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Spiral arms in thermally stratified protoplanetary discs

Spiral arms in thermally stratified protoplanetary discs MNRAS 474, L32–L36 (2018) doi:10.1093/mnrasl/slx182 Advance Access publication 2017 November 14 Spiral arms in thermally stratified protoplanetary discs ‹ ‹ Attila Juhasz ´ and Giovanni P. Rosotti Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK Accepted 2017 November 9. Received 2017 November 9; in original form 2017 November 1 ABSTRACT Spiral arms have been observed in nearly a dozen protoplanetary discs in near-infrared scattered light and recently also in the submillimetre continuum. While one of the most compelling explanations is that they are driven by planetary or stellar companions, in all but one cases such companions have not yet been detected and there is even ambiguity on whether the planet should be located inside or outside the spirals. Here, we use 3D hydrodynamic simulations to study the morphology of spiral density waves launched by embedded planets taking into account the vertical temperature gradient, a natural consequence of stellar irradiation. Our simulations show that the pitch angle of the spirals in thermally stratified discs is the lowest in the disc mid-plane and increases towards the disc surface. We combine the hydrodynamic simulations with 3D radiative transfer calculations to predict that the pitch angle of planetary spirals observed in the near-infrared is higher than in the submillimetre. We also find that in both cases the spirals converge towards the planet. This provides a new powerful observational method to determine if the perturbing planet is inside or outside the spirals, as well as map the thermal stratification of the disc. Key words: accretion, accretion discs – hydrodynamics – protoplanetary discs – circumstellar matter. NIR observations. Dong et al. (2015a) combined 3D hydrodynamic 1 INTRODUCTION simulation with 3D radiative transfer calculations to show that the With the dawn of extreme high resolution imaging capabilities of observed contrast and pitch angle of spirals are in fact in agreement current telescopes (e.g. SPHERE/VLT, GPI/Gemini, ALMA) more with the spirals being launched by massive (several M )planets Jup and more discs show spiral arms in near-infrared (NIR) scattered exterior to the spirals. light (e.g. Muto et al. 2012; Benisty et al. 2015;Wagneretal. 2015; However, planets are not the only explanations for the observed Stolker et al. 2016; Benisty et al. 2017) and recently also in the spirals. Multi-armed spiral structure can also form due to gravita- sub-millimetre (Perez ´ et al. 2016;Tobinetal. 2016). The observed tional instability (GI; see e.g. Rice et al. 2003) with the number of spirals have similar morphologies in most cases; two symmetric arms depending on the mass and temperature of the disc via the arms, shifted in azimuth by approximately 180 , and with a pitch Toomre Q parameter (Cossins, Lodato & Clarke 2009). For very ◦ ◦ angle of about 15 –30 . In most cases, there is also a gap/cavity just massive discs (M /M  0.25), GI can produce two armed spirals disc inwards of the spirals. in the disc, similar to the NIR scattered light observations (Dong It has been known for a long time that spiral density waves et al. 2015b). Meru et al. (2017) showed that GI could explain the are launched by planetary or stellar companions at Lindblad reso- spirals in the submillimetre ALMA images of Elias 2–27. nances (e.g. Goldreich & Tremaine 1979; Ogilvie & Lubow 2002; Additionally, if a marginally gravitationally stable disc is per- Rafikov 2002) both inwards and outwards of companion’s orbit. turbed by an embedded planet, the tidal interaction might tip the Therefore, a straightforward explanation for the observed spirals is balance and trigger GI in the disc (Pohl et al. 2015). This combined that they are launched by a so far undetected companion, possibly a mechanism may also produce spirals with pitch angles similar to forming embedded planet (e.g. Muto et al. 2012; Grady et al. 2013); the ones in observations. this would open up the exciting possibility of getting a direct in- Numerous theoretical studies have recently increased the level of sight into planet formation. Using 2D hydrodynamic simulations complexity in the modelling of spirals, showing the importance to and 3D radiative transfer computation, Juhasz ´ et al. (2015) showed, understand their origin. For instance, while tidally induced spiral however, that the pitch angle and contrast of the spirals driven by density waves have been studied for a long time in 2D, Zhu et al. planets orbiting inwards of the spirals are not compatible with the (2015) demonstrated the spiral amplitude and pitch angle (the com- plement of the angle between the radial direction and the tangent of the spiral) in the non-linear regime are higher in 3D than in 2D sim- E-mail: juhasz@ast.cam.ac.uk (AJ); rosotti@ast.cam.ac.uk (GPR) ulations. However, an important aspect of a realistic disc model has 2017 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society Downloaded from https://academic.oup.com/mnrasl/article-abstract/474/1/L32/4628072 by Ed 'DeepDyve' Gillespie user on 16 March 2018 Spiral arms in protoplanetary discs L33 