Question

Determine the minimum and maximum impact parameters bmin and bmax for the classical derivation of charged particle collisional stopping power. Explain the physical meaning of each.

Answer 1

Impact parameter b and scattering angle θ.

The impact parameter {\displaystyle b} is defined as the perpendicular distance between the path of a projectile and the center of a potential field {\displaystyle U(r)} created by an object that the projectile is approaching (see diagram). It is often referred to in nuclear physics (see Rutherford scattering) and in classical mechanics.

The impact parameter is related to the scattering angle {\displaystyle \theta } by^[1]

{\displaystyle \theta =\pi -2b\int _{r_{\mathrm {min} }}^{\infty }{\frac {dr}{r^{2}{\sqrt {1-(b/r)^{2}-2U/mv_{\infty }^{2}}}}}}

where {\displaystyle v_{\infty }} is the velocity of the projectile when it is far from the center, and {\displaystyle r_{\mathrm {min} }} is its closest distance from the center.

Let us consider a tidal disruption event of a star with mass MsMs and a radius RsRs by a black hole of mass MhMh. The tidal radius is defined as Rt≈Rs(Mh/Ms)1/3Rt≈Rs(Mh/Ms)1/3 and the dimensionless impact parameter is defined as β=Rt/Rpβ=Rt/Rp where RpRp is the periapse (i.e. the closest the star gets to the black hole). For very massive black holes the minimum periapse is given by the gravitational radius

Rp,min≈GMhc2Rp,min≈GMhc2

and the maximum dimensionless impact parameter is given by

βmax=RtRp,min=c2RsGMs(MhMs)−2/3βmax=RtRp,min=c2RsGMs(MhMs)−2/3

In the limit of a very small black hole, the minimal impact parameter is limited by the radius of the disrupted star

R′p,min≈RsRp,min′≈Rs

and the maximum dimensionless impact parameter in this case is given by

β′max=RsRt=(MhMs)1/3βmax′=RsRt=(MhMs)1/3

It is therefore clear that the maximum ββ is attained when

MhMs≈c2RsGMsMhMs≈c2RsGMs

The right hand side of the previous equation can be interpreted as the ratio between the actual radius of the star and the gravitational radius of the same mass. The corresponding maximal β is