Newtonian Fractional-Dimension Gravity and Rotationally Supported Galaxies
NNEWTONIAN FRACTIONAL-DIMENSION GRAVITY AND ROTATIONALLYSUPPORTED GALAXIES
Gabriele U. Varieschi ∗ Loyola Marymount University, Los Angeles, CA 90045, USA (Dated: November 11, 2020)We continue our analysis of Newtonian Fractional-Dimension Gravity (NFDG), an extension of thestandard laws of Newtonian gravity to lower dimensional spaces including those with fractional (i.e.,non-integer) dimension. We apply our model to three rotationally supported galaxies: NGC 7814(Bulge-Dominated Spiral), NGC 6503 (Disk-Dominated Spiral), and NGC 3741 (Gas-DominatedDwarf).As was done in the general cases of spherically-symmetric and axially-symmetric structures, whichwere studied in previous work on the subject, we examine a possible connection between NFDGand Modified Newtonian Dynamics (MOND), a leading alternative gravity model which explainsthe observed properties of these galaxies without requiring the Dark Matter (DM) hypothesis.In NFDG, the MOND acceleration constant a (cid:39) . × − m s − can be related to a naturalscale length l , namely a ≈ GM/l for a galaxy of mass M . Also, the empirical Radial AccelerationRelation (RAR), connecting the observed radial acceleration g obs with the baryonic one g bar , can beexplained in terms of a variable local dimension D . As an example of this methodology, we providedetailed rotation curve fits for the three galaxies mentioned above. PACS numbers: 04.50.Kd, 95.30.Sf, 95.35.+d, 98.62.DmKeywords: Newtonian Fractional-Dimension Gravity; Modified Gravity; Modified Newtonian Dynamics;Dark Matter; Galaxies ∗ E-mail me at: [email protected] ; Visit: http://gvarieschi.lmu.build
Typeset by REVTEX a r X i v : . [ g r- q c ] N ov I. INTRODUCTION
This work continues the analysis started in previous publications ([1, 2] - paper I and II, respectively, in thefollowing) of
Newtonian Fractional-Dimension Gravity (NFDG), an extension of the laws of Newtonian gravitationto lower dimensional spaces, including those with non-integer, “fractional” dimension (see also Ref. [3], for a generalintroduction to NFDG). This model is based on the methods of fractional mechanics/electromagnetism (see [4–6],and references therein) and on the general framework of fractional calculus (FC) [7, 8].Our NFDG as introduced in papers I-II is not a fractional theory, in the sense used by other models [9–11], sincethe NFDG field equations as outlined in the next section are of integer order; thus local, as opposed to non-local fieldequations based on fractional differential operators. In paper I, we based NFDG on a generalization of the gravitationalGauss’s law, replacing standard space integration over R with an appropriate Hausdorff measure over the space, whichwas related to Weyl’s fractional integrals. In this way, Newtonian gravity was extended to fractal spaces of non-integerdimension, with focus on spherically-symmetric galactic structures, such as homogeneous spheres, Plummer models,etc. In paper II, we extended our analysis to axially-symmetric structures, such as exponential thin/thick disk galaxies,Kuzmin models, etc., and produced a preliminary rotation curve fit for NGC 6503.Obviously, the goal of NFDG is to describe galactic dynamics without using the controversial DM component, inview of the absence of any direct detection of dark matter in recent years [12, 13]. As shown in papers I-II, NFDGmight also be connected with Modified Newtonian Dynamics (MOND) [14–16], although NFDG is an inherentlylinear theory due to the integer order of its field equations, while MOND is a fully non-linear theory. In addition,the empirical correlation between the observed gravitational acceleration g obs , traced by galactic rotation curves, andthe predicted acceleration g bar , based on the observed distribution of baryons (Radial Acceleration Relation - RAR[17, 18]) was explained by NFDG in terms of an effective variable dimension D , which characterizes each galacticstructure.In Sect. II, we will review the fundamental ideas of NFDG from our first two papers and their connections withMOND/RAR. In Sect. III, we will apply NFDG to three rotationally supported galaxies, which will be used asreference cases in this work: NGC 7814 (bulge-dominated spiral), NGC 6503 (disk-dominated spiral), and NGC 3741(gas-dominated dwarf). Also, in Sect. III D we will discuss how the analysis of these three reference galaxies mightbe generalized to all rotationally supported galaxies. Finally, in Sect. IV we will outline possible future work on thesubject and draw our conclusions. II. QUICK REVIEW OF NFDG
In this section we summarize the main ideas of NFDG and its applications to galactic dynamics. Full details canbe found in papers I-II [1, 2] and references therein.Newtonian Fractional-Dimension Gravity was introduced heuristically by extending Gauss’s law for gravitation to alower-dimensional space-time D + 1, where D ≤ l is needed toensure dimensional correctness of all expressions for D (cid:54) = 3; thus, it is convenient to use dimensionless coordinates inall formulas, such as the radial distance w r ≡ r/l or, in general, the dimensionless coordinates w ≡ x /l for the fieldpoint and w (cid:48) ≡ x (cid:48) /l for the source point. We also introduced a rescaled mass “density” (cid:101) ρ ( w (cid:48) ) = ρ ( w (cid:48) l ) l = ρ ( x (cid:48) ) l ,where ρ ( x (cid:48) ) is the standard mass density in kg m − , with d (cid:101) m ( D ) = (cid:101) ρ ( w (cid:48) ) d D w (cid:48) representing the infinitesimal mass ina D-dimensional space. The NFDG gravitational potential (cid:101) φ ( w ) was introduced as: (cid:101) φ ( w ) = − π − D/ Γ( D/ G ( D − l (cid:90) V D (cid:101) ρ ( w (cid:48) ) | w − w (cid:48) | D − d D w (cid:48) ; D (cid:54) = 2 (1) (cid:101) φ ( w ) = 2 Gl (cid:90) V (cid:101) ρ ( w (cid:48) ) ln | w − w (cid:48) | d w (cid:48) ; D = 2with (cid:101) φ ( w ) and g ( w ) connected by g ( w ) = −∇ D (cid:101) φ ( w ) /l , where the D-dimensional gradient ∇ D is equivalent to thestandard one, but the derivatives are taken with respect to the rescaled coordinates w . It is easy to check that allthe expressions above correctly reduce to the standard Newtonian ones for D = 3.In our previous papers I-II, we used φ ( w ) ≡ (cid:101) φ ( w ) /l to denote the NFDG potentials. Here we prefer to use (cid:101) φ , sothat the first line in Eq. (1) for D = 3 will yield the standard Newtonian potential. Since ∇ D is defined in terms of SI units will be used throughout this paper, unless otherwise noted. dimensionless coordinates, the physical dimensions for φ are the same as those for the gravitational field g , i.e., bothquantities are measured in m s − . On the contrary, the physical dimensions for the gravitational potential (cid:101) φ are nowthe same as those of the standard Newtonian potential (i.e., measured in m s − ).The gravitational potential in Eq. (1) was derived for a fixed value of the fractional dimension D , but we arguedthat it could be applicable also to the case of a variable dimension D ( w ), assuming a slow change of the dimension D with the field point coordinates. The scale length l was related to the MOND acceleration constant a (also denotedby g † [17, 18]): a ≡ g † = 1 . ± .
