早期型星系演化與基本性質之研究

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(1)國立臺灣師範大學地球科學研究所天文組碩士論文 Master Thesis Astronomy group, Graduate Institute of Earth Sciences National Taiwan Normal University. 指導教授：陳林文博士 Advisor: Professor L. W. Chen. 早期型星系演化與基本性質之研究 Investigating fundamental properties and evolution of early type galaxies. 研究生：王康華撰 by Kong-Hua Wang. 中華民國 107 年 8 月 August 2018.

(2) ABSTRACT Scaling relations of early-type galaxies (ETGs) connect galaxy properties. The ETG fundamental plane is one of the earliest examples that correlates effective radius Re with velocity dispersion σ. and surface brightness and is applied to estimate distance (Dressler 1987; Djorgovski and Davis 1987). Follow-up studies frequently utilize this scaling technique to find correlations among different properties of ETGs. In this study, based on the energy point of view, we construct our virial Fundamental Plane (vFP) with two variables of total mass density (ρ ~ M/V) and Virial Energy Density (VED ~ M2/ReV) to correlate properties of early-type galaxies, such as mass M, luminosity L, Re, σ and Age. Two sets of sample, core sample and extended sample, are included. The core sample includes 258 ETGs (z < 0.011), which are adopted from the ATLAS3D project (Cappellari et al. 2011, 2013b). The extended sample includes 3,128 ETGs (0.01< z < 0.1), which are selected from 14,034 galaxies (Nair and Abraham 2010), a subset of SDSS Data Release 4 (DR4). The results show that the properties of ETGs can be well constrained by the ρ -VED vFP, which could construct surfaces of σRe, σRe/8, Vcirc, L, E/L and Age. The vFP can define the MDM –surface, which offers a good solution to estimate the DM masses within one Re of ETGs. Simultaneously, the obtained DM masses could transform the distribution of Mstar –surface and the Mtotal –surface from a random type to a flat type with small thicknesses. We also note that both the stellar mass and dark matter mass are significantly correlated with ETG ellipticity ε, if it is not ignored in the galaxy volume estimate. The connections are Mstar ~ Mtotal(1−ε)0.5 and MDM ~ Re Mtotal0.5 (1−ε)0.75. Our results also reveal that the more massive ETGs are, the lower their mass density, which means if an ETG grows via multiple -merging evolution and keeps the original masses of the mergers, its volume becomes larger than the sum of the individual volumes of the mergers. On the other hand, the mass-to-light ratio and the stellar-to-dynamical mass ratio also evolve during the lifetime of ETGs, as our obtained correlations suggest Mstar ⁄L ~ Age (1/0.93~0.96) and Mtotal ⁄L ~ Age (1/1.00~1.02). Keywords: ETGs, fundamental plane (FP), vFP, VED, galaxy properties, property surface i.

(3) 摘. 要. 早期型星系之尺度關係 (scaling relations) 可用來連接星系的各種性質，其中又以基本面 (fundamental plane) 為最早的一個應用實例，它藉由有效半徑 Re、速度瀰散 σ 和平均表面亮度三者的關聯性，來估算星系的距離 (Dressler 1987; Djorgovski and Davis 1987)。後續的相關研究也經常地應用這種尺度關係技術，找出早期型星系各種性質間的關聯性。本論文，是以能量的觀點建構一個均功基本面 (virial Fundamental Plane, vFP), 它以質量密度 (ρ ~ M/V) 和均功能量密度 (VED ~ M2/ReV) 兩個變數去建立與其他早期型星系性質之間的關聯性，如質量 M，光度 L, Re, σ 和年齡 Age。本研究使用兩組樣本 --- 核心樣本和延伸樣本：核心樣本包括 258 個早期型星系 (z < 0.011), 採自 ATLAS3D 觀測計劃 (Cappellari et al. 2011, 2013b); 延伸樣本包括 3,128 個早期型星系 (0.01< z < 0.1), 選自 SDSS Data Release 4 中有較精準星系形態資訊的 14,034 個星系 (Nair and Abraham 2010)。. 本研究結果顯示：可與 ρ -VED vFP 構建出性質面的早期型星系性質包括 σRe,. σRe/8, Vcirc, L, E/L 和 Age。並且，由 vFP 建構的 MDM 面,可用來估計早期型星系在一個 Re 範圍內的暗物質質量，而這估算出來的暗物質質量可將 Mstar 和 Mtotal 數據點由隨機散佈的形態轉變成平面的分佈形態。本研究於分析計算星系體積 V 時，已將星系橢圓度 ε 考量進去，並獲得關係式 Mstar ~ Mtotal(1−ε)0.5 和 MDM ~ Re Mtotal0.5 (1−ε)0.75。同時，研究所得數據揭示：質量愈大的早期型星系具有較小的質量密度，這表示，若早期型星系藉由多次星系合併而演化，且其最後總質量為各合併星系質量之和，則其總體積卻大於原各合併星系個別體積之和。另一方面，由獲得結果得知質光比和星系演化年齡有著 Mstar ⁄L ~ Age 1/(0.93~0.96) 和 Mtotal ⁄L ~ Age 1/(1.00~1.02) 的關係。關鍵詞: 早期型星系 (ETGs), 基本面 (FP), 均功基本面 (vFP), 均功能量密度 (VED), 星系性質, 性質面. ii.

(4) ACKNOWLEDGEMENTS. The author would like to express his sincere gratitude to his supervisor, Professor L. W. Chen for his invaluable supervision, helpful suggestions and enlightening discussions throughout this project. Finally, my profound gratitude goes to my family (mom, my wife Katie and daughters Wenny and Alice), who may not always have understood but who have always been understanding and supportive.. iii.

(5) TABLE OF CONTENTS. ABSTRACT. i. ACKNOWLEDGEMENTS. iii. TABLE OF CONTENTS. iv. LIST OF FIGURES. v. LIST OF TABLES. vi. CHAPTER 1: Introduction. 1. CHAPTER 2: Methods and samples 2.1 Three-dimensional coordinate system for FPs. 6 6. 2.2 Virial Fundamental Plane 2.3 Sample selection. 7 11. CHAPTER 3: Results 3.1 Core sample (a) Velocity dispersion at one effective radius (b) Velocity dispersion at one-eighth effective radius (c) Circular velocity (d) Remaj-surface, Mtotal-surface and MDM-surface (e) L-surface, E/L-surface and Lx-surface (f) Appropriateness and application of E⁄L-surface 3.2 Extended sample (a) Estimating DM mass of each ETG of the extended sample (b) Other property surfaces. 15 16 16 20 22 24 26 29 31 31 35. CHAPTER 4: Discussions 4.1 From FP and MP to vFP 4.2 The confinement of the Mstar –surface 4.3 The E/L -surface and its application. 40 40 41 44. CHAPTER 5: Conclusions. 50. CHAPTER 6: Future work. 52. REFERENCES:. 53. iv.

(6) LIST OF FIGURES. Figure 1.1: An example of a conventional FP Figure 1.2: The correlations of twelve properties with Mtotal - Re FP Figure 1.3: The comb evolution scheme of galaxies. 2 3 4. Figure 2.1: The correlation of σRe with Mtotal and Re Figure 2.2: The virial fundamental plane (vFP) and property surface Figure 2.3: The distributions of Mtotal, ρ and VED of the core sample Figure 2.4: The distributions of Mtotal, ρ and VED of the extended sample. 7 8 12 14. Figure 3.1: Velocity-dispersion (σRe) plots in different PLANEs Figure 3.2: 3D plots of velocity-dispersion at one-eighth Re (σRe/8) Figure 3.3: 3D plots of maximum circular velocity (Vcirc) Figure 3.4: In parallel of σRe-surface, σRe/8-surface and Vcirc-surface Figure 3.5: 3D plots of Remaj-surface, Mtotal-surface and MDM-surface Figure 3.6: 3D plots of L-surface, E/L-surface and Lx-surface Figure 3.7: The distributions of individual deviation from the fitted surface of the ten propertie Figure 3.8: MDM-surface of the extended sample in the ρ-VED vFP Figure 3.9: Three views of MDM distribution in the ρ-VED vF Figure 3.10: Plots of fDM and Mstar distributions in different FPs Figure 3.11: 3D plots of Mtotal-surface, Remaj-surface and L-surface Figure 3.12: 3D plots of E/L- surface, σ-surface and Age-surface. 17 21 23 23 25 28. Figure 4.1: The core and the extended samples in the conventional FP Figure 4.2: Plots of the distributions of Mstar, Mtotal, MDM and fDM Figure 4.3: Plots of Mstar /MDM vs VED0.25, Mstar vs Mtotal(1−ε)0.5, MDM vs Re Mtotal0.5 (1−ε)0.75 and MDM vs (1-ε). 41 43. Figure 4.4: Plots of different propertie on the E/L-surface Figure 4.5: The relationship among Mtotal/L, E/L and Age Figure 4.6: Age distribution in PLANEs of Mtotal –L, Mstar –L and MDM –L. 47 48 49. v. 30 33 33 35 36 38. 44.

