Spiral arms have been observed in more than a dozen protoplanetary disks, yet the origin of nearly all systems is under debate. Multi-epoch monitoring of spiral arm morphology offers a dynamical way in distinguishing two leading arm formation mechanisms: companion-driven, and gravitational instability induction, since these mechanisms predict distinct motion patterns. By analyzing multi-epoch J-band observations of the SAO 206462 system using the SPHERE instrument on the Very Large Telescope (VLT) in 2015 and 2016, we measure the pattern motion for its two prominent spiral arms in polarized light. On one hand, if both arms are comoving, they can be driven by a planet at $86_{-13}^{+18}$ au on a circular orbit, with gravitational instability motion ruled out. On the other hand, they can be driven by two planets at $120_{-30}^{+30}$ au and $49_{-5}^{+6}$ au, offering a tentative evidence (3.0$σ$) that the two spirals are moving independently. The independent arm motion is possibly supported by our analysis of a re-reduction of archival observations using the NICMOS instrument onboard the Hubble Space Telescope (HST) in 1998 and 2005, yet artifacts including shadows can manifest spurious arm motion in HST observations. We expect future re-observations to better constrain the motion mechanism for the SAO 206462 spiral arms.
The disk surrounding SAO 206462, an 8 Myr-old Herbig Ae star, has recently been reported to exhibit spiral arms, an asymmetric dust continuum, and a dust-depleted inner cavity. By carrying out two-dimensional, two-fluid hydrodynamic calculations, we find that a planetary-mass companion located at the outer disk could be responsible for these observed structures. In this model, the planet excites primary and secondary arms interior to its orbit. It also carves a gap and generates a local pressure bump at the inner gap edge where a vortex forms through Rossby wave instability. The vortex traps r…
Disk Evolution Study Through Imaging of Nearby Young Stars (DESTINYS): Dynamical Evidence of a Spiral-Arm-Driving and Gap-Opening Protoplanet from SAO 206462 Spiral Motion
基于甚大望远镜对 SAO 206462 原行星盘 7 年的观测，测得双螺旋臂旋转速率，推断存在一颗位于 66 au 处的原行星同时驱动螺旋臂并刻划缝隙，为原行星成像提供了理想目标。
Recently, high angular resolution imaging instruments such as SPHERE and GPI have discovered many spiral-arm-like features in near-infrared scattered light images of protoplanetary disks. Theory and simulations have suggested that these arms are most likely excited by planets forming in the disks; however, a quantitative relation between the arm-to-disk brightness contrast and planet mass is still missing. Using 3D hydrodynamics and radiative transfer simulations, we examine the morphology and contrast of planet-induced arms in disks. We find a power-law relation for the face-on arm contrast (…
Spiral arms in protoplanetary discs are thought to be linked to the presence of companions. We test the hypothesis that the double spiral arm morphology observed in the transition disc MWC 758 can be generated by an $\approx 10$ M$_{\rm Jup}$ companion on an eccentric orbit internal to the spiral arms. Previous studies on MWC 758 have assumed an external companion. We compare simulated observations from three dimensional hydrodynamics simulations of disc-companion interaction to scattered light, infrared and CO molecular line observations, taking into account observational biases. The inner co…
Fragmentation of protoplanetary disks due to gravitational instabilities is a candidate of a formation mechanism of binary stars, brown dwarfs, and gaseous giant planets. The condition for the fragmentation has been thought that the disk cooling timescale is comparable to its dynamical timescale. However, some numerical simulations suggest that the fragmentation does not occur even if the cooling time is small enough, or the fragmentation can occur even when the cooling is inefficient. To reveal a realistic condition for fragmentation of self-gravitating disks, we perform two-dimensional numer…
Using 3D global hydro simulations coupled with radiative transfer calculations, we study the appearance of density waves induced by giant planets in direct imaging observations at near infrared wavelengths. We find that a 6 MJ planet in a typical disk around a 1 M_sun star can produce prominent and detectable spiral arms both interior and exterior to its orbit. The inner arms have (1) two well separated arms in roughly m=2 symmetry, (2) exhibit ~10-15 degrees pitch angles, (3) ~180-270 degrees extension in the azimuthal direction, and (4) ~150% surface brightness enhancement, all broadly consi…
