The effect of sculpting planets on the steepness of debris-disc inner edges

Debris discs are our best means to probe the outer regions of planetary systems. Many studies assume that planets lie at the inner edges of debris discs, akin to Neptune and the Kuiper Belt, and use the disc morphologies to constrain those otherwise-undetectable planets. However, this produces a degeneracy in planet mass and semimajor axis. We investigate the effect of a sculpting planet on the radial surface-density profile at the disc inner edge, and show that this degeneracy can be broken by considering the steepness of the edge profile. Like previous studies, we show that a planet on a circular orbit ejects unstable debris and excites surviving material through mean-motion resonances. For a non-migrating, circular-orbit planet, in the case where collisions are negligible, the steepness of the disc inner edge depends on the planet-to-star mass ratio and the initial-disc excitation level. We provide a simple analytic model to infer planet properties from the steepness of ALMA-resolved disc edges. We also perform a collisional analysis, showing that a purely planet-sculpted disc would be distinguishable from a purely collisional disc and that, whilst collisions flatten planet-sculpted edges, they are unlikely to fully erase a planet's signature. Finally, we apply our results to ALMA-resolved debris discs and show that, whilst many inner edges are too steep to be explained by collisions alone, they are too flat to arise through completed sculpting by non-migrating, circular-orbit planets. We discuss implications of this for the architectures, histories and dynamics in the outer regions of planetary systems.


INTRODUCTION
Debris discs, like the Solar System's Asteroid and Kuiper Belts, are circumstellar populations of sub-planet-mass objects (Wyatt 2008).These objects undergo destructive collisions that release observable dust, and such dust is detected around 20% of main-sequence stars (Hughes et al. 2018;Wyatt 2020).Extrasolar debris discs can be resolved in scattered light and thermal emission (e.g.Esposito et al. 2020;Lovell et al. 2021;Booth et al. 2023), and are invaluable probes of processes in the outer regions of planetary systems; modern planetdetection techniques are insensitive to mid-sized planets orbiting at 10s or 100s of au from stars, but such planets can be inferred from their influence on observed debris (e.g.Mouillet et al. 1997;Wyatt et al. 1999;Quillen 2006;Pearce & Wyatt 2014;Sefilian et al. 2021;Stuber et al. 2023).The launch of the James Webb Space Telescope (JWST) has prompted renewed interest in such hypothetical planets, because if they exist, then many should be detectable by this facility for the first time (Pearce et al. 2022).
Many theoretical studies have used the shapes and locations of ★ E-mail: tim.pearce@warwick.ac.uk debris discs to infer the properties of unseen planets.If such planets orbit just interior to debris discs, and are responsible for sculpting debris-disc inner edges, then the shapes and locations of these edges can be used to infer the planets' minimum-possible mass, maximum semimajor axis and minimum eccentricity (e.g.Pearce & Wyatt 2014;Faramaz et al. 2019;Pearce et al. 2022).However, this approach suffers from several key unknowns.First, whilst Neptune dominates the Kuiper Belt's inner edge (Malhotra 1993), it is unclear whether planets actually do reside just interior to extrasolar debris discs.It could be that the disc shapes and locations are instead set by other, non-planetary processes, such as debris-debris collisions or system formation.Second, even if exoplanets do sculpt debris-disc edges, it is unclear whether they do so on fixed orbits (as often assumed in planet-disc-interaction studies), or whether they migrate over time.Third, even if planetary sculpting occurs, it is unclear whether this process has finished in observed discs (as often assumed), or whether disc edges are still evolving under planet-debris interactions.These distinctions have significant implications for the morphology of debris discs, and the masses and locations of any responsible planets today.Such questions are difficult to answer using only debris-disc shapes and inner-edge locations.
However, we can gain more insight by also considering the radial surface-density profile of debris-disc edges.If an edge were sculpted by a non-migrating planet, then this profile would have a characteristic steepness that depends on planet mass (Mustill & Wyatt 2012;Rodigas et al. 2014;Quillen 2006;Chiang et al. 2009).Conversely, if the edge were set by non-planetary processes like collisions, then its steepness would be different (Wyatt et al. 2007;Löhne et al. 2008;Geiler & Krivov 2017;Imaz Blanco et al. 2023).Similarly, the profiles of edges sculpted by migrating planets would be distinct from those due to non-migrating planets (Friebe et al. 2022), and edges set by combinations of both planetary and non-planetary effects would also have different profiles (Thebault et al. 2012;Nesvold & Kuchner 2015).The profiles of debris-disc edges are therefore powerful tools for probing the architectures, histories and processes in the outer regions of planetary systems.
Recently, several studies used ALMA to resolve the radial profiles of debris-disc edges (e.g.Lovell et al. 2021;Faramaz et al. 2021;Marino 2021).These observations trace millimetre-sized grains that are unaffected by radiation forces (Hughes et al. 2018), so offer our best means to probe the distribution of larger, unseen planetesimals in the discs.Such studies show that some debris discs have flatter inner edges that are consistent with pure collisional evolution, whilst others have steeper profiles potentially indicative of planetary sculpting (Imaz Blanco et al. 2023).Our aim in this paper is to thoroughly investigate the dynamical effects of sculpting planets on the steepness of debris-disc inner edges, and establish whether any of the observed ALMA steepnesses are consistent with planetary sculpting.We will show that, whilst many inner edges are too steep to be explained through collisions alone, they are too flat to be purely the result of completed sculpting by non-migrating, circular-orbit planets.This potentially implies that different processes, or combinations of processes, are actually operating in these discs.
Several previous studies also explored the effect of sculpting planets on the steepness of debris-disc inner edges (e.g.Quillen 2006;Quillen & Faber 2006;Chiang et al. 2009;Mustill & Wyatt 2012;Rodigas et al. 2014;Nesvold & Kuchner 2015;Regály et al. 2018).Our study differs from these in several ways.First, since we do not know debris-disc masses or the sizes of the largest planetesimals (Krivov & Wyatt 2021), both of which are critical for quantifying collisional evolution, we choose to decouple the dynamical and collisional modelling in this paper.Instead, we initially model the interaction between a planet and a collisionless disc, to isolate the effect of a planet on the edge steepness.We then apply a collisional model, to demonstrate how collisions would change the steepness from the planet-only regime.Second, we explore a very broad parameter space in both the dynamical and collisional simulations, to provide general predictive models that are valid across a wide range of scenarios.Third, we parameterise our results in a form that directly relates to those fitted to ALMA data, to facilitate the application of our results to upcoming observations.Finally, we apply our predictive model to recent ALMA observations of debris discs.
The layout of this paper is as follows.Section 2 describes our body simulations of a circular-orbit planet and a collisionless debris disc.Section 3 provides a predictive model relating the steepness of a disc's edge to the parameters of a circular-orbit planet, and specifically Section 3.2.1 gives a step-by-step method demonstrating how to infer an unseen sculpting planet from a debris disc's inner edge.Section 4 addresses how collisions would affect planet-sculpted edges.In Section 5 we apply our results to observed debris discs, and in Section 6 we discuss our results (including what the observed-disc profiles may be telling us about planetary systems).We conclude in Section 7. A single planet (black circle) is initialised just interior to a massless debris disc, which spans 1 to 30 Hill radii exterior to the planet orbit.

N-BODY SIMULATIONS
We use a large suite of -body simulations to explore the dynamical interaction between a circular-orbit planet and a collisionless debris disc.The effects of debris-debris collisions will be assessed later (Section 4).We describe the circular-planet simulation setup in Section 2.1, discuss two example simulations in Section 2.2, and present the surface-density profiles of all simulated discs in Section 2.3.We measure the debris-excitation level at the simulated inner edges in Section 2.4, and show this to arise through MMR interactions with the circular-orbit planet.In addition, we also extend the -body analyses to eccentric planets in Appendix B.

Setup of 𝑛-body simulations
The initial setup of our -body simulations with circular-orbit planets is shown on Figure 1.Each simulation comprises one star, one planet, and a disc of 20,000 massless debris particles.We run over 300 simulations, each with a different combination of star mass, planet mass, planet semimajor axis and initial-disc excitation level.
Each debris particle is initialised with a semimajor axis between 1 and 30 Hill radii ( H ) exterior to the planet; where  p is the planet semimajor axis and  p and  * are the planet and star masses respectively.The inner edge of 1 Hill radius ensures that Trojans are omitted, which would otherwise bias the axisymmetric surface-density profiles that we fit later (we discuss Trojans in Section 6.1.9).The outer edge of 30 Hill radii ensures that the outer edge does not affect the inner-edge profile; a circular-orbit planet is expected to scatter all non-resonant material originating within approximately 3 Hill radii of its orbit (Gladman 1993;Ida et al. 2000;Kirsh et al. 2009;Malhotra et al. 2021;Friebe et al. 2022), so setting the outer edge at 30 H ensures that this is well beyond the inner region.Debris semimajor axes are drawn such that the initial surface-density distribution goes as approximately  −1.5 (where  is stellocentric distance), akin to the Minimum-Mass Solar Nebula (MMSN; Weidenschilling 1977;Hayashi 1981).Each debris particle has an initial eccentricity uniformly drawn between 0 and a maximum value  max,0 , and an initial inclination (relative to the planet's orbital plane) uniformly drawn between 0 and  max,0 /2 radians .Each particle's initial longitude of ascending node, argument of pericentre and mean anomaly are uniformly drawn between 0 and 360 • .
To ensure that scattering is essentially complete by the end of the simulations, we set each simulation end time depending on the diffusion timescale  diff .The value  diff characterises the scattering timescale, and is roughly the time taken for a planet to significantly scatter or eject 70 per cent of material originating within 3 Hill radii of its orbit (Costa, Pearce & Krivov, submitted).The diffusion time for a planet acting on a body with semimajor axis  is where  p is the planet's orbital period (Tremaine 1993).Equivalently, We run each simulation for at least 10 diff calculated at  =  p , to ensure that scattering is essentially complete by the end.Simulations are run in rebound (Rein & Liu 2012) using whfast (a symplectic Wisdom-Holman integrator; Rein & Tamayo 2015;Wisdom & Holman 1991), with a timestep of 1 per cent of the planet's orbital period 1 .Simulations are conducted and analysed in the centre-of-mass frame.
We run simulations with star masses of 1 to 2 M ⊙ , planet-to-star mass ratios of 3 × 10 −5 to 10 −1 , planet semimajor axes of 1 to 100 au, and maximum initial debris eccentricities of  max,0 = 0.001 to 0.3.In addition, we also re-weight debris particles in post-simulation analyses, so we can test different initial surface-density profiles; we consider initial surface-density profiles between  0 and  −1.5 .We assume the planet and debris to be point-like particles; this lets us apply simple scaling laws to our results, but also means that the potential removal of debris via accretion rather than ejection is ignored in our simulations (see Morrison & Malhotra 2015).

Example simulation results
Figures 2 and 3 show two example simulations after 10 diffusion timescales.Both have a solar-mass star and a 2 M Jup planet ( p / * = 2 × 10 −3 ), which is on a circular orbit at 10 au.The simulation on Figure 2 starts with an initially unexcited debris disc ( max,0 = 0.01), and that on Figure 3 has the same setup but an initially excited disc ( max,0 = 0.1).The main qualitative features of these simulations are typical of all our runs.
The left panels of Figures 2 and 3 show the semimajor axes and eccentricities of the simulated bodies, which demonstrate the two main effects of a circular-orbit planet on debris.The planet: (i) scatters and ejects most debris coming within ∼ 3 Hill radii of its orbit; this clears the region above the solid black lines on the left panels.
(ii) excites the eccentricities of surviving debris via mean-motion resonances (MMRs), particularly near the inner edge of the sculpted disc.
The role of MMRs in exciting debris eccentricity is clearest in the initially unexcited-disc simulation (Figure 2, left panel).Here, populations of debris in the 2:1 and 3:2 MMRs are clearly visible.For the initially excited disc (Figure 3), these MMR populations are similar to those in the initially unexcited simulation, only now they are less distinct due to the higher intrinsic eccentricity of the 1 rebound does not define several orbital parameters in the case of a perfectly circular orbit.To ensure correct behaviour, for our 'circular-planet' simulations we actually implement a planet eccentricity of 10 −4 .disc.The evolution of debris inclination is much less significant than eccentricity; there is some very small inclination excitation at specific resonances in these simulations, but the disc still remains thin across its entire width.
The right panels of Figures 2 and 3 show the azimuthally averaged surface density profiles of the simulated debris discs (thin blue lines).The planet imposes a profile on the disc inner edge, whilst the central and outer regions retain their initial profiles.Individual, strong MMRs can also impose additional structure, such as the 2:1 MMR at 15.9 au on Figure 2; in this example that MMR does not affect the inner-edge fit, but it can do in other simulations.The thick orange lines are parametric fits to the surface-density profiles, as detailed in Section 2.3.
Figure 4 shows the final positions of bodies in the initially unexcited-disc simulation (that on Figure 2).The disc is almost axisymmetric, with a slight asymmetry between the directions aligned and anti-aligned with the planet due to MMR structure (an effect noted by Tabeshian & Wiegert 2016, 2017).This leads to a small difference in the steepness of the inner-edge profile between the two sides of the disc, which we discuss further in Section 6.2.1.For the initially excited-disc simulation from Figure 3, the asymmetry is less pronounced.

Surface-density profile fitting
We aim to quantify how the properties of a sculpting planet affect the steepness and location of the debris disc's inner edge.To proceed, we use a parametric model to fit the radial surface-density profiles of debris in the -body simulations, and examine how the model parameters change as functions of system properties.Section 2.3.1 describes the model, Section 2.3.2 the dependence of the fitted models on system parameters, and Section 2.3.3 the timescale for a disc edge takes to reach its final state.

