2. The PA-inclination method
The 3-dimensional orientation of the SV of a galaxy is characterized by two angles: the polar angle between the galactic SV and a reference plane, and the azimuthal angle between the projection of the galactic SV on to this reference plane and the X-axis within this plane. When using the Local Supercluster (LSC) as reference, then and can be obtained from measurable quantities as follows (Flin & Godl owski 1986):
where i is the inclination angle, estimated with Holmberg's (1946) formula: with being the measured axial ratio. L, B and P represent the supergalactic longitude, latitude, and position angle, respectively.
Note, that for a given value of i, these expressions give two solutions for both and and hence 4 solutions for the angular momentum vector of the galaxy. However, for a large sample of galaxies it is hardly possible to determine - for each galaxy - which one is the physical correct one. As usual we have counted each of these possibilities independently.
Note further, that Eqs. (1) and (2) will give one single solution for both and when a galaxy is seen exactly face-on (). When seen edge-on (), the two solutions for both and differ just in sign.
More interestingly, the characteristics of the solutions for and are also strongly influenced by the used coordinate system. This was already noted by Flin & Godl owski 1986 (their Sect. 6.1). These authors suggested an analytical method to remove these selection effects due to positions. In the case of , both solutions converge when approaching the supergalactic pole or for galaxies with supergalactic PA . In the case of both solutions just differ in sign for . As an example, let a galaxy cluster of small angular size be filled with galaxies whose SVs are randomly distributed in space. The distribution of the calculated angles and will then depend on the position of that cluster on the celestial sphere. Consequently, when considering a larger region the expected distribution of and will be defined by the location of that region on the sky, but also by its boundaries.
Mathematically, the expected distribution of the calculated values of and , when applying the expressions (1) and (2), is determined by the assumed spatial distribution of the SVs of the galaxies and the interval from which the variables L, B, P, and i have been taken. The latter influence is what we call selection effect in the following.
Due to the special form of the expressions (1) and (2) these selection effects cannot in general be calculated analytically. In order to minimize the selection effects due to positions and inclination angles we have done numerical simulations.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001