The Dependence of Galaxy Shape on Luminosity and Surface Brightness Profile

32 Pages·2008·0.4 MB·English

The Dependence of Galaxy Shape on Luminosity and Surface
Brightness Profile
R. Anthony Vincent
Department of Physics, The Ohio State University, Columbus, OH 43210
5 [email protected]
0
0
2 and

n
a Barbara S. Ryden
J

2
Department of Astronomy, The Ohio State University, Columbus, OH 43210
1

1 [email protected]
v
1
2
2 ABSTRACT
1
0
5
For a sample of 96,951 galaxies from the Sloan Digital Sky Survey Data
0
/ Release3,westudythedistributionofapparentaxisratiosasafunctionofr-band
h
p absolute magnitude and surface brightness profile type. We use the parameter
-
o fracDeV to quantify the profile type (fracDeV = 1 for a pure de Vaucouleurs
r
t profile; fracDeV = 0forapureexponentialprofile). Whentheapparentaxisratio
s
a q is estimated from themoments of the light distribution, the roundest galaxies
: am
v
arevery bright (M ∼ −23) deVaucouleurs galaxies andtheflattest aremodestly
i r
X
bright (M ∼ −18) exponential galaxies. When the axis ratio q is estimated
r 25
r
a from the axis ratio of the 25 mag/arcsec2 isophote, we find that de Vaucouleurs
galaxies, at this low surface brightness, are flatter than exponential galaxies of
the same absolute magnitude. For a given surface brightness profile type, very
bright galaxies are rounder, on average, than fainter galaxies. We deconvolve
the distributions of apparent axis ratios to find the distribution of the intrinsic
short-to-long axis ratio γ, making the assumption of constant triaxiality T. For
all profile types and luminosities, the distribution of axis ratios is inconsistent
with a population of oblate spheroids, but is usually consistent with a population
ofprolatespheroids. Bright galaxieswithadeVaucouleursprofile(M ≤ −21.84,
r
fracDeV > 0.9) have a distribution of q that is consistent with triaxiality in
am
the range 0.4 . T . 0.8, with mean axis ratio 0.66 . hγi . 0.69. The fainter de
Vaucouleurs galaxies are best fit with prolate spheroids (T = 1) with mean axis
ratio hγi ≈ 0.51.
– 2 –
Subject headings: galaxies: elliptical and lenticular, cD, galaxies: fundamental
parameters, galaxies: photometry, galaxies: spiral, galaxies: statistics
1. INTRODUCTION
The galaxy classification scheme of Hubble (1926) has proved durably useful. As mod-
ified and extended by de Vaucouleurs (1959), it is still the standard method for classifying
low-redshift galaxies with high surface brightness. The Hubble classification scheme was
originally based on the appearance of galaxies on photographic plates. Elliptical galaxies
have smooth elliptical isophotes; spiral galaxies have spiral arms that wind outward from a
central bulgeorbar. It waslater discovered thatforluminous galaxies, the surfacebrightness
profile is strongly correlated with the Hubble type. If the surface brightness I is measured
alongthemajoraxisofagalaxy’simage, itisfoundthatbrightellipticalgalaxieshavesurface
brightness profiles that are well fit by a de Vaucouleurs, or R1/4 law, for which logI ∝ −R1/4
(de Vaucouleurs 1948). By contrast, the azimuthally averaged surface brightness profile of a
spiral galaxy, outside its central bulge, is typically well fit by an exponential law, logI ∝ −R
(Freeman 1970). It was also eventually realized that galaxies of different Hubble type have
different kinematic properties. The disks of spiral galaxies are rotationally flattened, with
stars and gas on nearly circular orbits with little random motion. Bright elliptical galaxies
(M . −20), by contrast, are slowly rotating and are supported mainly by their anisotropic
B
velocity dispersion.
