Planet-driven Spiral Arms in Protoplanetary Disks. II. Implications

An important feature in the planet-driven spiral arm formation mechanism is that the formation of both primary and additional arms can be understood as a linear process when the planet mass is sufficiently small. We can thus make use of linear wave theory to predict the phases of spiral arms. Here we provide a generalized analytic formula that can be used to fit observed spiral arms or to mimic spiral arms in models without carrying out planet–disk interaction simulations.

For low-mass planets (e.g., ≲gap-opening mass), the following phase equation provides the phases of both primary and additional arms as a function of radius:

$$\begin{eqnarray}{\phi }_{m,n}(r) & = & -\mathrm{sgn}(r-{r}_{p})\displaystyle \frac{\pi }{4m}+2\pi \displaystyle \frac{n}{m}\\ & & -{\displaystyle \int }_{{r}_{m}^{\pm }}^{r}\displaystyle \frac{{\rm{\Omega }}(r^{\prime} )}{{c}_{s}(r^{\prime} )}{\left|{\left(1-\displaystyle \frac{r{{\prime} }^{3/2}}{{r}_{p}^{3/2}}\right)}^{2}-\displaystyle \frac{1}{{m}^{2}}\right|}^{1/2}{dr}^{\prime} .\end{eqnarray} \tag{ 1 }$$

Here, r_p is the radius of the planet's circular orbit, ${r}_{m}^{\pm }\,={(1\pm 1/m)}^{2/3}\,{r}_{p}$ is the outer (with plus sign) and inner (with minus sign) Lindblad resonances, m is the azimuthal wavenumber, $n=0,1,\,\ldots ,\,m-1$ represents the individual wave modes, Ω is the disk rotational frequency, c_s is the sound speed, and r_p is the radius of the planet's circular orbit. As shown in Paper I, the phases of spiral arms follow the dominating azimuthal mode with $m\approx (1/2)(h/r{)}_{p}^{-1}$ well in the linear regime, where ${(h/r)}_{p}$ is the disk aspect ratio at the planet's orbit. The primary arm phase can be calculated with $m=2\pi {(h/r)}_{p}$ and n = 0. As can be inferred from Equation (1), one can simply shift the primary arm phase by $2\pi (n/m)$ in the azimuth for additional arms, where n = 1 and 2 for the secondary and the tertiary arms in the inner disk, and n = m−1 for the secondary arm in the outer disk. The derivation of Equation (1) can be found in Section 2 of Paper I.

Additional spiral arms form when wave modes having different azimuthal wavenumbers become in phase as they propagate. The propagation of wave modes depends on their azimuthal wavenumber but also the background disk temperature. For a given azimuthal wavenumber, a wave mode in a colder disk is more tightly wound and narrower in width. This raises the possibility that a larger number of additional arms can form. In addition, as shown in Paper I, the mass of the planet also affects the number of spiral arms.

In Figure 3, we present the number of spiral arms launched by planets in the inner ($0.2\,{r}_{p}\leqslant r\leqslant 1\,{r}_{p}$) and outer disk ($1\,{r}_{p}\leqslant r\leqslant 5\,{r}_{p}$) as a function of ${(h/r)}_{p}$ and M_p. Planets launch two or more (up to five in our parameter space) spiral arms interior to their orbits. At a given disk temperature, a smaller number of spiral arms form with a larger planet mass. Also, in general, a smaller number of spiral arms form in a disk with a larger ${(h/r)}_{p}$. As mentioned at the beginning of this section, this trend is expected because having a larger planet mass and/or a larger ${(h/r)}_{p}$ will make spiral arms (or wave modes) more open, and prevent more spiral arms forming. Our parameter study shows that, for $0.04\leqslant {(h/r)}_{p}\leqslant 0.15$, three or fewer spiral arms form interior to a planet's orbit when ${M}_{p}/{M}_{* }\,\gtrsim 3\times {10}^{-4}$, and two-armed spirals form when ${M}_{p}/{M}_{* }\gtrsim 3\,\times {10}^{-3}$. These masses correspond to a Saturn mass and 3 Jupiter masses, respectively, assuming a solar-mass central star.

