Are Elias 2-27's Spiral Arms Driven by Self-gravity, or by a Companion? A Comparative Spiral Morphology Study

Our hydrodynamic simulations are fully described in Meru et al. (2017), but for completeness we briefly reiterate some salient aspects. The simulations are performed using the 3D SPH code (sphNG) including heating due to work done and the radiative transfer of energy in the flux-limited diffusion limit (Whitehouse et al. 2005; Whitehouse & Bate 2006). A detailed description of the code can be found in Meru (2015) and Meru et al. (2017).

Our reference model consists of a $0.5\,{M}_{\odot }$ star, modeled as a sink particle, surrounded by a disk with an initial temperature (T) and surface density (Σ) that varies with radius (R) as

$$\begin{eqnarray}&&T(R)=13.4\,{\rm{K}}{\left(\displaystyle \frac{R}{200\mathrm{au}}\right)}^{-0.75},\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&{\rm{\Sigma }}(R)={{\rm{\Sigma }}}_{0}{\left(\displaystyle \frac{R}{200\mathrm{au}}\right)}^{-0.75}\end{eqnarray} \tag{ 3 }$$

respectively, between R_in = 10 au and R_out = 350 au. The first simulation is a gravitationally unstable disk (GI), with ${{\rm{\Sigma }}}_{0}\,=6\times {10}^{-7}\,{M}_{\odot }\,{\mathrm{au}}^{-2}$, giving a total disk mass of 0.24 ${M}_{\odot }$, and an initial Toomre parameter Q < 1 beyond ∼250 au.

In the second simulation (Companion), ${{\rm{\Sigma }}}_{0}=1.96\,\times {10}^{-7}\,{M}_{\odot }\,{\mathrm{au}}^{-2}$, giving a total disk mass of $0.078\,{M}_{\odot }$, and Q > 2 at all radii.

We note that for the GI case the disk does evolve, such that ${\rm{\Sigma }}\propto {R}^{-0.5}$. For the Companion simulation we model an $8\,{M}_{\mathrm{Jup}}$ companion initially located at 500 au that is allowed to freely interact with the disk, migrate, and grow. At the time when the simulation is analyzed the companion is $\approx 10\,{M}_{\mathrm{Jup}}$ and located at ≈425 au from the central star. The companion does not drive a gap, which is consistent with the gap opening criteria defined by both Lin & Papaloizou (1986) and Crida et al. (2006). We note that the presence or absence of a gap makes little difference to our results. Each disk is modeled using 250,000 SPH gas particles, and we assume that the gas and dust are well mixed (see Meru et al. 2017 for further details).

We use the tache code, which utilizes tensor-classification of the simulations to determine which SPH particles reside in spiral structures (Forgan et al. 2016a, 2018b). Briefly, we compute the velocity shear tensor of each particle:

$$\begin{eqnarray}&&{\sigma }_{{ij}}=-\displaystyle \frac{1}{2}\left(\displaystyle \frac{\partial {v}_{i}}{\partial {x}_{j}}+\displaystyle \frac{\partial {v}_{j}}{\partial {x}_{i}}\right).\end{eqnarray} \tag{ 4 }$$

And then compute the tensor's eigenvalues. The number of positive eigenvalues E encodes information about the dimensionality of the flow. For example, particles with E = 1 indicate their motion is planar, consistent with motion in the undisturbed disk. Particles with E = 2 indicate 2D filamentary motions (in this case, spiral structure). If the spiral structure is strong enough, particles near the center of the arm will possess E = 3 (3D collapse).

We can therefore identify particles belonging to the spiral using their E value, and discard the other particles. This allows us to trace the spine of the spiral structure (i.e., the location of maximum density), using a friends-of-friends-like algorithm, which yields a set of (x, y) points for each individual spiral. Each arm is then fitted separately via χ² minimization to a variety of spiral models (assuming a constant uncertainty of σ = 0.1 au for all points). We use Nelder–Mead (amoeba) optimization to obtain said minimum χ², implemented via scipy.optimize.minimize.

Logarithmic spirals are commonly found in isolated GI disks and in disks driven by encounters with a companion:

$$\begin{eqnarray}&&r={{ae}}^{b\theta },\end{eqnarray} \tag{ 5 }$$

where r is cylindrical radius, and θ is the azimuthal angle. The a parameter determines the initial distance of the spiral from the origin, and b determines the winding properties of the arm. The pitch angle of a logarithmic spiral

$$\begin{eqnarray}&&\phi ={\left|r\displaystyle \frac{d\theta }{{dr}}\right|}^{-1}=\arctan b.\end{eqnarray} \tag{ 6 }$$

Pure logarithmic spirals (where b = b₀ is a constant) are typically found in simulations of isolated GI disks, with a constant ϕ ∼ 10°–15° (for q < 0.5 Cossins et al. 2009; Hall et al. 2016; Forgan et al. 2018b). We will label model fits of this type as PURELOG.

