Using debris disk observations to infer substellar companions orbiting within or outside a parent planetesimal belt

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Astronomy & Astrophysics manuscript no. main ©ESO 2024

December 31, 2024

Using debris disk observations to infer substellar companions

orbiting within or outside a parent planetesimal belt

T. A. Stuber1, T. Löhne2, and S. Wolf1

1Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstr. 15, 24118 Kiel, Germany

e-mail: tstuber@astrophysik.uni-kiel.de

2Astrophysikalisches Institut und Universitätssternwarte, Friedrich-Schiller-Universität Jena, Schillergässchen 2–3, 07745 Jena,

Germany

Received 1 February 2022 / Accepted 2 October 2022

ABSTRACT

Context. Alongside a debris disk, substellar companions often exist in the same system. The companions inﬂuence the dust dynamics

via their gravitational potential.

Aims. We analyze whether the eﬀects of secular perturbations, originating from a substellar companion, on the dust dynamics can be

investigated with spatially resolved observations.

Methods. We numerically simulated the collisional evolution of narrow and eccentric cold planetesimal belts around a star of spectral

type A3 V that are secularly perturbed by a substellar companion that orbits either closer to or farther from the star than the belt. Our

model requires a perturber on an eccentric orbit (𝑒≳0.3) that is both far from and more massive than the collisionally dominated belt

around a luminous central star. Based on the resulting spatial dust distributions, we simulated spatially resolved maps of their surface

brightness in the 𝐾,𝑁, and 𝑄bands and at wavelengths of 70µmand 1300µm.

Results. Assuming a nearby debris disk seen face-on, we ﬁnd that the surface brightness distribution varies signiﬁcantly with observing

wavelength, for example between the 𝑁and 𝑄band. This can be explained by the varying relative contribution of the emission of the

smallest grains near the blowout limit. The orbits of both the small grains that form the halo and the large grains close to the parent

belt precess due to the secular perturbations induced by a substellar companion orbiting inward of the belt. The halo, being composed

of older grains, trails the belt. The magnitude of the trailing decreases with increasing perturber mass and hence with increasing

strength of the perturbations. We recovered this trend in synthetic maps of surface brightness by ﬁtting ellipses to lines of constant

brightness. Systems with an outer perturber do not show a uniform halo precession since the orbits of small grains are strongly altered.

We identiﬁed features of the brightness distributions suitable for distinguishing between systems with a potentially detectable inner or

outer perturber, especially with a combined observation with JWST/MIRI in the 𝑄band tracing small grain emission and with ALMA

at millimeter wavelengths tracing the position of the parent planetesimal belt.

Key words. planet-disk interactions – circumstellar matter – interplanetary medium – infrared: planetary systems – submillimeter:

planetary systems – methods: numerical

1. Introduction

Various close-by main-sequence stars with debris disks have been

found to host exoplanets, for example, 𝛽Pictoris (Lagrange et al.

2009,2010,2019;Nowak et al. 2020), 𝜖Eridani (Hatzes et al.

2000), or AU Microscopii (Plavchan et al. 2020;Martioli et al.

2021). Several techniques, direct andindirect, were used to detect

the aforementioned exoplanets: the exoplanets 𝛽Pic c and 𝜖Eri b

were inferred by measuring the radial velocity of the host star

(Struve 1952); AU Mic b and c were detected by measuring the

light curve of the host star while the planet transits the line of

sight (e.g., Struve 1952;Deeg & Alonso 2018); and 𝛽Pic b and

c were detected by direct imaging (Bowler 2016). The ﬁrst two

techniques are sensitive to planets orbiting relatively close to the

host star: for the radial velocity method, planets with long orbital

periods are diﬃcult to detect because the amplitude of the radial

velocity signal decreases with increasing distance between the

planet and the host star and due to the sheer time span required

to observationally cover an orbit (e.g., Lovis & Fischer 2010);

for the transit method, the larger the orbit of a planet, the smaller

the angular cross section the planet is blocking in front of the

star and the less likely a suﬃcient alignment of the host star,

the orbiting planet, and the observer is. The technique of direct

imaging is capable of ﬁnding substellar companions on larger

orbits, ≳100 au, but requires the planets to still be intrinsically

bright and to not have cooled down since formation. This, as such,

favors young systems as targets, that is, T Tauri and Herbig Ae/Be

stars as well as young moving group members (Bowler 2016);

older, already cooled planets are diﬃcult to detect. Astrometry

that uses the data obtained by Gaia (Gaia Collaboration et al.