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Exterior to their orbits, we find that planets launch only one or two spiral arms. The number of outer spiral arms appears to be more sensitive to the disk temperature than the planet mass. Fewer spiral arms form in the outer disk than in the inner disk because the propagation of wave modes become independent of the azimuthal wavenumber when $r\gg {r}_{p}$ (see Equation (1)). Therefore, additional spiral arms can form only when non-zero nth components become in phase within $r\lesssim {\rm{a}}\,\mathrm{few}\,{r}_{p}$, which can occur in cold disks. Our parameter study shows that two spiral arms form exterior to a planet's orbit only when ${(h/r)}_{p}\lesssim 0.06$.

In Figure 4, we present an example of which the largest number of spiral arms in our parameter study are launched: five in the inner disk and two in the outer disk. The parameters used in the example are ${(h/r)}_{p}=0.04$ and ${M}_{p}=0.3\,{M}_{\mathrm{th}}$. In addition to the primary arm directly attached to the planet, interior to the planet's orbit a secondary arm forms at $\sim 0.8\,{r}_{p}$, a tertiary arm forms at $\sim 0.6\,{r}_{p}$, a quaternary arm forms at $\sim 0.4\,{r}_{p}$, and a quinary arm forms at $\sim 0.3\,{r}_{p}$. Exterior to the planet's orbit, a secondary arm forms at $\sim 1.5\,{r}_{p}$ and no more spiral arms form beyond this radius. With a small planet mass of $0.3\,{M}_{\mathrm{th}}$ (=6.4 Earth mass assuming a solar-mass star), the level of perturbation driven by the spiral arms is low ($\delta {\rm{\Sigma }}/{{\rm{\Sigma }}}_{\mathrm{init}}\lesssim 0.1$). These spiral arms are therefore unlikely to be detectable in near-IR scattered light images (Dong & Fung 2017). While weak, however, these spiral arms may still be able to generate observable signatures preferentially in low-viscosity disks by opening gaps as they shock the disk gas, provided that sufficient time is allowed (Bae et al. 2017; see also Section 5 of the present paper).

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In Figure 5, we present the measured pitch angles of spiral arms from our simulations. The pitch angles β are measured using $\tan \beta =-{dr}/{rd}\phi$ as we follow each spiral arm in radius. In general, for any given disk temperature, larger planet masses result in larger pitch angles. So, in theory we can use the pitch angle to constrain the mass of an unseen planet. In practice, however, there are difficulties. First of all, we note that highly accurate temperature measurements are required. As shown in Figure 5, even a 30% uncertainty in temperature can change the mass estimates by an order of magnitude. Given the difficulty of measuring the actual midplane gas temperature—arising for instance from relating mm continuum emissions to gas temperatures, or from using molecular line ratios which tend not to probe exactly the same disk regions in height—making a one-to-one correlation between the pitch angle of a spiral arm to the planet mass seems challenging, at least for now. Even when the disk midplane temperature is accurately measured, an additional uncertainty in the mass constraints comes from the fact that we do not know the location of the planet and thus the exact ${(h/r)}_{p}$ value. Many of the spiral arms observed so far have fluctuating pitch angles (e.g., Reggiani et al. 2018), another challenge to use the pitch angle to constrain planet mass.

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On the other hand, as shown in Figure 5, the pitch angles of both the primary and additional spiral arms decrease away from the planet. This is expected from linear wave theory, and numerical simulations also show the trend. It is a particularly important characteristic because, if we measure the pitch angle of a spiral arm over a large enough radial range, the increasing or decreasing trend of the pitch angle can be used to distinguish whether the spiral arm is excited by a planet inside or outside of the spirals (see Section 4.1).

Previous numerical simulations showed that the separation between the primary and secondary arm increases with the planet mass (Fung & Dong 2015; Zhu et al. 2015; Lee 2016). Interestingly, a parameter study presented in Fung & Dong (2015) showed that the arm-to-arm separation is independent of the disk temperature profile. It is thus suggested that the arm-to-arm separation can be broadly used to infer the mass of an unseen planet.