not yet been investigated. Protoplanetary discs in which spirals have been detected are passively heated by the absorbed stellar radiation. As a consequence of this, they will have a positive vertical temper- ature gradient (Chiang & Goldreich 1997), resulting in a vertical thermal stratification in the disc. It is known that the pitch angle of the spirals depends on the local sound speed (Rafikov 2002), thus we expect vertical thermal stratification to affect the morphology of the spiral wake. (see e.g. Lee & Gu 2015). The goal of this study is to investigate the morphology of tidally induced spirals by embed- ded planets in 3D thermally stratified discs. We note, however, that our results would also be applicable to stellar companions as the tidal interaction between the disc and the companion is the same as for the planetary case. Figure 1. Dependence of the spiral pitch angle on the latitude in the strat- ified simulation. The red line shows the sound speed profile used in the 2 MODELLING hydrodynamic simulations. The large spread in the points close to the upper boundary comes from the difficulty of determining the spiral pitch angle in 2.1 Hydrodynamic simulations that region. We run 3D hydrodynamical simulations with the code PLUTO (Mignone et al. 2007) to investigate the disc response to the planet. −3.5 Draine 2001) for a grain size distribution of n(a) ∝ a between We include the FARGO algorithm (Mignone et al. 2012) to reduce a = 0.1 µmand a = 1 mm. With RADMC-3D, we first calculate the the computational cost and use the HLLC Riemann solver, linear dust temperature with a thermal Monte Carlo simulation and we reconstruction and second-order Runge–Kutta time integration. We then calculate images at wavelengths λ = 1.65 µmand λ = 1.3 mm employ a spherical grid with 400, 50, and 1024 cells, distributed in and for an inclination of 0 . Finally, we generate synthetic observa- the [0.4, 2.5], [1.22, π/2], and [0, 2π], intervals in the radial, polar, tions by convolving the calculated images with a Gaussian kernel and azimuthal directions. The planet was placed at the coordinates with an FWHM of 0.04 arcsec, typical of current state-of-the-art [1.0, 0, π/4]. In the radial and polar directions, we define ‘wave- NIR cameras such as SPHERE/VLT or GPI/Gemini and of ALMA killing zones’ where we damp disturbances, following de Val-Borro (ALMA Partnership et al. 2015). We assume a source distance of et al. (2006), extending up to 0.44 and from 2.25 in radius and un- 100 pc. til 1.27 in θ. We employ a locally isothermal equation of state, in which the sound speed c is a function of position only, and a physi- cal viscosity using the Shakura & Sunyaev (1973) prescription with 3 RESULTS −3 a value of α = 10 . We assume the planet mass to be 3 M . Jup We run two simulations, the first (subsequently called ‘isother- 3.1 Hydrodynamic simulations −1/4 mal’) in which c depends only on radius: c ∝ r ,withthe s s We show in Fig. 1 the pitch angle of the spiral in the stratified normalization set such that the disc aspect ratio at the location simulation as a function of the height above the mid-plane. We plot of the planet is 0.07, a typical value at tens of au. In the sec- with the colours shown in the legend the pitch angle at different ond simulation (stratified), we allow the sound speed to vary as radii, normalized to the value in the mid-plane. The red line shows a function of the vertical coordinate. To this end, we use the pre- how the sound speed varies with the vertical coordinate. We find scription commonly employed when fitting observations of Dartois, that the pitch angle increases with z at all radii; in addition, the Dutrey & Guilloteau (2003) and subsequently updated by Rosen- dependence of the pitch angle closely resembles the one of the feld et al. (2013). We assume that the temperature in the upper layer sound speed. of the disc is 2.5 times the value in the mid-plane (the same as in the This behaviour is the natural consequence of the temperature isothermal case), and that the transition between the cold mid-plane dependence of the pitch angle. The spirals are located where con- and the hot upper layer (the parameter z of Dartois et al. 2003) structive interference occurs between waves emitted from Lindblad happens at three scale heights. The initial surface density  of the −1 resonances in the disc (Ogilvie & Lubow 2002); this location de- disc follows  ∝ r . We solve for the hydrostatic equilibrium in pends on the wave propagation speed, and thus the disc temperature. the vertical direction to assign the initial density at every point and Indeed Rafikov (2002) showed that the pitch angle of a tidally in- we initialize the velocity so that the gas is initially in Keplerian duced spiral in a 2D Keplerian disc, in which the sound speed varies rotation. We evolve the simulations for 100 orbits to reach a steady −q with radius as c = c (r/r ) , depends on the sound speed: state in the structure of the spirals. s p p q+1 3/2 r r r p p −1 β = π/2 − tan − 1 , (1) h r r p p 2.2 Radiative transfer simulations We calculate images from the hydrodynamics simulations using the where β is the pitch angle, r is the orbital radius of the planet and 3D radiative transfer code RADMC-3D (http://www.ita.uni-heidelberg. h