M. Hendry1,2,
Simon P. Goodwin2 & John Gribbin2
1. Dept of Physics and Astronomy, University of Glasgow, UK
2. Astronomy Centre, University of Sussex, UK
Introduction
The use of galaxy diameters as a `standard ruler' distance indicator
has been a regular - if somewhat controversial - contributor
to the extragalactic distance scale debate for several
decades. While in principle a very simple and intuitive method -
relying only upon the
assumption that the intrinsic linear diameters of spiral galaxies
are approximately
constant - the application of this indicator to the determination
of the Hubble constant is undermined by several factors:
The intrinsic distribution of linear diameters has a large dispersion
Along with other `standard candle' techniques the distance indicator is highly
susceptible to observational selection effects
Until very recently the number of local calibrating galaxies, with
well-determined intrinsic diameters, was very small
Sandage (1993a,b) has attempted to overcome the first difficulty by
restricting
attention to galaxies of a particular morphological type. He derived
estimates of
H0= 43 ±11 kms-1 Mpc-1 and
H0= 45 ±12 kms-1 Mpc-1
by comparing the angular diameters of field galaxies in the
RSA catalogue which were
`lookalikes' of M101 and M31 respectively. These results have been
questioned, however,
on the grounds that M31 and M101 may not be representative of
the mean linear diameter
of the more distant field population (Ingram & Hogan, 1995).
The mean diameter of spiral galaxies and H0
The Hubble Space Telescope has now provided us with reliable Cepheid distances to more
than a dozen spiral galaxies in the Local Supercluster. This is a sufficiently large
sample to meaningfully determine a mean linear diameter for spirals of Hubble type
similar to the Milky Way.
In Goodwin, Gribbin & Hendry (1997a; hereafter GGH97a) we carry out such an analysis
for local spirals of Hubble type 2 < T < 6, and deduce a sample median linear diameter
(measured at the 25 B mag/arcsec isophote) of ~28 kpc. We find on this
basis that the Milky Way has a diameter very typical of galaxies of similar Hubble type,
while M31 and M101 are both larger than the average. (See Figs. 1 and 2 of GGH97a).
In Goodwin, Gribbin & Hendry (1997b; hereafter GGH97b) we use this sample of local
spirals in order to determine H0 by comparison with the angular diameters of a sample
of galaxies, with 2 < T < 6, extracted from the RC3 catalogue (de Vaucouleurs et al.,
1991). Our analysis involves the following model assumptions:
The intrinsic distribution of log linear diameter is a Gaussian with mean
L0 and dispersion {sigma}
The redshifts of the RC3 field galaxies reasonably approximate uniform
Hubble Flow
The apparent angular diameter selection function is well-described by a step
function, complete to 1 arcmin.
The presence of the (assumed) sharp angular diameter limit introduces an observational
selection bias, in that the distribution of log linear diameter for observable
galaxies is systematically different from the intrinsic population. A failure to properly
account for this effect will result in a systematic bias in the estimate of H0.
In order to ensure that our assumption of quiet Hubble Flow and a sharp angular
diameter selection function is reasonable, we impose in addition a selection function
on redshift, adopting a sharp lower and upper cut-off to our sample of RC3
galaxies. An important consequence of this redshift selection is, however, the
introduction of another selection bias (c.f. Gould, 1993), significantly
modifying the standard bias correction of Malmquist (1920). In particular, ignoring
the `Gould effect' and applying the Malmquist correction for angular diameter selection
only will result in a completely wrong bias correction.
We determine an estimate of H0 as follows:
Adopting a fiducial value of H0 (e.g. H0= 60) we infer the sampled
distribution of log linear diameter for observable galaxies from the RC3 catalogue.
Comparing this sampled distribution with that predicted from our model of the
intrinsic distribution and selection function, we deduce the mean and dispersion
of the intrinsic Gaussian distribution of log linear diameter.
We match the mean log linear diameter of the remote RC3 sample (with adopted
fiducial H0) to the mean of the local calibrators by adjusting the value of
H0 - which we then adopt as our estimate.
Results and discussion
We find that the distribution of log linear diameters for the local calibrators is
well-fitted by a Gaussian with mean value
<L_0>_local = 3.39±0.12
>From the observed distribution of log linear diameter for the remote RC3 sample,
we select 1388 galaxies with 1500 kms-1 < cz < 5000 kms-1. For these data
we infer an intrinsic mean log linear diameter (assuming H0= 60) of
<L_0>_RC3 = 3.25 ±0.02
after correction for angular diameter and redshift selection biases.
In order to match the means of the remote and nearby samples we require:
H_0 = 52 ±6 kms^-1 Mpc^-1
As a further means of placing limits on the likely value of H0 we
consider the order statistics of the modelled intrinsic
distribution of log linear diameters (c.f. Hendry, O'Dell & Collier-Cameron
1993). For the sample of 12 calibrating galaxies, which we are assuming to be
drawn from the same intrinsic population as the distant sample, we
pose the question of how likely it is - for a given value of H0 -
that one would obtain e.g. as small a galaxy as the galaxy with smallest
linear diameter, or conversely as large a galaxy as that with largest
linear diameter. We find on this basis that value of H0> 80 are
unlikely (with a probability less than 1%) since the largest galaxies in the
local sample would then be unreasonably large compared with the distant galaxies.
Values of H0< 35 are excluded with a similar level of confidence,
since then the smallest galaxies in the local sample would be unreasonably small
compared with the distant galaxies.