(7) LIST OF TABLES. Table 2.1 Reduced Chi-squared for evaluating the goodness of fit Table 2.2 Measurement error of the core and the extended samples Table 2.3 Selection criteria for the ATLAS3D sample Table 2.4 Main characteristics of the ATLAS3D parent sample Table 2.5 main properties of M87and M102 galaxies. 10 10 11 11 13. Table 3.1: The levels of appropriateness Table 3.2: Best-fit results of the σRe-surface Table 3.3: Best-fit results of the σRe/8 Table 3.4: Best-fit results of the Vcirc Table 3.5: Best-fit results of the Remaj-surface, Mtotal-surface and. 15 19 22 24. MDM-surface Table 3.6: Best-fit results of the L-surface, E/L-surface and Lx-surface Table 3.7: The X-ray luminosity observed by ROSAT and Chandra Table 3.8: The thickness and appropriateness of the core sample Table 3.9: Best-fit results of the extended and the core samples. 26 29 29 31 37. Table 4.1: The theoretical and obtained values of a and b Table 4.2: 4D Best-fit results of different property surfaces Table 4.3: Best-fit results of Age in PLANEs of Mtotal-L and Mstar-L. 40 47 49. vi.

(8) CHAPTER 1: Introduction Galaxies are building blocks of our universe. Because the structures of elliptical galaxies are simpler than those of spiral galaxies, many scaling relations have been explored for describing properties of elliptical galaxies. For example, the fundamental plane (FP) for early type galaxies (ETGs) suggested by Djorgovski & Davis (1987) and Dressler et al. (1987), is used to constrain effective radius (Re), velocity dispersion (σ), and average surface brightness (µe) within an area of Re, i.e. Re ~ σa µeb. This correlation is also a distance indicator. Typically, an FP is a bivariate (e.g. σ and µe) analysis, which conventionally involves a set of at least three (e.g. Re, σ and µe) independently observed or physical quantities/properties, such as mass, density, radii, isophotes, profile slopes and luminosities (Djorgovski, et al. 1996). Typical observed values of a and b in visible band are ≈ 1.3 and ≈ −0.8 respectively (Djorgovski et al. 1996). However, if calculated by using the ideal virial theorem, the theoretical values of a and b are 2 and −1. These differences may be due to the deviations of ETG surface brightness from the generally assumed de Vaucouleurs R1/4 law (“non-homology, Bertin, 2002; Prugniel and Simien, 1997; Djorgovski et al. 1996; Graham and Colless, 1997; and Novak et al. 2012). However, in the near infrared (K band), the observed values are 1.56 and −0.94 shown in Figure 1.1 (Djorgovski et al. 1996, Pahre et al. 1998), which seem better than those in the visible band, but still in discrepancy. D’Onofrio et al. (2013) reveal the main contributors of the ‘non-homology’ are structure of ETGs and their stellar population effects, e.g. ellipticity, stellar mass to light ratio, star formation history, and dark matter (DM) content (Renzini and Ciotti, 1993; Borriello et al. 2003; Blanton, and Moustakas, 2009). These factors make the analysis more sophisticated. Here a question which is raised, is whether the 1.

(9) selected properties to construct the FP are appropriate to constrain the properties of ETGs.. Figure 1.1: (a) a conventional FP of ETGs (b) relation of µK to Re, Panel (a) shows scatter reduced by adding the term of velocity dispersion (adopted from Djorgovski et al. 1996).. Hence, based on the virial theorem, some FP-like planes are introduced. These studies are using mass (stellar or total mass) to construct their FPs. e.g. stellar mass plane by Hyde and Bernardi (2009), mass-σ plane by Cappellari et al. (2013b) and mass-size plane by Cappellari et al. (2011, 2013a, 2013b, 2016). Through the ATLAS3D project (see Section 2.3), Cappellari and his colleagues propose a similar FP called mass - size plane (Mtotal – Re) instead of the conventional FP of Re ~ σa µeb. It is understandable to directly use Mtotal and Re, because of the relation of σ2 ~ M⁄⁄R, to construct their FP. They attempt to correlate different properties of ETGs with their FP. Both the Mtotal and the Re are basic physical properties of ETGs. The Re (in arcsec) can be observed directly, but the total mass (Mtotal) cannot, which includes dark and baryonic matters. Fortunately, the development of high-resolution and high-performance techniques of integral field spectroscopy (IFS), offers them much more accurate information, together with their Jeans Anisotropic Model (JAM) to estimate dynamical masses, that can be used to calculate the Mtotal (Cappellari et al. 2013a). 2.

(10) Figure 1.2 shows the twelve properties constrained by the FP of Mtotal –Re (adopted from Figure 22, Cappellari 2016). Obviously, all the distributions of the colors shown in the twelve panels give an identical increasing tendency (from blue to red then white) with the two vectors. This tendency is supportive of Cappellari’s suggestion (2016) of a comb evolution scheme of galaxies (shown in Figure 1.3), a modification of Hubble’s fork evolution scheme. This Mtotal –Re fundamental plane seems better than conventional σ - µe FP, and can analyze the properties of ETGs with more physical information. Consequently, selecting properties appropriately to construct an FP is important and that FP can be used to obtain precious information and valuable results.. Figure 1.2: The correlations of twelve properties with the FP of Mtotal - Re (adopted from Figure 22, Cappellari, 2016).. 3.

(11) Figure 1.3: The comb evolution scheme of galaxies (adopted from Figure 24, Cappellari, 2016).. Therefore, a better set of the properties is proposed in this study and the goals of this study are: i.. Based on a new point of view, energy aspect, to construct an FP, and investigate whether that the new FP can better correlate galaxy properties more accurately. The parameters applied to construct our FP are total energy density (potential energy U, plus kinematic energy K, divided by the volume V of a galaxy), and mass density (total mass including stellar mass and dark matter, divided by the volume V of a galaxy). Both parameters/ properties comprise basic properties of Mtotal, Re and ε (ellipticity).. ii. Explore the distribution and the role of DM mass/fraction in ETGs in our new FP, and understand the relationship between DM and ρ/VED. iii. Look for more correlative information about ETG evolution, such as age of ETGs, the property trends of merge, e.g. galaxy volume, mass and mass density. iv. Find the correlation of ellipticity (ε) with other properties, to understand the role of ε in ETGs. During the process of analyses, the high accurate results from ATLAS3d project (see Section 2.3) have been chosen as a template to verify the 4.

(12) appropriateness (see Table 3.1) of our new FP. After verifying the new FP, a set of larger and more complete sample from Sloan Digital Sky Survey (SDSS) is utilized and evaluated by using the same FP. Then more informative correlations of properties of ETGs with our FP are obtained. To keep the analyses as simple as possible, each of the ETGs analyzed is simplified as an axisymmetric oblate spheroid (Cappellari et al. 2013a). In the next chapter, the analysis methods applied in this study are described in detail. Chapter three shows obtained results of the core sample and the extended sample (see Section 2.3). A comparison of FP results from both samples is made. The ETG properties analyzed include mass, effective radius, velocity dispersion, luminosity and age. Chapter four discusses the obtained power indexes about virial theorem from using different FPs, the confinement of stellar mass on the new FP and the obtained information regarding ETG evolution, followed by chapters of conclusions and future work. A value of the Hubble constant H0 = 70 km s−1 Mpc−1 is assumed through out the thesis. Note that although the physical properties of the ATLAS3D sample adopted from literature are derived based on H0 = 72 km s−1 Mpc−1, the effect of the difference on the parameters of our interest is negligible due to all the sample being nearby (z < 0.011).. 5.

(13) CHAPTER 2: Methods and samples 2.1 Three-dimensional coordinate system for FPs An “FP analysis” regularly involves a three dimensional (3D) coordinate system to represent a parameter space, in which two properties are selected to construct the two coordinates of x-y plane, i.e. the two variables of FP, and generally the third property is allocated in the z coordinate of that coordinate system. The correlation of the third property with the FP (z ~ xayb) is commonly obtained by bivariate analysis of data fitting. The fitted result, obtained correlation of the set of the properties, usually is presented as a surface in the 3D parameter space, while the logarithmic scale is applied. The surface can be visually estimated by checking the orientations/slopes of the data distributed. A visual inspection of the data in 3D can help determine whether the ‘property surface’ exists. For example, by rotating the coordinate systems, the surface can be evaluated from two viewing angles of face-on (the normal line of the surface parallel to line of sight) and edge-on (the normal line of the surface perpendicular to line of sight) (Cappellari et al. 2013a). The dispersion of data in the edge-on view can be seen as the thickness of the fitted surface, and the distribution in the face-on view also provides information about the association of the third properties with the two variables/properties of an FP used. Choosing right variables/properties to construct an FP, frequently makes a fitted surface with a very thin thickness. After a satisfied visual inspection, a further quantitative analysis is followed to obtain the details of the correlation. The goodness of the built correlations is heavily affected by the variables/properties selected and applied in that FP. For example, Figure 2.1 shows a 3D coordinate system with Mtotal -Re –σRe properties corresponding to x–y–z coordinates 6.

