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Context. Over the past 20 yr, the quietest areas of the solar surface have revealed a weak but extremely dynamic magnetism occurring at small scales (<500 km), which may provide an important contribution to the dynamics and energetics of the outer layers of the atmosphere. Understanding this magnetism requires the inference of physical quantities from high-sensitivity spectro-polarimetric data with high spatio-temporal resolution. Aims. We present high-precision spectro-polarimetric data with high spatial resolution (0.4") of the very quiet Sun at 1.56 mu m obtained with the GREGOR telescope to shed some light on this complex magnetism. Methods. We used inversion techniques in two main approaches. First, we assumed that the observed profiles can be reproduced with a constant magnetic field atmosphere embedded in a field-free medium. Second, we assumed that the resolution element has a substructure with either two constant magnetic atmospheres or a single magnetic atmosphere with gradients of the physical quantities along the optical depth, both coexisting with a global stray-light component. Results. Half of our observed quiet-Sun region is better explained by magnetic substructure within the resolution element. However, we cannot distinguish whether this substructure comes from gradients of the physical parameters along the line of sight or from horizontal gradients (across the surface). In these pixels, a model with two magnetic components is preferred, and we find two distinct magnetic field populations. The population with the larger filling factor has very weak (similar to 150 G) horizontal fields similar to those obtained in previous works. We demonstrate that the field vector of this population is not constrained by the observations, given the spatial resolution and polarimetric accuracy of our data. The topology of the other component with the smaller filling factor is constrained by the observations for field strengths above 250 G: we infer hG fields with inclinations and azimuth values compatible with an isotropic distribution. The filling factors are typically below 30%. We also find that the flux of the two polarities is not balanced. From the other half of the observed quiet-Sun area similar to 50% are two-lobed Stokes V profiles, meaning that 23% of the field of view can be adequately explained with a single constant magnetic field embedded in a non-magnetic atmosphere. The magnetic field vector and filling factor are reliable inferred in only 50% based on the regular profiles. Therefore, 12% of the field of view harbour hG fields with filling factors typically below 30%. At our present spatial resolution, 70% of the pixels apparently are non-magnetised.
Meter-sized ground-based telescopes are frequently used today for the follow-up of extrasolar planet candidates. While the transit signal of a Jupiter-sized object can typically be detected to a high level of confidence with small telescope apertures as well, the shallow transit dips of planets with the size of Neptune and smaller are more challenging to reveal. We employ new observational data to illustrate the photometric follow-up capabilities of meter-sized telescopes for shallow exoplanet transits. We describe in detail the capability of distinguishing the photometric signal of an exoplanet transit from an underlying trend in the light curve. The transit depths of the six targets we observed, Kepler-94b, Kepler-63b, K2-100b, K2-138b, K2-138c, and K2-138e, range from 3.9 ppt down to 0.3 ppt. For five targets of this sample, we provide the first ground-based photometric follow-up. The timing of three targets is precisely known from previous observations, and the timing of the other three targets is uncertain and we aim to constrain it. We detect or rule out the transit features significantly in single observations for the targets that show transits of 1.3 ppt or deeper. The shallower transit depths of two targets of 0.6 and 0.8 ppt were detected tentatively in single light curves, and were detected significantly by repeated observations. Only for the target of the shallowest transit depth of 0.3 ppt were we unable to draw a significant conclusion despite combining five individual light curves. An injection-recovery test on our real data shows that we detect transits of 1.3 ppt depth significantly in single light curves if the transit is fully covered, including out-of-transit data toward both sides, in some cases down to 0.7 ppt depth. For Kepler-94b, Kepler-63b, and K2-100b, we were able to verify the ephemeris. In the case of K2-138c with a 0.6 ppt deep transit, we were able to refine it, and in the case of K2-138e, we ruled out the transit in the time interval of more than ±1.5 σ of its current literature ephemeris.