulsation and seismic information, the case of red giant stars is very illustrative of the power of this approach. From the observation of several tens of thousands of red giants by CoRoT and Kepler, we have learned to understand their oscillation spectra and to interpret them in terms of a few parameters, the so-called seismic indices (e.g. Mosser & Appourchaux 2009, Kallinger et al. 2010[START_REF] Bedding | Gravity modes as a way to distinguish between hydrogen-and heliumburning red giant stars[END_REF], Mosser et al. 2012):

-Δν: named the large separation, characterizes the first-order regularity in frequency of acoustic oscillation modes. This seismic index is tightly linked with the stellar mean density (Eddington 1919).

It is a very valuable piece of information about the individual stars, because, via the strong relation between radius and mass it imposes, it brings the possibility to characterize the stars in terms of mass, which is not measured with Gaia and only very poorly (via log g) in the spectroscopic analysis.

-νmax: the frequency of maximum pulsation amplitude is strongly linked with the surface gravity, and, to a lesser extent with the effective temperature [START_REF] Belkacem | The underlying physical meaning of the νmax -νc relation[END_REF]. It is another very valuable piece of information about the individual stars, because it gives accurate logg with a precision much better than spectroscopic analysis. When spectroscopic estimates of logg are known within a factor 2 or 3 for field stars, so-called 'seismic log g' are measured with a precision better than 2% [START_REF] Bruntt | Accurate fundamental parameters and detailed abundance patterns from spectroscopy of 93 solar-type Kepler targets[END_REF] The seismic indices Δν and νmax, combined with the spectroscopic measurements of effective temperature, are used to provide stellar masses and radii based on simple seismic scaling relations. For several thousands of these red giants, with long enough observations, it has been possible to measure a third seismic index [START_REF] Vrard | Period spacings in red giants. II. Automated measurement[END_REF]):

-ΔP is the period spacing for g-dominated oscillation dipole modes. It asymptotic value is measuring the size of the inner radiative core of red giants [START_REF] Montalbán | Testing Convective-core Overshooting Using Period Spacings of Dipole Modes in Red Giants[END_REF]. It is thus a very precise proxy of the evolution of the stellar structure with age, even if measured in non-asymptotic conditions (Mosser et al. 2011). It allows for instance a clear distinction between a horizontal branch red giants (or red clump, RC here after) having ignited the fusion of helium in a small convective core and a red giant branch stars still having an inert helium core [START_REF] Bedding | Gravity modes as a way to distinguish between hydrogen-and heliumburning red giant stars[END_REF] although both stars might have exactly the same radius and effective temperature.

The situation is still progressing in order to characterize both the core and the envelope of the red giants. Rotation of the red giant core can be measured [START_REF] Beck | Fast core-rotation in red-giant stars as revealed by gravity-dominated mixed modes[END_REF], Mosser et al. 2012[START_REF] Deheuvels | Seismic Evidence for a Rapidly Rotating Core in a Lower-giant-branch Star Observed with Kepler[END_REF]. Recently, [START_REF] Gehan | Core rotation braking on the red giant branch for various mass ranges[END_REF] could perform the automatic measurement of the rotation for about 900 red giants on the RGB. Several works are under way to make automatic and extend to a large number of stars the measurements of the pseudo-periodic perturbation of the pulsation frequency pattern characterizing the acoustic depth of the outer convective zone [START_REF] Vrard | Helium signature in red giant oscillation patterns observed by Kepler[END_REF]; in parallel, further information on the convection comes from the measurement of the mode lifetimes [START_REF] Vrard | Amplitude and lifetime of radial modes in red giant star spectra observed by Kepler[END_REF]) and of the luminosity bump [START_REF] Khan | The Red-giant Branch Bump Revisited: Constraints on Envelope Overshooting in a Wide Range of Masses and Metallicities[END_REF].

The potential of asteroseismology illustrated above on red giants can be extended to a large number and variety of stars. The measurement and use of Δν and ν max, for instance, can be straightforwardly applied to other solar-type pulsators (stars showing, as the Sun and red giants, pulsations excited by the near surface convection) which cover a large part of the HR diagram. This includes all mainsequence stars cooler than early F, subgiants, and AGB stars.

Beyond solar-type oscillations, many classical pulsators spanning various mass ranges and evolution stages in the HR diagramme provide in their pulsation spectra patterns like ∆ν or individual eigenfrequency values indicative of the mean star density. This is well known for Cepheids, RR Lyrae, but it is also potentially the case for main-sequence pulsators of different mass. For instance, δ Scuti stars are 1.5-2.5 stars in the main sequence and early subgiant phase. The analysis of their oscillation based on about 1800 of them observed with CoRoT has confirmed the existence of a regular pattern [START_REF] Garcia Hernández | An in-depth study of HD 174966 with CoRoT photometry and HARPS spectroscopy. Large separation as a new observable for δ Scuti stars[END_REF][START_REF] Paparo | Unexpected Series of Regular Frequency Spacing of δ Scuti Stars in the Nonasymptotic Regime. II. Sample-Echelle Diagrams and Rotation[END_REF][START_REF] Michel | What CoRoT tells us about delta Scuti stars -Existence of a regular pattern and seismic indices to characterize stars[END_REF]) and this frequency spacing, very similar to ∆ν has been shown to be a precise indicator of the mean density [START_REF] Garcia Hernández | Observational ∆ν-ρ̄ Relation for δ Sct Stars using Eclipsing Binaries and Space Photometry[END_REF], 2017). The effective extension of the seismic characterization is thus to large extent a matter of collecting precise photometric time series of large enough samples of these various pulsators in order to achieve the comprehensive analysis of their pulsation spectrum.

Rotation and activity: Several thousands of light-curves obtained with CoRoT and Kepler have revealed the signature of rotational modulation attributed to starspots [START_REF] Lanza | Photospheric activity, rotation, and radial velocity variations of the planethosting star CoRoT-7[END_REF][START_REF] De Medeiros | Overview of semi-sinusoidal stellar variability with the CoRoT satellite[END_REF], Garcia et al. 2013[START_REF] Bravo | Wavelets: a powerful tool for studying rotation, activity, and pulsation in Kepler and CoRoT stellar light curves[END_REF][START_REF] Leão | Rotation period distribution of CoRoT_ and Kepler Sun-like stars[END_REF]. In addition to providing a measurement of the rotational period free of the sin i projection factor, these light-curves revealed that the signature of activity is not only found in late F-G stars, for which magnetic fields and starspots were expected as in the Sun, but also in intermediate A and B type stars [START_REF] Savanov | Spots and Activity of A-type Stars[END_REF][START_REF] David-Uraz | Magnetic OB[A] Stars with TESS: probing their Evolutionary and Rotational properties (MOBSTER) -I. First-light observations of known magnetic B and A stars[END_REF][START_REF] Balona | Discovery of spots on hot stars requires major revision of stellar physics[END_REF]. This interpretation is supported by the discovery of the weak signature of magnetic field in A and B type stars [START_REF] Lignières | First evidence of a magnetic field on Vega. Towards a new class of magnetic A-type stars[END_REF][START_REF] Böhm | Discovery of starspots on Vega. First spectroscopic detection of surface structures on a normal A-type star[END_REF], Blazère et al. 2016). Since the surface rotation rate of a star is in first place a function of its mass and evolutionary state (at least for low-mass stars), gyrochronology (e.g. [START_REF] Barnes | Ages for Illustrative Field Stars Using Gyrochronology: Viability, Limitations, and Errors[END_REF]Meiböm et al. 2011) takes advantage of these measurements of the stellar rotation rate to produce an estimate of age.

