Resolving the Outer Ring of HD 38206 Using ALMA and Constraining Limits on Planets in the System

MNRAS 000, 000–000 (0000) Preprint 29 October 2020 Compiled using MNRAS LATEX style ﬁle v3.0

Resolving the outer ring of HD 38206 using ALMA and constraining limits on planets in the system

Mark Booth1?, Michael Schulz1, Alexander V. Krivov1, Sebastián Marino2,3, Tim D. Pearce1 and Ralf Launhardt2 1 Astrophysikalisches Institut und Universitätssternwarte, Friedrich-Schiller-Universität Jena, Schillergäßchen 2-3, 07745 Jena, Germany 2 Max Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany 3 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK

Accepted 2020 October 23. Received 2020 October 21; in original form 2020 September 1

ABSTRACT HD 38206 is an A0V star in the Columba association, hosting a debris disc first discovered by IRAS. Further observations by Spitzer and Herschel showed that the disc has two components, likely analogous to the asteroid and Kuiper belts of the Solar System. The young age of this star makes it a prime target for direct imaging planet searches. Possible planets in the system can be constrained using the debris disc. Here we present the first ALMA observations of the system’s Kuiper belt and fit them using a forward modelling MCMC approach. We detect an extended disc of dust peaking at around 180 au with a width of 140 au. The disc is close to edge on and shows tentative signs +0.10 of an asymmetry best fit by an eccentricity of 0.25−0.09. We use the fitted parameters to determine limits on the masses of planets interior to the cold belt. We determine that a minimum of four planets are required, each with a minimum mass of 0.64 MJ , in order to clear the gap between the asteroid and Kuiper belts of the system. If we make the assumption that the outermost planet is responsible for the stirring of the disc, the location of its inner edge and the eccentricity of the disc, then we can more tightly predict its eccentricity, mass and semimajor axis to +0.20 +0.5 +12 be ep = 0.34−0.13, mp = 0.7−0.3 MJ and ap = 76−13 au. Key words: circumstellar matter – planetary systems – submillimetre: planetary systems – stars: individual: HD 38206 – planet-disc interactions

1 INTRODUCTION In this paper we present and analyse the first ALMA image of the debris disc around HD 38206. HD 38206 is a star of As one of the key components of a planetary system, study- spectral type A0V. The debris disc around this star was first ing debris discs enables us to understand the current make identified using IRAS data by Mannings & Barlow(1998) up of a planetary system and its formation and evolution. In and has also been detected by Spitzer/MIPS (Rieke et al. recent years the Atacama Large Millimeter/submillimeter Ar- 2005), Spitzer/IRS (Morales et al. 2009), Gemini/T-ReCS ray (ALMA) has made it possible to image these discs at long (Moerchen et al. 2010) and Herschel/PACS (Morales et al. wavelengths in much finer detail than was previously possible 2016). By analysing both the resolved Herschel images and

arXiv:2010.14521v1 [astro-ph.EP] 27 Oct 2020 (for a recent review see Hughes et al. 2018). Of particular the full SED, Morales et al.(2016) demonstrate that the interest are systems where both debris discs and at least one system is seen close to edge on and has two belts at 11 planet have been observed. Analysis of such systems is of and 160 au. The system is thought to be a member of the prime importance for understanding the interaction between Columba association (Torres et al. 2008), giving it an age planets and the disc. Examples of such systems include Foma- of 42+6 Myr (Bell et al. 2015). The young age of this system lhaut (Kalas et al. 2008; Boley et al. 2012), HR 8799 (Marois −4 means that it is a prime candidate for direct imaging surveys, et al. 2010; Booth et al. 2016), β Pic (Lagrange et al. 2010; although no planets have been detected so far. Shannon et al. Dent et al. 2014) and HD 95086 (Rameau et al. 2013; Su et al. (2016) developed a model for the minimum mass of planets 2017). For systems where no planet has yet been directly im- required to clear a gap in a two belt debris disc system. They aged, strong constraints on where the outer planets in the use HD 38206 as an example case and show that the mini- system are can still be derived from studying the debris disc mum mass of planets is close to the upper limit possible from (e.g. Booth et al. 2017; Marino et al. 2018b, 2019). VLT/SPHERE observations. By analysing the ALMA data of this system we shall re-assess the limits on the masses of

? E-mail: [email protected]

