The Photometric Plane Relation for Elliptical Galaxies

David Riethmiller

2008 June 11

Abstract:

The Fundamental Plane relation is a well-established correlation among the velocity dispersion of a galaxy, its surface brightness, and its effective radius. However, recent evidence suggests that the correlation may still hold if the galaxy's velocity dispersion is replaced by its Sersic index n. If this latter correlation, termed the Photometric Plane by Graham 2002, proves correct, it may reveal a new fundamental property of elliptical galaxies, as well as reduce time and expense both at the telescope and during data reduction due to its independence of spectroscopic data. The goal of this project is to observe photometrically a sample of elliptical galaxies, obtaining a Sersic index n, a surface brightness μ_e, and an effective radius r_e for each, and to examine the correlation among these parameters. We find this correlation in the Photometric Plane to be described by log(n) = 0.541571 μ_e - 1.30641 log(r_e) - 10.8.

The Fundamental Plane is one of the tightest correlations known in astronomy. Formulated simultaneously by Dijorgovski & Davis (1987) and Dressler et al. (1987), it states that in the region of parameter space spanned by 1) the radius (r_e) at which half of a galaxy's total luminosity is enclosed , 2) the velocity dispersion (σ) of the matter in the galaxy, and 3) the galaxy's effective surface brightness (I_e), a measurement of these three values contributes to a plane, specified by the relation

$\displaystyle r_{e} \sim \sigma^{1.2} I_{e}^{-0.8}$

(1)

This relation produces a plot shown in Figure 1, taken from Dijorgovski & Davis (1987). The Fundamental Plane has been used as an alternative to the Faber-Jackson relation as a distance indicator to elliptical galaxies; by using the effective radius predicted by the Fundamental Plane and the angular distance subtended by this effective radius, we can estimate the physical distance to the galaxy via the small-angle approximation.

Figure 1. Fundamental Plane from Djorgovski & Davis (1987). The figure plots a linear combination of surface brightness and log velocity dispersion against log effective radius; thus, the 3-dimensional plane is rotated for an edge-on view.

1.2 The Photometric Plane

Recent evidence (Graham 2002, Khosroshanhi et al. 2000) suggests that an additional parameter may be added to the Fundamental Plane relation, and the correlation still holds: the Sersic index n, obtained from fitting the form:

(2)

to concentric isophotes of a galaxy, or logarithmically:

(3)

where I represents surface brightness, μ = log(I), r is the radial distance along the semi-major axis, n is the Sersic index as defined in Sersic 1968, and c_n = 2.5b_n and b_n = 0.868n - 0.142 for a range of 0.5 ≤ n ≤ 16.5 (Caon et al. 1993). Other parameters are identical to those in Equation 1.

According to de Vaucouleurs (1948), the radial brightness profile for normal elliptical galaxies is modeled remarkably well using a r^1/4 power law. However, subsequent observations of a large fraction of elliptical and spheroidal galaxies show that this power law is only a first-order approximation, fitting only a limited interval in the luminosity profile (Caon et al. 1993 and references therein). In fact, the Sersic index n was developed as a generalization of the de Vaucouleurs profile. Thus, variations in the Sersic index allow us to examine its correlation with effective radius and surface brightness.

This additional parameter may either be added as a fourth dimension to the Fundamental Plane (making it the Fundamental HyperPlane), or it may simply replace the velocity dispersion to create a Photometric Plane (Graham 2002). The clear advantage here is that observations concerning the Photometric Plane are independent of velocity dispersion and therefore require no spectroscopic data, which reduces time and expense both at the telescope and during data reduction.

Khosroshahi et al. (2000) constructed a Photometric Plane based on near infrared K band images of elliptical galaxies in the Coma cluster, shown in Figure 2, which plots the log Sersic index n as a function of a linear combination of the log effective radius of the galaxy and its central surface brightness. As in Figure 1, this plot is rotated to view the 3-dimensional plane edge-on.

Figure 2. Photometric Plane from Khosroshahi et al. (2000). Tight representation of the best-fit photometric plane. The filled circles represent elliptical galaxies, and the open circles represent bulges of disk galaxies (which are beyond the scope of this project). The line is a plane fit to the data such that log n = (0.173 ± 0.025) log r_e - (0.069 ± 0.007) μ_b(0) + (1.18 ± 0.05).

Khosroshahi's μ_b(0) represents a bulge component (hence the label "b") of surface brightness extrapolated back to the center, such that

(4)

Then we may obtain the extrapolated central surface brightness by assuming that the extent of the "bulge" for elliptical galaxies is similar to the predicted effective radius:

(5)

where b_n is the same as above.

2. Observations

Because of poor weather, the Great Ohio Telescope was not used; instead, the data for the project was obtained from Data Release Six of the Sloan Digital Sky Survey (Adelman-McCarthy et al. 2008) by searching for r-band images of elliptical galaxies, with the criterion that the galaxy is completely contained within a single FITS image (in order to avoid the necessity of mosaic images). Of over 500 locations of elliptical galaxies queried (taken mainly from Burnstein et al. 1987), only five Sloan r-band images were returned from the database: NGC 1052, NGC 4073, NGC 5846, NGC 6109, and UGC 4956. Each of these images was exposed for 53.907456 seconds.

