Surface Photometry and Dynamical Properties of Lenticular Galaxies: NGC3245 as Case Study ()
1. Introduction
Surface photometry is a bidimensional broadband technique to quantitatively describe the light distribution of extended objects like galaxies and HII regions. It is a technique rather than a distinct field of research. Reynolds [1] was the first one to try applying surface photometry on galaxies so it is considered as one of the oldest techniques in modern astronomy [2].
Surface photometry is extremely important since it helps us to get information on galactic colors and its implied ages and metallicity gradients [3], stellar populations [4-7], dust content and its extinction [5,8-11], and structure, formation and evolution of galaxies [12,13].
Surface photometry is usually based on fitting ellipses to the isophotes of galaxies especially for ellipticals and lenticulars whose their isophotes show little deviation from being perfect ellipses. Several software packages and tools can perform surface photometry; among them are GALPHOT [14], GASPHOT [15], GALFIT [16], GIM2D [17], and ISOPHOTE.
The package that concerns us here is the ISOPHOTE. The ISOPHOTE’s principle task is the ELLIPSE task which does the essential role of fitting the elliptical isophotes to the galaxy image. In addition to ELLIPSE, ISOPHOTE includes some “parameter set” tasks that control the process of ELLIPSE execution and other tasks that test the ELLIPSE performance by examining its results. The algorithm upon which ELLIPSE is based and how to deal with the various tasks is well described in [18] and [19]. The result of applying ELLIPSE task is a table containing the variation of many important quantities, like intensity, ellipse shape parameters, and Fourier coefficients which quantify the amount by which isophotes deviate from perfect ellipses, with semi-major axis. The most important parameter, on which we are interested here, is the intensity distribution. The first goal of this paper, is the establishment of a new relation to describe the intensity profile I(a), in contrast to the usual trials of describing I(r).
On the other hand, intensity profile I(r) plays an important role in finding the distributions density, mass, and potential which play a key role in the understanding of the galactic dynamics. We will derive these dynamical properties in terms of intensity profile I(r), which is the second goal of this paper. This is done by fitting it by a suitable model. A number of models have been put to describe the relation I(r), easily obtained from I(a), the most accepted ones are the Sérsic model for bulges and the exponential law for disks.
In the present paper, we applied the surface photometry on the g-band image of the galaxy NGC 3245 obtained from the Sloan Digital Sky Survey (SDSS). The resulted data are shown in Appendix B. Its intensity profile is well fitted by the Sérsic model at n = 2.9, then , we Substituted by Sérsic formula in our derivations for dynamical properties.
NGC 3245 (UGC 5663) is a late S01 galaxy. It is composed of an extremely bright, mildly active nucleus surrounded by a lower-surface brightness (but still very bright), smooth lens ending with a diffuse, faint outer envelope. The nucleus is spherical while both the lens and the envelope have an E5-like flattened structure [20-22]. Jian Hu [23] suggested the coexistence of a central classical bulge and outer boxy bulge in this galaxy since he found it as a boxy one with bulge Sérsic index ~4. Using measurements of surface brightness fluctuations, Tonry i [24] determined its distance modulus as 31.6 ± 0.20 (a distance of 20.9 Mpc) [25].
Section 2 describes the basic formulations including the linear least-square modeling of data and some basic statistics. Section 3 presents the new relation for I(a). Section 4 gives a brief description on the Sérsic model and the results of fitting. In Section 5, we derive in detail various dynamical quantities in terms of I(r). The conclusion is given in Section 6. Finally, statistical analysis of ELLIPSE output data is shown in Appendix A.
2. Basic Formulations
2.1. Linear Model Analysis of Observational Data in the Sense of Least-Squares Criterion
Let z be represented by the general linear model of the form 
where 
are linearly independent functions of x. Let c be the vector of the exact values of the c's coefficients and 
the least-squares estimators of c obtained from the solution of the normal equations of the form 
The coefficient matrix 
is symmetric positive definite ,that is , all its eigen values 
are positive. Let 
denotes the expectation of f and 
the variance of the fit, defined as:
(1)
where
(2)
N is the number of observations, y is the vector with elements 
and 
has elements
. The transpose of a vector or a matrix is indicated by the superscript “T”. According to the least-squares criterion, it could be shown that [26]
1) The estimators 
by the method of least-squares gives the minimum of
.
2) The estimators 
of the parameters c, obtained by the method of least-squares are unbiased; i.e. 
3) The variance–covariance matrix 
of the unbiased estimators 
is given by:
(3)
4) The average squared distance between 
and c is:
(4)
Finally it should be noted that if the precision is measured by the probable error 
then
(5)
2.2. Coefficient of Correlation
A coefficient of correlation is a statistical measure of the degree to which two variables x and y (say) are related to each other. In case of a linear equation, the coefficient of correlation is (e.g. [27])
(6)
where
indicates that the two variables are totally correlated, R = 0, no relationship between them, and 
indicates that there is a trend between the two variables x and y. The sign of R indicates whether y is increasing or decreasing when x increasing, while its magnitude indicates how well the linear approximation is.
2.3. Some Basic Statistics
For data analysis of the present paper we used some basic statistics of these are:
1) Descriptive statistic; 2) Location Statistics; 3) Dispersion statistics; 4) Shape statistics.
2.4. Autocorrelation
Autocorrelation is important in time series analysis. Let 
be the autocorrelation at lag k. An estimate of 
is [28]:
(7)
where 
is the mean of the data
.
3. New Relation between I and a of the Lenticular Galaxy NGC3245
In what follows, new relation describing the intensity profile along the semi-major axis, I(a) for the lenticular galaxy NGC3245 will be established in the sense of least–squares criterion of Section 2.1. The data used for this relation are the first two columns of Table I of Appendix B resulted by applying the IRAF ELLIPSE task on the SDSS g-band image of the galaxy.