2 ALMA OBSERVATIONS 100 2 The observation of HD 38206 was carried out by ALMA in band 6 during cycle 1 as part of the project 2012.1.00437.S 50 0 (PI: David Rodriguez). It was observed on the 7th March 2014 with a precipitable water vapour of 1.80 mm and 35.7 mins 0 µJy/beam on source time by 23 antennas in a compact configuration. Dec offset, " 2 These led to baselines between 15−365 m. The received data 50 was calibrated using the standard observatory calibration in 4 CASA version 4.7.74. The quasar J0609-1542 was used for bandpass and phase calibration and the active galactic nu- 100 cleus PKS 0521-36 as flux calibrator. The correlator was set 5.0 2.5 0.0 2.5 5.0 to provide four spectral windows, processing two polariza- RA offset, " tions in each of these. While one spectral window was centred at the CO J=2-1 line at 230.538 GHz, with 3840 channels of width 0.5 MHz, the other three had central frequencies of 213, Figure 1. Image of HD 38206 at 1.35 mm after processing with the 215 and 228 GHz and 128 channels with a width of 16 MHz CLEAN algorithm. The contours show the ±2, 4, 6 and 8σ levels. to study the dust continuum emission. The white cross marks the position on the star based on the Gaia The image of the disc is shown in Figure 1. This has been DR2 position and accounting for proper motion. created from the inversely Fourier transformed complex visibilities using natural weighting and multi-frequency synthesis, followed by processing with the CLEAN algorithm (Hög- Table 1. Overview of the stellar parameters that have been used bom 1974). We obtain a synthesised beam of size 0.9700×0.7500 for the analysis. ◦ and a beam position angle of −86.5 measured from North to Parameter Value Reference East. We measure the RMS to be σ = 23.5 µJy beam−1. The disc is seen to be edge-on. There are some signs of asymmetry d, pc 71.3±0.4 1 RA (J2000) 05h 43m 21.67s 2 with a peak in the emission of 0.18±0.02 mJy 2.100 east of the DEC (J2000) −18◦ 330 26.9100 2 star, whilst to the west the emission peaks at 2.900 from the Age, Myr 42+6 3 star, but with a lower flux density of 0.09±0.02 mJy. We do −4 R?,R 1.7±0.2 4 not expect this to be due to a pointing issue as the phase- +160 Teff, K 9610 2 centre location agrees well with the expected position of the −1740 L?,L 26 ± 7 4 star based on the Gaia DR2 (Gaia Collaboration et al. 2018) M?,M 2.4 ± 0.4 4 position after correcting for proper motion. log(g) 4.4±0.3 4 The total flux density within an ellipse surrounding the References. (1) Bailer-Jones et al.(2018), (2) Gaia Collaboration emission is 0.7±0.1 mJy. Morales et al.(2016) found the flux +6.5 et al.(2018), (3) Bell et al.(2015), (4) Stassun et al.(2018) density at 160 µm to be 188.9−6.5 mJy. By assuming a single power law between 160 µm and 1350 µm, typically formulated −(2+β) as Fν = λ , we find β = 0.6, a typical value for a debris bution in cylindrical coordinates, Σ(r, φ, z), as a Gaussian. disc (Holland et al. 2017). Given that the image shows some signs of asymmetry, we To check for CO J=2-1 emission, we also used CLEAN define the radial distribution as a Gaussian in terms of the to create a data cube of the channels around the expected semi-major axis, a(r, φ), rather than the radial distribution, −1 radial velocity of the star (25.3 km s Gontcharov 2006). r, where the semi-major axis is determined by No CO line emission was detected. By integrating over the pixels where continuum emission is detected (at > 3σ) we 1 − e cos(φ − ω) a(r, φ) = r 2 , (1) find the 3σ upper limit on an unresolved emission line to be 1 − e 1.4 × 10−22 W m−2 (using Equation A2 of Booth et al. 2019). where e is the eccentricity and ω is the argument of pericentre. Based on a model of gas production through the collisional The surface density distribution is then given by cascade, Kral et al.(2017) predicted that this system would (a − a )2 z2 have a CO J=2-1 level of 2.2×10−25 W m−2. Our upper limit 0 Σ(r, φ, z) ∝ exp − 2 − 2 2 , (2) is, therefore, consistent with this model and much deeper ob- 2σr 2h r servations will be necessary to detect any gas in this system. ∆a σr = √ , (3) 2 2 log 2

3 MODELLING where a0 is the mean of the Gaussian, ∆a is the full width half maximum (FWHM) of the Gaussian and h is the aspect 3.1 Disc setup ratio. Given the low resolution of these observations, we do The modelling procedure used here follows that of Marino not expect to vertically resolve the disc and so have arbitrarily et al.(2019). We model the dust radial and vertical distri- set h = 0.01.

MNRAS 000, 000–000 (0000) ALMA Observations of HD 38206 3

Table 2. Parameters for the best ﬁtting model. The uncertainties given for the parameters are the 16th and 84th percentiles of the posterior distribution. Ω is measured anti-clockwise from North, whilst ω is measured anti-clockwise from Ω.

◦ ◦ ◦ a0, au ∆a, au Mdust,M⊕ I, Ω, e ω, +19 +46 +0.016 +1.3 +1.2 +0.10 +22 184−17 143−36 0.105−0.015 83.3−1.3 84.3−1.2 0.25−0.09 49−25

100 5 1.0 75 4 4 4 50 0.5 2 2 25 3 Real, mJy 0 0

µJy 0 0.0

2 25 µJy/beam Dec offset, " 2 Dec offset, " 2 50 1 4 1 4 75 0 0 5.0 2.5 0.0 2.5 5.0 5.0 2.5 0.0 2.5 5.0 0 20 40 60 80 100 RA offset, " RA offset, " Imaginary, mJy Baseline distance, k

Figure 2. Left: Model image for the best-ﬁtting parameters of the model. Middle: Image residuals after subtracting the best-ﬁtting model. The contours show the 3σ residuals. Right: The deprojected and azimuthally averaged observed visibilities (points) and model visibilities (dashed line).