Surface-density profile model
To quantify the disc surface-density profile, we follow the approach of Rafikov (2023).They show that, for a low-eccentricity disc with a sharp semimajor-axis cutoff at the outer edge, the radial surfacedensity profile around that edge can be characterised as2 Here  o characterises the radial location of the outer edge,  o is the 'flatness' of the surface-density profile at that edge, and erf is the Gauss error function: Equation 4 is an 'S-shape' profile centered on  o , which is steeper for smaller  o and flatter for larger  o .The value of  o is strongly linked to debris eccentricity; if debris has a sharp cutoff in semimajor axis, and its root-mean-square (rms) eccentricity is  rms , then Rafikov 2023).This means that, for discs with sharp cutoffs in semimajor axis, lower rms eccentricities mean steeper Figure 2. Simulation of a circular-orbit planet interacting with an exterior debris disc, for an initially weakly excited disc.The simulation is of a solar-mass star, a 2 M Jup planet on a circular orbit at 10 au ( p / * = 2 × 10 −3 ), and 20,000 massless debris particles initially spanning 1 to 30 Hill radii exterior to the planet, with an initial surface-density profile of approximately  −1.5 .Initial debris eccentricities were each uniformly drawn between 0 and 0.01, and the plot shows the final state after 8.7 × 10 5 yr (10 diffusion timescales).The planet clears nearby debris through scattering, and excites the eccentricities of surviving material through MMRs.Left panel: semimajor axes and eccentricities of the planet (large circle) and debris (small points, coloured by initial semimajor axis).Grey points show debris at the start of the simulation.Solid black lines are the minimum eccentricities required for debris to come within 3 Hill radii of the planet, and dashed grey lines are nominal MMR locations.Right panel: radial surface-density profile of the disc, Σ ( ).The thin blue line shows the simulation data, and the thick orange line is the fit (Equation 6).Insets show the inner-edge region in more detail.The fitted model parameters are shown on the right panel;  was fixed to 1.5 in the fit.
Figure 3. Simulation with the same setup as that on Figure 2, except that the debris disc is initially more excited (maximum initial eccentricities of 0.1, rather than the value of 0.01 on Figure 2).The planet still excites debris through MMRs (left panel), but these populations are less significant against the background excitation level of the disc.The resulting inner edge is flatter than in the low-initial-excitation case.
edges (and smaller  o ).If the eccentricities are uniformly spread from 0 to  max , then  rms =  max / √ 3 and hence  o =  max / √ 6.We will later show that a planet-sculpted disc does not have a sharp cutoff in semimajor axis at the inner edge, so the corresponding relationship between the edge profile and debris eccentricity is slightly modified, but the two remain strongly linked.
To parameterise the surface density across a whole disc, we use profiles similar to Equation 4 at the two edges, combined with an  −  profile describing the surface density between the edges.By multiplying these three local profiles together, we arrive at the following model for the overall surface-density profile: where subscripts i and o denote terms characterising the inner and outer edges respectively, and Σ 0 is the the surface density at  ≈  i .
We numerically fit a radial profile of this form to each of our simulated discs, treating Σ 0 ,  i ,  o ,  i ,  o as free parameters.We typically fix  to the initial surface-density index of the disc (i.e. = 1.5 for an initially MMSN profile).The fitting procedure is described in Appendix A, and the orange lines on the right panels of Figures 2 and 3 show the resulting fitted profiles for those simulations.2).The asterisk, filled black circle and solid line denote the star, planet and planet orbit respectively, and small points are debris (coloured by initial semimajor axis).The disc is almost axisymmetric, with a slight asymmetry between the left and right sides due to MMRs.This causes a small difference in the inner-edge steepness between the two sides; the right side (aligned with the planet position) has  i = 0.0286, whilst the value for the opposite side is 0.0354.

Dependence of the final inner-edge profile on system parameters
Having fitted surface-density profiles to each of our -body discs, we now assess how the profile of a planet-sculpted inner edge depends on system parameters.The inner-edge profiles are fully characterised by  i and  i , and in this section we show how those fitted parameters vary with simulation setup.
Figure 5 shows how the location of the inner edge of a planetsculpted disc,  i , scales with the planet-to-star mass ratio, planet semimajor axis, and the initial disc-excitation level.The location  i is just exterior to the 'chaotic zone' around the planet's circular orbit, which can be defined as either ∼ 3 Hill radii or via the Wisdom overlap criterion (Wisdom 1980).The value  i is larger for discs with higher initial-excitation levels, and kinks occur when  i is close to the nominal location of a strong MMR; this is especially true for discs with low initial eccentricities.Note that  i / p is independent of the planet's semimajor axis, and so depends only on the planet-to-star mass ratio and the disc's initial-excitation level.
Figure 6 shows how the flatness of the inner-edge profile,  i , scales with the planet-to-star mass ratio and the initial disc-excitation level.It is independent of all other parameters.The inner edges are generally steeper if sculpted by lower-mass planets, and flatter for higher-mass planets.The inner edges are also flattest for discs with the highest initial-excitation levels, although there is less of a dependence for discs with low initial-excitation levels.The relationships between  i and  p / * are also not smooth, but show complicated spikes.These spikes are not numerical effects, because they are replicated across simulations with the same mass ratios but different planet semimajor axes and star masses; the spikes actually occur when the inner edge is near the nominal location of a strong MMR.
The values of  i and  i on Figures 5 and 6 were fitted for discs with initial surface-density profiles of  −  , where  = 1.5, but they are actually independent of  for realistic setups.This is because the planet-induced edges are typically much steeper than the overall disc profiles.To check this, we re-scaled the simulated discs to have initially flat surface-density profiles (i.e. = 0) and re-fitted the edges.This resulted in values of  i and  i that are within a few percent of the  = 1.5 values, and hence the profiles of planet-sculpted edges do not strongly depend on the broader disc profiles.We further checked this analytically, and show that planet-sculpted inner edges should be much steeper than the overall disc provided that the initial-disc profile is flatter than  −4 (and also  4 ); this calculation is presented in Appendix C. Most measured debris-disc profiles are shallower than this, so the steepness of a planet-sculpted edge should typically not depend on the overall disc profile.

Edge-sculpting timescale
The timescale for a circular-orbit planet to sculpt the disc inner edge is expected to scale with the diffusion timescale (Equations 2 and 3), which quantifies how quickly a planet scatters and ejects debris.We find that this is reflected in our simulations; after some number of diffusion times, the inner edges settle into their final configurations.However, the number of diffusion timescales required appears to have some dependence on the planet-to-star mass ratio, with different regimes for ratios above and below 10 −2 .For planet-to-star mass ratios below 10 −2 , we find that the inner edge settles into its final shape within 1 diffusion timescale; therefore, for the majority of realistic planets, Equations 2 and 3 are reasonable estimates of the sculpting timescale.However, for mass ratios above 10 −2 , more diffusion timescales are needed; such planets appear to take closer to 10 diffusion times to sculpt the disc.For this reason, we ran all simulations that had  p / * ≥ 10 −2 until a time of 100 diff , whilst all simulations with  p / * < 10 −2 were run for 10 diff .Regardless, despite the difference in the number of diffusion times required, large planets still sculpt discs much faster than small planets, because the diffusion timescale strongly decreases with increasing planet mass.
There may be two reasons for this behaviour change around  p / * = 10 −2 .First, it may mark the transition between a starplanet interaction, where  p / * ≪ 1, to a binary interaction, where the two bodies have comparable mass.In the latter case, dynamical definitions like the Hill radius break down, so the interaction dynamics may fundamentally change.
The second reason relates to the ratio of  diff to the planet's orbital period.For  p / * = 10 −2 the diffusion time is approximately 100 planet periods, and for  p / * = 10 −1 the diffusion time is just 1 planet period.For such mass ratios the approximations used to define the diffusion timescale start to break down.For example, a planet with  p / * = 10 −1 must take more than one period (and hence more than one  diff ) to clear unstable material, because it can only eject material that passes close to it; any material that is nearly co-orbital with the planet but located on the opposite side of the star must therefore take several orbital periods (and hence several diffusion times) to pass close to the planet and get ejected.Conversely, for a small mass ratio the diffusion time is much longer than the planet period, so such effects are less important.

Debris eccentricity at the disc inner edge
To understand the profiles of the simulated inner edges, we must understand the dynamical processes occurring there.Like Marino (2021) and Rafikov (2023), we expect the edge steepness to be related to debris eccentricity, so understanding the planet-induced eccentricity is vital for understanding how planets shape disc inner edges.In this section we quantify the eccentricity excitation at the simulated inner edges (Section 2.4.1), and show that this excitation is caused by MMR interactions (Section 2.4.2).Later, in Section 3, we will use these results to produce a predictive model relating planet properties to inner-edge profiles.

Measuring debris eccentricity at the simulated inner edges
We first directly measure the eccentricities of inner-edge debris at the final snapshot of each of our -body simulations.To do this, we fit the erf-powerlaw surface-density model (Equation 6) to each simulated disc, then use this to define the inner-edge region; the inner edge has a characteristic width of several times  i  i , so we define the 'inneredge region' as that centred on  i with an arbitrary full width 3 i  i .We then calculate the rms eccentricity of all debris bodies that are instantaneously located in this radial range, which we define as the rms eccentricity of the debris-disc inner edge,  i,rms .As an example, for this definition the simulation on Figure 2 has an inner-edge region spanning from 12.5 to 13.9 au, with  i,rms = 0.0433.
Figure 7 shows the rms eccentricities at the inner edges of each of our simulated discs, calculated using the above method.These rms eccentricities depend only on the planet-to-star mass ratio and the initial-disc excitation level.For discs with low initial-excitation levels, the eccentricity imparted by the planet increases with the planet-to-star mass ratio, up to mass ratios of ∼ 3.3 × 10 −3 ; above this, the inner-edge eccentricity is roughly independent of mass ratio.Conversely, if the disc's initial eccentricity exceeds the level that would be imparted on the edge by the planet, then the resulting edge eccentricity is essentially independent of the planet, and remains close to the initial level throughout the simulation.

Mean-motion resonances as the edge-excitation mechanism for circular-orbit planets
We now identify the mechanism that excites inner-edge debris.Despite the significant eccentricities of this material (Figure 2, left panel), this excitation is not caused by planet-debris scattering, because the semimajor axes of this debris remains essentially unchanged.Excitation is also not due to secular interactions, because the low-eccentricity planet cannot sufficiently excite debris through secular interactions.This leaves MMRs as the only possible excitation mechanism, in agreement with previous studies (Quillen 2006;Quillen & Faber 2006).The idea that MMRs are responsible is also supported by the presence of debris populations with similar excited eccentricities near nominal MMR locations (e.g. the 2:1 MMR on Figure 2).Note that a single MMR does not usually dominate all surviving debris at the disc edge; rather, several nearby MMRs excite debris across a range of semimajor axes.
Given the resonant nature of debris at the inner edge, we can dynamically explain the dependence of  i,rms on planet-to-star mass ratio.The specific MMRs at the edge depend on the mass ratio, because as the mass ratio increases, the nominal MMR locations remain unchanged but the width of the chaotic zone around the planet increases.However, for low mass ratios the excitation level appears similar to that expected from the 2:1 or 3:2 MMRs, even if these are not the specific resonances at the edge.To show this, the grey region on Figure 7 is a theoretical prediction for the rms debris eccentricity expected for a population of bodies in the 2:1 or 3:2 MMRs, found by integrating the equations of motion for those MMRs (Pearce et al., in prep.).This is similar to the ( p / * ) 1/3 scaling from Petrovich et al.Root-mean-square eccentricities of surviving debris at the disc inner edges, at the end of our -body simulations.Lines and symbols are defined on Figure 5. Dotted horizontal lines are the values expected if the discs maintain their initial-excitation levels; these should be compared to simulation lines of the matching colour.The shaded region is a theoretical prediction for the rms eccentricity acquired by bodies in the planet's 3:2 or 2:1 MMRs; its width is the spread of these values.The planet excites surviving debris through MMRs; the black line is a rough fit to the eccentricity it imparts on debris at the inner edge (Equation 7).If this planet-induced eccentricity is smaller than the initial-disc excitation level, then the planet does not significantly increase the excitation level above that of the pre-interaction disc.
results for lower planet-to-star mass ratios.Specifically, if a circularorbit, non-migrating planet sculpts a debris-disc inner edge, then the rms eccentricity at that edge increases with planet mass, provided  p / * ≲ 3.3 × 10 −3 .For planet-to-star mass ratios above ∼ 3.3 × 10 −3 , the inner-edge excitation is below that expected of 2:1 and 3:2 MMRs; this is because those MMRs would now lie inside the chaotic zone, and higher-order MMRs at semimajor axes outside the nominal 2:1 location are weaker and less effective at exciting debris (Figure 2).Hence for planet-tostar mass ratios above ∼ 3.3 × 10 −3 , the debris eccentricity at the disc edge does not further increase with mass ratio.
Given this behaviour, we can roughly quantify the eccentricity that a planet on a circular orbit induces on initially unexcited debris at the disc inner edge: This is the black line on Figure 7, which is a good match to the simulations where the planet-induced excitation dominates over the intrinsic disc excitation (e.g. the purple lines for mass ratios above 10 −4 ).Conversely, since the initial-disc excitation would dominate over the planet-induced excitation if the former were high enough, we can generally predict the eccentricity of surviving debris at the inner edge of a planet-sculpted debris disc as where  i,rms,0 is the 'intrinsic' rms eccentricity at the disc edge (i.e. .Schematic of our simple model for predicting the disc's inner-edge profile (Section 3).The black circle is the planet, and the red region surviving debris.Debris has eccentricities between 0 and  max .The planet eventually ejects all debris originating within its chaotic zone, which is any debris above the solid line (Equations 9 and 10).The value  1 is the smallest semimajor axis of any surviving particle (a circular orbit just exterior to the chaotic zone), and  2 is the largest semimajor axis of any unstable particle (an orbit with the highest eccentricity  max , whose pericentre is at the outer edge of the chaotic zone).
the pre-interaction eccentricity), and the planet-induced eccentricity  i,rms,p is given by Equation 7.