Combining 3D smoothed-particle hydrodynamics and Monte Carlo radiative transfer calculations, we examine the morphology of spiral density waves induced by gravitational instability (GI) in protoplanetary disks, as they would appear in direct images at near-infrared (NIR) wavelengths. We find that systems with disk-to-star-mass ratios q=M_disk/M_star that are ~0.25 or more may produce prominent spiral arms in NIR imaging, remarkably resembling features observed in the MWC 758 and SAO 206462 systems. The contrast of GI-induced arms at NIR wavelengths can reach a factor of ~3, and their pitch ang…
Recent VLT/SPHERE near-infrared imaging observations revealed two spiral arms with a near m=2 rotational symmetry in the protoplanetary disk around the ~1.7 solar mass Herbig star HD 100453. A ~0.3 solar mass M dwarf companion, HD 100453 B, was also identified at a projected separation of 120 AU from the primary. In this Letter, we carry out hydrodynamic and radiative transfer simulations to examine the scattered light morphology of the HD 100453 disk as perturbed by the companion on a circular and coplanar orbit. We find that the companion truncates the disk at ~45 AU in scattered light image…
More than a dozen young stars host spiral arms in their surrounding protoplanetary disks. The excitation mechanisms of such arms are under debate. The two leading hypotheses -- companion-disk interaction and gravitational instability (GI) -- predict distinct motion for spirals. By imaging the MWC 758 spiral arm system at two epochs spanning ${\sim}5$ yr using the SPHERE instrument on the Very Large Telescope (VLT), we test the two hypotheses for the first time. We find that the pattern speeds of the spirals are not consistent with the GI origin. Our measurements further evince the existence of…
We use two-dimensional Fourier transformation to measure the pitch angle ($φ$) of the dominant spiral Fourier mode of well-defined spiral arms in 13 protoplanetary disks, making use of near-infrared scattered-light images of AB Aur, SAO 206462, MWC 758, V1247 Ori, HD 142527, DZ Cha, LkH$α$ 330, and HD 100453, and ALMA millimeter continuum images of Elias 2-27, IM Lup, AS 205, and HT Lup. We find that the measured pitch angle correlates strongly with disk mass ($M_{D}$), such that more massive protoplanetary disks have smaller pitch angles, following $|φ| = -(7.8\pm1.7)\log(M_{D}/M_{\odot})+(2.…
Protoplanetary disk simulations show that a single planet can excite more than one spiral arm, possibly explaining recent observations of multiple spiral arms in some systems. In this paper, we explain the mechanism by which a planet excites multiple spiral arms in a protoplanetary disk. Contrary to previous speculations, the formation of both primary and additional arms can be understood as a linear process when the planet mass is sufficiently small. A planet resonantly interacts with epicyclic oscillations in the disk, launching spiral wave modes around the Lindblad resonances. When a set of…
High-resolution imaging of protoplanetary disks has unveiled a rich diversity of spiral structure, some of which may arise from disk-planet interaction. Using 3D hydrodynamics with $β$-cooling to a vertically-stratified background, as well as radiative-transfer modeling, we investigate the temperature rise in planet-driven spirals. In rapidly cooling disks, the temperature rise is dominated by a contribution from stellar irradiation, 0.3-3% inside the planet radius but always <0.5% outside. When cooling time equals or exceeds dynamical time, however, this is overwhelmed by hydrodynamic PdV wor…
Formed in protoplanetary disks around young stars, giant planets can leave observational features such as spirals and gaps in their natal disks through planet-disk interactions. Although such features can indicate the existence of giant planets, protoplanetary disk signals can overwhelm the innate luminosity of planets. Therefore, in order to image planets that are embedded in disks, it is necessary to remove the contamination from the disks to reveal the planets possibly hiding within their natal environments. We observe and directly model the detected disk in the Keck/NIRC2 vortex coronagrap…
Spiral arms are observed in numerous protoplanetary discs. These spiral arms can be excited by companions, either on bound or unbound orbits. We simulate a scenario where an unbound perturber, i.e. a flyby, excites spiral arms during a periastron passage. We run three-dimensional hydrodynamical simulations of a parabolic flyby encountering a gaseous protoplanetary disc. The perturber mass ranges from $10\, \rm M_J$ to $1\, \rm M_{\odot}$. The perturber excites a two-armed spiral structure, with a more prominent spiral feature for higher mass perturbers. The two arms evolve over time, eventuall…
HD 100453AB is a 10+/-2 Myr old binary whose protoplanetary disk was recently revealed to host a global two-armed spiral structure. Given the relatively small projected separation of the binary (1.05", or ~108 au), gravitational perturbations by the binary seemed to be a likely driving force behind the formation of the spiral arms. However, the orbit of these stars remained poorly understood, which prevented a proper treatment of the dynamical influence of the companion on the disk. We observed HD 100453AB between 2015-2017 utilizing extreme adaptive optics systems on the Very Large Telescope …