02 (random) ± .
24 (syst) × − m s − , (2)which represents the acceleration scale below which MOND corrections are needed.MOND [14–16] proposed modifications of Newtonian dynamics in terms of modified inertia (MI), or modified gravity(MG) [19]. Respectively: mµ ( a/a ) a = F as MI, since the mass m is replaced by mµ ( a/a ), and µ ( g/a ) g = g N asMG, since the observed gravitational field g can differ from the Newtonian one g N . While there is now limited evidence[20, 21] that MG might be favored over MI, the two formulations are practically equivalent, although conceptuallydifferent: the former modifies Newton’s laws of motion, while the latter modifies Newton’s law of universal gravitation.Following Eq. (1) above, NFDG can be also considered a modification of gravity and not of inertia, since we assumethat test objects, such as stars in galaxies, will still move in a (classical) 3 + 1 space-time and obey standard laws ofdynamics.The interpolation function, µ ( x ) ≡ µ ( a/a ) for the MI case or µ ( x ) ≡ µ ( g/a ) for the MG case, was introducedby MOND as: µ ( x ) → x (cid:29) µ ( x ) → x for x (cid:28) µ ( x ) (or its inverse function ν ( y ), see papers I-II for details), with the particularchoice ν ( y ) = (cid:2) − exp (cid:0) − y / (cid:1)(cid:3) − as the favorite interpolation function in the literature [17, 18]. This function isequivalent to the Radial Acceleration Relation - RAR: g obs = g bar − e − √ g bar /g † , (3)where g † corresponds numerically to the MOND acceleration scale a , reported in Eq. (2) above.The RAR relates the radial acceleration g obs traced by rotation curves with the radial acceleration g bar predictedby the observed distribution of baryonic matter in galaxies [17, 18] and was originally obtained by analyzing datafrom 175 galaxies in the Spitzer Photometry and Accurate Rotation Curves (SPARC) database [22]. It was confirmedin more recent work [18, 23] by adding early-type-galaxies (elliptical and lenticular) and dwarf spheroidal galaxies tothe same database.In papers I-II, we proposed a possible connection between the scale length l and the MOND acceleration a as: a ≈ GMl , (4)where M is the total mass (or a reference mass) of the system being studied. It was shown that the main consequencesof the MOND theory could be recovered in NFDG by considering the Deep-MOND limit equivalent to reducing thespace dimension to D ≈
2. In particular, the asymptotic or flat rotation velocity V f ≈ √ GM a exhibited by galacticrotation curves, the “baryonic” Tully-Fisher relation-BTFR: M bar ∼ V f , and other fundamental MOND predictionswere recovered in NFDG for the case D ≈ (cid:101) ρ ( w (cid:48) r ) and proved that the gravitationalfield g ( w r ), in a fractal space of dimension D ( w r ) depending on the radial distance w r = r/l , can be computed as: g obs ( w r ) = − πGl w D ( w r ) − r (cid:90) w r ˜ ρ ( w (cid:48) r ) w (cid:48) D ( wr ) − r dw (cid:48) r (cid:98) w r , (5)for 1 ≤ D ≤
3. In the last equation, the gravitational field is denoted as the “observed” one, g obs , while the “baryonic” g bar is considered to be the one for fixed dimension D = 3: g bar ( w r ) = − πGl w r (cid:90) w r ˜ ρ ( w (cid:48) r ) w (cid:48) r dw (cid:48) r (cid:98) w r . (6)Therefore, the connection with MOND and the RAR was simply obtained in NFDG by considering the ratio betweenthe previous two equations: (cid:18) g obs g bar (cid:19) NF DG ( w r ) = w − D ( w r ) r (cid:90) w r (cid:101) ρ ( w (cid:48) r ) w (cid:48) D ( wr ) − r dw (cid:48) r (cid:90) w r (cid:101) ρ ( w (cid:48) r ) w (cid:48) r dw (cid:48) r , (7)and by comparing it with the similar ratio coming from the RAR in Eq. (3): (cid:0) g obs g bar (cid:1) MOND ( w r ) = − e − √ gbar ( wr ) /g † .This procedure was indeed successful for several different forms of spherically-symmetric mass distributions analyzedin paper I. In each case, we computed numerically the dimension functions D ( w r ) by comparing directly the ratios (cid:0) g obs g bar (cid:1) NF DG and (cid:0) g obs g bar (cid:1) MOND above. The variable dimension D ( w r ) assumed values D ≈ D ≈ ∇ D to obtain the field. For the general NFDG potential in thefirst line of Eq. (1), (cid:101) φ ∼ / | w − w (cid:48) | D − (sometimes referred to as the Euler kernel ), we used the following expansion[24] in (rescaled) spherical coordinates: 1 | w − w (cid:48) | D − = ∞ (cid:88) l =0 w lr< w l + D − r> C ( D − ) l (cos γ ) , (8)where w r< ( w r> ) is the smaller (larger) of w r and w (cid:48) r , γ is the angle between the unit vectors (cid:98) w and (cid:98) w (cid:48) , and C ( λ ) l ( x )denotes Gegenbauer polynomials (see paper I or Ref. [25] for general properties of these special functions).This expansion was easily adapted to the case of cylindrical coordinates ( w R , ϕ , w z ). In the case of thin disks, inthe w z = w (cid:48) z = 0 plane and in the ϕ = 0 direction, the angle γ is replaced by ϕ (cid:48) and the radial spherical coordinate w r ≡ r/l with the cylindrical w R ≡ R/l :1 | w − w (cid:48) | D − = ∞ (cid:88) l =0 w lR< w l + D − R> C ( D − ) l (cos ϕ (cid:48) ) , (9)while, for thick disks with w (cid:48) z (cid:54) = 0 (but still in the w z = 0 plane and ϕ = 0 direction), we used the following coordinatetransformations: w r = w R (10) w (cid:48) r = (cid:113) w (cid:48) R + w (cid:48) z cos γ = w (cid:48) R cos ϕ (cid:48) (cid:112) w (cid:48) R + w (cid:48) z , and modified the original expansion in Eq. (8) accordingly.The general NFDG potential in the first line of Eq. (1) was then computed using the techniques for multi-variableintegration over a fractal metric space W ⊂ R (see again papers I-II), combined with the expansions of the Eulerkernel outlined above. For thin/thick