A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution

We present a class of functions $\mathcal{K}$ in $C^0(\R)$ which is variant of the Knopp class of nowhere differentiable functions. We derive estimates which establish \mathcal{K} \sub C^{0,\al}(\R) for 0<\al<1 but no $K \in \mathcal{K}$ is pointwise anywhere improvable to C^{0,\be} for any \be>\al. In particular, all $K$'s are nowhere differentiable with derivatives singular distributions. $\mathcal{K}$ furnishes explicit realizations of the functional analytic result of Berezhnoi. Recently, the author and simulteously others laid the foundations of Vector-Valued Calculus of Variations in $L^\infty$ (Katzourakis), of $L^\infty$-Extremal Quasiconformal maps (Capogna and Raich, Katzourakis) and of Optimal Lipschitz Extensions of maps (Sheffield and Smart). The "Euler-Lagrange PDE" of Calculus of Variations in $L^\infty$ is the nonlinear nondivergence form Aronsson PDE with as special case the $\infty$-Laplacian. Using $\mathcal{K}$, we construct singular solutions for these PDEs. In the scalar case, we partially answered the open $C^1$ regularity problem of Viscosity Solutions to Aronsson's PDE (Katzourakis). In the vector case, the solutions can not be rigorously interpreted by existing PDE theories and justify our new theory of Contact solutions for fully nonlinear systems (Katzourakis). Validity of arguments of our new theory and failure of classical approaches both rely on the properties of $\mathcal{K}$.Comment: 5 figures, accepted to SeMA Journal (2012), to appea

A spectral survey of an ultra-hot Jupiter: Detection of metals in the transmission spectrum of KELT-9 b

Context: KELT-9 b exemplifies a newly emerging class of short-period gaseous exoplanets that tend to orbit hot, early type stars - termed ultra-hot Jupiters. The severe stellar irradiation heats their atmospheres to temperatures of $\sim 4,000$ K, similar to the photospheres of dwarf stars. Due to the absence of aerosols and complex molecular chemistry at such temperatures, these planets offer the potential of detailed chemical characterisation through transit and day-side spectroscopy. Studies of their chemical inventories may provide crucial constraints on their formation process and evolution history. Aims: To search the optical transmission spectrum of KELT-9 b for absorption lines by metals using the cross-correlation technique. Methods: We analyse 2 transits observed with the HARPS-N spectrograph. We use an isothermal equilibrium chemistry model to predict the transmission spectrum for each of the neutral and singly-ionized atoms with atomic numbers between 3 and 78. Of these, we identify the elements that are expected to have spectral lines in the visible wavelength range and use those as cross-correlation templates. Results: We detect absorption of Na I, Cr II, Sc II and Y II, and confirm previous detections of Mg I, Fe I, Fe II and Ti II. In addition, we find evidence of Ca I, Cr I, Co I, and Sr II that will require further observations to verify. The detected absorption lines are significantly deeper than model predictions, suggesting that material is transported to higher altitudes where the density is enhanced compared to a hydrostatic profile. There appears to be no significant blue-shift of the absorption spectrum due to a net day-to-night side wind. In particular, the strong Fe II feature is shifted by $0.18 \pm 0.27$ km~s$^{-1}$, consistent with zero. Using the orbital velocity of the planet we revise the steller and planetary masses and radii.Comment: Submitted to Astronomy and Astrophysics on January 18, 2019. Accepted on May 3, 2019. 26 pages, 11 figure

DREAM: III.A helium survey in exoplanets on the edge of the hot Neptune desert with GIANO-B@TNG