ANALYTIC MODEL RELATING THE INNER-EDGE PROFILE TO THE SCULPTING PLANET
In Section 2 we showed that, if a circular-orbit planet sculpts a collisionless debris disc, then the steepness and relative location of the disc's inner edge depend only on the planet-to-star mass ratio and the disc's initial-excitation level.We also quantified how the planet excites debris eccentricities at the inner edge.Using these results, we now produce a simple analytical model to infer the properties of sculpting planets from inner-edge profiles.Section 3.1 details our analytical model, and Section 3.2 demonstrates how to use it to infer the properties of an unseen sculpting planet from an observed inneredge profile.

Model setup and predictions
We assume a simplified model of the inner-edge region, as shown on Figure 8.We consider a planet on a circular orbit, and a debris disc that initially extends from the planet's semimajor axis out to some larger distance.The debris particles have initial eccentricities uniformly distributed between 0 and some  max ; for this model, the origin of these eccentricities is unimportant.To predict the location and steepness of such a planet-sculpted disc edge, we consider which particles would be scattered by the planet, and which would survive.
In our model, any particle whose orbit comes within the planet's chaotic zone would be scattered and eventually ejected.This means that, at late times, no particles should occupy the parameter space above the solid line on Figure 8; this line is the eccentricity  q () that results in a particle's pericentre coinciding with the outer edge of the chaotic zone, defined by Here  is the semimajor axis, and  is the half-width of the chaotic zone, taken as 3 Hill radii for a circular-planet orbit: Since the planet eccentricity is zero, any non-scattered debris would retain its initial semimajor axis and eccentricity.Hence any debris with initial eccentricity above the Equation 9 line would eventually get ejected, whilst any with eccentricity below the line would remain unperturbed.So in our model, at late times debris only occupies the shaded region of Figure 8.
In the following sections, we use this model to infer the location and steepness of the planet-sculpted inner edge.Before doing so, we must first define two final parameters: the semimajor axes  1 and  2 , as shown on Figure 8.The span  1 to  2 roughly defines the inner-edge region in semimajor-axis space.The values of  1 and  2 correspond to the semimajor axes of orbits at the edge of the chaotic zone;  1 corresponds to a circular orbit, and  2 to an orbit with eccentricity equal to the maximum debris eccentricity  max .Expressions for  1 and  2 are given in Appendix D1.

Predicted radial location of the disc inner edge, 𝑟 i
We now use the simple model on Figure 8 to predict  i , the characteristic location of a planet-sculpted inner edge.A full derivation of the  i prediction is presented in Appendix D2; our basic method is to first calculate the time-averaged radial location of all debris at a single semimajor axis , and then average these for all debris with  1 ≤  ≤  2 .This yields our prediction: where we used  i,rms =  max / √ 3 for a uniform  distribution to convert  max into  i,rms .
Equation 11 predicts that the edge location depends on planet semimajor axis, planet mass, and debris eccentricity.The origin of this eccentricity is unimportant, provided that the eccentricity-semimajor axis distribution resembles that on Figure 8.We can therefore evaluate the eccentricity term in Equation 11 using Equations 7 and 8.If the debris eccentricity is set by the intrinsic (i.e.pre-interaction) eccentricity of the disc, which would occur if the intrinsic eccentricity is higher than that induced by the planet, then  i,rms in Equation 11can simply be replaced by the intrinsic disc eccentricity  i,rms,0 .Alternatively, for a disc with sufficiently low pre-interaction eccentricity, we can substitute  i,rms =  i,rms,p from Equation 7to predict the location of the planet-sculpted inner edge: where we omit higher-order terms in  p / *  11.The horizontal grey line denotes unity; all other symbols and lines are defined on Figure 5.Our predicted edge locations are generally similar to those from the simulations, with the predictions typically within 10 per cent of the simulated values for planets below 1 per cent of the stellar mass and initial debris-excitation levels of  max,0 < 0.3.Predictions are worse for larger mass ratios, and also for simulations with both high initial-debris eccentricities and low-mass planets.
Having made a prediction for  i , we now compare this prediction to the results of our -body simulations.For each simulation, we predict  i by evaluating Equation 11; to do this, we take  p and  p from the simulation setup, and also use the value of  i,rms measured directly from the simulation.The results are shown on Figure 9.We see that the prediction generally works well; the predicted value of  i −  p for simulations with  max,0 < 0.3 is typically within 10 per cent of that from simulations.However, the prediction is less accurate for discs with very high intrinsic eccentricities ( max,0 = 0.3) that are interacting with low-mass planets ( p / * ≲ 10 −4 ); for such high eccentricities there is considerable overlap of debris orbits, so our simple  i approximation using only the average debris position probably no longer holds.There is also a noticeable divergence for planetto-star mass ratios above 10 −2 , which could be due to a fundamental shift in the dynamics; this is discussed in Section 2.3.3.Nonetheless, Equation 11 holds across the large majority of our explored parameter space, so offers a reasonable means to predict the location of a planet-sculpted inner edge.

Predicted steepness of the disc inner edge, 𝜎 i
In this section we use our simple model to predict the shape of the planet-sculpted inner edge, as quantified by  i .Near the inner edge, our fitted profile (Equation 6) is essentially an erf function; this has characteristic width  i  i , where  is a scalar of order unity.We hypothesise that this width should be roughly equivalent to the distance between the pericentre of the innermost stable particle and the apocentre of the outermost unstable particle, i.e.  i  i ≈  2 (1 +  max ) −  1 from Figure 8. Substituting expressions for  i ,  1 and  2 yields  i ∝  max to first order in  max , i.e. the steepness of a planet-sculpted inner edge depends entirely on the eccentricity of debris around that edge.Marino (2021) and Rafikov (2023) reached a similar conclusion for edges with sharp cutoffs in semimajor axis, but we show that this proportionality is also expected from the smooth distribution arising from scattering.Relating the steepness of sculpted-disc inner edges to the debris eccentricities at those edges, from our -body simulations.Smaller  i values correspond to steeper edges.Our simple model predicts  i and  i,rms to be proportional, in agreement with the simulations.The horizontal grey line and shaded region show the value and uncertainty of the prefactor 1.2 ± 0.2 in Equation 13, which is empirically measured from the simulations.All other lines and symbols are defined on Figure 5.The breakdown in the fit at planetto-star mass ratios between 10 −3 and 10 −2 for low-eccentricity simulations is due to the 2:1 MMR imposing additional structure on the edge (discussed in Section 2.3.2).
Since we expect  i ∝  max , we use our -body simulations to directly relate  i to debris eccentricity.Figure 10 shows the rms eccentricities measured at the inner edges of our simulated discs, divided by the  i values fitted to those simulations.This ratio is roughly constant for all simulations, verifying our prediction that  i depends only on  max (and hence  i,rms ).Taking the median of this ratio from our simulations, we find that the flatness of the inner edge of a planet-sculpted disc depends on the eccentricity of debris at that edge as Here the uncertainty on the empirical prefactor is defined from the inter-quartile range on  i,rms / i from our simulations.

Inferring the properties of a sculpting planet from the location and shape of a debris-disc inner edge
Section 3.1 related the disc inner edge to the properties of a sculpting planet.Since it has historically been easier to resolve a cold debris disc than to detect a distant planet, it is common to use observed debris discs to infer the properties of unseen planets (e.g.Pearce et al. 2022).In this section we provide a method to constrain a sculpting planet on a circular orbit from the shape and location of a debris-disc inner edge.
To demonstrate the method, we apply it to the simulation on Figure 2 as a example.We assume that the simulated disc has been observed, but that the sculpting planet is undetectable.We will constrain the properties of the unseen planet from the disc alone, then compare these to the known parameters of the simulated planet to gauge the effectiveness of the method.The various steps in the calculation are shown on Figure 11, and described below.For this example we assume that the planet has finished sculpting the disc by the time the observations are made.Example application of our results: using a debris disc's inneredge profile to infer the properties of an unseen sculpting planet.The plot shows the disc from the -body simulation on Figure 2, for which we attempt to constrain the planet using the step-by-step method from Section 3.2.1.The disc has a fitted inner-edge radius  i = 13.2 au and flatness  i = 0.0361.The dashed vertical line is  i , and the light brown region around it is the approximate full width of the inner edge (3 i  i ).The large black circle shows the planet properties we infer, and the red diamond is the actual planet in the -body simulation.The white square is the planet's minimum-possible mass and maximum semimajor axis found using the alternative method of Pearce et al. ( 2022), which accounts for the edge location but not its steepness.Line 1 shows the planet properties required to set the edge at  i (Equation 14), and Line 2 is the minimum planet mass required to have sculpted the disc edge within the age of the star (Equation 15); for this example the age is arbitrarily taken as 100 Myr.

Step-by-step method
The first step is to fit the disc surface-density profile with some function that quantifies the location and steepness of the inner edge.
In this paper we use an erf-powerlaw function (Equation 6), which yields inner-edge location  i and flatness  i ; other parameterisations could alternatively be used, and in Appendix E we provide equations to convert the outputs of common parametric models into  i and  i values.For our example simulation, the fitted inner-edge profile has  i = 13.2 au and  i = 0.0361.The second step, now that we have the edge profile, is to infer the sculpting-planet mass as a function of its semimajor axis.This step makes the implicit assumption that the planet has a circular orbit and has not migrated, but it otherwise applies regardless of whether the planet or some other process is responsible for debris eccentricities.Equation 11 relates planet mass and semimajor axis to the location and rms eccentricity of the disc inner edge; using this equation, and noting the relation between  i,rms and  i (Equation 13), we can infer the sculpting-planet mass using This is Line 1 on Figure 11 (noting that  * = 1 M ⊙ in our example).The next step is to put a lower bound on the planet mass, assuming that the planet has finished sculpting the observed-disc edge.This means that the sculpting timescale must be shorter than the star age  * .In Section 2.3.3 we showed that the sculpting timescale is ap-proximately the diffusion time if  p / * ≲ 10 −2 , and ∼ 10 diffusion times otherwise; we can therefore rearrange Equation 3 and evaluate it at  =  i to show that  p ≳ 0.105 where  ≈ 1, if  p / * ≲ 10 −2 ; 10, else. (16) For our example we ascribe an arbitrary stellar age of 100 Myr, which results in Line 2 on Figure 11.
The final step is to use the edge shape to break the degeneracy between planet mass and semimajor axis.Unlike previous steps, this step requires the implicit assumption that the planet is solely responsible for exciting debris.It is also only valid if  i ≲ 0.05 (i.e. i,rms ≲ 0.06); this is the flattest profile that a non-migrating, circular-orbit planet could impart on an initially low-eccentricity disc (Figure 7), so if  i ≳ 0.05 then some other process must have excited debris and this step cannot be applied.If  i ≲ 0.05, then rearranging Equation 7 yields the planet mass as we can then substitute this mass into Equation 14to yield the planet's semimajor axis as For our example, this yields a planet mass of 1.24 M Jup and semimajor axis 10.3 au; these are in good agreement with the actual values of 2 M Jup and 10 au from the simulation, as shown on Figure 11.Note that caution must be applied if  i ∼ 0.05 (i.e. i,rms ∼ 0.06), because such an edge profile could be imparted by any planet with  p / * ≳ 3.3 × 10 −3 (Figure 7); in this case, the mass estimate from Equation 17 should be interpreted as a lower bound rather than a single value.

Accuracy of planet parameters inferred from disc edges
The above example showed that planet parameters can be well inferred from inner-edge profiles in at least some setups.We now repeat the above process for a large fraction of our simulations, to test how well the method applies in general.We omit simulations where the initial disc-excitation level is larger than the expected excitation generated by the planet, because for those simulations, Equation 17 cannot be used to infer planet mass.The results are shown on Figure 12.The top panel shows that the inferred debris-excitation level  i,rms , calculated from the edge profile using Equation 13, agrees with the actual simulation values in almost all cases; the largest discrepancy is for planet-to-star mass ratios around 5 × 10 −3 , which is where additional structure from the strong 2:1 MMR coincides with the disc edge.The bottom panel shows that the planet masses inferred using Equation 17 agree with the actual planet masses for low mass ratios, Planet properties inferred from simulated-disc inner edges using the process in Section 3.2, compared to the actual planets.Top panel:  i,rms values inferred from our fitted inner-edge profiles (Equation 13), which agree well with the actual values.Bottom panel: planet masses inferred from the edge steepnesses (Equation 17), which agree well with the actual values for planet-to-star mass ratios below 3.3 × 10 −3 (equivalent to 3.5 M Jup around a Sun-like star).For higher ratios, Equation 17 can significantly underpredict the planet mass, because the edge excitation level becomes constant with planet mass (Figure 7).Note that this degeneracy only affects planet masses that were directly inferred from the edge steepness (black circle on Figure 11); regardless of this, all planets would still lie along the mass-semimajor axis relation (Line 1 on Figure 11).Blue circles and red squares are simulations where the inferred debris excitation  i,rms at the disc inner edge is below and above 0.06, respectively (where  i,rms ≈ 0.06 corresponds to an edge profile with  i ≈ 0.05).
but that the two diverge if the actual planet has  p / * ≳ 3.3 × 10 −3 .For the highest mass ratios, Equation 17 can underpredict the planet mass by over one order of magnitude.The reason for this is that, for planets with mass ratios above 3.3 × 10 −3 , the level of eccentricity excitation imparted on the disc edge is independent of planet mass (Figure 7); this can cause Equation 17 to significantly underpredict the planet mass if debris at the disc edge has an excitation level of  i,rms ≈ 0.06.Note that this degeneracy only affects the use of Equation 17; regardless of the mass ratio, and whether or not the planet was responsible for exciting debris, a sculpting planet should still lie close to the mass-semimajor axis relation from Equation 14(Line 1 on Figure 11).

