
mass ratio µ = 3· 10^{6}  e = 0 

alpha = 0.008  c/v_{K} = 0.025 

resol.: 0.01 x 0.01  t_{max}= 100 P 

negligible  (by assumpt.) 

da/dt = 2.5 · 10^{5} v_{K} (inward)  t_{a}= 75 kyr 

v= 7.5 · 10^{6} v_{K}  t_{visc}= 250 kyr 

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 nonlinear
interaction of m>1 waves that form a wake is the explanation.
(coordinates in # of grid points, i.e. not a uniform
mapping!).

mass ratio µ = 10^{5}  e = 0.01 

alpha = 0.008  c/v_{K} = 0.025 

resol.: 0.011 x 0.005  t_{max}= 100 P 

0  (by assumption) 

+2.0 · 10^{5} v_{K} (outward!)  t_{a}= 94 kyr 

de^{2}/dt = 2 · 10^{6} /t_{dyn}  t_{e}= 190 yr 
Comments: Eccentricity damped and t_{e}
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= 2.64e4 (inner disk) + (2.43e4)
(outer disk)= +2.14e5 (whole disk) for t= 9699 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 onesided effect on da/dt, accompanied by
increased
differential migration as the simulation progresses (dip/gap achieving
equilibrium).

mass ratio µ = 10^{5}  e = 0.01 

alpha = 0.008  c/v_{K} = 0.05 

resol.: 0.011 x 0.0055  t_{max}= 45 P (120 lower res. ver.) 

0  (by assumption) 

1.0 · 10^{5} v_{K} (0.5 · 10^{5} in low res.)  t_{a}= 188 kyr (380 kyr in lower res. ver.) 

v=3.0· 10^{5} v_{K}  t_{visc}= 63 kyr 

de^{2}/dt = 5 · 10^{7} /t_{dyn}  t_{e} = 750 yr 
Comments: Eccentricity is damped as expected, although
not necessarily at the analytically predicted rate; The t_{e} is
very short. Both inner and outer disk damp e equally.
The migration breakdown: da/dt= 7e5 (inner disk) +
(8e5) (outer disk) = 1e5 (whole disk) in the t=4043 P interval.

mass ratio µ = 10^{5}  e = 0.01 

alpha = 0.004  c/v_{K} = 0.1 

resol.: 0.020 x 0.015  t_{max}= 75 P 

0  (by assumption) 

0.45 · 10^{5} v_{K} (inward)  t_{a}= 420 kyr 

v=6 · 10^{5} v_{K}  t_{visc}= 31 kyr 

de^{2}/dt = 1.2 · 10^{7} /t_{dyn}  t_{e}= 3.1 kyr 
Comments: Eccentricity damped as expected, although
not at the analytically predicted rate; The t_{e} is very short,
contributions from inner and outer disk of the same order.
The migration breakdown: da/dt= +1.22e5 (inner disk)
+ (1.68e5) (outer disk) = 0.45e5 (whole disk) in the t= 7073 P interval.

mass ratio µ = 3 · 10^{5}  e = 0 

alpha = 0.006  c/v_{K} = 0.025 

resol.: 0.010 x 0.004  t_{max}= 100 P 

0  (by assumption) 

2.3· 10^{5} v_{K}  t_{a}= 84 kyr 

v=0.56· 10^{5} v_{K}  t_{visc}= 330 kyr 

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 r_{L} away
from the planet are used here. In this calculation, interestingly, migration
is reduced slightly if material between ½ and 1 r_{L}
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/v_{K})^{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/v_{K})^{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!