Helioseismology: a fantastic tool to probe the interior of the Sun

Helioseismology, the study of global solar oscillations, has proved to be an extremely powerful tool for the investigation of the internal structure and dynamics of the Sun. Studies of time changes in frequency observations of solar oscillations from helioseismology experiments on Earth and in space have shown, for example, that the Sun's shape varies over solar cycle timescales. In particular, far-reaching inferences about the Sun have been obtained by applying inversion techniques to observations of frequencies of oscillations. The results, so far, have shown that the solar structure is remarkably close to the predictions of the standard solar model and, recently, that the near-surface region can be probed with sufficiently high spatial resolution as to allow investigations of the equation of state and of the solar envelope helium abundance. The same helioseismic inversion methods can be applied to the rotational frequency splittings to deduce with high accuracy the internal rotation velocity of the Sun, as function of radius and latitude. This also allows us to study some global astrophysical properties of the Sun, such as the angular momentum, the grativational quadrupole moment and the effect of distortion induced on the surface (oblateness). The helioseismic approach and what we have learnt from it during the last decades about the interior of the Sun are reviewed here.Comment: 36 page

Abstract densities and ideals of sets

Abstract upper densities are monotone and subadditive functions from the power set of positive integers to the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density. We answer a question posed by G. Grekos in 2013, and prove the existence of translation invariant abstract upper densities onto the unit interval, whose null sets are precisely the family of finite sets, or the family of sequences whose series of reciprocals converge. We also show that no such density can be atomless. (More generally, these results also hold for a large class of summable ideals.