Recent observations of protoplanetary disks, as well as simulations of planet-disk interaction, have suggested that a single planet may excite multiple spiral arms in the disk, in contrast to the previous expectations based on linear theory (predicting a one-armed density wave). We re-assess the origin of multiple arms in the framework of linear theory, by solving for the global two-dimensional response of a non-barotropic disk to an orbiting planet. We show that the formation of a secondary arm in the inner disk, at about half of the orbital radius of the planet, is a robust prediction of lin…
DR Tau has been noted for its unusually high variability in comparison with other T Tauri stars. Although it is one of the most extensively studied pre-main sequence stars, observations with millimeter interferometry have so far been relatively limited. We present NOEMA images of $^{12}$CO, $^{13}$CO, C$^{18}$O, SO, DCO$^+$, and H$_2$CO toward DR Tau at a resolution of $\sim0.5''$ ($\sim100$ au). In addition to the protoplanetary disk, CO emission reveals an envelope, a faint asymmetric outflow, and a spiral arm with a clump. The $\sim1200$ au extent of the CO arm far exceeds that of the spira…
Some circumstellar disks are observed to show prominent spiral arms in infrared scattered light or (sub-)millimeter dust continuum. The spirals might be formed from self-gravity, shadows, or planet-disk interactions. Recently, it was hypothesized that massive vortices can drive spiral arms in protoplanetary disks in a way analogous to planets. In this paper, we study the basic properties of vortex-driven spirals by the Rossby Wave Instability in 2D hydrodynamics simulations. We study how the surface density contrast, the number, and the shape of vortex-driven spirals depend on the properties o…
During the evolution of protoplanetary disks into planetary systems we expect to detect signatures that trace mechanisms such as planet-disk interaction. Protoplanetary disks display a large variety of structures in recently published high-spatial resolution images. However, the three-dimensional morphology of these disks is often difficult to infer from the two-dimensional projected images we observe. We spatially resolve the disk around HD 34282 using VLT/SPHERE in polarimetric imaging mode. We retrieve a profile for the height of the scattering surface to create a height-corrected deproject…
Direct imaging observations of protoplanetary disks at near-infrared (NIR) wavelengths have revealed structures of potentially planetary origin. Investigations of observational signatures from planet-induced features have so far focused on disks viewed face-on. Combining 3D hydrodynamics and radiative transfer simulations, we study how the appearance of the spiral arms and the gap produced in a disk by a companion varies with inclination and position angle in NIR scattered light. We compare the cases of a $3M_{\rm J}$ and a $0.1M_{\odot}$ companion, and make predictions suitable for testing wi…
We examine whether various characteristics of planet-driven spiral arms can be used to constrain the masses of unseen planets and their positions within their disks. By carrying out two-dimensional hydrodynamic simulations varying planet mass and disk gas temperature, we find that a larger number of spiral arms form with a smaller planet mass and a lower disk temperature. A planet excites two or more spiral arms interior to its orbit for a range of disk temperature characterized by the disk aspect ratio $0.04\leq(h/r)_p\leq0.15$, whereas exterior to a planet's orbit multiple spiral arms can fo…
One of the striking discoveries of protoplanetary disc research in recent years are the spiral arms seen in several transitional discs in polarised scattered light. An interesting interpretation of the observed spiral features is that they are density waves launched by one or more embedded (proto-)planets in the disc. In this paper we investigate whether planets can be held responsible for the excitation mechanism of the observed spirals. We use locally isothermal hydrodynamic simulations as well as analytic formulae to model the spiral waves launched by planets. Then H-band scattered light im…
Spiral arms have been observed in nearly a dozen protoplanetary discs in near-infrared scattered light and recently also in the sub-millimetre 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 nat…