disk structures, the rescaled mass distributions were taken, respectively, as: (cid:101) ρ ( w (cid:48) R , w (cid:48) z ) = (cid:101) Σ ( w (cid:48) R ) δ ( w (cid:48) z ) (11) (cid:101) ρ ( w (cid:48) R , w (cid:48) z ) = (cid:101) Σ ( w (cid:48) R ) (cid:101) ζ ( w (cid:48) z )where the surface mass distribution (cid:101) Σ ( w (cid:48) R ) can be related to an exponential model, a Kuzmin model, or simplyobtained by interpolating SPARC surface luminosity data and transforming them into surface mass distributions,using appropriate mass-to-light ratios.For thin disks, the vertical density is simply described by the delta function δ ( w (cid:48) z ), while for thick disks we typicallyuse an exponential function (cid:101) ζ ( w (cid:48) z ) = H z e − w (cid:48) z /H z , where the rescaled parameter H z = h z /l is connected with theoriginal vertical scale height h z . We also adopted the standard relation [22, 26], ( h z / kpc) = 0 .
196 ( R d / kpc) . ,between the vertical scale height h z and the radial scale length R d (available from SPARC data), properly rescaledby using our dimensionless variables.For exponential and Kuzmin thin disks, the NFDG potential can be obtained analytically and full results werereported in paper II, while similar analytic results for spherical models were reported in both papers I and II. In thiswork, we will analyze instead galaxies whose mass distributions are obtained directly from the SPARC luminositydata. In general, SPARC data include three types of luminosity distributions: a spherically-symmetric bulge , a stellar disk component, and a gas disk component (in the following: bulge, disk, and gas components, for short).For the disk and gas components, the SPARC surface luminosities Σ ( L ) disk ( R ) and Σ ( L ) gas ( R ), can be turned into(rescaled) surface mass distributions, (cid:101) Σ disk ( w (cid:48) R ) and (cid:101) Σ gas ( w (cid:48) R ), by using appropriate mass-to-light ratios [27]: Υ disk (cid:39) . M (cid:12) /L (cid:12) , Υ gas (cid:39) . M (cid:12) /L (cid:12) (this value for Υ gas includes also the helium gas contribution). We then sum thesedistributions together, (cid:101) Σ ( w (cid:48) R ) = (cid:101) Σ gas ( w (cid:48) R ) + (cid:101) Σ disk ( w (cid:48) R ), and combine this total surface distribution (cid:101) Σ ( w (cid:48) R ) with thevertical exponential function (cid:101) ζ ( w (cid:48) z ) described above, into the second line of Eq. (11).For the bulge component, SPARC data are also available in terms of a surface luminosity Σ ( L ) bulge ( R ) which can beturned into a (rescaled) surface mass distribution (cid:101) Σ bulge ( w (cid:48) R ) by using Υ bulge (cid:39) . M (cid:12) /L (cid:12) [27] and then convertedinto a spherically-symmetric mass distribution ˜ ρ ( w (cid:48) r ) by applying Eq. (1.79) in Ref. [28]. In this way, for each galaxystudied in this paper, we will have bulge, disk, and gas mass distributions based directly on the experimental SPARCdata, instead of using pre-determined models, such as exponential, Kuzmin for disk galaxies, and Plummer or othermodels for spherical components.For the combined disk and gas components, the NFDG potential from Eq. (1) can be obtained by applying theexpansion in Eqs. (8)-(10) and by performing a triple integration over w (cid:48) R , ϕ (cid:48) , and w (cid:48) z . As discussed in paper II, wealso need to connect the overall space dimension D with the respective fractional dimensions - α R , α ϕ , α z - of eachsub-space, so that D = α R + α ϕ + α z . As in paper II, we assume α z = 1, i.e., no fractional dimension is needed inthe z (cid:48) direction, as opposed to α R = α ϕ = α = D − , so that the radial and angular coordinates will share the same(variable) fractional dimension. Therefore, we will have: D = 2 α + 1 ≤
3, and the results will depend only on theoverall dimension D of the space.With these assumptions, and using the techniques outlined in papers I-II for multi-variable integration over a fractalmetric space, we then obtain the total thick-disk potential in the w z = 0 plane as: (cid:101) φ ( w R ) = − √ π Γ ( D/ G ( D − (cid:2) Γ (cid:0) D − (cid:1)(cid:3) l ∞ (cid:88) l =0 (cid:90) ∞ (cid:101) Σ ( w (cid:48) R ) w lR< w l + D − R> w (cid:48) D − R dw (cid:48) R (cid:90) ∞ (cid:101) ζ ( w (cid:48) z ) c l,D ( w (cid:48) R , w (cid:48) z ) dw (cid:48) z (12)= − √ π Γ ( D/ G ( D − (cid:2) Γ (cid:0) D − (cid:1)(cid:3) l ∞ (cid:88) l =0 (cid:40) (cid:90) w R (cid:101) Σ ( w (cid:48) R ) (cid:16)(cid:112) w (cid:48) R + w (cid:48) z (cid:17) l w l + D − R w (cid:48) D − R dw (cid:48) R (cid:90) ∞ (cid:101) ζ ( w (cid:48) z ) c l,D ( w (cid:48) R , w (cid:48) z ) dw (cid:48) z + (cid:90) ∞ w R (cid:101) Σ ( w (cid:48) R ) w lR (cid:16)(cid:112) w (cid:48) R + w (cid:48) z (cid:17) l + D − w (cid:48) D − R dw (cid:48) R (cid:90) ∞ (cid:101) ζ ( w (cid:48) z ) c l,D ( w (cid:48) R , w (cid:48) z ) dw (cid:48) z (cid:41) In the previous equation the integrals in w (cid:48) R and w (cid:48) z will need to be computed numerically, while we have denotedwith c l,D ( w (cid:48) R , w (cid:48) z ) the results of the angular integrations for D >
1, i.e.: c l,D ( w (cid:48) R , w (cid:48) z ) = (cid:90) π | sin ϕ (cid:48) | D − | cos ϕ (cid:48) | D − C ( D − ) l (cid:18) w (cid:48) R cos ϕ (cid:48) (cid:112) w (cid:48) R + w (cid:48) z (cid:19) dϕ (cid:48) (13) c ,D = − − D π / sec (cid:2) π (1+ D )4 (cid:3) Γ (cid:0) − D (cid:1) Γ (cid:0) D (cid:1) ; c ,D = 2 − D π / ( D − (cid:2) ( D − w (cid:48) R − w (cid:48) z (cid:3) Γ (cid:0) D − (cid:1) ( w (cid:48) R + w (cid:48) z ) Γ (cid:0) D (cid:1) ; ...c ,D = c ,D = ... = 0where these functions are identically zero for odd values of l , while they can be computed analytically for all evenvalues of l . All the numerical computations (and some of the analytical ones) in this work, were performed with Mathematica, Version 12.1.1,Wolfram Research Inc.
