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Pebble accretion is recognized as a significant accelerator of planet formation. Yet only formulae for single-sized (monodisperse) distribution have been derived in the literature. These can lead to significant underestimates for Bondi accretion, for which the best accreted pebble size may not be the one that dominates the mass distribution. We derive in this paper the polydisperse theory of pebble accretion. We consider a power-law distribution in pebble radius, and we find the resulting surface and volume number density distribution functions. We derive also the exact monodisperse analytical pebble accretion rate for which 3D accretion and 2D accretion are limits. In addition, we find analytical solutions to the polydisperse 2D Hill and 3D Bondi limits. We integrate the polydisperse pebble accretion numerically for the MRN distribution, finding a slight decrease (by an exact factor 3/7) in the Hill regime compared to the monodisperse case. In contrast, in the Bondi regime, we find accretion rates 1–2 orders of magnitude higher compared to monodisperse, also extending the onset of pebble accretion to 1–2 orders of magnitude lower in mass. We find megayear timescales, within the disk lifetime, for Bondi accretion on top of planetary seeds of masses 10⁻⁶to 10⁻⁴M_⊕, over amore »significant range of the parameter space. This mass range overlaps with the high-mass end of the planetesimal initial mass function, and thus pebble accretion is possible directly following formation by streaming instability. This alleviates the need for mutual planetesimal collisions as a major contribution to planetary growth.

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With the growing demand for locating services in a variety of commercial applications, positioning techniques have been considered as a vital part in cellular networks. With the evolution from 2G to 5G, the positioning techniques have been enhanced in various aspects. In this paper, we summarize the evolution of positioning standards in the Third Generation Partnership Project (3GPP) and briefly introduce the new positioning standards in 5G NR (New Radio), which include new positioning requirements, the general positioning structure, new positioning reference signals, and general positioning methods.

Rings and gaps are commonly observed in the dust continuum emission of young stellar discs. Previous studies have shown that substructures naturally develop in the weakly ionized gas of magnetized, non-ideal MHD discs. The gas rings are expected to trap large mm/cm-sized grains through pressure gradient-induced radial dust–gas drift. Using 2D (axisymmetric) MHD simulations that include ambipolar diffusion and dust grains of three representative sizes (1 mm, 3.3 mm, and 1 cm), we show that the grains indeed tend to drift radially relative to the gas towards the centres of the gas rings, at speeds much higher than in a smooth disc because of steeper pressure gradients. However, their spatial distribution is primarily controlled by meridional gas motions, which are typically much faster than the dust–gas drift. In particular, the grains that have settled near the mid-plane are carried rapidly inwards by a fast accretion stream to the inner edges of the gas rings, where they are lifted up by the gas flows diverted away from the mid-plane by a strong poloidal magnetic field. The flow pattern in our simulation provides an attractive explanation for the meridional flows recently inferred in HD 163296 and other discs, including both ‘collapsing’ regions where themore »gas near the disc surface converges towards the mid-plane and a disc wind. Our study highlights the prevalence of the potentially observable meridional flows associated with the gas substructure formation in non-ideal MHD discs and their crucial role in generating rings and gaps in dust.

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