Granulation:

The light-curves of red giants have also unveiled the presence of a signal characterizing the stellar granulation, which is a superficial signature of convection inside the stellar envelope. As for the seismic indices, it is possible to extract in rather easy way the characteristic parameters of stellar granulation. These parameters also obey scaling relations, which in addition to seismic scaling relations, provide information about the surface layers of the star. In particular, the CoRoT and Kepler observations have revealed the existence of a universal scaling relation between the characteristic time-scale of stellar granulation and surface gravity (Kallinger et al. 2010, Mathur et al. 2011, Bastien et al. 2013, Kallinger et al. 2016); in absence of detectable oscillations, this scaling provides an estimate of star surface gravity.

2b) Specific observational constraints:

A practical difficulty exists however to design a unique large sky coverage survey for seismic characterization of stars in our Galaxy. It resides in the different pulsation time scales and amplitudes between these evolution stages. The Main Sequence objects for instance are the more demanding in noise level, with amplitudes of the order of 1 ppm µHz -1/2 and in sampling cadence (~1min or less), but a couple of months are enough to resolve their eigenfrequency pattern and extract seismic indices. At the other end of the evolution sequence, AGB stars have more comfortable amplitudes (~100 ppm µHz -1/2 ) and sampling cadence (a few days sufficient) but the frequency resolution necessary to measure their large separation ∆ν and νmax requires long durations, of the order of 7 to 10 years (see [START_REF] Soszynski | The Optical Gravitational Lensing Experiment. Small Amplitude Variable Red Giants in the Magellanic Clouds[END_REF][START_REF] Soszynski | The Optical Gravitational Lensing Experiment. The OGLE-III Catalog of Variable Stars. IV. Long-Period Variables in the Large Magellanic Cloud[END_REF][START_REF] Mosser | Period-luminosity relations in evolved red giants explained by solar-like oscillations[END_REF][START_REF] Takayama | On the pulsation modes of OGLE small amplitude red giant variables in the LMC[END_REF] This is why it is reasonable to plan a few distinct surveys, optimized each for a given domain of the HR diagramme.

As we will show in Sect.4, PLATO represents presently the best opportunity to lead a substantial survey on Main Sequence solar-type pulsators on about half the sky. At the other extremity of the HR diagramme, LSST is very promising for a survey of AGB stars on about half the sky too.

In the present white paper, we consider the intermediate domain between these two extremes. We focus on what we can learn about our Galaxy with a survey able to characterize seismically all stars from subgiants, to red giants and early AGB in a few kpc around our Sun.

3) How our understanding of our Galaxy will increase with a fast cadence long duration variability survey of red giant stars over a few kpc

The understanding of our Galactic surrounding and of our Galaxy as a whole is experiencing an unprecedented progress under the influence of the In order to illustrate in a practical way the interest of the information coming from variability for the study of our Galaxy, we focus here on what can be done with red giant stars and give a non-exhaustive list of breakthrough studies that could be conducted with a survey like Chronos (described in Section 4).

Generally speaking, understanding our Galaxy is relying on a precise census of the positions, kinematics, chemical composition and age of its stars. The ongoing Gaia programme is building such a census with measurements of unprecedented precision for over a billion of stars over a large part of our Galaxy. The mass and age estimate however is presently the weak link of the chain. Most stellar masses and ages used in Galactic astronomy are still based on fitting stellar positions in CMD diagrams with stellar evolution model isochrones (e.g., Sanders & Das, 2018;[START_REF] Gallart | Uncovering the birth of the Milky Way through accurate stellar ages with Gaia[END_REF]. The Gaia survey and the complementary large spectroscopic surveys will bring refined luminosity (L) and effective temperature (T eff) and chemical composition estimates to make this comparison. However, there are fundamental limitations to this approach:

-one limitation resides in the reliability of the theoretical evolution sequences used for comparison. This is a reason why improving our understanding of stellar structure and evolution is important and one of the ways to do it right now is stellar ensemble asteroseismology. This is why time domain surveys like CoRoT, Kepler, K2, TESS or PLATO always represent an opportunity to improve the understanding of stellar structure and evolution and thus improve theoretical evolution sequences. This objective is included in PLATO scientific objectives and we can foresee great advances on the reliability of theoretical sequences for the Main Sequence cooler than early F and for the subsequent subgiant and red giant phase.

-the second limitation comes from the fact that, depending on the region considered in the HR diagramme, Teff and L values characterizing evolution sequences show more or less dependency on age, mass, and chemical composition. The case of red giants is very illustrative of this point. The red giant parts of the evolution sequences are clustering in a small range of Teff. Gaia brings very precise measurement of L (thus of R, for a known Teff), but the Teff measurement, as precise as it can be, cannot bring a strong constraint on the mass estimate even if the chemical composition is assumed to be perfectly known. As a result, the constraint on the age is very poor, since for these red giants, mass is the predominant key factor for age on the RGB [START_REF] Miglio | PLATO as it is : A legacy mission for Galactic archaeology[END_REF].

As commented in Sect.2, one of the first information brought by seismology is the star mean density. This is true for various types of pulsators and especially for red giants where the combination of the two seismic indices ∆ν and νmax with an estimate of Teff (which does not even have to be very precise, since the dependency in Teff is small) leads straightforwardly to the mass estimate which in turn provides a precise age estimate for several tens of thousands of red giants already.

A non-exhaustive list of breakthrough Galactic studies that could be conducted with a survey like Chronos in a few kpc around the Sun is the following:

-Chemical evolution of the Milky Way. Detailed relations between age -chemical abundance ratios of a variety of different elements (e.g., [START_REF] Snaith | Reconstructing the star formation history of the Milky Way disc(s) from chemical abundances[END_REF]) have proved to be fundamental for constraining the chemical evolution of our Galaxy, but we are just at the beginning of using them, because of the difficulty to obtain ages with the useful precision. Yet, even with the precision provided by stellar isochrone age dating, these relations show remarkably tight correlations between age and chemical element abundance ratios that need to be confirmed with better ages, in particular for stars older than 8 Gyr (e.g., [START_REF] Haywood | The age structure of stellar populations in the solar vicinity. Clues of a twophase formation history of the Milky Way disk[END_REF], Nissen et al., Spina et al. 2017). These relations in turn can be used to constrain the star formation history of the Milky Way, its gas accretion history, nucleosynthetic yields, but also dynamical mechanisms such as stellar radial migration triggered by a combination of the effect of external and internal perturbations (such as the bar and spiral arms, e.g., Minchev & Famaey 2010).