The observed radial acceleration g obs , in the w z = 0 plane, can be obtained directly from Eqs. (12)-(13) bydifferentiation, i.e., g obs ( w R ) = − dφdw R (cid:98) w R . This radial acceleration g obs can be compared with the standard baryonic g bar , obtained with the same procedure but with a fixed D = 3 value, or directly using the total baryonic rotationalvelocity, V bar ( R ), available from the SPARC data for each galaxy and with g bar ( R ) = V bar ( R ) /R . In Sect. III, wewill use this second option to compute g bar , since it allows for a more direct comparison with the Newtonian behavior.Finally, in order to include also the spherical bulge in our NFDG analysis we recall from paper II that the NFDGpotential expansion in Eq. (8) is better suited to spherical coordinates, rather than cylindrical. Aligning the fieldvector w in the direction of the w (cid:48) z axis and using (rescaled) spherical coordinates ( w (cid:48) r , θ (cid:48) , ϕ (cid:48) ) for the source vector w (cid:48) , we can use directly the expansion (8) with the angle γ replaced by θ (cid:48) .To compute the gravitational potential (cid:101) φ ( w r ) for a spherically-symmetric mass distribution (cid:101) ρ ( w (cid:48) r ), we can still useour main Eq. (1), but the volume integration must be performed in spherical coordinates. The full details can befound in our paper II; here we present again the final results obtained by assuming that the fractional dimension willapply to all three coordinates equally, i.e., α r = α θ = α ϕ = D/
3, so that the appropriate angular integrals in thiscase are: c l,D = (cid:90) π | sin θ (cid:48) | D − | cos θ (cid:48) | D − C ( D − ) l (cos θ (cid:48) ) dθ (cid:48) (14) c ,D = π csc (cid:0) πD (cid:1) Γ (cid:0) D (cid:1) Γ (cid:0) − D (cid:1) Γ (cid:0) D (cid:1) ; c ,D = π (cid:0) D − (cid:1) csc (cid:0) πD (cid:1) Γ (cid:0) D (cid:1) Γ (cid:0) − D (cid:1) Γ (cid:0) D − (cid:1) ; ... (cid:90) π | sin ϕ (cid:48) | D − | cos ϕ (cid:48) | D − dϕ (cid:48) = 2 π csc (cid:0) πD (cid:1) Γ (cid:0) D (cid:1) Γ (cid:0) − D (cid:1) Γ (cid:0) D (cid:1) . The general potential for spherically-symmetric mass distributions can be written as: (cid:101) φ ( w r ) = − π Γ (cid:0) D − (cid:1) G Γ (cid:0) D (cid:1) Γ (cid:0) D (cid:1) l ∞ (cid:88) l =0 , , ,... c l,D (cid:90) ∞ (cid:101) ρ ( w (cid:48) r ) w lr< w l + D − r> w (cid:48) D − r dw (cid:48) r (15)= − π Γ (cid:0) D − (cid:1) G Γ (cid:0) D (cid:1) Γ (cid:0) D (cid:1) l ∞ (cid:88) l =0 , , ,... c l,D (cid:18)(cid:90) w r (cid:101) ρ ( w (cid:48) r ) w (cid:48) lr w l + D − r w (cid:48) D − r dw (cid:48) r + (cid:90) ∞ w r (cid:101) ρ ( w (cid:48) r ) w lr w (cid:48) l + D − r w (cid:48) D − r dw (cid:48) r (cid:19) and the observed radial acceleration g obs ( w r ) can be obtained again directly by differentiation of the last equation.The standard radial acceleration g bar can be simply obtained from Eqs. (14)-(15) with D = 3, or derived directlyfrom SPARC data as already remarked above for the disk case.For galaxies such as NGC 7814 (see Sect. III A) whose mass distribution consists of a spherical bulge plus cylindricaldisk and gas components, we will simply add together the NFDG potentials from Eqs. (12) and (15), and then obtain g obs ( w r ) by differentiation. On the contrary, NGC 6503 (see Sect. III B) and NGC 3741 (see Sect. III C) do notpossess a spherical bulge component, so they will be modeled using just the NFDG potential in Eq. (12).As a final consideration, we note that all our formulas for the gravitational potential (cid:101) φ (and, thus, also for thegravitational field g obs ) require summations over all non-zero terms for l = 0 , , , ... (terms for odd values of l beingidentically zero). As in paper II, we found that these series of functions converge rather quickly over most of therange for w R > w r > w R .All the results that will be presented in the following sections were computed by summing typically the first sixnon-zero terms (for l = 0 , , , , ,
10) of our NFDG expansions. For each case, we also checked numerically thatour NFDG expansions reduce to the standard g bar values from SPARC data, again by summing only the first fewnon-zero terms of our ( D = 3) expansions. Therefore, we are confident that our NFDG formulas can be used todescribe accurately the physical reality of galactic structures in spaces of dimension D ≤ III. GALACTIC DATA FITTING
In the following sub-sections, we will apply NFDG to three notable examples of rotationally supported galaxiesfrom the SPARC database: NGC 7814, NGC 6503, and NGC 3741. The choice of these galaxies is simply due to thefact that they were used as the main examples of the RAR in the seminal paper by McGaugh, Lelli, and Schombert[17]. The detailed luminosity data for the gas, disk, and bulge components of these three galaxies were obtained fromthe SPARC database administrator [27] and from the publicly available data [22].
A. NGC 7814
We will start with the case of NGC 7814, a spiral galaxy approximately 40 million light-years away in the con-stellation Pegasus. This galaxy, also known as the “Little Sombrero” for its similarities with M104 (the “SombreroGalaxy”), has a bright central bulge and a bright gas halo extending outward in space, as seen edge-on from Earth inimages taken by the NASA/ESA Hubble Space Telescope [29].This is a bulge dominated galaxy, in which all three components (bulge, disk, and gas) are present, with thespherically-symmetric bulge being the most prominent. Following the general discussion of NFDG methods outlined inSect. II, we converted the SPARC surface luminosity data for the three components into equivalent mass distributions (cid:101)