For the dust grains we assume a differential size distribu- adjusted to each wavelength assuming a power law change in tion n ∝ sα, with the power-law index α = − 3.5, following the flux density with a β = 0.6 for the disc and β = 0 for the Dohnanyi(1969). Following the spectral energy distribution star (see section 2). The resultant image output from RADMC- analysis of Morales et al.(2016), we set the minimum grain 3D is then multiplied by the primary beam, Fourier trans- radius to Smin = 5 µm. The maximum grain radius is set to formed and then sampled at the same points in the Fourier an arbitrarily large value of Sdust = 1 cm. The total mass of plane as the original data in order to compare the two. The 2 dust in the disc is defined by Mdust. For the optical proper- χ statistic is then calculated using: ties, astrosilicate grains were assumed (Draine 2003) as these Nvis produced the best fit to the photometry (Morales et al. 2016). 2 X 2 χ = (Re (Dj ) − Re (Mj )) wj We note that, since we are modelling a resolved image at a j=1 single wavelength, these parameters have little effect on the 2 final results, with the exception of the dust mass that defines + (Im (Dj ) − Im (Mj )) wj . (4) the total flux density. where Dj and Mj are the observed and model visibilities The stellar parameters are given in Table 1. To calculate respectively and wj are the weights. the stellar flux we have used a Kurucz stellar template spec- A known problem with data taken during the early cycles trum (Kurucz 1979) with values closest to those for HD 38206 of ALMA, such as the data analysed here, is that the weights (Teff=9500 K and log(g)=4.5). are correct relative to each other but not necessarily correct To create the final image of the disc we define the inclina- in an absolute sense1. Using the uncorrected weights will still tion with respect to face-on, I, and the position angle anti- result in determining the best fit model but the uncertainties clockwise from North, Ω, before running a radiative transfer on the fitted parameters may be incorrect. To account for this computation using RADMC-3D (Dullemond et al. 2012). we need to determine the factor, f, that the uncertainties are To summarise, our model has seven free parameters – a0, underestimated by. This could be left as a free parameter, ∆a, Mdust, I, Ω, e and ω. however, since the SNR per visibility is much less than 1, we can make a very good estimate of f by calculating the χ2 for p 2 a null model. Through this method f = χ /(2Nvis), where 3.2 MCMC fit Nvis is the number of visibilities used in the fit and we are To find the best fitting parameters we make use of the emcee using 2Nvis because the real and imaginary components are module (Foreman-Mackey et al. 2013). It is a Python imple- independent (see e.g. Section 6.2.2 of Thompson et al. 2017). wj mentation of an Affine Invariant Markov chain Monte Carlo The weights are then adjusted so that wj → . f 2 (MCMC) Ensemble Sampler (Goodman & Weare 2010). In Assuming Gaussian uncertainties, the likelihood is then: this paper we fit directly to the visibility data, which has been 2 averaged to one channel per spectral window. This means ln L ∝ −χ /2. (5) that we are fitting images at four different wavelengths, which We used uniform priors for all free parameters with limits of is done since the differences between the spectral windows 20 au < r < 350 au, 10 au < ∆r < 2r , 60◦ < I < 90◦, are large enough that simply averaging to one frequency 0 0 could distort the final visibilities. However, this means that the model must be compared at four different wavelengths. 1 https://casaguides.nrao.edu/index.php/ Rather than create four separate models, one is created and DataWeightsAndCombination

MNRAS 000, 000–000 (0000) 4 M. Booth et al.

can easily be converted to a flux density by summing the flux in the best fit image. Through this process we find that F1350 µm =0.81±0.12 mJy, which is consistent with that found from the CLEAN image in section 2.

4 DISCUSSION When analysing the spatial distribution of a debris disc, the natural question to ask is what does this mean for the planetary system? From our and prior observations we know that: 150 • the very fact that we see dust at all requires that the planetesimals must have been stirred up enough to initiate a

