31. Efficient radiative transfer techniques in hydrodynamic
simulations
Mercer et al. Accepted by MNRAS

https://arxiv.org/pdf/1805.09568.pdf

The aim of this paper is to compare how the above methods (Stamatellos et al. 2007b;
Lombardi et al. 2015) behave when applied in actual hydrodynamic simulations.
To calculate the optical depth through which
the gas heats and cools:(i) Stamatellos method uses thegravitational potential and the density(ii) Lombardi method instead uses thepressure scale-height.

(a)a low-mass relaxed disc(b)a high-mass disc(c)a disc with spirals arms(d) a disc with dense clumps(e) a disc with an embedded 1.4 MJ planet(f) a disc with an embedded higher-mass 11 MJ planet[X®]Lombardi method is more accurate
when considering protostellar discs.
Stamatellos methodopticaldepth,+  overestimate (cooling ]§-> Lombardi method&(+E cRelaxed low-mass discHigh-mass discMps = 1M

Mps = 0.8M

Mdisk = 0.01M

Mdisk = 0.2M

N ⇡ 2 ⇥ 106 SPH particle (5 au < R < 100 au)

N ⇡ 2 ⇥ 106 SPH particle (5 au < R < 100 au)

T (R) / R 1/2 , (T0 = 250 K at 5 au)

T (R) / R 1/2 , (T0 = 250 K at 5 au)⌃(R) / R 1<latexit sha1_base64="QU65RPcqLrsFLOWu1P3hDmVhtLE=">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</latexit>⌃(R) / R 1<latexit sha1_base64="bqkz1O3aMtlcYgdYq41Rz7ZN2h0=">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</latexit>

overestimate

overestimate
underestimate
optically thick

optically thin

Fig.*(d),(e)-> 34-36 auBM )@K
_SF*optical depthcooling ratevertical profile

High-mass disc with spiral arms
Fig.3¥V_High-mass disc with clumpsdisk}Lombarli methodrate,& Y{$*)clumpwXcooling

Disc with an embedded 1.4 MJ planet

Disc with an embedded 11 MJ planet

Mps = 1M

Mps = 1M

Mdisk = 0.005M

The planet migrates
slightly inwards (0.1 au)

Mdisk = 0.005M

N ⇡ 106 SPH particle

N ⇡ 106 SPH particle

T (R) / R 3/4 , (T0 = 250 K at 1 au)

T (R) / R 3/4 , (T0 = 250 K at 1 au)⌃(R) / R 1/2<latexit sha1_base64="6QsA6g1PQ0NfvPhbKtXz3mmaV8E=">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</latexit>

The planet migrates
inwards (14 au)

1MJ is embedded at 5.2 au

Fig.6d,esolid lines:
planet gap (4-6au)
dashed lines:
interior gap (3-4au)⌃(R) / R 1<latexit sha1_base64="ThcH3xAXVknT/LnF6XWptximZVc=">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</latexit>-> gap: 4-6 au
Mpl ~ 1.4MJ

1MJ is embedded at 50 au

Fig.7d,esolid lines:
planet gap (33-37au)
dashed lines:
interior gap (23-27au)

(a)a low-mass relaxed disc(b)a high-mass disc(c)a disc with spirals arms(d) a disc with dense clumps(e) a disc with an embedded 1.4 MJ planet(f) a disc with an embedded higher-mass 11 MJ planet(a), (b)-> βeff > 20 (inner disc), βeff ~ 3 (outer region)(c)-> βeff ~ 1 (spiral region) -> cooling is efficient-> may be prone to gravitational collapse(d)-> βeff ~ 200 (clump) <- being extremely
optically thick(e)-> βeff is high (outer regions), low (gap)(f)-> βeff ~ 1 (high-density spiral), βeff > 20 (gap),
βeff < 1 (around planet)Due to a generally higher estimation of thecooling rate, the Lombardi method allows
the disc to cool more efficiently anddevelop stronger spiral arms.

Stamatellos method

Lombardi method