2016) is expected to add thousands of exoplanet detections, but

the orbital period of the planets discovered is limited by the

mission lifetime of approximately ten years (Casertano et al.

2008;Perryman et al. 2014;Ranalli et al. 2018). An exoplanet

huntingmethodwithoutthesebiasesis gravitational microlensing

(e.g., Mao & Paczynski 1991;Gould & Loeb 1992;Mao 2012;

Tsapras 2018), but with this method systems at distances on

the order of kiloparsecs are probed, too distant to be spatially

resolved. In summary, we lack a planet hunting method to ﬁnd

old,and hence intrinsicallydark, far-out planetsin close-bystellar

systems.

In addition to exoplanets, stars have often been found to host

debris disks (e.g., Maldonado et al. 2012,2015;Marshall et al.

2014), a common component in stellar systems beyond the proto-

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arXiv:2210.05315v2 [astro-ph.EP] 29 Dec 2024

A&A proofs: manuscript no. main

planetary phase(e.g.,Su et al.2006;Eiroaet al. 2013;Montesinos

et al. 2016;Sibthorpe et al. 2018). They are produced and contin-

uously replenished by mutually colliding planetesimals that grind

themselves down to dust in a collisional cascade and are charac-

terized as being optically thin (for recent reviews, see Matthews

et al. 2014;Hughes et al. 2018;Wyatt 2020). The disks are usu-

ally observed in the near-infrared via the stellar light scattered oﬀ

the dust and in the mid-infrared, far-infrared, and (sub)millimeter

wavelength range via the thermal emission of the dust itself.

Planets orbiting in a debris disk system have an impact on

the planetesimal and dust grain dynamics via their gravitational

potential. Therefore, by observing the dust emission one can po-

tentially draw conclusions regarding the possibly unseen planets

orbiting the central star (e.g., Wyatt et al. 1999;Wolf et al. 2007;

Krivov 2010;Lee & Chiang 2016). The strength of the perturbing

eﬀect that a substellar companion has on the orbits of planetesi-

mals and dust primarily depends on the distance of the perturber

to the perturbed objects. Therefore, the spatial dust distribution

produced by planetesimal collisions can be a signpost of old and

far-out planets as well. Hence, analyses of spatially resolved ob-

servations of debris disks potentially serve as a planet hunting

method that is complementary to the well-established methods

that make use of stellar radial velocity, transits, and direct imag-

ing to ﬁnd exoplanets in close-by stellar systems (e.g., Pearce

et al. 2022).

Narrow, eccentric debris ringsare particularlypromisingtrac-

ers of long-term planetary perturbations. The deviation from cir-

cularity suggests that perturbations have happened, while the

narrowness excludes violent short-term causes such as stellar

ﬂybys (e.g., Larwood & Kalas 2001;Kobayashi & Ida 2001)

or (repeated) close encounters with planets (e.g., Gomes et al.

2005). For long-term perturbations, where timescales are much

longer than individual orbital periods, the orbits of belt objects

are aﬀected more coherently, with little spread in orbital ele-

ments. In contrast, instantaneous positions along the orbits are

important in short-term perturbation events, resulting in a wider

spread in orbital elements and wider disks. A narrow yet eccen-

tric disk can only be compatible with a disk-crossing planet if

the thus-excited wide disk component is subsequently removed

(Pearce et al. 2021). The belts around Fomalhaut (e.g., Kalas

et al. 2005,2013;Boley et al. 2012;MacGregor et al. 2017),

HD 202628 (Krist et al. 2012;Schneider et al. 2016;Faramaz

et al. 2019), HD 53143 (MacGregor et al. 2022), and the younger

system HR 4796 A (e.g., Moerchen et al. 2011;Kennedy et al.

2018) are well-resolved examples of narrow, eccentric disks.

