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Astron. Astrophys. 322, 86-88 (1997)

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2. The orientation of the warps of the Milky Way, M 31 and M 33

To characterize the position of a warp in space, we have two possibilities. One of them is the direction of the line of nodes itself. The other one is the "orientation" vector. Let us precisely define the "orientation" vector of a warp. Suppose an untwisted line of nodes. Let [FORMULA] be a unitary vector contained in the line of nodes. Its direction is conventionally assigned as pointing to a region from which the warps are seen as N-shaped (not S-shaped), i.e. from which, due to the observed bending, the galaxy appears to have a clockwise gyration. Let [FORMULA] be a unitary vector with the direction of a rotation axis. We define the orientation direction as that containing either [FORMULA] defined as

[FORMULA] or [FORMULA].

From an observational point of view [FORMULA] and [FORMULA] are undistinguishable. From a theoretical point of view, whatever the interpretation is assumed, the distinction between [FORMULA] and [FORMULA] is probably meaningless.

[FIGURE] Fig. 1. Definition of the orientation vectors

In the figure, we show the direction [FORMULA], [FORMULA] in a galaxy observed as edge-on. We define [FORMULA], [FORMULA], in this way, because under the interpretation of warps made in the magnetic model by Battaner et al. (1990), [FORMULA], [FORMULA] would coincide with the direction of the intergalactic field. But it is emphasized that we present this work as completely free from any theoretical interpretation. By defining the orientation in this way we can easily compare it with our previous determinations of other galaxies, too.

The information given by both, the direction of the line of nodes and the orientation vector, are not equivalent. For instance, if the galaxy is gyrated about the line of nodes, its direction would not be modified, but the orientation vector could change. Therefore we have made the calculation about the coherence for both vectors.

Let us calculate [FORMULA] for M 31, M 33 and the Milky Way, using the same coordinate system. The orientation vector defined in the system centred in the plane of the considered galaxy must be subject to the following transformations:

a) From [FORMULA] referred to the plane of the galaxy, we obtain [FORMULA] referred to the plane of the sky, defined by the vectors [FORMULA] ([FORMULA] in the line of the sight direction; [FORMULA] and [FORMULA] in the tangent plane, [FORMULA] northwards and [FORMULA] eastwards)

[FORMULA]

b) From [FORMULA] referred to the plane of the sky, we obtain [FORMULA] referred to the plane of the equator

[FORMULA]

where [FORMULA] are the inclination, position angle and equatorial coordinates, respectively.

The same transformations must be taken into account to obtain the direction of the line of nodes.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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