We compute arm-to-arm separations from our simulations. Specifically, we measure the angular separation between spiral arms at each radius. We then average it around r = 0.2, 0.3, 0.4, 0.5, and 0.6 r_p for the primary-to-secondary separation and around r = 0.1, 0.2, and 0.3 r_p for the secondary-to-tertiary separation, with a radial bin of ${\rm{\Delta }}r=0.1{r}_{p}$. Focusing on ${(h/r)}_{p}=0.1$ models first, we present the primary-to-secondary separation ${\rm{\Delta }}{\phi }_{p-s}$ and the secondary-to-tertiary separation ${\rm{\Delta }}{\phi }_{s-t}$ as a function of the planet mass in Figure 6. For both ${\rm{\Delta }}{\phi }_{p-s}$ and ${\rm{\Delta }}{\phi }_{s-t}$, there is a general trend that the arm-to-arm separation increases with the planet mass, consistent with previous studies (Fung & Dong 2015; Zhu et al. 2015). For ${\rm{\Delta }}{\phi }_{p-s}$, it converges to ∼180° beyond ${M}_{p}\gtrsim 3\,{M}_{\mathrm{th}}$, for which planet mass only two spiral arms form. For the planet masses in a range of ${M}_{p}\geqslant 0.1\,{M}_{\mathrm{th}}$ (${M}_{p}/{M}_{* }\geqslant {10}^{-4}$), the primary-to-secondary arm separation at $0.2\,{r}_{p}\leqslant r\leqslant 0.6\,{r}_{p}$ is best fitted with ${\rm{\Delta }}{\phi }_{p-s}=101^\circ {({M}_{p}/{M}_{\mathrm{th}})}^{0.20}$. However, we note that the arm-to-arm separation increases more steeply with the planet mass at smaller radii. The best-fit slopes for individual radial bins increase from 0.11 at 0.6 r_p to 0.22 at 0.3 r_p. Similarly, the slope in ${\rm{\Delta }}{\phi }_{s-t}$ also increases more steeply at smaller radii. It is therefore helpful to know an approximate planet location for more accurate mass estimates with arm-to-arm separation.

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There are a few other things worth pointing out from Figure 6. First, compared at a given radius, ${\rm{\Delta }}{\phi }_{p-s}$ is larger than ${\rm{\Delta }}{\phi }_{s-t}$ regardless of the planet mass. Second, both ${\rm{\Delta }}{\phi }_{p-s}$ and ${\rm{\Delta }}{\phi }_{s-t}$ flatten out for sufficiently low-mass planets, suggesting that spiral arms cannot be infinitely close to each other in the linear regime. Last, for a given planet mass, ${\rm{\Delta }}{\phi }_{p-s}$ increases as a function of radius for low-mass planets (i.e., ${M}_{p}\lt 0.3\,{M}_{\mathrm{th}}$), while the separation decreases as a function of radius for high-mass planets (i.e., ${M}_{p}\gt 0.3\,{M}_{\mathrm{th}}$).

In Figure 7, we present ${\rm{\Delta }}{\phi }_{p-s}$ and ${\rm{\Delta }}{\phi }_{s-t}$ measured in simulations with ${(h/r)}_{p}=0.05$ and ${(h/r)}_{p}=0.15$. Comparing Figure 7 with Figure 6, we note that the general trends seen in the fiducial case with ${(h/r)}_{p}=0.1$ are commonly seen with ${(h/r)}_{p}=0.05$ and 0.15. Most importantly, the arm-to-arm separation increases more steeply with the planet mass at smaller radii with other ${(h/r)}_{p}$ values. It is worth noting that this trend is more clearly seen with larger ${(h/r)}_{p}$ values because the interference and/or merging between spiral arms tends to occur more frequently with smaller ${(h/r)}_{p}$ values.

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In Figure 8(a), we present ${\rm{\Delta }}{\phi }_{p-s}$ computed in all our models differing M_p and ${(h/r)}_{p}$. The data points are color-coded to represent ${\rm{\Delta }}{\phi }_{p-s}$ at different radial bins. Note that the x axis of the plot is now planet-to-star mass ratio ${M}_{p}/{M}_{* }$, different from Figures 6 and 7. We fit the data points ${M}_{p}/{M}_{* }\geqslant {10}^{-4}$, having chosen similarly to Fung & Dong (2015) for a comparison. The entire data points are best fitted with Δϕ_p−s = 106° (q/0.001)^0.21 where $q\equiv {M}_{p}/{M}_{* }$, which is in excellent agreement with Fung & Dong (2015). However, as in Figures 6 and 7, we find there is a general trend that the arm-to-arm separations increase more steeply at smaller radii. In Figure 8(b), data points are now color-coded to represent different ${(h/r)}_{p}$ in models. Again, we do find a trend that the arm-to-arm separation increases more steeply with smaller ${(h/r)}_{p}$ values. In Table 1, we provide the best fits to the ${\rm{\Delta }}{\phi }_{p-s}-{M}_{p}/{M}_{* }$ relation. A fit given for a certain radius uses the data points with all ${(h/r)}_{p}$ values, whereas a fit given for a certain ${(h/r)}_{p}$ value uses the data points at all different radii. In Table 2 of the Appendix, we provide best fits for each ${(h/r)}_{p}$ values and radial bins.