One shortcoming of the Hubble classification scheme, imposed by necessity, is that
elliptical galaxies are classified by their apparent two-dimensional shape, seen in projection
on the sky, rather than their intrinsic three-dimensional shape. Consider an idealized galaxy
whose surfaces of constant luminosity density are concentric, coaxial, similar ellipsoids, with
principal axes of lengths a ≥ b ≥ c; the shape of the galaxy can then be described by
the two axis ratios β ≡ b/a and γ ≡ c/a. Equivalently, the shape can be described by
the two numbers γ and T, where the triaxiality parameter T is given by the relation T ≡
(1−β2)/(1−γ2). If the ellipsoidal galaxy is seen in projection, though, its isophotes will be
concentric, coaxial, similar ellipses. The shape of the projected image can then be described
by the single axis ratio q ≡ B/A, where A and B are the major and minor axis length of
any isophote.
Although knowing the apparent axis ratio q is not, by itself, sufficient to determine
the intrinsic axis ratios β and γ, the three-dimensional shape of galaxies is not beyond all
conjecture. Two approaches to determining the three-dimensional shape of galaxies have
been used. First, the intrinsic shape of an individual galaxy can be modeled if kinematic
– 3 –
data are available in addition to photometric data (Binney 1985; Franx, Illingworth, & de
Zeeuw 1991). However, accurate galaxy modeling requires either multiple long-slit position
angles (Statler 1994) or integral-field spectroscopy (Statler et al. 2004), and is best applied
to systems with high surface brightness and large angular size. The second approach, which
can be used in the absence of kinematic information, is to make statistical estimates of
the distribution of β and γ for a large sample of galaxies. Early estimates of the intrinsic
shape distribution made the assumption that elliptical galaxies were oblate spheroids, with
β = 1 (Hubble 1926;Sandage, Freeman, & Stokes 1970). More recent studies, using accurate
surface photometry, reveal that the distribution of apparent shapes for ellipticals cannot be
explained by a population of randomly oriented oblate spheroids; it can easily be explained,
however, by a population of triaxial ellipsoids (Fasano & Vio 1991; Lambas, Maddox, &
Loveday 1992; Ryden 1992; Tremblay & Merritt 1995; Alam & Ryden 2002).
In this paper, we use the photometry-only approach to constraining the intrinsic shapes
of galaxies, using the Sloan Digital Sky Survey Data Release 3 (SDSS DR3) as our source of
galaxy photometry. Previous studies using the SDSS Early Data Release and Data Release
1 studied the axis ratios of galaxies best fit by de Vaucouleurs profiles (Alam & Ryden 2002)
and those best fit by exponential profiles (Ryden 2004). In this paper, we more closely
examine the relation between surface brightness profile and intrinsic shape. In addition,
we determine, for each profile type, the dependence of intrinsic shape on galaxy luminosity.
For elliptical galaxies, the distribution of apparent shapes is known to depend on absolute
magnitude. Elliptical galaxies with M . −20 arerounder onaverage thanfainter ellipticals
B
(Tremblay & Merritt 1996); for a typical elliptical galaxy color of B−V ≈ 0.9 (Roberts &
Haynes 1994), the dividing luminosity corresponds to M ≈ −20.9. In this paper, given the
V
large sample size provided by the SDSS DR3, we can look for a similar shape dichotomy
not only among galaxies with de Vaucouleurs profiles, but among galaxies with a variety of
surface brightness profile types.
In §2 of this paper, we describe the SDSS and the methods by which we determine the
apparent axis ratio of the galaxies in our sample. In §3, we examine how the apparent axis
ratios depend on the galaxy profile type and the galaxy luminosity, then use nonparametric
kernel estimators to determine the distribution of apparent axis ratios for different samples
of galaxies, subdivided by luminosity and profile type. In §4, we invert the distribution of
apparent axis ratiosto find the distribution of intrinsic axis ratios, assuming galaxies all have
the same trixiality parameter T. In addition to looking at purely oblate galaxies (T = 0)
and purely prolate galaxies (T = 1), we also examine the results for triaxial galaxies with
T = 0.2, T = 0.4, T = 0.6, and T = 0.8. in §5, we discuss the implications of the observed
galaxy shapes for different scenarios of galaxy formation and evolution.
