A comprehensive analysis using 9 dark matter halo models on the spiral galaxy NGC 4321
IIndian Journal of Physics manuscript No. (will be inserted by the editor)
A comprehensive analysis using 9 dark matter halomodels on the spiral galaxy NGC 4321
Tan Wei Shen ∗ · Zamri Zainal Abidin · Norsiah Hashim · Received: date / Accepted: date
Abstract
This paper addressed the dark matter analysis on the spiral galaxyNGC 4321 (M100) by considering the nine different dark matter profiles, sofar lacking in the scientific literature, i.e. Pseudoisothermal, Burkert, NFW,Moore, Einasto, core-modified, DC14, coreNFW and Lucky13 profiles. In thispaper, we analyzed the rotation curve analysis on the galaxy NGC 4321 byusing nonlinear fitting of star, gaseous and dark matter halo equations withselected VLA HI observation data. Among the nine dark matter profiles, fourdark matter profiles (DC14, Lucky13, Burkert and Moore profiles) showeddeclining features and hence not suitable for this galaxy. This is concluded tobe mainly due to the characteristics of those dark matter profiles and also thevarying levels of problems within the inner region fittings. For the remainingfive accepted dark matter profiles, we further conducted the analysis by usingreduced chi-square test. Four out of the five accepted dark matter profiles liewithin the range of 0.40 < χ red < χ red nearestto 1, mainly due to its linearity in the inner region and flatness at large radii. Keywords
Cosmology · Dark matter · Radioastronomy · Spiral galaxy
PACS · · · Tan Wei Shen Radio Cosmology Research Lab, Physics Dept., Faculty of Science, University of Malaya,Kuala Lumpur, MalaysiaE-mail: [email protected] Zainal Abidin Radio Cosmology Research Lab, Physics Dept., Faculty of Science, University of Malaya,Kuala Lumpur, MalaysiaE-mail: [email protected] Hashim Mathematics Section, Centre for Foundation Studies in Science, University of Malaya,Kuala Lumpur, MalaysiaE-mail: [email protected] a r X i v : . [ a s t r o - ph . GA ] F e b Tan Wei Shen ∗ et al. Dark matter is one of the most important investigations since there are stilla lot of mysteries related to it. In galaxies, the distribution of the baryoniccomponents cannot justify the observed profiles or the amplitudes of the mea-sured circular velocities [1]. Furthermore, the profiles of the rotation curveimply that the distribution of light does not match the distribution of masswithin the galaxies [2]. The explanation of the lacking distribution leads to thesuggestions of a presence of an additional invisible mass component [3]. Thisis usually solved by adding an extra mass component, i.e. the dark matterhalo [4].Though initial evidence for dark matter came from rotation curves of galax-ies, more compelling evidence for non-baryonic matter exists. In 1980, VeraRubin and Kent Ford present the observations of a set of spiral galaxies thatorbital velocities of stars in galaxies were unexpectedly high at large distancesfrom the nucleus by using the new sophisticated optical spectrograph thatthey developed [5]. This unexpected result indicated that the falloff in lumi-nous mass with distance from the centre is balanced by an increase in non-luminous mass. Although initially met with skepticism, Rubin’s result of theexistence of dark matter became scientifically accepted after the subsequentdecades by studying more than 200 galaxies and enough documented dataproved that the universe was virtually 90 percent undiscovered matter [3].NGC 4321 (also known as M100) is a grand design galaxy located in theVirgo cluster. NGC 4321 is an SAB(s)bc galaxy that has two symmetric well-defined spiral arms [6]. M100 has receive much observational and theoreticalattention, not only because it is one of the closest galaxies in Virgo cluster,but also because its bar of moderate strength gives rise to a particularly clearresonant circumnuclear structure [7]. Its relative proximity and moderate in-clination make it suitable to study the content, distribution and kinematics ofthe neutral hydrogen gas in both its molecular (CO) and atomic (HI) formsof its interstellar medium [8].Rotation of spiral galaxies is measured by spectroscopic observations ofemission lines such as H α , HI and CO lines. In these lines, the velocity disper-sion is negligibly small compared to rotational velocity. This implies that thepressure term in the Virial theorem is also negligible so that the mass can becalculated in sufficient accuracy by the dynamical balance between the grav-itational and centrifugal forces [9]. HI rotation curves are most often derivedfrom velocity fields. A velocity field aims to give a compact and accurate short-hand description of the dynamics of a galaxy by assigning a typical velocity toevery spatial position. That is, for every position, one uses the velocity thatmost accurately represents the circular motion of the bulk of the quiescentcomponent of the gas as it moves around the center of the galaxy [10]. Fit-ting the inner part of the rotation curve is challenging. For the case of M100,the existence of intermediate bar [11] and the known starburst activity in thecenter of this galaxy [12] may induced non circular motions. ark matter analysis on galaxy M100 3 A large number of HI observation studies in the broader literature haveexamined this galaxy, namely, HI rotation curves for this galaxy were derivedby Vera Rubin [1,3], HI observation by VLA and Nobeyama radio observatoryto study large-scale star-formation processes [13], HI rotation curves to see thebehavior of the approaching and receding sides of NGC 4321 by Knapen [14]and study of environmental effects on HI gas properties of