Theoretical model of the outer disk of TW Hya presently forming in-situ planets and comparison with models of AS 209 and HL Tau

TL;DR

This paper presents a theoretical isothermal oscillatory density model of the TW Hya disk, comparing its properties with HL Tau and AS 209, revealing similarities and differences in disk structure and stability.

Contribution

The study introduces a detailed isothermal density model for TW Hya's outer disk, highlighting its stability and comparing it with other protostellar disks, which is a novel approach.

Findings

TW Hya's disk has a power-law density profile with index -0.2.

TW Hya's disk is more dense centrally than HL Tau by a factor of 5.7.

The disk shows stability with minimal centrifugal support.

Abstract

We fit an isothermal oscillatory density model to the outer disk of TW Hya in which planets have presumably already formed and they are orbiting within four observed dark gaps. At first sight, this 52 AU small disk does not appear to be similar to our solar nebula; it shows several physical properties comparable to those in HL Tau (size $R_{\rm max}=102$ AU) and very few similarities to AS 209 ($R_{\rm max}=144$ AU). We find a power-law density profile with index $k=-0.2$ (radial densities $\rho(R) \propto R^{-1.2}$) and centrifugal support against self-gravity so small that it virtually guarantees dynamical stability for millions of years of evolution to come. Compared to HL Tau, the scale length $R_0$ and the core size $R_1$ of TW Hya are smaller only by factors of $\sim$2, reflecting the disk's half size. On the opposite end, the Jeans frequency $\Omega_J$ and the angular velocity $\Omega_0$ of the smaller core of TW Hya are larger only by factors of $\sim$2. The only striking difference is that the central density ($\rho_0$) of TW Hya is 5.7 times larger than that of HL Tau, which is understood because the core of TW Hya is only half the size ($R_1$) of HL Tau and about twice as heavy ($\Omega_J$). In the end, we compare the protostellar disks that we have modeled so far.

Tables

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

Gap	AS 209	Gap	HL Tau	Gap	TW Hya
Name	$R\left(AU\right)$	Name	$R\left(AU\right)$	Name	$R\left(AU\right)$
D9	8.69	D14	13.9	D1	1
D24	23.84	D34	33.9	D26	25.62
D35	35.04	D44	44	D32	31.5
D61	60.8	D53	53	D42	41.64
D90	89.9	D67	67.4	D48	48
D105	105.5	D77	77.4
D137	137	D96	96

Table 2. Table 2: Comparison of the protostellar disk model of TW Hya against AS 209 and HL Tau

Property	Property	AS 209	HL Tau	TW Hya
Name	Symbol (Unit)	Best-Fit Model	Best-Fit Model	Best-Fit Model
Density power-law index	$k$	$0.0$	$0.0$	$-0.2$
Rotational parameter	${\beta }_0$	0.0165	0.00562	0.00401
Inner core radius	$R_1$ (AU)	6.555	52.04	28.67
Outer flat-density radius	$R_2$ (AU)	68.96	90.55	$\cdots$
Scale length	$R_0$ (AU)	0.01835	0.009813	0.004100
Equation of state	$c_0^2/{\rho }_0$ (${\mathrm{cm}}^5{g}^{-1}{s}^{-2}$)	$6.32\times {10}^{16}$	$1.81\times {10}^{16}$	$3.15\times {10}^{15}$
Minimum core density for $T=10$ K, $\overline{\mu }=2.34$	${\rho }_0$ (g cm-3)	$5.62\times {10}^{-9}$	$1.97\times {10}^{-8}$	$1.13\times {10}^{-7}$
Isothermal sound speed for $T=10$ K, $\overline{\mu }=2.34$	$c_0$ (m s-1)	188	188	188
Jeans gravitational frequency	${\Omega }_J$ (rad s-1)	$4.9\times {10}^{-8}$	$9.1\times {10}^{-8}$	$2.2\times {10}^{-7}$
Core angular velocity	${\Omega }_0$ (rad s-1)	$8.0\times {10}^{-10}$	$5.1\times {10}^{-10}$	$8.7\times {10}^{-10}$
Core rotation period	$P_0$ (yr)	249	390	228
Maximum disk size	$R_{\max }$ (AU)	144	102	52

Full text

1 Introduction

In previous work (Christodoulou & Kazanas, 2019a), 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., 2018a; Pérez et al., 2018; Kudo et al., 2018; Long et al., 2018; Pineda et al., 2018; van der Marel et al., 2019).

The ALMA/DSHARP observations show many circular protostellar disks with circular dark gaps presumably carved out by protoplanets that have already been formed at a time long long before accretion/dispersal processes will dissipate their disks. Very few disks show asymmetries and spiral arms, signs of setting-in instabilities (Pérez et al., 2018; Huang et al., 2018a; van der Marel et al., 2019). Motivated by these observations, we have produced models of the disks of AS 209 (seven gaps), HL Tau (seven gaps), and RU Lup (4 gaps) (Christodoulou & Kazanas, 2019b, c). In this work, we apply the same theoretical model to the observed disk of TW Hya, a prototypical young system observed by ALMA/DSHARP (ALMA Partnership, 2015; Huang et al., 2018a, b).

TW Hya is unusual in that it shows a inner dark gap at 1 AU, and the next gap, D26, lies at 25.62 AU. It is obvious that the inner disk is not resolved, but then the outer disk shows clearly four dark gaps in a configuration that is not easy to model: gap D42 is 10 AU away from D32, but the next outer gap, D48, lies 4 AU closer to D42. The disk of TW Hya is relatively small in extent ($R_{\rm max}=52$ AU). It seems that the observations managed to find only the dark gaps in the outer disk (beyond 25 AU). We think that there may be more planets between 1AU and 25 AU, but we have no way of modeling them presently. This also means that the core region of TW Hya will be large, making room for the yet undetected planets. So we ignore the 1 AU gap, and we focus our models to the outer disk with its four pronounced dark gaps.

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 (2019a) for the solar nebula. Here, we apply in § 2 the same model to the four outer dark gaps of TW Hya, and we compare the best-fit results against AS 209 and, more importantly, HL Tau (Christodoulou & Kazanas, 2019b). In § 3, we summarize our results and we discuss the ALMA-observed disks that we have modeled so far.

2 Physical Models of the TW Hya Protostellar Disk

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. The boundary conditions for the oscillatory density profiles are, as usual, $\tau(0)=1$ and $[d\tau/dx](0)=0$, where $\tau$ and $x$ are the dimensionless values of the density and the radius, respectively.

2.1 Best-Fit models of TW Hya

The radii of the five known dark gaps in TW Hya are shown in Table 1, but the 1 AU gap will be ignored in what follows. In Fig. 1, we show the best optimized fit to the four outer dark gaps of TW Hya. In these models, we have used three free parameters ($k$, $\beta_{0}$, and $R_{1}$) because the disk is relatively small and it does not need a flat-density outer region. The mean relative error of the fit is 5.1% and it comes from gaps D26 and D42 (Table 1).

The physical properties of the best-fit TW Hya model are listed in Table 2 along with the best models of AS 209 and HL Tau. It is obvious that TW Hya is more similar to HL Tau. TW Hya and AS 209 share effectively only two common properties in their values of $k$ and $\Omega_{0}$. The agreement in $\Omega_{0}$ is a coincidence as the core of AS 209 is much smaller in size (by a factor of 4.4). A detailed comparison between the disks of TW Hya and HL Tau is discussed below.

2.2 Comparison between the best-fit models of TW Hya and HL Tau

The TW Hya power-law density profile with index $k=-0.2$ is close to the value of $k=0.0$ of the other two disks (Table 2). Centrifugal support against self-gravity is extremely low ($\beta_{0}\simeq 0.004$) for TW Hya, guaranteeing the disk’s long-term dynamical stability (Christodoulou et al., 1995). The TW Hya model compares well to the best-fit HL Tau model: Its scale length $R_{0}$ and its core size $R_{1}$ are smaller only by factors of 1.8-2.4. On the other hand, the Jeans frequency $\Omega_{J}$ and the angular velocity $\Omega_{0}$ of the smaller core of TW Hya are larger by factors of 1.7-2.4. These scalings do make sense.

There is, however, a striking difference in the central density $\rho_{0}$ of TW Hya; it is about 6 times larger than that of HL Tau, and this is understood as follows: the disk of TW Hya is only half the size of HL Tau ($R_{\rm max}=52$ AU versus 102 AU) and 2.4 times heavier (see the $\Omega_{J}$ values in Table 2).

3 Summary

In § 2, we presented our best-fit isothermal differentially-rotating protostellar models of TW Hya observed by ALMA/DSHARP (ALMA Partnership, 2015; Huang et al., 2018a, b; Guzmán et al., 2018; Zhang et al., 2018). This model shows four dark gaps in the outer disk (Table 1), and it is widely believed that protoplanets have already formed and curved out these gaps in the observed outer disk. The best-fit model is depicted in Figure 1 and a comparison of its physical properties versus AS 209 and HL Tau is shown in Table 2. The physical properties of TW Hya are much closer to those of HL Tau than those of AS 209.

In Fig. 2, we show a schematic diagram of dark gaps (dots) in the ALMA-observed disks that we have modeled so far and we also included our solar nebula. We have also included the empty density peaks predicted by some best-fit models and these are denoted in the figure by crosses. No dark gaps have been observed in these positions yet, so these comparisons are very much model dependent. The picture that we have formed is as follows:

The gap arrangement of HL Tau is roughly similar to that of AS 209 than any other disk. In fact, the two disks also share several physical properties (Table 2), except for the inner core radius $R_{1}$. 2. 2.

The TW Hya gaps, including the two empty density peaks, appear to have a similar arrangement to the gaps of RU Lup (Fig. 2), but in physical properties, only the cores ($R_{1}$ and $\Omega_{0}$) of the disks are similar. On the other hand, several of the physical properties of TW Hya are much more similar to those of HL Tau (Table 2). 3. 3.

RU Lup definitely looks like our outer solar system, in particular, the region between Jupiter and Pluto (Christodoulou & Kazanas, 2019c). Both planetary arrangements also share about the same size ($R_{\rm max}\simeq 50$ AU) and most other physical properties. Only the cores ($R_{1}$ and $\Omega_{0}$) of the disks are different, but this is because the core of RU Lup is not adequately resolved by ALMA (whereas the core of our solar nebula is populated with the four terrestrial planets).

From this comparison, we have two ALMA-observed disks in which the dark gap arrangements in their outer regions appear to be similar to the outer planets in our solar system: RU Lup and TW Hya (but the physical properties of RU Lup are much closer to those of the solar system). This is a good start in our efforts to identify extremely young exoplanetary systems similar to our own much older solar system.