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Table 2:

Summary of the analytic recipe to obtain the impact radii $b_\sigma $ and approach velocities $v_{\rm a}$.

1. Calculate dimensionless parameters: 		$\zeta _{\rm w}$ (headwind velocity) Eq. (15)  
  $\alpha_{\rm p}$ (planet size) Eq. (16)  
  ${\rm St} = t_{\rm s} \Omega$ (Stokes number) Eq. (12)  

2. Calculate impact radii: 		$\tilde{b}_{\rm set}$ Eq. (27), Eq. (32)  
  $b_{\rm hyp}$ Eq. (28)  
  $b_{\rm 3b}$ Eq. (31)  

3. Determine regime: 		${\rm St}<\min(1, 12/\zeta_{\rm w}^3)$   ${\rm St}>{\rm max}(\zeta _{\rm w}, 1)$
  Settling Hyperbolic Three body

4. Results 		     
Impact radius (accretion), $b_\sigma $: ${\rm max}(\tilde{b}_{\rm set}, b_{\rm geo}$) ${\rm max}(\tilde{b}_{\rm set}, b_{\rm hyp})$ ${\rm max}(b_{\rm 3b}, b_{\rm geo})$
Approach velocity $v_{\rm a}$: $3b_\sigma/2 + \zeta_{\rm w}$ Eq. (29) 3.2
Approach radius $b_{\rm app}$: $b_\sigma $ $b_\sigma $ 2.5

Notes. Description of impact radii: $b_{\rm geo}$, geometrical impact radius ( $=\alpha_{\rm p}$); $b_{\rm set}$ impact radius in settling regime; $\tilde{b}_{\rm set}$, modified $b_{\rm set}$ (to cover the transition regime); $b_{\rm hyp}$ impact radius in the hyperbolic regime; $b_{\rm 3b}$ drag-enhanced impact radius for the 3-body regimes; $b_{\rm app}$, approach distance.


Source LaTeX | All tables | In the text

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