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Table 1.  Best Fits to Δϕ_p−s − M_p/M_* Relation

Data	Best Fits
Entire Data Points	Δϕp−s = 106° (q/0.001)0.21
r = 0.2 rp	Δϕp−s = 110° (q/0.001)0.25
r = 0.3 rp	Δϕp−s = 104° (q/0.001)0.26
r = 0.4 rp	Δϕp−s = 109° (q/0.001)0.22
r = 0.5 rp	Δϕp−s = 104° (q/0.001)0.19
r = 0.6 rp	Δϕp−s = 104° (q/0.001)0.14
${(h/r)}_{p}=0.04,0.05,0.06$	Δϕp−s = 117° (q/0.001)0.29
${(h/r)}_{p}=0.07,0.08$	Δϕp−s = 102° (q/0.001)0.24
${(h/r)}_{p}=0.1$	Δϕp−s = 101° (q/0.001)0.20
${(h/r)}_{p}=0.12$	Δϕp−s = 101° (q/0.001)0.16
${(h/r)}_{p}=0.15$	Δϕp−s = 119° (q/0.001)0.10

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To summarize, we confirm that the arm-to-arm separation increases with the planet mass, in agreement with previous studies (Fung & Dong 2015; Zhu et al. 2015; Lee 2016). Our best fit of primary-to-secondary separation is ${\rm{\Delta }}\phi =106^\circ {(q/0.001)}^{0.21}$. However, we find that, in the inner disk, the arm-to-arm separation increases more steeply with the planet mass at smaller radii and with smaller ${(h/r)}_{p}$ values.

First of all, secondary and tertiary arms can create stronger density perturbations/contrast than the primary arm at a given radius (e.g., Figure 4; see also Fung & Dong 2015). In addition, the brightness/intensity of spiral arms can significantly differ from their intrinsic brightness/intensity because of interaction with background disk structures (e.g., a vortex; Bae et al. 2016b) and/or distortion via the spiral wave instability (Bae et al. 2016a). In the case of near-IR scattered light observations, shadows cast by structures located closer to the central star may also affect the brightness of spiral arms located far away (Stolker et al. 2016). We thus caution that the observed intensity or brightness of spiral arms is not an ideal characteristic to determine whether a certain spiral arm is a primary or a secondary/tertiary, but also to provide planet mass constraints. Variations in the brightness of spiral arms seen in multi-epoch scattered light observations (e.g., Stolker et al. 2016, 2017) also support this conclusion.

MWC 758 is a $3.5\pm 2$ Myr old Herbig Ae star (Meeus et al. 2012) at a distance of ${151}_{-8}^{+9}$ pc (Gaia Collaboration et al. 2016). Two spiral arms are detected in near-IR scattered light observations using the Subaru Telescope High Contrast Instrument with Adaptive Optics in K_s- and H-band (Grady et al. 2013) and the VLT with Spectro-Polarimetric High-contrast Exoplanet Research in Y-band (Benisty et al. 2015). Recently, using L'-band observations with NIRC2 at the Keck II telescope, Reggiani et al. (2018) reported three spiral arms (two previously reported and one new) spanning in between ∼025 and ∼06 (∼40–90 au) from the central star. In addition, a point-like source is detected at a separation of 011 (∼20 au) from the central star (Reggiani et al. 2018), making the disk a very interesting object to test planet–disk interaction theories.

To examine a potential planetary origin of the spiral arms in the MWC 758 disk, we first assume an external companion at 100 au and $h/r=0.12$ at the radius, similar to the model in Dong et al. (2015), but rescaled with the distance based on the Gaia Collaboration et al. (2016). Assuming that the external companion is responsible for all three spiral arms, our parameter study suggests that the unseen planet should have a mass less than 3 thermal mass; a larger-mass planet would excite only two spiral arms. Adopting a 1.5 solar-mass central star (Isella et al. 2010; Reggiani et al. 2018) gives an upper limit of 7.8 Jupiter mass. An interesting feature regarding the arm-to-arm separation is that the separation between the arm S1 and S3 decreases with radius (see Figure 7 of Reggiani et al. 2018). Although a radially decreasing arm-to-arm separation is not completely unexpected as Figure 6 shows (see M_p = 3 and 10 M_th, for example), such a small arm-to-arm separation of ∼45° as well as a rapid drop over a short radial distance is not what is typically seen our simulations—assuming again a companion at 100 au as above, the arm-to-arm separation decreases from 140° at $\sim 0.25\,{r}_{p}$ to 45° at $\sim 0.63\,{r}_{p}$. One possibility to explain these features might be interference between the arms (Paper I), but our understanding of the phenomenon is too incomplete to arrive at a conclusion.