-Measuring the Milky Way Star Formation History. The measurement of the Milky Way star formation history is not reflected in the age distribution of the stars in the solar vicinity, because most old stellar populations are confined to the inner Galaxy (e.g. Bovy et al., 2012), beyond the reach of presently available stellar age dating methods. Today, our best prospect to measure the SFH of our Galaxy is to measure the stellar mass density of populations as a function of alpha abundance, and to relate these abundances to a well-calibrated age scale, determined locally.

Ongoing or upcoming large spectroscopic surveys (see above) will provide deep spectroscopic sampling of all stellar populations throughout the Milky Way, on distances of several kpc, allowing the determination of stellar densities as a function of (alpha) abundances, or mono-abundance populations (Bovy et al. 2012[START_REF] Mackereth | The age-metallicity structure of the Milky Way disc using APOGEE[END_REF]. Because of the relations linking age to alpha element abundances, chemical abundances can be used as proxy for stellar age on very large distances, allowing us to determine the amount of stellar mass as a function of age in the Galaxy, or the Star Formation History. Ultimately, the validity of the determination of the Star Formation History will however depend on how accurately chemical abundances can be linked to an accurate age-scale provided by the proposed survey. It will also allow us to better understand how the Milky Way disk was built (inside-out?) and crucially to compare the Milky Way history with other galaxies of the same mass.

-Understanding the early phases of our Galaxy. Precise ages, or high temporal resolution, are also necessary to understand the sequence of events that built the early Galaxy. Recent Gaia results suggest for instance that a large fraction of the oldest stars in the Galaxy have hot orbits aligned with the Galactic plane [START_REF] Sestito | Tracing the formation of the Milky Way through ultra metal-poor stars[END_REF], and that the stellar halo is almost fully accreted (e.g., Di [START_REF] Matteo | The Milky Way has no in-situ halo but it has a thick disc[END_REF]). Moreover, it appears that these accretion events might in turn have played an important role in shaping the in-situ disk components of the Galaxy, in particular the thick disk component (e.g., [START_REF] Helmi | The merger that led to the formation of the Milky Way's inner stellar halo and thick disk[END_REF]. The formation of the first stars, the rapid early chemical enrichment, the dissipation of the gas in a disk configuration, the accretion and merger of satellite galaxies, all these event occurred within a few Gyr, and a high temporal resolution, complemented with spectroscopic kinematic information is thus needed to disentangle the different components (in situ or accreted) and reconstruct the early history of our Galaxy. This can be done with Chronos with red giants in a few kpc around the Sun.

-Kinematics of the Galactic disk. At magnitude brighter than G~ 12, Gaia provides the best performance in terms of stellar radial velocities and proper motions, yielding extremely detailed kinematics of the solar vicinity (d<2-3kpc for red giants). While accretion events might have played an important role in the early shaping the in-situ disk components of the Galaxy, in particular the thick disk component (see above), it appears that more recent accretions are still influencing disk dynamics today. One of the main findings of Gaia has been a phase-spiral when looking at the phase-space surface of section of height vs. vertical velocity for local stars [START_REF] Antoja | A dynamically young and perturbed Milky Way disk[END_REF], which could not be revealed before due to uncertainties on parallaxes and proper motions. This clear out-of-equilbrium signature in the disk kinematics has been suggested to be the consequence of the passage of the Sagittarius dwarf through the plane of the Milky Way [START_REF] Antoja | A dynamically young and perturbed Milky Way disk[END_REF][START_REF] Laporte | Footprints of the Sagittarius dwarf galaxy in the Gaia data set[END_REF]), but other explanations based on the buckling of the Galactic bar have also been proposed [START_REF] Khoperskov | The echo of the bar buckling: Phase-space spirals in Gaia Data Release 2[END_REF], although the most likely explanation is a complex interplay between such external and internal mechanisms. What is certain is that it means that the hypothesis of equilibrium for modelling the Galactic disk can only be used as a basis for perturbation theory. A crucial observable allowing to disentangle the various proposed scenarios is the shape of this phase-spiral as a function of stellar ages (e.g., [START_REF] Laporte | Footprints of the Sagittarius dwarf galaxy in the Gaia data set[END_REF], which Chronos will provide. Moreover, the current structure of the Galactic bar itself is also constrained from Gaia data on disk kinematics in the Solar vicinity (e.g., [START_REF] Monari | Signatures of the resonances of a large Galactic bar in local velocity space[END_REF]), but again, the dynamical effects of non-axisymmetric perturbations on stellar populations of various ages would allow to disentangle better the effect of the bar and spiral arms, and also to constrain the age of the Galactic bar itself, which is still largely unknown.

The Chronos sample would thus allow a detailed study of the kinematics of the disk as a function of age, and would contribute to the understanding of the various effects of external and internal perturbations, such as resonances of the bar and spiral arms and their signature in the solar vicinity. Coupled with available spectroscopy, it would also permit to study the effect of radial migration (e.g., Minchev & Famaey 2010), which intensity and effect on stars in our neighbourhood is still largely debated (e.g., [START_REF] Minchev | Radial migration in galactic disks caused by resonance overlap of multiple patterns: Self-consistent simulations[END_REF][START_REF] Hallé | Quantifying stellar radial migration in an N-body simulation: blurring, churning, and the outer regions of galaxy discs[END_REF]. More generally, a detailed characterization of the phase-space distribution of red giant stars as a function of age and chemical composition should allow to help re-constructing the global secular evolution of the Galaxy by making use of a dressed Fokker-Planck approach as presented in, e.g., [START_REF] Fouvry | Self-gravity, Resonances, and Orbital Diffusion in Stellar Disks[END_REF].

The above four examples of ground-breaking studies in the context of Galactic dynamics and archaeology are limited to red giants for which we already have developed a sufficient understanding to build precise seismic diagnostics and seismic characterization. Beyond red giants, as we explain in Sect.2, we can foresee natural extension of the application of seismic characterization on large portions of the sky to main sequence solar-type pulsators (with PLATO) and to very luminous AGB stars (with LSST). We also mentioned in Sect.2 the interest and possibility to extend this approach to main sequence classical pulsators. Because their pulsations are not so different in characteristic time scales, in amplitude and in necessary resolution frequency, these objects would also be covered by the survey considered in Sect.4.

Applying seismic characterization to pulsating stars with mass between 1.5 and 15 on the Main Sequence and subgiant branch would enrich the study of our Galactic neighbourhood (not as deep as for red giants which are intrinsically very luminous, but on a few hundreds of pc and open new perspective also because of the much shorter time scales of evolution toward high masses.

4)

Chronos -An all-sky long-term seismic survey to characterize half a million red giant stars up to 1.7 kpc around the Sun

Here we show that it is possible to plan an all-sky, fast cadence, long duration photometric survey satisfying the needs of the scientific objective introduced in Sect.3. The concept presented in Sect.4a is of course only illustrative and possible ameliorations are left aside for a later phase. As it is, we would expect this project to match the requirements for a medium size ESA mission.

a. A possible instrument and mission profile Mission concept -The instrument and the spinning platform:

A platform is carrying eight identical elementary cameras, each covering a 11.25x11.25 deg 2 field of view (FOV). Cameras are grouped by two, so that at any time they are staring at four fields on the sky, 22.5x11.25 deg 2 each (see Fig. 1). The platform is spinning (Fig. 1) around the same rotational axis during Dc (here we take Dc=3.75 months) and the detectors "sweep" regularly a 22.5x360 deg 2 corona on the sky. The rotation period Prot (here Prot=8 min) ensures that each star of this corona is observed regularly with a sampling time Ts=Prot/4 (here Ts=2 min). The image of each star is crossing the focal plane in 15 s and thus one CCD in 7.5 s. The CCDs are read in TDI mode (time delay integration mode, as for Gaia, where images cross the CCD in 4.5s), the charges being drifted at the same speed as the star images.