COLLISIONS
Sections 2 and 3 assessed how a circular-orbit planet would sculpt the inner edge of a debris disc, if the interaction were purely body dynamics.For that scenario we showed that the edge steepness could be directly related to the planet mass.However, in a real disc there are also collisions between debris bodies, which would alter the edge profile.In this section we consider the collisional evolution, and answer two basic questions: could a collision-dominated disc edge be differentiated from a planet-dominated edge, and how would collisions change the profile of a planet-sculpted edge?
4.1 Could a purely collisional disc be differentiated from a purely planet-sculpted disc?
Destructive collisions cause debris discs to lose mass over time; material grinds down to dust, which is eventually removed by radiation forces (Wyatt et al. 1999;Wyatt 2008;Krivov 2010).The rate at which this process occurs depends on several main factors: the number density of debris bodies, their size distribution, and their relative velocities.
For an initially broad debris disc, these factors result in welldefined collisional evolution.At early times the surface density is expected to decrease with distance, for example going as  −1.5 in the MMSN.Relative velocities are also higher in the inner regions, and this combination makes the initial collisional depletion faster closer to the star.However, this means that the surface density closer to the star drops faster than that further out, reducing the inner collision rate.The result is that the initially negative surface-density slope gradually transforms into a positive one; a 'collision front' moves outwards across the disc, where the region interior to the front tends towards a positive surface-density slope, whilst the exterior region still has its original profile.In the collisionally processed region, this profile is well-characterised as a powerlaw   i , with index  i = 7/3 = 2.3 in Wyatt et al. (2007) and Kennedy & Wyatt (2010), or  i = 2 in more-recent models (Löhne et al. 2008;Imaz Blanco et al. 2023).
Conversely, if a planet sculpts a collisionless disc, then the inneredge steepness is set by the planet mass.To answer whether such a planet-dominated disc can be differentiated from a collisiondominated disc, we must compare the inner-edge slopes.Using Equation E5 to relate an erf-fitted slope to a powerlaw slope, we see that a planet-dominated disc will have a steeper inner edge than a collisiondominated disc if  i < 0.40 (assuming  i = 2 for the collisional disc; if  i = 7/3 were used instead, then  i < 0.35 for the sculpted edge to be steeper).Figure 6 shows that these conditions are satisfied by the end of all of our circular-planet simulations, meaning that the inner edge of a purely planet-sculpted disc is steeper than that of a purely collisional disc if sculpting has completed.In some cases it would be much steeper; our steepest -body discs have  i ∼ 10 −2 , which would correspond to  i > 10 in   i notation.So we should be able to differentiate purely planet-sculpted discs from purely collisiondominated discs, with the former having much steeper inner edges.

How would collisions affect a planet-sculpted edge?
The previous section compared a collisionless, planet-sculpted disc to a collisional disc without a planet.However, in reality a planetsculpted disc would also undergo collisions.In this section we consider collisions in more detail, to assess how they would affect a planet-sculpted disc.
A fully self-consistent model, where a disc undergoes -body interactions whilst simultaneously collisionally evolving, is difficult to properly implement.In particular, there are several parameters that are critical for quantifying the collisional evolution, but that are fundamentally unknown for real debris discs; these include the debris-disc mass, and the sizes of the largest planetesimals (Krivov & Wyatt 2021).For this reason we chose to decouple the dynamical and collisional modelling in this paper.Instead, we will take the disc morphologies arising from our -body simulations, and then input these into a collisional model to estimate how collisions would modify a planet-sculpted disc edge.This approach is not fully selfconsistent, but would be valid if the planet-sculpting timescale were much shorter than the collisional timescale; we will later show that this condition holds for many plausible scenarios.

Collisional model
To model collisions, we use a mathematica implementation of the Löhne et al. (2008) collisional model.The model takes an axisym-metric surface-density profile, and splits it into radial bins.Each bin is then treated separately, and the dust mass in each bin is determined after some time.We then use the masses in each radial bin to calculate the dust surface-density profile.In this section we briefly describe the model and the parameters we use; we refer the reader to Löhne et al. (2008) for a much more detailed description.
We assume the eccentricity  and inclination  of the debris population are related by  ≈ /2, and treat  as a free parameter.Following Löhne et al. (2012), Schüppler et al. (2016) and Krivov et al. (2018), we assume a critical fragmentation energy of with  0 = 3 km s −1 ,  s =  g = 5 × 10 6 erg g −1 ,  s = −0.12 and  g = 0.46.The planetesimal-collision speed  col is given by where G is the gravitational constant and The value of  * D is dominated by the material strength for smaller bodies, and gravity for larger bodies.The transition between these can be defined either as the size at which the strength and gravity terms in Equation 19 are equal, or as the size at which  * D is minimised.Following Löhne et al. (2008), here we use the former definition.For our parameters, the strength-gravity transition then occurs at the "breaking" radius  b = 232 m.
We assume that debris in each bin follows a three-slope size distribution as in Löhne et al. (2008), with slope  p = 1.87 for primordial bodies,  g = 1.68 in the gravity-dominated quasi-steady state and  s = 1.89 in the strength-dominated quasi-steady state.We choose minimum and maximum dust-grain radii of  min = 2 m and  dust = 1 mm, and treat the largest-planetesimal radius  max as a free parameter.We use a bulk density of solids  = 3.5 g cm −3 .
According to the model, there are two key timescales that determine the collisional evolution.The shorter,  b , is the time when the weakest bodies begin to collide.At this time, the collisional decay of the dust density sets in (if  p ≤  s ) or at least speeds up (if  p >  s ); see Equation 43 and Figure 9 in Löhne et al. (2008).The value of  b is given by Equation 31 in Löhne et al. (2008): where  0 is the total initial mass of solids in the bin,  is the radial width of the bin, and We arbitrarily use / = 0.1 in our code, but this later cancels out so its value does not affect any results.The longer timescale is  max : the collisional lifetime of the largest planetesimals.This is Equation 42 in Löhne et al. (2008) 3 : We will later show that  max is the most important factor for determining the collisional behaviour.
We provide an open-access python code to calculate  max given the assumptions in this paper 4 , and in Section 4.2.3 we write  max in an alternative form to make its dependencies more explicit.
Given the above setup, the model yields the dust mass  dust in each radial bin after some time .This is Equation 43 in Löhne et al. (2008): for  b <  <  max , where T = / b if  <  max or  max / b otherwise 5 .For  <  b ,  g and  g should be replaced by  s and  s respectively.
For each collisional simulation, we initialise the disc with a surface-density profile similar to the outcomes of our -body simulations.This is a powerlaw with a truncated inner edge: where a disc with  m = 1 and  = 3/2 would have the MMSN profile beyond the inner edge.This initial profile sets  0 in the above equations.The initial values of  m and  are free parameters that we vary between models.After setting up the disc, we use Equation 26to determine the dust mass in each radial bin after some time.Using these bin masses, we compute the dust surface-density profile.Finally, we fit this profile with an erf-powerlaw function similar to Equation 27, to measure  i ,  i and  at that time.

Collision results
The previous section described the collisional model.In this section we apply the model to assess the impact of collisions on planetsculpted disc edges.Solid lines show the edge at various times; collisions flatten the edge, tending towards an  2 profile (dashed line).Note that our simulated collisions only deplete material; the lines are scaled to the same value at 13.2 au for gradient comparison, giving the impression that the surface density increases with time at some radii, but in reality it decreases at each successive snapshot and unscaled lines would not cross (the lines are scaled because the surface density drops considerably with time, and a logarithmic plot would skew the gradient comparison).The edge undergoes little evolution until the time becomes comparable to  max ; after that, the edge starts to flatten.
We first apply it to the -body disc from Figure 2 as an example.We initialise the disc with  m = 1 and  = 3/2, with an inner edge defined by  i = 13.2 au and  i = 0.0361, and set the largest-planetesimal radius to  max = 50 km.We also set the debris eccentricity to 0.043, the rms eccentricity at the disc inner edge in the -body simulation.This results in  max = 1.1 Gyr and  b = 0.14 Myr at 13.2 au (Equations 22 and 25).We then use the collisional model to calculate the edge profile at various times up to 10 Gyr, the lifetime of the solar-type star.The first snapshot is made shortly after  b , to ensure there is sufficient dust in the system.Figure 13 shows the results; the inner edge is largely unchanged for the first 100 Myr, but from 1 Gyr collisions begin to make the edge flatter.We quantify this flattening using the erf-powerlaw function; at 0.2, 10, 100, 10 3 and 10 4 Myr, the respective fitted values are  i = 13.1, 13.0, 13.0, 13.0 and 13.2 au, and  i = 0.034, 0.042, 0.043, 0.064 and 0.11.Note that the erf function is an increasingly poor fit to the edge shape as  max is approached, due to the complex edge profile.
Figure 13 demonstrates the importance of  max in setting the collisional evolution.Before  max , collisions have little effect on the inner-edge profile.However, once  max is reached, collisions start to have a significant effect, making the edge flatter.This collisional profile is expected to eventually tend towards  2 , the profile for a broad collisional disc, as shown by the dashed line on Figure 13.However, in this example the profile is never reached within the stellar lifetime; in fact, the inner edge assumes two slopes, with an erf-like profile interior to  i and a flatter, positive profile beyond this.This means The black line is the reference simulation from Figure 13, and other lines show simulations that differ from the reference setup by the parameters listed in the key.At early times collisions make the edge slightly flatter (i.e. increase  i ), before making it significantly flatter as the time approaches  max ( i,0 ).The local peak around  max should be ignored; it arises because the erf-powerlaw function used to fit the profile struggles with the complex edge shape at time.that, in this example, the effect of planet sculpting would still be visible on the disc's inner-edge even at very late times6 .
Figure 13 was for one example setup.Figure 14 shows how the inner-edge steepness collisionally evolves in a number of different setups, demonstrating that this evolution is both qualitatively and quantitatively similar across a broad parameter space.Starting with the setup from Figure 13 as a reference, we vary the input parameters and re-run the collisional simulation multiple times.We see that all of the setups have the same general collisional evolution.First, between  b and  max the inner edge becomes slightly flatter, with  i increasing by a factor of ∼ 1.2.This remains constant, until the time approaches  max ; around this time the edge becomes much flatter, and tends towards the  2 collisional profile.The plot shows the importance of  max in setting the collisional evolution of the sculpted-disc edge.

Implications of collisions for planet-sculpted debris discs
The previous sections showed that collisions have a very small effect on the inner-edge profile, until the time  max .After this, a planetsculpted edge would become significantly flatter.Consideration of  max , the stellar lifetime and the planet-sculpting timescale  diff are therefore vital for assessing the effect of collisions on the edge profile.
We can use the above timescales to identify three possibilities for the evolution of the debris-disc inner edge, depending on  diff and  max : •  diff ≪  max : the planet sculpts the disc, producing a sharp inner edge like our -body simulations.This edge then gradually gets flatter due to collisions, but does not change significantly until the time approaches  max .Then, if left long enough, it may eventually assume an  2 profile.
•  diff ∼  max : unknown outcome.In this regime planetary sculpting and collisions are equally important at all times, and our simulations are insufficient to model this.
•  diff ≫  max : the disc collisionally evolves almost as if no planet were present.The disc tends toward an  2 profile.So if  diff and  max are known, then the relative importance of collisions to planetary sculpting can be ascertained, and the evolution of the disc edge predicted.However, whilst  diff is straightforward, Equations 23 to 25 show that  max is a complicated function depending on many parameters, several of which are interdependent and unknown.
We can gain deeper insight by rewriting  max , to make its dependencies more explicit.Equations 23 to 25 give  max ,  and  c in their general forms; we can simplify these by substituting our assumed form for the initial Σ(), as well as  * D ,  , / etc.We also assume  max ≫  min .For the region beyond the inner edge, this yields where and Expressing the equations in this form demonstrates that  max is inversely proportional to the unknown  m , which is related to the initial mass of the debris disc in MMSN units.We use this to better visualise  max .
Figure 15 shows  max multiplied by  m , for a Solar-type star and a disc with an initial surface-density profile of  3/2 .The figure shows that, for reasonable debris parameters,  max is typically extremely long, unless  max is very small or  m very large.However,  max is unlikely to be smaller than a few kilometers, otherwise it would violate planetesimal-formation models and the statistics of discs of various ages (Krivov & Wyatt 2021, and refs therein).A 100 au disc with debris eccentricity 0.01 would therefore have  m  max ∼ 10 10 yr.Furthermore,  m > 10 is unlikely because such debris discs would have masses exceeding those of solids in the preceding, protoplanetary stage (Krivov & Wyatt 2021).As a result, for many observed debris discs  max may be longer than the ∼ 10 7 to 10 9 yr system age.In fact,  max can even be infinite, which means that collisions never disrupt the largest bodies; this occurs in the large regions of parameter space where  c ( max ) ≥ 1 (for which  ≤ 0).Rearranging Equation 24,  max becomes infinite if Plots of  max (collisional lifetime of the largest debris) multiplied by  m (initial surface density relative to the MMSN), for a Solar-type star and a disc with an initial surface-density profile of  3/2 .The plots show that  max strongly depends on distance,  max and debris eccentricity, and can sometimes be very long.It is unlikely that  m > 10, so  max can be much longer than the ∼ 10 7 to 10 9 yr ages of debris-disc systems.If the largest bodies lie above the solid line (Equation 31), then they would never be destroyed and  max becomes infinite.Note that  m and  max cannot be arbitrarily varied for an observed debris disc; only certain combinations are allowed, since they set the observed-dust mass at the system age (Equation 26).Increasing or decreasing the star mass respectively raises or lowers the contours slightly, but does not have a large effect.
which yields the solid lines on Figure 15.Conversely, calculating  diff (Equation 2) shows that a Jupitermass planet located within 100 au of a Solar-mass star would finish sculpting a disc within < 10 Myr.So if such planets were sculpting debris discs, then they would likely do so long before collisions had any effect on the inner edges.This would make the inner-edge profiles long lived, and since even 10 max is insufficient to fully erase the planet's signature in the example on Figure 13, it is likely that a disc sculpted by a massive planet would maintain a distinctive shape even if collisions were fully accounted for.For such cases, the planet can be constrained using the process in Section 3.2.For smaller, Neptunemass planets the sculpting timescale is 100 times longer, in which case both collisions and planets could simultaneously sculpt the discs.Such systems may not have reached their final configurations by the time they are observed, meaning that the disc edges would be somewhere between the very steep values expected for such planets, and the  2 profile expected from collisions.This is discussed further in Section 6.1.