– 4 –
2. DATA
The Sloan Digital Sky Survey (York et al. 2000; Stoughton et al. 2002) will, when
complete, provide a map of nearly one-fourth of the celestial sphere. A CCD mosaic camera
(Gunn et al. 1998) images the sky in five photometric bands (ugriz; Fukugita et al. (1996);
Smith et al. (2002)). The Main Galaxy Sample (MGS) of the SDSS (Stoughton et al. 2002)
will eventually contain ∼ 106 galaxies with r ≤ 17.77; the mean redshift of galaxies in the
MGS is ∼ 0.1, determined from a follow-up spectroscopic survey. The SDSS Data Release
3, issued to the astronomical community in 2004 October, contains 5282 square degrees of
imaging data and 4188 square degrees of spectroscopic data (Abazajian et al. (2005); see
also Stoughton et al. (2002), Abazajian et al. (2003), and Abazajian et al. (2004)).
The SDSS DR3 data processing pipeline provides a morphological star/galaxy separa-
tion, with extended objects being classified as ‘galaxies’ and point-like objects being clas-
sified as ‘stars’. For each galaxy, in each photometric band, a pair of models are fitted to
the two-dimensional galaxy image. The first model has a de Vaucouleurs surface profile (de
Vaucouleurs 1948):
I(R) = I exp −7.67[(R/R )1/4 −1] , (1)
e e
(cid:0) (cid:1)
truncated beyond 7R to go smoothly to zero at 8R . The second model has an exponential
e e
profile:
I(R) = I exp(−1.68[R/R −1]) , (2)
e e
truncated beyond 3R to go smoothly to zero at 4R . For each model, the apparent axis
e e
ratio q and the phase angle ϕ are assumed to be constant with radius. The parameters
m m
q , ϕ , R , and I are varied to give the best χ2 fit to the galaxy image, after convolution
m m e e
with a double-Gaussian fit to the point spread function.
The SDSS DR3 pipeline also takes the best-fit exponential model and the best-fit de
Vaucouleurs model and finds the linear combination of the two that best fits the galaxy
image. The fraction of the total flux contributed by the de Vaucouleurs component is the
parameter fracDeV, which is constrained to lie in the interval 0 ≤ fracDeV ≤ 1. The
fracDeV parameter delineates a continuum of surface brightness profile types, from the pure
de Vaucouleurs (fracDeV = 1) to the pure exponential (fracDeV = 0).
The SDSS DR3 databases provide many different measures of the apparent axis ratio q
of each galaxy in each of the five photometric bands. In this paper, we will use the r band
data, at an effective wavelength of 6165˚A. (We also repeated our analysis at g and i, the
other two bands with highsignal-to-noise, anddid not find significantly different results from
our r band analysis.) Two measures of the apparent axis ratio are given by the best-fitting
axis ratios q for the de Vaucouleurs and exponential models. However, the algorithm for
m
– 5 –
fitting the models introduces quantization in the distribution of q . Because of this artificial
m
quantization, we do not use the model axis ratios as estimates of the true apparent shapes
of galaxies.
A useful measure of the apparent shape in the outer regions of galaxies is the axis ratio
of the 25 mag arcsec−2 isophote. The SDSS DR3 data pipeline finds the best fitting ellipse
to the 25 mag arcsec−2 isophote in each band; the semimajor axis and semiminor axis of
this isophotal ellipse are A and B . The isophotal axis ratio q ≡ B /A then provides
25 25 25 25 25
a measure of the apparent galaxy shape at a few times the effective radius. For galaxies in
our sample with fracDeV = 1, the mean and standard deviation of A /R are 3.12±1.00;
25 e
for galaxies with fracDeV = 0, the mean and standard deviation are A /R = 2.40±0.36.