(14) respectively. All three panels display the same dataset of σRe, Mtotal and Re (data from Cappellari et al. 2013a). Through the use of Mtotal -Re FP, it is clear that the correlation of σRe with Mtotal and Re. is heavily connected and quantified.. (a). (b). (c). Figure 2.1: The correlation of σRe with Mtotal and Re. (a) Front view, the FP of Mtotal -Re is displayed parallel to this paper and σRe is perpendicular to the paper, colors represent the values of log σRe (1.80 in blue ~ 2.35 log km/s in red) divided into four groups. (b) Face-on view, showing the distribution of the ETGs. (c) Edge-on view, indicating the thickness of the fitted surface (data from Cappellari et al. 2013a).. 2.2 Virial Fundamental Plane This study comprises two stages. The first is using the results from ATLAS3D database to create a virial Fundamental Plane (vFP) with two variables of total mass density (ρ) and Virial Energy Density (VED) as x-coordinate and y-coordinate of a coordinate system, (hereafter x-y plane is called “PLANE”). Then, a selected property (P) of interest of individual galaxies is used to form the 3rd (z) coordinate and then the property is plotted into the 3D parameter space. For each property, a best-fit surface, named a property surface i.e. P ~ ρa × VEDb, is obtained (see Figure 2.2) by using multiple linear regression technique, and the returned values of a and b contain one sigma uncertainty. The equation of the obtained best-fit surface represents the correlation among the three properties (ρ - VED - P) of the galaxies. For verifying the effectiveness of the vFP, all the results are compared with the 7.

(15) original data obtained from Jeans Anisotropic Model (JAM) developed by the ATLAS3D project (Cappellari et al. 2013a, 2016).. (a). (b). (c). (d). Figure 2.2: The virial fundamental plane (vFP) and property surface. The property surface shown in panel (c) is velocity-dispersion (σRe) surface. Panel (a) is the front view of the 3D plot (c), (b) the edge-on view of plot (a) being rotated by a certain angle, to see the thinnest view visually (d) the fluctuation indicating the thickness of the property surface.. The VED indicates the energy content within a range of one Re of an ETG in an ideal equilibrium state. For a virialized elliptical galaxy, its total energy is proportional to GM2/Re, where G is gravitational constant, M is the total mass (stellar mas plus dark matter mass) and Re is the effective radius of the galaxy. However, there is no way to obtain the exact total energy of a galaxy. In order to achieve an accurate estimation, a coefficient of k is added (the applied initial value of k is 1/16). This value makes the total energy of a galaxy become GM2/16Re, which is combining the potential energy (U=GM2/8Re) and kinematic energy (K=GM2/32Re). 8.

(16) VED represents the energy density of the system, and equals to GM2/(16RemajV), where V = 4π(Remaj)3(1−ε)/3 = 4π(Remaj)3(a/b)/3, is the axisymmetric (Cappellari, et al. 2013a）volume within semi major axis of effective isophote, ε is the ellipticity of a galaxy and a/b denotes the ratio of minor to major axis of an elliptic galaxy. In the second stage, 3,128 SDSS ETGS are selected (see next section) from SDSS catalog (Nair & Abraham, 2010). Their properties are analyzed through the same process as that of the first stage. However, the M applied in the x-coordinate and y-coordinate of the vFP PLANE is total mass of a galaxy. The lack of dark matter mass from source catalog obstructs the process to construct the vFP. Steps listed are followed for obtaining the DM mass. (i) convert the DM fraction of each core ETG to its DM mass (in the first stage). excluding those with zero DM fraction. (ii) correlate the MDM with ρ and VED (in the first stage) to obtain the fitted surface MDM –surface. (iii) use the MDM -surface as a template to calculate DM mass of each ETG of the extended sample. An iteration loop is utilized in the above steps to find the numerical solution of the dark matter mass of each ETG. Once the ρ -VED vFP is constructed, we can proceed with the next steps of analyzing, estimating and predicting other properties of the extended ETGs data set, these properties are not observed easily or directly. In both stages, a reduced chi-squared statistic is used to evaluate the goodness of fit. Both Equations (2.1) and (2.2) shown in Table 2.1 are applied simultaneously. Equation (2.1) uses iterations to search appropriate εz (the intrinsic scatter in the z coordinate), by using the least squares algorithm and Brent’s method (Press et al. 2007, sec. 9.3), and removes the outliers of the ETGs distribution, and then re-fits the remaining data (Cappellari personal webpage). On the other hand, iterations and removing outliers are not applied in Equation (2.2), it simply uses all the data to calculate and fit the property surfaces. 9.

(17) Table 2.1 Reduced chi-squared for evaluating the goodness of fit ! 𝐶ℎ𝑖!"_! =. ! !! !!! !! !! !!! !!!!! ! ! !!! !∆! ! ! !∆! ! ! ∆! ! !! ! ! ! ! !. ∕ 𝐷𝑂𝐹. (2.1)§. where the best fitted surface plane z = a(x − x0) + b(y − y0)+c of N ETGs, systematic errors of ∆x, ∆y and ∆z for x, y and z coordinates respectively, reference points x0 and y0 are median of the xj and yj values, εz is the intrinsic scatter in the z coordinate, and DOF, degrees of freedom = N – 3. Equation (2.1) is utilized by Cappellari et al. (2013a). Similar formulae are used by Jorgensen et al. (1996) and Pahre et al. (1998). ! 𝐶ℎ𝑖!" =. !!! !!!! !!!!! ! ! !!! !∆! ! ! !∆! ! ! ∆! ! ! ! !. ∕ 𝐷𝑂𝐹. (2.2). where the best fitted surface z = ax + by + c of N ETGs, measurement errors of ∆x, ∆y and ∆z for x, y and z coordinates respectively, and DOF, degrees of freedom = N – 2. This equation is generally used for FP fitting.. In this study, the measurement errors of basic properties used for Equations (2.1) and (2.2) are listed in Table 2.2. Some of the properties are using error propagation to estimate their errors. The errors of the core sample are adopted from Cappellari et al. (2013a, 2016). The errors of the extended sample are given by the estimation of current measurement techniques. Table 2.2: Measurement error of the core and the extended samples Properties Mtotal, Mstar MDM Re, Remaj, L, ε Lx σRe, σRe/8, Vcirc E/L Age. ρ. VED. Error %. Error %. (258ETGs). (3,128ETGs). 6* 32* 10* 20 5* 16** 15 21** 24**. 20 20 15 10 35 20 36 44. *: Adopted from Cappellari et al. (2013a, 2016) **: Calculated by error propagation §. : The author would like to express his appreciation to M. Cappellari, for posting his program. (lts_planefit.pro) on his webpage (http://www-astro.physics.ox.ac. uk/~mxc/software/), which was used in calculation of Equation (2.1). 10.

(18) 2.3 Sample selection Two sets of sample, core sample and extended sample, are included in this study. The core sample includes 258 ETGs, which excludes two galaxies without values of dark matter fraction from ATLAS3D project (Cappellari et al. 2013b). The 260 ETGs studied in ATLAS3D project are selected from its parent sample of 871 galaxies with 68 ellipticals (8%, T ≤−3.5, from HyperLeda, Gavrilovic et al., 2006 ), 192 S0 galaxies (22%, T > −3.5) and 611 spirals and irregulars (70%). The parent sample is coming from Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006). The selection criterion and characteristics are listed in the Tables 2.3 and 2.4. The histogram shows of the total mass, mass density (ρ) and VED of the core sample respectively in panels (a) to (c) of Figure 2.3. Panel (d) displays the correlation of mass with ρ and VED. The colors represent the log (Mtotal/M¤) masses of the 258 ETGs from 9.6 (dark blue) to 11.8 (red and white). The symbols Table 2.3 Selection criteria for the ATLAS3D sample (reproduced from Cappellari et al. 2011) Distance: D < 42 Mpc Galaxy mag: MK < −21.5mag Observability: |δ − 29°| < 35° Galaxy zone of avoidance: | b| > 15° Table 2.4 Main characteristics of the ATLAS3D parent sample (adopted from Cappellari et al. 2013) Survey volume: V = 1.16 ×105 Mpc3 Galaxy K-band luminosity: LK > 8.2 ×109 L¤K Galaxy stellar mass: M* ≥ 6 × 109 M¤ Galaxy B-band total mag: MB ≤ −18.0mag Galaxy SDSS r-band total mag: Mr ≤ −18.9mag Total number of galaxies: 871 Spiral and irregular galaxies: 611 (70%). 11.

(19) (a). (b). (c). (d). Figure 2.3: The distributions of the total mass, mass density (ρ) and VED of the core sample are shown respectively in panels (a) to (c). Panel (d) shows the correlation of the mass with ρ and VED. Colors in blue to red, then to white represent the gradually increasing mass. of “+” and “×” plotted in the middle of the panel are M87 (NGC4486) and M102 (NGC5866) respectively, which are two of the 258 ETGs and are used as references for easy understanding and sensing the quantities of ρ and VED. Table 2.5 lists the main properties of the two galaxies. The well known galaxy of M87 is an elliptical galaxy with ellipticity close to zero, i.e. a perfect spheroidal. On the other hand, M102 is a typical lenticular galaxy with ellipticity of 0.579, and owns a VED value near the median of the core sample.. 12.