APPLICATION TO OBSERVED DEBRIS DISCS
Finally, we now compare the inner-edge slopes we predict from planetary sculpting to the inner edges of observed debris discs.We consider seven ALMA-resolved, extrasolar debris discs, each of which has a literature measurement of its inner-edge steepness.The systems are listed in Table 1, and were analysed by Lovell et al. (2021), Faramaz et al. (2021) and Imaz Blanco et al. (2023).
The literature works used various parameterisations to quantify disc profiles, which we must first convert to erf-powerlaw functions for comparison with our simulations.The literature works generally fitted the inner edges using a radial powerlaw, with some turnover occurring further out; this turnover could be the disc outer edge, or a gap within the disc.For our analyses the nature of the turnover is unimportant, but its location is needed to define the 'inner' region of the disc.Following Lovell et al. (2021), we quantify the surface densities around the inner edges of the literature discs using a double powerlaw: where  i is the slope of the inner edge,  t is the radial location of the turnover,  o is the slope exterior to the turnover, and  sets how sharp the turnover is.The assumed values for each disc are listed in Table 1; for fits where  was undefined or unconstrained, we assumed a value of 2 as in Lovell et al. (2021).We next convert these powerlaw fits into erf-powerlaw profiles for comparison with our simulations.This is not strictly valid as we should really fit our erf-powerlaw profile to the underlying data, but it should be sufficient for this simple analysis.We consider the profile which is equivalent our Equation 6 near the inner edge.We set  o to the literature values describing the region just beyond the edge, noting that  o is typically negative.We then use Equation C1 to convert the inner-edge powerlaw index  i to an equivalent flatness  i , which approximates how the inner edge would appear if fitted with an erf function instead.Finally, we deduce  i , which characterises the location of the inner edge in the erf model, such that Equations 32 and 33 peak at the same radial location.A comparison of the erf-powerlaw and double-powerlaw profiles for an example system is shown on Figure E1, and the inferred values of  i and  i are listed for all systems in Table 1.In some cases  i could not be fitted because the region beyond the turnover was so steep that Equation 33does not  turn over; however, this does not affect the edge flatness  i , which is the parameter of interest in this section.
Figure 16 shows the inferred  i values for the observed discs, which exhibit a range of inner-edge slopes.At least two have relatively flat slopes consistent with pure collisional evolution, which would yield powerlaw indices of  i ≈ 2 as described in Section 4 (equivalent to  i ≈ 0.4, shown as the dashed line on Figure 16).Several have much steeper inner edges, which Imaz Blanco et al. (2023) argued are indicative of planetary sculpting.However, whilst those edges are indeed steeper than expected from collisions alone, they are still flatter than our simulations predict for in-situ sculpting of loweccentricity discs by planets on circular orbits.The shaded region on Figure 16 shows the maximum  i expected if a  p / * ≤ 10 −3 planet sculpts a disc with  max,0 ≤ 0.1; all of the observed inner edges are flatter than this 7 .In Section 6.1 we discuss the potential implications of these flatter edges, which could teach us about architectures, dynamical processes and histories in the outer regions of planetary systems.

DISCUSSION
In Section 6.1 we discuss the possible implications of observed discs having flatter-than-expected edge profiles, and in Section 6.2 we compare the analyses in this paper to previous studies.

Why are observed inner edges flatter than expected from sculpting planets?
In Section 5 we showed that, whilst the edges of several ALMAresolved discs are steeper than expected from collisions alone, they are flatter than expected from planetary sculpting of an initially loweccentricity disc.Here we discuss several possible reasons for this, and what these could teach us about planetary systems.

Debris discs have high intrinsic-excitation levels
One possibility is that planetesimals in debris discs have higher intrinsic excitation levels than is often assumed.For example, if debris at a disc inner edge had  i,rms ≈ 0.13, then that edge would have 7 Whilst a planet with mass  p / * ≈ 0.07 could drive  i to required values of 0.11 (Figure 6), we disfavour this possibility because it seems unlikely that planets with such specific masses exist in two of our seven systems.1).Blue circles and orange triangles are  i values and upper limits respectively.Several discs (49 Ceti, AU Mic and HD 206893) have flat inner-edge slopes consistent with pure collisional evolution, as shown by the dashed horizontal line.Other discs (q 1 Eri, HD 92945 and HD 107146) have much steeper profiles, but these are still flatter than expected from in-situ sculpting of low-eccentricity discs by circular-orbit planets.
i ≈ 0.11, which is similar to the steepest discs in Table 1.This value of  i,rms is higher than the maximum of 0.06 that could be imparted by a non-migrating, circular-orbit planet (Figure 7), raising the possibility that these eccentricities arise through non-planetary processes.Examples of possible excitation mechanisms include self stirring by the debris disc's self gravity, or processes that occurred during system formation.Marino (2021) used the outer edges of several observed discs to infer the debris-excitation levels, and found these to be consistent with  i,rms ≳ 0.1; that sample included HD 92945 and HD 107146, which have the steepest edges in Table 1.It is therefore plausible that in-situ planets sculpted the inner edges of some observed discs, but that this debris has separately been excited by some non-planetary processes.
One means to test this would be to measure scale heights at the inner edges of debris discs.We find that MMRs of a coplanar planet excite debris eccentricity but not inclination (Section 2.2), so if in situ, coplanar planets sculpt debris-disc inner edges, then these edges would be more excited radially than vertically.

Planets are present at the inner edges, but sculpting has not yet finished
Planets take time to eject debris and fully sculpt a debris-disc inner edge.The required time is characterised by 1 to 10 diffusion timescales (Section 2.3.3).Until this time, the disc's inner edge could have almost any profile, which would depend on its initial configuration.It is therefore possible that planets reside just interior to debris discs, but the planets have not yet finished sculpting the edges into steep profiles.This is particularly plausible for young systems, or those with low-mass planets.
The observed discs in Table 1 with the flattest inner-edge profiles are also the youngest, with ages less than 100 Myr.These are 49 Ceti, AU Mic and HR 8799.Equation 15shows that, if planets with masses below 0.1 to 1 M Jup lie at the inner edges of those discs, then these systems would be younger than one diffusion time and hence planetary sculpting would not yet have finished.It is therefore plausible that inner-edge profiles are flatter than expected not because sculpting planets are absent, but because sculpting has not yet finished.If true, then care must be taken when inferring planet properties from debris discs, because the hypothesised planet-disc interactions may not have reached their final state.This may be a prevalent problem because the discs most favourable for observation are generally the youngest, for which the dust mass (and hence brightness) is highest.

Discs are sculpted by migrating planets
In this paper we modelled planets on non-evolving orbits, but a migrating planet would impose a very different morphology on the disc inner edge.Specifically, a migrating planet causes MMR sweeping, where the nominal locations of MMRs move across the debris disc and trap large numbers of bodies in resonance.During this process, resonant debris is excited to increasingly high eccentricities (e.g.Wyatt 2003;Reche et al. 2008;Friebe et al. 2022), which would result in a flatter edge profile.
An example of this occurred in the Solar System.Neptune is located at the inner edge of the Kuiper Belt, but the orbits of Kuiper-Belt objects (KBOs) show that the planet migrated outwards in the past (Malhotra 1993).As a result, KBOs at the Kuiper Belt's inner edge have relatively high eccentricities, which make the Belt's inner edge flatter than would be expected from perturbations by a non-migrating Neptune.In an upcoming paper (Morgner et al., in prep.), we take the observed KBOs and, using a method similar to Vitense et al. (2010), de-bias these observations to estimate the true KBO population distribution.We then fit the surface density of this de-biased population with Equation 33, to get the steepness of the Kuiper-Belt inner edge.This yields  i = 0.076, which is much flatter than would be expected for a non-migrating Neptune; Neptune has  p / * = 5 × 10 −5 , so the inner edge should have  i < 0.02 if Neptune's orbit never changed (Figure 6).The reason for this difference is that Neptune excited KBOs through MMR sweeping as it migrated, resulting in eccentricities at the Kuiper Belt's inner edge of  i,rms ∼ 0.2, compared to just 0.01 expected from excitation by a non-migrating Neptune (Equation 7).
It is possible that some extrasolar discs were also sculpted by migrating planets, which made their edge profiles flatter than our insitu model predicts.Dust clumps that may be indicative of planetary migration are inferred in debris discs (Lovell et al. 2021;Booth et al. 2023), andPearce et al. (2022) argue that migration may be required to relate the location of debris-disc edges to system-formation theory.These results, combined with debris-disc edges being flatter than expected for sculpting by non-migrating planets, could imply that planetary migration is common in the outer regions of debris-disc systems.Such migration could be caused by planetesimal scattering (e.g.Friebe et al. 2022), planet-gas interactions in the protoplanetary disc phase, or planet-planet scattering.
It may be possible to test exoplanet-migration scenarios in the near future.JWST will search for planets in the outer regions of debrisdisc systems, and should detect many of the inferred planets if they exist (Pearce et al. 2022).If a planet were found just interior to a debris disc, and that disc had a flatter edge profile than would be expected from in-situ planetary sculpting, then it could be evidence that the planet migrated outwards in the past, exciting debris as it did so.

Collisions have flattened the planet-sculpted edges
Debris-disc edges could initially have been sculpted by planets, and these edges could since have undergone significant collisional evolution.The steepest edges of the Table 1 discs have  i ≈ 0.11, which is at least 1.3 times flatter than expected from sculpting by nonmigrating planets (Figure 6).However, Figure 14 shows that collisions can quickly increase  i by a factor of 1.2, long before the largest bodies start colliding at  max .This could explain the steepest inner edges in Table 1; in these cases, the system ages could be longer than  b but shorter than  max .After  max , collisions would flatten the edge further, increasing  i by a factor of 3 to 6 within 10 max ; this could potentially reproduce some of the flatter edges in Table 1, if those systems are already older than  max .Since some observed edges are too steep to be explained by collisions alone, yet too flat to be explained by non-migrating planets alone, this could suggest that both processes play a significant role in some systems.

Planets are present at debris-disc inner edges, but the edges are set by planet formation rather than scattering
A circular-orbit planet would eject debris that initially lies within 3 Hill radii of its orbit (Figure 5).Therefore, if a newly formed system had a planetesimal disc extending down to less than 3 Hill radii exterior to a planet's orbit, then that planet would eventually impart a sharp inner edge on the disc.This is the scenario we modelled in our simulations.However, a real planet would also have accreted material from around its orbit as it formed.If the forming planet's feeding zone were wider than 3 Hill radii, then much of the debris within the scattering radius would already have been accreted by the time the planet formed.Also, other processes in the protoplanetary disc phase could further affect planetesimal profiles, such as the forming and trapping of material in planet-induced pressure bumps near a planet.Therefore, it is possible that a debris disc's edge profile is set by processes occurring during planet formation, rather than by pure planetary scattering; in this case, its edge profile could differ from those that we model (e.g.Eriksson et al. 2020).

Debris-disc edges are set by planetesimal formation alone
Planets are often invoked to explain the shapes and locations of debris-disc inner edges, motivated in part by Neptune dominating the inner edge of the Kuiper Belt.However, an alternative possibility is that planetesimals naturally form in distinct radial zones; in this case debris-disc edges may be completely unrelated to planets, but instead mark locations were planetesimal formation transitioned from being inefficient to efficient.Such formation zones are predicted by some streaming-instability models (Carrera et al. 2017), as well as models involving snowlines (Ida & Guillot 2016;Schoonenberg & Ormel 2017;Drążkowska & Alibert 2017;Schoonenberg et al. 2018;Izidoro et al. 2022;Morbidelli et al. 2022).This possibility can soon be tested because, if sculpting planets exist, then many should be detectable by JWST (Pearce et al. 2022); the absence of such detections would potentially imply that other processes set disc-edge locations.

