We investigate the effects of a passing stellar encounter on a planetesimal disk through analytical calculations and numerical simulations , and derive the boundary radius ( a _ { planet } ) outside which planet formation is inhibited by disruptive collisions with high relative velocities .
Ida , Larwood , and Burkert ( 2000 .
ApJ .
528 , 1013-1025 ) suggested that a stellar encounter caused inhibition of planet formation in the outer part of a protoplanetary disk .
We study orbital eccentricity ( e ) and inclination ( i ) of planetesimals pumped up by perturbations of a passing single star .
We also study the degree of alignment of longitude of pericenter and ascending node to estimate relative velocities between the planetesimals .
We model a protoplanetary system as a disk of massless particles circularly orbiting a host star , following Ida et al . ( 2000 ) .
The massless particles represent planetesimals .
A single star as massive as the host star encounters the protoplanetary system .
Numerical orbital simulations show that in the inner region at semimajor axis a \lesssim 0.2 D where D is pericenter distance of the encounter , e and i have power-law dependence on ( a / D ) as e \propto ( a / D ) ^ { 5 / 2 } and i \propto ( a / D ) ^ { 3 / 2 } and the longitudes are aligned , independent of the encounter parameters .
In the outer region a \gtrsim 0.2 D , the radial gradient is steeper , and is not expressed by a single power-law .
The longitudes are not aligned .
Since planet accretion is inhibited by e as small as 0.01 , we focus on the weakly perturbed inner region .
We analytically reproduce the power-law dependence and explicitly give numerical factors of the power-law dependence as functions of encounter parameters .
We derive the boundary radius ( a _ { planet } ) of planet forming region as a function of dynamical parameters of a stellar cluster , assuming the protoplanetary system belongs to the stellar cluster .
Since the radial gradient of e is so steep that the boundary is sharply determined .
Planetesimal orbits are significantly modified beyond the boundary , while they are almost intact inside the boundary .
This tendency is strengthened by reduction of relative velocity due to the longitude alignment in the inner region .
We find a _ { planet } \sim 40 -60AU in the case of D \sim 150 -200AU . D \sim 200AU may be likely to occur in a relatively dense cluster .
We point out that the size of planetary systems ( a _ { planet } ) born in a dense cluster may be necessarily restricted to that comparable to the size of planet region ( \sim 30 -40AU ) of our Solar system .