(20) Table 2.5 main properties of M87 (NGC4486) and M102 (NGC5866) galaxies Galaxy M87. Mass. Remaj. (log M/M¤). (kpc). 11.727. 6.888. 11.001. 4.303. (NGC4486). M102 (NGC5866). ε. ρ. VED 3. (kg/m ). (ergs/m3). 0.083. -19.542. -2.224. 0.579. -19.317. -2.520. The reason why choose the sample of ATLAS3D survey as our core sample is they have much more accurately estimated dynamical masses obtained by using the techniques of Integral Field Spectroscopy (IFS, The SAURON integral field spectrograph mounted at the William Herschel Telescopeat on La Palma, see Cappellari, et al. 2011 and references therein) with high-resolution and high-performance Charge-Coupled Device (CCD). This technique brings enough observed data to build 3D structures of ETGs. Together with multi-wavelength (optical and from radio to millimeter) observations (Cappellari, et al. 2011) and theoretical modeling, the results from ATLAS3D survey become valuable. Following the sprit of the core sample, the extended sample includes 3,128 ETGs, which are selected with the values of T-Types smaller than −1 (equivalent to S0 and E types in Hubble's classification scheme, see Nair and Abraham, 2010), and stellar mass larger than 6×109 solar mass, from the 14,034 SDSS Data Release 4 (DR4) spectroscopically targeted galaxies (0.01< z < 0.1 and Mg <16 mag) with morphology information classified visually by Nair and Abraham (2010). All the obtained results are compared with those of the core sample. The histogram shows of the mass, mass density (ρ) and VED of the extended sample respectively in panels (a) to (c) of Figure 2.4. Panel (d) displays the correlation of total mass with ρ and VED. The colors represent the total masses of each of the 3,128 ETGs from 9.9 (dark blue) to 12.0 log M/M¤ (red and white). The same references as the core sample are also plotted. 13.

(21) (a). (b). (c). (d). Figure 2.4: The distributions of the total mass, mass density (ρ) and VED of the extended sample are shown respectively in panels (a) to (c). Panel (d) shows the correlation of the mass with ρ and VED. Colors in blue to red, then to white represent the gradually increasing mass.. 14.

(22) CHAPTER 3: Results By using the concept of FP, the Mass Plane (MP) of Mtotal - Remaj PLANE is used by Cappellari et al. (2016), where Remaj is the semi major axis of Re isophote. For describing various properties (P) of the core sample in the MP, Equation log(P) = a × log(Mtotal) + b ×log(Remaj) + c. (3.1). where a and b are coefficients, and c is a constant. Pinp denotes the observed or calculated values of the property, and Pfit is the value of best-fit result. is used to define the best-fit surfaces of the properties. Meanwhile, Equation log(P) = a × log(ρ) + b ×log(VED) + c. (3.2). is also applied to show the distribution of the sample in the new ρ-VED vFP. The fluctuations of both best-fit surfaces are calculated by using Equations Fluc_rms = rms(Pinp – Pfit ) × 100. (3.3). Fluc_wt = Fluc_rms ⁄ (max(Pinp) − min( Pinp)). (3.3a).. The fluctuations indicate deviation between the input values and the best-fit surfaces, and also specify the status of the distribution. Fluc_rms is the root-mean-square (rms) fluctuation of a fitted surface, and Fluc_wt is weighted fluctuation equivalent to Fluc_rms divided by total interval (the difference between maximum and minimum) of Pinp. The Fluc_wt is an indicator and also called the thickness of the best-fit surface. An appropriateness level can be defined based on the Fluc_wt value to denote how the data points are close to the fitted surface. Table 3.1 lists the different levels of appropriateness and the values of the thickness. Table 3.1: The levels of appropriateness Thickness (Fluc_wt). ≤1. ≤5. ≤ 10. ≤ 15. ≤ 20. > 20. Level. 1 perfect. 2 strong. 3 good. 4 weak. 5 bad. 6 no. 15.

(23) 3.1 Core sample (a) Velocity dispersion at effective radius In Figure 3.1, Panel (a) shows the velocity dispersion at one effective radius (σRe) of the core sample (260 ETGs) in the MP of Mtotal-Remaj, replicated from Cappellari et al. (2013a, 2016). Panels (b) to (l) show 258 ETGs of the same sample, excluding two galaxies without values of dark matter fraction. Panels (a) to (c) are two-dimensional (2D) plots while others are three- dimensional (3D). The applied PLANEs in the panels are stated in the figure caption. Colors in blue to red, then to white represent the gradually increasing values of log σRe. The difference in color between panels (a) to (c) and other panels is due to different scaling factors being used for displaying 2D and 3D plots. Panel (a) displays the trend of σRe by colors, and implies the association of total mass (M*dyn used in Cappelarri et al. 2016) with semi major axis of effective radius (Remaj), indicating the larger the values of σRe, the bigger the Remaj and the more massive the galaxy. By changing the PLANE from Mtotal-Re maj to Mtotal-VED, the VED coordinate (=GM2/16RemajV) of the 258 ETGs makes the distribution with a clearer gradient of the color image shown in panel (b). Furthermore, by using ρ-VED vFP, the values of ρ of the ETGs enhance the association of σRe with ρ and VED. Panel (c) shows the plotted image is re-allocated to a narrow triangle band. This infers a tighter correlation between the σRe and the two properties of ρ and VED, than that of the Mtotal-Re maj or Mtotal-VED.. 16.

(24) (a). (d). (b). (e). (c). (f). (g). (h). (i). (j). (k). (l). Figure 3.1: The velocity-dispersion (σRe) plots in different PLANEs, different colors represent different values of σRe. Panels (a) to (c) are 2D plots. Panel (a) is adopted from Cappellari (2016) in the Mtotal-Remaj PLANE. (b) plot in a PLANE of Mtotal-VED, and (c) in ρ-VED vFP. Colors in panels (b) and (c) indicate values from 1.55 (dark blue) to 2.45 log km/s (red and white). Panels (d) to (f) are 3D plots of panels (a) to (c) with the 3rd (z) coordinate of σRe. Panels (g) to (i) and (j) to (l) are panels (d) to (f) after being rotated by certain angles, to exhibit the face-on and edge-on views respectively of the fitted surfaces. 17.

(25) We can deproject the plots in panels (a) and (b) by adding an extra coordinate (in z direction, perpendicular to this paper) of σRe, to find out the exact correlation between the property and ρ-VED vFP, thus panels (d) to (f) have the same PLANE as panels (a) to (c) respectively. Certainly, while viewing from the front (z direction, perpendicular to the PLANE of ρ-VED), the appearance of each image is exactly the same as those of panels (a) to (c). The only difference is the displayed colors, because of different scaling factors in plotting 2D and 3D figures. However, the same trend can be obviously seen. The 3D plots have an advantage of inspecting data distribution from any point of view and easily identifying data clustering visually. For example, by rotating the plots to face-on views, panels (g) to (i) reveal the scattered data points turn out to be three surfaces, which are located within their own 3D energy spaces. The constructed surface is named velocity-dispersion surface, one of the property surfaces. The tightness of the correlations is presented by their fluctuations or thicknesses of the surfaces. Shown in panels (j) to (l), the edge-on views of the surfaces, panel (l) displays a thinnest thickness, which implies the vFP is the best ability (smallest fluctuation) to represent the property. In this study, multiple linear regression algorithm is chosen for 3D analysis to compute the best-fit surfaces. The returned tolerance values (∆a and ∆b) from the IDL “regress” subroutine contain one-sigma uncertainties, which indicate the levels of fluctuations. The returned values are affected by the measurement errors (∆x, ∆y and ∆z) stated in Equations (2.1 and 2.2), and will be discussed in Chapter 4. Furthermore, the values calculated by Equations (3.3) and (3.3a), are also utilized together with visual images to assess the convergence of the fitted property surfaces. Table 3.2 lists the fitting results of the σRe-surface in the above three PLANEs. The best-fit result with a minimum thickness (Fluc_wt) of 3.719, is 18.

(26) in the Mtotal-VED PLANE. However, its power indexes of a and b, are 0.261 and 0.120. Too small to conclude that a good correlation exists. The number shown in parentheses in the last column of Table 3.2 is the number of removed outlier. Table 3.2: Best-fit results of the σRe-surface. Because the Velocity dispersion (σRe) of an ETG is a phenomenon of its kinematic energy, and heavily associated with gravitational energy, the PLANE of ρ - VED is more appropriate than M-Remaj PLANE for defining property surface of σRe. The correlation between σRe and ρ -VED listed in the third row of Table 3.2, has the best values of power indexes and can be simplified as following. σRe. ~ ρa × VEDb. (3.4). ~ ρ-0.5 × VED0.5. (3.5). ~ (M/V)-0.5 × (M2/(Re maj× V))0.5. (3.6). ~ (M/ Remaj)0.5. (3.7). are exactly fit the virial theorem. Equation (3.5) is much simpler than the description of M*dyn ~ σRe2.3 and Remaj ~ (M*dyn)0.12, stated by the Cappellari’s data (2016), obtained from their Jeans Anisotropic Model (JAM) developed by the ATLAS3D project (Cappellari et al., 2013a). The thickness (Fluc_wt) of M-Remaj PLANE is 5.408 (row 1, Table 3.2), and can be improved to 4.488, (row 4, Table 3.2), if log scale is considered in the fitting calculation. However, their power indexes, 0.482 and −0.440 are still not good enough to meet the virial theorem. 19.