cluster galaxies [15].Besides the usage of HI observational data to measure the galaxy’s rotationcurves, CO and H α have also been previously used for this purpose since 1980,namely, central CO rotation curves of this galaxy derived by Sofue [16], H α rotation curves and Position-Velocity diagram by Daigle [17] and H α rotationcurves of the inner disc by Morales [18] and CO rotation curve to study thedark matter in the central region by using ALMA [19].Previous CO rotation curve studies by Ali [19] have exclusively focused onthe dark matter in the central region of this galaxy, which is defined to be upto the radius 0.7 kpc with NFW dark matter profile. The unexpected findingsof the dark matter in the central region signal the need for additional studiesto understand more deeply about the overall dark matter distribution in thisgalaxy. To fill this literature gap, this paper will employ HI observed dataup to radius 10 kpc to identify the overall dark matter distribution by usinga rotation curve with the nine different dark matter profiles. The nine darkmatter profiles that we used are Pseudoisothermal, Burkert, NFW, Moore,Einasto, core-modified, DC14, coreNFW and Lucky13 dark matter profiles.More detailed information about the nine dark matter profiles will be discussedin Methodology section. This section is organized as follows: In subsection 2.1 we introduce the massmodeling of rotation curve technique and then each parameter in the subse-quent subsection. The HI observed data processing is explained in subsection2.2. In subsection 2.3, we present the observed data calculation by using thetilted-ring method. The rotation curve comparison with previous works is ex-plained in subsection 2.4. In the last few subsections, we discuss the massmodel parameter of the nine dark matter profiles, stellar, and gas in subsec-tion 2.5, 2.6, and 2.7, respectively.2.1
Mass modeling rotation curve
The measurement of the rotation curves of disc galaxies is a powerful tool toinvestigate the nature of dark matter [20]. Rotation curve has been used toassess the existence, the amount and the distribution of this dark component[4]. Most galaxies have rotation curves that show sharp rising or high velocityin the very centre, following by a slowly rising or constant velocity rotation inthe outer parts. The flatness of the rotation curves in the outer part implies
Tan Wei Shen ∗ et al. Parameter ValueObservation date 25th March 2003Total observation time 12910 secondsTotal antennas used 27Antennas diameter (each) 25.0mTotal data recorded 855738Polarizations LL and RRConfigurations DBand LRA 12:22:54Dec +15:49:20Systemic velocity 1575 kms − Position Angle -26 ◦ Rest Frequency 1420.41 MHzTotal bandwidth observation 1538.1 kHzDivided channel 63Each channel frequency 24.41 kHzRestoring beam (major, minor) (52.95”, 47.66”)Velocity resolution 10 kms − Primary calibrator 1331+305Secondary calibrator 1221+282
Table 1: HI observational parameter of VLA on galaxy NGC 4321 that galaxies contain large amounts of dark matter [21]. The contributionof each component to the rotation curve is computed by using Mathematicasoftware. By summing in quadrature the contribution of the three components(disc, gas, dark matter) in all possible combinations and get the best-fittingto the observed data, we will obtain the most accurate value of star, gas anddark matter [22]. The rotation curve can be represented as a model, which isthe sum of the contribution from the star, halo and gas components [23] asfollows: V rot = V gas + V star + V DM (1)2.2 VLA (Very Large Array) HI dataThe observed data is NGC 4321 HI (neutral atomic hydrogen) 21 cm linefrom VLA archive data with project number AS0750 D030325. The data re-duction for inspection, flagging, bandpass calibration, gain calibration, con-tinuum subtraction, cleaning, imaging, moment mapping and PV (Position-Velocity) Diagram generating is processed by using CASA (Common Astron-omy Software Application) software. The distance adopted is 17.1 Mpc [24].The overall HI observational parameters of VLA on galaxy NGC 4321 aredetailed in Table 1.Figure 1 shows the velocity channel map of HI in the central region of NGC4321 with restoring beam of 52 . × .
66” was produced by using the Briggsweighting. The HI emission is detected in 28 channel maps with a velocitywidth of 10 kms − from 1436 kms − to 1706 kms − . The pixel size is set toCELL = 15” and IMSIZE = 256. Then, the channel map is used as input togenerate the integrated intensity HI zeroth moment (mom0) map as shown in ark matter analysis on galaxy M100 5 Figure 2. The outcome of Figure 1 and Figure 2 is comparable with the previ-ous study (refer to [25] Figure 2 and Figure 3). Next, the integrated-intensityHI map is used to generate PV diagram and obtain the HI velocity of thegalaxy as shown in Figure 3.2.3
Data calculation by using tilted-ring method
Next, the tilted-ring method is used in order to obtain the rotation velocity.By using the tilted-ring method, the rotation velocity, radial velocity andinclination angle in a galactic disk are coupled to each other [26] as below: V r ( r, θ ) = V obs ( r, θ ) − V sys = V rot ( r ) cos θ sin i (2)where V r ( r, θ ) is the radial velocity due to rotational motion, V obs ( r, θ ) is theobserved radial velocity, r is the radius from the center of the galaxy, θ is theazimuth angle in the disk of a measured point from the major axis, V sys isthe systemic velocity of the galaxy, V rot ( r ) is the rotation velocity and i is theinclination angle of the galaxy.Any coupling of rotation velocity and the inclination can be solved byusing the tilted-ring method if a velocity field is observed [1, 27–29]. This isbecause of the functional shape of the variation of V r ( r, θ ) /V r ( r,