Σ ( w (cid:48) R ) (disk plus gas) and (cid:101) ρ ( w (cid:48) r ) (bulge). We then entered these functions into our general NFDG potentials in Eqs.(12)-(15), from which we then obtained g obs ( w R ) by differentiation.Following our NFDG assumptions, we can define ( g obs /g bar ) NF DG as the ratio of g obs defined above and theNewtonian g bar from the SPARC data (after converting the total baryonic rotational velocity V bar ( R ) into g bar ( R ) = V bar ( R ) /R ): (cid:18) g obs g bar (cid:19) NF DG ( w R ) = g obs ( w R , D ( w R )) g bar ( w R ) . (16)The gravitational field in the last equation is denoted as g obs ( w R , D ( w R )) to indicate that the dimension D isnow considered a function of the field point radial coordinate, i.e., D = D ( w R ), but not a function of the angularcoordinates due to the symmetry of the galactic structure (spherical symmetry of the bulge and cylindrical symmetryof disk/gas components).As was done in paper II, the NFDG ratio (cid:0) g obs g bar (cid:1) NF DG ( w R ) above, can be compared with the MOND (RAR) ratio (cid:0) g obs g bar (cid:1) MOND ( w R ) = − e − √ gbar ( wR ) /g † , or directly with the ratio of the experimental SPARC data for NGC 7814, (cid:0) g obs g bar (cid:1) SP ARC ( w R ), as obtained from the rotational velocity data [22, 27] transformed into radial accelerations. In theformer case, we are able to determine D ( w R ) based on the general MOND (RAR) model, while in the latter case thevariable dimension D ( w R ) is based directly on the SPARC - NGC 7814 experimental data, thus allowing for a moredirect validation of NFDG methods.Figure 1 shows the main NFDG results for NGC 7814. While in papers I-II most of the figures were plotted interms of rescaled distances w r or w R , in this work we prefer to use the astrophysical radial distance R in kiloparsec,while rotational circular velocities v circ are measured in km s − .The top panel illustrates the variable dimension D ( R ) obtained by equating the NFDG ratio ( g obs /g bar ) NF DG inEq. (16) with the similar MOND (RAR) and SPARC ratios, respectively. The resulting curves (blue-solid for MOND,green-solid for SPARC) show D ≈ . − . D ≈ . − . D = 2 reference value (red-dashed), since we argued in Sect.II that this dimension value represents the deep-MOND limit.At the smallest radial distances, our NFDG results are extrapolated and, therefore, not fully reliable. We alreadymentioned in paper II that our numerical computation method, based on the expansion in Eqs. (8)-(10), has someconvergence issues at low radial distances. The minimum distance below which our results were extrapolated isindicated by vertical gray-thin lines in both panels in Fig. 1 at R min (cid:39) . kpc (the same considerations will applyto the similar figures for the other galaxies in Sects. III B-III C). Other vertical gray-thin lines at R max (cid:39) . kpc limit our plots at large distances, corresponding to the radial distance of the last SPARC data-point for this galaxy.The low-distance results are also affected by the uncertainties in the galaxy mass distributions at low radii, since weused extrapolations of the SPARC luminosity distributions at low distances. As a consequence, the decreasing valuesfor D ( R ) at the lowest distances in the top-panel graphs are probably unphysical. Apart from this low-R behavior,the dimension plots for NGC 7814 are very similar to those obtained in papers I-II for general models with sphericalsymmetry (for example, see the Plummer model in Fig. 3 of paper I and Fig. 4 of paper II). In these previouslypublished figures, we assumed D = 3 at the origin ( w R = 0), while in this paper we do not force the Newtonianbehavior at the galactic center.Since NGC 7814 is dominated by its spherical bulge, it is not unexpected to obtain plots for D ( R ) similar topreviously analyzed spherical cases, in which D ≈ D decreases slowly to smaller values atlarger distances. For these structures, the D ≈ R in kiloparsec, whilerotational circular velocities v circ are measured in km s − . NFDG ( MOND ) NFDG ( SPARC ) NFDG ( D = ) [ kpc ] D NFDG ( MOND ) NFDG ( SPARC ) NFDG ( D = ) Newtonian Newtonian - disk Newtonian - gasNewtonian - bulge V flat SPARC data [ kpc ] v c i r c [ k m s - ] FIG. 1. NFDG results for NGC 7814. Top panel: NFDG variable dimension D ( R ) for MOND (RAR) interpolation function,or based directly on SPARC data. Bottom panel: NFDG rotation curves (circular velocity vs. radial distance) compared tothe original SPARC data (black circles). Also shown: Newtonian rotation curves (total and different components - gray lines)and corresponding SPARC data (gray circles). In particular, in this panel we show the NFDG (MOND) and NFDG (SPARC) curves (blue-solid and green-solid,respectively) which have been extrapolated below R min (cid:39) . kpc , due to the convergence issues already mentioned.These curves are also limited by R max (cid:39) . kpc , which is the radial distance of the last SPARC data-point. Again,these limits are shown as vertical thin-gray lines in the figure. In addition, a third NFDG curve (red-dashed) is shownfor a fixed value ( D = 2) of the space dimension. This curve can be computed even at very low radial distances,because it is based on the logarithmic potential in the second line of Eq. (1), so it does not suffer from the numericallimitations at low- R of the other NFDG curves.For this panel we assumed a total mass M = 8 . × kg (obtained by integrating the interpolated total massdistribution), with l ≈ (cid:113) GMa (cid:39) . × m, and disk scale length R d = 2 .