A first optical sketch has been drawn considering, for each camera, a dioptric (6-lenses) system with a 30 cm objective, illuminating a focal plane with four CCDs (4.5k x 4.5k pixels, 18μm/px, similar to PLATO CCDs) and a point spread function within 4 pixels anywhere on the square FOV 11.25x11.25 deg 2 .

Chronos main figures Sizing

Entrance

Mission nominal lifetime 5 years

Tab.1 Summary of the main figures of the mission concept in its present version. *The effective pupil takes into account that each star of a given corona is observed by a camera regularly but during 1/8 of the time.

Mission concept -orbit and observational strategy:

We considered a platform located at Sun-Earth L2, observing the same corona on the sky during Dc=3.75 months and change from one corona to the next. With a rotation axis chosen to point toward the Sun at Dc/2, the Sun is kept away from the FOV by a guard angle GA (here GA=22.5 deg).

Phase 1 (2.5 years): by shifting every Dc (3.75 months) the rotation axis by 112.5 deg (112.5=3.75x30 being the angle described by the Sun in 3.75 months), it is possible to cover the whole sky with 8 successive coronas within 30 months. For comparison, covering the whole sky by shifting every 3 months a pointed instrument with a FOV of 1000 deg 2 requires more than 10 years. We notice that beside this, the two regions of the sky where the 8 coronas overlap will be observed a longer time (up to the total length of the mission for the continuous viewing zones CVZ, two symmetric disks of diameter 22.5 deg centred at the ecliptic poles).

Phase 2 (+30 months → 5 years, nominal duration of the mission): with 16 runs of duration Dc, the whole sky will be observed twice and each target observed two times Dc, separated by 7 Dc (here 2x3.75 months separated by ~2 years and 2 months).

Phase 3 (possible extension: +30 months → 7 years and 6 months): with 24 runs of duration Dc, the whole sky will be observed 3 times Dc, separated by 7 Dc (here 3x3.75 months separated by ~2.years and 2 months each)

Performances in terms of seismic indices and seismic diagnostics performances:

These estimates result from tests and scalings based on CoRoT and Kepler data. These tests show that seismic indices of red giants could be measured on discontinuous periods 2xDC or 3xDC separated by ~2 years as considered here with the following performances (cf Tab.2).

In phase 2 (5 years, nominal mission): we measure seismic indices Δν and νmax for all red giants (RGB, RC, early AGB) and up to 1.7 kpc (mV ≤11) and for part of the subgiants up to ~200pc (mV ≤10), i.e. approximately 500 000 stars spread over the whole sky. We also measure indices ΔP for all the red giants around the RC, where this index is of crucial interest to disentangle stars on the RGB from stars on the RC.

In phase 3 (7.5 years, extended mission): Δν and νmax and ΔP are measure with an increased precision for all red giants and for subgiants with 8 ≤ mV ≤ 11. .

The Continuous viewing zone:

The survey of Chronos gives access to a continuous viewing zone (CVZ), two cones of angular diameter 22.5 deg, about the size of the of TESS CVZ (24 deg) and in the same region as TESS (Eccliptic pôles). The scientific yield for stars observed 5 years in the CVZ will outpass all projects, with many more objects to be observed much longer in all regions of the HR diagram. For these stars, the most precise seismic analysis will be performed: analysis of the signatures of helium, rotation and convection for main-sequence stars; core properties and rotation profiles for evolved stars.

Observation As mentioned previously, the need for photometric surveys covering a large part of the sky over long duration has been taken into account as much as possible in various existing and planned space and ground-based projects. However, there are strong practical difficulties to optimize observation for such a large range of contradictory constraints. We show here that none of these projects is suited for the objectives fixed in our present scientific proposal. This is understandable since projects usually had to optimize in first place other scientific objectives with other instrumental constraints, like the search for exoplanets.

CoRoT, Kepler and K2: CoRoT and Kepler have brought wonderful data for red giant stars, because the duration of the runs was at least several months and up to years in the case of Kepler. But this long duration was paid in both cases by a limited coverage of the sky. The K2 extension of the Kepler mission allowed to enlarge the sky coverage considerably, (although it was still not complete) but the duration of the runs was limited to 80 days, which is not enough to derive the seismic indices needed in the present project.

TESS: With 86% of the all-sky accessible, TESS [START_REF] Ricker | Transiting Exoplanet Survey Satellite (TESS)[END_REF] is the project offering presently the largest sky coverage. However, the primary goal of TESS is to discover planets smaller than Neptune that transit stars bright enough to enable follow-up spectroscopic observations that can provide planet masses and atmospheric compositions, with short revolution periods and most of the sky (63%) will be observed only 27 days within the 2-year nominal mission and only 23% will be observed 54 days or more. This duration is too short to measure seismic indices for red giant stars.

PLATO: PLATO mission profile [START_REF] Rauer | The PLATO mission[END_REF]) is organized around 2 long duration fields totalizing 4400 deg 2 to be observed during 2 to 3 years each. Then, it is planned to complete the observations with a few step and stare fields of shorter duration in order to cover about half the sky. Considering the scientific objective considered here, the two long duration field will bring valuable data for red giants, but on a limited fraction of the sky. The step and stare fields, if they have to increase this sky coverage significantly, say to half the sky, will hardly reach more than 2.5 months each. This would be a great survey in terms of solar-type Main Sequence pulsators, but this duration is too short to properly estimate seismic indices like ∆P for red giants.

LSST:

The LSST survey (LSST Science Collaboration et al. 2009) will cover about half the sky (southern hemisphere) with two consecutive 15s measurements every 3 days at best. Such a sampling rate is not suited for the red giant stars considered in the present project. On the other hand, it will be very efficient for AGB stars or very large red giants near the tip of the red giant branch, which require very long duration observations (7-10 years) as planned for LSST.

Chronos:

The present project, offering an all-sky survey with long duration (2x3.75 months within the 5-year nominal mission) will bridge the gap left between LSST and PLATO, bringing seismic indices for solar-type pulsators from subgiants to early AGB stages (~half a million red giants and subgiants), plus classical pulsators on the Main Sequence (δ Scuti, γ Doradus, β Cephei and SPB stars).

Project

5) Possible instrumental improvements and technology challenges

The technology considered here to support the Chronos Project is inherited from the various photometric surveys realized or designed over the last decades (CoRoT, Kepler, TESS, PLATO) and from the Gaia mission for the spinning platform and the time delay integration (TDI) readout of the CCDs. We thus are reasonably confident in the possibility to reach the performances anticipated, even if there is probably room to optimize the instrumental concept considered here.