25 e
Another measure of the apparent shape is q , the axis ratio determined by the use
am
of adaptive moments of the galaxy’s light. The method of adaptive moments determines
the nth order moments of a galaxy image, using an elliptical weight function whose shape
matches that of the image (Bernstein & Jarvis 2002; Hirata & Seljak 2003). The SDSS DR3
adaptive moments use a weight function w(x,y) that is a Gaussian matched to the size and
ellipticity of the galaxy image I(x,y). The adaptive first order moments,
xw(x,y)I(x,y)dxdy
x = (3)
0 R w(x,y)I(x,y)dxdy
R
and
yw(x,y)I(x,y)dxdy
y = , (4)
0 R w(x,y)I(x,y)dxdy
tell us the ‘center of light’ of the galaxRy’s image. With this knowledge, we can compute the
adaptive second order moments:
(x−x )2w(x,y)I(x,y)dxdy
0
M = , (5)
xx R w(x,y)I(x,y)dxdy
R
and so forth. The SDSS DR3 provides for each image the values of the parameters τ =
Mxx +Myy, e+ = (Mxx −Myy)/τ, and e× = 2Mxy/τ. The adaptive second moments can be
converted into an axis ratio using the relation
1/2
1−e
q = , (6)
am
(cid:18)1+e(cid:19)
where e = (e2 +e2)1/2.
+ ×
The adaptive moments axis ratio q , computed in the manner given above, is not
am
corrected for the effects of seeing. The SDSS DR3 also provides the fourth order adap-
tive moments of the galaxy image, and the adaptive moments τpsf, e+,psf, and e×,psf of the
– 6 –
point spread function at the galaxy’s location. These moments can be used to correct for
the smearing and shearing due to seeing; such corrections are essential for studying the
small shape changes resulting from weak lensing (Bernstein & Jarvis 2002; Hirata & Seljak
2003). However, in this paper we will only look at well-resolved galaxies for which the seeing
corrections are negligibly small.
Our full sample of galaxies consists of those objects in the SDSS DR3 spectroscopic
sample which are flagged as galaxies and which have spectroscopic redshifts z > 0.004
(to eliminate contaminating foreground objects) and z < 0.12 (to reduce the possibility
of weak lensing significantly distorting the observed shape). To ensure that galaxies were
well resolved, we also required τ > 7τ . The absolute magnitude M of each galaxy was
psf r
computed fromthe Petrosian r magnitudeand thespectroscopic redshift, assuming a Hubble
constant H = 70kms−1Mpc−1 in a flat universe with Ω = 0.3 and Ω = 0.7. No
0 m,0 Λ,0
K-correction was applied to the data; for galaxies with normal colors at low redshift the
K-correction in the r band is small – less than 0.2 mag for an average elliptical galaxy at
z = 0.12, and even less for galaxies at smaller redshifts or with bluer colors (Fukugita,
Shimasaku, & Ichikawa 1995).
The total number of galaxies in our full sample, selected in this way, is N = 96,951.
gal
Of the full sample, only 919 galaxies have fracDeV = 1, and 26,994 have fracDeV = 0.
The remainder, constituting 71% of the full sample, are best fit by a combination of a de
Vaucouleurs and exponential model, with 0 < fracDeV < 1.
3. APPARENT SHAPES
It remains to be demonstrated that fracDeV is a useful parameter for describing the
surface brightness profile type of galaxies. After all, a linear combination of the best de
Vaucouleurs fit and the best exponential fit will not, in general, be the best possible de
Vaucouleurs plus exponential fit. An alternative method of describing surface brightness
profiles is by fitting a S´ersic profile (S´ersic 1968):
logI ∝ −(R/R )1/n , (7)
e
where n can be an arbitrary number. This fitting method was used, for instance, by Blanton
et al. (2003) in their study of SDSS galaxies. If, in fact, galaxies are well described by S´ersic
profiles, then fracDeV is a useful surrogate for the S´ersic index n. Consider, for instance,
a galaxy whose surface brightness is perfectly described by a S´ersic profile of index n = 2
and effective radius R = X. The best-fitting exponential model for this galaxy (fitting in
e
the radial region 0.1X ≤ R ≤ 10X) has R = 1.10X; the best-fitting de Vaucouleurs model
e
– 7 –
has R = 0.80X. Combining the models gives fracDeV = 0.51 for this n = 2 S´ersic galaxy.
e
A similar fit to an n = 3 S´ersic galaxy yields fracDeV = 0.83. In general, if galaxies have
S´ersic profiles with 1 ≤ n ≤ 4, then fracDeV, as computed by the SDSS DR3 pipeline, is a
monotonically increasing function of the S´ersic index n, and thus can be used as a surrogate
for n.