Debris discs do not have sharp cutoffs in semimajor axis
Debris-disc inner edges could be broad in semimajor-axes space, meaning that the number density of debris gradually increases with semimajor axis.The radial profile of such a disc would mimic that of a higher-eccentricity disc with a sharper semimajor-axis distribution.Shallow semimajor-axis distributions could be a natural outcome of system formation (Section 6.1.6),arise through MMR sweeping during planet migration (Friebe et al. 2022), or result from debris diffusion during self stirring (Ida & Makino 1993).Since the average eccentricities and inclinations would be related in a relaxed debris disc, the disc vertical thickness would test this; Marino (2021) argued that the outer region of the AU Mic disc is more extended radially than vertically, which could imply low eccentricities and hence a slowly changing semimajor-axis profile.

Debris discs are sculpted by planets with low but non-zero eccentricities
Eccentric planets can excite debris to higher eccentricities than circular-orbit planets, due to additional secular perturbations (eccentric planets are considered in Appendix B).This would result in flatter disc inner edges.Combining Equations 13 and B3, we can show that a Neptune-mass planet orbiting a solar-type star with eccentricity 0.10 could induce an inner-edge steepness of  i ≈ 0.11, like the sharpest observed edges.Such a planet would make the inner edge elliptical with global eccentricity 0.052 (Equation B5), which is high enough to detect with modern observations (e.g.Lovell et al. 2021).Therefore, it should be possible to rule out an eccentric planet as being responsible for a flat disc edge if the disc's global eccentricity were robustly measured as negligible.
However, an intriguing possibility arises if the star position is poorly constrained.In this scenario, a disc with a low global eccentricity could masquerade as axisymmetric, owing to a poorly constrained stellar offset.Were this the case, then an eccentric planet could flatten the disc inner edge, whilst no global disc eccentricity would be detected.One way to rule out this scenario would be to measure the azimuthal variation in the disc profile, because a planet sculpting a truly eccentric disc would make the inner edge flatter at pericentre than apocentre (Section B2).Depending on the disc width and the intrinsic debris excitation, the eccentric planet may also flatten the outer edge, which could be observable (compare Figures 2  and B1).

Unresolved Trojans are present interior to the inner edges
Unresolved Trojans could be present at the disc inner edges.In our simulations, we purposefully truncated the initial-disc edge at one Hill radius exterior to the planet, to omit any Trojans and ensure that our fitted  i values describe the 'true' inner-edge slopes.However, if sculpting planets are present at the inner edges of real discs, but there are also unresolved Trojans co-orbiting with the planets, then the Trojans would flatten the azimuthally averaged inner-edge profiles.The degree of flattening would depend on the surface density of Trojans relative to the disc.

Dust transport is important
Dust-transport processes could be operating in addition to planetary sculpting, which would make the disc edges more radially extended and their profiles flatter.Poynting-Robertson (PR) drag can be significant for small grains, causing discs to be radially extended in scattered light; whilst the larger, ALMA-imaged grains we consider should be unaffected by PR drag, they could be affected in a similar way by stellar winds.Winds would be particularly important for discs around late-type stars, and would drag material inwards from the planetesimal belt (e.g.Plavchan et al. 2005;Reidemeister et al. 2011;Schüppler et al. 2015).This would flatten the inner-edge profile.Similarly, CO gas is detected in many debris discs around early-type stars, which could also cause radial drift through gas drag (e.g.Krivov et al. 2009;Marino et al. 2020;Pearce et al. 2020).Again, this would flatten the inner edges.

More complex disc-planet interactions occur in some debris discs
The inner edges of some specific systems in Table 1 could be excited by more complex planet-disc interactions.There are gaps in the discs of both HD 92945 and HD 107146, which are the discs with the steepest inner edges.Friebe et al. (2022) showed that the simplest and most self-consistent way to explain the morphology of HD 107146 is if a planet has migrated across the gap; such migration would cause sweeping by mean-motion resonances, which would excite debris at the disc inner edge (their Figure 9, lower-right panel).In that model the planet would now lie just exterior to the inner edge of the gap, and may have swept up a Trojan population which could resemble the additional gap features in Imaz Blanco et al. (2023).It is therefore possible that planets lying further out in more-complex discs could excite debris at the inner edge, leading to flatter edge profiles.Alternatively, planets could excite inner-edge debris through secular resonances rather than mean-motion resonances, which could lead to higher debris eccentricities and thus flatter edges.A secular resonance occurs where the apsidal precession rate of debris (due to the planet and disc self-gravity) matches that of the planet (due to the disc or other planets), and such resonances can drive up debris eccentricities and even open remote gaps in broad discs (Pearce & Wyatt 2015;Yelverton & Kennedy 2018;Sefilian et al. 2021Sefilian et al. , 2023)).If a secular resonance were located near the disc inner edge, then it could dominate over scattering and MMRs, and hence produce a different edge location and profile to those in our simulations (e.g.Smallwood 2023).Given the very large uncertainties on both debrisdisc masses and orbital architectures in the outer regions of systems, we cannot state for certain whether this secular-resonance excitation occurs, but we can estimate the disc masses that could lead to this scenario.
In a coplanar system with a single planet and an external, selfgravitating disc, a secular resonance occurs at semimajor axis  res .The location of  res depends on the total disc mass  d relative to that of the planet, as well as the planetary semimajor axis.A general expression describing this relationship is derived in Sefilian et al.
(2021) (their Equation 19) which, assuming a power-law disc with surface density ∝  −1.5 spanning semimajor axes  in to  out , with  p ≈  in , reads as where  1 () is a factor of order unity (see also Sefilian et al. 2023).A secular resonance will hence lie near the disc inner edge if  d / p ≳ 0.89 ( out / in ) 1/2 ; here we arbitrarily define 'near' as  res < 1.2 in , where  1 (1.2 in ) ≈ −1.07 (computed using Equation 18in Sefilian & Rafikov 2019, assuming  = 0.05 for the disc scale height8 ).This implies that, for disc-to-planet mass ratios of roughly unity or higher, secular resonances could play a role in shaping the disc inner edge.Since debris discs could have masses up to 1000 M ⊕ (Krivov & Wyatt 2021), this effect could be significant even for Jupiter-mass planets.We will further investigate the effect of secular resonances on debris-disc edges in a future work (Sefilian et al., in prep.).

Comparison to literature studies
Several literature studies also assessed the effect of a sculpting planet on debris-disc inner edges.These used various different approaches, and yielded various different results.In this Section we compare our paper to literature works, specifically focussing on the -body results (Section 6.2.1), collisional modelling (Section 6.2.2) and the planetinferring technique of Pearce et al. (2022), which uses the locations of debris-disc inner edges but not their steepnesses (Section 6.2.3).

Gravitational effects
The fact that MMRs of planets on circular orbits excite debris at disc inner edges, and that more-massive planets cause higher excitation and hence flatter edge profiles, was also identified in previous works.Mustill & Wyatt (2012) used encounter maps to demonstrate this effect on planetesimals (their Figure 4), and showed that a similar effect arises in the prescription of Wisdom (1980).Chiang et al. (2009) and Rodigas et al. (2014) consider small dust released from a population of planet-sculpted planetesimals, and show that this decrease in edge steepness with increasing planet mass is also expected in scattered-light observations (their Figures 3 and 1b respectively).Quillen (2006) and Quillen & Faber (2006) used MMR theory to predict that a low-eccentricity planet should impose an eccentricity dispersion at the disc inner edge that is proportional to ( p / * ) 3/7 , which is slightly steeper than our empirical ( p / * ) 1/3 for a circularorbit planet (Equation 7).However, we note that the zero-planeteccentricity simulations of Quillen & Faber (2006) appear more consistent with our flatter index of 1/3 (filled pentagons on their Figure 2), and this index is also predicted theoretically by Petrovich et al. (2013) (their Equations 19 and 34).We therefore argue that ( p / * ) 1/3 is a better estimate of debris eccentricities at the inner edge of a disc sculpted by a circular, non-migrating planet, provided that  p / * ≲ 3.3 × 10 −3 .Above this mass ratio, we observe a turnover where increasing the planet mass no longer excites debris to higher eccentricities; this is due to the inner edge of the planet's chaotic zone extending beyond the location of strong MMRs, and was not observed by the above authors because their simulations did not extend to such high mass ratios.Tabeshian & Wiegert (2016, 2017) modelled the interaction between a planet and a planetesimal disc, and showed that the MMRs of a circular-orbit planet can impose asymmetric structure on an external disc (Figure 5 in Tabeshian & Wiegert 2016).In particular, they show that MMRs can carve crescent-shaped gaps in the disc, and that the inner-edge profile can vary with azimuth.We find the same result; in the simulation on Figure 4, a crescent-shaped gap can be seen aligned with the planet, and the inner-edge steepness differs slightly on the left and right sides of the disc.However, we do not investigate this asymmetry in more detail, because the difference is small enough that it is unlikely to significantly affect any inferredplanet parameters.Also, Tabeshian & Wiegert (2016) show that the degree of asymmetry scales with the planet-to-star mass ratio, and this asymmetry is already small for the comparatively high mass ratio on Figure 4; for smaller mass ratios, the asymmetry would be even less pronounced.

Collisional effects
We find that collisions make the disc edge flatter, and move its characteristic radius outwards (Figures 13 and 14).This is in qualitative agreement with Figure 3 of Nesvold & Kuchner (2015), which shows that the same effects manifest in the smack collision prescription; that prescription simultaneously models the collisional and n-body dynamical evolution of debris (Nesvold et al. 2013).Nesvold & Kuchner (2015) also find that it takes 10 to 100 collisional timescales for this process to occur (their Figure 4).However, our collisional models, and our definitions of collisional timescales, are fundamentally different.
We employ the analytical model of Löhne et al. (2008), where the most relevant collisional timescale is  max , the collisional lifetime of the largest bodies.This predominantly depends on disc location, debris eccentricity, the disc's initial mass and the size of the largest bodies.The latter two are unknown, but for physically plausible values, a disc with inner edge at 50 au would have  max of at least ∼ 10 7 yr, and potentially much longer (Figure 15).This means that  max should be at least comparable to the ages of debris-disc stars (Table 1), and may be much longer; based on this, we argue that collisions should not significantly reduce the steepness of planetsculpted edges in many observed discs.
This conclusion differs from Nesvold & Kuchner (2015), who suggest that collisions typically would have a significant effect on the observed edges of planet-sculpted discs.They model the collisional cascade in detail and self-consistently with the dynamical evolution, unlike our simulations.However, the difference between our conclusions is unlikely to be due to our separation of dynamics and collisions, but rather differences between our collisional assumptions and those of Nesvold & Kuchner (2015).
There are two main reasons for this difference.First, the maximum grain radius in Nesvold & Kuchner (2015) is  max = 10 cm (their Table 1); these pebbles are depleted much faster than the ≳ 1 km planetesimals in our model, so their collisional erosion occurs on much shorter timescales.Second, the amount of dust in the Nesvold & Kuchner (2015) models is higher because they assume a greater minimum grain size; extrapolating from their  min = 1 mm to the typical  min = 1 to 10 m expected from radiation pressure blowout corresponds to vertical optical depths that are higher than ours by factors of 10 to 30, assuming a size distribution index of 3.5.
Accounting for the respective differences in  min and  max each increase the ratio between the collision timescales of smallest dust (their Eq. 3) and that of the biggest objects (which govern the longterm evolution).Nesvold & Kuchner (2015) define smallest-grain collisional timescales of 10 3 to 10 6 yr for a sample of observed discs (their Section 4); instead of the factors of 10 to 100 in their Figure 4, planetesimals would evolve collisionally on timescales that are longer by several orders of magnitude, in line with our results.So this difference in timescales appears to be the cause of our different conclusions regarding the relative effects of planets and collisions on the disc inner edge.
Nonetheless, the Nesvold & Kuchner (2015) model captures additional physics that is omitted in our simpler prescription.In particular, our model would not capture the expected collisional enhancement in resonant populations (Stark & Kuchner 2009); this means we probably underestimate collision rates at the MMR-dominated inner edges, and therefore overestimate collisional timescales.This effect could be significant in some cases, because the inner-edge eccentricities in our low-eccentricity planet simulations are determined by MMRs.Detailed, self-consistent modelling is therefore required in future, to properly assess the balance between collisions and planet-debris interactions at the inner edges of debris discs.Pearce et al. (2022) planet predictions Pearce et al. (2022) give a general model to infer the minimum mass, maximum semimajor axis and minimum eccentricity of a perturbing planet based on the shape and location of a debris disc's inner edge.Those predictions are based on Pearce & Wyatt (2014), and do not consider the steepness of the disc edge; as a result they have a degeneracy in planet mass and semimajor axis, because they cannot distinguish a low-mass planet near the disc from a high-mass planet away from the disc.However, in this paper we make predictions that also include information about the edge steepness; this can break the degeneracy, because edge steepness can be directly related to planet mass (at least in the collisionless regime).