An orbit allowing longer individual continuous runs (beyond 3.75 months):

One interesting possible significant improvement would be to find an orbit allowing uninterrupted observations longer than the 3.75 months considered in the present version. Compared with the present 2 x 3.75 months sequences, an uninterrupted sequence would allow not only to measure ∆P but also ∆π1, the asymptotic period spacing which is a very sensitive indicator for evolution but requires the measurement of g-dominated mixed modes with long lifetime. These modes appear as coherent pulsations and are unresolved in the Fourier spectrum even for long durations of several years. In such a case, continuous runs of e.g. 7.5 months would be better than discontinuous (2 x 3.75 months).

Increase the duration would require a change in the present concept that is challenging. Among possibilities we could think of:

-an orbit eclipsing the Sun: the length of individual runs DC* (Dc*=3.75 months in the present setup) is determined by the will not to have the Sun coming closer than GA=22.5 deg from the observed corona. This constraint could be relaxed (and Dc* increased without limitation) if it were possible to operate an orbit where the platform is in the shadow of the earth (or of an artificial shield) at least during the 1-2 months when the Sun crosses the corona (possibly all the time).

-positioning the platform further from the Sun. For instance, at Mars L2 point (or any where at Mars distance from the Sun): this would naturaly increase Dc* by a factor ~2 (year_Mars/Year_Earth=1.9) without changing the mission observational strategy.

An adaptation of the TDI mode allowing an increase of the angular size of the coronas:

In the present design, corona are limited to 22.5 deg width. Enlarging this width (on the side opposite to the Sun) would improve the sky coverage performances. This increase however would lead stars images to describe curved trajectories on the CCDs instead of staying in the same ~4 columns. The solution consisting in increasing the number of columns considered for each star is a possibility but rapidly becomes unsatisfying (overlap of the images of different stars).

A possibility would be:

-adapt the TDI readout mode to take into account the curvature of the trajectory of star images when crossing the CCD. This would mean attributing to each star the photoelectrons collected in different columns as time varies. This approach has to be tested, but the fact that we stay on one given corona for several months should help since the trajectories are not expected to change at first order. We note that this option would also relax the constraints on the alignement of the CCDs (1' in the present design to keep the trajectory within the same 5 columns) on the stability of the orientation of the rotation axis (also 1' in the present design).

6) Summary and conclusion in a broad scientific perspective

The period 2035-50 considered for the Voyage ESA long-term plan lies at the crossroad of several major progresses in the characterization of the stellar content of our Galaxy. Gaia and the various spectroscopic large surveys are building up an unprecedented census in terms of astrometric, kinematic and chemical properties of the Galactic stellar populations. Within a decade, precise measurements of these properties will thus be available for hundreds of millions of stars over a large part of our Galaxy. Meanwhile, stellar time domain surveys initiated by CoRoT and Kepler and continued by space missions like TESS and PLATO or ground based projects like LSST will have brought stellar seismology and stellar seismic characterization to a high level of maturity. The synergy between seismic information and the astrometric one and the great interest for Galactic population studies has already been illustrated with tens of thousands of red giant stars and we can foresee its extension to other types of stars. This is why we are convinced that an all-sky, high-cadence, long-duration variability survey will be a top scientific priority in the 2035-50 period.

The Chronos concept presented here, constitutes a time-domain extension of Gaia. It will allow mass and age estimates for half a million red giants within 1.7 kpc from the Sun and shed a new light on our understanding of the Galactic dynamics and archaeology.

In terms of characteristic time scales, Chronos will bridge up the gap between PLATO and LSST. It will outpass all previous time-domain projects in terms of sky coverage (100%) and duration (2 x 3.75 months within the 5-year nominal mission).

Fig. 1

 1 Fig.1 left: The 22.5x360 deg 2 sky corona swept by the instrument every Ts=2 minutes (rotation period of the platform 8 min). Right: the same corona is observed during Dc, keeping the Sun away by the guard angle GA.

  Gaia mission(Gaia Collab. et al. 2018a) combined with ongoing and soon-to-start large spectroscopic surveys, such as APOGEE[START_REF] Majewski | The Apache Point Observatory Galactic Evolution Experiment (APOGEE)[END_REF], WEAVE[START_REF] Dalton | WEAVE: the next generation wide-field spectroscopy facility for the William Herschel Telescope[END_REF]) or 4MOST (de[START_REF] Jong | 4MOST: Project overview and information for the First Call for Proposals[END_REF]. In this context and in the 2035-50 perspective, we illustrate how this understanding can be dramatically improved with information about mass, age, rotation, etc…brought by a variability survey(Anders et al. 2017ab, Rendle et al. 2019, Valentini et al. 2019).

Programmatic context: Comparison with existing and planned space and ground based projects.

  Tab.2 limit mV magnitude for Δν and νmax measurement, for different observation duration and different regions of the HR diagram(Mosser et al. 2019). Green indicates the matching with the 8-11 mV range observable with Chronos in its present version.

		duration	MS	MS	Subgiant	Early RGB Red clump Early AGB
			type G	type F
		1x	8.0	8.8	9.5	10.5	12.0	11.1
	3.5 months	2x	8.6	9.5	10.2	11.0	13.3	12.4
		3x	9.2	10.2	10.8	11.5	14.6	13.7
	5 years	CVZ	11.3	12.4	12.9	13.5	16.0	15.1

b.

  Comparison of Chronos main figures with other past and expected projects

		Sky coverage	Observation length and sky fraction	Cadence of observation and
		(%)			number of observed stars
			Min	Max	SC	LC
	CoRoT	0.20	1 month	5 months	32 sec	8.5 min
					~ 1k	~ 150k
	Kepler	0.25	1 month	4 years	2 min	30 min
					~ 1k	~ 150k
	K2	2.5	80 days	~ 5k	~ 2M
	TESS	86	27days (63%)	1 year (CVZ ~ 2 %)	2 min	30 min
			54days (23%)		~ 200k	~1M
	LSST	~ 50	10 years	~ 3 days
	PLATO	~ 50	2 months (50%)	3 years (10%)	1 min	10 min
						~1M
	Chronos	100	2 x 3.75 months	> 5 years	2 min
			(100%)	(CVZ ~ 2 %)	1M
	Tab.3


Simon Gravelle

Christophe Ybert

Flow-Induced Shift of the Donnan Equilibrium for Ultra-Sensitive Mass Transport Measurement Through a Single Nanochannel

Despite mass flow is arguably the most elementary transport associated to nanofluidics, its measurement still constitutes a significant bottleneck for the development of this promising field. Here, we investigate how a liquid flow perturbs the ubiquitous enrichment -or depletion-of a solute inside a single nanochannel. Using Fluorescence Correlation Spectroscopy to access the local solute concentration, we demonstrate that the initial enrichment -the so-called Donnan equilibrium-is depleted under flow thus revealing the underlying mass transport. Combining theoretical and numerical calculations beyond the classical 1D treatments of nanochannels, we rationalize quantitatively our observations and demonstrate unprecedented flow rate sensitivity. Because the present mass transport investigations are based on generic effects, we believe they can develop into a versatile approach for nanofluidics.