is the pressure scale height (which depends on the sound speed de/dullemond/software/radmc-3d/), using the same 3D spherical via h = c / ). The resemblance between the dependence with the p p p mesh of the hydrodynamic simulations. We assume that the cen- vertical coordinate of the pitch angle and the sound speed profile tral star is a Herbig Ae star (T = 9500 K, M = 2.5 M , R may imply that this relation holds also in a 3D disc, provided that eff = 2.0 R ), a constant gas-to-dust ratio of 100 and we calculate the local temperature is used in the relation. We leave a detailed the dust opacity assuming astronomical silicate (Weingartner & analysis of the pitch angle in 3D to further studies and conclude MNRASL 474, L32–L36 (2018) Downloaded from https://academic.oup.com/mnrasl/article-abstract/474/1/L32/4628072 by Ed 'DeepDyve' Gillespie user on 16 March 2018 L34 A. Juhasz ´ and G. P. Rosotti (a) (b) (c) (d) Figure 2. Synthetic images in NIR scattered light (Panels a and c) and in submillimetre (Panels b and d) based on 3D hydrodynamic simulations assuming a vertically isothermal (Panels a and b) and stratified (Panels c and d) temperature structure. The scattered light images are scaled with the square of the radial distance to the star. In the NIR, the isothermal simulation shows higher contrast spirals with lower pitch angle compared to the stratified simulation. At submillimetre wavelengths, the pitch angle and the amplitude of the spirals are rather similar in the vertically stratified and isothermal simulations. that the vertical temperature gradient increases the pitch angle of disc mid-plane, in the submillimetre case (Figs 3b and d) the peak- the spirals in the upper layers of the disc. to-peak intensity variation and the width of the spiral peak agree within 10 per cent. In the NIR, scattered light images (Figs 3aand c) instead the spiral contrast is higher in the isothermal simulations by 3.2 Simulated observations 49 per cent and 45 per cent in the inner and outer arms, respectively. This is caused by the higher temperature in the upper layers in the We show the simulated NIR scattered light and submillimetre con- stratified simulations; the enhanced pressure forces tend to wash out tinuum images calculated from vertically isothermal (see Figs 2a more the spiral wake and reduce its amplitude. While the azimuthal and b) and thermally stratified (see Figs 2c and d) hydrodynamic width of the spirals in the two simulations agrees to 4 per cent in- simulations. In terms of the general structure, the isothermal and wards of the planet, for the outer spirals in the isothermal models stratified simulations look largely similar, with the only significant the azimuthal width is a factor 3.5 times larger compared to the difference that the inner part of the disc in the stratified simulation stratified simulations. is brighter with respect to the isothermal simulation. This is be- cause the inner edge of the disc in the stratified simulation is more ‘rounded off’, increasing the flaring index of the disc locally and consequently the brightness of the disc. 3.2.2 Spiral wake In Fig. 4, we present the spiral wakes we derived from the synthetic images by extracting the positions of the local maxima in the radial 3.2.1 Amplitude of the spirals surface brightness distribution. In submillimetre images, probing In Fig. 3, we compare the azimuthal surface brightness profiles of the disc mid-plane, the spirals have similar pitch angles in the locally the two models at two radii, 1.7 r and 0.58 r .Fromthisplot, we isothermal simulations and in the vertically stratified simulations p p measure the peak-to-peak variation of the intensity in the azimuthal (see Figs 4a and d) since the mid-plane temperatures in the two direction and the full width at half-maximum (FWHM) of the spi- simulations (locally isothermal versus stratified) are identical. ral peak (for the inner spiral we considered the stronger, primary Spirals in scattered light images computed from stratified sim- arm). Since the two models employ the same temperature in the ulations (see Fig. 4e) instead are more open compared to the MNRASL 474, L32–L36 (2018) Downloaded from https://academic.oup.com/mnrasl/article-abstract/474/1/L32/4628072 by Ed 'DeepDyve' Gillespie user on 16 March 2018 Spiral arms in protoplanetary discs L35 (a) (b) (d) (c) Figure 3. Comparison of the azimuthal surface brightness profile of the disc in synthetic images calculated from locally isothermal and stratified simulations. The comparison is shown for inner (a, c) and outer spirals (b, d) and both in NIR scattered light (a, b) and in the submillimetre continuum (c, d). The surface ◦ ◦ brightness profiles are normalized to the mean value in the [200 , 300 ] interval. In the NIR, the amplitude and the azimuthal width of the spirals are significantly larger in isothermal simulations compared to the stratified one, while the vertical structure has little effect on the submillimetre continuum observations. (a) (b) (c) (d) (e) (f) Figure 4. Panels a and b show the spiral wake in the vertically isothermal simulation inwards and outwards of the planetary orbit, respectively. Panel c shows the difference in the azimuthal position of the spirals between submillimetre and NIR