Secular evolution of MHD wind-driven discs: analytical solutions in the expanded {\alpha}-framework

TL;DR

This paper develops analytical solutions for the evolution of protoplanetary discs driven by MHD winds, extending classical viscous models to include wind effects and exploring their impact on disc dispersal and evolution.

Contribution

It introduces an analytical framework for wind-driven disc evolution using an alpha-like parameter, extending Lynden-Bell and Pringle solutions to include magnetic wind effects.

Findings

Wind-driven accretion causes faster disc mass decline than viscous models.

Discs can disperse after finite time if magnetic field decay is slow.

Disc evolution depends on wind and turbulence parameters, affecting observable correlations.

Abstract

The evolution of protoplanetary discs and the related process of planet formation is regulated by angular momentum transport and mass-loss processes. Over the past decade, the paradigm of viscosity has been challenged and MHD disc winds appear as a compelling scenario to account for disc accretion. In this work, we aim to construct the equivalent of the widely used analytical description of viscous evolution for the MHD wind case. The transport of angular momentum and mass induced by the wind is parameterized by an $\alpha$-like parameter and by the magnetic lever arm parameter $\lambda$. Extensions of the paradigmatic Lynden-Bell and Pringle similarity solutions to the wind case are presented. We show that wind-driven accretion leads to a steeper decrease in the disc mass and accretion rate than in viscous models due to the absence of disc spreading. If the decline of the magnetic field strength is slower than that of the gas surface density, the disc is dispersed after a finite time. The evolution of the disc in the $\dot{M}_*-M_D$ plane is sensitive to the wind and turbulence parameters. A disc population evolving under the action of winds can exhibit a correlation between $\dot{M}_*$ and $M_D$ depending on the initial conditions. The simplified framework proposed in this work opens to a new avenue to test the effectiveness of wind-driven accretion from the observed disc demographics and constitutes an important step to include wind-driven accretion in planet population synthesis models.