We note that the point-like source detected in Reggiani et al. (2018) is unlikely to be the perturber exciting three spiral arms exterior to its orbit based on our parameter study. However, it is possible that the point-like source excites one of the three spiral arms and a yet unseen external companion excites the other two. In fact, we find some supporting features that S3 could be an outer spiral of a planet interior to S3. First, the decreasing pitch angle of S3 as a function of radius (Figure 5 of Reggiani et al. 2018) supports the idea that S3 is an outer spiral arm launched by an internal perturber (see Section 3.2). This also helps to resolve the arm-to-arm separation problems discussed in the previous paragraph. In addition, it is known from three-dimensional simulations that inner spiral arms produce significant vertical motion while the outer spiral arms induce little vertical motion (Zhu et al. 2015). If S3 is an outer spiral arm it is likely that the pressure scale height along the arm does not increase significantly. This can explain the non-detection of S3 at a shorter wavelength in the Y-band (1.04 μm; Benisty et al. 2015), as this wavelength traces upper layers of the disk than the L'-band-band does. This scenario is consistent with a suggestion made in Reggiani et al. (2018) and also supports the suggestion by Juhász et al. (2015) who observed that spiral arms might be the result of pressure scale height changes. While no apparent physical connection between the point-like source and S3 is found from the scattered light image, it is possibly because the point-like emission traces a protoplanet located near the midplane whereas S3 traces the surface. This explanation is consistent with the inclination and PA of the disk. In the case where an unseen external planet excites two of the three spiral arms, the external companion has to have more mass than a thermal mass (see Section 3.1). We are thus left with a fairly narrow companion mass range of 2.6–5 Jupiter mass: the lower limit from our parameter study and the upper limit from Reggiani et al. (2018).

When the three spiral arms do not share the same origin, distinguishable orbital motions among the spirals are expected in monitoring observations. If spiral arm S3 is excited by the point-like source at 20 au, the spiral pattern will rotate 49 per year in the deprojected plane. Assuming an external planet at ≳100 au is responsible for the other two spiral arms, the two spirals will rotate ≲04 per year in the deprojected plane. Therefore, observations in 2018 may reveal more than >10° of difference in the orbital motions between spiral arm S3 and the other two compared with the first epoch observation of Reggiani et al. (2018) taken in 2015 October.

Elias 2–27 is a young (∼1 Myr; Luhman & Rieke 1999), low-mass (0.6 solar-mass; Andrews et al. 2009) star in the ρ-Ophiuchus star-forming region. This object is particularly interesting because spiral arms are detected at mm wavelengths with ALMA (Pérez et al. 2016). The emission at 1.3 mm is optically thin, so the continuum emission traces down to the midplane of the disk (Pérez et al. 2016), in contrast to near-IR scattered light imaging, which traces spiral arms near the disk surface.

Both spiral arms in the Elias 2–27 disk appear to be well fit with two symmetric logarithmic spirals, i.e., a constant pitch angle of 79 ± 04 (Pérez et al. 2016). This is an interesting characteristic because a constant pitch angle is unlikely for planet-driven spiral arms (Figure 5; see also Zhu et al. 2015). One possibility is that the disk has a steeper temperature gradient than the constraints made in Pérez et al. (2016). In Figure 9 we plot the predicted pitch angles as a function of radius based on the linear wave theory, assuming ${(h/r)}_{p}\,=0.15$. As shown, the steeper the disk temperature gradient is, the more slowly the pitch angle varies in the inner disk. If it is the temperature gradient alone though, the planet probably has to be located at a very large distance (≳500 au) to help its inner spiral arms have a constant pitch angle. More accurate temperature measurements in the future could help to test whether this is the case. If the two spiral arms are driven by a planet exterior to the spirals, our parameter study suggests that the planet has to have more mass than a Jupiter mass, assuming the midplane temperature constrained by Pérez et al. (2016). This planetary mass is consistent with the constraints made by previous observations and numerical simulations (see Meru et al. 2017 and references therein). The disk has $h/r\,\geqslant 0.06$ beyond ∼10 au, so it is very unlikely that an internal planet excites the two spiral arms.