0) is againstthe position angle on the sky. In this situation, V r ( r,
0) is the maximum valueof V r along an initially chosen ring [26]. For θ = 0, V rot ( r ) = V r ( r, i , where V r ( r,
0) = V obs ( r,
0) - V sys . The systemic velocity and inclination angleof M100 is 1575 kms − [7] and 27 ◦ [14] respectively. The complete rotationvelocity calculation of M100 is shown in Table 2.In Table 2, column 1 illustrates the radius, r of M100, column 2 revealsthe observed velocity, V obs of M100, obtained from Figure 3 VLA PV diagram.Column 3 demonstrates the radial velocity, V r of M100 by using observedvelocity, V obs minus systemic velocity, V sys (1575 kms − ). While column 4shows the rotation velocity, V rot of M100 by using radial velocity, V r dividessin i (27 ◦ ). Then, a graph of the total rotation velocity against the radius ofM100 is drawn as shown in Figure 4. In Figure 4, the error bars of the observeddata are obtained from the rms values of the VLA data, which the root meansquare is often use as a synonym for standard deviation of a signal from agiven baseline.2.4 Rotation curve comparison
Our rotation curve is then compared with Rubin et al. [3] and Knapen etal. [14] rotation curves. Rubin and Knapen only showed rotation velocity anddid not decompose it into separate contribution of gas, star and dark matter.Hence, we were only able to compare the total rotation velocities betweenRubin, Knapen and our rotation curves.For Rubin rotation curve, the values of the rotation velocities are availablein their paper, we have included their rotation velocities in the Table 3. For
Tan Wei Shen ∗ et al.ark matter analysis on galaxy M100 7 Tan Wei Shen ∗ et al. Fig. 1
Velocity channel map of HI in the central region of NGC 4321
Fig. 2
Integrated intensity (mom0) map of the HI 21cm line emission of NGC 4321 ark matter analysis on galaxy M100 9
Fig. 3
NGC 4321 PV diagram
Knapen rotation curve, they did not include the values of rotation velocitiesfrom their paper, so we will trace the rotation velocity value from the figure intheir paper. We standardized all the relevant units from the two said papersand our manuscript. The total rotation velocity against the radius of Rubin,Knapen and our rotation curve is hence shown in Table 3. The three rotationcurves are plotted together in Figure 5. ∗ et al.Radius, r (kpc) V obs ( kms − ) V obs − V sys ( kms − ) V rot = ( V obs − V sys ) /sin i ( kms − )0.02 1586.42 11.42 25.150.41 1603.5 28.50 62.780.83 1614.27 39.27 86.501.24 1622.81 47.81 105.311.66 1631.35 56.35 124.122.07 1637.49 62.49 137.652.49 1643.8 68.80 151.552.90 1649 74.00 163.003.31 1652.71 77.71 171.173.73 1656.42 81.42 179.344.14 1660.5 85.50 188.334.56 1664.96 89.96 198.154.97 1668.3 93.30 205.515.39 1671.27 96.27 212.055.80 1674.24 99.24 218.596.21 1675.73 100.73 221.886.63 1678.33 103.33 227.607.04 1679.81 104.81 230.867.46 1681.66 106.66 234.947.87 1684.71 109.71 241.668.29 1686.55 111.55 245.718.70 1688.38 113.38 249.749.11 1690.21 115.21 253.779.53 1691.44 116.44 256.489.94 1693.27 118.27 260.51Table 2: The radius, observed velocity, radial velocity and rotation velocity of galaxy M100 ( kpc ) R o t a t i o n v e l o c i t y ( k m / s ) Fig. 4
The total rotation velocity against the radius of M100 ark matter analysis on galaxy M100 11Radius (kpc) Total rotation velocity ( kms − )Rubin et al. 1980 [3] Knapen et al. 1993 [14] Our rotation curve0 0 0 01 133 159 962 124 205 1373 158 211 1684 182 216 1855 188 229 2066 190 231 2207 193 233 2308 197 247 2439 199 259 25210 201 265 262Table 3: The comparison of the rotation velocity obtained by Rubin et al., Knapen et al.and our rotation curve HI OUR RUBINKNAPEN ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) Fig. 5
The total rotation velocity comparison between Rubin, Knapen and ourrotation curve. The red line represents Rubin rotation curve, the blue linerepresents Knapen rotation curve and the green line represents our rotationcurve.
By referring to Figure 5, the reasons of Rubin rotation curve to be differentfrom Knapen and our rotation curves are due to the differences of systemic ve-locity, position angle and inclination angle that were applied. The comparisonis shown in Table 4.The rotation curve of Vera Rubin was observed in the HII regions emis-sion from the Kitt Peak 4 m RC spectrograph in 1978 [3], while the rotationcurve of Knapen and ours were observed in the 21 cm line of neutral hydrogenwith Very Large Array (VLA) in 1990 and 2003, respectively. In Table 4, thedifference in position angle will affect the Position-Velocity diagram obtained ∗ et al.Parameter Rubin et al. 1980 [3] Knapen et al. 1993 [14] This paperSystemic Velocity( kms − ) 1545 ±
25 1570 . ± . ◦ ) 140 153 ± ◦ ) 35 27 27Table 4: The comparison of parameter adopted between Rubin et al, Knapen et al. and inthis paper during data reduction. It is worth noting that the differences in systemic veloc-ity and inclination angle will affect the calculation for rotation velocity. Thisis because the tilted ring method formulation involves both systemic velocityand inclination angle. The uncertainty of Rubin is far larger than the uncer-tainty of Knapen. The reason is that different instrument is used and normallyinstruments improved in their uncertainty calculations over the years.Furthermore, the measurement methods vary between Knapen and ours.They produced maps with a resolution of 12 . × .
3” uniform weighting and31 . × .
75” natural weighting by using AIPS software [14]. However, weproduced images with a resolution of 52 . × .
