table of contents next: superplanet growth

2. Gap opening by planets

Model 53: 25 Earth-mass protoplanet in a standard nebula (c=0.04 vK)

  • Planet:
    		• mass ratio µ = 7.5 · 10-5 e = 0.01
  • Disk:
    		•  alpha = 0.006 c/vK = 0.04
  • Grid (phi,r): 244 x 220
    		• resol.: 0.0091 x 0.0049 tmax= ? P
  • Accretion:
    		•  dm/dt= 1.43 × (dm/dt)0 tremove = 1.8 P
  • Unpert. disk accretion:
    		• (dm/dt)0 = 3 pi Sigma nu nu = 0.96 × 10-5
    - (dm/dt)0 = 3.2 · 10-8 Msun/yr -
  • Doubling time:
    		•  tm = 1.6 kyr (or more if tremove shorter)
  • Migration (by gravity):
    		• da/dt = -6 · 10-5 vK (inward) ta= 31 kyr
  • Migration (by advection):
    		• da/dt = +0.17 · 10-5 vK (outward) adv << grav
  • Unpert. disk flow:
    		• v = -1.4 · 10-5 vK tvisc= 130 kyr
  • Eccentricity (by gravity) :
    		• de2/dt < 10-7 grav << adv
  • Eccentricity (by advection) :
    		• de2/dt = -4 × 10-7 (damped) te = 940 yr
  • Standard gap opening criteria:
    		• viscous, µ > 130 µEarth (violated) thermal, µ > 65 µEarth (violated)

    Comments: Eccentricity damped mainly by advection, and here even weakly excited by gravity. Of course it's diufficult to disentangle the two since the wakes and streams of matter contribute to both, with different signs.
    The gap is cleared, although it isn't empty: has density of up to 0.1 $times the unperturbed density.
    Standard thermal gap opening criterion: mass ratio>µ = 3 (c/vK)3 = 4.7 · 10-5, is violated by a factor of 2.5.
    Standard viscous gap opening criterion: mass ratio>µ = 40 alpha (c/vK)2 = 1.5 · 10-4, is also violated by a factor of 5.
    Advection is, however, weaker than gravity in causing migration. To some extent, it is so because we restrict the e-folding removal time of gas from the inner 2/3 of the Roche lobe to be more than 2 orbital periods of the planet. (This estimate corresponds e.g. to a circumplanetary disk with the same temperature and viscosity as the nebula.

    Model 01: 1 Jupiter-mass protoplanet in a standard nebula

  • Planet:
    		• mu = 10-5 e = 0.01
  • Disk:
    		•  alpha = 0.008 c/vK = 0.025
  • Grid (phi,r): 200 x 200
    		• resol.: 0.011 x 0.005 tmax= 100 P
  • Accretion:
    		• 0 (by assumption)
  • Migration (by gravity):
    		• 2.0 · 10-5 vK ta= 94 kyr
  • Eccentricity: damped
    		•  de2/dt=-2· 10-6 /tdyn te= 188 yr
    Here is a snapshot from a very high-resolution Jupiter simulation with the same disk parameters as above (but zero eccentricity):

    table of contents next topic: superplanet growth