, 0 collisional cascade • the radial distribution of the outer belt and the presence of an inner belt implies gap clearing in between the two • there are tentative signs of asymmetry that imply dy- 150 namical interactions 0 0.4 0.8 150 150 In the following subsections we shall discuss whether or not e , these features necessitate the presence of planets in the system and make predictions for the properties of of these po- tential planets. Figure 3. Posterior distribution for the eccentricity and the argument of pericentre. Whilst a non-zero eccentricity is clearly preferred, a circular ring is still consistent within the 3σ uncertainties 4.1 Self-stirring (defined by the outermost contour). In order to create a debris disc, the planetesimals remaining in a system after the gas of the protoplanetary disc has dis- ◦ ◦ 50 < Ω < 130 , 0.0001 M⊕ < Mdust < 0.3 M⊕, 0 < e < sipated must be stirred so that the planetesimals can reach ◦ ◦ 0.9 and −180 < ω < 180 . The MCMC was run with 200 the relative velocities necessary to initiate a collisional cas- walkers and for 1000 steps. cade. The most likely ways for this to happen are via the gravitational effects of nearby planets (planet stirring, e.g. 3.3 Results Mustill & Wyatt 2009) or large planetesimals within the disc (self-stirring, e.g. Kenyon et al. 2008; Krivov & Booth 2018). The resulting best fitting parameters are given in Table 2 First we consider whether the disc around HD 38206 can be whilst the best fitting model, residuals and deprojected visi- self-stirred during the age of the system. If large planetesi- bilities are shown in Figure 2. The clearest difference between mals form quickly in the disc (e.g. by streaming instability), our results and the analysis of the Herschel data by Morales Krivov & Booth(2018) show that et al.(2016) is that we find the disc to be very wide whereas they assumed a narrow ring (of ∼20 au) and found it to be Ts = 9.3 Myr 00 consistent with their data, which had a resolution of ∼ 6 . 1 ρ −1 v 4 S −3 × frag max Nonetheless, our finding that the peak in the emission is at γ 1 g cm−3 30 m s−1 200 km +19 184−17 au is consistent with their narrow ring location. Our M −1/2 a 7/2 ∆a/a M −1 higher resolution data also clearly shows that this disc is close × ? 0 0 disc (6) to edge on. Morales et al.(2016) found a lower inclination of M 100 au 0.1 100 M⊕ ◦ 60 ± 4 , but this may also be a result of their use of a narrow where γ is a constant factor between 1 and 2, ρ is the den- ring model. sity, vfrag is the velocity required for destructive collisions, We find that a non-zero eccentricity is preferred by our fit, Smax is the maximum radius of planetesimals in the colli- with a value of 0.25+0.10. However, we cannot completely rule −0.09 sional cascade and Mdisc is the total mass in the disc. For out a circular ring since that is still within the 3σ uncertainty the composition we are using, ρ = 3.5 g cm−3. From subsec- region (see Figure 3). The apocentre location on the west tion 3.3 we find that a0 = 184 au and ∆a = 143 au. We side of the disc allows this model to explain the extension can estimate Mdisc by extrapolating the dust mass assuming to the west noted in section 2. It does not, however, explain a single power law size distribution from dust grains up to the brightness asymmetry as seen by the 3σ residual in the Smax (although note that estimating the total disc mass is residuals image (Figure 2). This is not surprising as we do highly uncertain due a lack of knowledge in the form of the not expect to see any pericentre glow at these wavelengths size distribution and its maximum extent; see Krivov & Wy- (e.g. Pan et al. 2016). Since the residuals image also shows att 2020, for further discussion on the total mass of debris a number of other 3σ sources away from the disc, it is likely discs) that these are simply peaks in the noise, although it is also possible that they are background sources. S4+α − S4+α M = max min M . (7) Our model uses the dust mass as a free parameter. This disc 4+α 4+α dust Sdust − Smin

MNRAS 000, 000–000 (0000) ALMA Observations of HD 38206 5

This results in Mdisc = 420 M⊕. For all other parameters 2018a). However, none of this is the case for HD 38206, so we use the standard values used by Krivov & Booth(2018): that we can only use the gap radius to constrain parameters −1 γ = 1.5, vfrag = 30 m s and Smax = 200 km. This results in of the possible planets in this system. This makes the prob- Tstir = 13 Myr. lem highly degenerate, since the same gap could be cleared HD 38206 has previously been reported as a member of by many different planet configurations. the Columba association (Torres et al. 2008). We first check In view of this degeneracy, we make a number of assump- that this is still the case using the latest Gaia data and the tions. All of them are arbitrary and do not have to hold in bayesian inference association membership code BANYAN reality, yet they allow us to make specific estimates. Again by Σ(Gagnéet al. 2018). This results in a 99.9% chance that analogy with our Solar system, we confine ourselves to a sim- this star is a member of the Columba association. According ple scenario in which the presumed planets in the gap are all to Bell et al.