66” Briggs weighting by us-ing CASA software. Natural weighting is applied to all visibilities equally andresults in maximum point-source sensitivity in images. However, this weight-ing produces poor synthesized beam-shape and side-lobe levels. The uniformweighting gives the visibilities a weight that is inversely proportional to thesampling density function. This causes a reduction in the side-lobes of the PSF.The uniform weighting scheme provides better resolution but lowers the sensi-tivity of the image [30]. Both natural and uniform weightings have some weakpoints but the use of Briggs weighting presents a way to overcome these weak-nesses. Briggs weighting creates a PSF that smoothly varies between naturaland uniform weightings based on the signal-to-noise ratio of the measurementsand a tunable parameter that defines a noise threshold [31], which producedbetter image. Furthermore, for Knapen rotation curve central region, theydid mention that their curve rises more steeply near the center. This is be-cause their profile was measured by a high full width at half maximum withina beam of half-power width, which corresponds to a rapidly rising rotationcurve [14]. Beyond 2 kpc radius (0.4 arcmin), the rapidly rising rotation veloc-ity has slowed down and, eventually, starting from 6 kpc radius, the rotationvelocity of Knapen rotation curve is comparable with the rotation velocity ofour rotation curve.2.5
Dark Matter Profile
Researchers expand the dark matter study with a larger theoretical frameworkto provide better ways of directly investigating it, since then a lot of darkmatter profiles have been derived. In our research of rotation curve-fitting onthe galaxy NGC 4321, we will consider the nine dark matter profiles including ark matter analysis on galaxy M100 13 seven cored and two cuspy profiles to analyze the distribution and mass ofdark matter halo. Cored profiles that we used are Pseudoisothermal, Burkert,Einasto, core-modified, DC14, coreNFW and Lucky13 profiles. While for cuspyprofiles, we used Navarro, Frenk and White (NFW) and Moore profiles.
Pseudoisothermal profile is a singular density profile that approaches a powerlaw at the centre [32] and the mass distribution for larger radii would divergeproportional to the radius. For Pseudoisothermal profile, the density, mass andvelocity equations of dark matter [33] are as follows: ρ Iso ( r ) = ρ r/r s ) (3) M Iso ( r ) = 4 πρ r s ( r − r s tan − ( rr s )) (4) V Iso ( r ) = 4 πGρ r s (1 − r s r tan − ( rr s )) (5)where parameter G is the universal gravitational constant, ρ is the scaledensity, r s is the scale radius and r is the radius from the centre of the galaxy.These parameter definitions are the same for all others profiles except Einastoand DC14 profiles. Hence, these parameter definitions will not be mentionedagain in the following profiles. Burkert profile revises density law of Pseudoisothermal profile in the innerregion. This represents the mass profile for larger radii that diverge loga-rithmically with increasing radius. This is in agreement with the predictionsof cosmological CDM calculations [34]. For Burkert profile, the density [34],mass [35] and velocity equations [36] of dark matter are as follows: ρ Bur ( r ) = ρ r s ( r + r s )( r + r s ) (6) M Bur ( r ) = 6 . ρ r s [ln(1 + rr s ) − tan − ( rr s )) + 0 . rr s ) )] (7) V Bur ( r ) = 6 . Gρ r s r (ln[(1 + rr s )(1 + ( rr s ) ) . ] − tan − ( rr s )) (8) ∗ et al. NFW profile is a traditional benchmark profile motivated by N-body simula-tions, which is shallower than Pseudoisothermal near the centre, and steeperthan Pseudoisothermal in the outer regions [37]. For NFW profile, the den-sity [37], mass [38] and velocity equation [36] of dark matter are as follows: ρ NF W ( r ) = ρ r s r (1 + rr s ) − (9) M NF W ( r ) = 4 πρ r s (ln(1 + rr s ) − rr s + r ) (10) V NF W ( r ) = 12 . Gρ r s r (ln(1 + rr s ) − rr s + r ) (11) Moore profile behaves similarly to the NFW profile at large radii but is steeperthan NFW profile at smaller radii [39]. For Moore profile, the density [40],mass [41] and velocity equations of dark matter are as follows: ρ Moo ( r ) = ρ ( rr s ) . (1 + ( rr s ) . ) (12) M Moo ( r ) = 83 πρ r s (ln(1 + ( rr s ) . )) (13) V Moo ( r ) = 8 . Gρ r s r (ln(1 + ( rr s ) . )) (14) Einasto profile is emerging as a better fit for more recent numerical simula-tions and provides the most accurate description of dark matter haloes [42].Compared to other dark matter profiles use two free parameters, Einasto pro-file uses three free parameters to describe the halo mass profile instead andhence significantly improves the accuracy of the fitting to the inner densityprofiles of simulated haloes [43]. For Einasto profile, the density, mass [44] andvelocity equations of dark matter are as follows: ρ Ein ( r ) = ρ − exp [ − n (( rr − ) /n − M Ein ( r ) = 4 πnr − ρ − e n (2 n ) − n γ (3 n, rr − ) (16) V Ein ( r ) = 4 πGnr − ρ − r e n (2 n ) − n γ (3 n, rr − ) (17)where γ (3 n, x ) = (cid:82) x e − t t n − dt , parameter G is the universal gravitationalconstant, r − is the radius where the density profile has a slope of -2, ρ − is ark matter analysis on galaxy M100 15 the local density at that radius and r is the radius from centre of the galaxy.While other dark matter models are described by two free parameters, r s and ρ , a characteristic scale and a characteristic density at that radius, Einastomodel involves a third parameter, n , the Einasto index which describes theshape of the density profile [44]. The NFW profile is singular at the galactic center. To avoid the singularity,Brownstein proposed the core-modified profile. The core-modified profile is aprofile with constant density in the central core [45]. The density, mass andvelocity [46] equations of core-modified profile are as follows: ρ com ( r ) = ρ r s r + r s (18) M com ( r ) = 43 πρ r s [ln( r + r s ) − ln( r s )] (19) V com ( r ) = 43 πGρ r s r [ln( r + r s ) − ln( r s )] (20) DC14 profile considers the baryonic feedback on the halo due to the su-pernovae, and hence modifies the halo profiles [47]. Cintio et al. establishedthe DC14 model, whose profile is defined in terms of the model class ( α , β , γ ) [48], [49]: ρ αβγ ( r ) = ρ ( rr s ) γ [1 + ( rr s ) α ] ( β − γ ) /α (21)where β and γ are the inner and outer slopes, respectively, and α describesthe transition between the inner and outer regions. These parameters are rep-resented by the equations below: α = 2 . − log(10 ( X +2 . − . + 10 ( X +2 . . ) ,β = 4 .
