Astron. Astrophys. 339, 409-422 (1998)

Appendix A: The strip brightness method to derive the luminosity density of an axisymmetric ellipsoid

The strip density method was first introduced by Schwarzschild (1954) to evaluate the gravitational potential of the Coma cluster. In the following we describe the method and extend it to the more general situation of an oblate, axisymmetric, ellipsoidal distribution of mass (and luminosity). This technique offers a quite simple and straightforward way to recover the spatial density distribution, and the potential and rotation curve of such systems, especially when they are not described parametrically but rather observed as projected distributions on the sky.

Let us consider an ellipsoidal distribution, centered at O, with oblate rotational symmetry about z, inclined by an angle i to the line of sight [Fig. 6b]. The isodensity surfaces can be parametrized by their semimajor axis, that is their equatorial radius a:

where the constant is the intrinsic eccentricity. The projected distribution on the plane of the sky () has elliptical symmetry with an apparent eccentricity . The surface brightness is then

Fig. 6a and b. Geometry of the ellipsoid a as seen projected on the sky plane, and b  from a lateral viewpoint. The symmetry axis z is inclined with respect to the line of sight and the axes x and coincide with the line of nodes, interception of the equatorial plane with the sky plane . The strip brightness is the integral of the surface brightness along a strip perpendicular to the line of nodes, as sketched by the dashed lines in panel a .

We now define the strip brightness at distance from the image minor axis [Fig. 6a] as

On the other hand is the luminosity of a section perpendicular to x at of the ellipsoid. Such a section will have again elliptical symmetry with eccentricity . Any elliptical isodensity contour of this section can be described by:

Then,

and

In practice, the observed image is divided in strips normal to and integrated in the direction. This gives for spanning from 0 to the outermost radius of the image. Eq. A6 is then used to derive , and with it the potential and circular speed. From a numerical point of view the method is easy to implement and was tested on a number of analytical solutions. Due to the strip integration prior to the differentiation in Eq. A6, the noise introduced is low and hardly appreciable even in the outer regions.

© European Southern Observatory (ESO) 1998

Online publication: October 21, 1998
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