To accomplish the task of using spatially resolved observa-

tions of dust emission to infer exoplanets and their properties, two

key ingredients are necessary: ﬁrst, the planet-disk interaction,

thatis, howperturbingplanets shape thespatial dustdistributions,

and second, how these dust distributions appear in observations.

With such a framework, in addition to searching for hints of exo-

planets, we can use the known exoplanet–debris disk systems as

test beds to better constrain debris disk properties such as col-

lisional timescales, planetesimal stirring (e.g., Marino 2021), or

self-gravitation (Seﬁlian et al. 2021) as well as planetesimal and

dust material properties such as the critical energy for fragmen-

tation, 𝑄★

D(e.g., Kim et al. 2018).

This paper is organized as follows: In Sect. 2we present the

numerical methods applied to collisionally evolve planetesimal

belts secularly perturbed by a substellar companion and discuss

theresultingspatialgraindistributions fordiﬀerentperturber–belt

combinations. Based on these results, we show in Sect. 3how we

simulated ﬂux density maps and explore the relative contribution

of diﬀerent grain sizes to the total radiation as well as how the

halo of small grains on very eccentric orbits can be investigated

observationally. In Sect. 4we search for observable features to

distinguishbetween systemswitha substellar companion orbiting

inside or outside a parent planetesimal belt and present our results

in a simple decision tree. Lastly, in Sect. 5we discuss the results

and in Sect. 6brieﬂy summarize our ﬁndings.

2. ACE simulations

We used the code Analysis of Collisional Evolution (ACE,Krivov

et al. 2005,2006;Vitense et al. 2010;Löhne et al. 2017;Sende &

Löhne 2019) to evolve the radial, azimuthal, and size distribution

of the material in debris disks. ACE follows a statistical approach,

grouping particles in category bins according to their masses, 𝑚,

and three orbital elements: pericenter distances, 𝑞, eccentricities,

𝑒, and longitudes of periapse, 𝜛= Ω +𝜔.

Mutual collisions can lead to four diﬀerent outcomes in ACE,

depending on the masses and relative velocities of the colliders.

At the highest impact energies, both colliders are shattered and

the fragments dispersed such that the largest remaining fragment

has less than half the original mass. Below the energy threshold

for disruption and dispersal, a larger fragment remains, either

as a direct remnant of the bigger object or as a consequence

of gravitational re-accumulation. A cloud of smaller fragments

is produced. If neither of the two colliders is shattered by the

impact, both were assumed to rebound, resulting in two large

remnants and a fragment cloud. In the unreached case of an

impact velocity below ∼1m/s, the colliders would stick. These

four regimes assumed in ACE represent a simpliﬁcation of the

zoo of outcomes mapped by Güttler et al. (2010). In addition

to the collisional cascade, we took into account stellar radiation

and wind pressure, the accompanying drag forces, and secular

gravitational perturbation by a substellar companion.

In the following subsections we describe the recent improve-

ments made to the ACE code, motivate and detail the simulation

parameters, and present the resulting distributions.

2.1. Improved advection scheme

In Vitense et al. (2010) and subsequent work (e.g., Reidemeister

et al. 2011;Schüppler et al. 2014;Löhne et al. 2017) we used

the upwind advection scheme to propagate material through the

𝑞–𝑒grid of orbital elements under the inﬂuence of Poynting–

Robertson (PR) and stellar wind drag. Sende & Löhne (2019)

applied that scheme to the modeling of secular perturbations,

where 𝑞,𝑒, and 𝜛are aﬀected while the semimajor axes are

constant. In the following we refer to both PR drag and secular

perturbations as transport.