(2015), this means that it has a likely age of in nearly-circular, nearly-coplanar orbits (e.g., Su & Rieke +6 42−4 Myr, longer than the timescale needed for self-stirring 2014), although alternative scenarios exist, such as diver- and demonstrating that planets are not required to stir the gent planetesimal-driven migration of pairs of planets (Mor- disc. rison & Kratter 2018) or sweeping secular resonances with However, we must note that there is a lot of uncertainty one ‘lonely’ giant planet in an eccentric orbit (Zheng et al. in many of these values. For instance, even a small change in 2017). For simplicity, we also assume a log-uniform spacing the slope of the size distribution can result in a large change and equal masses for all planets in the gap. Finally, we as- in total disc mass. The timescale for stirring is even more sume that the alleged planets are dynamically stable against strongly dependent on vfrag and Smax. Laboratory experi- mutual perturbation over the age of the central star. ments demonstrate that there is an uncertainty of about an Many studies derived stability criteria for both lower mass order of magnitude on vfrag (see Blum & Wurm 2008, and planets (e.g., Chambers et al. 1996; Faber & Quillen 2007; references therein), whilst Smax is dependent on the assump- Zhou et al. 2007; Smith & Lissauer 2009, among others) and tion that the planetesimals formed through the process of massive ones (Morrison & Kratter 2016). One way of apply- pebble concentration described in Johansen et al.(2015); Si- ing these criteria to the gap clearing was proposed by Faber & mon et al.(2016); Sch äfer et al.(2017); Simon et al.(2017). Quillen(2007). First they estimate the maximum number of Bearing all this in mind, there is at least a couple of orders planets Np in relation to their mass mp that would still lead of magnitude uncertainty on the self-stirring timescale and to a (marginally) stable system. Since the planets themselves so we should also consider the possibility that planets are would already be close to becoming unstable, they argue that required to stir the disc. planetesimals would already be completely removed from the planetary region, ensuring the gap to be devoid of any debris material. They then also estimate the minimum number of 4.2 Gap clearing by planets planets. To do this, they place them at separations that are Like many other debris discs, the one around HD 38206 has twice as large as those that correspond to a marginally sta- a two-component structure – in addition to the belt on the ble system. In this case, the expectation is that a significant outer edge of this system imaged by ALMA, we know from the fraction of planetesimals residing between the orbits of neigh- SED (Morales et al. 2009, 2016) that there is also warm emis- bouring planets would be removed from the system. It is only sion likely originating from an asteroid belt analogue. Both the bodies orbiting exactly half-way between the planets that the origin of the broad gap between the outer and the inner would have a chance to survive. The planetary system would rings and the nature of the warm component in such disks are then certainly be stable, while the gap would be nearly free of a matter of debate. It is possible that planetesimals failed to debris. In order to calculate these extremes, Faber & Quillen form at intermediate distances, creating a two-belt architec- (2007) provide the following formulae: ture by the time of gas dispersal (e.g., Carrera et al. 2017), log10(ain/aw) although this depends strongly on the planetesimal formation Np = (8) log (1 + gδmin) model. Another conceivable explanation for two-component 10 discs would be a swarm of planetesimals in eccentric orbits 1/4 with apocentres in the outer belt and pericentres in the inner µ −7 δmin = (log10(τ/yr) + 1 + log10(µ/10 )) (9) one (Wyatt et al. 2010), although this implies high eccentric- 3.7 ities that would be difficult to explain. where ain is the inner edge of the outer belt, aw is the loca- Nevertheless, by analogy with the Solar system with its tion of the inner belt (which is assumed to be narrow), g is a Kuiper and asteroid belts and giant planets in between, it parameter that equals 1 for calculating the maximum num- is natural to expect that the gap between the two rings was ber of planets and 2 for calculating the minimum number of carved by as yet undiscovered planets. It would be interesting planets, µ is the mass ratio mp/M? and τ is the age of the to put some constraints on the number, masses, and location system. The relationship between Np and mp using the pa- of the alleged planets in the cavity. Such constraints would be rameters for the HD 38206 system is shown in Figure 4 with tighter if the disc exhibited strong asymmetries (e.g., Lee & the grey region representing the allowed parameters. Chiang 2016;L öhne et al. 2017), or if an accurate radial pro- Based on these results, we might expect between 4 and 7 file of the inner edge of the outer disc could be inferred from Jupiter-mass planets. A larger number of lower mass planets the resolved images (e.g., Nesvold & Kuchner 2015; Booth would also do. Note that there is no stringent lower limit et al. 2017). Also, for systems with the observed hot dust on the mass of a single planet in the model, except that the close to the star, constraints can be derived by assuming this underlying stability criterion was only established by Faber dust to stem from comets scattered into the inner system by & Quillen(2007) by numerical integrations within a certain a chain of planets interior to the outer belt (e.g., Marino et al. range of masses and may fail outside that range.

