A reaction-diffusion model for describing the ring/gap structure in disks surrounding individual young stars

Abstract

The embedded disks surrounding individual Class 0 protostars are structureless. Disks surrounding Class I stars may be continuous or have a ring-gap substructure, whereas all disks around Class II stars have a ring-gap substructure that gradually disappear as the disks evolve into debris disks. This common sequence in young lone stars requires an explanation. This study aims to show that the physical model Reaction-Diffusion Systems with Moving Reaction Front can be used to describe and classify protostellar disks according to their structure. A comprehensive review of observations made with the ALMA radio telescope shows: first, that the protostar-disk system presents a geometry analogous to that of an reaction-diffusion system with two separate compartments, namely, protostar and disk. Second, that in the protostar, matter is processed at high temperature, resulting in a chemical composition different from that of the disk. Third, that the equatorial outflow emitted by the protostar, rich in highly reactive trihydrogen cation, acts as a moving reaction front, MRF, that triggers the formation of molecules and nuclei in the disk. The time lag of nucleation with respect to the passage of the MRf would be the cause of the formation of the gaps between the rings of particles that form in the disk. The MRF is a transient phenomenon and its passage causes the transformation of a continuous disk, Class 0, into a disk with a ring-gap structure, Class II, whose temporal evolution begins at the interface of the star and moves outwards.

Figures (9)

• Figure 1: A) Distance travelled by the MRF and position of the NF as a function of time. The distance r is measured from the interface between reactants A$_i$ and B$_i$, which is chosen as r$_0=0$, while the time of emission of the MRF is chosen as t$_0=0$. The passage of the reaction front triggers the nucleation and the formation of bands of particles at positions r$_n$, which are represented with coloured dots. The formation of bands is not instantaneous but takes place over a time interval. The time lag between the MRF and the NF is $\tau_1$ whereas the time lag between the NF and the formation of bands is $\tau_2$. (B) Micro particle bands in an RDS-MRF approximating a two-dimensional system. Reagent A (Na$_2$HPO$_4$) is placed within the circular cavity in the centre from where diffuses to find reactant B (CaCl$_2$) that is dissolved in a gel (99.5 per cent $w/w$ is water).
• Figure 2: Sketch of the RDS-MRF model including a protostar, an interface, an optical thick disk (grey), and an optical thin equatorial outflow (magenta and azure) all viewed edge-on. The interface will be used as the origin of the reference system to measure distances in the disk, The distribution of ions in the disk and its atmosphere is taken from Aikawa et al aikawa2021molecules.
• Figure 3: Gas and dust disks around young stars. The top row shows zeroth-moment maps of $^{12}$CO (2-1) and the bottom row, the 220 GHz continuous images showing the variety of continuous structures present in the MAPS samples observed at the same spatial resolution. The synthesized beam and a scale bar indicating 50 au are shown in the lower left and right corners, respectively, of each panel. Upper panels are adapted from oberg2021molecules and reproduced with Öberg permission. Bottom panels are adapted from law2021molecules and reproduced with Law permission. Copyright AAS.
• Figure 4: (A) Schematic showing the high-velocity bipolar outflow (between 30 and 100 km s$^{-1}$) represented by two cones and arrows, and the low-speed equatorial outflow (18 km s$^{-1}$) represented by a toroidal surface in the Radio source I system in the BK/KL region in Orion. Left panel is adapted from greenhill1998coexisting and is copyright Springer Nature. Reused under license number 5667730238292. (B) Images obtained at ALMA of: (1) H$_2$CO and (2) C$^{18}$O emission lines observed in the Class 0 protostar IRAS 15398-3359. Both emissions show a bipolar outflow extending from northeast to southwest (position angle, P.A. 220°) and an equatorial outflow extending from southeast to northwest (P.A. 130º-140º). The dashed lines represent the disk envelope axis (P.A.130º). Left panel is from okoda2021faust and reproducedused under the terms of the Creative Commons Attribution 4.0 licence
• Figure 5: Radii of the outermost gas (11 square dots) and dust rings (14 circular dots) of some protostellar disks as a function of stellar age. From youngest to oldest, the data obtained from van der Marel et al.,van2019protoplanetary correspond to: Elías 24, HL Tau, GY 91, AS 209, Sz 98, RXJ1615, HD 97048, DM Tau, HD 163296, TW Hya, V1094 Sco, HD 100546, HD 135344B, HD 169142. The ages of stars often vary by a factor of 2 or more between different studies, so their average age has been calculated. Data for AA Tau and V1247 Ori are not represented because only the lower limit of R$_{gas}$ ($>$ 220 and $>$ 280 au respectively) has been reported for these disks. For the three youngest stars (HL Tau, Elias 24 and GY 91) there is no gas ring beyond the outer dust ring and there are only 11 R$_{gas}$ data in the figure.