Theoretical models of the protostellar disks of AS 209 and HL Tau presently forming in-situ planets

TL;DR

This paper presents isothermal oscillatory density models fitted to ALMA observations of the protoplanetary disks of AS 209 and HL Tau, revealing differences in core size and density that relate to planet formation regions.

Contribution

It introduces a new application of isothermal oscillatory density models to observed disks, highlighting differences in core properties and stability conditions.

Findings

HL Tau's core is significantly larger and denser than AS 209's core.

Both disks exhibit stable density profiles with low centrifugal support.

Four dark gaps are within HL Tau's core, none in AS 209.

Abstract

We fit an isothermal oscillatory density model to two ALMA/DSHARP-observed disks, AS 209 and HL Tau, in which planets have presumably already formed and they are orbiting within the observed seven dark gaps in each system. These large disks are roughly similar to our solar nebula, albeit they exhibit milder radial density profiles and they enjoy lower centrifugal support. We find power-law density profiles with index $k=0.0$ (radial densities $\rho(R) \propto R^{-1}$) and centrifugal support against self-gravity so small that it guarantees dynamical stability for millions of years of evolution. The scale lengths of the models differ only by a factor of 1.9, but the inner cores of the disks are very different: HL Tau's core is 8.0 times larger and 3.6 times denser than the core of AS 209. This results in four dark gaps having formed within the core of HL Tau, whereas no dark gap is found in the core of AS 209. On the other hand, the Jeans frequencies and the angular velocities of the cores are comparable to within factors of 1.9 and 1.6, respectively.

Tables

Table 1. Table 1: Radii of dark gaps in AS 209 and HL Tau (from Table 1 of Huang et al. 2018 )

Gap	AS 209	Gap	HL Tau
Name	$R\left(AU\right)$	Name	$R\left(AU\right)$
D9	08.69	D14	13.9
D24	23.84	D34	33.9
D35	35.04	D44	44
D61	60.8	D53	53
D90	89.9	D67	67.4
D105	105.5	D77	77.4
D137	137	D96	96

Table 2. Table 2: Comparison of the protostellar disk models of AS 209 and HL Tau

Property	Property	AS 209	HL Tau
Name	Symbol (Unit)	Best-Fit Model	Best-Fit Model
Density power-law index	$k$	$0.0$	$0.0$
Rotational parameter	${\beta }_0$	0.0165	0.00562
Inner core radius	$R_1$ (AU)	6.555	52.04
Outer flat-density radius	$R_2$ (AU)	68.96	90.55
Scale length	$R_0$ (AU)	0.01835	0.009813
Equation of state	$c_0^2/{\rho }_0$ (${\mathrm{cm}}^5{g}^{-1}{s}^{-2}$)	$6.32\times {10}^{16}$	$1.81\times {10}^{16}$
Minimum core density for $T=10$ K, $\overline{\mu }=2.34$	${\rho }_0$ (g cm-3)	$5.62\times {10}^{-9}$	$1.97\times {10}^{-8}$
Isothermal sound speed for $T=10$ K, $\overline{\mu }=2.34$	$c_0$ (m s-1)	188	188
Jeans gravitational frequency	${\Omega }_J$ (rad s-1)	$4.9\times {10}^{-8}$	$9.1\times {10}^{-8}$
Core angular velocity	${\Omega }_0$ (rad s-1)	$8.0\times {10}^{-10}$	$5.1\times {10}^{-10}$
Core rotation period	$P_0$ (yr)	249	390
Maximum disk size	$R_{\max }$ (AU)	144	102

Full text

1 Introduction

In previous work (Christodoulou & Kazanas 2019), we presented isothermal models of the solar nebula capable of forming protoplanets long before the protosun is actually formed by accretion processes. This entirely new “bottom-up” formation scenario is currently observed in real time by the latest high-resolution ($\sim$1-5 AU) observations of many protostellar disks by the ALMA telescope (ALMA Partnership 2015; Andrews et al. 2016; Ruane 2017; Lee et al. 2017, 2018; Macías et al. 2018; Avenhaus et al. 2018; Clarke et al. 2018; Keppler et al. 2018; Guzmán et al. 2018; Isella et al. 2018; Zhang et al. 2018; Dullemond et al. 2018; Favre et al. 2018; Harsono et al. 2018; Huang et al. 2018; Pérez et al. 2018; Kudo et al. 2018; Long et al. 2018; Pineda et al. 2018; van der Marel et al. 2019). In this work, we apply the same theoretical model to the observed disks of AS 209 and HL Tau, two young systems resolved by ALMA/DSHARP observations (ALMA Partnership 2015; Huang et al. 2018). We assume that the observations have captured all forming protoplanets and there are no other protoplanets orbiting inside the observed bright rings. That could be proven wrong by future observations, so the present models should be considered as preliminary models based entirely on the current state-of-the-art observations.

An inspection of the ALMA/DSHARP brightness profiles gives the impression that the disks of these systems appear to be similar in structure (ALMA Partnership 2015; Huang et al. 2018). They both extend to more than 100 AU, and each disk exhibits seven pronounced dark gaps or annuli. The goal of our modeling effort is to quantify the physical properties of these two disks and to search for differences, based on the arrangements of their dark gaps that are widely believed to already host orbiting protoplanets. The models show only two striking differences, larger than a factor of 2.9 (their $\beta_{0}$ ratio): the inner core radius of HL Tau is 8.0 times larger than that of AS 209, and the central density of HL Tau is 3.6 times larger than that of AS 209.

