4.2. Mass-to-Luminosity Ratios and Projection Factors
It is known that galaxies vary in both luminosity and mass over ranges of hundreds to thousands. This cannot be said about the ratio of the integrated mass of a galaxy to its integrated luminosity, which depends only weakly on the absolute magnitude of the objects and is confined to a relatively small range of values. The data fully support the proposition that, for double galaxies as well, the ratio of the total mass of the components to their total luminosity, f0 = (M1 + M2) / (L1 + L2), is limited to a rather small range of values.
By way of contrast to f0, the estimator of the ratio of orbital mass to luminosity
![]() |
(4.15) |
exhibits marked variation from pair to pair due to the role of the
projection factor,
.
The distribution of 585 double systems according to estimates of f is
shown on a logarithmic scale in figure 21.
We will use these data to estimate the distribution of pairs according
to the true mass-to-luminosity ratio, f0.
![]() |
Figure 21. |
For sufficiently large samples of double galaxies with random orientations to the direction of observation, an estimate of f may be viewed as the convolution of two independent quantities,
![]() |
(4.16) |
substituting the integration over the entire number of pairs for the
integration over viewing angles. For mutually uncorrelated quantities
and
f0,
the following relation may be obtained between the n-th moments
of the distribution
![]() |
(4.17) |
If the motion of double galaxies is circular, these moments of the projection factors become Gamma functions of the given argument:
![]() |
(4.18) |
In the case of purely radial motion of the galaxies, it is easy to show that
![]() |
(4.19) |
Analogous expressions for the moments of the projection factor may be obtained from (4.11) and (4.12) for the general case of elliptical motion with fixed orbital eccentricity e.
Therefore, equation (4.17) allows one to obtain the moments of the true distribution of the physical quantity f0 from the moments of f, computed from observations, if the type of the orbital motion of double galaxies is known. Knowing the moments f0n it is possible to establish the standard deviation of the true distribution p(f0).
However, the relation (4.17) between moments of observed and true mass-to-luminosity ratios describes only an idealized situation in which errors of measurement of the radial velocities of the galaxies are negligibly small. In reality the differential radial velocity of the components of a pair may be viewed as the sum of the true difference, y0, and random errors of measurement u:
![]() |
(4.20) |
Assuming that, for the total sample of pairs, y0 and u are mutually independent random variables, we have from (4.15) and (4.20)
![]() |
(4.21) |
where L = L1 + L2 is the total luminosity of the pair components. Together with the simple relation (4.17), we now have a relation between adjacent moments:
![]() |
(4.22) |
It is obvious that the projected separation between double galaxies and the orientation factor of the pair are not, in fact, independent quantities. In addition, the pair luminosity L and the mass-to-luminosity ratio f0 may be correlated with one another and with the relative separation of the components, due to the physical properties of double galaxies, such as extensive envelopes around them or, in fact, the previously discussed selection effects. Therefore, the practical application of expression (4.22) is a complex matter.
The distribution of errors in radial velocity may be accurately described as a symmetric function (i.e. <u> = 0) and for its first moment, from (4.22), we have
![]() |
(4.23) |
where
fu
indicates the correction to the mass-to-luminosity ratio
required by errors in the measured radial velocity.
To go further, we introduce the limiting assumption that the quantities
X, L,
, and
f0 are mutually uncorrelated.
Then, the recurrence relation (4.22) becomes
![]() |
(4.24) |
Thus, having calculated the sample moments of the estimator f for the
double galaxies and incorporating the correction
fu
for errors
in radial velocities, we may finally establish from (4.24) the mean value,
dispersion, skewness, and kurtosis of the true quantities
f0 for
a given type of orbital motion, and from this establish the form of the
distribution p(f0).
Before moving on to this question, we will deal with the problem of possible selection in the distribution of pairs according to viewing angle. Turner (1976b) and Faber and Gallagher (1979) showed that the assumption of a mean projection factor established according to (4.7) requires a significant correction for the effects of a non-random orientation of pairs. This comes from the fact that the basic selection of double galaxies from their relative angular separations operates against the selection of wide pairs. According to the estimates of this effect by these authors, neglect of this correction reduces the mean derived orbital mass by factors of a few.
This assumption about anisotropic selection may be
examined using the model pairs.
For each M-pair, both the true separation r and its projection
X are known.
For a uniform random orientation of pairs and uniform selection according
to angular separation, the basic relation is
<X / r> =
/ 4
0.785.
For the sample of M-pairs satisfying the isolation criterion, the mean
value of this ratio is in fact 0.763, which satisfies the prediction,
within the statistical errors.