54 kpc = 7 . × m [22] (rescaled length W d = R d /l = 0 . v circ = (cid:112) ( g obs ) NF DG ( R, D ( R )) R/ (cid:2) kms − (cid:3) ,with the dimension functions D ( R ) as plotted in the top panel, while the Newtonian speeds are computed as v circ = (cid:112) ( g bar ) SP ARC ( R ) R/ (cid:2) kms − (cid:3) , with ( g bar ) SP ARC ( R ) representing the functions obtained by interpolating theSPARC data for the different components and the total Newtonian one.The three NFDG curves can be compared with the SPARC data-points (black circles) and related error barsobtained from the published data [22, 27], and also with the flat rotation velocity V f = 218 . ± . (cid:2) km s − (cid:3) [22],represented by the horizontal gray lines/band. For completeness, we also show the SPARC data for the Newtoniancases (disk, gas, bulge, and total Newtonian - gray circles) together with the Newtonian curves (in gray for thedifferent components; black-dashed for the total Newtonian) from the interpolated mass distributions (derived fromthe original luminosity distributions [27]). It is evident from the Newtonian curves of the three components that thisgalaxy is bulge-dominated, while the disk and gas contributions are less important, even at larger radial distances.For this bulge-dominated spiral galaxy, the flattening effect of the observed rotation curve is evident over mostof the radial range. Our NFDG (SPARC) curve (green-solid line) can perfectly model the published data over theapplicable range ( R min , R max ) and even at the lower radial values, where the NFDG results have been extrapolated.Again, this green-solid curve is obtained by assuming the variable dimension D ( R ) as described by the correspondinggreen-solid curve in the top-panel of Fig. 1, i.e., assuming that NFG 7841 behaves as a fractal medium whose fractionaldimension is described by this function D ( R ).The NFDG (MOND) curve (blue-solid line) is less effective than the previous one in modeling the SPARC data,but still close to the gray band of the flat rotation velocities over most of the radial range. As already remarked,this curve corresponds to the general RAR, which is an empirical fit to all SPARC data-points and, therefore, lessaccurate in the analysis of an individual galaxy. This shows that, from the NFDG point of view, the RAR is only anapproximation, while the correct way to model the transition from the Newtonian to the MOND regime is to assumea variable dimension D ( R ) which is a distinctive characteristic of each galaxy being studied.In the bottom panel, we also plot the third NFDG curve (red-dashed) for a fixed value ( D = 2) because this canbe easily computed even at low values of R and represents the deep-MOND general limit. This D = 2 curve is noteffective at all in describing NGC 7814 (although remarkably flat over most of the radial range). This is due to thedominant spherical symmetry of this galaxy, which is not consistent with a dimension D ≈
2, except perhaps atdistances beyond the last SPARC data point.We interpret the above results as a possible indication that, for bulge-dominated galaxies like NGC 7814, the fractaldimension function D ( R ) essentially follows the general behavior of the spherically-symmetric models as studied inpaper I: near the galactic center Newtonian regime ( D (cid:39)
3) is present, with the dimension D slowly decreasingat larger densities toward the deep-MOND regime D (cid:39) . (cid:46) D (cid:46)
3, due to the slow decrease of D typical of these spherical structures. Thedisk and gas components, which are characterized by dimension values D (cid:46) g obs ) vs. log ( g bar ) plots, similar to thoseused in the literature (see [17], [18], and also papers I-II) to illustrate the validity of the general MOND-RAR relationin Eq. (3). Compared to the Line of Unity (black-dashed), representing the purely Newtonian case, in this figurewe show the log-log plots obtained with our NFDG models, using g obs ( w R ) based on the dimension functions D ( R )shown in the top panel of Fig. 1 (blue-solid based on MOND-RAR, green-solid based on SPARC data) and with g bar ( w R ) based on Newtonian SPARC data.The former log-log plot (blue-solid) is equivalent to the general RAR, representing the most general empirical relationfor all the galaxies of the SPARC database. The latter log-log plot (green-solid) represents the actual log ( g obs ) vs.log ( g bar ) curve for this particular galaxy. The upper-right portion of the figure corresponds to the low-R region, butwe have excluded from this plot the extrapolated region (0 < R < R min ), because it is not fully reliable.From the central part of the figure, toward the lower-left corner (high-R region), the two log-log curves show similarbehavior, once they become separated from the Newtonian Line of Unity. However, there are differences between thegeneral NFDG (MOND) plot and the particular NFDG (SPARC) one. Once again, this shows that the RAR is a0 NFDG ( MOND ) NFDG ( SPARC ) Line of Unity - - - - - - - - - - ( g bar ) [ m s - ] l og ( g ob s )[ m s - ] FIG. 2. NFDG log-log plots for NGC 7814. The NFDG (MOND) curve (blue-solid) represents the general MOND-RARrelation, while the NFDG (SPARC) curve (green-solid) represents the particular case of NGC 7814, since it is based directlyon the SPARC data for this galaxy. Also shown: Newtonian behavior-Line of Unity (black-dashed diagonal line). good general empirical approximation, but each galactic case can be explained in NFDG in terms of its particulardimension function D ( R ).Comparing these first two figures with the similar ones in papers I-II, we note that this time we did not plotthe corresponding curves based directly on the MOND (RAR) or on the SPARC data, without any use of NFDGpredictions (which appeared as dotted curves in all the similar plots of papers I-II). These dotted curves were used inpapers I-II to double-check our NFDG results and, in general, they were completely consistent with the solid curvesof the NFDG predictions. We checked this perfect correspondence also for the situations analyzed in this paper, butwe omitted these additional curves as they were not providing any additional information to our analysis. B. NGC 6503