Moreover, some basic statistics of independent variable (a) and the dependent variable (I) of this relation are given in Appendix A.
The relation and its error analysis are:
3.1. The Fitted Equation
.
3.2. The c’s Coefficients and Their Probable Errors
.
3.3. The Probable Error of the Fit
.
3.4. The Average Squared Distance between c and 
.
3.5. The Observed and Computed Data
4. Sérsic Model
The intensity distribution along the equivalent radius can be expressed by Sérsic model [29,30]
where re is the effective radius, the radius encloses half of the whole of the galaxy, Ie is the intensity at this radius, and bn can be given by the expression
Detailed deduction and approximation of bn is discussed in Graham & Driver [31] ([32]).
Starting by the above expression of I(r), Caon et al. [33] converted Sérsic’s equation to the following formula that describes the surface brightness distribution
by using the formula
hence, by definition, µe is the surface brightness at re.
By fitting the intensity profile of the SDSS g-band image of NGC3245 by Sérsic model we found that the profile is well fitted at n = 2.9.
Using the first three columns of Table I of Appendix B with 
we get for fitting the intensity profile I(r) by Sérsic model in the sense of least-squares criterion of Section 2.1. the following:
4.1. The Fitted Equation
.
4.2. The c’s Coefficients and Their Probable Errors
4.3. The Probable Error of the Fit
4.4. The Average Squared Distance between c and 
4.5. The Observed and Computed Data
From the values of c1 and c2, we get the effective radius re = 18.95″, whereas µe = 17.5578 mag/arcsec2.
5. Dynamical Properties
The dynamical properties of galaxies can be easily obtained if the intensity profile along radius I(r) is available. As I(r) is the main output from the surface photometry technique, distributions of properties like density, mass, potential, escape and circular velocities can be found as follows:
5.1. Density Distribution
The density distribution is given by
where γ is the mass to light ratio.
Sérsic defined I(r) as
If we defined k = bn/re1/n and A = ebn, Sérsic equation can be written as
Then, ρ(r) can be written as
5.2. Mass Distribution
The mass enclosed by a given radius r is
.
Substituting by the formula of ρ(r)
.
Putting
, and
, then M(r) becomes
.
Let us consider the general integral
.
Integrating, by parts, we get.
Substituting for Q0, Q1, Q2
, we can deduce the relation
going back to the initial values of R and m, the final mass descriptive equation is resulted.
If n is a positive fraction , then we have to consider 
where [t] is the greatest integer 
(for our case n is taken as 3).
5.3. Distribution of Potential
The potential in terms of ρ(r) is given by
where G is the gravitational constant. 
can be regarded as
where
by following similar integration steps as that done for mass, 
(r) equals
substituting by both expressions of 
and
, the final equation of 
is
5.4. Distribution of Escape Speed
The escape speed 
is given by
substitution by 
results in
5.5. Distribution of Circular Speed
by substituting for the expression of 
and differentiation, we get
6. Conclusions
In this paper, surface photometry is applied on the lenticular galaxy NGC 3245 as a case study to resulting in the new relation Log I(a) = c1 + c2 a0.44 , we hope to generalize this relation in the future by applying a large descriptive sample of galaxies.
Since the intensity profile I(r) is well fitted by the Sérsic r1/n model where n = 2.9, we derived relations for various dynamical properties in terms of Sérsic model, strengthening the usefulness of the surface photometry technique.
Both relations, I(a) and I(r), are accurate as judged by a given precision criteria based on linear least-squares fitting criterion. Correlation coefficients between some parameters of the isophotes are also computed.
Appendix A
Analysis of the Semi-Major Axis a and the Intensity I of the Lenticular Galaxy NGC3245
I-Coefficients of correlation between ellipse task parameters Correlation coefficient between a & I= –0.430894 Correlation coefficient between a & e= 0.201083 Correlation coefficient between a & Pa = –0.240178
Correlation coefficient between a & e = –0.81832 Correlation coefficient between a & Pa = 0.860549 Correlation coefficient between a & Pa = –0.662681 II-Statistics of the semi-major axis a II-1-Basic Descriptive Statistics
• The mean = 148
• The median (central value) = 148
• The variance = 7276.67 II-2-Location Statistics
• The geometric mean = 109.9175238162870485
• The harmonic mean = 47.0803
• The root mean square = 170.751 II-3-Dispersion Statistics
• The Variance of sample mean = 24.6667
• The standard error of sample mean = 4.96655
• The coefficient of variation = 0.576374
• The mean deviation = 73.7492
• The median deviation = 74
• Sample range = 294 II-4-Shape Statistics
• Skewness = 0
• The Pearson skewness 2 = 0
• The kurtosis = 1.79997
• The kurtosis excess = –1.20003 III-Statistics of the intensity I III-1-Basic descriptive statistics
• The mean = 269.584
• The median (central value) = 31.7
• The Variance = 842620 III-2-Location Statistics
• The Geometric mean = 38.8341456925116324
• The harmonic mean 13.3765
• The root mean square = 955.217 III-3-Dispersion statistics
• The variation of sample mean = 2856.34
• The standard error of sample mean = 53.4447
• The coefficient of variation = 3.40503
• The mean variation = 375.968
• The median variation 27.12
• Sample range = 8644.82 III-4-Shape statistics
• Skewness = 6.23942
• The Pearson skewness 2 = 0.777447
• The kurtosos = 46.9564
• The kurtosos excess = 43.9564 IV-Antocorrelation of the semi-major axis V-Autocorrelation of the intensity
Table 1. Autocorrelation of the semi-major axis.
Table 2. Autocorrelation of the intensity.
Appendix B