All functions are (locally) ss-harmonic (up to a small error) - and applications

The classical and the fractional Laplacians exhibit a number of similarities, but also some rather striking, and sometimes surprising, structural differences. A quite important example of these differences is that any function (regardless of its shape) can be locally approximated by functions with locally vanishing fractional Laplacian, as it was recently proved by Serena Dipierro, Ovidiu Savin and myself. This informal note is an exposition of this result and of some of its consequences

Spin-Orbit and Tensor Forces in Heavy-quark Light-quark Mesons: Implications of the New Ds state at 2.32 GeV

We consider the spectroscopy of heavy-quark light-quark mesons with a simple model based on the non-relativistic reduction of vector and scalar exchange between fermions. Four forces are induced: the spin-orbit forces on the light and heavy quark spins, the tensor force, and a spin-spin force. If the vector force is Coulombic, the spin-spin force is a contact interaction, and the tensor force and spin-orbit force on the heavy quark to order $1/m_1m_2$ are directly proportional. As a result, just two independent parameters characterize these perturbations. The measurement of the masses of three p-wave states suffices to predict the mass of the fourth. This technique is applied to the $D_s$ system, where the newly discovered state at 2.32 GeV provides the third measured level, and to the $D$ system. The mixing of the two $J^P=1^+$ p-wave states is reflected in their widths and provides additional constraints. The resulting picture is at odds with previous expectations and raises new puzzles.Comment: 6 pages, 1 figur

A Triple Protostar System Formed via Fragmentation of a Gravitationally Unstable Disk

Binary and multiple star systems are a frequent outcome of the star formation process, and as a result, almost half of all sun-like stars have at least one companion star. Theoretical studies indicate that there are two main pathways that can operate concurrently to form binary/multiple star systems: large scale fragmentation of turbulent gas cores and filaments or smaller scale fragmentation of a massive protostellar disk due to gravitational instability. Observational evidence for turbulent fragmentation on scales of $>$1000~AU has recently emerged. Previous evidence for disk fragmentation was limited to inferences based on the separations of more-evolved pre-main sequence and protostellar multiple systems. The triple protostar system L1448 IRS3B is an ideal candidate to search for evidence of disk fragmentation. L1448 IRS3B is in an early phase of the star formation process, likely less than 150,000 years in age, and all protostars in the system are separated by $<$200~AU. Here we report observations of dust and molecular gas emission that reveal a disk with spiral structure surrounding the three protostars. Two protostars near the center of the disk are separated by 61 AU, and a tertiary protostar is coincident with a spiral arm in the outer disk at a 183 AU separation. The inferred mass of the central pair of protostellar objects is $\sim$1 M$_{sun}$, while the disk surrounding the three protostars has a total mass of $\sim$0.30 M_{\sun}. The tertiary protostar itself has a minimum mass of $\sim$0.085 M$_{sun}$. We demonstrate that the disk around L1448 IRS3B appears susceptible to disk fragmentation at radii between 150~AU and 320~AU, overlapping with the location of the tertiary protostar. This is consistent with models for a protostellar disk that has recently undergone gravitational instability, spawning one or two companion stars.Comment: Published in Nature on Oct. 27th. 24 pages, 8 figure

A dust and gas cavity in the disc around CQ Tau revealed by ALMA

The combination of high-resolution and sensitivity offered by ALMA is revolutionizing our understanding of protoplanetary discs, as their bulk gas and dust distributions can be studied independently. In this paper we present resolved ALMA observations of the continuum emission (\u3bb = 1.3 mm) and CO isotopologues (12CO, 13CO, C18O, J = 2 12 1) integrated intensity from the disc around the nearby (d = 162 pc), intermediate-mass (M = 1.67 M) pre-main-sequence star CQ Tau. The data show an inner depression in continuum and in both 13CO and C18O emission. We employ a thermo-chemical model of the disc reproducing both continuum and gas radial intensity profiles, together with the disc spectral energy distribution. The models show that a gas inner cavity with size between 15 and 25 au is needed to reproduce the data with a density depletion factor between 3c10 121 and 3c10 123. The radial profile of the distinct cavity in the dust continuum is described by a Gaussian ring centred at Rdust = 53 au and with a width of \u3c3 = 13 au. Three-dimensional gas and dust numerical simulations of a disc with an embedded planet at a separation from the central star of 3c20 au and with a mass of 3c6\u20139 MJup reproduce qualitatively the gas and dust profiles of the CQ Tau disc. However, a one-planet model appears not to be able to reproduce the dust Gaussian density profile predicted using the thermo-chemical modeling

Circumstellar discs: What will be next?

This prospective chapter gives our view on the evolution of the study of circumstellar discs within the next 20 years from both observational and theoretical sides. We first present the expected improvements in our knowledge of protoplanetary discs as for their masses, sizes, chemistry, the presence of planets as well as the evolutionary processes shaping these discs. We then explore the older debris disc stage and explain what will be learnt concerning their birth, the intrinsic links between these discs and planets, the hot dust and the gas detected around main sequence stars as well as discs around white dwarfs.Comment: invited review; comments welcome (32 pages

The newborn planet population emerging from ring-like structures in discs

ALMA has observed a plethora of ring-like structures in planet forming discs at distances of 10-100 au from their host star. Although several mechanisms have been invoked to explain the origin of such rings, a common explanation is that they trace new-born planets. Under the planetary hypothesis, a natural question is how to reconcile the apparently high frequency of gap-carving planets at 10-100 au with the paucity of Jupiter mass planets observed around main sequence stars at those separations. Here, we provide an analysis of the new-born planet population emerging from observations of gaps in discs, under the assumption that the observed gaps are due to planets. We use a simple estimate of the planet mass based on the gap morphology, and apply it to a sample of gaps recently obtained by us in a survey of Taurus with ALMA. We also include additional data from recent published surveys, thus analysing the largest gap sample to date, for a total of 48 gaps. The properties of the purported planets occupy a distinctively different region of parameter space with respect to the known exo-planet population, currently not accessible through planet finding methods. Thus, no discrepancy in the mass and radius distribution of the two populations can be claimed at this stage. We show that the mass of the inferred planets conforms to the theoretically expected trend for the minimum planet mass needed to carve a dust gap. Finally, we estimate the separation and mass of the putative planets after accounting for migration and accretion, for a range of evolutionary times, finding a good match with the distribution of cold Jupiters