3 Statistical description of structure

The evolution of structure as it moves out with the flow can be usefully characterised by a number of statistical quantities. In this paper we mainly use the clumping factor, the velocity dispersion and the correlation coefficient, defined as

$$f_{\rm cl} = \frac{<\rho^2>}{<\rho>^2},$$ (6)

$$v_{\rm disp} = \sqrt{<v^2>-<v>^2} {\rm\;\;and}$$ (7)

$$C_{v\log\rho} = \frac{<v\log\rho >-<v><\log\rho>}{v_{\rm disp} \sqrt{<(\log\rho)^2>-<\log\rho>^2}},$$ (8)

respectively. The symbol <> denotes a time average and all of the above quantities may depend on radius.

The clumping factor describes how mass is distributed; it becomes larger as mass is concentrated in a smaller volume. The velocity dispersion is the standard deviation of the velocity and as such describes the typical velocity variation around the mean flow speed. The correlation coefficient gives the correlation between density and velocity variations and has values between -1 (complete anti-correlation) and 1 (complete correlation, i.e. velocity and density are in phase). Positive (negative) correlation indicates a net preponderance of structure that propagates outward (inward) relative to the mean outflow. In particular, for variations due to shocks, a positive (negative) correlation indicates a net preponderance of forward (reverse) shocks.

Diagnostics sensitive to the square of the density (e.g., emission in Balmer lines or the infrared and radio continua) will tend to overestimate the mass loss rate of a clumped wind by a factor of $\sqrt{f_{\rm cl}}$. In this context, an alternative quantity commonly used to describe wind structure is the volume filling factor f. It is defined as the fractional volume of the dense gas, assuming that the gas is made up of two components: dense clumps and rarefied interclump material. For a constant filling factor, the clumping factor can then be expressed (Abbott et al. 1981) as

$$f_{\rm cl} =\frac{f+(1-f) x^2}{[f+x(1-f)]^2},$$ (9)

where x is the ratio of low to high density gas. If most of the mass is concentrated in the clumps ( $f \gg (1-f)x$), the clumping factor is the inverse of the volume filling factor, and equals the density contrast of the clumps with respect to the mean density.