A plot of the mean axis ratio hqi as a function of the parameter fracDeV is shown
in Figure 1. The mean adaptive moments axis ratio hq i, indicated by the filled circles,
am
shows an increasing trend with fracDeV, from hq i = 0.59 for galaxies with purely expo-
am
nential profiles (fracDeV = 0) to hq i = 0.74 for galaxies with pure de Vaucouleurs profiles
am
(fracDeV = 1). However, thetrendinhqiisnotalinearone; forgalaxieswithfracDeV . 0.5,
the value of hqi is nearly constant at hqi ≈ 0.6; it is only at fracDeV & 0.5 that hqi shows an
increasing trend with fracDeV. The mean isophotal axis ratio hq i, indicated by the open
25
circles in Figure 1, shows less of a trend with fracDeV. Except in the case of nearly pure de
Vaucouleurs profiles (fracDeV & 0.9), the axis ratio of a galaxy in its outer regions doesn’t
seem to depend on its surface brightness profile.
For convenience in analysis, we have divided our galaxy sample into four classes, based
on the value of fracDeV. Galaxies with fracDeV ≤ 0.1, corresponding to a S´ersic index
n . 1.2, are called ‘ex’ galaxies; there are N = 44,289 ‘ex’ galaxies in our sample. Galaxies
ex
with 0.1 < fracDeV ≤ 0.5, corresponding to 1.2 . n . 2.0, are labeled ‘ex/de’ galaxies
(N = 36,645). Galaxies with 0.5 < fracDeV ≤ 0.9, corresponding to 2.0 . n .
ex/de
3.3, are labeled ‘de/ex’ galaxies (N = 13,780). Finally, galaxies with fracDeV > 0.9,
de/ex
corresponding to n & 3.3, are called ‘de’ galaxies. The small fraction of ‘de’ galaxies in our
sample (N = 2237) is partly due to the fact that the centrally concentrated de Vaucouleurs
de
galaxies are less likely to satisfy our resolution criterion, and partly due to the fact that
galaxies with high S´ersic indices are intrinsically rare; Blanton et al. (2003) estimated that
only ∼ 5% of the SDSS galaxies in their sample had S´ersic index n > 3.
The ‘de’ galaxies are rounder in their central regions than in their outer regions: hq −
am
q i = 0.083. This is consistent with the ‘de’ galaxies being relatively bright elliptical
25
galaxies, for which the isophotal axis ratios tend to decrease with increasing semimajor axis
length (Ryden, Forbes, & Terlevich 2001). By contrast, the ‘ex’ galaxies are actually slightly
flatter in their central regions than in their outer regions: hq −q i = −0.017.
am 25
For a given surface brightness profile type, there exists a relation between absolute
magnitudeandapparentaxisratio. Contourplotsofmeanapparentaxisratioasafunctionof
both fracDeV and M are given in Figure 2. (To give a feel for the absolute magnitude scale,
r
fitting a Schechter function to the luminosity function of SDSS galaxies yields M∗,r ≈ −21.4
(Nakamura et al. 2003).) The upper panel of the figure shows that the trend in hq i runs
am
– 8 –
fromtheflattestgalaxiesatfracDeV ≈ 0andM ≈ −18, wherehq i ≈ 0.52,totheroundest
r am
galaxies at fracDeV ≈ 1 and M ≈ −23, where hq i ≈ 0.83. This result is not surprising,
r am
since moderately bright galaxies with exponential profiles are intrinsically flattened disk
galaxies, while extremely bright galaxies with de Vaucouleurs profiles are intrinsically nearly
spherical giant elliptical galaxies (Tremblay & Merritt 1996). More surprising are the results
shown in the bottom panel of Figure 2, which shows hq i, the mean isophotal axis ratio.