(27) By using the 2D data shown in panel (a), it is not easy to build a single correlation between σRe and Remaj, or σRe and Mtotal accurately, because σRe is not only associated with Re, but also associated with Mtotal. Thus the introduction of 3D analysis can improve the obtained correlations of σRe with Remaj and Mtotal. Meanwhile, choosing properties appropriately to construct an FP, also plays an important role. From the values of power indexes (a and b) listed in Table 3.2, we can see how significant an FP influences on the obtained results.. (b) Velocity dispersion at one-eighth effective radius Figure 3.2 displays the plots of central velocity dispersion at one-eighth effective radius (σRe/8) of the core sample, and Table 3.3 lists the fitting results. Both show the same distribution trend as those of σRe, but the power index is about 0.535 instead of 0.5, see Equations. σRe/8 ~ ρ-0.536±0.023 × VED0.535±0.016 ≈ (VED/ρ)0.535 σRe/8 ~ (M/ Remaj)0.535. (3.8) (3.9). The power index is obviously larger than 0.5. This does not seem to have made an error from measurements. The fitting is executed again by changing the Remaj to log scale, and the obtained exponent is about 0.530, still bigger than 0.5 (see row 4 of Table 3.3). This finding is not reported in the papers of Cappellari et al. (2013a, 2016). It is speculated that the effect maybe affected by the different mass distribution between the core region (within Re/8) and the outer region (within Re). The thickness of σRe/8-surface is 7.972, level 3, a good appropriateness, but a little worse than 4.596 of the σRe-surface.. 20.

(28) Furthermore, the ratio of σRe/8 to σRe, ranging from 0.670 to 1.528 (Cappellari et al. 2013a), indicates that the dispersion in the central region is not always larger than that in the region of effective radius. This gives indirectly evidence of different mass distribution in the ETGs. For making a comparison with the results obtained by Capperllari (2016), the data of σRe/8⁄σRe of 258 ETGs are plotted in the ρ–VED vFP (not shown in here). The results show the best-fit thickness of the. σRe/8⁄σRe–surface is 13.517. All data points are located randomly to certain extent. At the same time, by deriving the fitting results of σRe/8⁄σRe = 10(−6.756− (− 6.059). ×(M/ Remaj)0.535-0.5 = 10−0.697×(M/ Remaj)0.035 = 0.201 ×(M/ Remaj)0.035, the. power index of 0.035 is so small that the correlation is not so well to associate this property (σRe/8⁄σRe) with ρ and VED, or the relationship between σRe/8 and σRe may be complicate.. (a). (b). (c). (d). (e). (f). Figure 3.2: The 3D plots of velocity-dispersion at one-eighth effective radius (σRe/8), different colors represent the T-Type of the core sample (Cappelarri et al. 2011). All panels have values of σRe/8 in z direction (perpendicular to this paper in panels (a) to (c)). Panels (a) and (d) have the MP of M-Remaj. Panels (b) and (e) have M-VED PLANE. Panels (c) and (f) have the vFP of ρ-VED. Panels (a) to (c) are front views. Panels (d) to (f) show the edge-on images of each surface. 21.

(29) Note: The color trend displayed in Figure 22, panel b (Cappellari 2016) shows only 159 ETGs, the other 101 ETGs are excluded by some unknown reasons, maybe they are outliers removed in their fitting process, see Section 2.1. In this study, an effort tries to reproduce the color trend of the 159 ETGs, but vainly.. Table 3.3: Best-fit results of the σRe/8. (c) Circular velocity Figure 3.3 shows the plots of maximum circular velocity (Vcirc) of the core sample. It also has the same distribution as that of σRe. Table 3.4 lists the fitting results. The thickness of the Vcirc-surface is 4.620, a strong appropriateness. The result of row 3 can be written as Vcirc. ~ ρ-0.5 × VED0.5. = (VED/ρ)0.5. (3.10). Both results of σRe and Vcirc analyses conform that the ρ-VED vFP is an appropriate scheme to describe the properties of σRe, σRe/8 and Vcirc. In other words, the three fitted property surfaces, σRe-surface, σRe/8-surface and Vcirc-surface, can be used to predict the three properties of other galaxies, if their masses, effective radii and ellipticities are known. Because of the similarity of Equations (3.6), (3.7), (3.9) and (3.10), the three property surfaces should be in parallel to each other, and as shown in Figure 3.4. Compared to the values of Vcirc/σRe measured by Cappellari (2016), which is between 1.51 and 1.76, the fitting result in this study is 1.32 (=10(−5.938−(−6.059))) obtained by calculating the. 22.

(30) difference between two constants (power indexes c). This is consistent with the result done by Cappellari (2016).. (a). (b). (d). (e). (c). (f). Figure 3.3: The 3D plots of maximum circular velocity (Vcirc) obtained by Cappellari et al. (2013a), different colors represent the T-Type of the galaxies (Cappelarri et al. 2011). All panels have values of Vcirc in z direction (perpendicular to this paper in panels (a) to (c)). Panels (a) and (d) have an MP of M-Remaj. Panels (b) and (e) have M-VED PLANE. Panels (c) and (f) have a vFP of ρ-VED. Panels (a) to (c) are front views, and panels (d) to (f) show edge-on images of each surface.. Figure 3.4: In parallel of σRe-surface (upper right), σRe/8-surface (upper left) and Vcirc-surface (lower). Different colors represent the T-Type of the galaxies (Cappelarri et al. 2011). 23.

(31) Table 3.4: Best-fit results of the Vcirc. The reasons why the obtained best-fit results of above three surfaces in the ρ -VED vFP are better than those results in M-Remaj PLANE could be: (i) the virial energy and mass based vFP is physically more related to σRe, σRe/8 and Vcirc than previous FP, and (ii) the galaxy volume definition in this study involves ETG ellipticity, ε. Although the correlation between ε and σ or surface brightness is rarely reported in the literature (eg, Djorgovski and Zeeuw, 1986), ε turns out to play a non-negligible role in our analyses with vFP, and our results give clear correlations, which almost exactly meet the virial theorem.. (d) Remaj-surface, Mtotal-surface and MDM-surface Both panels (a) and (b) in Figure 3.5 show the Remaj-surface from face-on and edge-on views respectively. Panels (c) and (d) show the total-mass (Mtotal) surface, and panels (e) and (f) show the MDM surface. Table 3.5 lists fitting results of the correlations about the three surfaces. The colors represent the T-Types of the galaxies. Because ρ (~ Mtotal/V) and VED (~ Mtotal2/RemajV) are used to construct the vFP, both Remaj –surface and Mtotal –surface need to be investigated firstly. The Remaj -surface shown in panel (b) has a little bit more deviation with a thickness of 6.004. This surface is a bit wide but still in the level of good. The Mtotal -surface exhibits the smaller thickness of 3.331. Why both thicknesses are not close to 24.

(32) zero? This is caused by applying the formula of volume, V = 4π(Remaj)3 (1-ε)/3. Different galaxies have different values of Remaj and ε. Hence, both thicknesses cannot be zeros.. (a). (b). (c). (d). (e). (f). maj. Figure 3.5: The Re -surface, Mtotal-surface and MDM-surface All panels have the vFP of ρ-VED. Panels (a) and (b) have Remaj in z direction, and panels (c) and (d) have Mtotal in z direction, while (e) and (f) have MDM. Panels (a), (c) and (e) are face-on views and the others are edge-on views.. Panels (e) and (f) present a shape of MDM-surface. The number of the galaxies used for calculating the MDM-surface is 197, and the other 61 ETGs with zero DM fraction are not included. The surface has a wide thickness of 10.830. This is a reasonable value for estimating dark matter in the present day. This surface is 25.

(33) used as a template to estimate the DM mass within one Re of each ETG of the extended sample (see Sections 2.2 and 3.2). Table 3.5: Best-fit results of the Remaj-surface, Mtotal-surface and MDM-surface. (e) L-surface, E/L-surface and Lx-surface Luminosities of ETGs are always used for photometric analyses, and X-ray luminosity is observed to estimate hot gas around ETGs. Both are important properties in astrophysics. Figure 3.6 displays the distributions of L, E/L and Lx -surfaces in the vFP. Panels (a) and (b) show the L-surface of face-on and edge-on views, the input total luminosity is in the r band (616.5nm). Panels (c) and (d) show the E⁄L-surface (ratio of total energy to luminosity), and panels (e) and (f) show the X-ray luminosity. The colors represent the T-Types of the galaxies. Table 3.6 lists fitting results of the three surfaces. It is obvious that L-surface shown on the bottom part of panel (b) in Figure 3.6 has a small region with a little more deviation from the surface, where a few galaxies are located in the area of small ρ and low VED. These are galaxies with masses smaller than 1010M¤. The thickness of L-surface is 7.364, a good appropriateness. The definition of E/L, ratio of total energy to luminosity, is the total energy of a galaxy divided by its luminosity (r band). There is no doubt that mass to light (M/L) is an important property and has been studied in astronomy for a long time. 26.

(34) In this study, the E/L is also explored and analyzed to see any findings that are not revealed before. The thickness of the fitted E/L-surface is 4.024, a strong level of appropriateness. In ETGs, X-ray-emitting gas plays a role in sustaining their corona of hot. The amount of the hot gas around these ETGS is one of the important properties. Regarding panels (e) and (f), two sets of data (Sarzi et al. 2013) are input for mapping the Lx-surface. Both sets are sub groups of the core sample. By using currently best distance estimates (Cappellari et al. 2011), the first set includes rescaled X-ray luminosity of 46 galaxies from the catalogue of O’Sullivan et al. (2001), which are observed by ROSAT and Einstein. The second set includes 19 galaxies from Boroson et al. (2011) observed by Chandra. Fourteen galaxies (listed in Table 3.7) are observed in both observations.. Because of the different resolution, the obtained values of the14 objects observed by Chandra are smaller than those observed by ROSAT and Einstein, mostly because the contributions from AGN, the ICM or low mass X-ray binaries are excluded (Sarzi et al. 2013, Kim et al. 2008). Table 3.7 reveals that the exclusion percentage (difference, see last column of the Table) of X-ray luminosities between those by ROSAT and by Chandra of the 14 galaxies. The percentage is so high that 11 of the 14 galaxies are higher than 50%.. Without showing the distribution of the 19 ETGs (better resolution, measured by Chandra) in Figure 3.6, panels (e) and (f) display the distribution of the 46 ETGs measured by ROSAT. They are scattered randomly and do not present a good shape of a surface. Table 3.6 lists the fitting results of the 46 and the 19 galaxies respectively (row 3 and 4). It seems the latter has a thickness of 13.833, a little bit better than 14.945 of the former. But both are in the range of weak level 27.