scattered light images as a function of radius. Panels d–f are forthe vertically stratified simulation. In a vertically isothermal disc, the shape of the spiral wake is largely the same at both wavelengths. For the stratified simulation, the spirals are more open in the NIR than in the submillimetre both inwards and outwards of the planetary orbit. In the stratified simulation, the difference in the azimuthal position of the spirals between NIR and submillimetre increases with increasing distance from the planet, i.e. the spirals converge towards the planet. ones computed from locally isothermal simulations (see Fig. 4b), tered light and submillimetre images as a function of stellocentric consistently with the analysis of the simulation in Section 3.1. For radius in Figs 4c and f for the locally isothermal simulations and example, while we find that outer spirals in NIR scattered light for the stratified simulations, respectively. In the locally isother- images have a low pitch angle (7.5 at 1.5r ) in the isothermal sim- mal simulation, the difference in the azimuthal position of the ◦ ◦ ulations, the pitch angle is higher in the stratified simulation (13 spiral wake is not larger than 8 for the inner arms at all radii ◦ ◦ at 1.5r ); the effect is smaller for the inner spiral (15 versus 13 at with a weak trend of increasing difference towards smaller radii. 0.7r ). Dong et al. (2015a) suggested that a low pitch angle could be For the outer spirals, the difference fluctuates between −19 and used to identify the spiral as an outer one, but it is not clear whether −6 with no clear dependence on the radius. In case of the strat- this method still applies to the stratified case. ified simulations, there is a clear trend for the azimuthal posi- To better illustrate the difference in opening angle, we show the tion difference of the spirals in NIR and submillimetre. The az- difference in azimuth angles between the spirals derived from scat- imuthal difference in the spiral position increases both inwards and MNRASL 474, L32–L36 (2018) Downloaded from https://academic.oup.com/mnrasl/article-abstract/474/1/L32/4628072 by Ed 'DeepDyve' Gillespie user on 16 March 2018 L36 A. Juhasz ´ and G. P. Rosotti outwards from the position of the planet. The difference in the ACKNOWLEDGEMENTS spiral position in NIR and submillimetre images is nearly −20 This work has been supported by the DISCSIM project, grant agree- for the inner arms at 0.55r in the stratified simulations com- pl ment 341137 funded by the European Research Councilunder ERC- paredtothe −8 in the locally isothermal simulations. For the 2013-ADG. We thank the referee, Ruobing Dong, for the insightful outer arms, the azimuthal position difference is about 30 at 2r in pl review that improved our paper. the stratified simulations while only −15 in the locally isothermal simulations. REFERENCES 4 DISCUSSION AND CONCLUSIONS ALMA Partnership et al., 2015, ApJ, 808, L3 Benisty M. et al., 2015, A&A, 578, L6 Direct detection of young planets is notoriously difficult due to the Benisty M. et al., 2017, A&A, 597, A42 contrast limitation of the observations. Due to their larger spatial Chiang E. I., Goldreich P., 1997, ApJ, 490, 368 extent, spiral density waves excited by young planets are easier to Cossins P., Lodato G., Clarke C. J., 2009, MNRAS, 393, 1157 detect. Thus, spiral arms could, in principle, be used as signposts of Dartois E., Dutrey A., Guilloteau S., 2003, A&A, 399, 773 planets (see e.g. Dong et al. 2015a). However, there is ambiguity in de Val-Borro M. et al., 2006, MNRAS, 370, 529 whether the planet is inside or outside the spirals. In the previous Dong R., Fung J., 2017, ApJ, 835, 38 section, we have shown that the spiral arms in the submillimetre and Dong R., Zhu Z., Rafikov R. R., Stone J. M., 2015a, ApJ, 809, L5 in scattered light converge towards the location of the planet due to Dong R., Hall C., Rice K., Chiang E., 2015b, ApJ, 812, L32 Goldreich P., Tremaine S., 1979, ApJ, 233, 857 the different temperature in the mid-plane compared to the upper Grady C. A. et al., 2013, ApJ, 762, 48 layers of the disc. Therefore, we suggest to use the direction of Juhasz ´ A., Benisty M., Pohl A., Dullemond C. P., Dominik C., Paardekooper convergence of the spirals in NIR and in submillimetre continuum S.-J., 2015, MNRAS, 451, 1147 images to infer whether the planet is inwards or outwards of the Lee W.-K., Gu P.-G., 2015, ApJ, 814, 72 spirals. Meru F., Juhasz ´ A., Ilee J. D., Clarke C. J., Rosotti G. P., Booth R. A., 2017, Observationally, this requires having high-resolution observa- ApJ, 839, L24 tions of the same disc in NIR scattered light, probing the upper Mignone A., Bodo G., Massaglia S., Matsakos T., Tesileanu O., Zanni C., layers of the disc, and in submillimetre continuum, probing the disc Ferrari A., 2007, ApJS, 170, 228 mid-plane. Although not explored in this paper, we note that opti- Mignone A., Flock M., Stute M., Kolb S. M., Muscianisi G., 2012, A&A, cally thin gas observations also trace the disc mid-plane and could 545, A152 Muto T. et al., 2012, ApJ, 748, L22 therefore be used in place of the submillimetre continuum. Ogilvie G. I., Lubow S. H., 2002, MNRAS, 330, 950 We also note that, if we know that a planet is responsible for the Perez ´ L. M. et al., 2016, Science, 353, 1519 perturbation, the dependence of the spiral pitch angle on the local Pohl A., Pinilla P., Benisty M., Ataiee S., Juhasz ´ A., Dullemond C. P., Van temperature could be used as a diagnostic to map the vertical thermal Boekel R., Henning T., 2015, MNRAS, 453, 1768 stratification of the disc. This method requires using different tracers Rafikov R. R., 2002, ApJ, 569, 997 (e.g. NIR scattered light, various CO isotopologues, submillimetre Rice W. K. M., Armitage P. J., Bate M. R., Bonnell I. A., 2003, MNRAS, continuum) coming from different heights above the mid-plane. 