25
Here, we find that the apparently flattest galaxies, as measured by q , are not exponential
25
galaxies, but galaxies with fracDeV ≈ 0.7 (corresponding to S´ersic index n ≈ 2.5) and M ≈
r
−20.5; these galaxies have hq i ≈ 0.53. The roundest galaxies, measured by hq i, are not
25 25
bright de Vaucouleurs galaxies, but bright exponential galaxies; the maximum value of hq i
25
is ≈ 0.74, at fracDeV ≈ 0, M ≈ −22.5. Note also that for bright galaxies (M . −21), the
r r
contours of constant hq i in Figure 2 are nearly horizontal. That is, among bright galaxies,
25
the flattening of the outer isophotes doesn’t depend strongly on the surface brightness profile
type.
A view of the dependence of hqi on absolute magnitude for each of our four profile types,
‘ex’, ‘ex/de’, ’de/ex’, and ‘de’, is given in Figure 3. In the upper panel, which shows hq i,
am
note that for each profile type, there is a critical absolute magnitude M at which hq i
crit am
is at a minimum. This critical absolute magnitude ranges from M ∼ −20.6 for the ‘de’
crit
galaxies to M ∼ −19.4 for the ‘ex’ galaxies. At M < M , the value of hq i increases
crit r crit am
relatively rapidly with increasing luminosity; at M > M , the value of hq i increases less
r crit am
rapidly with decreasing luminosity. At a fixed absolute magnitude, the average axis ratio of
‘de’ galaxies is always greater than that of ‘ex’ galaxies; however, for M . −20, the flattest
r
galaxies, on average, at a given absolute magnitude are not the ‘ex’ galaxies, but those with
the mixed ‘de/ex’ and ‘ex/de’ profile types. The bottom panel of Figure 3 shows hq i versus
25
M for the different profile types. In the interval −20 . M . −22, we see that ‘ex’ galaxies
r r
have larger values of hq i than galaxies of other profile types.
25
Tremblay & Merritt (1996) divided elliptical galaxies into two classes; galaxies brighter
than M ≈ −20 are rounder on average than fainter ellipticals. Using a typical color for
B
elliptical galaxies of B−V ≈ 0.9, this corresponds to an absolute magnitude in the r band
of M ≈ −21.2, using the transformation r = B−1.44(B−V)+0.12 (Smith et al. 2002). In
r
our data, we do not see an abrupt jump in hq i for ‘de’ galaxies at M = −21.2, but rather
am r
a more gradual increase between M ≈ −20.6 and M ≈ −22.7. In addition, it is true for
r r
any profile type that highly luminous galaxies are rounder, on average, than less luminous
galaxies.
Knowing the mean shape as a function of M for each profile type (‘de’, ‘de/ex’, ‘ex/de’,
r
’ex’) does not tell us the complete distribution of axis ratios, f(q). In order to compare the
– 9 –
distribution of axis ratios for highly luminous galaxies with that for less luminous galaxies,
we determined, for each profile type, the dividing absolute magnitude M such that the
0
distribution f(q ) for galaxies with M ≤ M is maximally different from the distribution
am r 0
f(q ) for galaxies with M > M . We measure the difference between the distributions
am r 0
by applying a Kolmogorov-Smirnov (K-S) test, and finding the value of M that minimizes
0
the K-S probability P . The values of the dividing magnitude M for each profile type,
KS 0
and the associated probability P , are given in Table 1. (Using a Student’s t-test for the
KS
significance of the difference of the means yields similar values of M , differing by less than
0
0.2 mag from the values found using the K-S test.)
To estimate the distribution of apparent axis ratios f(q) for each of our galaxy subsam-
ples, we use a nonparametric kernel technique (Vio et al. 1994; Tremblay & Merritt 1995;
Ryden 1996). Given a sample of N axis ratios, q , q , ..., q , the kernel estimate of the
1 2 N
frequency distribution f(q) is
N
1 1 q −q
ˆ i
f(q) = K , (8)
N h (cid:18) h (cid:19)
Xi=1 i i
where K(x) is the kernel function, normalized so that
∞
K(x)dx = 1 . (9)
Z−∞
To ensure thatourestimate fˆissmooth, we use aGaussiankernel, withK(x) ∝ exp(−x2/2).