Comparison to
Section 3 describes our simple analytic model to predict planet properties from an observed-disc edge.These predictions are comparable to those of Pearce et al. (2022), with three important differences.First, we have an extra dependence on the edge profile, and hence the debris eccentricity  i,rms ; our predicted inner-edge location (Equation 11) is increased by a factor of (1 + 2 i,rms / √ 3) relative to that in Pearce et al. (2022).
Second, Pearce & Wyatt (2014) use a stricter definition of the sculpting timescale; they assume sculpting takes 10 diffusion times, which is the time it takes an eccentric planet to eject 95 per cent of unstable material.However, we find that an edge sculpted by a circular-orbit planet with mass  p / * < 10 −2 takes just one diffusion timescale to assume roughly its final configuration (Equation 15).This is because it takes a planet exponentially longer to clear increasingly large fractions of unstable debris; a circular-orbit planet removes ∼ 70 per cent of unstable debris within 1 diffusion time, but takes 10 diffusion times to remove ∼ 90 per cent (Costa, Pearce & Krivov, submitted).Hence our minimum-allowed planet mass can be a factor of √ 10 ≈ 3 times smaller than Pearce et al. (2022).Finally, we take the width of the chaotic zone around the planet's orbit to be 3 Hill radii, whilst Pearce et al. (2022) used 5.The reason for this is that Pearce & Wyatt (2014) studied eccentric planets, for which the innermost stable semimajor axis is 5 eccentric Hill radii exterior to planet apocentre (their Figure 9), whilst for circular-orbit planets a value of 3 provides a better fit (Gladman 1993;Ida et al. 2000;Kirsh et al. 2009;Malhotra et al. 2021;Friebe et al. 2022).It is unclear at what planet eccentricities the transition from 3 to 5 Hill radii occurs, though Figure B1 shows that a planet eccentricity of 0.2 is already high enough for 5 Hill radii to provide a better estimate.This is consistent with Regály et al. (2018), who find the transition occurs at planet eccentricities somewhere between 0 and 0.2 (their Figure 5c).
The result of these three differences is that any planets predicted using our model will differ slightly from the Pearce et al. (2022) model.However, the mass predictions should typically be within an order of magnitude or so of each other.For the example on Figure 11, the lower bound on planet mass predicted by the Pearce et al. (2022) model is 10 times smaller than the mass predicted from our model; the two predictions still lie in a similar region of parameters space, along Line 1 on that figure.

Application to narrow discs
We only considered broad discs in our simulations, to isolate the inner edge from the outer edge.This ensured that our inner-edge profiles were dominated by planet-disc interactions, rather than the initial profile of the disc.However, this means that our quantitative results may not hold for very narrow discs, where the outer-and inner-edge profiles could overlap.
In Section 2.4 we argued that the characteristic inner-edge width is several times  i  i .Hence our results should hold if the disc is sufficiently wide for the edges to be well separated, i.e.
Equation 13 shows that  i is proportional to the eccentricity of inneredge debris, and a similar relation holds for the outer edge.Therefore, if the eccentricity distribution were constant across the disc, then Equation 35 implies that our results should hold provided the disc fractional width is larger than the rms debris eccentricity; this is in agreement with Marino (2021).Our full criterion (Equation 35) is slightly more complicated, because the debris-eccentricity level would vary across a planet-sculpted disc.

Range of explored mass ratios
We explored planet-to-star-mass ratios ranging from 3 × 10 −5 to 10 −1 .We did not consider lower-mass planets, because in many cases the dynamical timescales would then become comparable to the stellar lifetimes.For example, an Earth-mass planet at 10 au from a Solar-type star ( p / * = 3 × 10 −6 ) would have a diffusion timescale of 40 Gyr, which would exceed the stellar lifetime of ∼ 10 Gyr.Hence our results would not hold for very low mass ratios, because the interaction we study would not have time to occur.Our upper limit of  p / * = 10 −1 represents the upper end of what could realistically be called a 'planet'.For example, this could be a 10 M Jup super-Jupiter orbiting a 0.1 M ⊙ M6 dwarf.In many cases, such high-mass companions would be detectable via imaging, radial velocity or astrometry, particularly in younger systems (e.g.Carter et al. 2023).
Were a companion detected, then our results could be used to infer its evolutionary history.For example, if a debris disc in the system had an edge steepness consistent with sculpting by the observed companion, but the companion were detected far interior to the disc, then this could be evidence that the detected companion sculpted the disc historically but has since migrated inward.This scenario has been suggested as a way to explain non-detections of planets at the inner edges of debris discs (Pearce et al. 2022).

CONCLUSIONS
We perform a dynamical investigation into the effect of planets on the profiles of debris-disc inner edges.We consider both planetdisc interactions and debris collisions, and explore the interaction across a broad parameter space.We quantify our simulated surfacedensity profiles using an erf function, as in Rafikov (2023), for direct comparison with ALMA-resolved inner edges.Our main conclusions are as follows: (i) For a non-migrating, circular-orbit planet, in the case where collisions are negligible, the steepness of the disc inner edge is set by the planet-to-star mass ratio and the initial-disc excitation level.Lower-mass planets lead to steeper inner edges, with the edge width proportional to ( p / * ) 1/3 (Equations 7, 8 and 13).
(ii) There is a maximum eccentricity that a planet on a circular, non-migrating orbit can generally impart on the population of planetesimals at a debris-disc inner edge, which is a root-mean-squared eccentricity of 0.06 (Equation 7).
(iii) Considering the steepness of a debris disc's inner edge when inferring the properties of unseen sculpting planets can break the degeneracy between planet mass and semimajor axis.We provide a step-by-step method for inferring such planets from disc inner edges in Section 3.2.1.
(v) The inner edge of a purely planet-sculpted debris disc is much steeper than that of a purely collisional debris disc.
(vi) Collisions flatten the profile of a planet-sculpted inner edge.The effect of collisions is small before the largest bodies start to collide; after this time, collisions start to significantly flatten the inner edge.
(vii) In most cases, collisions would never fully erase the signature of a sculpting planet on a debris-disc inner edge.
(viii) Whilst the inner edges of many ALMA-resolved debris discs are too steep to be caused by collisions alone, they are also too flat to arise through pure sculpting by non-migrating, circular-orbit planets, unless sculpting is still ongoing.Possible implications of this for the outer regions of planetary systems are discussed in Section 6.1.

APPENDIX A: SURFACE-DENSITY PROFILE FITTING
We fit surface-density profiles to our simulations through several rounds of  2 minimisation, using the following process.First, we perform an initial  2 minimisation, which typically provides a good fit to the broad disc as a whole.However, this generally produces poor fits at the edges, because the edges make up a small fraction of the overall profile; the inner edge on Figure 2 spans just 25 radial bins, compared to the 400 bins across the entire disc, so even a poor edge fit would not significantly impact the overall  2 .To properly fit the inner edge, we then repeat the fitting using the parameters of the first fit as our initial guess, except for  i for which we use a value between 10 −8 and 10 2 .We repeat this several times for different initial guesses of  i , and the fit with the lowest  2 is taken as the new reference fit.Finally, this whole process is repeated for the outer edge, using the new reference fit as our initial guess and testing different guesses for  o .The fit with the lowest  2 after this process is the fit we use.This technique avoids local minima, and ensures that the edge profiles are properly fitted.It is also considerably faster than performing a full MCMC fit to each of our simulations.The fitting is performed using the python module scipy.optimize.

APPENDIX B: ECCENTRIC PLANETS
We showed that a planet on a circular orbit excites debris eccentricities through MMRs, and that these eccentricities set the steepness of the disc inner edge.However, an eccentric planet would also drive up eccentricities through secular interactions.Whilst we focus on circular-orbit planets in this paper, in this section we briefly consider how planet eccentricity would affect the inner-edge profiles.We will show that a low-to moderate-eccentricity planet can excite debris more than a circular-orbit planet, resulting in a flatter inner edge, and that the planet eccentricities required to do this can be low enough that they do not necessarily impose a clear asymmetry on the disc.

B1 N-body simulation with an eccentric planet
To demonstrate the effect of an eccentric planet, we re-run the simulation from Figure 2, but with the planet eccentricity increased to 0.2.We also initialise the disc inner edge to be slightly further outwards than before; we still place it one Hill radius exterior to the planet's apocentre, but since the planet is now eccentric, we switch to the definition of the eccentric Hill radius at apocentre from Pearce & Wyatt (2014): where  p is the planet eccentricity.We do this because the Hill radius of an eccentric planet varies around the planet orbit; the above equation is the Hill radius at the planet's apocentre, and the equivalent at planet pericentre is found by changing  p to − p in this equation.We run the eccentric-planet simulation for 16.5 Myr, which is much longer than the 0.868 Myr for the circular case.We do this because an eccentric planet would drive spiral density waves in a massless disc through secular interactions, in addition to sculpting the inner edge through scattering.Pearce & Wyatt (2014) show that such a disc would settle into its final state after at least 10 secular times have elapsed at its outer edge (by which time the spirals are tightly wound and indistinct), and at least 10 diffusion times have elapsed at the inner edge (by which time scattering would be largely complete).We compute the secular timescale using Equation 17 in Pearce & Wyatt (2014), and run our simulation to 10 times that value.
Figure B1 shows the simulation with an eccentric planet, for comparison with the equivalent circular-planet simulation of Figure 2. As well as ejecting debris, the eccentric planet drives the disc into a broad, eccentric structure aligned with the planet orbit, as described in Pearce & Wyatt (2014) and Faramaz et al. (2014).Debris eccentricities are much higher than in the circular-planet case (Figure B1 middle panel, compared to Figure 2 left panel), with implications for the inner-edge profile as discussed below.Another difference is that the inner edge is truncated at a semimajor axis corresponding to 5 eccentric Hill radii outside the planet's apocentre, rather than 3 Hill radii as in the circular case; this behaviour is discussed in Pearce & Wyatt (2014).The eccentric-planet simulation also has resonant debris surviving on planet-crossing orbits (e.g the 3:2 MMR), and such populations can be stable even for highly eccentric planets (Pearce et al. 2021).
We now assess the profile of the eccentric-disc inner edge.In our circular-planet simulations we simply azimuthally averaged the entire disc to produce radial surface-density profiles, but this approach is insufficient for asymmetric discs.Instead, we follow Tabeshian & Wiegert (2016, 2017) and divide the disc into several sectors.We define one sector as everything 45 • either side of the planet's pericentre direction, and another as everything 45 • either side of the planet's apocentre.These sectors are coloured orange and green respectively on the left panel of Figure B1.For each of these sectors we azimuthally average the particles within them, to produce one radial surface-density profile for the region around pericentre and another for the region around apocentre.These profiles are shown on the right panel of Figure B1.We fit each profile with Equation 6 as before, again fixing the powerlaw index  to 1.5.The result is that the inner edge has a different steepness on the apocentre and pericentre sides of the disc; we fit  i = 0.137 on the pericentre side, compared to 0.106 on the apocentre side.Both are much flatter than the circularplanet case, which had  i = 0.0361.In the next section we quantify the general impact of planet eccentricity on the inner-edge profile.

B2 Effect of planet eccentricity on the inner-edge profile
The inner-edge steepness is set by the eccentricity of surviving debris.
A circular-orbit planet excites debris eccentricities through MMRs, whilst an eccentric planet also excites eccentricities through secular interactions.If the secular excitation were greater than the MMR excitation, then the eccentric planet would produce a flatter edge than a circular-orbit planet.In this section we calculate how eccentric a planet can be before the inner-edge profile deviates from the circularplanet case, and the effect of an eccentric planet on the inner edge.Secular interactions with an eccentric planet cause debris eccentricities to oscillate, whilst their semimajor axes remain constant.Specifically, debris initialised on a circular orbit with semimajor axis  would oscillate in eccentricity between zero and twice the forcing eccentricity  forced , where 2 forced () ≈ 5 2  p   p (B2) (Murray & Dermott 1999).This is shown on the middle panel of Figure B1, where the dotted line is 2 forced .We can use this to derive the maximum eccentricity of secular debris at the disc inner edge; the innermost stable semimajor axis is about 5 eccentric Hill radii beyond planet apocentre, i.e.  in ≈  p (1 +  p ) + 5 Hill,Q (Pearce & Wyatt 2014), so we can use Equation B1 to get the innermost stable semimajor axis, then substitute this into Equation B2.The result is that, to first order in planet eccentricity, the maximum debris eccentricity from secular effects at the disc inner edge is  max,sec ≈ 2 forced ( in ) ≈ 5 2  p 1 + 3.47 For the scenario on Figure B1 ( p = 0.2,  p / * = 1.9 × 10 −3 ), Equation B3 yields  max,sec ≈ 0.35, in good agreement with the simulation.
We can use Equation B3 to determine how eccentric a planet must be for the inner-edge profile to differ significantly from the circular-planet case.Equation 7 gives the rms eccentricity at the inner edge arising from MMRs from a circular-orbit planet; if an eccentric planet excited debris to higher values through secular interactions, then the edge profile would be flatter than in the circular-planet case.Comparing Equations 7 and B3, the disc inner edge would be flatter than the circular-planet case if the planet has eccentricity For the examples on Figures 2 and B1 with a 2 M Jup planet orbiting a solar-type star,  p / * = 1.9 × 10 −3 and so Equation B4 predicts a critical planet eccentricity of 0.05 (equal to the eccentricity of Jupiter).So for this example, if the planet eccentricity were below 0.05 then the inner-edge profile would be similar to the circular-planet case (Figure 2), whilst higher planet eccentricities would result in a flatter inner edge (Figure B1).The actual inner-edge steepnesses on the pericentre and apocentre sides are comparable to that predicted by Equation 13; for the example on Figure B1, inserting  i,rms ≈  max,sec / √ 3 = 0.17 into Equation 13 predicts  i ≈ 0.144, comparable to the simulation values of 0.137 and 0.106 on the pericentre and apocentre sides respectively 9 .
Generally, the inner edge of an eccentric disc would be steeper on the apocentre side than the pericentre side, due to how the innermost debris evolves.Pearce & Wyatt (2014) showed that the inner edge is set by debris with the innermost stable semimajor axis  in , which oscillates in eccentricity between 0 and 2 forced .This debris also evolves in orientation, being maximally aligned with the planet orbit when its eccentricity is high, and minimally aligned when low.The result is that the disc inner edge assumes an eccentric shape; combining Equations 5 and 6 in Pearce & Wyatt (2014) shows that an ellipse fitted to the disc inner edge will have eccentricity which is  forced ( in ) to first order 10 .This means that at apocentre the inner edge is a superposition of low-eccentricity orbits at the innermost stable semimajor axis, so its profile is relatively sharp; conversely, at pericentre the inner edge is a diffuse superposition of higher-eccentricity orbits and is flatter.There may also be additional surface-density features that would manifest in particularly narrow and/or eccentric discs (Pearce & Wyatt 2014).