Since our "observations" involved more "data gathering" than actual observing, it is necessary that we list here some additional facts about the targets. Because both the Fundamental and Photometric planes depend on the physical distance of the effective radius, we must know the distance to each of the targets in order to convert between angular size and physical size. Table 1 lists these numbers along with the sources from which they came.

Target Name Distance to Target Source Method of Determination

NGC 1052
22.6 Mpc
Kalder et al. 2003

NGC 4073
99 Mpc
Jenson 2001

NGC 5846
24.9 Mpc
Tonry et al. 2001

NGC 6109
123.3 Mpc
NED Query
H = 73 ± 5 km/sec/Mpc

UGC 4956
68.5 Mpc
NED Query
H = 73 ± 5 km/sec/Mpc

Table 1. List of distance to targets.

3. Reductions

The Sloan data came already fully reduced (i.e. bias-corrected, flat-fielded, dark-corrected, etc.). It was, however, necessary to convert Sloan raw counts to actual magnitudes via the equations

(6)

and

(7)

where f / f₀ may be taken as the flux rate, aa represents the photometric zero point, and kk is the extinction coefficient. A more detailed description of this process is laid out in Lupton et al. 2001 and Lupton et al. 2003.

3.1 Background Sky Subtraction

The background subtraction methods suggested for this type of modeling proved to be inadequate. The first method called for the definition of a simple rectangular region of blank sky, from which the average count rate is determined and then subtracted from the entire image; this was unacceptable because of the density of stars in the field, and also because such a small rectangular region was not sufficiently representative of the entire background sky. The second method required the combination of two or more blank sky exposures, which we did not have.

Instead, for accurate sky subtraction, we used an IDL star-finding routine to mask out the stars in the image, and also excluded the galaxy itself. The median count of the remaining sky area was subtracted from the entire image, resulting in a more accurate background subtraction than simply averaging the count over a small blank sky area determined by eye. A sample graphic depiction of the excluded area may be seen in Figure 3.

Figure 3. Area excluded from the background determination of NGC 1052. Regions circled in red are not considered the background count. The image dimensions are 2048 x 1490 pixels (13.65 x 9.93 arcminutes).

3.2 Isophotal Fitting and Radial Profile

We used the IRAF task "ELLIPSE" (from "STSDAS.ISOPHOT") to fit isophotes to the elliptical galaxies, allowing the position angle and ellipticity to fluctuate but keeping the center fixed. We also masked out prominent point sources to ensure their exclusion from the isophote fit. Then we employed the IDL package "mpfit" (provided by Craig B. Markwardt, NASA/GSFC, available at http://cow.physics.wisc.edu/~craigm/idl/idl.html) to fit a Sersic profile (Equation 3) to the resulting radial brightness profile. Since we expect the most accurate data to come from the intermediate region of the galaxy, as the inner region is dominated by seeing effects and the outer region is more sensitive to our background subtraction, we applied this fit only to the intermediate region (see below). For each segment, the "mpfit" procedure returns the best-fit parameters (Sersic index, effective radius, and surface brightness at the effective radius), along with weighted errors on each.

3.3 Systematic Errors

Because the radial profile's fitting region is defined somewhat arbitrarily, this uncertainty must be factored in as a systematic error. To this effect, we iterate the "mpfit" procedure while varying independently the inner and outer boundaries of the fitting region within 30 percent of their chosen values, generating fit parameters with each iteration; we believe 30 percent sufficiently represents the uncertainty on the boundary locations. Then the standard deviation in the fit parameters represents the systematic error in our fitting procedure. We incorporate both the statistical and systematic errors into the total error on each of the three fit parameters, for all five targets.

4. Results

Figure 4 below shows each of the 5 targets (left) along with their radial brightness profiles (right). Although the "ELLIPSE" task was able to identify point sources lying along the fitting rings, it was less efficient at identifying extended sources; thus, the red boxes overlaid on the images indicate regions that have been masked out manually from the isophote fit. Each figure caption gives the semi-major axis distance of the largest isophote ring in arcseconds, for scale. The fitting range was chosen by eye with the intent to fit only the intermediate regions of the galaxies, making sure to include at least the region from 0.1 r_e up to the effective radius itself.