MNRAS 000, 000–000 (0000) 6 M. Booth et al.

12 3 planets, each with a mass of 1.4 MJ , are required to clear the gap. 10 In their case, ain was determined solely from a ﬁt to the SED and assumption of a narrow belt. From our results, whilst we do not resolve the inner edge of the disc, we do re- 8 solve the disc well enough to determine that it must be much wider than previously assumed. For our purposes, we shall 6 approximate the edges of the disc with the FWHM of the ﬁt- ted Gaussian. In other words, ain = a0 −∆a/2 = 110±30 au. 4 Whilst follow-up, higher resolution observations may deter-

Number of planets mine that the real distribution is different to this, the un- 2 certainties on the inner edge measured here are still quite large and so likely to contain the real inner edge2. In ad- 0 dition we also update the other two parameters, now using 10 1 100 101 +6 τ = 42−4 Myr (Bell et al. 2015) and aw = 8.7 au (Morales Mass of each planet, MJ et al. 2016) From this we find (assuming Gaussian uncertainty propagation) Np = 3.3 ± 1.4 and mp = 0.6 ± 0.3MJ for K = 20 and Np = 4.8 ± 2.0 and mp = 0.28 ± 0.14MJ for Figure 4. Constraints on the possible number of planets in the gap K = 16 (large filled circle and cross, respectively, in Fig.4). and their masses. The grey shaded region represents the bounds In other words, since our data allows us to determine that based on the model of Faber & Quillen(2007) (see Equations8 and the disc is wide – something that was not possible with the 9). The cross and the dot represent the minimum number and mass based on the model of Shannon et al.(2016) and using K = 16 Herschel data – we find that the gap is smaller than the gap and K = 20 respectively (see Equations 10 and 11). The horizon- assumed by Shannon et al.(2016) and so the masses required tal dashed line represents the minimum number below which the to clear the gap are lower than given by their analysis. equations of Shannon et al.(2016) are invalid (see Equation 12). However, these results should be treated with caution. Shannon et al.(2016) showed that at some point Equation 10 breaks down, and increasing the planet mass beyond this Another approach is to put the suspected planets in a con- point no longer reduces the clearing time (their Figures 2 figuration that is spaced widely enough to ensure stability, and 4). Systems are particularly susceptible to this effect if place planetesimals between the planetary orbits, and to in- they are young, or have a distant outer belt. Specifically this voke numerical integrations and analytic arguments to see happens when: whether, and on which timescales, these planetesimals will be ain removed from the system. This idea was employed by Shan- Np/2 − 1 > log . (12) 10 a non et al.(2016) who found the clearing timescale of the w chaotic zone as a function of the planet mass and semimajor For our parameters we find that at least 4 planets are nec- axis using N-body simulations. From these simulations they essary to satisfy this condition. This criterion is, therefore, derived a lower limit on the mass of each planet, mp, inside satisfied when we assume K = 16, but not when we assume the gap, assuming equal mass planets with a separation of K K = 20. This implies that for a wide spacing of planets, it mutual Hill radii. Modifying their equations such that K is is not possible for any planetary system to clear the gap in left as a free parameter, we find that: such a short time, no matter how massive they are. Nonethe- less, even in this case, there are reasonable uncertainties on 3/2 1/2 κ ain M? several parameters in Equations 10 and 11, and planets with mp = m⊕ , (10) τ 1 au M masses at the lower end of the uncertainty interval would and also the minimum number of planets that are needed to satisfy Equation 12 and could therefore clear the gap within clear the gap within the age of the system: the system age. In summary, if multiple equal-mass planets are to have cleared the gap, then this implies that either the ain location of the inner edge of the outer disc lies at the lower log10 a N = 1 + w , (11) end of our calculated uncertainty interval, that the system p  1/3 −1/3  mp M? age and/or stellar mass is at the upper end of the allowed 1 + 0.006K m M log  ⊕  range, or that the planets have a spacing that is smaller than 10  1/3 −1/3  1 − 0.006K mp M? 20 mutual Hill radii. m⊕ M These estimates are very conservative compared to those where κ is a parameter dependent on K. They first assumed obtained with the Faber & Quillen(2007) method, provid- a wide spacing, with separation of K = 20 mutual Hill radii ing a very lower limit on the total number of planets in the between the neighbouring planet orbits, which is close to a typical separation in Kepler multiplanet systems. In this case κ = (4 ± 1) × 106 yrs. They also checked a slightly tighter 2 For example, if the real distribution was closer to a boxcar func- packing, with K = 16, and found the results to be similar. In 6 tion with sharp edges, then it can be demonstrated that fitting this case κ = (2 ± 0.2) × 10 yrs. a Gaussian function to this results in an inner edge located at Shannon et al.(2016) applied their formulae to the a0 − (0.85∆a)/2 = 123 au. Therefore, in this example the esti- HD 38206 system using the parameters known at the time mates of the mass and number of planets would be closer to the (τ = 30 Myr, aw = 15 au, ain = 180 au). They found that upper limit of the predictions we have made.

MNRAS 000, 000–000 (0000) ALMA Observations of HD 38206 7

impact that the outermost planet will have on the inner edge of the disc. This is a problem that has been studied by multiple authors over the years (see e.g., Wisdom 1980; Mustill & Wyatt 2012; Pearce & Wyatt 2014; Nesvold & Kuchner 2015). Here we use the formulas of Pearce & Wyatt(2014), which take into account the eccentricity of the planet. Com- bining their equations 9 and 10, we ﬁnd: ) J 3

M 1 ain

, m = − 1 (3 − e )M (14) p 0.0 p p ? 125 ap(1 + ep) m ( g o l Thus we now have two equations, with three unknowns. 2.5 One of the unknowns is the planet eccentricity. If the planet

120 is eccentric, then this will aﬀect the disc eccentricity. There-

u fore our constraints on the disc eccentricity can provide con- a

, p straints on the planet eccentricity. The two can be related a 60 through the equation (Mustill & Wyatt 2009):

60 4 a 0.4 0.8 2.5 0.0 120 ep = e. (15) ep log(mp, MJ) ap, au 5 ap

In our modelling we have assumed a single eccentricity for the Figure 5. Posterior distribution for the eccentricity, mass and semi- disc. In fact, as can be seen from this equation, the influence major axis of the predicted outermost planet based on the assump- of a planet on a planetesimal’s eccentricity is stronger the tion that it is responsible for stirring the disc, clearing material out to the inner edge and forcing the eccentricity of the disc particles. closer the planetesimal is to the planet. Therefore, for a wide The three equations (13, 14 and 15) are then solved simultaneously disc, such as that of HD 38206, the disc will have a greater for each of the sets of disc parameters given by the MCMC samples eccentricity at its inner edge than at its outer edge. However, in section 3. the resolution of our observations is not good enough to detect such a variation in the eccentricity and so we assume a single eccentricity throughout and set a = a . cavity. The reason is a wide spacing of K = 16 or 20 is as- 0 Therefore, if we make the assumption that the outermost sumed by Shannon et al.(2016). For comparison, the Faber planet is simultaneously responsible for the stirring of the & Quillen(2007) curve for the maximum number of planets disc, the clearing of the inner edge and the eccentricity of corresponds to values of K between 6 and 7, and the curve the disc, then we can solve Equations 13, 14 and 15. Due to for the minimum number of planets to values of K between the large uncertainty in our knowledge of the disc parame- 12 and 14. ters, we make use of all of the samples from our MCMC run, solving the simultaneous equations for each set of disc param- 4.3 Constraints on the outermost planet eters. Doing so we determine that such a planet would have +0.20 +0.5 +12 ep = 0.34−0.13, mp = 0.7−0.3 MJ and ap = 76−13 au. The full As noted in subsection 4.1, an alternative possibility to the posterior distribution is shown in Figure 5. disc being stirred by the largest planetesimals is that the disc An attempt to find planets in the system has been made is stirred by planets in the system. This is most likely to be by the NaCo Imaging Survey for Planets around Young stars due to secular perturbations from the outermost planet. The (ISPY) using the Very Large Telescope (Launhardt et al. timescale for planet stirring is defined by Mustill & Wyatt 2020). No planets were detected, nonetheless they find that (2009) as the most stringent limits are for planets at distances of 2 3/2 9/2 3 (1 − ep) aout &70 au, for which they rule out planets of masses & 7 MJ tp ≈ 1.53 × 10 ep 10 au (Launhardt et al., in prep.). Interior to this the upper limits are less constraining. It is also important to note that, given M 1/2 m −1 a −3 × ? p p yr (13) that the disc is close to edge on, this only applies to planets M M 1 au that are close to the disc ansae. Assuming that any planets where aout = a0 + ∆a, ep is the eccentricity of the planet and are coplanar with the disc, planets more massive than these ap is the semimajor axis of the planet. Therefore, if we as- limits could have evaded detection, particularly if they are on sume that this timescale is equivalent to the age of the system parts of their orbits that place them inside the inner working and the outermost planet in the system is solely responsible angle of the observations. An improvement in mass detection for stirring the disc out to its outer edge, we can find a fam- sensitivity of about an order of magnitude will be necessary ily of solutions for its mass, semi-major axis and eccentricity. (which requires an improvement of roughly 5 orders of magni- Strictly speaking, these are all lower limits since a more mas- tude in contrast) in order to be certain to detect the expected sive, more distant or more eccentric planet could stir the disc planets in the system and, given the high inclination, a low on a timescale shorter than the age of the system. inner working angle will also be necessary to avoid occluding In addition, following subsection 4.2, we can consider the them when they are not near the disc ansae.