The upwind scheme moves material across the borders from

one bin to its neighbors based on the coordinate velocities and

amount of material in that bin. The scheme is linear in time and

has the advantage that the transport gains and losses can be added

simply to the collisionalones. For the PR eﬀect, where drag leads

to smooth and monotonous inward spread and circularization

from the parent belt and halo, this scheme is suﬃcient. However,

a narrow, eccentric parent belt under the inﬂuence of (periodic)

secular perturbations requires the translation of sharp features

in 𝑞,𝑒, and 𝜛across the grid. The upwind scheme smears the

sharp features out too quickly, inducing an unwanted widening

and dynamical excitation of the disk (Sende & Löhne 2019). To

reduce the eﬀect of this numerical dispersion, we introduced an

operator splitting to the ACE code, where collisions and transport

(caused by drag and secular perturbation) are integrated one after

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T. A. Stuber et al.: Infer substellar companions from debris disk observations

the other for every time step. The transport part is integrated using

a total variance diminishing (TVD) scheme (Harten 1983) with

thesuperbee ﬂux limiter (Roe1986). Thecontributionsfrom PR

drag and secular perturbation to the change rates ¤𝑞,¤𝑒, and ¤𝜛are

summed up. For each time step Δ𝑡, the ﬂow in three dimensions

is again subdivided into ﬁve stages: Δ𝑡/2in 𝑞,Δ𝑡/2in 𝑒,Δ𝑡in 𝜛,

Δ𝑡/2in 𝑒, and Δ𝑡/2in 𝑞. A comparison of the resulting amounts

of numerical dispersion in the TVD and the upwind schemes is

shown in Appendix A.

2.2. Common parameters

The distribution of observable dust grains is determined by a

range of parameters, including not only parameters of the dust,

the disk, and the perturber, but also of the host star. The dust

material was not varied in our study as we deem the discussed

tracers of planetary perturbations unaﬀected. We chose a mate-

rial blend of equal volume fractions of water ice Li & Greenberg

(1998) and astronomical silicate (Draine 2003), assuming a bulk

density of 2.35 g cm−3for the blend. The refractive indices are

combined with the Maxwell–Garnett mixing rule (Garnett 1904).

Radiation pressure eﬃciency as well as absorption and scattering

cross sections were calculated assuming compact spheres, using

the Mie theory (Mie 1908) algorithm miex (Wolf & Voshchin-

nikov 2004). Below a limiting grain radius 𝑠bo, which depends

on the dust material and the stellar luminosity, radiation pres-

sure overcomes the gravitational pull and removes the grains on

short timescales (e.g., Burns et al. 1979). We assumed the same

critical speciﬁc energy for disruption and dispersal, 𝑄∗

D, as in

Löhne et al. (2017, their Eq. 12): a sum of three power laws for

strength- and gravity-dominated shocks and a pure gravitational

bond, respectively.

The grid of object radii extended from 0.36 µmto 481 m. At

the upper end, a factor of 2.3separated neighboring size bins.

This factor reduced to 1.23 near the blowout limit. The according

mass grid follows Eq. (26) of Sende & Löhne (2019). Material

in the initial dust belt covered only radii that exceeded 100 µm,

with a power-law index of −3.66, close to the steady-state slope

expected in the strength regime (O’Brien & Greenberg 2003).

The lower size bins were ﬁlled in the subsequent evolution. The

logarithmic grid of pericenter distances had 40 bins between

40 au and 160 au. The eccentricity grid was linear for 0.4≲𝑒≲1

and logarithmic outside of this range, following Eq. (29) of Löhne

et al. (2017). The 36 bins in the grid of orbit orientations, 𝜛, were

each 10◦wide.

The belts were assumed to start unperturbed but pre-stirred;

initially circular with a maximum free eccentricity 𝑒max =0.1.

The distributions in 𝑞,𝑒, and 𝜛were jointly initialized from

a random sample of uniformly distributed semimajor axes, ec-

centricities, and longitudes of ascending nodes. This random

sample was then propagated for a time 𝑡pre under the sole in-

ﬂuence of transport, that is, PR drag and secular perturbation.