Nanofluidics, the fluidic transport in nanoscale confined systems, is encountered in a wide range of situations spanning from biological systems to filtration and colloidal science [START_REF] Spiegler | Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient membranes[END_REF][START_REF] Gross | Membrane Transport Characteristics of Ultrafine Capillaries[END_REF]. Despite its ubiquity and longlasting history, it has experienced a strong and fast growing interest over the past few years, thanks to our increasing ability to fabricate and characterize nanoscale systems [START_REF] Bocquet | Nanofluidics, from bulk to interfaces[END_REF][START_REF] Bocquet | Physics and technological aspects of nanofluidics[END_REF][START_REF] Segerink | Nanofluidics in point of care applications[END_REF]. This led notably to the identification of striking phenomena, among which giant permeability enhancement in carbon nanotubes [START_REF] Holt | Fast mass transport through sub-2-nanometer carbon nanotubes[END_REF][START_REF] Falk | Molecular Origin of Fast Water Transport in Carbon Nanotube Membranes: Superlubricity versus Curvature Dependent Friction[END_REF][START_REF] Secchi | Massive radius-dependent flow slippage in carbon nanotubes[END_REF], original energy transduction pathways [START_REF] Pennathur | Energy conversion in microsystems: is there a role for micro/nanofluidics?[END_REF][START_REF] Van Der Heyden | Power generation by pressure-driven transport of ions in nanofluidic channels[END_REF][START_REF] Siria | Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube[END_REF] or complex electrokinetic transports [START_REF] Daiguji | Nanofluidic diode and bipolar transistor[END_REF][START_REF] Karnik | Rectification of ionic current in a nanofluidic diode[END_REF][START_REF] Deng | Overlimiting current and shock electrodialysis in porous media[END_REF].

At the roots of these phenomena usually lies one major property of nanofluidics that originates from the enhanced surface effects: the intimate interplay between transports and drivings of different nature: electrical, hydrodynamic, solutal or thermal. So far however, only ionic transport is routinely captured at the scale of a single nanochannel [START_REF] Branton | The potential and challenges of nanopore sequencing[END_REF], while disentanglement of the physical mechanisms at stake would require independent access to the different transport currents. Surprisingly, the flow rate measurement at the scale of single nanofluidic object -arguably the most basic fluidic transport -still constitutes a serious bottleneck for quantitative studies.

In order to meet the sensitivity required for single nanochannel flow rate measurements, significant efforts have been made over the past few years. This includes electrical measurement approaches such as cross-correlation for nanochannels [START_REF] Mathwig | Electrical cross-correlation spectroscopy: Measuring picoliter-per-minute flows in nanochannels[END_REF] or Coulter counting for transmembrane nanopores [START_REF] Gadaleta | Ultra-sensitive flow measurement in individual nanopores through pressure-driven particle translocation[END_REF]. In addition to electrical approaches, important advances have been made using direct monitoring of volume change in reservoirs [START_REF] Sharma | A direct sensor to measure minute liquid flow rates[END_REF] or optical velocimetry approaches. For pipette-like geometries, the peculiar properties of the Landau-Squire nanojet have been used to access exit flow rates via optical-tweezer based vorticity maps [START_REF] Laohakunakorn | A Landau-Squire nanojet[END_REF] and more recently by PIV flow maps [START_REF] Secchi | Massive radius-dependent flow slippage in carbon nanotubes[END_REF] or chemical plumes [START_REF] Secchi | The Landau-Squire plume[END_REF]. Finally, for the benchmark geometry of nanochannels, a milestone in sensitivity, around a few fL/s, has been reached by some of the present authors by imposing fluorescent dye gradients along the system [START_REF] Lee | Osmotic flow through fully permeable nanochannels[END_REF]. Despite such advances, no approach has currently emerged that could address the various needs in terms of experimental configuration or required sensitivity. Exploring alternative routes on mass transport signature in nanofluidic systems thus remains an important issue both from the fundamental point of view or from the practical perspective of developing the versatility of the nanoscale toolbox.

In this context, we consider here a generic property of solutes in confined systems, that are either enriched or depleted with respect to their bulk reservoir concentration. This enrichment (or depletion), called Donnan equilibrium, is due to specific solute-surface interactions whose characteristic scales are typically nanometric [START_REF] Bocquet | Nanofluidics, from bulk to interfaces[END_REF] and thus comparable with the confining scale. In the case of electrolytes under strong confinement, Debye layers overlap and only the counter-ion contributes to the ion concentration in the channel. This limit gives rise to a nanochannel ion-selectivity at the origin of striking nanofluidic properties [START_REF] Schoch | Transport phenomena in nanofluidics[END_REF]. Among these, ion concentration polarization effects, that develop under an electric field forcing, promise huge practical applications such as water desalination [START_REF] Kim | Direct seawater desalination by ion concentration polarization[END_REF] and purification [START_REF] Deng | Water purification by shock electrodialysis: Deionization, filtration, separation, and disinfection[END_REF]. Physically, such concentration gradients appear because conservation laws encompassed in the Nernst-Planck equation

∇ • (-D∇c -ZcD∇φ + uc) = 0, (1)
where D is the diffusion coefficient, k B T the thermal energy, φ = eV /(k B T ) the reduced electrical potential, u the velocity field and Z the species valence, have to be enforced at the junction between media with different properties -the reservoirs and the ion-selective nanochannel.

In the following, we examine a complementary configuration where a nanochannel responds to a pressure forcing rather than an electric one. Using Fluctuation Correlation Spectroscopy (FCS) to access the local dye concentration within a nanochannel, we show that convection induces a concentration depletion (or enrichment) that we refer to as a shift in the Donnan equilibrium. This concentration depletion/enrichment can be used to extract information about mass fluxes. This effect holds even far from the Debye layer overlap regime, and we present theoretical and numerical analysis to go beyond the classical 1-dimensional (1D) approaches adapted to highly confined systems. Overall quantitative agreement is achieved between experiments and calculations from which ultra-small single nanochannel flow rates can be deduced. Based on generic effects, we believe that both the physical mechanisms and the technique -highly sensitive and involving very low dye levels throughout the entire system-can open perspectives for future investigations in the field of nanofluidics.

Practically, measurements are performed inside slit nanochannels of height h 150 nm, width w = 5 µm and length L = 25 µm or 500 µm, engraved inside silicon chips as shown in Fig. 1 A (see Supp. Mat. for fabrication details). The system is filled with a potassium chloride (KCl) solution of concentration c s ranging from 10 -4 to 10 -2 M supplemented with c 0 = 10 -100 nM of Rhodamine 6G fluorescent dye (R6G). Under such conditions, the silicon oxide nanochannel walls are expected to bear a negative charge as experimentally verified in similar systems [START_REF] Lee | Osmotic flow through fully permeable nanochannels[END_REF], with wall potential in the range [-90, -50] mV. Because R6G is a cationic dye, a surface excess develops close to solid walls leading to a global concentration enhancement in the confined nanochannel. A pressure forcing ∆p is imposed between the ends of the nanofluidic system using either a liquid column or a pressure controller (Elveflow). The generated flow is deduced from the Hagen-Poiseuille relationship Q = ∆p wh 3 /(12ηL), with η the water viscosity, with a mean flow velocity u = Q/(hw).