[Full abstract in the paper] In recent years, protoplanetary disks with spiral structures have been detected in scattered light, millimeter continuum, and CO gas emission. The mechanisms causing these structures are still under debate. A popular scenario to drive the spiral arms is the one of a planet perturbing the material in the disk. However, if the disk is massive, gravitational instability is usually the favored explanation. Multiwavelength studies could be helpful to distinguish between the two scenarios. So far, only a handful of disks with spiral arms have been observed in both scatte…
We analyze two-armed global spiral density wave modes generated by gravitational instability in razor-thin, non-viscous, self-gravitating protoplanetary disks to understand the dependence of spiral arm morphology (pitch angle $α$ and amplitude) on various disk conditions. The morphologies of the resulting spiral density wave modes closely resemble observations. Their pitch angles and pattern speeds are insensitive to the boundary conditions adopted. Gaussian disks exhibit more tightly wound spirals (smaller pitch angle) than power law disks under the same conditions. We find that at a fixed di…
Gravitational forces are expected to excite spiral density waves in protoplanetary disks, disks of gas and dust orbiting young stars. However, previous observations that showed spiral structure were not able to probe disk midplanes, where most of the mass is concentrated and where planet formation takes place. Using the Atacama Large Millimeter/submillimeter Array we detected a pair of trailing symmetric spiral arms in the protoplanetary disk surrounding the young star Elias 2-27. The arms extend to the disk outer regions and can be traced down to the midplane. These millimeter-wave observatio…
Context. The SAO 206462 (HD 135344B) disk is one of the few known transitional disks showing asymmetric features in scattered light and thermal emission. Near-infrared scattered-light images revealed two bright outer spiral arms and an inner cavity depleted in dust. Giant protoplanets have been proposed to account for the disk morphology. Aims. We aim to search for giant planets responsible for the disk features and, in the case of non-detection, to constrain recent planet predictions using the data detection limits. Methods. We obtained new high-contrast and high-resolution total intensity im…
Azimuthally asymmetric structures have been discovered in millimeter continuum emission from many protoplanetary disks. One hypothesis is that they are vortices produced by the Rossby wave instability, for example at edges of planet-opened gaps or deadzones. Confirming the vortex nature of these structures will have profound implications to planet formation. One way to test the hypothesis is to compare the observed morphology of vortex candidates in near-infrared scattered light with theoretical expectations. To this end, we synthesize the appearance of vortices in H-band polarized light by co…
High-resolution imaging of some protoplanetary disks in scattered light reveals presence of the global spiral arms of significant amplitude, likely excited by massive planets or stellar companions. Assuming that these arms are density waves, evolving into spiral shocks, we assess their effect on the thermodynamics, accretion, and global evolution of the disk. We derive analytical expressions for the direct (irreversible) heating, angular momentum transport, and mass accretion rate induced by the disk shocks of arbitrary strength. We find these processes to be very sensitive to the shock amplit…
Recent spatially-resolved observations of protoplanetary disks revealed a plethora of substructures, including concentric rings and gaps, inner cavities, misalignments, spiral arms, and azimuthal asymmetries. This is the major breakthrough in studies of protoplanetary disks since Protostars and Planets VI and is reshaping the field of planet formation. However, while the capability of imaging substructures in protoplanetary disks has been steadily improving, the origin of many substructures are still largely debated. The structured distributions of gas and solids in protoplanetary disks likely…
Warps survive beyond fly-by encounters in protoplanetary disks. RW Aur A as a case study
Università degli Studi di Milano（米兰大学）；Heidelberg University（海德堡大学）；Universidad de Santiago de Chile（智利圣地亚哥大学）；Millennium Nucleus on Young Exoplanets and their Moons (YEMS)（年轻系外行星及其卫星千年核）；Center for Interdisciplinary Research in Astrophysics Space Exploration (CIRAS)（天体物理学空间探索跨学科研究中心）
Recent observations of protoplanetary disk have reported spiral structures that are potential signatures of embedded planets, and modeling efforts have shown that a single planet can excite multiple spiral arms, in contrast to conventional disk-planet interaction theory. Using two and three-dimensional hydrodynamics simulations to perform a systematic parameter survey, we confirm the existence of multiple spiral arms in disks with a single planet, and discover a scaling relation between the azimuthal separation of the primary and secondary arm, $ϕ_{\rm sep}$, and the planet-to-star mass ratio …