After this propagation the resulting sample was injected into the

grid. From this time on, the distribution was allowed to settle into

collisional equilibrium for 𝑡settle =20 Myr. Only after 𝑡pre +𝑡settle

passed would the combined simulation of transport and colli-

sions begin, lasting for a time 𝑡full. The procedure ensures that

(a) the numerical dispersion is further reduced by solving the ini-

tial perturbation before the discretization of the grid is imposed

and (b) the size distribution of dust grains has time to reach a

quasi-steady state. Figure 1illustrates the constant orbit of the

planet and the mean belt orbit at the diﬀerent stages of the sim-

ulation runs. Stages 𝑡pre and 𝑡full were tuned such that the mean

belt orientation (and eccentricity) was the same for all simulation

𝑡pre

𝑡full

𝑡settle (a)

(b)

(c)

perturber

Fig. 1: Schematic representation of the belt orbits (solid gray and

black) and the planet orbit (red) at the diﬀerent stages of the ACE

simulations: (a) the initial, circular, unperturbed belt; (a–b) the

belt perturbed by the planet, but not modiﬁed by collisions; (b)

the belt modiﬁed by collisions, but not by the perturber; (b–c)

the belt modiﬁed by both the perturber and collisions.

runs. This is to mimic the normal case of an observed disk of

given eccentricity and orientation and an unseen perturber with

unknown mass and orbital parameters.

Our simulations resulted in snapshots of narrow belts that

have not reached a dynamical equilibrium with the perturber yet,

undergoing further evolution. If not prevented by the self-gravity

of the belt, diﬀerential secular precession between belt inner and

outer edges would widen the eccentricity distribution to a point

where it is only compatible with the broad disks that are in the

majority among those observed (e.g., Matrà et al. 2018;Hughes

et al. 2018). The combined collisional and dynamical steady state

of broad disks was simulated by Thebault et al. (2012). Kennedy

(2020) discussed the potential dynamical history of both broad

and narrow disks.

We assumed the perturber to be a point mass on a constant,

eccentric orbit closer to or further away from the star than the

belt.The diskwasassumed not to exert secular perturbation itself,

neglecting the eﬀects of secular resonances internal to the disk

or between disk and planet (cf. Pearce & Wyatt 2015;Seﬁlian

et al. 2021). The model is only applicable to systems where the

perturber is more massive than the disk. Because the estimated

total masses of the best-studied, brightest debris disks can exceed

tens or hundreds of Earth masses (Krivov et al. 2018;Krivov &

Wyatt 2021), we limited our study to perturber masses on the

order of a Jupiter mass or above.

The stellar mass and luminosity determine orbital velocities

and the blowout limit. For stars with a higher luminosity-to-mass

ratio, the blowout limit is at larger grain radii. Barely bound

grains slightly above that limit populate extended orbits to form

the disk halo. In addition, the lower size cutoﬀ induces a char-

acteristic wave in the grain size distribution (Campo Bagatin

et al. 1994;Thébault et al. 2003;Krivov et al. 2006) that can

translate to observable spectro-photometric features (Thébault

Article number, page 3 of 27

A&A proofs: manuscript no. main

Table 1: Assumed parameters for the parent belt.

Id. 𝑀bΔ𝑎b𝑡settle Description

[𝑀⊕][au] [Myr]

n0.09 10 20 reference

w0.09 20 20 wide

m2 0.28 10 6.4 high mass

m3 1.410 1.3 very high mass

Notes. 𝑀bis the total belt mass in objects with radii 𝑠 < 500 m,Δ𝑎bthe

spread in orbital semimajor axes, and 𝑡settle the time during which the

size distribution is allowed to settle to a collisional equilibrium before

collisions and perturbations were modeled jointly. See text for details.

& Augereau 2007). Both the spectral ranges at which the halo

and the wave in the size distribution are most prominent are de-

termined by the stellar spectral type. However, that inﬂuence is

well understood and mostly qualitative. The diﬀerences from one

host star to another at a constant wavelength (of light) are simi-

lar to the diﬀerences from one wavelength (of light) to another

for a single host star. We therefore modeled only one speciﬁc

host star with a mass of 1.92 M⊙and a luminosity of 16.6 L⊙,

roughly matching the A3 V star Fomalhaut. We assumed the

spectrum of a modeled stellar atmosphere with eﬀective temper-

ature 𝑇eﬀ =8600 K, surface gravity log10 (𝑔[cm s−2]) =4.0, and

metallicity [Fe/H]=0.0(Hauschildt et al. 1999). The results

are insensitive to the last two parameters. The corresponding

blowout limit is 𝑠bo ≈4µm. For main-sequence stars of mass

lower than the Sun, radiation pressure is too weak to produce

blowout grains (e.g., Mann et al. 2006;Kirchschlager & Wolf

2013;Arnold et al. 2019). The lack of a natural lower size cut-oﬀ

for late-type stars can lead to transport processes (e.g., Reide-

meister et al. 2011) and nondisruptive collisions (Krĳt & Kama

2014;Thebault 2016) becoming more important, potentially re-

sulting in observable features that are qualitatively diﬀerent from

the ones presented here. We do not cover this regime here.