(35) of appropriateness. Maybe this is caused by lack of enough data to represent a good surface.. (a). (b). (c). (d). (e). (f). Figure 3.6: The L-surface, E/L-surface and Lx-surface All panels have the vFP of ρ-VED. Panels (a) and (b) have values of luminosity (r band) in z direction, and panels (c) and (d) have E/L, while (e) and (f) have Lx.. 28.

(36) Table 3.6: Best-fit results of the L-surface, E/L-surface and Lx-surface. Table 3.7: The X-ray luminosity observed by ROSAT and Chandra NGC. LX_CHANDRA. 2768 4261 4278 4365 4374 4382 4472 4473 4526 4552 4621 4649 4697 5866. 40.08 40.82 39.4 39.82 40.78 40.05 41.32 39.24 39.49 40.39 38.61 41.09 39.26 39.36. LX_ROSAT Difference exclu. % 40.42 41.19 40.33 40.58 40.96 40.43 41.49 40.09 39.9 40.7 39.96 41.35 39.87 39.8. 0.34 0.37 0.93 0.76 0.18 0.38 0.17 0.85 0.41 0.31 1.35 0.26 0.61 0.44. 54.29 57.34 88.25 82.62 33.93 58.31 32.39 85.87 61.10 51.02 95.53 45.05 75.45 63.69. Unit: log LX¤. (f) Appropriateness and application of E⁄L-surface Summarizing the thicknesses of the ten property surfaces, the values listed in Table 3.8 are between about 3 and 15. Because of the lack of enough accurate data, three (Lx, Lx_19 and MDM surfaces) of the ten surfaces belong to the weak level of the appropriateness. Four property surfaces (σRe, Vcirc, E/L and Mtotal surfaces) have strong appropriateness. The best ones are E/L-surface and Mtotal-surface with thickness of 4.024 and 3.331 respectively. Figure 3-7 displays the deviation distributions to the ten fitted surfaces individually. The deviation is defined as Diff. ⁄ int.=(Pinp–Pfit ) ⁄ (max(Pinp)−min( Pinp))×100%, see Equations (3.3) and (3.3a). It is clear that the values of thicknesses listed in Table 3.8 are well corresponded to the distribution of the plots in Figure 3.7.. 29.

(37) Figure 3.7: The distributions of individual deviation to the fitted surface of the ten properties. The obtained fitted surfaces are crossed at the zero, parallel to the horizontal axes of the plots, and perpendicular to the paper.. 30.

(38) Table 3.8: The thickness and appropriateness of the core sample Properties. Appropriateness. Thickness (Fluc_wt). Fluc_rms. Vcirc L E/L Lx Lx_19 Remaj Mtotal MDM. 2 strong 3 good 2 strong 3 good 2 strong 4 weak 4 weak 3 good 2 strong 4 weak. 4.596 7.972 4.620 7.364 4.024 14.945 13.833 6.004 3.331 10.830. 4.127 6.098 3.946 12.363 11.676 58.735 55.608 7.288 7.288 34.019. σRe σRe/8. 3.2 Extended sample The extended sample of 3,128 ETGs is adopted and selected from DR4 SDSS, no duplication with the core sample. The Mstar, Re and ratio of semi minor axis to major axis (b/a) of the individual galaxies are released from the SDSS catalogue, but either the dark matter fraction or the dark matter mass is not provided in the catalogue. In order to apply the same analysis as previous section, the dark matter mass must be known.. (a) Estimating DM mass of each ETG of the extended sample The difficulty of obtaining total mass of a galaxy is caused by the unknown mass fraction of dark matter. Once the MDM-surface is built in the vFP, the total mass of a galaxy can be easily found by adding MDM and stellar mass together. Then, the values of the ρ and VED of each galaxy can be calculated directly, and the other galaxy properties can also be analyzed for their possible surfaces in the vFP. In this study, the dark matter mass of each of the 3,128 galaxies is computed by using the fitting results of MDM-surface (row 3 of Table 3.5) as Equation log(MDM) = -2.014 × log(ρ) + 1.265 × log(VED) – 25.646. 31. (3.11).

(39) Coming from the model of ATLAS3D project (Cappellari et al. 2013a), the value of semi major axis (Remaj) used for computing volume is transferred by Equations Remaj = Re / (-0.3117 × (1-b/a)2 – 0.4823 × (1-b/a) + 1.0016) or. Remaj = Re / (-0.3117 × ε2 – 0.4823 × ε + 1.0016). (3.12) (3.12a). where a, b are radii of semi major and minor axes of an ETG.. Figure 3.8 shows the MDM-surface of the extended sample. Panel (a) shows the data points scattered in the 3D energy space. Panel (a) shows the face-on view, and panel (b) displays the edge-on view with a shape of a line, which indicates the surface is really a flat plane (in a log-log-log scale) without fluctuation. This is not a surprise, because the plane is defined by ρ and VED (Equation (3.11)). Zero fluctuation and zero thickness are expected. However, the fitted MDM-surface has a thickness of 0.027 (see row 2 in Table 3.9), which is not a true thickness. Because MDM –surface, built by fitting ETGs of the core sample, is used as a reference to predict dark matter masses of the extended sample. During the process (see Section 2.2), a numerical iteration algorithm is executed for obtaining DM mass. The value of 0.027 is the difference between Equation (3.11) and the numerical solution. Ideally there is no thickness, and it should be zero. In order to compare the results with literature (Cappellari et al. 2013a), the formula they use is shown in Equation fDM ≈ 0.13 + 0.24 × (log Mstar -10.6)2. (3.13). is applied to the extended sample to predict the dark-matter fraction (fDM) of a galaxy, if stellar mass, Mstar (M/M¤) is known. 32.

(40) (a). (b). Figure 3.8: MDM-surface of the extended sample in the ρ-VED vFP.. Shown in Figure 3.9, three panels present the same predicted MDM distribution of the 3,128 ETGs in the ρ-VED vFP from three different points of view. All the three plots reveal a two-group distribution. One is concentrated on a narrow surface and the other is a swarm of data points around the surface.. (a). (b). (c). Figure 3.9: Three views of MDM distribution in the ρ-VED vFP predicted by Equation (3.13). Figure 3.10 provides two hints to explain the two groups. For comparison, the published 2D figure is shown in panel (a) (Cappellari et al. 2013a). The curve inside the panel is the fitted arc of fDM of their ETG sample (the core sample). Equation (3.13) in Figure 3.10 (a) represents the fitted arc. Panel (b) is a 3D plot of fDM distribution in PLANE of Mstar-VEDstar. It is a curved-surface distribution 33.

(41) obtained by Equation (3.13). Panel (c) is a re-plot of panel (b) by changing PLANE to MCAPP -VEDCAPP. The subscript CAPP denotes the total mass which is the sum of Mstar and MDM, and MDM is estimated by Equation (3.13). This demonstrates that the two sub-groups shown in panel (c) are coming from the fDM estimation. This 2D estimation is too simple to be used, because obviously a large number of data points are deviated far away from the fitted arc (see panels (a) and (c)). Therefore, applying this equation to estimate the fDM must result in big errors. Furthermore, the second hint: Panels (d) to (f) in Figure 3.10 show the stellar masses (Mstar) of 3,128 ETGs from SDSS catalogue, in three different vFPs of. ρstar -VEDstar, ρCAPP -VEDCAPP and ρtotal -VEDtotal respectively. An obvious change in Mstar distribution from a two-group random type to a flatten type, can be seen in panels (d) to (f). Panel (d) exhibits the two groups of Mstar, which is directly coming from the data of SDSS catalogue. Panel (e) still keeps the shape of the two groups without much improvement. However, panel (f) shows a mixed result of the two groups, and displays the distribution of Mstar in a shape of a flat surface. In this study, using the built MDM-surface to predict the MDM of the extended sample can mix the two groups into a surface, which is much better than that of 2D model published previously. Consequently, the MDM –surface provides a good solution to predict the dark matter masses of ETGs. Does our universe for ETGs really exist an MDM –surface or/and an Mstar – surface in the energy space? By reviewing the data obtained from ATLAS3D project (Cappellari et al. 2012, 2013a), we have state-of-the-art prediction of dark matter, which reveals a high possibility.. 34.