We also consider models where the pitch angle varies with radius, as is expected if the spiral is being driven by a companion (Goodman & Rafikov 2001; Rafikov 2002; Muto et al. 2012). For low-mass companions ($M\lessapprox 1\,{M}_{\mathrm{Jup}}$ in low-mass disks (${M}_{d}/{M}_{* }\lessapprox 0.1$), logarithmic spirals are typically found with the following function for b (Zhu et al. 2015),

$$\begin{eqnarray}&&b(r)={h}_{p}{\left(\displaystyle \frac{{r}_{p}}{r}\right)}^{1+\eta }\displaystyle \frac{{r}^{\alpha }}{{r}^{\alpha }-{r}_{p}^{\alpha }},\end{eqnarray} \tag{ 7 }$$

where r_p is the orbital radius of the companion ${\rm{\Omega }}\propto {r}^{-\alpha }$, ${c}_{s}\propto {r}^{-\eta }$, and h_p is the aspect ratio of the disk at the location of the companion.

We find that this function gives a very poor fit for both cases, as the spiral arms are deeply non-linear, due to the massive, self-gravitating nature of both disks (and the relatively high mass companion). Despite the lack of theoretical guidance in this regime, we still consider the possibility that the pitch angle does vary with radius. Instead, we fit a simpler function that describes a pitch angle that increases relatively slowly with radius (VARLOG),

$$\begin{eqnarray}&&b(r)={b}_{0}+c{\left(\displaystyle \frac{r-a}{a}\right)}^{n},\end{eqnarray} \tag{ 8 }$$

where b = b₀ at r = a, and the PURELOG solution is recovered for c = 0. We also check specifically whether a linear dependence of b with r is sufficient by running fits with n = 1 (VARLOGN). Finally, to double check that the spirals do indeed possess a logarithmic form, we also fit a power spiral (POW) function, where

$$\begin{eqnarray}&&r=a{\theta }^{n}\end{eqnarray} \tag{ 9 }$$

which also includes a radius-dependent pitch angle of the form

$$\begin{eqnarray}&&\phi =n{\left(\displaystyle \frac{r}{a}\right)}^{-1/n},\end{eqnarray} \tag{ 10 }$$

where for the special cases of n = ±1, the power spiral reduces to the Archimedean and hyperbolic spirals, respectively (modulo constants describing the spiral origin).

We find that for the isolated GI disk, pure logarithmic spirals of constant pitch angle (PURELOG) deliver a very good fit to the data. Other spiral functions yield reasonable fits to the arms (Figure 2), but all yield poorer χ² values than PURELOG for the upper arm, and deliver little change on the lower arm. POW fits produce quite poor fits at the inner and outer regions, strongly indicating that the arm's pitch angles are indeed constant with radius, as is the case for a PURELOG spiral. VARLOG/VARLOGN fits require extra parameters and fail to yield better fits, again indicating that a constant pitch angle is the simplest and most effective model fit to these arms.

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Such a result is in accordance with our expectation that spiral arms in GI disks are density waves, and should hence be pure logarithmic spirals (Cossins et al. 2009; Hall et al. 2016). It is also in accord with tache's previous applications to GI disks with disk-to-star mass ratios less than 0.5 (Forgan et al. 2018b). Note that the derived pitch angles for each arm are within 02 (1.4%) of each other. A hallmark of isolated GI disks are arms with extremely similar (if not identical) pitch angles.

A naive conclusion might therefore be that Elias 2-27 is an isolated GI disk, as it too possesses symmetric arms with low pitch angle. However, we must note that companion-driven simulations yield unsharp mask images that also appear to possess symmetric, tightly wound arms (Meru et al. 2017), and that Elias 2-27's arm symmetry may be an artifact of the fitting procedure (see Section 3.3). As we will see in the following section, the "ground-truth" morphology of companion-driven spirals is markedly different.

We can see from Table 1 that PURELOG fits for the companion-driven arms are significantly poorer compared to the VARLOG/VARLOGN fits. The minimum χ² solution for PURELOG achieves a reasonable fit at intermediate radii, but at the cost of poorly fitting points in the inner and outer disk (see Figure 3). This is heavily indicative of the fact that the pitch angle does indeed vary with radius, as is expected for companion-driven arms (Goodman & Rafikov 2001; Rafikov 2002; Muto et al. 2012; Zhu et al. 2015).

Notably, while the companion-hosting arm is poorly fitted, the secondary arm is much better fitted, with both arms possessing larger pitch angles than the GI run. The companion arm has a pitch angle of almost 17°, which is (just) beyond the 10°–15° range expected for isolated GI disks (Cossins et al. 2009). The secondary arm, at ϕ ≈ 15°, therefore appears similar to a typical GI density wave, making it less easy to distinguish.

It is worth noting that analyzing subsequent timesteps of the companion simulation yields a range of derived pitch angles for PURELOG, in some cases as large as ∼30° for the companion arm. We find throughout that the difference in pitch angles between the companion and secondary arm are much larger than determined for pairs of arms in the GI disk (Section 3.1).