23 + 1 . X + 0 . X ,γ = − .
06 + log 10 ( X +2 . − . + 10 X +2 . (22)where X = log( M star /M halo ) , is the stellar-to-halo mass (SHM) ratio in log-arithm space. The mass [50] and velocity of DC14 profile are as follows: ∗ et al. M DC ( r ) = 4 πρ r s α ( B [ a, b + 1 , (cid:15) ] + B [ a + 1 , b, (cid:15) ]) (23) V DC ( r ) = 4 πGρ r s r α ( B [ a, b + 1 , (cid:15) ] + B [ a + 1 , b, (cid:15) ]) (24)where B ( a, b, x ) = (cid:82) x t α − (1 − t ) b − dt is the incomplete Beta function and a = (3 − γ ) /α , b = ( β − /α and (cid:15) = ( r/r s ) α (1+( r/r s ) α ) . This equation only worksfor the SHM ratio within -4.1 < X < -1.3, due to the fact that this is therange where the supernovae feedback is significant and dominant [47]. At X < -4.1, the energy released by supernova is insufficient to modify the initialcuspy profile, while at X > -1.3, the feedback due to active galactic nucleimight start to dominate [50]. A coreNFW halo [51] is essentially a NFW halo which transforms an inner cuspinto a finite central core by a spherically symmetric function f n that modelsthe effects of supernova feedback [52]. The mass of coreNFW is defined as: M cNF W ( < r ) = M NF W ( < r ) f n ( r ) (25)with f ( r ) = [tanh( rr s )] (26)The strength of the core is determined by the parameter n , which rangesbetween 0 < n ≤
1. The equation of n is as follows: n = tanh( κ t SF t dyn ) (27)where κ is a tunning parameter and t SF is the star formation time of thegalaxy. We set κ = 0.04 and t SF = 14 Gyrs as suggested by the simulations ofRead et al. [51]. The dynamical time, t dyn is the duration of 1 circular orbitat the scale radius in the unmodified NFW halo: t dyn = 2 π (cid:115) r s ( GM NF W ( r s )) (28)Hence, the mass and velocity of coreNFW profile are defined as: M cNF W ( < r ) = 4 πρ r s (ln(1 + rr s ) − rr s + r ) f n ( r ) (29) V cNF W ( < r ) = 4 πGρ r s r (ln(1 + rr s ) − rr s + r ) f n ( r ) (30) ark matter analysis on galaxy M100 17 Lucky13 is a new semi-empirical profile constructed from Equation (21), the( α , β , γ ) models by Li et al. [50]. They considered the transition parameter α = 1, γ = 0 to reach a finite core and β = 3 to get the same decreasing rate asthe NFW profile at large radii. The density, mass and velocity of Lucky13 [50]are as follows: ρ = ρ [1 + ( rr s )] (31) M ( r ) = 4 πρ r s [ln(1 + rr s ) + 2(1 + rr s ) − rr s ) −
32 ] (32) V ( r ) = 4 πGρ r s r [ln(1 + rr s ) + 2(1 + rr s ) − rr s ) −
32 ] (33)2.6
Star Velocity
Most of the luminous mass is occupied by stars, and the rest (i.e. less than 10%) is filled with interstellar gases. Hence, the star disc luminosity distributionroughly represents the luminous mass distribution [53]. To estimate the stellarstar luminosity, we use the velocity of star disc derived by Freeman [54, 55],which is described by the equation below: V star = GM D x R D ( I o K o − I K ) (34)where G is the universal gravitational constant, I n and K n are the modifiedBessel functions of the first and second kinds computed at x / x = rR D , M D is the star mass and R D is the star scale length with value in kpc. Thestar scale length for NGC 4321 that we adopted is 75 arcsec [56], which isequivalent to 6.22 kpc.The bulge in M100 is small compared to the disks hence no bulge-diskdecomposition was adopted [57]. The bulge of M100 does not contribute muchto the surface brightness of the disk and the surface brightness have a linearcorrelation with stellar mass [58]. This indicates that the bulge of M100 doesnot contribute much to the stellar mass and can be omitted.2.7 Gas velocity
Taking account of the gas component and if its mass is significant, it is possibleto use photometry to calculate the rotation curve of the baryonic disc com-ponent by matching it with observations to estimate the effect of dark matter ∗ et al. ( kpc ) ∑ G a s Su r f a ce D e n s i t y , σ ( r )( M ⊙ ) Fig. 6 (cid:80)
Gas surface density graph on the disc dynamics [59]. We compute the mass of gas from the equation [60]that is given below: M gas ( r ) = 2 π (cid:90) r rσ ( r ) dr (35)where r is the radius from centre of the galaxy and σ ( r ) is the surface density.From Equation (35), we applied the non-linear fitting method to the curveof total gas surface density. The steps of obtaining the gas velocity by usingsoftware Mathematica are as below:i Adopt the surface density profile total gas data from T. Wong and L.Blitz [61] as shown in Figure 6.ii Plot the best fitting curve that computed from several trial values of theparameter and equation as shown in Figure 7.iii Calculate the goodness of fit, χ to indicate the best-fitting curves and itwill show the best model of Σ gas surface density, σ ( r ).iv Apply the best model of Σ gas surface density, σ ( r ) to Equation (35) tocalculate the mass of gas for each radius.v From the mass of gas for each radius, then derive the velocity for eachradius by considering the equation v = (cid:114) GM ( < r ) r (36)and the result is as shown in Figure 8. ark matter analysis on galaxy M100 19 ( kpc ) L og ∑ G a s Su r f a ce D e n s i t y , σ ( r )( M ⊙ ) Fig. 7
The best-fitting graph with χ = 0 . . The red dot point represents gassurface density and the blue line represents the best fitting line. ( kpc ) G a s V e l o c i t y ( k m / s ) Fig. 8