2.3. Varied parameters

We varied a total of ﬁve physically motivated parameters in our

study: the disk mass 𝑀b, the belt widths, Δ𝑎b, as well as the

perturber mass, 𝑀p, semimajor axis, 𝑎p, and eccentricity, 𝑒p.

The parameter combinations assumed for belts and perturbers

are summarized in Tables 1and 2, respectively, together with the

collisional settling time, 𝑡settle, the initial perturbation time, 𝑡pre,

and the period of full simulation of perturbations and collisions,

𝑡full. We use the abbreviations given in the ﬁrst columns of these

tables to refer to individual model runs. For example, the run that

combined the wider parent belt with an inner high-mass perturber

is denoted w-i-M3, while the combination of a narrower belt

with the low-eccentricity, high-mass inner perturber is denoted

n-i-M3-le.

Theeﬀectsofsecular perturbationbya single perturber canbe

reduced to two main quantities: the timescale and the amplitude.

To leading orders in orbital eccentricities and semimajor axes, the

time required for a full precession cycle of grains launched from

parent bodies on near-circular orbits at 𝑎bis given by (Sende &

Löhne 2019)

𝑇prec ≈4

𝑀∗

𝑀p𝑃b









𝑎b

𝑎p2(1−𝛽)4

(1−2𝛽)7/2∝𝑎7/2

b𝑎−2

p𝑀−1

p(inner)

𝑎p

𝑎b3(1−2𝛽)3/2

1−𝛽∝𝑎−3/2

b𝑎3

p𝑀−1

p(outer)

(1)

for perturbers distant from the belt, where 𝑀∗=1.92 𝑀⊙is the

mass of the host star, 𝑃bthe orbital period of the parent bod-

ies, and 𝛽the radiation-pressure-to-gravity ratio of the launched

grains. Hence, the perturbation timescale is determined by a

combination of perturber semimajor axis, perturber mass, and

belt radius.

The amplitude of the perturbations is controlled by the forced

orbital eccentricity that is induced by the perturber,

𝑒f≈5

4𝑒p(𝑎p

𝑎b

1−2𝛽

1−𝛽∝𝑎−1

b𝑎p𝑒p(inner)

𝑎b

𝑎p

1−𝛽

1−2𝛽∝𝑎b𝑎−1

p𝑒p(outer),(2)

around which the actual belt eccentricity evolves. This ampli-

tude is determined by belt radius, perturber semimajor axis, and

perturber eccentricity.

With the perturbation problem being only two-dimensional,

we reduced the set of varied parameters by ﬁxing the mean radius

of the belt at 100 au, a typical value for cold debris disks observed

in the far-infrared and at (sub)millimeter wavelengths (e.g., Eiroa

et al. 2013;Pawellek et al. 2014;Holland et al. 2017;Sibthorpe

et al. 2018;Matrà et al. 2018;Hughes et al. 2018;Marshall

et al. 2021;Adam et al. 2021). For 𝑎b=const, which is a given

parent belt, the timescale is constant for 𝑀p∝𝑎−2

pand an inner

perturber, or 𝑀p∝𝑎3

pand an outer perturber. The amplitude

is constant for 𝑒p∝𝑎−1

pand an inner perturber, or 𝑒p∝𝑎p

and an outer perturber. We expect degenerate behavior for some

parameter combinations even in our reduced set.