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Alternatively, it might be that a constant pitch angle is a generic characteristic of GI-driven spiral arms. In fact, the GI-driven spiral arms seen in the numerical simulations of Bae et al. (2014) have a constant pitch angle over a broad range of radius, in particular, inward of the gravitationally unstable region.⁵ If the shear rate of the disk rotation is what determines the pitch angle of the GI-driven spiral arms, the potential universality of a constant pitch angle among GI-driven spiral arms might be explained. Pérez et al. (2016) calculated the Toomre Q parameter based on the dust continuum emission and suggested that the disk can be gravitationally unstable at the radii where the spiral arms are detected, when the dust opacity is reduced by a factor of ≳4 than what is typically assumed in literature. More accurate observational constraints on the gas surface density, as well as improvements in our understanding of GI-driven spiral arms, are desired to better understand the origin of the spiral arms in the Elias 2–27 disk.

In order for spiral arms to be observable, they have to (1) have a large enough pitch angle to overcome a finite spatial resolution (see e.g., Figure 4 of Kanagawa et al. 2015); and (2) produce a sufficient level of perturbations (Dong & Fung 2017). For a given disk property the two conditions can be more readily met with a large planet mass, since spiral arms from a planet with larger mass have a larger pitch angle (Figure 5) and produce a larger level of perturbations (Figure 2). While Dong & Fung (2017) noted that assessing the detectable planet mass limit generally requires a case-by-case study, their empirical scaling relation between arm-to-disk contrast and planet mass suggests that multi-Jupiter-mass planets are required to produce the arm-to-disk contrast of the observed spiral arms (e.g., MWC 758, SAO 206462). In agreement with the arm-to-disk contrast argument, hydrodynamic modeling of observed spiral arms suggests that an order of 10 Jupiter-mass planets are required to reproduce the morphology of nearly axisymmetric m = 2 spiral arms: $9\,{M}_{\mathrm{Jup}}$ for MWC 758 (Dong et al. 2015); $10\,{M}_{\mathrm{Jup}}$ for SAO 206462 (Bae et al. 2016b); and Elias 2–27 (Meru et al. 2017).

The disk viscosity can also affect the observability of spiral arms versus rings and gaps. The morphology of planet-driven spiral arms and the arm-to-disk contrast are shown to be not very sensitive to disk viscosity (Dong & Fung 2017). On the other hand, spiral arms are not able to open gaps when the mass transport by the background disk turbulence exceeds that by spiral shocks. While the parameter space has not been fully explored yet, the results in Bae et al. (2017) suggest that a viscosity of $\alpha \lesssim {10}^{-3}$ is required for a Jupiter-mass planet to create multiple rings and gaps. Thus, rings and gaps can be observed preferentially in disks with low viscosity.

The method through which substructures are probed is also important. Probing disk surface layers using near-infrared scattered light or optically thick line emissions will offer a higher chance of detecting spiral arms, because significantly larger perturbations are expected from spiral arms at the disk surface than at the disk midplane (Zhu et al. 2015). In contrast, (sub-)mm dust continuum traces deep in the disk near the midplane, where the perturbations driven by spiral arms are intrinsically smaller. In addition, as millimeter-sized particles experience more significant aerodynamic drag than μm-sized particles, they can be more efficiently collected in pressure bumps rather than collected at the spiral arm front.

Lastly, the timescale also matters. Spiral arms launch and reach a quasi-steady state within a sound-crossing time, which is typically a few planetary orbital time. On the other hand, rings and gaps require a much longer time to fully develop (≳100 orbital times, Bae et al. 2017).

Meteorites contain calcium–aluminum-rich inclusions (CAIs), millimeter- to centimeter-sized particles believed to be formed very early in the solar nebula (e.g., Connelly et al. 2012). Interestingly, meteorite parent body accretion is known to have continued for 2–4 Myr after the formation of CAIs (Connelly et al. 2012; Kita & Ushikubo 2012; Budde et al. 2016b). Millimeter- to centimeter-sized particles would experience very rapid inward drift through aerodynamic drag. It has been suggested that one way to store CAIs in the solar nebula for millions of years is if they were trapped in local pressure maxima until they accumulate in to meteorite parent bodies (Desch et al. 2017). Moreover, there are different types of meteorites, each of which has distinct chemical compositions and CAI sizes and abundances, suggesting that there might have been multiple spatially separated reservoirs of solid particles in the solar nebula (Warren 2011; Budde et al. 2016a; Kruijer et al. 2017).