(42) (a). (d). (b). (e). (c). (f). Figure 3.10: (a) A 2D plot of fDM distribution of 260 ETGs (core sample) based on the model of ATLAS3D with Mstar in horizontal coordinate (adopted from Cappellari et al. 2013a), (b) 3D fDM distribution, the extended sample of 3,128 ETGs in the Mstar -VEDstar PLANE, based on Equation (3.13), (c) the 3D plot with fDM distribution of the extended sample in the ρCAPP -VEDCAPP PLANE, subscript CAPP denotes total mass calculated via Equation (3.13) (Cappellari et al. 2013a). Panels (d) to (f) are plots of Mstar distributions in different vFP: panel (d) ρstar -VEDstar, panel (e) ρCAPP -VEDCAPP, and panel (f) ρtotal -VEDtotal.. (b) Other property surfaces Once the dark matter masses of individual galaxies are obtained, the total masses of each ETG are promptly known by simply adding their stellar masses, then the properties of the galaxies of the extended sample can be plotted in the ρ – VED vFP. Table 3.9 lists fitting results of property surfaces. The formula used to analyze the correlations between the vFP and ETG properties is Equation (3.2). Figure 3.11 shows the Mtotal-surface, Remaj-surface and L-surface respectively, while MDM –surface is displayed in Figure 3.8. All of the panels give a flat-shape surface, but with different thicknesses. Their thicknesses (Fluc_wt) are 2.073, 3.234 and 5.781 (see Table 3.9). 35.

(43) (a). (b). (c). (d). (e) (f) Figure 3.11: Panels (a) and (b): Mtotal-surface, (c) and (d): Remaj-surface, (e) and (f): L-surface. All panels in the vFP of ρtotal –VEDtotal. Panels (a), (c) and (e) show the face-on views. Panels (b), (d) and (f) show the edge-on views. The colors represent the T-types of the galaxies. In addition, listed in Table 3.9, the power indexes of a and b of Remaj –surface and Mtotal –surface express a strong relationship between each other. By deriving Mtotal ⁄ Remaj, it is found that Mtotal and Remaj are dependent, because both are directly involved to construct the vFP. Table 3.9 shows the fitting results of the extended and the core samples. The thicknesses of Re –surfaces (0.411 of extended and 1.044 of core sample) are in the level of perfect appropriateness. The input Re values of the core sample are 36.

(44) obtained by Re ≡ (Ae ⁄ π)0.5, where Ae is the area of the effective isophote (Cappellari et al. 2013a). Those of the extended sample are directly coming from SDSS catalogue, and are calculated by Blanton et al. (2003 & 2005). Both thicknesses exhibit a clear correlation, which can be shown approximately as Equation Re ~ VED0.5 ⁄ ρ. (3.14). Re is directly proportional to the square root of VED, and is inversely proportional to mass density, ρ. A denser galaxy has a smaller Re, and a stronger total-energy-density galaxy owns a larger Re. For example, if an ETG with a constant Mtotal, have a lower mass density it implies it has a larger volume i.e. a larger Re. Table 3.9: Best-fit results of the extended sample and the core sample (The change of coefficient k stated in Section 2.2, does have influence on the results of c, but not on the other results listed in this Table.). Extended sample (3,128 ETGs) VED const Fluc_wt Fluc_rms Chi2Re Chi2Re_C b c. P Property. ρ. Mstar MDM Mtotal Remaj Re L E/L. -1.938±0.036 -2.014±0.035 -1.948±0.036 -0.948±0.016 -1.007±0.017 -1.550±0.028 -1.398±0.032 -0.534±0.012 -0.424±0.012. 1.489±0.031 1.265±0.030 1.445±0.031 0.445±0.013 0.507±0.014 1.074±0.024 1.371±0.028 0.470±0.010 0.434±0.010. -2.004±0.072 -2.014±0.089 -1.907±0.068 -0.907 ± 0.031 -1.014±0.034 -1.542±0.054 -1.365 ± 0.060 -0.492±0.021 -0.416±0.026. 1.538±0.051 1.265±0.063 1.438±0.048 0.438±0.022 0.505±0.024 1.064±0.038 1.374±0.042 0.491±0.015 0.448±0.019. σ Age. a. -22.884 -25.646 -23.118 -16.688 -17.727 -16.913 -9.489 -6.879 2.649. 2.499 0.027 2.073 3.234 0.411 5.781 4.247 8.530 10.250. 5.311 0.060 4.370 4.370 0.537 10.966 11.197 7.870 13.798. 0.016 0.000 0.011 0.057 0.001 0.115 0.090 0.310 1.013. 0.010(-139) 0.000(-23) 0.007(-126) 0.036(-133) 0.000(-473) 0.073(-113) 0.061(-108) 0.182(-83) 0.415(-108). 16.035 34.019 7.288 7.288 1.405 12.363 11.676 4.127 20.265. 0.438 1.671 0.101 0.499 0.015 0.458 0.336 0.339 5.284. 0.138(-20) 0.958(-5) 0.082(-3) 0.376(-3) 0.001(-94) 0.321(-8) 0.273(-5) 0.296(-2) 0.996(-8). Core sample (258 ETGs) Mstar MDM Mtotal Remaj Re L E/L. σ Age. -24.008 -25.646 -22.319 -15.890 -17.878 16.659 -8.679 -6.059 3.084. 37. 5.406 10.830 3.331 6.004 1.044 7.364 4.024 4.596 16.182.

(45) For E/L –surface, σ-surface and Age-surface, Figure 3.12 shows the three surfaces of the 3,128 galaxies in the ρ -VED vFP respectively. Colors represent the T-types of the ETGs. All panels show a flat shape with different thickness. The thinnest one is the E/L-surface with a value of 4.247 (see row 7 in Table 3.9). Further analyses and obtained results are discussed in Section 4.3.. (a). (b). (c). (d). (e). (f). Figure 3.12: Panels (a) and (b): E/L- surface, (c) and (d): σ-surface, and (e) and (f): Age-surface. All panels have a ρ -VED vFP. The colors represent the T-types of the galaxies.. 38.

(46) The fitted σ -surface of the extended sample has the same distribution type as Equation (3.7), but with a thickness of 8.530, about twice of 4.596 of the core sample. It is reasonable because of the number of galaxies increases twelve times. The thickness of Age –surface is 10.250, a weak level of appropriateness. It is not so well related to the ρ -VED vFP. The fitted Age –surface is discussed in the next chapter.. 39.

(47) CHAPTER 4: Discussions 4.1 From FP and MP to vFP The conventional FP correlates effective radius with velocity dispersion and surface brightness, i.e. Re ~ σa µeb. In the status of virial equilibrium, the index a is 2 and b is −1. However the observed a is about 1.3 and b is about −0.8 (Djorgovski & Davis, 1987; Djorgovski et al. 1996). Similar results obtained by other studies are listed in Table 4.1. The FP of Re versus observed Re plots for the cases of our core and extended samples are shown in Figure 4.1. On the other hand, by using M instead of µe, the MP (M, σe, Re) proposed by Cappellari et al. (2013a, 2016) defines the correlation of M ~ σea Reb. Because of directly using the form of virial theorem (σea ~ M Re-b), the obtained values (#5 in Table 4.1) are so close to those of the theoretical values of 2 and 1., where M is the total mass (stellar plus dark). Table 4.1: The theoretical and obtained values of a and b. #. Correlation a. b. Theoretical values. a. b. Observed values. a. b. Conventional FP Re ~ σ µe 1.3 −0.8 ATLAS3D 260ETGs 1.063 −0.765 −1 a b 2 ATLAS3D 258 ETGs Re ~ σ Σe 0.984 −0.739 SDSS 3,128 ETGs 0.893 -0.657 3D a b 5 ATLAS 260ETGs M ~ σe Re 1 1.928 0.964 6 vFP 258 ETGs −0.492 0.491 σe ~ ρa VED b −0.5 0.5 7 vFP 3,128 ETGs −0.534 0.470 Note: 1: Djorgovski & Davis, 1987 and Djorgovski et al. 1996. 2: the 260 ATLAS3D ETGs, analyzed by Cappellari et al. (2013a), although the surface brightness of Σe= L/2πRe2, instead of the surface brightness (µe), is used to calculate their σ, where L is the luminosity of a galaxy. 3: by using the same Re ~ σaΣe b, re-analyses of the 258 ETGs core sample. 4: by using Re ~ σaΣe b, to analyze the 3,128 ETGs extended sample. 5: 260 ATLAS3D ETGs, analyzed by using M ~ σea Reb. 6: 258 ETGs of the core sample analyzed by using vFP. 7: 3,128 ETGs of the extended sample analyzed by using vFP. 1 2 3 4. 40.

(48) (a). (b). Figure 4.1: “The conventional FP”. Panel (a) 258 ETGs of the core sample (b) 3,128 ETGs of the extended sample. Colors in blue to red, then to white represent the gradually increasing values of the difference between log Re and log Re = a×log σe+b×log Σe+c, where a,b and c are obtained by Re ~ σa Σeb, see #3 and #4 in Table 4.1. In this study, we apply the vFP to analyze σe. The correlation is σe ~ ρa VEDb. The a and b values (named index_ab) of virial theorem are −0.5 and 0.5, see Equations (3.4) to (3.7). The obtained index_ab of the 258 ETGs, are −0.492 and 0.491 (see #6 in Table 4.1), which are so close to the values of the theorem. This is because σe ~ (M/Re)0.5 and M/Re is contained in the ρ (M/V) and VED (M2/ReV). With the same reason, the index_ab of the 3,128 ETGs, are −0.534 and 0.470 (see #7 in Table 4.1), which are a little bigger than those of the core sample, but still accurate, because the large number of the observed ETGs increases the dispersion and uncertainty.. 4.2 The confinement of the Mstar –surface Panels (d) to (f) of Figure 3.10 are plots of Mstar distributions. A dramatic change has been shown between panel (d) in a vFP of ρstar -VEDstar, and panel (f) in a vFP of ρtotal -VEDtotal. The change reveals the data points of Mstar can distribute so different while using the two vFPs respectively. Panel (d) uses only stellar mass to construct the vFP, while panel (f) using total mass. This obviously indicates the MDM –surface provides a good solution to predict the dark matter 41.