APPENDIX C: CRITERION FOR PLANET-SCULPTED EDGES TO BE STEEPER THAN THE OVERALL DISC
In Appendix E1 we show that the steepness of the erf function is equivalent to that of a powerlaw   if and so the inner edge is much steeper than the overall disc profile if  i ≪ √︁ 2//.Since all of our simulated inner edges have  i < 0.2 (Figure 6), the planet-sculpted edges should be much steeper than the overall disc provided that the initial-disc profile is flatter than  −4 (and also  4 ).

APPENDIX D: DERIVING INNER-EDGE PARAMETERS FROM OUR SIMPLE SCATTERING MODEL
Here we use our simple scattering model to predict inner-edge parameters, yielding the results in Sections 3.1.1and 3.1.2.These parameters are derived with reference to Figure 8. 9 Strictly the rms eccentricity of a secular population scales with  max,sec by a factor √ 2 rather than √ 3, since the eccentricity distribution is not uniform, but for this simple estimate we neglect this difference. 10The equivalent equation for the eccentricity of the outer edge is  out =  forced ( out )/[1 +  forced ( out ) ], where  out is the outermost semimajor axis; similarly, this reduces to  out ≈  forced ( out ) to first order.D1 Semimajor axes  1 and  2 , and the fraction of surviving debris with semimajor axis We define to be the semimajor axis where  q ( 1 ) = 0, i.e. the semimajor axis of a circular orbit at the outer edge of the chaotic zone (recalling that  q () is given by Equation 9).Then we define as the semimajor axis where  q =  max , i.e. the semimajor axis of an orbit with pericentre at the outer edge of the chaotic zone and eccentricity equal to the maximum eccentricity  max .
Next, we assume that the fraction of surviving particles with semimajor axis  transitions from 0 per cent for  ≤  1 to 100 per cent for  ≥  2 .Since we assume that all debris with initial eccentricity above  q () is ejected, for particles with initial eccentricities uniformly distributed between 0 and  max the probability density function of surviving particles goes as () ∝  q ()  max (D3) for  1 ≤  ≤  2 .Similarly, () = 0 for  ≤  1 and is constant for  ≥  2 .

D2 Inner-edge location 𝑟 i
For an orbit with semimajor axis  and eccentricity , the timeaveraged value of some parameter  over a single orbit is where  and  are the radial distance and true anomaly respectively (e.g.Murray & Dermott 1999).Hence the time-averaged radial distance of a single body is For a collection of bodies with the same semimajor axis but with eccentricities uniformly drawn between 0 and  q , we can integrate Equation D5 to get the average radial position of the bodies in this group; this is ⟨  ⟩ = ∫  q 0 ⟨ , ⟩()d/ q , which yields Next, we integrate Equation D6 to find the average position of all particles with semimajor axes between  1 and  2 , which we hypothesise to be similar to the characteristic inner-edge radius  i .This is  i ≈ ∫  2  1 ⟨  ⟩()()d.Finally, we replace  max in the above equations with √ 3 i,rms for a uniform eccentricity distribution, and expand to first order in eccentricity.This yields our prediction for the characteristic radius of the sculpted disc's inner edge:

Figure 1 .
Figure 1.Initial setup of our -body simulations with a circular-orbit planet.A single planet (black circle) is initialised just interior to a massless debris disc, which spans 1 to 30 Hill radii exterior to the planet orbit.

Figure 4 .
Figure 4. Positions of bodies from the simulation with a low-eccentricity disc (Figure2).The asterisk, filled black circle and solid line denote the star, planet and planet orbit respectively, and small points are debris (coloured by initial semimajor axis).The disc is almost axisymmetric, with a slight asymmetry between the left and right sides due to MMRs.This causes a small difference in the inner-edge steepness between the two sides; the right side (aligned with the planet position) has  i = 0.0286, whilst the value for the opposite side is 0.0354.

Figure 5 .Figure 6 .
Figure 5. Location of the sculpted disc's inner edge  i (in terms of planet semimajor axis  p ), from fits to surface-density profiles at the end of our -body simulations.Coloured points and lines show simulations with different initial setups, as denoted in the key.Grey solid lines denote different definitions of the outer edge of the chaotic zone; after the planet has ejected unstable debris, the disc's inner edge is typically just outside this chaotic zone.Dashed horizontal lines denote where  i coincides with the nominal location of a strong MMR; these resonances can significantly affect the edge location.
Figure7.Root-mean-square eccentricities of surviving debris at the disc inner edges, at the end of our -body simulations.Lines and symbols are defined on Figure5.Dotted horizontal lines are the values expected if the discs maintain their initial-excitation levels; these should be compared to simulation lines of the matching colour.The shaded region is a theoretical prediction for the rms eccentricity acquired by bodies in the planet's 3:2 or 2:1 MMRs; its width is the spread of these values.The planet excites surviving debris through MMRs; the black line is a rough fit to the eccentricity it imparts on debris at the inner edge (Equation7).If this planet-induced eccentricity is smaller than the initial-disc excitation level, then the planet does not significantly increase the excitation level above that of the pre-interaction disc.
Figure8.Schematic of our simple model for predicting the disc's inner-edge profile (Section 3).The black circle is the planet, and the red region surviving debris.Debris has eccentricities between 0 and  max .The planet eventually ejects all debris originating within its chaotic zone, which is any debris above the solid line (Equations 9 and 10).The value  1 is the smallest semimajor axis of any surviving particle (a circular orbit just exterior to the chaotic zone), and  2 is the largest semimajor axis of any unstable particle (an orbit with the highest eccentricity  max , whose pericentre is at the outer edge of the chaotic zone).

Figure 9 .
Figure9.Comparison of our predicted inner-edge location to the results of -body simulations.The vertical axis is the difference between the planet orbit and the inner-edge location  i fitted from the simulations, divided by the equivalent value predicted by Equation11.The horizontal grey line denotes unity; all other symbols and lines are defined on Figure5.Our predicted edge locations are generally similar to those from the simulations, with the predictions typically within 10 per cent of the simulated values for planets below 1 per cent of the stellar mass and initial debris-excitation levels of  max,0 < 0.3.Predictions are worse for larger mass ratios, and also for simulations with both high initial-debris eccentricities and low-mass planets.
Figure10.Relating the steepness of sculpted-disc inner edges to the debris eccentricities at those edges, from our -body simulations.Smaller  i values correspond to steeper edges.Our simple model predicts  i and  i,rms to be proportional, in agreement with the simulations.The horizontal grey line and shaded region show the value and uncertainty of the prefactor 1.2 ± 0.2 in Equation13, which is empirically measured from the simulations.All other lines and symbols are defined on Figure5.The breakdown in the fit at planetto-star mass ratios between 10 −3 and 10 −2 for low-eccentricity simulations is due to the 2:1 MMR imposing additional structure on the edge (discussed in Section 2.3.2).
Figure 11.Example application of our results: using a debris disc's inneredge profile to infer the properties of an unseen sculpting planet.The plot shows the disc from the -body simulation on Figure2, for which we attempt to constrain the planet using the step-by-step method from Section 3.2.1.The disc has a fitted inner-edge radius  i = 13.2 au and flatness  i = 0.0361.The dashed vertical line is  i , and the light brown region around it is the approximate full width of the inner edge (3 i  i ).The large black circle shows the planet properties we infer, and the red diamond is the actual planet in the -body simulation.The white square is the planet's minimum-possible mass and maximum semimajor axis found using the alternative method ofPearce et al. (2022), which accounts for the edge location but not its steepness.Line 1 shows the planet properties required to set the edge at  i (Equation14), and Line 2 is the minimum planet mass required to have sculpted the disc edge within the age of the star (Equation 15); for this example the age is arbitrarily taken as 100 Myr.
Figure12.Planet properties inferred from simulated-disc inner edges using the process in Section 3.2, compared to the actual planets.Top panel:  i,rms values inferred from our fitted inner-edge profiles (Equation13), which agree well with the actual values.Bottom panel: planet masses inferred from the edge steepnesses (Equation17), which agree well with the actual values for planet-to-star mass ratios below 3.3 × 10 −3 (equivalent to 3.5 M Jup around a Sun-like star).For higher ratios, Equation 17 can significantly underpredict the planet mass, because the edge excitation level becomes constant with planet mass (Figure7).Note that this degeneracy only affects planet masses that were directly inferred from the edge steepness (black circle on Figure11); regardless of this, all planets would still lie along the mass-semimajor axis relation (Line 1 on Figure11).Blue circles and red squares are simulations where the inferred debris excitation  i,rms at the disc inner edge is below and above 0.06, respectively (where  i,rms ≈ 0.06 corresponds to an edge profile with  i ≈ 0.05).

Figure 13 .
Figure 13.Collisional evolution of the planet-sculpted disc from Figure 2, according to the Löhne et al. (2008) model (Section 4.2).We initialise the disc with  i = 13.2 au and  i = 0.0361, and assume  * = 1 M ⊙ ,  m = 1 and  max = 50 km.This yields  max = 1.1 Gyr and  b = 0.14 Myr at 13.2 au.Solid lines show the edge at various times; collisions flatten the edge, tending towards an  2 profile (dashed line).Note that our simulated collisions only deplete material; the lines are scaled to the same value at 13.2 au for gradient comparison, giving the impression that the surface density increases with time at some radii, but in reality it decreases at each successive snapshot and unscaled lines would not cross (the lines are scaled because the surface density drops considerably with time, and a logarithmic plot would skew the gradient comparison).The edge undergoes little evolution until the time becomes comparable to  max ; after that, the edge starts to flatten.

Figure 14 .
Figure14.General collisional evolution of the inner edge of a planet-sculpted debris disc.The plot shows the evolution of the inner-edge profile over time, where time is expressed relative to  max ( i,0 ), the collisional lifetime of the largest planetesimals at the initial inner edge.The simulations start at time  b .The black line is the reference simulation from Figure13, and other lines show simulations that differ from the reference setup by the parameters listed in the key.At early times collisions make the edge slightly flatter (i.e. increase  i ), before making it significantly flatter as the time approaches  max ( i,0 ).The local peak around  max should be ignored; it arises because the erf-powerlaw function used to fit the profile struggles with the complex edge shape at time.

Figure
Figure15.Plots of  max (collisional lifetime of the largest debris) multiplied by  m (initial surface density relative to the MMSN), for a Solar-type star and a disc with an initial surface-density profile of  3/2 .The plots show that  max strongly depends on distance,  max and debris eccentricity, and can sometimes be very long.It is unlikely that  m > 10, so  max can be much longer than the ∼ 10 7 to 10 9 yr ages of debris-disc systems.If the largest bodies lie above the solid line (Equation31), then they would never be destroyed and  max becomes infinite.Note that  m and  max cannot be arbitrarily varied for an observed debris disc; only certain combinations are allowed, since they set the observed-dust mass at the system age (Equation26).Increasing or decreasing the star mass respectively raises or lowers the contours slightly, but does not have a large effect.

Figure 16 .
Figure16.Flatness of the inner edges of observed discs (Table1).Blue circles and orange triangles are  i values and upper limits respectively.Several discs (49 Ceti, AU Mic and HD 206893) have flat inner-edge slopes consistent with pure collisional evolution, as shown by the dashed horizontal line.Other discs (q 1 Eri, HD 92945 and HD 107146) have much steeper profiles, but these are still flatter than expected from in-situ sculpting of low-eccentricity discs by circular-orbit planets.

Figure B1 .
Figure B1.Simulation with an eccentric planet.The simulation has the same setup as that on Figure2, except here the planet has eccentricity 0.2, and the initial disc lies further out (see Section B1).The eccentric planet excites debris through secular interactions, resulting in higher eccentricities and flatter edges than the circular-planet case.Left panel: positions at the end of the simulation.Orange and green points are debris within 45 • of the planet's pericentre and apocentre directions respectively.Middle panel: the dotted black line is 2 forced (), the maximum debris eccentricity expected from secular interactions (EquationB2).The solid lines define the chaotic region, which is 5 (rather than 3) times the eccentric Hill radius either side of the eccentric planet's orbit.Right panel: separate surface-density profiles for material within 45 • of the planet's pericentre and apocentre.The thick and thin lines are the simulation data and fitted models respectively, relative to the peak of the pericentre model.The fit to the pericentre side has  i = 16.6 au,  i = 0.137,  o = 33.9au and  o = 0.0625, whilst that to the apocentre side has  i = 20.0 au,  i = 0.106,  o = 37.7 au and  o = 0.0666; the peak fitted surface density on the apocentre side is 0.973 times that on the pericentre side.All remaining lines and symbols are defined on previous figures.

Table 1 .
Blanco et al. 2023 inner edges of several axisymmetric, extrasolar debris discs.The parameters  t ,  i ,  o and  are from literature fits to the surface-density profiles in the inner-edge regions (Equation32), and  i and  i are the equivalent parameters in our erf-powerlaw model for comparison with our simulations (Equation33).Note that  i and  o are often 0.5 larger than the values provided in disc-imaging papers, because we consider the surface-density slope whilst those papers often give the intensity slope (e.g.ImazBlanco et al. 2023).