Here we examine the possibility that Jupiter's growing core could create multiple pressure maxima in the solar nebula, which could have acted as solid particle reservoirs. In Bae et al. (2017), we showed that spiral arms driven by a planet (or a planetary core) can create multiple gaps in the disk. Pressure maxima (i.e., rings) can develop between those gaps, potentially trapping solid particles.

Following the conclusion of Kruijer et al. (2017) that Jupiter's core had to grow quickly to 20 M_⊕ within 1 Myr, making it the oldest planet of the solar system, we carry out a numerical simulation with a 20 M_⊕ core located at 5 au. We adopt a minimum mass solar nebula surface density profile (Weidenschilling 1977; Hayashi 1981) and a passively heated, stellar irradiation-dominated temperature profile: $T(r)\,=180\,{\rm{K}}{(r/1\mathrm{au})}^{-1/2}$. With this disk model ${(h/r)}_{p}=0.04$ at 5 au, and the 20 M_⊕ Jupiter's core is 94% of a thermal mass. We use the inner and outer boundaries of 0.5 and 25 au and adopt 1024 logarithmically spaced grid cells in radius and 1640 uniformly spaced grid cells in azimuth.

The radial gas pressure distribution after 200 kyr (∼18,000 orbits) of evolution is shown in Figure 10. The core excites five spiral arms interior to its orbit and two spiral arms exterior to its orbit. Each spiral arm opens a gap through shock dissipation, and pressure maxima form in between the gaps. In total, the core creates four pressure bumps interior to its orbit and two pressure bumps exterior to its orbit.

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To test whether these pressure bumps are strong enough to trap solid particles, we follow the analytic approach presented in Zhu et al. (2012). To briefly summarize, in order for a particle to be trapped in a pressure bump the drift velocity toward the pressure maximum has to be greater than two velocities: (1) the diffusion velocity by gas turbulence; and (2) the enhanced radial gas velocity within the gap caused by the deficit of gas. The second requirement assumes a constant mass accretion rate across the ring and gap. One assumption that has to be made to use this analytic approach is that the shape of the additional gaps are similar to the main gap within which the planet orbits, as described by Equation (24) in Zhu et al. (2012).

Using ${T}_{s,0}={\rho }_{p}s\pi /(2{{\rm{\Sigma }}}_{g,0})$, Equation (27) of Zhu et al. (2012) can be rearranged as

$$\begin{eqnarray}&&{\alpha }_{\mathrm{diff}}=\displaystyle \frac{{\rho }_{p}s\pi }{2{{\rm{\Sigma }}}_{g}}{\left[\mathrm{ln}\left(\displaystyle \frac{{\gamma }_{0}}{\gamma }\right)\right]}^{-1},\end{eqnarray} \tag{ 2 }$$

where ${\rho }_{p}$ is the solid particle density, s is the radius of the solid particle, ${{\rm{\Sigma }}}_{g}$ is the gas surface density, γ₀/γ is the dust depletion factor. Now, let us consider a mm-sized particle placed right outside of the inner tertiary gap at ∼2.3 au to examine whether it will be dragged into the inner disk or dragged outward to be trapped at the inner secondary pressure bump at ∼2.8 au, as an example. Considering a spherical solid particle with a density of ${\rho }_{p}=3\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and a radius of $s=1\,\mathrm{mm}$, and adopting a disk gas density of ${{\rm{\Sigma }}}_{g}=500\,{\rm{g}}\,{\mathrm{cm}}^{-2}$, a disk viscosity of ${\alpha }_{\mathrm{diff}}\gtrsim 1.4\times {10}^{-4}$ is required for the particle to penetrate the tertiary gap via turbulent diffusion assuming a dust depletion factor of ${\gamma }_{0}/\gamma =1000$ as in Zhu et al. (2012):

$$\begin{eqnarray}{\alpha }_{\mathrm{diff}} & = & 1.4\times {10}^{-4}\left(\displaystyle \frac{{\rho }_{p}}{3\,{\rm{g}}\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{s}{1\,\mathrm{mm}}\right){\left(\displaystyle \frac{{{\rm{\Sigma }}}_{g}}{500{\rm{g}}{\mathrm{cm}}^{-2}}\right)}^{-1}\\ & & \times {\left(\displaystyle \frac{\mathrm{ln}({\gamma }_{0}/\gamma )}{\mathrm{ln}(0.001)}\right)}^{-1}.\end{eqnarray} \tag{ 3 }$$

As shown in the above equation, ${\alpha }_{\mathrm{diff}}$ increases inversely proportional to the gas surface density for a given solid particle density. Therefore, when $\alpha \lesssim 1.4\times {10}^{-4}$ all the pressure bumps beyond 2 au satisfy this first condition to trap solid particles with sizes of a mm or greater.