The parameter h in equation (8) is the kernel width; too small a width introduces noise into
i
the estimate, while too great a kernel width produces excessive smoothing. We choose the
kernel width h by using the standard adaptive two-stage estimator of Abramson (1982). In
i
ˆ
this technique, an initial estimate f is made using a global fixed kernel width
0
h = 0.9A/N0.2 , (10)
with A = min(σ,Q /1.34), where σ is the standard deviation and Q is the interquartile
4 4
range of the axis ratios. For samples that are not strongly skewed, this formula minimizes
ˆ
the mean square error (Silverman 1986; Vio et al. 1994). The final estimate f is then
computed, using at each data point q the kernel width
i
1/2
ˆ
h = h G/f (q ) , (11)
i 0 i
h i
ˆ
where G is the geometric mean of f over all values of i.
0
ˆ
The axis ratio for a galaxy must lie in the range 0 ≤ q ≤ 1. To ensure that f = 0 for
q < 0 and q > 1, we impose reflective boundary conditions at q = 0 and q = 1. In practice,
– 10 –
this means replacing the simple Gaussian kernel with the kernel
q −q −q −q 2−q −q
i i i
K = K +K +K . (12)
ref
(cid:18) h (cid:19) (cid:18) h (cid:19) (cid:18) h (cid:19)
i i i
ˆ
Although use of reflective boundary conditions ensures that f remains within bounds, it
ˆ
imposes the possibly unphysical constraint that df/dq = 0 at q = 0 and q = 1.
ˆ
The estimated distributions f(q ) are shown in the left column of Figure 4, for the
am
ˆ
‘de’, ‘de/ex’, ‘ex/de’, and ‘ex’ galaxies; the equivalent distributions f(q ) for the isophotal
25
shapes are shown in the right column. In each panel of Figure 4, the heavier line indicates
the shape distribution for the bright galaxies, and the lighter line indicates the distribution
for the fainter galaxies. The dotted lines indicate the 98% error intervals found by bootstrap
resampling. In our bootstrap analysis, we randomly selected N data points, with substi-
ˆ
tution, from the original sample of N axis ratios. We then created a new estimate f from
the bootstrapped data. After doing 1000 bootstrap estimates, we then determined the 98%
error intervals: in Figure 4, 1% of the bootstrap estimates lie above the upper dotted line,
and 1% of the bootstrap estimates lie below the lower dotted line.
NotefromtheupperleftpanelofFigure4thatthebright‘de’galaxieshaveadistribution
that peaks at q ≈ 0.84, similar to the result found for relatively bright elliptical galaxies
am
(Benacchio & Galletta 1980; Fasano & Vio 1991; Franx, Illingworth, & de Zeeuw 1991;
Lambas, Maddox, & Loveday 1992). The faint ‘de’ galaxies have a flatter modal shape
(q ≈ 0.70) and a broader distribution of shapes. Both the bright and faint ‘de’ galaxies
am
have a relative scarcity of galaxies with q = 1; this is the usual sign that the galaxies
am
in the population cannot be purely oblate. For ‘de’ galaxies, the difference between the
ˆ
adaptive moments shapes and the isophotal shapes is intriguing. Note that f(q ) for the
25
‘de’ galaxies, shown in the upper right panel of Figure 4, peaks at q ≈ 0.76 for the bright
25
galaxies. For the fainter ‘de’ galaxies, the distribution of isophotal axis ratios, though it
ˆ
peaks at q ≈ 0.74, is very broad. An axis ratio distribution f that is nearly constant
25
over a broad range of q is the signature of a population of flattened disks. The distribution
fˆ(q ) drops at q . 0.2, indicating that the disks are not infinitesimally thin, and at q & 0.9,
25
indicating that the disks are not perfectly circular.
The distribution of shapes for ‘de/ex’ galaxies are seen in the second row of Figure 4.
As with all profile types, the bright galaxies are rounder on average than the fainter galaxies.
Both bright and faint ‘de/ex’ galaxies have a scarcity of nearly circular galaxies. Particularly
ˆ
noteworthy isthe distinct difference between f(q ) for thebright ‘de’galaxies, very strongly
am
ˆ
peaked at q ≈ 0.84, and the equivalent distribution f(q ) for the bright ‘de/ex’ galaxies,
am am
which is much broader and peaks at the flatter shape of q ≈ 0.67.
am