Gas velocity graph ∗ et al.Dark Matter Profile M D ( M (cid:12) ) ρ ( ρ − for Einasto)( M (cid:12) kpc − ) r s ( r − forEinasto) (kpc)Pseudoisothermalprofile (5 . ± . × (1 . ± . × . ± . . ± . × (1 . ± . × . ± . . ± . × (1 . ± . × . ± . . ± . × (1 . ± . × . ± . . ± . × (3 . ± . × . ± . . ± . × (5 . ± . × . ± . . ± . × (7 . ± . × . ± . . ± . × (6 . ± . × . ± . . ± . × (0 . ± . × . ± . This section is organized as follows: In subsection 3.1 we present the freeparameter obtained and nonlinear fitting rotation curve. Then we illustratethe chi-square test and the mass of dark matter in subsection 3.2. The ninedark matter profiles analysis will be discussed one by one in subsections 3.1and 3.2.3.1
Free parameter and nonlinear fitting rotation curve
By referring to Equation (1), where V rot = V HI from VLA data, V gas derivedfrom total gas surface density data, while for star and dark matter velocitywe will use nonlinear fitting rotation curve method to find the most accuratefree parameter. For star velocity V star , we will use Equation (34) while fordark matter velocity V DM , we will use Equations (5), (8), (11), (14), (17),(20), (24), (30), (33) for Pseudoisothermal, Burkert, NFW, Moore, Einasto,core-modified, DC14, coreNFW and Lucky13 profiles respectively.We calculated the free parameters from the best fit that we have achieved.The free parameters of star mass M D , scale density ρ and scale radius r s areobtained using the Pseudoisothermal, Burkert, NFW, Moore, core-modified,DC14, coreNFW and Lucky13 profiles. Meanwhile the radius where densityprofile has a slope of -2, r − and the local density at that radius, ρ − isobtained using the Einasto profile. The third free parameter of Einasto profileis obtained using Einasto index that is fitted to n = 0 . ± . < X < -1.3 [47]. During the fitting process, we used X as a free parameter and ark matter analysis on galaxy M100 21 PSEUDOISOTHERMAL
STAR PSEUDOISOTHERMAL _ HALOTOTALGAS ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) BURKERT
STAR BURKERT _ HALOTOTALGAS ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) NFW
STAR NFW _ HALOTOTALGAS ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) MOORE
STAR MOORE _ HALOTOTALGAS ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) EINASTO
STAR EINASTO _ HALOTOTALGAS ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) COREMODIFIED
STAR COREMODIFIED _ HALOTOTALGAS ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) DC14
STAR DC14 _ HALOTOTALGAS ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) CORENFW
STAR coreNFW _ HALOTOTALGAS ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) LUCKY13
STAR LUCKY13 _ HALOTOTALGAS ( kpc ) R o t a t i o n V e l o c i t y ( k m / s ) Fig. 9
The nonlinear rotation curve of NGC 4321 by using the nine dark matterprofiles. The black and red color represents the total rotation velocity andbest fitting line, while the grey, orange and blue color represent gas, starand dark matter halo velocities respectively. ∗ et al. performed the fitting within the range of -4.1 and -1.3 but none of the valueswere able to fit the rotation curve well. We then calculated the SHM ratioby applying the relation X = log (Mstar/Mhalo) and the masses of the starand dark matter from the five accepted dark matter profiles (Table 7). Thecalculation for all five accepted profiles showed that the limit of X > -1.3 forthis galaxy, which was found to be beyond the limit of the SHM ratio range.Meanwhile, for the Lucky13 profile, this new semi-empirical profile changesthe value of alpha, beta and gamma of the ( α, β.γ ) model. The gamma setto 0 to reach a finite core, beta set to 3 to get the same decreasing rate asthe NFW profile at large radii and transition parameter alpha is set to 1 [50].These changes within the new semi-empirical profile are not suitable for thisgalaxy.According to the previous researches, the Burkert profile is statisticallymore suitable for the dark matter-dominated dwarf galaxies [34]. Burkert pro-file revises the density law of Pseudoisothermal profile in the inner region anddiverge logarithmically for larger radii. By referring to Figure 9, the Burkertprofile reveals the existence of high dark matter velocity 138 kms − within theinner region close to 0 kpc and increases gradually until the large radii of thisgalaxy. However, the HI observation data shows the total rotational velocityis only 25 kms − within the inner region close to 0 kpc, hence the mismatchbetween observation data and Burkert profile suggests that Burkert profile isnot suitable for this spiral galaxy.Moore profile behaves similarly to NFW profile at large radii but it issteeper than NFW profile at smaller radii [39]. By referring to Figure 9, thedark matter velocity of the Moore profile has the steepest rise in the innerregion compare to other profiles (except DC14 and Lucky13 profiles). Thesteepness in the Moore profile leads to its dark matter velocity to be foundas higher than the total rotational velocity in the inner region. Even as itssteepness reduces after radius 0.5 kpc, the dark matter velocity is still higherthan the total rotational velocity below the radius 2.0 kpc. After radius 2.0kpc, the dark matter velocity of Moore profile is continuously lower than thetotal rotational velocity until the large radii of the galaxy. However, our resultfor the dark matter velocity of Moore profile that is found to be higher thanthe total rotational velocity before radius 2.0 kpc suggests that Moore profileis not suitable for this galaxy too.For the remaining five accepted dark matter profiles, it is difficult to deter-mine which dark matter model is better by looking at the figure only. Hence,we will do a chi-square test calculation first before making dark matter modelanalysis.3.2 Chi-square test and the mass of dark matter