As our second galaxy, we revisit the case of NGC 6503 which has been already studied in a preliminary way inour paper II [2]. Here we expand our previous analysis of this galaxy and compare the NFDG results with the othercases studied in this paper. NGC 6503 is a field dwarf spiral galaxy approximately 17 million light-years away in the1constellation of Draco. This spiral galaxy shows bright blue regions typically related to star formation, bright redregions of gas along its spiral arms, and dark-brown dust areas in the galaxy’s center and arms, as seen in imagestaken by the NASA/ESA Hubble Space Telescope [30].This is a disk dominated galaxy, in which only two of the three components are present - disk and gas - withthe axially-symmetric stellar disk component being the most prominent. Following the general discussion of NFDGmethods outlined in Sect. II, we converted the SPARC surface luminosity data for the two components into anequivalent mass distribution (cid:101)
Σ ( w (cid:48) R ) (disk plus gas) and entered this function into our general NFDG thick-diskpotential in Eqs. (12)-(13), from which we then obtained g obs ( w R ) by differentiation. Since there is no sphericalbulge for this galaxy, we did not include any contribution from Eqs. (14)-(15) in this case.Figure 3 summarizes the main NFDG results for NGC 6503, in the same way of the similar Fig. 1 in the previoussubsection. The top panel shows the variable dimension function D ( R ) computed using our two main options (basedon the RAR or on SPARC data) over the range for the radial distance R consistent with available observational data.The bottom panel shows the NFDG circular speeds for our two main options (blue-solid and green-solid) as well asthe curve for fixed D = 2 (red-dashed), compared with SPARC data and with the NGC 6503 flat rotation velocity V f = 116 . ± . (cid:2) km s − (cid:3) [22] represented by the horizontal gray lines and gray band in the figure. Newtonian dataand curves are also included, for completeness.For this galaxy, we assumed a total mass M = 1 . × kg with l ≈ (cid:113) GMa (cid:39) . × m, and disk scalelength R d = 2 .
16 kpc = 6 . × m (rescaled length W d = R d /l = 0 . R min (cid:39) . kpc , due to the convergence issues already mentioned, although we improved onthese results at low-R in this work, compared to those presented in our paper II. These curves are also limited by R max (cid:39) . kpc , which is the radial distance of the last SPARC data-point. The limits are shown as usual by thevertical thin-gray lines in the two panels of Fig. 3.As already remarked in paper II, the top panel shows D ( R ) (cid:39) D (cid:39) D (cid:39)
2. Thick disk galaxies, with a dominant stellar disk component, are probablymore likely to show this behavior, as opposed to bulge and gas dominated galaxies characterized by different dimensionfunctions.The bottom panel in Fig. 3 confirms our analysis above, by showing a perfect agreement of our main NFGD(SPARC) fit (green-solid) with the SPARC data, even in the low-R extrapolated range. The more general NFDG(MOND) curve (blue-solid) also fits most data reasonably well, showing that the RAR is more effective for this typeof galaxies. The fixed D = 2 NFDG curve, which does not suffer from any extrapolation at low-R, is able to fit wellthe low distance data, while is less effective at higher distances, but still remarkably flat. In summary, NFDG appliedto NGC 6503 shows a perfect example of MOND behavior explained by an almost constant D (cid:39) g obs ) vs. log ( g bar ) plots for NGC 6503, similar to those shown in Fig. 2 for NGC7814. As usual the blue-solid curve represents the general MOND-RAR case, while the green-solid curve is peculiar tothe individual galaxy being studied. Apart from the results shown in the top-right corner of this figure (representingthe low-R region, still excluding the extrapolated part for 0 < R < R min ), there is strong agreement between thegeneral MOND-RAR curve and the particular NFDG-SPARC one. This shows again that disk-dominated galaxies areprobably closer to the general behavior described by the RAR relation, as compared with bulge and gas-dominatedgalaxies. C. NGC 3741
In our third and final case we consider NGC 3741, an irregular galaxy discovered by J. Herschel in 1828. It is agas-dominated spiral galaxy approximately 11 million light-years away in the constellation Ursa Major [31]. Onlytwo of the three components are present - disk and gas - with the axially-symmetric gas component being the mostprominent. Following again the general discussion of NFDG methods in Sect. II, we converted the SPARC surfaceluminosity data for the two components into an equivalent mass distribution (cid:101)
