table of contents  next: gap opening

1. Before the gap opening

Model 29: Earth in a thin nebula

  • Planet: 
    		•  mass ratio µ = 3· 10-6 e = 0 
  • Disk:
    		•  alpha = 0.008  c/vK = 0.025
  • grid (phi,r): 222 x 200 
    		•  resol.: 0.01 x 0.01 tmax= 100 P 
  • Accretion: 
    		•  negligible  (by assumpt.) 
  • Migration (by gravity): 
    		•  da/dt = -2.5 · 10-5 vK (inward) ta= 75 kyr 
  • Unpert. disk accretion: 
    		•  v= -7.5 · 10-6 vK tvisc= 250 kyr 
  • Eccentricity: 
    		•  const.  - 

    Comments: This simulation shows PPM can handle such small planets. Wave amplitude of order 10%.
    Interesting wake structure is shown below. Despite the rather thin disk NO LINDBALD RESONACES SEEN directly. Maybe strong non-linear interaction of m>1 waves that form a wake is the explanation.

    Additional models:

    Here is an interesting shape of a wake behind (and preceding) an Earth-mass planet in a very thin disk (c=0.01 vK). The wake consistes of multiple, closely-spaced shock waves. A bit like a keel-water wake of a ship.


    (coordinates in # of grid points, i.e. not a uniform mapping!).
     

    Model 27: 3.3 Earth-mass protoplanet in a thin nebula

  • Planet: 
    		•  mass ratio µ = 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 (outward!) ta= 94 kyr 
  • Eccentricity: damped
    		•  de2/dt = -2 · 10-6 /tdyn te= 190 yr 

    Comments: Eccentricity damped and te is very short.
    Notice how deep a through does this little planet create in the disk. Inner disk pushes stronger, this is a bit unexpected.
    The result is: da/dt= 2.64e-4 (inner disk) + (-2.43e-4) (outer disk)= +2.14e-5 (whole disk) for t= 96-99 P period.
    There is negligible noise in this result (which was checked by a separate test run with influence of planet on disk forced to be zero; but there is a slow decay of one-sided effect on da/dt, accompanied by increased differential migration as the simulation progresses (dip/gap achieving equilibrium). 

    Model 28: 3.3 Earth-mass protoplanet in a standard solar nebula

  • Planet: 
    		•  mass ratio µ = 10-5 e = 0.01 
  • Disk:
    		•  alpha = 0.008  c/vK = 0.05
  • Grid (phi,r): 200 x 200 
    		•  resol.: 0.011 x 0.0055 tmax= 45 P (120 lower res. ver.) 
  • Accretion: 
    		•  0  (by assumption) 
  • Migration (by gravity): 
    		•  -1.0 · 10-5 vK (-0.5 · 10-5 in low res.) ta= 188 kyr (380 kyr in lower res. ver.)
  • Unpert. disk accretion: 
    		•  v=-3.0· 10-5 vK tvisc= 63 kyr 
  • Eccentricity: damped
    		•  de2/dt = -5 · 10-7 /tdyn te = 750 yr 

    Comments: Eccentricity is damped as expected, although not necessarily at the analytically predicted rate; The te is very short. Both inner and outer disk damp e equally.
    The migration break-down: da/dt= 7e-5 (inner disk) + (-8e-5) (outer disk) = -1e-5 (whole disk) in the t=40-43 P interval. 

    Model 23: 3.3 Earth-mass protoplanet in a thick solar nebula

  • Planet: 
    		•  mass ratio µ = 10-5 e = 0.01 
  • Disk:
    		•  alpha = 0.004  c/vK = 0.1
  • Grid (phi,r): 100 x 100 
    		•  resol.: 0.020 x 0.015 tmax= 75 P 
  • Accretion: 
    		•  0  (by assumption) 
  • Migration (by gravity): 
    		•  -0.45 · 10-5 vK (inward) ta= 420 kyr 
  • Unpert. disk accretion: 
    		•  v=-6 · 10-5 vK tvisc= 31 kyr 
  • Eccentricity: damped
    		•  de2/dt = -1.2 · 10-7 /tdyn te= 3.1 kyr 

    Comments: Eccentricity damped as expected, although not at the analytically predicted rate; The te is very short, contributions from inner and outer disk of the same order.
    The migration break-down: da/dt= +1.22e-5 (inner disk) + (-1.68e-5) (outer disk) = -0.45e-5 (whole disk) in the t= 70-73 P interval. 

    Model 41: 10 Earth-mass protoplanet in a thin nebula

  • Planet: 
    		•  mass ratio µ = 3 · 10-5 e = 0 
  • Disk:
    		•  alpha = 0.006  c/vK = 0.025
  • Grid (phi,r): 200 x 180 
    		•  resol.: 0.010 x 0.004 tmax= 100 P 
  • Accretion: 
    		•  0  (by assumption) 
  • Migration (by gravity): 
    		•  -2.3· 10-5 vK ta= 84 kyr 
  • Unpert. disk accretion: 
    		•  v=-0.56· 10-5 vK tvisc= 330 kyr 
  • Eccentricity: 
    		•  const.  - 

    Comments: Eccentricity damped as expected, although not necessarily at the analytically predicted rate; the same with migration. Technically, torques and dE/dt from a region outside 1 rL away from the planet are used here. In this calculation, interestingly, migration is reduced slightly if material between ½ and 1 rL is also considered. Usually the innermost region boosts the interaction from further out.
    Notice how deep a through does this planet create in the disk. It's beginning to look like a gap (density on the opposite side w.r.t. the planet < ½ that in the surrounding disk).
    Standard thermal gap opening criterion: mass ratio>µ = 3 (c/vK)3 = 4.7 · 10-5, is violated , i.e. the gap starts to open disregarding that Roche lobe is smaller than the disk scale height.
    Standard viscous gap opening criterion: mass ratio>µ = 40 alpha (c/vK)2 = 1.5 · 10-4, is also violated , i.e. the gap starts to open disregarding that the viscous torques are supposedly too week to counteract disk viscosity.
    Of course, the gap is admittedly open only marginally. Nevertheless, consider a 20 or 30 Earth mass planet: the viscous criterion will still be violated, and the gap will then be unambiguously open!



    This is a new simulation  I did with a different PPM code in Nov. 2000.  It uses a different (inertial, nonuniform square) grid.  The disk has no alpha-viscosity, therefore you see a gap starting to openin already at the simulated planet  mass of 10 Earth.  The time in this snapshot is  t=150 P. The density on the grid was in the range 0.55 to 2.81 at that point of time, i.e. the 10 Earth object created in its vicinity a local drop in density of 45%, a beginning of a gap or axisymmetric trough of ~20% underdensity, the spiral waves were ~20% density contrast, and the protoplanet (with softened gravity) was surrounded by gas peaking at 2.8 times the ambient disk density in one computational cell.
     
     

    table of contents  next topic: gap opening