We implemented the chi-square test to test the goodness-of-fit of the rotationcurve fitting. The goodness-of-fit parameter, χ to be minimized is [62]: ark matter analysis on galaxy M100 23Dark Matter Profile Model χ red Cored Pseudoisothermal profile 1.25Einasto profile 0.52core-modified profile 3.29coreNFW profile 1.64Cuspy NFW profile 0.49Table 6: χ red for five dark matter profiles χ = N (cid:88) i =1 σ i [ y i − y ( x i , a, b, c )] (37)where N are observed data points, σ i is the uncertainty in y i , y i is the observedrotation curve, y ( x i , a, b, c ) are the values of the model function calculated at x i , x i is the radius from the galactic centre and a, b, c are the fit parameters.Then we performed the goodness-of-fit test by implementing the reducedchi-square: χ red = χ v (38)where v = N − N c , N is the number of data points and N c is the number offit parameters. In principle, a value of χ red = 1 indicates that the extent ofthe match between the observations data and the value of the estimates is inaccord with the error variance [62].All reduced chi-square, χ red for the five accepted dark matter profiles areshown in Table 6. Generally, we achieved 0.40 < χ red < χ red closestto 1, which is Pseudoisothermal profile. For cored profiles, the core-modifiedprofile achieved the highest χ red with 3.29 and followed by coreNFW profileswith χ red of 1.64. While the Einasto and cuspy NFW profile achieved similarlowest χ red with 0.52 and 0.49, respectively.Next, we will continue our five accepted dark matter model analysis byreferring to the results from the rotation curve in Figure 9 rotation curve andthe reduced chi-square test in Table 6.Core-modified is a profile with constant density in the central core to avoidsingularity in the galactic center [45]. However, the constant density in the cen-tral core presented very little flexibility for this profile during the fitting processin the rotation curve modeling. During rotation curve fitting, we found thatthis profile has certain limitations where less fitting changes that can be madeespecially in the central core of the galaxy. For the large radii of the galaxy,the increasing rate of dark matter velocity is declining and becomes almostconstant after 7.5 kpc radius. The dark matter velocity is normally supposedto continue increasing as the total rotational velocity increases throughout.The increment of dark matter velocity in the large radii of the galaxy can beseen in NFW, coreNFW, Einasto and Pseudoisothermal profiles. This darkmatter profile is found to be able to fit the galaxy and the calculated dark ∗ et al. matter velocity is reasonably better when compare to the other four rejecteddark matter profiles. However, the limited flexibility in the core and the con-stant dark matter velocity in the large radii make the core-modified profile isworse than other four accepted dark matter profiles and only achieved a χ red of 3.29.A coreNFW halo is essentially a NFW halo which transforms an inner cuspinto a finite central core [51]. The change of inner cusp into a finite centralcore of this profile have brought improvement in the rotation curve fitting inthe inner region of this galaxy. By referring to the inner region of this profile,the increasing dark matter velocity is less steep than the cuspy NFW profile.The steepness of dark matter velocity has made this profile to have good fitwith total rotational velocity in the inner region of the galaxy. However, theincreasing rate of dark matter velocity is found to be declining beyond 8 kpcradius, causing unsuitability of fitting with the continuously increasing totalrotational velocity. This issue causes the coreNFW profile to be considered asnot as good as Peudoisothermal profile and achieved a χ red of 1.64.The NFW profile is called ‘universal’ because it works for a large varietyof halo masses, same shape of initial density fluctuation spectrum, from indi-vidual galaxies to the halos of galaxy clusters [63] and this leads to the NFWprofile that fits well to this galaxy as well. However, the NFW profile has thecuspy halo problem that increases steeply at small radii [64]. This problemcan be seen in Figure 9, the inner region of the NFW profile is steeper thanthe coreNFW, Pseudoisothermal and Einasto profiles. From the inner region0 kpc to 1 kpc, the steepness of the NFW profile causes the rotational veloc-ity of halo, star and gas to mismatch with the total rotational velocity. Thismismatch causes the fitting of NFW profile to be achieved χ red of 0.49, whichis slightly not as good as the case for the Einasto profile.High-resolution N-body CDM simulations indicate that nonsingular three-parameter models such as the Einasto profile perform better than the singulartwo-parameter models such as Pseudoisothermal and NFW profiles, providingan excellent fit to a wide range of dark matter haloes [65]. The Einasto profileinvolves a third parameter, n, the Einasto index, which describes the shape ofthe overall profile distribution, larger values of n results in steeper inner profilesand shallower outer profiles [66]. In order to have the best-fitting with the totalrotational velocity of this galaxy, the inner profile needs to be shallower andthe outer profile needs to be steeper. In our case for this paper, the thirdparameter, n, acts a very important role to fulfill the criteria of the fitting,where we are able to use smaller values of n to make the inner profile shallowerand the outer profile steeper (Figure 9). Furthermore, these three parametersmodel allows the profile to be tailored to each individual halo, thereby yieldingimproved fits [67]. This can be seen in Figure 9 where each individual halo iswell fitted with total rotational velocity. However, the addition of the thirdparameter and the profile tailored to each individual halo causes the Einastoprofile to ultimately overfits the data with the value χ red of 0.52.Pseudoisothermal profile is a commonly used model for dark matter haloanalysis and is often seen to better fit the galactic rotation curve than the ark matter analysis on galaxy M100 25Dark MatterProfile Model Mass of star( M (cid:12) ) Mass of dark matter( M (cid:12) )Cored Pseudoisothermal (3 . ± . × (1 . ± . × Einasto (3 . ± . × (1 . ± . × core-modified (3 . ± . × (1 . ± . × coreNFW (3 . ± . × (1 . ± . × Cuspy NFW (3 . ± . × (1 . ± . × Table 7: Total dark matter mass within radius 10 kpc for the dark matter profiles