MNRAS 000, 000–000 (0000) 8 M. Booth et al.

5 CONCLUSIONS Blum J., Wurm G., 2008, ARA&A, 46, 21 Boley A. C., Payne M. J., Corder S., Dent W. R. F., Ford E. B., In this paper we present archival ALMA observations, taken Shabram M., 2012, ApJ, 750, L21 at a wavelength of 1.3 mm, of the debris disc around the Booth M., et al., 2016, MNRAS, 460, L10 A0V star HD 38206. The data show the disc to be clearly Booth M., et al., 2017, MNRAS, 469, 3200 resolved and highly inclined, close to edge-on. We also note Booth M., et al., 2019, MNRAS, 482, 3443 some signs of asymmetry, with the disc extending further to Carrera D., Gorti U., Johansen A., Davies M. B., 2017, ApJ, 839, the west than it does to the east. Using an MCMC analysis, 16 +19 Chambers J. E., Wetherill G. W., Boss A. P., 1996, Icarus, 119, we determine the dust to peak at a distance of 184−17 au +46 261 with a width of 143−36 au and find that the asymmetry is best fit by an eccentricity of 0.25+0.10. Dent W. R. F., et al., 2014, Science, 343, 1490 −0.09 Dohnanyi J. S., 1969, J. Geophys. Res., 74, 2531 The extreme width of the disc naturally leads to the ques- Draine B. T., 2003, ApJ, 598, 1017 tion of whether the disc can be self-stirred or whether plan- Dullemond C. P., Juhasz A., Pohl A., Sereshti F., Shetty R., Pe- ets are necessary to initiate the collisional cascade. Using the ters T., Commercon B., Flock M., 2012, RADMC-3D: A multi- equations of Krivov & Booth(2018) and their canonical pa- purpose radiative transfer tool (ascl:1202.015) rameters, we find that ∼16 Myr is necessary for self-stirring, Faber P., Quillen A. C., 2007, MNRAS, 382, 1823 which is less than the estimated age of the system, demon- Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, strating that self-stirring should suffice to stir the disc out PASP, 125, 306 to its outer edge within the age of the system. Nonetheless, GagnéJ., et al., 2018, ApJ, 856, 23 planets may well be required to explain the gap between the Gaia Collaboration et al., 2018, A&A, 616, A1 cold dust and the warm dust detected by Spitzer. Based on Gontcharov G. A., 2006, Astronomy Letters, 32, 759 Goodman J., Weare J., 2010, Communications in Applied Mathe- the geometry of the system and the equations of Shannon matics and Computational Science, 5, 65 et al.(2016), we determine that a minimum of four planets Högbom J. A., 1974, A&AS, 15, 417 are required, each with a mass of 0.28±0.14 MJ . Holland W. S., et al., 2017, MNRAS, 470, 3606 By making the assumption that the outermost planet is Hughes A. M., Duchêne G., Matthews B. C., 2018, ARA&A, 56, responsible for the features of the disc – its stirred nature, 541 inner edge and eccentricity – we predict that it would need Johansen A., Mac Low M.-M., Lacerda P., Bizzarro M., 2015, Sci- +0.20 +0.5 +12 to have ep = 0.34−0.13, mp = 0.7−0.3 MJ and ap = 76−13 au. ence Advances, 1, 1500109 To detect such a planet requires an order of magnitude im- Kalas P., et al., 2008, Science, 322, 1345 provement in mass detection sensitivity over current observ- Kenyon S. J., Bromley B. C., O’Brien D. P., Davis D. R., 2008, ing capabilities. Formation and Collisional Evolution of Kuiper Belt Objects. Univ. Arizona Press, Tucson, pp 293–313 Kral Q., MatràL., Wyatt M. C., Kennedy G. M., 2017, MNRAS, 469, 521 ACKNOWLEDGEMENTS Krivov A. V., Booth M., 2018, MNRAS, 479, 3300 Krivov A. V., Wyatt M. C., 2020, MNRAS, MB, AK and TP acknowledge support from the Deutsche Kurucz R. L., 1979, ApJS, 40, 1 Forschungsgemeinschaft through projects Kr 2164/13-2, Kr Lagrange A.-M., et al., 2010, Science, 329, 57 2164/14-2 and Kr 2164/15-2. The authors thank the referee Launhardt R., et al., 2020, A&A, 635, A162 for their constructive report. Lee E. J., Chiang E., 2016, ApJ, 827, 125 ALMA is a partnership of ESO (representing its member Löhne T., Krivov A. V., Kirchschlager F., Sende J. A., Wolf S., states), NSF (USA) and NINS (Japan), together with NRC 2017, A&A, 605, A7 (Canada), MOST and ASIAA (Taiwan), and KASI (Republic Mannings V., Barlow M. J., 1998, ApJ, 497, 330 of Korea), in cooperation with the Republic of Chile. The Marino S., Bonsor A., Wyatt M. C., Kral Q., 2018a, MNRAS, 479, 1651 Joint ALMA Observatory is operated by ESO, AUI/NRAO Marino S., et al., 2018b, Monthly Notices of the Royal Astronom- and NAOJ. In addition, publications from NA authors must ical Society, 479, 5423 include the standard NRAO acknowledgement: The National Marino S., Yelverton B., Booth M., Faramaz V., Kennedy G. M., Radio Astronomy Observatory is a facility of the National MatràL., Wyatt M. C., 2019, Monthly Notices of the Royal Science Foundation operated under cooperative agreement Astronomical Society, 484, 1257 by Associated Universities, Inc. Marois C., Zuckerman B., Konopacky Q. M., Macintosh B., Bar- man T., 2010, Nature, 468, 1080 Moerchen M. M., Telesco C. M., Packham C., 2010, ApJ, 723, 1418 Morales F. Y., et al., 2009, ApJ, 699, 1067 DATA AVAILABILITY Morales F. Y., Bryden G., Werner M. W., Stapelfeldt K. R., 2016, ApJ, 831, 97 This paper makes use of the following ALMA data: Morrison S. J., Kratter K. M., 2016, ApJ, 823, 118 ADS/JAO.ALMA#2012.1.00437.S. Data resulting from this Morrison S. J., Kratter K. M., 2018, MNRAS, 481, 5180 work are available upon reasonable request. Mustill A. J., Wyatt M. C., 2009, MNRAS, 399, 1403 Mustill A. J., Wyatt M. C., 2012, MNRAS, 419, 3074 Nesvold E. R., Kuchner M. J., 2015, ApJ, 798, 83 REFERENCES Pan M., Nesvold E. R., Kuchner M. J., 2016, ApJ, 832, 81 Pearce T. D., Wyatt M. C., 2014, MNRAS, 443, 2541 Bailer-Jones C. A. L., Rybizki J., Fouesneau M., Mantelet G., An- Rameau J., et al., 2013, ApJ, 772, L15 drae R., 2018, AJ, 156, 58 Rieke G. H., et al., 2005, ApJ, 620, 1010 Bell C. P. M., Mamajek E. E., Naylor T., 2015, MNRAS, 454, 593 Schäfer U., Yang C.-C., Johansen A., 2017, A&A, 597, A69