Protoplanetary disks with large inner dust cavities are thought to host massive planetary or substellar companions. These disks show asymmetries and rings in the millimeter continuum, caused by dust trapping in pressure bumps, and potentially vortices or horseshoes. The origin of the asymmetries and their diversity remains unclear. We present a comprehensive study of 16 disks for which the gas surface density profile has been constrained by CO isotopologue data. We compare the azimuthal extents of the dust continuum profiles with the local gas surface density in each disk, and find that the as…
Protoplanets can produce structures in protoplanetary disks via gravitational disk-planet interactions. Once detected, such structures serve as signposts of planet formation. Here we investigate the kinematic signatures in disks produced by multi-Jupiter mass ($M_{\rm J}$) planets using 3D hydrodynamics and radiative transfer simulations. Such a planet opens a deep gap, and drives transonic vertical motions inside. Such motions include both a bulk motion of the entire half-disk column, and turbulence on scales comparable to and smaller than the scale height. They significantly broaden molecula…
JWST/NIRCam Imaging of Young Stellar Objects. II. Deep Constraints on Giant Planets and a Planet Candidate Outside of the Spiral Disk Around SAO 206462
We report Gemini Planet Imager H band high-contrast imaging/integral field spectroscopy and polarimetry of the HD 100546, a 10 $Myr$-old early-type star recently confirmed to host a thermal infrared bright (super)jovian protoplanet at wide separation, HD 100546 b. We resolve the inner disk cavity in polarized light, recover the thermal-infrared (IR) bright arm, and identify one additional spiral arm. We easily recover HD 100546 b and show that much of its emission originates an unresolved, point source. HD 100546 b likely has extremely red infrared colors compared to field brown dwarfs, qualit…
We explore whether protoplanetary disks with self-shadowing from puffed up inner rims exhibit observable features in scattered light images. We use both self-consistent hydrostatic equilibrium calculations and parameterized models to produce the vertically puffed up inner rims. We find that, in general, the transition between the shadowed and flared regions occurs in a smooth manner over a broad radius range, and no sudden jump exists at the outer edge of the shadow in either the disk temperature or density structures. As a result, a puffed up rim cannot create sharp ring/arc/spiral-arm-like f…
Spiral arm structures seen in scattered light observations of protoplanetary disks can potentially serve as signposts of planetary companions. They can also lend unique insights into disk masses, which are critical in setting the mass budget for planet formation but are difficult to determine directly. A surprisingly high fraction of disks that have been well-studied in scattered light have spiral arms of some kind (8/29), as do a high fraction (6/11) of well-studied Herbig intermediate mass stars (i.e., Herbig stars $> 1.5M_\odot$). Here we explore the origin of spiral arms in Herbig systems …
Companions at sub-arcsecond separation from young stars are difficult to image. However their presence can be inferred from the perturbations they create in the dust and gas of protoplanetary disks. Here we present a new interpretation of SPHERE polarised observations that reveal the previously detected inner spiral in the disk of HD 100546. The spiral coincides with a newly detected 12CO inner spiral and the previously reported CO emission Doppler-flip, which has been interpreted as the signature of an embedded protoplanet. Comparisons with hydrodynamical models indicate that this Doppler-fli…
Large-scale spiral arms have been revealed in scattered light images of a few protoplanetary disks. Theoretical models suggest that such arms may be driven by and co-rotate with giant planets, which has called for remarkable observational efforts to look for them. By examining the rotation of the spiral arms for the MWC 758 system over a 10-yr timescale, we are able to provide dynamical constraints on the locations of their perturbers. We present reprocessed Hubble Space Telescope (HST)/NICMOS F110W observations of the target in 2005, and the new Keck/NIRC2 $L'$-band observations in 2017. MWC~…
Direct imaging observations have revealed spiral structures in protoplanetary disks. Previous studies have suggested that planet-induced spiral arms cannot explain some of these spiral patterns, due to the large pitch angle and high contrast of the spiral arms in observations. We have carried out three dimensional (3-D) hydrodynamical simulations to study spiral wakes/shocks excited by young planets. We find that, in contrast with linear theory, the pitch angle of spiral arms does depend on the planet mass, which can be explained by the non-linear density wave theory. A secondary (or even a te…