		Figure 4a. NGC 1052. Left: Isophote fitting, with largest ring at semi-major axis a= 136 arcsec. Red boxes indicated masked regions. Right: Radial profile (white), fitted between 1 and 70 arcsec (green).
		Figure 4b. NGC 4073. Left: Isophote fitting, with largest ring at semi-major axis a= 93 arcsec. Red boxes indicated masked regions. Right: Radial profile (white), fitted between 0.7 and 60 arcsec (green).
		Figure 4c. NGC 5846. Left: Isophote fitting, with largest ring at semi-major axis a= 136 arcsec. Red boxes indicated masked regions. Right: Radial profile (white), fitted between 1.3 and 60 arcsec (green).
		Figure 4d. NGC 6109. Left: Isophote fitting, with largest ring at semi-major axis a= 40 arcsec. Red boxes indicated masked regions. Right: Radial profile (white), fitted between 1.2 and 20 arcsec (green).
		Figure 4e. UGC 4956. Left: Isophote fitting, with largest ring at semi-major axis a= 112 arcsec. Red boxes indicated masked regions. Right: Radial profile (white), fitted between 2 and 40 arcsec (green).

From several of the plots in Figure 4, it is apparent that some elliptical galaxies may have multiple r^1/n laws even within the "intermediate region" (i.e., UGC 4956 exhibits a "double hump" nature which does not fit well with an r^1/4 law). To this extent, we attempt to fit the region which is farther away from the saturated bright center; this also results in a more stable value of χ² for the fit.

Table 2 presents the numerical results to the fitting shown in Figure 4. Errors include both statistical and systematic uncertainties.

Sersic Fit

Target Name Fitting Range (arcsec) Fitting Range (kpc) μ_e n
r_e (arcsec) χ²/_dof

NGC 1052
1.0 - 70.0
0.109 - 7.670
23.240 ± 0.220
5.364 ± 0.249
62.005 ± 7.353
1.101

NGC 4073
0.7 - 60.0
0.336 - 28.798
24.935 ± 0.393
5.406 ± 0.412
49.016 ± 10.938
0.780

NGC 5846
1.3 - 60.0
0.157 - 7.243
22.792 ± 0.172
3.414 ± 0.155
51.706 ± 4.800
1.468

NGC 6109
1.2 - 20.0
0.717 - 11.956
23.140 ± 0.318
3.510 ± 0.393
13.413 ± 2.310
0.973

UGC 4956
2.0 - 40.0
0.664 - 13.284
22.140 ± 0.240
2.870 ± 0.234
33.011 ± 6.142
0.474

Table 2. Best fit values for Sersic fitting within the intervals specified. The uncertainties listed here include both statistical and systematic errors. We attempted to fit similar regions for all 5 targets; in some cases this was not possible. We were successful, however, in fitting the region up to and including the effective radius.

As we stated earlier, both the Fundamental and Photometric Planes depend on the physical effective radius rather than the angular effective radius. Thus, we use the galactic distances listed in Table 1 to convert effective radii in arcseconds into kilo-parsecs. We use a linear combination of the log effective radius along with the surface brightness as the abscissa, and the log Sersic index as the ordinate for the resulting plot of the Photometric Plane (Figure 5). As in Figure 2, note that the horizontal axis exhibits a dependence on a linear combination of surface brightness and log effective radius; this plot is a two dimensional "slice" of the three dimensional plane, rotated for an edge-on view. From the rotation, we find the best fit equation to be log(n) = 0.541571 μ_e - 1.30641 log(r_e) - 10.8.

Figure 5. Photometric Plane, rotated for an edge-on view such that

log(n) = 0.541571 μ_e - 1.30641 log(r_e) - 10.8.

Errors on effective radius and surface brightness are combined via an appropriate error-propagation formula to produce the horizontal error bars here.

5. Discussion

5.1 Comparison with Published Data

We attempt to compare our data with the Photometric Plane defined in Khosroshahi et al. (2000), seen in Figure 2 above. Although Khosroshahi does not give a table with data (i.e. such that we can plot their data), they do provide a best fit equation. We extrapolate our effective surface brightness back to the central surface brightness using Equation 5, in order to match Khosroshahi's prescription. Then in Figure 6, by overlaying our data sample along their fit equation, we attempt to verify visually our data against theirs.

Figure 6. Our data (white) overlaid along the best fit line (red) defined by Khosroshahi et al. (2000).

Unfortunately, the visual inspection of this comparison indicates a lack of agreement (or insufficient data) between the two definitions of the Photometric Plane.

5.2 Conclusions and Future Improvements

We succeeded in reconstructing a Photometric Plane using the Sersic index, the effective radius, and the surface brightness at the effective radius as measured for five elliptical galaxies. However, our reconstruction was inconsistent with published papers on the Photometric Plane.

This project suffered from a lack of available data; both the failure in acceptable weather conditions for the GOT and the limited supply of information from the Sloan Digital Sky Survey contributed to a scarcity in data to reduce. Perhaps if more data were added to that presented here, our Photometric Plane reconstruction might show a more obvious agreement with published data. For similar studies in the future, we recommend that targets be chosen from several surveys as well as the GOT, so that the maximum number of points are available for analysis.

Acknowledgements

A special word of thanks to my classmates and the TAC committee for their useful comments and advice; to Tom Statler for helping us learn how the GOT works, how to mine data archives, and for general guidance through this project; and to George Eberts for allowing us to use his back yard for observing.

...and required by the Sloan Digital Sky Survey:

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.

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