2.2.6 The FP as a Distance Indicator

If the FP is universal (or the deviations from universality are known to be within certain limits), it can be used to determine distances.

If the FP is established for a given cluster of galaxies and in addition the distance to the cluster is known from some other method, the intrinsic FP zero point can be calculated, cf. Eq. (). A recent example of this is Hjorth & Tanvir (1997), who calibrated the intrinsic FP zero point using the observed FP zero point for 5 E and S0 galaxies in the Leo-I group and the HST cepheid distance to the Leo-I galaxy M96.

Without knowing the intrinsic FP zero point ,
the FP can be used to determine *relative* distances.
For example, if we have two clusters HydraI and Coma,
it follows from Eq. () that
their relative distance is related to their observed FP zero point difference as

In order to compare different zero points , i.e. to calculate a meaningful , the same values of and should be used, since is very sensitive to the choice of and . This follows from the fact that is the intersection of the fundamental plane with the axis (cf. Eq. ), and that the galaxies are not symmetrically distributed around this axis, but rather displaced somewhat to the side. The zero point

The subscript ``A'' on the distances *d*
in Eq. ()
indicates that they are so-called
angular diameter distances, cf. Weinberg (1972).
is defined as
,
where *D* is the linear diameter
and
is the angular diameter of the object.
Another distance is the luminosity distance
,
which is defined as
,
where *L* is the (intrinsic) luminosity
and *l* is the apparent luminosity of the object.
In Euclidian geometry the two distances agree with each other
and with the true distance.
In an expanding universe (here given by the Robertson-Walker metric),
this is not the case.
Rather,
and
are related through the redshift *z* as

(Weinberg 1972). The luminosity distance is needed to calculate the distance modulus,

In the absence of peculiar motions (i.e. deviations from the pure Hubble expansion), can be calculated from the redshift

(Weinberg 1972), neglecting terms of higher than second order in

Later (in Sect. ) we will
determine the distance to Coma and HydraI in the following way.
The distance to Coma will be derived from the redshift,
assuming Coma to be at rest in the CMB frame.
The distance modulus can then be found from
Eq. () and ().
The distance to HydraI relative to Coma
will be calculated from the observed FP zero point difference.
By combining
Eq. (), (), and (),
we get the following equation for

where is the CMB redshift for HydraI if HydraI has zero peculiar velocity. Equation () can be solved numerically. The distance modulus can then be found from Eq. () and (). In the same framework, the conversion of from arcsec to kpc becomes

using Eq. (), (), and (), and for HydraI in addition Eq. (). The identity enables us to insert

Properties of E and S0 Galaxies in the Clusters HydraI and Coma

Master's Thesis, University of Copenhagen, July 1997

Bo Milvang-Jensen (`milvang@astro.ku.dk`)