The runs di, with an inner perturber closer to the belt, and

o-M3, with an outer perturber, were constructed as degeneracy

checks. Their outcomes should be as close to the reference run,

n-i-M2, as possible. The perturbation timescales and amplitudes

in the middles of the respective belts, given in Eqs. (1) and (2),

are the same for all three parameter sets. For n-di the parameters

listed in Tables 1and 2imply that diﬀerential precession acts on

exactly the same timescale and with the same amplitude as in run

n-i-M2 throughout the whole disk. The outcomes of the di runs,

which had an inner perturber closer to the belt, should therefore

be fully degenerate with the equivalent M2 runs. For the runs

with an outer perturber, o-M3, the degeneracy applies only to the

belt center because the sense of the diﬀerential perturbation is

inverted as the exponent to 𝑎bchanges from +7/2to −3/2. The

dependence on 𝛽is inverted too: the 𝛽-dependent term in Eq. (1)

increases with increasing 𝛽for an inner perturber and decreases

with increasing 𝛽for an outer perturber.

While 𝑀paﬀects only the perturbation timescale, 𝑎pand 𝑒p

aﬀect the overall amplitude of the perturbations. In runs le we

therefore lowered the perturber eccentricities to the more mod-

erate value of 𝑒p=0.3(from the reference value, 𝑒p=0.6). In

an initially circular belt at 100 au, a perturber with 𝑒p=0.3at

20 au induces a maximum belt eccentricity that amounts to ap-

proximately twicethe forced eccentricity given by Eq. (2), that is,

≈2×5/4×0.3×20/100 =0.15 (compared to 0.30 for 𝑒p=0.6).

While planetary orbital eccentricities around 0.3 are common

among long-period exoplanets, eccentricities around 0.6 are rare

(Bryan et al. 2016). Likewise, belt eccentricities around 0.15 are

closer to the maximum of what is derived for observed disks,

as exempliﬁed by the aforementioned disks around Fomalhaut,

HD 202628, and HR 4796 A. Therefore, our reference case pro-

videsclearer insights intothe expected qualitative behavior,while

runs le are closer to observed disks.

The perturber determines the rate at which the secular per-

turbation occurs, while the collisional evolution in the belt deter-

Article number, page 4 of 27

T. A. Stuber et al.: Infer substellar companions from debris disk observations

Table 2: Parameter combinations for the perturbers.

Id. 𝑀p𝑎p𝑒p𝑡pre 𝑡full Description

[𝑀Jup][au] [Myr] [Myr]

i-M1 0.5 20 0.625 15 inner, low mass

i-M2 2.5 20 0.65 3 inner, medium mass, reference

i-M3 12.5 20 0.61 0.6 inner, high mass

i-M4 62.5 20 0.60.2 0.12 inner, very high mass

o-M2 2.5 500 0.625 15 outer, medium mass

o-M3 12.5 500 0.65 3 outer, high mass

o-M4 62.5 500 0.61 0.6 outer, very high mass

-le — — 0.3— — low eccentricity

di 0.49 40 0.31 — — degenerate, inner

p0 — — — — — no precession

p1 — — — — — no ongoing prec.

Notes. Where no values are given, the corresponding reference values apply.

mines the rate at which the small-grain halo is replenished and

realigned. Collisions occur on a timescale given by the dynamical

excitation, spatial density, and strength of the material. Instead

of varying all these quantities, we varied only the disk mass (in

runs m2 and m3) as a proxy for the spatial density. An increased

disk mass and a reduced perturber mass are expected to yield

similar results, as in both cases, the ratio between the timescales

for collisional evolution and for secular perturbation is reduced.

The total disk masses given in Table 1may seem low because the

simulation runs are limited to object radii ≲0.5km. However,

when extrapolating to planetesimal radii ∼100 km with a typical

power-law index of -3.0. . . -2.8, as observed in the Solar System

asteroid and Kuiper belts, the total masses increase by factors of

(100/0.5)1.0...1.2(i.e., by 2 to 2.5 orders of magnitude).

The belt width was varied explicitly in runs w. Not only do

the belts in runs whave lower collision rates but also increased

diﬀerential precession and potentially a clearer spatial separation

of observable features.

Finally, we set up two runs that allow us to diﬀerentiate

between the eﬀects of the mean belt eccentricity, the diﬀerential

precession of the belt, and the ongoing diﬀerential precession of

the halo with respect to the belt. Only the last would be a sign

of a currently present perturber. The ﬁrst two could, for example,

be the result of an earlier, ceased perturbation. In runs p0, we

assumed belts with the same mean eccentricity and orientation

as M2, but without any diﬀerential secular perturbation, that is,

no twisted belt or halo. Such a conﬁguration could result if the

perturbations have ceased some time ago or the eccentricity was

caused by another mechanism, such as a single giant breakup.

In runs p1, the belts were initially twisted to the same degree as

for n-i-M1 to n-i-M4, but no ongoing precession was assumed

to drive the twisting of the halo. Ceased perturbation could be a

physical motivation for that scenario as well. However, the main

purpose of p0 and p1 was to help interpret the causes of and act

as a baseline for features in the other simulation runs.

2.4. Grain distributions for an inner perturber

Figure 2shows distributions of normal geometrical optical thick-

ness for grains of diﬀerent sizes in run n-i-M2, our reference for

subsequent comparisons. These maps resulted from a Monte-

Carlo sampling of the 𝑞–𝑒–𝜛phase space as well as mean

anomaly. The maps show the contribution per size bin. The to-

tal optical thickness peaks at ≈3×10−5in run n-i-M2, the

total fractional luminosity at a similar value, which is a moder-

ate value among the minority of mass-rich disks with brightness

above current far-infrared detection limits (e.g., Eiroa et al. 2013;

Sibthorpe et al. 2018) and a low value among those above mil-

limeter detection limits (e.g., Matrà et al. 2018).

The big grains in Fig. 2d represent the parent belt, which

started out circular and then completed almost half a counter-

clockwise precession cycle. The higher precession rate of the

inner belt edge caused the left side to be diluted and the right side

to be compressed, resulting in an azimuthal brightness asym-

metry. This geometric eﬀect of diﬀerential precession is notable

only when the width of the belt is resolved. In wider belts, dif-

ferential precession can create spiral density variations (Hahn

2003;Quillen et al. 2005;Wyatt 2005a). The eﬀect becomes in-

creasingly prominent over time or reaches a limit set by the belt’s

self-gravity.

For smaller grains, the eﬀects described in Löhne et al. (2017)

come into play. Figure 2c shows the distribution of grains with

radii 𝑠≈9µm. These grains are produced in or near the parent

belt, but populate eccentric orbits as a result of their radiation

pressure-to-gravity ratio 𝛽≈0.23. Those grains born near the

parent belt’s pericenter inherit more orbital energy and form

a more extended halo on the opposite side, beyond the belt’s

apocenter, in the lower part of the panel. At the same time, the

alignment of their orbits creates an overdensity of mid-sized

grains near the belt pericenter. The yet smaller grains in panels

(a) and (b) tend to become unbound when launched from the

parent belt’s pericenter, while those launched from the belt’s

apocenter have their apocenters on the opposite side, forming a

halo beyond the belt’s pericenter, in the upper parts of the panels.

These eﬀects are purely caused by the parent belt being eccentric.

The ongoing diﬀerential precession causes a misalignment of

the halo with respect to the parent belt (Sende & Löhne 2019).

This misalignment is more clearly seen in panels (b) and (c),

which show a clockwise oﬀset of the outer halo with respect to

the belt, resulting from the wider halo orbits precessing at a lower

rate. The population of halo grains is in an equilibrium between

steady erosion and replenishment. Erosion is caused by colli-

sions and PR drag, while replenishment is caused by collisions

of somewhat bigger grains closer to the belt. The population of

freshly produced small grains forms a halo that is aligned with

the belt, while the older grains in the halo trail behind. The ratio

of grain lifetime and diﬀerential precession timescale determines

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Astronomy&Astrophysicsmanuscriptno.main©ESO2024December31,2024UsingdebrisdiskobservationstoinfersubstellarcompanionsorbitingwithinoroutsideaparentplanetesimalbeltT.A.Stuber1,T.Löhne2,andS.Wolf11InstitutfürTheoretischePhysikundAstrophysik,Christian-Albrechts-UniversitätzuKiel,Leibnizstr.15,24118Kiel,...

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Using debris disk observations to infer substellar companions orbiting within or outside a parent planetesimal belt

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