338, 227 Mapping the vertical thermal structure of the disc with the spiral Rosenfeld K. A., Andrews S. M., Hughes A. M., Wilner D. J., Qi C., 2013, pitch angle has the advantage over other alternative methods (e.g. ApJ, 774, 16 modelling of the dust continuum emission, molecular rotational Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 Stolker T. et al., 2016, A&A, 595, A113 transitions), that the pitch angle depends only on the dynamics and Tobin J. J. et al., 2016, Nature, 538, 483 is not affected by uncertainties in chemistry and opacities. While Wagner K., Apai D., Kasper M., Robberto M., 2015, ApJ, 813, L2 the absolute value of the pitch angle might be weakly affected by Weingartner J. C., Draine B. T., 2001, ApJ, 548, 296 the planet mass or viscosity (Dong & Fung 2017), the differential Zhu Z., Dong R., Stone J. M., Rafikov R. R., 2015, ApJ, 813, 88 variation of it as a function of height is likely independent of these factors. A detailed mapping will require a non-linear theory of spiral density waves in thermally stratified discs, which has yet to This paper has been typeset from a T X/LT X file prepared by the author. E E be developed. MNRASL 474, L32–L36 (2018) Downloaded from https://academic.oup.com/mnrasl/article-abstract/474/1/L32/4628072 by Ed 'DeepDyve' Gillespie user on 16 March 2018 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society Letters Oxford University Press

Spiral arms in thermally stratified protoplanetary discs

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© 2017 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society
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Abstract

MNRAS 474, L32–L36 (2018) doi:10.1093/mnrasl/slx182 Advance Access publication 2017 November 14 Spiral arms in thermally stratified protoplanetary discs ‹ ‹ Attila Juhasz ´ and Giovanni P. Rosotti Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK Accepted 2017 November 9. Received 2017 November 9; in original form 2017 November 1 ABSTRACT Spiral arms have been observed in nearly a dozen protoplanetary discs in near-infrared scattered light and recently also in the submillimetre continuum. While one of the most compelling explanations is that they are driven by planetary or stellar companions, in all but one cases such companions have not yet been detected and there is even ambiguity on whether the planet should be located inside or outside the spirals. Here, we use 3D hydrodynamic simulations to study the morphology of spiral density waves launched by embedded planets taking into account the vertical temperature gradient, a natural consequence of stellar irradiation. Our simulations show that the pitch angle of the spirals in thermally stratified discs is the lowest in the disc mid-plane and increases towards the disc surface. We combine the hydrodynamic simulations with 3D radiative transfer calculations to predict that the pitch angle of planetary spirals observed in the near-infrared is higher than in the submillimetre. We also find that in both cases the spirals converge towards the planet. This provides a new powerful observational method to determine if the perturbing planet is inside or outside the spirals, as well as map the thermal stratification of the disc. Key words: accretion, accretion discs – hydrodynamics – protoplanetary discs – circumstellar matter. NIR observations. Dong et al. (2015a) combined 3D hydrodynamic 1 INTRODUCTION simulation with 3D radiative transfer calculations to show that the With the dawn of extreme high resolution imaging capabilities of observed contrast and pitch angle of spirals are in fact in agreement current telescopes (e.g. SPHERE/VLT, GPI/Gemini, ALMA) more with the spirals being launched by massive (several M )planets Jup and more discs show spiral arms in near-infrared (NIR) scattered exterior to the spirals. light (e.g. Muto et al. 2012; Benisty et al. 2015;Wagneretal. 2015; However, planets are not the only explanations for the observed Stolker et al. 2016; Benisty et al. 2017) and recently also in the spirals. Multi-armed spiral structure can also form due to gravita- sub-millimetre (Perez ´ et al. 2016;Tobinetal. 