Derivation of the day: alpha

The quantity alpha links the Keplerian orbital dynamics to the interactions between two planetesimals.

alpha = Rsun/a

where Rsun is the solar radius and a is the semi-major axis. alpha is approximately 10^(-4) in the Kuiper belt region.

Previously we derived:

vesc = sqrt(GM/R)

where vesc is the escape velocity from a large planetesimal with mass M and radius R

vH = RH Omega = sqrt(GM/RH)

where vH is the hill velocity, RH is the hill radius of the large planetesimal, and Omega is the orbital frequency.

RH = (M/Msun)^(1/3) a

where Msun is the mass of the sun

We can rewrite:

Msun = rho_sun Rsun^3

where rho_sun is the mean solar density

and

M = rho R^3

where rho is the mean density of the large planetesimal. We will approximate that rho = rho_sun. Thus the relationship between RH and R is:

RH = R/Rsun a = R/alpha

The relationship between vesc and vH is:

vH/vesc = sqrt(R/RH) = alpha^(1/2)