For M 31, from Brinks and Burton (1984) we know that the warp forms an angle of about with the line of sight, when measured clockwise in the main plane of M 31. On the other hand, the warp points towards the observer in the NW part, so for we obtainFor M 33, following the study by Rogstad, Wright and Lockhart (1976), the warp points in a direction contained in the plane of M 33 which forms (clockwise) with the major axis. Furthermore, the warp points towards the observer at its closest part. Then for we obtain For the Milky Way, taking the above-mentioned measurements by Burton (1996) and the figure in Battaner, Florido and Sánchez-Saavedra (1990) we obtain We have also calculated the direction of the line of nodes and the angles between them. This is important because in the early paper by Kahn and Woltjer (1959), a relation between the line of nodes and the motion of the galaxy is predicted. Unfortunately, little was known about the motion of galaxies in the Local Group, with the exception of the Magellanic Clouds (Lin, Jones and Klemola, 1995). However, this motion cannot be irregular and a common direction of the line of nodes, specially of those of M 31 and M 33 is to be expected.
For the direction of the line of nodes, , we obtain the following table
and the angles between them are[M 31, M 33]=
[M 31, Milky Way]=
[M 33, Milky Way]= showing a noticeable degree of alignment.
However, the degree of alignment is much higher when we use the vector to characterize the orientation of the warp. The results obtained are summarized in the following table
The angles between these warp orientations are[M 31, M 33]=
[M 31, Milky Way]=
[M 33, Milky Way]=
Consider any two random warp orientations. It is easily calculated, taking into account that is the same orientation as - , that the probability of obtaining an angle between them, less than , is
Therefore the probabilities of having obtained angles as small as the above ones are 0.14, 0.18 and 0.0025, respectively: the highest degree of alignment corresponds to the pair formed by the Milky Way and M 33.
There is therefore a noticeable coincidence. The three largest galaxies in the Local Group seem to be warped in a coherent form.
This fact gives some additional support to the magnetic hypothesis and to the theory by Kahn and Woltjer (1959) (see for current theories about warps the reviews by Binney (1992), Combes (1994), Battaner (1995) and others). It is however difficult to obtain firm conclusions, due to the very small size of the sample and the difficulties in interpreting the warp of our disc. Even so, the calculation carried out is a necessary one to have all data at hand, to physically interpret the phenomenon of warps.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998