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planets and satellites: dynamical evolution and stability, planetary systems

The paper ‘Cross-sections for planetary systems interacting with passing stars and binaries’ was published in MNRAS, 2015, 448, 344 (Li & Adams 2015). In this contribution, we discuss some scaling relations and additional issues regarding the cross-sections and interaction rates presented in the paper.

To calculate the cross-sections, we assume the velocity of stars in the cluster follows the Maxwellian distribution of the form
(1)
where s is the one-dimensional velocity dispersion. Next, we want to sample the incoming velocities (vrel) of the stars, where vrel represents the relative velocity of the stars. The relative velocity dispersion between stars in the cluster is larger than the velocity dispersion of the stellar cluster by a factor of |$\sqrt{2}$| (for further discussions, see Binney & Tremaine 2008). As a result, the distribution of relative velocities, denoted here as f(vrel), should have the form
(2)
where |$s_{{\rm rel}}=\sqrt{2}s$|⁠. However, the paper (Li & Adams 2015) set |$s_{{\rm rel}} = s/\sqrt{2}$|⁠.
In addition, the paper (Li & Adams 2015) presents results for the interaction rates using the product of s and the averaged cross section (the quantity 〈σ〉s), instead of the averaged product of incoming velocity and cross-section (〈σvrel〉). We thus want to find the relationship between the quantity 〈σvrel〉, calculated using the correct distribution with |$s_{{\rm rel}} = \sqrt{2} s$|⁠, and the related quantity 〈σ〉s calculated using the distribution with |$s_{{\rm rel}} = s/\sqrt{2}$|⁠, as used in the original paper. This scaling can be determined when the cross-sections have velocity dependence of the power-law form |$\sigma \propto v_{{\rm rel}}^{-\gamma }$|⁠, as found in the original paper. Specifically, this ratio becomes
(3)
(4)

According to the fitting formulae of the original paper (see equation 19), the index γ = 7/5 for the regime of interest. In this case, we find 〈σv〉/(〈σ〉s) = Γ(13/10)/(22/5Γ(4/5)) ≈ 0.584. As a result, the velocity-weighted cross-section (〈σv〉/s), including the relative cluster dispersion, is smaller than the averaged cross-section (〈σ〉) listed in Table 2 of the original paper by a factor of ∼0.6. To take this scaling correction into account, the corresponding interaction rates in the paper should also be decreased by a factor of ∼0.6. The correction is thus of order unity. Finally, we note that the cross sections have additional (independent) uncertainties due to other assumptions of the calculation (e.g., choice of the initial mass function of cluster stars, the allowed range of impact parameters, etc., as outlined in Li & Adams 2015).

The authors would like to thank Scott Tremaine for helpful discussions.

REFERENCES

Binney
J.
Tremaine
S.
2008
Galactic Dynamics
Princeton Univ. Press
Princeton

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Li
G.
Adams
F. C.
2015
MNRAS
448
344

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© 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society
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