The population of close-in exoplanets features a desert of hot Neptunes whose origin is uncertain. These planets may have lost their atmosphere, eroding into mini-Neptunes and super-Earths. Direct observations of evaporating atmospheres are essential to derive mass-loss estimates and constrain this scenario. The metastable 1083.3nm HeI triplet represents a powerful diagnostic of atmospheric evaporation since it traces the hot gas in extended exoplanet atmospheres, is observable from the ground, and is weakly affected by interstellar medium absorption. We conducted a uniform HeI transmission spectroscopy survey, focusing on 9 planets located at the edges of the Neptunian desert, aiming to gain insights into the role of photo-evaporation in its formation. We observed one transit per planet using the high-resolution, near-infrared spectrograph GIANO-B on the Telescopio Nazionale Galileo. We focused our analysis on the HeI triplet by computing high-resolution transmission spectra. We then employed the p-winds model to interpret the observed transmission spectra. We found no sign of planetary absorption in the HeI triplet in any of the investigated targets. We thus provided 3sigma upper-limit estimations on the thermosphere absorption, temperature, and mass loss, and combined them with past measurements to search for correlations with parameters thought to be drivers in the formation of the HeI triplet. Our results strengthen the importance of performing homogeneous surveys and analyses to bring clarification in the HeI detection and hence in the Neptunian desert origin. Our findings corroborate the literature expectations that the HeI absorption signal correlates with the stellar mass and the received XUV flux. However, these trends seem to disappear in terms of mass-loss rates; further studies are essential to shed light on this aspect and to understand better the photo-evaporation process.Comment: 23 pages, 13 figures, accepted for publication in A&A, after language editin

Neutral Iron Emission Lines From The Day-side Of KELT-9b -- The GAPS Programme With HARPS-N At TNG XX

We present the first detection of atomic emission lines from the atmosphere of an exoplanet. We detect neutral iron lines from the day-side of KELT-9b (Teq $\sim$ 4, 000 K). We combined thousands of spectrally resolved lines observed during one night with the HARPS-N spectrograph (R $\sim$ 115, 000), mounted at the Telescopio Nazionale Galileo. We introduce a novel statistical approach to extract the planetary parameters from the binary mask cross-correlation analysis. We also adapt the concept of contribution function to the context of high spectral resolution observations, to identify the location in the planetary atmosphere where the detected emission originates. The average planetary line profile intersected by a stellar G2 binary mask was found in emission with a contrast of 84 $\pm$ 14 ppm relative to the planetary plus stellar continuum (40 $\pm$ 5$\%$ relative to the planetary continuum only). This result unambiguously indicates the presence of an atmospheric thermal inversion. Finally, assuming a modelled temperature profile previously published (Lothringer et al. 2018), we show that an iron abundance consistent with a few times the stellar value explains the data well. In this scenario, the iron emission originates at the $10^{-3}$-$10^{-5}$ bar level.Comment: Accepted for publication on ApJL; 19 pages, 4 figures, 3 table

Automatic model-based telluric correction for the ESPRESSO data reduction software. Model description and application to radial velocity computation

Ground-based high-resolution spectrographs are key instruments for several astrophysical domains. Unfortunately, the observed spectra are contaminated by the Earth's atmosphere. While different techniques exist to correct for telluric lines in exoplanet atmospheric studies, in radial velocity (RV) studies, telluric lines with an absorption depth of >2% are generally masked, which poses a problem for faint targets and M dwarfs as most of their RV content is present where telluric contamination is important. We propose a simple telluric model to be embedded in the ESPRESSO DRS. The goal is to provide telluric-free spectra and enable RV measurements, including spectral ranges where telluric lines fall. The model is a line-by-line radiative transfer code that assumes a single atmospheric layer. We use the sky conditions and the physical properties of the lines from HITRAN to create the telluric spectrum. A subset of selected telluric lines is used to robustly fit the spectrum through a Levenberg-Marquardt minimization algorithm. When applied to stellar spectra from A0- to M5-type stars, the residuals of the strongest H2O lines are below 2% for all spectral types, with the exception of M dwarfs, which are within the pseudo-continuum. We then determined the RVs from the telluric-corrected ESPRESSO spectra of Tau Ceti and Proxima. We created telluric-free masks and compared the obtained RVs with the DRS RVs. In the case of Tau Ceti, we identified that micro-telluric lines introduce systematics up to an amplitude of 58 cm/s and with a period of one year. For Proxima, the gain in spectral content at redder wavelengths is equivalent to a gain of 25% in photon noise. This leads to better constraints on the semi-amplitude and eccentricity of Proxima d. We showcase that our model can be applied to other molecules, and thus to other wavelength regions observed by other spectrographs, such as NIRPS.Comment: 18 pages, 18 figures, accepted to A&