One can do the same experiment for the second condition about the enhanced gas velocity. Rearranging Equation (30) of Zhu et al. (2012), we obtain

$$\begin{eqnarray}&&{\alpha }_{\mathrm{egv}}=\left|\mathrm{ln}\left(\displaystyle \frac{{{\rm{\Sigma }}}_{g}}{{{\rm{\Sigma }}}_{g,0}}\right)\right|{\left(\displaystyle \frac{{\mu }^{2}}{9\pi }\right)}^{1/4}{\left(\displaystyle \frac{{\rho }_{p}s\pi }{{{\rm{\Sigma }}}_{g,0}}\right)}^{3/4}{\left(\displaystyle \frac{h}{r}\right)}^{-1/2},\end{eqnarray} \tag{ 4 }$$

where $\mu ={M}_{p}/{M}_{* }$. Assuming again canonical values for a mm-sized particle placed right outside of the inner tertiary gap at ∼2.3 au with a 10% of gas depletion in the gap (i.e., ${{\rm{\Sigma }}}_{g}/{{\rm{\Sigma }}}_{g,0}=0.9$), a disk viscosity of ${\alpha }_{\mathrm{egv}}\gtrsim 1.8\times {10}^{-5}$ is required for the particle to penetrate the tertiary gap:

$$\begin{eqnarray}{\alpha }_{\mathrm{egv}} & = & 1.8\times {10}^{-5}\left(\displaystyle \frac{| \mathrm{ln}({{\rm{\Sigma }}}_{g}/{{\rm{\Sigma }}}_{g,0})| }{| \mathrm{ln}(0.9)| }\right){\left(\displaystyle \frac{\mu }{6\times {10}^{-5}}\right)}^{1/2}\\ & & \times {\left(\displaystyle \frac{{\rho }_{p}}{3{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{3/4}{\left(\displaystyle \frac{s}{1\mathrm{mm}}\right)}^{3/4}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{g,0}}{500{\rm{g}}{\mathrm{cm}}^{-2}}\right)}^{-3/4}\\ & & \times {\left(\displaystyle \frac{h/r}{0.033}\right)}^{-1/2}.\end{eqnarray} \tag{ 5 }$$

Based on Equations (3) and (5), we can infer that solid particles with sizes of $\geqslant 1\,\mathrm{mm}$ can remain trapped in the inner primary and secondary and the outer primary and secondary pressure bumps when the disk viscosity is sufficiently low: $\alpha \lesssim {10}^{-5}$. We note, however, that this is an α value associated with the radial mass transport only, because the analysis presented above considers only the radial movement of gas and dust. Since disk turbulence can be anisotropic such that vertical stress is significantly stronger than the radial stress (e.g., Stoll et al. 2017), we caution that this value should not be taken as a representative value for the total disk turbulence.

It has to be noted that the simulation presented in this section assumes zero kinematic disk viscosity.⁶ Using a non-zero kinematic viscosity will result in less prominent pressure bumps. On the other hand, the core will generate stronger spiral arms as it grows to a full Jupiter-mass and can create strong enough pressure bumps to trap solid particles even in the presence of a moderate viscosity (e.g., $\alpha \sim {\rm{a}}\,\mathrm{few}\times {10}^{-4};$ Bae et al. 2017).

Having multiple pressure bumps helps prevent solid particles from rapidly drifting inward, but their different strengths and locations in the disk can naturally explain the range of sizes and abundances of CAIs (and chondrules) seen in meteorites. Follow-up studies, which incorporate growth of the core with more realistic disk thermal evolution and gas–dust interaction, are certainly required to further examine this scenario. Nevertheless, the formation of multiple pressure bumps by a growing Jupiter seems to be an intriguing possibility to explain the meteoritic property in the solar system and deserves further consideration.