(49) masses of ETGs. Then the ρtotal -VEDtotal vFP confines the Mstar –surface in a strong level with a thickness (Fluc_wt) of 2.499. Reviewing the fitting results listed in the row 1 and 2 of Table 3.9 and panels (a) to (c) of Figure 4.2, we can roughly simplify the Mstar –surface and MDM – surface as the two Equations below. Mstar ~ ρ−2 VED1.5. (4.1). MDM ~ ρ−2 VED1.25. (4.2). From Equations (4.1) and (4.2), the ratio of Mstar to MDM is proportional to VED0.25, i.e. Mstar /MDM ~ VED0-.25. The ratio directly reveals that the relationship between Mstar and MDM is linked by energy density, VED0.25. This result is shown in the panel (f) of Figure 3.10. In the space of vFP, it indicates the Mstar depends on VED is slightly stronger than MDM. Panel (e) of Figure 3.10, in contrast, shows a different result while using prediction model (Equation 3.13) of DM fraction for ATLAS3D (Cappellari et al. 2013a), by which a confined Mstar –surface cannot be found in the vFP space. All these phenomena strongly exhibit the influence of DM mass. The causes are still not clear. Panel (d) of Figure 4.2 displays a good correlation of DM fraction with ρ and VED. Equation (4.1) can be reduced to Mstar ~ (Mtotal/Re3(1−ε))−2 (Mtotal2/Re4(1−ε))1.5 = Mtotal(1−ε)0.5. (4.3). It indicates that the Mstar is only related to Mtotal and ε, unlike MDM related to Re as well, MDM ~ (Mtotal/Re3(1−ε))−2 (Mtotal2/Re4(1−ε))1.25 = Re Mtotal0.5 (1−ε)0.75. (4.4). This equation is derived from Equation (4.2). It indicates an ETG with a bigger Re has more DM mass. The ratio of MDM / Mstar is ~ Re (1−ε)0.25/ Mtotal0.5. For an ETG 42.

(50) with ε ≈ 0, its Re is ~ Mtotal0.5 MDM / Mstar = (MDM+Mstar)0.5×MDM / Mstar. This gives a constraint on Re, stellar and DM masses of ETGs. Panel (a) in Figure 4.3 shows the ratio of Mstar to MDM versus VED0.25. It indicates the higher ratio (i.e. smaller fDM) the larger VED. Panel (b) displays the Mstar distribution depends heavily on total mass and ellipticity. Panel (c) exhibits a different distribution from Panel (b), which shows a line-type distribution. It exactly corresponds to Equation (3.11). This seems that the effect of ellipticity is cancelled out by Remaj. Meanwhile, Panel (d), displaying the distribution of MDM versus (1-ε), also indicates that the ellipticity plays an insignificant role on MDM. Compared to the results of Deur (2014), at the same luminosities the rounder elliptical galaxies have less dark. (a). (b). (c) (d) Figure 4.2: The distributions of stellar mass, total mass and DM mass of the 3,128 ETGs, shown in panels (a) to (c) respectively. Panel (d) slows the DM fraction. Colors in blue to red, then to white represent the gradually increasing values of the masses or the DM fraction in the corresponding panels respectively. 43.

(51) matter than those of flatter ones, our results do not show a consistent trend. However, Results of the study by Tortora et al. (2009) show dark matter increases for the luminosities with brighter ETGs. This is consistent with our results shown in Panel (d).. (c). (d). Figure 4.3: Panels (a) to (c) show Mstar /MDM versus VED0.25, Mstar versus Mtotal(1−ε)0.5 and MDM versus Re Mtotal0.5 (1−ε)0.75respectively. All the panels are in log scale. Colors represent the increasing values of (1-ε) from 0.22 (dark blue) to 0.998 (red and white). Panel (d) displays MDM versus (1-ε). Colors represent the luminosity from 9.47 (dark blue) to 11.08 log L/L¤ (red and white).. 4.3 The E/L -surface and its application Reviewing the fitted results of Age –surfaces of the extended and the core samples, we can see weak and bad appropriateness of Fluc_wt (10.250 and 16.182 listed in rows 9 and 18 respectively in Table 3.9). Both imply the 44.

(52) properties involved to correlate Age are not suitable. Following the results of E/L –surface shown in panels (a) and (b) of Figure 3.12, we can further investigate how E/L –surface can be connected to other properties as shown in Figure 4.4. All panels are plotted in the ρ -VED vFP together with the E/L –surface (the 3rd coordinate). The 4th property by means of colors is projected onto the E/L – surface. Panel (a) in Figure 4.4 is a plot of Mtotal distribution on E/L-surface, colors represent the values of Mtotal. (b) is viewed from the direction perpendicular to the. ρ –E/L (xz plane), and (c) is from the direction perpendicular to the VED – E/L (yz plane). These color plots show a clear trend of the mass is increasing as E /L increases. Moreover, while the masses of ETGs increase, the mass density decrease. This infers the size of mergers after merging is larger than the total volume of the individual galaxies before merging, and the values of E/L of the galaxies increase gradually, see panel (d). This indicates that the ETGs are getting more massive, lower density and larger values of E/L. Thus, E/L can be an indicator of ETG evolution. Panels (e) to (g) in Figure 4.4 also exhibit the trend of luminosity distribution (e), σ distribution (f), and Age distribution (g) respectively. Roughly they all display the same tendency as panels (a) to (c), but with different levels of appropriateness. After visual examination, a four-dimensional (4D) fitting process is used for obtaining their best-fit correlations. log(P) = a × log(ρ) + b × log (VED) + c × log(E/L) + d. (4.5). which is an extension of Equation (3.2), and can be used to find the correlations of properties with the E/L and vFP. Obtained results are shown in Table 4.2. Except L and Age, the power indexes c of other properties are too small to conclude that there have correlations between these properties and E/L. Roughly, The obtained values of 45.

(53) power indexes a and b of Age are much different from those listed in Table 3.9, and the correlation can be presented as Age ~ [ρ × (E/L) ⁄ VED]0.87 ~ 0.95. (4.6). Based on Equation (4.6), it seems a toy-model for ETGs could be suggested by this study: The initial state of two mergers before merging can be treated as two isolated systems. After merging, mergers become one mixed system, and the total energy could be the direct sum of the individual energy of the two galaxies, if energy is conservative. In a short time, the luminosity will increase because of the growth of the star formation rate. However, from the long-term view, the luminosity will decline because the system is in equilibrium and runs out the useable fuel. Thus, through multiple merge process, the ratio of E to L is gradually increasing, like entropy is always increasing. The E /L can be applied as an indicator of ETG evolution and merging history. The power index c of Age ~ (E/L)c is 0.954 (see row 9 in Table 4.2), which constrains the correlation to the E /L. This is one piece of supportive evidence. Further analysis and discussions are followed. Judge by thicknesses, an improvement on the appropriateness, from weak level, 10.250 (see row 9,Table 3.9) to good level, 6.492 (see row 9, Table 4.2) can be seen. This improvement indicates that the Age not only related to ρ and VED but also related to E/L. Reducing Equation (4.6), we obtain Age ~. (M/L)fitted 0.87 ~0.95. (4.6a). where M is total mass. It is clear that M, L and E/L are key factors to affect the Age of ETGs. Figure 4.5 displays the relationship between Mtotal/L, E/L and Age. Obviously these three properties are correlated heavily. It shows that the older galaxies tend to have both higher M/L and E/L simultaneously. 46.

(54) (a). (b). (d). (c). (e). (f) (g) Figure 4.4: Different properties are projected onto the E/L-surface. Colors in panels (a) to (c) show Mtotal distribution of the extended sample from three different viewing angles, panel (d) volume (log kpc3) distribution, (e) luminosity distribution, (f) σ distribution and (g) Age distribution. Except panels (b) and (c), all other panels are face-on views.. Table 4.2: 4D Best-fit results of different properties P Property Mstar MDM Mtotal Remaj Re L E/L. σ Age. ρ a. VED b. E/L c. -1.780±0.093 1.334±0.090 0.113±0.062 -2.014±0.097 1.265±0.094 0.000±0.065 -1.813±0.093 1.312±0.090 0.097±0.062 -0.813±0.040 0.312±0.038 0.097±0.026 -1.023±0.050 0.523±0.048 -0.012±0.033 -2.813±0.154 2.312±0.149 -0.903±0.103 0.000±0.042 0.000±0.041 1.000±0.028 -0.546±0.037 0.482±0.035 -0.009±0.024 0.909±0.066 -0.873±0.064 0.954±0.044. 47. const d -21.808 -25.646 -22.201 -15.772 -17.838 -25.485 0.000 -6.960 11.698. Fluc_wt Fluc_rms Chi2Re 2.427 0.027 2.009 3.134 0.398 2.232 0.000 8.530 6.492. 5.157 0.060 4.234 4.234 0.520 4.234 0.000 7.869 8.739. 0.018 0.000 0.004 0.065 0.001 0.004 0.000 0.266 0.099.