2016). The observed tional instability (GI; see e.g. Rice et al. 2003) with the number of spirals have similar morphologies in most cases; two symmetric arms depending on the mass and temperature of the disc via the arms, shifted in azimuth by approximately 180 , and with a pitch Toomre Q parameter (Cossins, Lodato & Clarke 2009). For very ◦ ◦ angle of about 15 –30 . In most cases, there is also a gap/cavity just massive discs (M /M  0.25), GI can produce two armed spirals disc inwards of the spirals. in the disc, similar to the NIR scattered light observations (Dong It has been known for a long time that spiral density waves et al. 2015b). Meru et al. (2017) showed that GI could explain the are launched by planetary or stellar companions at Lindblad reso- spirals in the submillimetre ALMA images of Elias 2–27. nances (e.g. Goldreich & Tremaine 1979; Ogilvie & Lubow 2002; Additionally, if a marginally gravitationally stable disc is per- Rafikov 2002) both inwards and outwards of companion’s orbit. turbed by an embedded planet, the tidal interaction might tip the Therefore, a straightforward explanation for the observed spirals is balance and trigger GI in the disc (Pohl et al. 2015). This combined that they are launched by a so far undetected companion, possibly a mechanism may also produce spirals with pitch angles similar to forming embedded planet (e.g. Muto et al. 2012; Grady et al. 2013); the ones in observations. this would open up the exciting possibility of getting a direct in- Numerous theoretical studies have recently increased the level of sight into planet formation. Using 2D hydrodynamic simulations complexity in the modelling of spirals, showing the importance to and 3D radiative transfer computation, Juhasz ´ et al. (2015) showed, understand their origin. For instance, while tidally induced spiral however, that the pitch angle and contrast of the spirals driven by density waves have been studied for a long time in 2D, Zhu et al. planets orbiting inwards of the spirals are not compatible with the (2015) demonstrated the spiral amplitude and pitch angle (the com- plement of the angle between the radial direction and the tangent of the spiral) in the non-linear regime are higher in 3D than in 2D sim- E-mail: juhasz@ast.cam.ac.uk (AJ); rosotti@ast.cam.ac.uk (GPR) ulations. However, an important aspect of a realistic disc model has 2017 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society Downloaded from https://academic.oup.com/mnrasl/article-abstract/474/1/L32/4628072 by Ed 'DeepDyve' Gillespie user on 16 March 2018 Spiral arms in protoplanetary discs L33 not yet been investigated. Protoplanetary discs in which spirals have been detected are passively heated by the absorbed stellar radiation. As a consequence of this, they will have a positive vertical temper- ature gradient (Chiang & Goldreich 1997), resulting in a vertical thermal stratification in the disc. It is known that the pitch angle of the spirals depends on the local sound speed (Rafikov 2002), thus we expect vertical thermal stratification to affect the morphology of the spiral wake. (see e.g. Lee & Gu 2015). The goal of this study is to investigate the morphology of tidally induced spirals by embed- ded planets in 3D thermally stratified discs. We note, however, that our results would also be applicable to stellar companions as the tidal interaction between the disc and the companion is the same as for the planetary case. Figure 1. Dependence of the spiral pitch angle on the latitude in the strat- ified simulation. The red line shows the sound speed profile used in the 2 MODELLING hydrodynamic simulations. The large spread in the points close to the upper boundary comes from the difficulty of determining the spiral pitch angle in 2.1 Hydrodynamic simulations that region. We run 3D hydrodynamical simulations with the code PLUTO (Mignone et al. 2007) to investigate the disc response to the planet. −3.5 Draine 2001) for a grain size distribution of n(a) ∝ a between We include the FARGO algorithm (Mignone et al. 2012) to reduce a = 0.1 µmand a = 1 mm. With RADMC-3D, we first calculate the the computational cost and use the HLLC Riemann solver, linear dust temperature with a thermal Monte Carlo simulation and we reconstruction and second-order Runge–Kutta time integration. We then calculate images at wavelengths λ = 1.65 µmand λ = 1.3 mm employ a spherical grid with 400, 50, and 1024 cells, distributed in and for an inclination of 0 . Finally, we generate synthetic observa- the [0.4, 2.5], [1.22, π/2], and [0, 2π], intervals in the radial, polar, tions by convolving the calculated images with a Gaussian kernel and azimuthal directions. The planet was placed at the coordinates with an FWHM of 0.04 arcsec, typical of current state-of-the-art [1.0, 0, π/4]. In the radial and polar directions, we define ‘wave- NIR cameras such as SPHERE/VLT or GPI/Gemini and of ALMA killing zones’ where we damp disturbances, following de Val-Borro (ALMA Partnership et al. 2015). We assume a source distance of et al. (2006), extending up to 0.44 and from 2.25 in radius and un- 100 pc. til 1.27 in θ. We employ a locally isothermal equation of state, in which the sound speed c is a function of position only, and a physi- cal viscosity using the Shakura & Sunyaev (1973) prescription with 3 RESULTS −3 a value of α = 10 . We assume the planet mass to be 3 M . Jup We run two simulations, the first (subsequently called ‘isother- 3.1 Hydrodynamic simulations −1/4 mal’) in which c depends only on radius: c ∝ r ,withthe s s We show in Fig. 1 the pitch angle of the spiral in the stratified normalization set such that the disc aspect ratio at the location simulation as a function of the height above the mid-plane. We plot of the planet is 0.07, a typical value at