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In recent years, a number of eccentric debris belts have been observed in extrasolar systems. The most common explanation for their shape is the presence of a nearby eccentric planetary companion. The gravitational perturbation from such a companion would induce periodic eccentricity variations on the planetesimals in the belt, with a range of precession frequencies. The overall expected shape is an eccentric belt with a finite minimum width. However, several observed eccentric debris discs have been found to exhibit a narrower width than the theoretical expectation. In this paper, we study two mechanisms that can produce this small width: (i) the protoplanetary disc can interact with the planet and/or the planetesimals, slowly driving the eccentricity of the former and damping the eccentricities of the latter; and (ii) the companion planet could have gained its eccentricity stochastically, through planet–planet scatterings. We show that under appropriate conditions, both of these scenarios offer a plausible way to reduce the minimum width of an eccentric belt exterior to a perturbing planet. However, the effects of protoplanetary discs are diminished at large separations (a > 10 au) due to the scarcity of gas and the limited disc lifetime. These findings suggest that one canmore »use the shape and width of debris discs to shed light on the evolution of extrasolar systems, constraining the protoplanetary disc properties and the prevalence of planet–planet scatterings. Further observations of debris-harbouring systems could confirm whether thin debris belts are a common occurrence, or the results of rare initial conditions or evolutionary processes.

Planets migrating in their natal discs can be captured into mean-motion resonance (MMR), in which the planets’ periods are related by integer ratios. Recent observations indicate that planets in MMR can be either apsidally aligned or anti-aligned. How these different configurations arise is unclear. In this paper, we study the MMR capture process of migrating planets, focusing on the property of the apsidal angles of the captured planets. We show that the standard picture of MMR capture, in which the planets undergo convergent migration and experience eccentricity damping due to planet–disc interactions, always leads to apsidal anti-alignment of the captured planets. However, when the planets experience eccentricity driving from the disc, apsidally aligned configuration in MMR can be produced. In this configuration, both planets’ resonance angles circulate, but a ‘mixed’ resonance angle librates and traps the planets near the nominal resonance location. The MMR capture process in the presence of disc eccentricity driving is generally complex and irregular, and can lead to various outcomes, including apsidal alignment and anti-alignment, as well as the disruption of the resonance. We suggest that the two resonant planets in the K2-19 system, with their moderate eccentricities and aligned apsides, have experienced eccentricity drivingmore »from their natal disc in the past.