Exploring the conditions for forming planetesimals via the streaming instability and planetary systems via pebble accretion

¹ Center for Star and Planet Formation, Globe Institute, University of Copenhagen, Øster Voldgade 5-7, 1350 Copenhagen, Denmark
² Division of Astrophysics, Department of Physics, Lund University, Box 118, 22100 Lund, Sweden
³ New Mexico State University, Department of Astronomy, PO Box 30001 MSC 4500, Las Cruces, NM 88001, USA

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 26 December 2025
Accepted: 28 April 2026

Abstract

The streaming instability and pebble accretion are two physical mechanisms with demonstrated potentials to drive, respectively, the formation of planetesimals and the growth of planetary systems containing a diverse range of planetary types. Here we explore the protoplanetary disc conditions in terms of turbulence strength, Stokes number, and initial disc size that are needed to (i) form planetesimals by the streaming instability, (ii) form gas giant planets in cold orbits, (iii) form super-Earths and sub-Neptunes close to the star, and (iv) form rocky planet embryos in temperate orbits. We identify an optimum Stokes number range between St = 0.01 and St = 0.03 where all three planetary classes form and where the streaming instability is triggered for a slightly elevated pebble metallicity. Cold gas giants require a turbulence strength of at most δ = 10⁻⁴ and furthermore need large initial disc sizes to benefit from a prolonged pebble flux; super-Earths and rocky planet embryos tolerate higher turbulence strengths similar to those measured for the vertical shear instability. A higher Stokes number of St = 0.1 is detrimental to the formation of cold gas giants due to the short-lived pebble flux. For Stokes numbers below St = 0.003, extremely low values of turbulence (δ < 10⁻⁵) are required to form cold gas giants. We highlight how loss of gas to disc winds, reduction in the migration speed by thermal or dynamical torques, or the presence of pressure bumps in the outer disc could increase the parameter space for the formation of cold gas giants. We derive analytically that the mass of the largest planetesimals formed by the streaming instability is of similar magnitude to the threshold mass beyond which pebble accretion becomes efficient, if planetesimals form in the earliest phases of protoplanetary disc evolution.