The nanofluidic chip is located onto the stage of an inverted microscope (Nikon TE-2000U) with a water immersion objective (60x, NA=1.2). A parallel laser beam (CNI, 532 nm) filling the back pupil of the objective generates a diffraction-limited spot in the sample (diameter d ∼ 500 nm). A confocal excitation and detection pathway is used to collect the dye fluorescence intensity signal I(t), with the signal filtered through a confocal pinhole to reject out-of-focus contributions before being sent towards two avalanche photo-diode detectors (Perkin Elmer), see raw signal in Fig. 1 A. The collected raw intensity is analyzed through the autocorrelation function G(τ ) which contains information about the probe concentration and dynamics [START_REF] Magde | Fluorescence correlation spectroscopy. II. An experimental realization[END_REF][START_REF] Digman | Lessons in Fluctuation Correlation Spectroscopy[END_REF][START_REF] Koppel | Statistical accuracy in fluorescence correlation spectroscopy[END_REF] (Fig. 1 B).

Classical FCS velocimetry would rely on analyzing characteristic decay times in G(τ ) [START_REF] Palmer | Theory of sample translation in fluorescence correlation spectroscopy[END_REF][START_REF] Yordanov | Direct studies of liquid flows near solid surfaces by total inter-nal reflection fluorescence cross-correlation spectroscopy[END_REF]. In the present context however, such approach is meaningless as single point FCS velocimetry has a "poor" sensitivity threshold set by the -small-focal spot size u > D/d ∼ 1 mm/s. Accordingly, normalized autocorrelations obtained with or without pressure-driven flows superimpose perfectly showing no evolution in time scales, as represented in the inset of Fig. 1 B. Note that this master shape is rendered complex by the existence of reversible adsorption that gives rise to a slow time scale ∼ 10 ms atop the fast freediffusion one ∼ 10 µs [START_REF] Durand | Direct observation of transitions between surface-dominated and bulk diffusion regimes in nanochannels[END_REF][START_REF] Daniels | Dye diffusion at surfaces: Charge matters[END_REF][START_REF] Fowlkes | Single-molecule mobility in confined and crowded femtolitre chambers[END_REF][START_REF] Patel | Effect of surface interactions on diffusion of molecules: Fluorescence correlation spectroscopic investigation[END_REF] (see Supp. Mat. for further discussion).

Despite the flow having no measurable impact on time scales, it has a strong and straightforward effect on the measured autocorrelation functions inside the nanochannel, as can be seen in Fig. 1 B: the stronger the flow, the larger the amplitude of G(τ ). This effect is a signature of a flow-induced decrease of the dye concentration at the -fixed-measurement location within the nanochannel [START_REF] Krichevsky | Fluorescence correlation spectroscopy: the technique and its applications[END_REF]. Indeed the local concentration in the probed volume is known to obey reaches a constant value. Qualitatively, under quiescent conditions the nanochannel has a higher concentration c m set by the equilibrium between the confining slit and the neighboring reservoirs at c 0 . Under flow, the solute influx from reservoir is thus J = c 0 Q while being c m Q > J inside the nanochannel. This mismatch will induce polarization effects to arise in the system and overall the nanochannel solute content will decrease under flow. Under strong flows, incoming solute concentration will set the one in the nanochannel hence to c 0 . More quantitatively, the inset of Fig. 1 C presents the same concentration data, normalized according to (c -c 0 )/(c m -c 0 ) and in semi-log scale. As can be clearly seen, the experimental concentration decreases exponentially toward the reservoir concentration c 0 upon pressure difference increase.

c ∝ [G(τ → 0) -1] -1 .
Going further into the characterization, Fig. 2 shows how these flow-induced effects depend on the experimental parameters. Changing the nanochannel length from L = 25 µm to L = 500 µm, we recover the same qualitative concentration decay (Fig. 2 A). Quantitatively, the proper scale for concentration decay proves not to be the imposed pressure gradient ∆p/L, which sets the flow velocity u , but rather the absolute pressure difference ∆p (inset of Fig. 2 A). This suggests the Peclet number over the channel length to be the relevant variable Pe = u L/D with D = 4 • 10 -6 cm 2 /s the dye diffusion coefficient, yielding a sensitivity orders of magnitude (L/d) higher than for the standard point-like FCS velocimetry. Note that as will be shown hereafter, the location x ∈ [-L/2, L/2] where the concentration is probed enters as a normalizing factor α 1D = (1/2 -x/L) of the Peclet. Now changing the salt background concentration at fixed L, the exponential concentration decay with ∆p is preserved but not the data collapse, with faster probe concentration decay occurring under smaller salt concentration (Fig. 2 B). Such effect demonstrates that Peclet number alone is not sufficient to rationalize the full behavior of the nanosystems.

To progress further in the understanding, we now develop a theoretical description of the problem. As a first step, we consider the classical assumption of a 1D situation where all physical quantities only depends on the x position along the channel axis. The system considered is made of a nanochannel of length L connected to two reservoirs of length L R L. Inside the nanochannel, we assume a homogeneous Donnan electric potential V D , originating from the channel wall charges. From the reservoirs to the nanochannel, we consider that the electric potential linearly builds from zero to V D over thin transition regions of extent L, thus yielding a system made out of five different regions (see Supp. Mat. for details). Such 1D approach is in principle restricted to strong overlap conditions 2λ D /h 1, but is classically used beyond and has often proven effective [START_REF] Dydek | Overlimiting current in a microchannel[END_REF]. The solute concentration c 0 is imposed at the far ends x = ±(L R + + L/2) of the reservoirs, and a flow velocity u is imposed all along the system. Due to the difference in diffusivities -about one order of magnitude [START_REF] Harned | The Diffusion Coefficient of Potassium Chloride in Dilute Aqueous Solution[END_REF][START_REF] Culbertson | Diffusion coefficient measurements in microfluidic devices[END_REF] -, we will neglect convection effects for the salt, with electric potential profiles thus keeping their at-equilibrium characteristics. Formally this corresponds to restricting to moderate Peclet numbers such that Pe ≤ D/D s (see Supp. Mat. for numerical validation and details).

The 1D transport equation for a dye solute of charge +e follows from Eq. ( 1): uc -Dc ∂ x φ -D∂ x c = J 0 , with J 0 the solute current, independent of the x location. Such equation is readily solved in each five regions of the system and assuming concentration continuity between regions, the solution for the dye solute concentration is obtained analytically. In the limit where the reservoirs are large enough to avoid finite-size effects, i.e. (L R /L)Pe 1, the concentration inside the nanochannel takes a simple expression

c ≈ c 0 + c 0 e -φ D -1 e -uL/(2D) e ux/D . (2)
Following the concentration normalization introduced previously for experimental data, the theoretical prediction eventually reads

c(x) -c 0 c m -c 0 = e -α 1D (x)Pe , (3)
where the prefactor α 1D (x) = (1/2-x/L) accounts for the measurement location, and where the enriched nanochannel concentration without flow c m is given by the Donnan equilibrium with reservoirs c m = c 0 e -φ D . As can be seen in Fig. 3 A (middle row), this theoretical prediction Eq. ( 3) is in perfect agreement with a finite element calculation of the full problem (see Supp. Mat.) in the limit of strong Debye layer overlap 2λ D /h = 10, for which the 1D approximation is expected to stand. Coming back to experiments, Eq. ( 3) predicts the observed exponential decay with the Peclet number shown in Fig. 1 and2, and probed by varying either the driving pressure or the channel length. In addition, Eq. (3) accounts for the influence of the measuring location involved in Fig. 2 A. Moreover, for the lowest salt concentration c s = 0.1 mM for which the 1D assumption is approached (2λ D /h ∼ 0.5), Eq. ( 3) satisfactorily describes experimental data with no free parameter (Fig. 4 A).

