Radiative effects in gap opening by planets in protoplanetary disks

Alexandros Ziampras (Uni Tübingen), Wilhelm Kley (Uni Tübingen), Cornelis P. Dullemond (ITA, Heidelberg)

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ALMA observations and past numerical modeling

The DSHARP survey revealed a variety of substructures such as rings and gaps and opened possiblities to test the planet–disk interaction theory. These structure-rich systems show resolved features at the 50–100 au scale, where the disks are likely passive, optically thin, and starlight-heated. This suggests short cooling timescales and makes modeling these systems with a locally isothermal equation of state reasonable. For AS 209, the gap structure of the resolved disk could be reconstructed with a single planet exceptionally well assuming locally isothermal conditions.


Young planets inside a gaseous disk can interact gravitationally with the disk, launching spirals that carry angular momentum away from the planet. This mechanism can lead to gap opening, and the formation of bright rings in continuum emission. Miranda & Rafikov (2019) showed that, in disks with embedded planets, the angular momentum flux carried by spirals depends on the cooling time per orbital period $\beta$ such that a disk can be considered "locally isothermal" for $\beta < 10^{-3}\text{–}10^{-2}$ depending on planet mass. At the same time, ALMA observations also suggest short cooling timescales around this range.

So the question is: How short is "short enough"?

We would like to examine the planet–disk interaction scenario if we are to attribute the observed ring/gap structures to young planets and understand the impact of the cooling timescale. To that end, we aim to test the locally isothermal assumption by modeling radiative effects such as heating and cooling and comparing the two equations of state to each other.


We model HD 163296 and AS 209 with PLUTO to reconstruct their ring/gap structures. We run both locally isothermal and radiative models with embedded planets, with initial profiles based on DSHARP. Our radiative models include a treatment for viscous/shock/irradiation heating and thermal cooling.

Simulations (click on an image to enlarge!)

A comparison between locally isothermal and radiative disk models of HD 163296 with a single, $0.5 M_\text{J}$ planet at 48$\,$au. A secondary gap around 26 au and a stronger spiral arm contrast are evident in the locally isothermal model.

A comparison between locally isothermal and radiative disk models of AS 209 with a single, $0.1 M_\text{J}$ planet at 99$\,$au. A secondary gap is clearly visible around 50 au, and the primary gap structure is vastly different in the locally isothermal model.

A radiative model of HD 163296 with 3 planets embedded at 48, 86 and 145 au. The innermost 60 au show the same structure as our single-planet runs.

An attempt to reconstruct the locally isothermal limit by tweaking our parameters. The tweaked models closely resemble the locally isothermal results.

What's happening?

Even though the presence of the planet doesn't affect the disk's temperature profile, radiative effects can still impact the planet's gap opening capacity. That is because, while the short cooling timescale prevents shocks from heating up the disk, it's still long enough to affect the angular momentum flux of the planet's spirals. Incidentally, these two systems show a cooling timescale profile that hovers around the range suggested by Miranda & Rafikov in the inner 50 and 120 au for HD and AS, respectively. As a result, the two equations of state show different results at those regions!


We find that locally isothermal models overestimate the angular momentum flux carried away by spirals, and therefore enhance planet-generated features with respect to radiative models:

  • gaps are deeper for locally isothermal disks compared to radiative ones,
  • a single planet can open multiple gaps within its orbit, and
  • spirals have slightly higher contrast, enhancing pressure bumps.

All of the above points support dropping the locally isothermal assumption in favor of a parameterized cooling prescription or a self-consistent model of gas cooling.


In general, the locally isothermal assumption proves to be dangerous even at the range of tens of au regarding planet–disk interaction and should therefore be avoided in favor of an adiabatic equation of state with a prescription for radiative cooling in the disk. The results between different equations of state converge in the limit of very short cooling timescales (0.1%-1% of the orbital timescale depending on planet mass), in good agreement with Miranda & Rafikov (2019).

Our results are consistent with previous studies and imply that a single planet cannot always explain the existence of multiple gaps. With that in mind, different possibilities can be explored to explain the origin of multiple gaps in the systems observed by ALMA.


ALMA Partnership (2018). ADS, DSHARP data
Zhang, S., et al. ApJ, 869, L47 (2018). ADS
Miranda, R. & Rafikov, R. R., ApJ, 878, L9 (2019). ADS
Miranda, R. & Rafikov, R. R., ApJ, 892, 1 (2020). ADS

More info

Our paper on this!
Model details: physics and numerics
Model details: contribution of shock heating
Results details: spiral arms
Extra details: an estimate of the cooling timescale
The fine print