It follows that the selection of double galaxies on the basis of their
isolation does not introduce a noticeable anisotropy in the apparent
orientation of the pair, and systematic errors in estimates of the mean
orbital mass should then be negligible.
In order to establish the density distribution of pairs according to their
mass-to-luminosity ratio, p(f0), it is
necessary to exclude false double systems from the sample.
The desired rule for this, apparently, does not exist.
As shown by the methods above using random distributions, the number of
optical and non-isolated double systems in the catalogue is about 43%.
Optical pairs having formal estimates of
f = 103 - 106 are
easy to distinguish from physical pairs, but
the problem of dealing with members of groups and clusters is
much more difficult even when they satisfy the isolation criteria for
the catalogue. For characteristic separations ~ 50 kpc and
peculiar velocities ~ 400 km/s, pseudo-pairs that are close
in projection have values occupying the rather large
range f = 100 - 1000.
Therefore, for false, non-isolated pairs we apply the criterion
f > f* = 100.
On this basis, we may exclude from the catalogue
98/585 17%
of the objects, which is 2.5 times fewer than the fraction predicted
from the modelling.
We deal further with only this sub-sample, consisting of 487 pairs with
f < 100, realising, however, that it may contain a significant
contribution from non-isolated systems.
(If we further restrict the adopted boundary between physical and
false pairs, say f* = 50, then we risk
excluding from the analysis
actual massive pairs which would satisfy the same selection.)
The distribution of these 487 pairs with f < 100
is presented in figure 22.
The histogram shows the distribution of f without regard to
errors in the radial velocities.
The points indicate the distribution of the corrected estimate
fc = f -
fu.
Note that the displayed quantity fc includes pairs for
which the errors in measurement of the radial velocity difference
approach the absolute value of the measured difference
(
y >
y).
Comparison of these distributions shows a significant effect of the error
y on
estimates of the mass.
As already remarked in discussing the episodic measurement of radial
velocities for pairs performed by various authors, considering the known
errors it is just barely possible to establish the function
p(f0).
![]() |
Figure 22. |
Proceeding to pare down and improve the observational material, we further
exclude from the analysis pairs for which the orbital mass-to-luminosity
ratio exhibits a large expected error.
Among 487 pairs we identified 286 having a correction
fu
< 2 (sub-sample A).
As a control we also examined sub-samples B and C which are limited
respectively by the criteria
fu
< 5 and
fu
< 10.
The small corrections resulting from errors in measured velocities
weaken the role of any possible correlations between random
changes in expression (4.22), permitting the use in its place of
the simplified expression (4.24),
which does not incorporate mixed moments.
The basic characteristics of these three sub-samples are presented in
Table 8, which lists n, the number of
pairs;
fu,
the mean value of the correction resulting from velocity errors; <
f >, the mean
corrected estimate of the orbital mass-to-luminosity ratio;
<f2*> /
<f2>, the characteristic dispersion;
1(f )=
<f3*>2 /
<f*2>3, the
normalised index of skewness; and
2(f )=
<f4*> /
<f*2>2, the
normalised index of kurtosis, where
<fk*> indicates the central
limit of the k-th moment.
To estimate the moments
<f0k> we proceeded as follows.
Using (4.11) and (4.12) for a series of values of the orbital
eccentricity e (0.05, 0.15, 0.25 ... 0.95), we generated
a distribution of projection factors using Monte Carlo methods.
For each value of e, we generated 105 trial values for
the moments <k>.
Note that for purely circular and purely radial motion the moments
of the projection factor have the analytical forms (4.18) and (4.19).
Together with (4.24) the sample moments <fk> and
fku
were finally transformed into moments of the true mass-to-luminosity ratio
<f0k>.
Analytical expressions for the distribution
p(f0) were produced with the aid of Pearson
diagrams
(Gunn and Shapiro, 1967)
incorporating the sample values of the skewness and kurtosis indices
1 and
2.
The results of these calculations are presented in
Table 9 and in Figure 23.
The Pearson diagram, figure 23, requires some
explanation.
Two of its coordinates are the dimensionless indices of skewness
1
and kurtosis
2
which uniquely characterize the form of the distribution.
For example, uniform and normal distributions appear on the
(
1,
2)
diagram as the points (0, 1.8) and (0, 3).
An exponential distribution on this diagram appears at point (4, 9).
A gamma distribution has the form of a straight line, shown as the dot-dash
line, extending beyond the two last points.
A log-normal distribution is the light solid line below this.
The shaded part of the Pearson diagram indicates a critical zone
[
1
< 0 and
2
<
1
+ 1], where no actual distribution of the desired form can exist.