tens of au. In the sec- with the colours shown in the legend the pitch angle at different ond simulation (stratified), we allow the sound speed to vary as radii, normalized to the value in the mid-plane. The red line shows a function of the vertical coordinate. To this end, we use the pre- how the sound speed varies with the vertical coordinate. We find scription commonly employed when fitting observations of Dartois, that the pitch angle increases with z at all radii; in addition, the Dutrey & Guilloteau (2003) and subsequently updated by Rosen- dependence of the pitch angle closely resembles the one of the feld et al. (2013). We assume that the temperature in the upper layer sound speed. of the disc is 2.5 times the value in the mid-plane (the same as in the This behaviour is the natural consequence of the temperature isothermal case), and that the transition between the cold mid-plane dependence of the pitch angle. The spirals are located where con- and the hot upper layer (the parameter z of Dartois et al. 2003) structive interference occurs between waves emitted from Lindblad happens at three scale heights. The initial surface density  of the −1 resonances in the disc (Ogilvie & Lubow 2002); this location de- disc follows  ∝ r . We solve for the hydrostatic equilibrium in pends on the wave propagation speed, and thus the disc temperature. the vertical direction to assign the initial density at every point and Indeed Rafikov (2002) showed that the pitch angle of a tidally in- we initialize the velocity so that the gas is initially in Keplerian duced spiral in a 2D Keplerian disc, in which the sound speed varies rotation. We evolve the simulations for 100 orbits to reach a steady −q with radius as c = c (r/r ) , depends on the sound speed: state in the structure of the spirals. s p p q+1 3/2 r r r p p −1 β = π/2 − tan − 1 , (1) h r r p p 2.2 Radiative transfer simulations We calculate images from the hydrodynamics simulations using the where β is the pitch angle, r is the orbital radius of the planet and 3D radiative transfer code RADMC-3D (http://www.ita.uni-heidelberg. h is the pressure scale height (which depends on the sound speed de/dullemond/software/radmc-3d/), using the same 3D spherical via h = c / ). The resemblance between the dependence with the p p p mesh of the hydrodynamic simulations. We assume that the cen- vertical coordinate of the pitch angle and the sound speed profile tral star is a Herbig Ae star (T = 9500 K, M = 2.5 M , R may imply that this relation holds also in a 3D disc, provided that eff = 2.0 R ), a constant gas-to-dust ratio of 100 and we calculate the local temperature is used in the relation. We leave a detailed the dust opacity assuming astronomical silicate (Weingartner & analysis of the pitch angle in 3D to further studies and conclude MNRASL 474, L32–L36 (2018) Downloaded from https://academic.oup.com/mnrasl/article-abstract/474/1/L32/4628072 by Ed 'DeepDyve' Gillespie user on 16 March 2018 L34 A. Juhasz ´ and G. P. Rosotti (a) (b) (c) (d) Figure 2. Synthetic images in NIR scattered light (Panels a and c) and in submillimetre (Panels b and d) based on 3D hydrodynamic simulations assuming a vertically isothermal (Panels a and b) and stratified (Panels c and d) temperature structure. The scattered light images are scaled with the square of the radial distance to the star. In the NIR, the isothermal simulation shows higher contrast spirals with lower pitch angle compared to the stratified simulation. At submillimetre wavelengths, the pitch angle and the amplitude of the spirals are rather similar in the vertically stratified and isothermal simulations. that the vertical temperature gradient increases the pitch angle of disc mid-plane, in the submillimetre case (Figs 3b and d) the peak- the spirals in the upper layers of the disc. to-peak intensity variation and the width of the spiral peak agree within 10 per cent. In the NIR, scattered light images (Figs 3aand c) instead the spiral contrast is higher in the isothermal simulations by 3.2 Simulated observations 49 per cent and 45 per cent in the inner and outer arms, respectively. This is caused by the higher temperature in the upper layers in the We show the simulated NIR scattered light and submillimetre con- stratified simulations; the enhanced pressure forces tend to wash out tinuum images calculated from vertically isothermal (see Figs 2a more the spiral wake and reduce its amplitude. While the azimuthal and b) and thermally stratified (see Figs 2c and d) hydrodynamic width of the spirals in the two simulations agrees to 4 per cent in- simulations. In terms of the general structure, the isothermal and wards of the planet, for the outer spirals in the isothermal models stratified simulations look largely similar, with the only significant the azimuthal width is a factor 3.5 times larger compared to the difference that the inner part of the disc in the stratified simulation stratified simulations. is brighter with respect to the isothermal simulation. This is be- cause the inner edge of the disc in the stratified simulation is more ‘rounded off’, increasing the flaring index of the disc locally and consequently the brightness of the disc. 3.2.2 Spiral wake In Fig. 4, we present the spiral wakes we derived from