MNRAS 000, 000–000 (0000) ALMA Observations of HD 38206 9

Shannon A., Bonsor A., Kral Q., Matthews E., 2016, MNRAS, 462, L116 Simon J. B., Armitage P. J., Li R., Youdin A. N., 2016, ApJ, 822, 55 Simon J. B., Armitage P. J., Youdin A. N., Li R., 2017, ApJ, 847, L12 Smith A. W., Lissauer J. J., 2009, Icarus, 201, 381 Stassun K. G., et al., 2018, AJ, 156, 102 Su K. Y. L., Rieke G. H., 2014, in Booth M., Matthews B. C., Graham J. R., eds, IAU Symposium Vol. 299, Exploring the Formation and Evolution of Planetary Systems. pp 318–321 (arXiv:1307.1735), doi:10.1017/S1743921313008764 Su K. Y. L., et al., 2017, AJ, 154, 225 Thompson A. R., Moran J. M., Swenson George W. J., 2017, In- terferometry and Synthesis in Radio Astronomy, 3rd Edition. Springer, Cham, doi:10.1007/978-3-319-44431-4 Torres C. A. O., Quast G. R., Melo C. H. F., Sterzik M. F., 2008, Handbook of Star Forming Regions: Volume II, The Southern Sky. Astronomical Society of the Paciﬁc, p. 757 Wisdom J., 1980, AJ, 85, 1122 Wyatt M. C., Booth M., Payne M. J., Churcher L. J., 2010, MN- RAS, 402, 657 Zheng X., Lin D. N. C., Kouwenhoven M. B. N., Mao S., Zhang X., 2017, ApJ, 849, 98 Zhou J.-L., Lin D. N. C., Sun Y.-S., 2007, ApJ, 666, 423

This paper has been typeset from a TEX/LATEX ﬁle prepared by the author.

MNRAS 000, 000–000 (0000)