How reasonable is the assumption that the local calibrators are indeed a
representative sample drawn from the distant intrinsic population? One
immediate advantage of having a larger calibrating sample than the single
object calibrators used by Sandage (1993a,b) is that one can compute the
sample variance (or equivalently sample dispersion) of linear diameter for
the local galaxies and compare it with the variance of the intrinsic
population - as deduced from the remote sample of observed galaxies after
correction for angular diameter and redshift selection. If the local sample
is assumed to be drawn from the intrinsic population, but
perhaps systematically from e.g. the upper half of the log linear diameter
distribution, one may pose the question of how much greater the mean of the
local sample might be compared to the mean of the intrinsic distribution, and
yet still have a sample variance compatible with the value observed for
the real data.
We have attempted to answer this question via Monte Carlo simulations, and
determine an estimate of 8 kms-1 Mpc-1 for the maximum
possible systematic error on H0 from asymmetric sampling of the diameter
distribution, still consistent with the observed sample dispersion. Thus we
estimate H0 from our analysis to be:
H_0 = 52 ±6 ±8 kms^-1 Mpc^-1
Of course this analysis is purely statistical and takes no account
of what prior information we have on both the strategy of the HST Key
Project and also the physical conditions of the Local Supercluster.
In particular, our purely statistical assessment of the systematic error,
based on the observed sample variance, makes no distinction between the
likelihood of selecting local galaxies which are preferentially too large
or too small. From the fact that one of the principal goals of the Key
Project was to calibrate the Tully-Fisher relation (Kennicutt, Freedman
& Mould 1995) we know that the systematic error in reality is more likely
to select galaxies which are preferentially larger than the average
- so called ``Grand
Design spirals''. One could, e.g., attempt to understand more precisely the
galaxy selection procedure adopted by the Key project. We will address
this issue in more detail in future work, but for the moment consider
our purely statistical estimate of the systematic error in our determination
of H0 to be an adequate first approximation - subject to the caveat that the
true systematic error is more likely to increase the value H0
because of the reasons outlined above.
Finally we considered the impact on our results of changing the model for the
selection function. We adopted a model where the selection function is, in
general, not a step function but `fades out' as one approaches an angular
diameter of 1 arcmin. We allowed two parameters to vary: the value of
the selection function at an angular diameter of 1 arcmin, and the value of
the angular diameter at which the selection function falls below unity - i.e.
the angular diameter, {theta}comp, above which the catalogue is complete.
We also considered two different cases for the shape
of the selection function between {theta}= {theta}comp and
{theta}= 1 arcmin: a `fade-out' which was linear in log{theta} and
a smoother fade-out described by a half-Gaussian.
In general we found that the goodness of fit of the observed
distribution of log linear diameter to the predicted distribution
was not sensitive to the parameters
of the selection function, thus indicating that our estimate of H0
from the particular model of a sharp selection function was robust.
In particular we found that the rejection of H0> 80 - identified
at the 1% level by consideration of the order statistics of the local
sample and also by the poor fit to the predicted cumulative distribution of
log linear diameter for observable galaxies - was robust to all
attempted variation of selection function parameters.
Conclusions
The results presented in this report strongly suggest that values of H0
signficantly higher than 75 kms-1 Mpc-1 or below 40 kms-1 Mpc-1
are ruled out. Clearly this is not a dramatically new conclusion, but
should serve to underline that the simple idea of comparing the linear size
of local and distant galaxies is certainly a plausible method for
estimating H0. A value of H0 in the range 50 - 55 kms-1 Mpc-1
gives a good fit between the mean log diameter of the local
and distant samples and
renders the statistical properties of the largest and smallest local galaxies
easily compatible with those predicted from the distant sample. This result
is also consistent with a number of independent recent estimates from
other distance indicators.
We are currently developing an improved, iterative, method which will
determine simultaneously and in a self-consistent way estimates of the
diameter selection function parameters and the moments of the intrinsic
distribution of log linear diameter. This ongoing work is in recognition that
the width of the diameter distribution function - even for galaxies of similar
Hubble type - requires both a sizeable sample of local calibrators and
a careful treatment of observational selection effects if a meaningful estimate
of H0 is to be obtained. Nevertheless, it seems to us that galaxy diameters
as a straightforward `standard ruler' method still has something to offer as a useful
adjoint to more precise distance indicator techniques used to estimate H0 in
recent literature.
Full versions of GGH97a and GGH97b can be found on the astro-ph preprint server:
Both papers have been somewhat modified, however, since their submission to astro-ph in
the light of referee's comments. Updated versions are available by anonymous ftp
on the server neptune.astro.gla.ac.uk. Users should login in as `anonymous' and
change directory to `pub/martin'. GGH97a is archived as the uuencoded file
`milky_way.uu';
GGH97b is archived as the uuencoded file `hubble.uu'.
Acknowledgements
MAH would like to thank the organisers of the workshop `How Far Can You Go?' for
a most enjoyable, informative and stimulating meeting.
References
Gould A 1993, ApJ, 412, 55
Goodwin S. P., Gribbin J., Hendry M. A. 1997a, MNRAS,
submitted, LANL preprint no. astro-ph/9704216 (GGH97a)
Goodwin S. P., Gribbin J., Hendry M. A. 1997b, AJ,
submitted, LANL preprint no. astro-ph/9704289 (GGH97b)
Hendry M. A., O'Dell M. A., Collier-Cameron A. 1993,
MNRAS, 265, 983
Ingram D. R., Hogan C. J. 1995, AJ, 110,, 634
Malmquist K. G. 1920, Medd. Lund. Astron. Obs., 20
Sandage A. 1993a, ApJ, 402, 3
Sandage A. 1993b, ApJ, 404, 419
de Vaucouleurs G., de Vaucouleurs A., Corwin H. G.,
Buta R. J., Paturel G., Fouque P. 1991, Third Reference Cataloge
of Bright Galaxies (RC3).