We verify that the spiral form is indeed logarithmic by considering the POW fits. We find that we cannot improve our fits (indeed, they are slightly worse). Again, we find that the best-fit power spiral fails to correctly track the location of the companion.

The best fit to the companion spirals are from the VARLOG/VARLOGN fits. For both fixed n and varying n, we find fits to the companion arm that have χ² two to three times smaller than the PURELOG and POW fits. Allowing b to vary constrains the inner spiral much more effectively than the PURELOG fit, and from this analysis it seems clear that the companion arms are indeed logarithmic spirals, not power spirals. However, in contrast to the GI case, the logarithmic spirals are better fitted with a varying pitch angle as opposed to a constant pitch angle.

It is notable that for the VARLOG/VARLOGN fits our functional form for b results in a minimum in θ(r):

$$\begin{eqnarray}&&\theta (r)=\displaystyle \frac{1}{b(r)}\mathrm{ln}\left(\displaystyle \frac{r}{a}\right),\end{eqnarray} \tag{ 11 }$$

which occurs at

$$\begin{eqnarray}&&\displaystyle \frac{d\theta }{{dr}}=\displaystyle \frac{1}{{br}}-\displaystyle \frac{\theta }{b}\displaystyle \frac{{db}}{{dr}}=0.\end{eqnarray} \tag{ 12 }$$

As

$$\begin{eqnarray}&&\displaystyle \frac{{db}}{{dr}}={nc}{\left(\displaystyle \frac{r-a}{a}\right)}^{n}\displaystyle \frac{1}{r-a}=n\displaystyle \frac{b-{b}_{0}}{r-a}.\end{eqnarray} \tag{ 13 }$$

We therefore find a minimum in θ(r) at

$$\begin{eqnarray}&&\theta ={\left(r\displaystyle \frac{{db}}{{dr}}\right)}^{-1}=\displaystyle \frac{r-a}{r}\displaystyle \frac{1}{n(b-{b}_{0})}.\end{eqnarray} \tag{ 14 }$$

Hence we see that beyond this minimum, the fitted spiral function turns away from the companion. Note that attempts to fit this function on subsequent timesteps delivers a good fit to the companion arm, and not the secondary arm. Essentially, the length of the arm determines whether VARLOG/VARLOGN fits will capture the entire arm, or turn away at $\tfrac{d\theta }{{dr}}=0$.

This is an important issue for attempts to fit spiral structure driven by companions. Despite tache being able to roughly locate the companion inside the spiral structure, attempts to fit the structure with typical spiral functions uniformly fail to locate the companion correctly. We therefore urge caution when attempting to determine the location of an unseen companion using spiral structure alone. If the companion mass is much less than ∼0.02 times the stellar mass, the two arms can be significantly asymmetric, and the companion location can be determined using the two spiral arms alone (Fung & Dong 2015).

Both the GI and Companion simulations, when observed synthetically and subjected to the same unsharp mask imaging techniques as carried out by Pérez et al. (2016), broadly reproduce the observed spiral morphology of Elias 2-27 (Meru et al. 2017). However, our results identify key discriminators between the "ground-truth" spiral morphologies of the two cases.

If Elias 2-27 is an isolated, gravitationally unstable protostellar disk, then our simulations predict symmetric, logarithmic spiral arms of constant pitch ϕ ∼ 12°; these are consistent with expectations from density wave theory (Lin & Shu 1964; Bertin & Lin 1996), which are appropriate as for this case the disk-to-star mass ratio q ⪅ 0.5 (Cossins et al. 2009; Forgan et al. 2011; Hall et al. 2016). This is slightly larger than the measured pitch of ∼8° from Pérez et al. (2016), but we should be encouraged by the fact that our simulations produce a similar pitch angle without a great deal of tuning (as the simulations by Meru et al. were not intended to reproduce the exact ALMA image, but were testing which scenarios produced morphologies that were consistent with the observations).

If Elias 2-27 is stable against GI, but undergoing encounters with an external companion, our simulations show it should produce asymmetric arms with pitch angles that vary with radius, and a larger mean pitch overall. Due to the relatively large disk mass in both simulations (and the relatively massive companion in the Companion simulation), we find that the typical expressions for companion-driven arms (e.g., Zhu et al. 2015) are a poor fit for the spirals. We also find that other spiral functions (such as the power spiral) are generally a worse fit to the data.

The current ALMA observations indicate that Elias 2-27 does possess tightly wound, symmetric logarithmic spiral structure, suggesting that it is in fact a GI disk. However, we note that Pérez et al. (2016)'s fits to the arms assume symmetry, precluding the study of an important observable for determining the origin of spiral structure. We recommend that future observations of Elias 2-27 (and other disks with midplane-driven spiral arms) conduct fitting of individual arms rather than a single, simultaneous fit to all arms.