Between this critical line
2
=
1
+ 1 and the line
corresponding to gamma functions is the region of beta distributions.
This consists of zones for the U-like or bi-modal, and J-like or
uni-modal, distributions.
The last of these is shown as the short dashed curve in
figure 23.
![]() |
Figure 23. |
In the lower right corner of figure 23, the
three open circles indicate the
properties of the distributions p(f ) for sub-samples A, B
and C, using the data from Table 8.
All of the values
1
and
2
for the projection factors
(second and third lines of Table 9) occupy a
uniform sequence in the Pearson diagram, shown as the narrow dashed line.
The points along it indicate the values expected for various orbital
eccentricities.
As will be shown later, the skewness and kurtosis of the distribution
of
rapidly
increase with the value of the eccentricity, e.
The function p
(
) for any
desired e will form a J-type or beta distribution.
This distribution for circular orbits was presented in figure 18.
We remark on one final property. For
e
0,
the parameters
1
and
2
asymptotically approach the values for pure circular motion.
However, for
e
1,
elliptical motion does not have as its
asymptotic case the values for pure radial motion.
The distribution of projection factors for radial motion forms a point at
coordinates
1
= 3.66 and
2
= 6.40.
Such a `degeneracy' for radial motion is characteristic only of the higher
order moments, since, according to (4.13) and (4.14), the first two moments
do not show such a distinction.
We return to the question of the form of the true distribution p
(f0).
Its trajectory on the Pearson diagram is shown by the sequence of points
for various values of e, connected by the dashed line.
To avoid cluttering figure 23 we present this
information only for the basic sub-sample, A.
The trajectories of sub-samples B and C may easily be produced from the data
in Table 9. The values of
1
and
2
for e = (0.0 - 0.30) lie in the
region of the J-like beta distributions.
On going to more elliptical orbits, e = (0.30 - 0.55), the
distribution lies in the region which is U-shaped, and for e >
0.55 it appears in the critical region.
In other words, a hypothesis of orbital motion with e > 0.55
does not have any imaginable physical significance.
From the data in Table 9 it is apparent that
inclusion in the samples
of objects with poorly measured radial velocities only serves to narrow
the permitted range of orbital eccentricities. For pairs with
fu
< 5, the function p(f0) has
a U-form (except for e < 0.35), and for the sub-sample with
fu
< 10, all possible values of the eccentricity lie in the
forbidden region. Therefore, the observed data point towards orbital
motions of double galaxies
characterized by small values of the eccentricity (e < 0.55).
To assign a definite value and to limit discussion of the M/L
distribution of pairs, we will assign them the value for almost circular
orbits, with e < 0.30.
For calculation we will adopt the assumption presented above of a pure
circular motion of galaxies in pairs. Then, after calculation of the
parameters
1
and
2,
the derived distribution p (f0) may be
presented as
![]() |
(4.25) |
where c is a normalisation constant and
f0max = 30.0.
Thus, the summed mass-to-luminosity ratios for double galaxies exhibit a
distribution in the interval [0-30] with mean value
<f0> = 7.75 and standard deviation
(f0) =
6.47. A graphical representation of (4.25) is given as the short dashed
line in figure 22.
We will note here one important circumstance.
For a distribution of the form (4.25), technically the estimate of
f for double systems may be shown to have a maximal value
fmax = (32 /
3)
f0max = 102, and this value
practically matches the limit f* = 100 which
was selected as
the basic boundary condition between physical and false pairs.
However, the analytical description of the function p (f0) is not absolute. For the sample moments, a description of these distributions may only be approached in iterative steps. The real value of expressions for p (f0) is limited not only by statistical fluctuations but also by inclusion in the sample of false pairs. Further, the form of the derived distribution will be affected by errors in measured radial velocities.
Ever-increasing interest has been evident in the calculation
of the distribution of individual mass-to-light ratios
p (fin)
for the components of pairs, with the help of (4.25) for
the ratio of integrated mass to luminosity for double galaxies.
We will illustrate this with some examples.
Let pair members have identical luminosities and let their
mass-to-luminosity ratios be uncorrelated.
Then, the skewness and kurtosis indices of the distributions of
f0 and fin will be given by
1(fin) =
2
1(f0) and
2(fin) =
2
2(f0). Having adopted
1(f0) = 0.75 and
2(f0) = 2.88
(sub-sample A, circular orbits), we find for
p (fin) values
1
and
2
agreeing with the log-normal distribution. According to
Shaw and Reinhart (1973)
just such a form is shown by the
distribution of selected galaxies in their individual mass-to-light ratios.
The question of comparing orbital and individual masses of galaxies will be
examined in detail in section 4.4.