Resolved ALMA and VLA observations indicate the existence of two dust traps in the protoplanetary disc MWC 758. By means of 2D gas+dust hydrodynamical simulations post-processed with 3D dust radiative transfer calculations, we show that the spirals in scattered light, the eccentric, asymmetric ring and the crescent-shaped structure in the (sub)millimetre can all be caused by two giant planets: a 1.5-Jupiter mass planet at 35 au (inside the spirals) and a 5-Jupiter mass planet at 140 au (outside the spirals). The outer planet forms a dust-trapping vortex at the inner edge of its gap (at ~85 au)…
Aims: Response of a protoplanetary disk to luminosity bursts of various duration is studied with the purpose to determine the effect of the bursts on the strength and sustainability of gravitational instability in the disk. A special emphasis is paid to the spatial distribution of gas and grown dust (from 1 mm to a few cm) during and after the burst. Methods: Numerical hydrodynamics simulations using the FEOSAD code were employed to study the dynamics of gas and dust in the thin-disk limit. Dust-to-gas friction including back reaction and dust growth were also considered. Bursts of various dur…
We investigate how the detectability of signatures of self-gravity in a protoplanetary disc depends on its temporal evolution. We run a one-dimensional model for secular timescales to follow the disc mass as a function of time. We then combine this with three-dimensional global hydrodynamics simulations that employ a hybrid radiative transfer method to approximate realistic heating and cooling. We simulate ALMA continuum observations of these systems, and find that structures induced by the gravitational instability (GI) are readily detectable when $q=M_\mathrm{disc}/M_*\gtrsim 0.25$ and $R_\m…
Photoevaporation and magnetically driven winds are two independent mechanisms to remove mass from protoplanetary disks. In addition to accretion, the effect of these two principles acting concurrently could be significant and the transition between those two has not been extensively studied and quantified in the literature yet. In order to contribute to the understanding of disk winds, we present the phenomena emerging in the framework of two-dimensional axisymmetric, non-ideal magnetohydrodynamic simulations including EUV-/ X-ray driven photoevaporation. Of particular interest are the examina…
Spiral arms, rings and large scale asymmetries are structures observed in high resolution observations of protoplanetary disks, and it appears that some of the disks showing spiral arms in scattered light also show asymmetries in millimeter-sized dust. HD~135344B is one of these disks. Planets are invoked as the origin of these structures, but no planet has been observed so far. We investigate the nature of the asymmetric structure in the HD~135344B disk in order to understand the origin of the spirals and of the asymmetry seen in this disk. Ultimately, we aim at understanding whether or not o…
Spiral structures are a common feature in scattered-light images of protoplanetary disks, and of great interest as possible tracers of the presence of planets. However, other mechanisms have been put foward to explain them, including self-gravity, disk-envelope interactions, and dead zone boundaries. These mechanisms explain many spirals very well, but are unable to easily account for very loosely wound spirals and single spiral arms. We study the effect of light travel time on the shape of a shadow cast by a clump orbiting close (within ${\sim}1\,$au) of the central star, where there can be s…
We carry out three dimensional SPH simulations to show that a migrating giant planet strongly suppresses the spiral structure in self-gravitating discs. We present mock ALMA continuum observations which show that in the absence of a planet, spiral arms due to gravitational instability are easily observed. Whereas in the presence of a giant planet, the spiral structures are suppressed by the migrating planet resulting in a largely axisymmetric disc with a ring and gap structure. Our modelling of the gas kinematics shows that the planet's presence could be inferred, for example, using optically …
(shortened) Planet forming discs are believed to be very weakly turbulent in the regions outside of 1 AU. For this reason, it is now believed that magnetized winds could be the dominant mechanism driving accretion in these systems. However, there is today no self-consistent way to describe discs subject to a magnetized wind, in a way similar to the $α$ disc model. In this article, I explore in a systematic way the parameter space of wind-driven protoplanetary discs and present scaling laws which can be used in reduced models à la alpha-disc. Methods: I compute a series of self-similar wind sol…
