We analyze the formation and migration of an already formed proto-Jovian companion embedded in a circumstellar disk .
We use two dimensional ( r, \theta ) Throughout this paper and its companion , Paper II , we use ‘ \theta ’ to denote the azimuth coordinate rather than the more usual variable ‘ \phi ’ in order to avoid confusion between references to the coordinate and to components of the planet ’ s gravitational potential , \phi _ { m } , common throughout Paper II . hydrodynamic simulations using a ‘ Piecewise Parabolic Method ’ ( PPM ) code to model the evolutionary period in which the companion makes its transition from ‘ Type I ’ migration to ‘ Type II ’ migration .

The results of our simulations show that spiral waves extending several wavelengths inward and outward from the planet are generated by the gravitational torque of the planet on the disk .
Their effect on the planet cause it to migrate inward towards the star , and their effect on the disk cause it to form a deep ( low surface density ) gap near the planet .
We study the sensitivity of the planet ’ s migration rate to the planet ’ s mass and to the disk ’ s mass .
Until a transition to slower Type II migration , the migration rate of the planet is of order 1 AU/10 ^ { 3 } yr , and varies by less than a factor of two with a factor twenty change in planet mass , but depends near linearly on the disk mass .
Although the disk is stable to self gravitating disk perturbations ( Toomre Q > 5 everywhere ) , implying the effects of gravity should be insignificant , migration is faster by a factor of two or more when disk self gravity is suppressed .
Migration is equally sensitive to the disk ’ s mass distribution within 1–2 Hill radii of the planet , as demonstrated by our simulations ’ sensitivity to the planet ’ s assumed gravitational softening parameter , and which also crudely models the effect of the disk ’ s extent into the third ( z ) dimension .

Deep gaps form within \sim 500 yr after the beginning of the simulations , but migration can continue much longer : the formation of a deep gap and the onset of Type II migration are not equivalent .
The gap is several AU in width and displays very nearly the M _ { pl } ^ { 2 / 3 } proportionality predicted by theory .
Beginning from an initially unperturbed 0.05 M _ { \odot } disk , planets of mass M _ { pl } > 0.3 M _ { J } can open a gap which is deep and wide enough to complete the transition to slower Type II migration .
Lower mass objects continue to migrate rapidly for the duration of the simulation , eventually impacting the inner boundary of our grid .
This transition mass is much larger than that predicted as the ‘ Shiva mass ’ discussed in Ward & Hahn ( 51 ) , making the survival of forming planets even more precarious than they would predict .