NFW model [68, 69] and this works for this spiral galaxy as well. By lookingat Figure 9, the overall Pseudoisothermal profile distribution fits very well andmatches with Equation (1). In addition, the Pseudoisothermal rotation curvehas a linear growth at the inner region then becomes flat at large radii [70].This trend can be seen on this galaxy too, where the rotational velocity ofthe Pseudoisothermal profile in the inner region below 1 kpc increases linearlyand the rotational velocity becomes flat at large radii. The linearity at theinner region and the flatness at the large radii of the fitting characteristic arevery suitable for this galaxy. This causes the Pseudoisothermal to achieve thebest fitting among the nine dark matter profiles with χ red of 1.25, which isthe nearest to 1.Based on the free parameters obtained as shown in Table 5, we calculatedthe total star mass and dark matter mass within radius 10 kpc for each darkmatter profile as shown in Table 7. We obtained the star velocity from Equation(34) and subsequently calculated the star mass with Equation (36), while themass of dark matter is calculated by applying the Equations (4), (10), (16),(19), (29) for Pseudoisothermal, NFW, Einasto, core-modified and coreNFWprofiles, respectively.The total dark matter mass for the five accepted dark matter profiles withinradius 10 kpc in this galaxy is in the range from (1 . ± . × M (cid:12) to(1 . ± . × M (cid:12) . By taking consideration of the χ red , the Pseudoisother-mal, Einasto, coreNFW and NFW profiles achieved the χ red within the rangeof 0.40 < χ red < . ± . × M (cid:12) to (1 . ± . × M (cid:12) . For this galaxy, we analyzed the dark matter of the galaxy NGC 4321 by usingthe nine dark matter profiles. The nine dark matter profiles consist of sevencore and two cuspy profiles. The core profiles are namely Pseudoisothermal,Burkert, Einasto, core-modified, DC14, coreNFW and Lucky13 profiles. Thecuspy profiles are NFW and Moore profiles. We analyzed in detail how eachdark matter profile performs in this galaxy. We rejected four dark matterprofiles while five dark matter profiles are accepted for this galaxy. DC14profile is rejected due to the stellar-to-halo mass ratio that is found to be outof the range of the profile condition. Lucky13 profile is rejected due to the ∗ et al. unsuitability of the galaxy model setting of this new semi-empirical profile.The revised density law of Burkert profile causes the dark matter velocity tomismatch with regards to the total rotational velocity starting from the innerregion of the galaxy. Further analysis showed that the Moore profile has thesteepest rise in the inner region among the five dark matter profiles and thiscauses the dark matter velocity in the inner region to be higher than the totalrotational velocity below the radius 2.0 kpc. These reasons suggest that thesefour dark matter profiles are not suitable for this galaxy.For the remaining five accepted dark matter profiles, we analyzed the darkmatter velocity together with the reduced chi-square test. The constant coreof the core-modified profile causes the limited flexibility on fitting and henceproducing a large value of χ red . The NFW profile cuspy halo has a weakness,where the inner region increases steeply at small radii, resulting in the totalrotational velocity of halo, star and gas to be higher than the total rotationalvelocity by HI observed data at the inner region of this galaxy. The change ofinner cusp into a finite central core of the coreNFW profile improved the NFWprofile to become less steep in the inner region of the galaxy and this resulted ina better rotation curve fitting. The additional third parameter of the Einastoprofile brings the flexibility of adjustment in the inner and outer radii of thegalaxy and this enabled the Einasto profile to generate a better overall fitting.The Pseudoisothermal profile produced linear rotational velocity in the innerregion and flat rotational velocity at the large radii of this galaxy. This fittingcharacteristic suits this galaxy very well by producing the best fitting amongthe nine dark matter profiles with a χ red of 1.25.Generally, the NFW, coreNFW, Einasto and Pseudoisothermal profilesachieve relatively good fittings and the mass of dark matter for these fourprofiles is calculated to be within the range from (1 . ± . × M (cid:12) to(1 . ± . × M (cid:12) . Ultimately, our results showed that the Pseudoisother-mal profile achieved the best fitting among the nine dark matter profiles andthe dark matter mass of this galaxy obtained through the use of this profile iscalculated as (1 . ± . × M (cid:12) . Acknowledgements
The authors would like to acknowledge the funding by the Universityof Malaya (FG033-017AFR).ark matter analysis on galaxy M100 27
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