The analytic (intrinsic) and numerical (oscillatory) solutions of the isothermal Lane-Emden equation (Lane 1869; Emden 1907) with differential rotation, and the resulting model of the midplane of the gaseous disk have been described in detail in Christodoulou & Kazanas (2019) for the solar nebula. Here, we apply in § 2 the same model to the dark gaps of AS 209 and HL Tau, and we compare the best-fit results in these two cases. In § 3, we summarize our results.

2 Physical Models of the AS 209 and HL Tau Protostellar Disks

The numerical integrations that produce oscillatory density profiles were performed with the Matlab ode15s integrator (Shampine & Reichelt 1997; Shampine et al. 1999) and the optimization used the Nelder-Mead simplex algorithm as implemented by Lagarias et al. (1998). This method (Matlab routine fminsearch) does not use any numerical or analytical gradients in its search procedure which makes it extremely stable numerically, albeit somewhat slow.

2.1 Best-Fit models of AS 209 and HL Tau

The radii of the seven dark gaps in both protostellar systems are shown in Table 1. In Figures 1 and 2, we show the best optimized fits to the dark gaps of AS 209 and HL Tau, respectively. In these models, we have used all four available free parameters ($k$, $\beta_{0}$, $R_{1}$, and $R_{2}$). The mean relative errors are 7.2% and 8.9%, respectively, and they all come from the second and third dark gaps in both cases. Because of this discrepancy, we suspect that there may be more dark gaps, yet undetected, in the inner regions of these disks.

The physical properties of these best-fit models are listed in Table 2. We can see from this table several similarities and just two differences between the two disks: The inner core of HL Tau appears to be 8.0 times larger and 3.6 times denser than the disk of AS 209. Furthermore, the centrifugal support in these models is so low, that it practically guarantees their dynamical stability to nonaxisymmetric self-gravitating instabilities (the critical value for the onset of dynamical instabilities is $\beta_{*}\simeq 0.50$; Christodoulou et al. 1995).

The outer flat-density regions starting at radius $R_{2}$ are roughly comparable between the two systems ($\simeq$70 AU and 96 AU, for AS 209 and HL Tau, respectively). This parameter was used in an attempt to obtain a better fit to the outer dark gaps in each system. When we discarded parameter $R_{2}$, we found significantly worse best-fits for the two systems.

2.2 Comparison between the models of AS 209 and HL Tau

We show a comparison between the physical parameters of the best-fit models of AS 209 and HL Tau in Table 2. Obviously, these two protoplanetary disks are similar in many of their physical properties. In particular, the power-law index $k=0.0$ is the same, and most physical parameters are to within factors of 1.0-2.9. The larger differences in $\rho_{0}$ and $R_{1}$ were already mentioned in § 2.1 above. Another striking difference stems from the discrepancy in $R_{1}$ values: the small core region of AS 209 ($\simeq$6.6 AU) is empty (Fig. 1), whereas the enormous core of HL Tau ($\simeq$52 AU) hosts four of the seven observed dark gaps (Fig. 2).

The power-law index $k=0.0$ found in both models was unexpected. It implies a roughly uniform surface density profile for these two disks and a volume density profile of the form $\rho(R)\propto R^{-1}$, unlike those observed so far in protostellar systems (Andrews & Williams 2007; Hung et al. 2010; Lee et al. 2018). This parameter may certainly change substantially if more planets will be discovered in these disks by higher resolution observations.

3 Summary

In § 2, we presented our best-fit isothermal differentially-rotating protostellar models of two young systems, AS 209 and HL Tau, recently observed by ALMA/DSHARP (ALMA Partnership 2015; Huang et al. 2018; Guzmán et al. 2018; Zhang et al. 2018). These models show seven dark gaps each (Table 1), and it is widely believed that protoplanets have already formed and curved out these dark gaps in the observed otherwise monotonic surface brightness profiles of the disks. The best-fit models are depicted in Figures 1 and 2, respectively, and a comparison of their physical properties is shown in Table 2.

The two disks appear to be similar in the ALMA/DSHARP observations (e.g., ALMA Partnership 2015; Huang et al. 2018). We found only two pronounced differences in their physical properties: HL Tau’s inner core is 8.0 times larger and 3.6 times denser than the core of AS 209.

The orbital periods of the cores turn out to be 249 yr and 390 yr for AS 209 and HL Tau, respectively. In our solar system, these values fall roughly within the Kuiper belt (Trujillo & Brown 2003; Brown & Pan 2004). Furthermore, both models appear to be extremely stable and long-lived (their $\beta_{0}$ values are extremely small), so we believe that AS 209 and HL Tau will continue to evolve in a nonviolent fashion (i.e., they will be subject to no planet-planet resonant interactions, no planet-gas disk interactions, and no orbital migrations), allowing for the observed well-organized planet formation to conclude within the observed dark gaps. These results continue to support strongly a “bottom-up” scenario in which planets form first in $<0.1$ Myr (Greaves & Rice 2010; Harsono et al. 2018), followed by the formation of the central star after the gaseous disk has cleared out by accretion and dispersal processes.