## 1. IntroductionThe determination of distances is one of the most important
problems in extragalactic astronomy and cosmology. Distances between
two objects X and Y depend on their redshifts
and , the Hubble
constant , the cosmological constant
, the density parameter and
the inhomogeneity parameter .
We stress the fact that the inhomogeneity can be as important as
the other cosmological parameters, both in the field of more
traditional cosmology and in the case of gravitational lensing, where,
e.g. in the case of the time delay between the different images
of a multiply imaged source, the inhomogeneity cannot be neglected in
a thorough analysis (Kayser & Refsdal 1983). For an example
involving a more traditional cosmological test, Perlmutter et al.
(1995) (see also Goobar & Perlmutter (1995)) discuss using
supernovae with -0.5 to determine
; for The plan of this paper is as follows. In Sect. 2the basics of Friedmann-Lemaître cosmology are briefly discussed; this also serves to define our terms, which is important since various conflicting notational schemes are in use. (For a more thorough discussion using a similar notation see, e.g., Feige (1992).) Sect. 3defines the various distances used in cosmology. In Sect. 4our new differential equation is derived. Similar efforts in the literature are briefly discussed. Sect. 5briefly describes our numerical implementation and gives the details on how to obtain the source code for use as a 'black box' (which however can be opened) for use in cosmology and extragalactic astronomy. The symmetry properties of the angular size distance, analytic solutions and methods of calculating the volume element are addressed in three appendices. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |