Stress-driven two-phase integral elasticity for Timoshenko curved beams

In this research, the size-dependent static behaviour of elastic curved stubby beams is investigated by Timoshenko kinematics. Stress-driven two-phase integral elasticity is adopted to model size effects which soften or stiffen classical local responses. The corresponding governing equations of nonlocal elasticity are established and discussed, non-classical boundary conditions are detected and an effective coordinate-free solution procedure is proposed. The presented mixture approach is elucidated by solving simple curved small-scale beams of current interest in Nanotechnology. The contributed results could be useful for design and optimization of modern sensors and actuators.

Closed-formed solutions of geometric albedos and phase curves of exoplanets for any reflection law

The albedo of a celestial body is the frac-tion of light reflected by it. Studying the albe-dos of the planets and moons of the Solar Sys-tem dates back at least a century [1, 2, 3, 4, 5]. Of particular interest is the relationship between the albedo measured at superior conjunction (full phase), known as the “geometric albedo”, and the albedo considered over all phase angles, known as the “spherical albedo” [2, 6, 7]. Modern astronom-ical facilities enable the measurement of geomet-ric albedos from visible/optical secondary eclipses [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and the inference of the Bond albedo (spherical albedo measured over all wavelengths) from in-frared phase curves [21, 22, 23, 24, 25] of transit-ing exoplanets. Determining the relationship be-tween the geometric and spherical or Bond albe-dos usually involves complex numerical calculations [26, 27, 28, 29, 30, 31, 32] and closed-form solu-tions are restricted to simple reflection laws [33, 34]. Here we report the discovery of closed-form solu-tions for the geometric albedo and integral phase function that apply to any law of reflection. The integral phase function is used to obtain the phase integral, which is the ratio of the spherical to the geometric albedos. The generality of the solu-tions stems from a judicious choice of the coor-dinate system in which to perform different parts of the derivation. The closed-formed solutions have profound implications for interpreting obser-vations. The shape of a reflected light phase curve and the secondary eclipse depth may now be self-consistently inverted to retrieve fundamental phys-ical parameters (single-scattering albedo, scatter-ing asymmetry factor, cloud cover). Fully-Bayesian phase curve inversions for reflectance maps and si-multaneous light curve detrending may now be per-formed, without the need for binning in time, due to the efficiency of computation. We demonstrate these innovations for the hot Jupiter Kepler-7b, inferring a revised geometric albedo of 0.12 ± 0.02, a Bond albedo of 0.18 ± 0.03 and a phase integral of 1.5 ± 0.1, which is consistent with isotropic scatter-ing. The scattering asymmetry factor is 0.04±0.15, implying that the aerosols are small compared to the wavelengths probed by the Kepler space tele-scope. In the near future, one may use the closed-form solutions discovered here to extract funda-mental parameters, across wavelength, from multi-wavelength phase curves of both gas-giant and ter-restrial exoplanets measured by the James Webb Space Telescope.

Phase integral of asteroids

The values of the phase integral q were determined for asteroids using a numerical integration of the brightness phase functions over a wide phase-angle range and the relations between q and the G parameter of the HG function and q and the G1, G2 parameters of the HG1G2 function. The phase-integral values for asteroids of different geometric albedo range from 0.34 to 0.54 with an average value of 0.44. These values can be used for the determination of the Bond albedo of asteroids. Estimates for the phase-integral values using the G1 and G2 parameters are in very good agreement with the available observational data. We recommend using the HG1G2 function for the determination of the phase integral. Comparison of the phase integrals of asteroids and planetary satellites shows that asteroids have systematically lower values of q.