Despite capturing the global form of the flow-induced solute concentration polarization inside the nanochannel, the previous theoretical description fails to encompass the effect of increasing salt concentration, thus pointing to a limitation of the 1D assumption, see Fig. 3 A (lower row) and Fig. 4 A. In order to capture this effect, inhomogeneous distributions of all physical quantities across the nanochannel height z need to be incorporated in the model. Starting from Eq. ( 1), with c, u, φ and J 0 now including a z-dependency, we followed a perturbation approach of the 2D problem looking for first order solution c c (0) + Pe c (1) of a Peclet development, in the limit of Pe 1.

Such approach provides the z-averaged concentration inside the nanochannel

c z = c 0 e -φ z + c 0 Pe ue -φ z u z -1 x L , (4)
where the Peclet number is now defined according to the z-averaged flow profile u z and where φ(z) is the x-invariant reduced electric potential without flow (see Supp. Mat. for details). Proceeding similarly for the 1D problem we find that the 2D solution can be matched onto the 1D expression providing Pe is replaced by α 2D Pe, with

α 2D = (c -c 0 )u z c -c 0 z u z , (5)
where c 0 is the reservoir concentration, and where we have used for the Donnan potential e -φ D = e -φ z . For larger Peclet number, we thus propose an ansatz solution which accounts for 2D effects in non-overlapping confinements, and that is based on the 1D Eq. ( 3):

c z -c 0 c m z -c 0 = e -α 1D (x)α 2D Pe . (6)
For the mapping factor α 2D , a simple estimate can be obtained by assuming weak surface potentials together with the additivity of opposite-wall contributions. From Eq. ( 5) and with u(z) = (z 2 -(h/2) 2 )∆p/(2Lη), we obtain α 2D 3[ h coth( h) -1]/ h2 that only depends on the reduced thickness h = h/(2λ D ).

As can be seen in Fig. 3, where our ansatz Eq. ( 6) is compared to finite elements results, the exponential decay in Pe is perfectly recovered whatever the overlapping parameter. Moreover, the simple estimate for the mapping factor α 2D well captures the effect of Debye layer confinement. Note however a moderate shift between analytical prediction and finite element results for 2λ D /h < 1 (Fig. 3 C). This deviation may result from some of the effects neglected in the present treatment such as: couplings between x and z directions, which are likely to occur at the channel entrances but were neglected in the theoretical calculations; a different flow field in the reservoirs as compared with the model assumption due to the large reservoirs height; etc.

Combining the theoretical description given by Eq.( 6), together with α 2D as estimated from finite element calculations, we are now able to extract an experimental flow velocity u exp from the measured flow-induced concentration polarization presented in Fig. 2. Comparing with the expected velocity u th , as estimated from the Poiseuille-Hagen relationship for known imposed pressures, we obtain an excellent agreement evidenced in Fig. 4 B. As mentioned, accounting for 2D effects across the channel height is essential for reliable flow measurements (Fig. 4 A)

In terms of flow rate sensitivity the present results reach sub-fL/s limits with Q 0.4 fL/s, making of the flowinduced shift in Donnan equilibrium an ultra-sensitive probe of mass transport in single nanochannels. Note that because effects actually depends on the flow velocity, reducing the channel width down to the focused spot size would push resolution down a decade without modifying the signal to noise ratio. Similarly, thought at as an integrated sensor in series with other nanofluidic systems, the system sensitivity can be further increased by varying the nanoslit length. Based on ubiquitous concentration polarization effects, its minimal probe requirements and sensitivity to transport processes, we believe that this work can contribute to the fast development of nanofluidics both for fundamental aspects and towards high impact applications.

FIG. 1 .

 1 FIG. 1. (A) Experimental setup: nanofluidic silicon chip (left); fluorescence detection configuration (middle); raw fluorescenceintensity fluctuations I(t) (right). (B) Fluorescence intensity auto-correlation function G(τ ) -1 under different pressuredriven flows (∆p = 0, 6 and 15 kPa, respectively Pe = 0, 30 and 70). Inset: same data normalized by the value at vanishing lagtime (τ = 10 -6 s). (C) Measured dye concentration vs pressure differences (L = 25 µm; cs = 10 mM). Inset: same data normalized (see text) and in semi-log scale.

Fig. 1 CFIG. 2 .

 12 FIG. 2. (A) Normalized probe concentration as a function of pressure gradient ∆p/L, for two nanochannel lengths and for cs = 0.1 mM. ( ): L = 25 µm probed in its middle x = 0; ( ): L = 500 µm probed at its entrance x = -L/2. Inset: same curves as a function of pressure difference (1/2 -x/L)∆p, see text for the position-dependent scaling factor. The full lines: prediction from Eq. (3). (B) Same as (A), for different salt concentrations and thus Debye layer overlap ratio 2λD/h, (L = 25 µm probed in their middle x = 0). ( ): cs = 1 mM salt concentration; (•): cs = 10 mM salt concentration. Inset: same data in semi-log scale.

FIG. 3 .

 3 FIG. 3. Finite element calculations. (A) z-averaged potential (up) and dye concentration profiles within the channel (top and middle: 2λD/h = 10; low: 2λD/h = 0.3). (-): equilibrium profiles under no convection; blue symbols: profiles under convection (Pe = 3). (--): 1D theoretical prediction Eq. (3); (-): 2D theoretical prediction Eq. (6). (B) z-averaged dye concentration at the middle location (x = 0) as a function of the Peclet number for various confinements (2λD/h ≈ 0.03, 0.1, 0.3, 1, 3, 10 and 30). (--): same as in (A). (C) Twodimensional correction α2D as a function of the confinement ratio 2λD/h. (-): analytical prediction (see text).

FIG. 4 .

 4 FIG. 4. Experimental velocities uexp from flow-induced concentration polarization measurements against Poiseuille-Hagen expectation u th . (A) Experimental velocities extracted from 1D theory Eq. (3). L = 500 µm: cs = 0.1 mM ( ); L = 25 µm: cs = 0.1 mM; ( ), cs = 1 mM ( ), cs = 10 mM (•). Full lines indicate uexp. ∝ u th. scaling. (B) Same as (A) with experimental velocities extracted from 2D theory Eq. (6).

ACKNOWLEDGMENTS

The authors thank R. Fulcrand, C. Lee and P. Joseph for Chips nanofabrication; L. Joly, C. Cottin-Bizonne and A.-L. Biance for fruitful discussions; L. Joly and M. Stojanova for comments on the manuscripts. S. G. acknowledge financial support from L. Bocquet through the European Research Council program Micromegas project.