the synthetic images by extracting the positions of the local maxima in the radial 3.2.1 Amplitude of the spirals surface brightness distribution. In submillimetre images, probing In Fig. 3, we compare the azimuthal surface brightness profiles of the disc mid-plane, the spirals have similar pitch angles in the locally the two models at two radii, 1.7 r and 0.58 r .Fromthisplot, we isothermal simulations and in the vertically stratified simulations p p measure the peak-to-peak variation of the intensity in the azimuthal (see Figs 4a and d) since the mid-plane temperatures in the two direction and the full width at half-maximum (FWHM) of the spi- simulations (locally isothermal versus stratified) are identical. ral peak (for the inner spiral we considered the stronger, primary Spirals in scattered light images computed from stratified sim- arm). Since the two models employ the same temperature in the ulations (see Fig. 4e) instead are more open compared to the MNRASL 474, L32–L36 (2018) Downloaded from https://academic.oup.com/mnrasl/article-abstract/474/1/L32/4628072 by Ed 'DeepDyve' Gillespie user on 16 March 2018 Spiral arms in protoplanetary discs L35 (a) (b) (d) (c) Figure 3. Comparison of the azimuthal surface brightness profile of the disc in synthetic images calculated from locally isothermal and stratified simulations. The comparison is shown for inner (a, c) and outer spirals (b, d) and both in NIR scattered light (a, b) and in the submillimetre continuum (c, d). The surface ◦ ◦ brightness profiles are normalized to the mean value in the [200 , 300 ] interval. In the NIR, the amplitude and the azimuthal width of the spirals are significantly larger in isothermal simulations compared to the stratified one, while the vertical structure has little effect on the submillimetre continuum observations. (a) (b) (c) (d) (e) (f) Figure 4. Panels a and b show the spiral wake in the vertically isothermal simulation inwards and outwards of the planetary orbit, respectively. Panel c shows the difference in the azimuthal position of the spirals between submillimetre and NIR scattered light images as a function of radius. Panels d–f are forthe vertically stratified simulation. In a vertically isothermal disc, the shape of the spiral wake is largely the same at both wavelengths. For the stratified simulation, the spirals are more open in the NIR than in the submillimetre both inwards and outwards of the planetary orbit. In the stratified simulation, the difference in the azimuthal position of the spirals between NIR and submillimetre increases with increasing distance from the planet, i.e. the spirals converge towards the planet. ones computed from locally isothermal simulations (see Fig. 4b), tered light and submillimetre images as a function of stellocentric consistently with the analysis of the simulation in Section 3.1. For radius in Figs 4c and f for the locally isothermal simulations and example, while we find that outer spirals in NIR scattered light for the stratified simulations, respectively. In the locally isother- images have a low pitch angle (7.5 at 1.5r ) in the isothermal sim- mal simulation, the difference in the azimuthal position of the ◦ ◦ ulations, the pitch angle is higher in the stratified simulation (13 spiral wake is not larger than 8 for the inner arms at all radii ◦ ◦ at 1.5r ); the effect is smaller for the inner spiral (15 versus 13 at with a weak trend of increasing difference towards smaller radii. 0.7r ). Dong et al. (2015a) suggested that a low pitch angle could be For the outer spirals, the difference fluctuates between −19 and used to identify the spiral as an outer one, but it is not clear whether −6 with no clear dependence on the radius. In case of the strat- this method still applies to the stratified case. ified simulations, there is a clear trend for the azimuthal posi- To better illustrate the difference in opening angle, we show the tion difference of the spirals in NIR and submillimetre. The az- difference in azimuth angles between the spirals derived from scat- imuthal difference in the spiral position increases both inwards and MNRASL 474, L32–L36 (2018) Downloaded from https://academic.oup.com/mnrasl/article-abstract/474/1/L32/4628072 by Ed 'DeepDyve' Gillespie user on 16 March 2018 L36 A. Juhasz ´ and G. P. Rosotti outwards from the position of the planet. The difference in the ACKNOWLEDGEMENTS spiral position in NIR and submillimetre images is nearly −20 This work has been supported by the DISCSIM project, grant agree- for the inner arms at 0.55r in the stratified simulations com- pl ment 341137 funded by the European Research Councilunder ERC- paredtothe −8 in the locally isothermal simulations. For the 2013-ADG. We thank the referee, Ruobing Dong, for the insightful outer arms, the azimuthal position difference is about 30 at 2r in pl review that improved our paper. the stratified simulations while only −15 in the locally isothermal simulations. 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Journal

Monthly Notices of the Royal Astronomical Society Letters – Oxford University Press

Published: Nov 14, 2017

Keywords: accretion, accretion discs; hydrodynamics; protoplanetary discs; circumstellar matter

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