Σ ( w (cid:48) R ) (disk plus gas) and entered thisfunction into our general NFDG thick-disk potential in Eqs. (12)-(13), from which we then obtained g obs ( w R ) bydifferentiation.Figure 5 summarizes the main NFDG results for NGC 3741, in the same way of the similar figures in the previoussubsections. The top panel shows the variable dimension function D ( R ) computed using our two main options,while the bottom panel shows the NFDG circular speeds for the same options (blue-solid and green-solid) as well as2 NFDG ( MOND ) NFDG ( SPARC ) NFDG ( D = ) [ kpc ] D NFDG ( MOND ) NFDG ( SPARC ) NFDG ( D = ) Newtonian Newtonian - diskNewtonian - gas V flat SPARC data [ kpc ] v c i r c [ k m s - ] FIG. 3. NFDG results for NGC 6503. Top panel: NFDG variable dimension D ( R ) for MOND (RAR) interpolation function,or based directly on SPARC data. Bottom panel: NFDG rotation curves (circular velocity vs. radial distance) compared tothe original SPARC data (black circles). Also shown: Newtonian rotation curves (total and different components - gray lines)and corresponding SPARC data (gray circles). NFDG ( MOND ) NFDG ( SPARC ) Line of Unity - - - - - - - - ( g bar ) [ m s - ] l og ( g ob s )[ m s - ] FIG. 4. NFDG log-log plots for NGC 6503. The NFDG (MOND) curve (blue-solid) represents the general MOND-RARrelation, while the NFDG (SPARC) curve (green-solid) represents the particular case of NGC 6503, since it is based directlyon the SPARC data for this galaxy. Also shown: Newtonian behavior-Line of Unity (black-dashed diagonal line). the curve for fixed D = 2 (red-dashed), compared with SPARC data and with the NGC 3741 flat rotation velocity V f = 50 . ± . (cid:2) km s − (cid:3) [22] represented by the horizontal gray lines and gray band in the figure. Newtonian dataand curves are also included, for completeness.For this galaxy, we assumed a total mass M = 5 . × kg with l ≈ (cid:113) GMa (cid:39) . × m, and disk scale length R d = 0 .
20 kpc = 6 . × m (rescaled length W d = R d /l = 0 . R min (cid:39) . kpc and also limited by R max (cid:39) . kpc , which is the radial distance of the lastSPARC data-point. The limits are shown as usual by the vertical thin-gray lines in the two panels of Fig. 5.The top panel shows D ( R ) (cid:39) . − . < D (cid:46)
2, which is notably different from the first two cases where 2 (cid:46) D ≤
3. Thickdisk galaxies with a dominant gas component might be characterized by a lower fractal dimension and might attain4
NFDG ( MOND ) NFDG ( SPARC ) NFDG ( D = ) [ kpc ] D NFDG ( MOND ) NFDG ( SPARC ) NFDG ( D = ) Newtonian Newtonian - disk Newtonian - gas V flat SPARC data [ kpc ] v c i r c [ k m s - ] FIG. 5. NFDG results for NGC 3741. Top panel: NFDG variable dimension D ( R ) for MOND (RAR) interpolation function,or based directly on SPARC data. Bottom panel: NFDG rotation curves (circular velocity vs. radial distance) compared tothe original SPARC data (black circles). Also shown: Newtonian rotation curves (total and different components - gray lines)and corresponding SPARC data (gray circles). D ≈
2) regime only at very large radial distances, beyond the observed range.The bottom panel in Fig. 5 supports our analysis, by showing a perfect agreement of our main NFGD (SPARC)fit (green-solid) with the SPARC data, except in the very low-R extrapolated range, where the NFDG D = 2 curvefits the first experimental point better. The more general NFDG (MOND) curve (blue-solid) also fits most data inthe mid-high range well, but is much less effective at low-R, showing that the RAR is just an approximation for thistype of galaxies.The fixed D = 2 NFDG curve, which does not suffer from any extrapolation at low-R, is unable in this case to fitthe data (except the first two points), since the flattening of the circular velocity data only happens at high-R values,for the last four data points. In summary, NFDG applied to NGC 3741 shows a more complex behavior, comparedto the previous two cases, probably due to the fractal effects of the gas component.In figure 6, we show the log ( g obs ) vs. log ( g bar ) plots for NGC 3741, similar to those for the previous galaxies.Again, the blue-solid curve represents the general MOND-RAR case, while the green-solid curve is peculiar to NGC3741. The results shown in the top-right corner of this figure represent the low-R region, and the MOND and SPARCplots do not match in this region. The two NFDG curves agree well only in the left part of the figure (high-R region),generally showing that the RAR is not a good approximation for this gas-dominated galaxy. D. Discussion
Following the three cases analyzed in the previous sub-sections, we can attempt to generalize our NFDG results toall cases of rotationally supported galaxies. While the RAR is an effective empirical law which works for most of thegalaxies in the SPARC database and others, we argued that it is only an approximation of the fractal behavior ofthese galaxies. This behavior is characterized by the variable fractional dimension function D ( R ), as observed in thegalactic plane ( z = 0) and thus depending only on the radial distance, for symmetry reasons.This dimension function D ( R ) is an individual feature of each galaxy that can be derived directly from the observedcircular velocities and luminosity distributions, properly converted into mass distributions for each galaxy beingconsidered. In general, the fractional dimension range is 1 < D ( R ) ≤
3, as seen in the three cases studied in thiswork. Once the full dimension function has been determined, the NFDG formulas for the gravitational potentials andfields are able to produce accurate fits to the observed circular speed data, without the need of any dark matter orother supplemental hypotheses.Although the dimension function D ( R ) is peculiar to each galaxy analyzed, it is likely that the interplay of thethree main components - bulge, disk, and gas - will play a significant role in determining this dimension function.We have argued that if one of these three components is dominant, as in the galaxies studied in this work, it mightdetermine the overall shape of the D ( R ) function.Bulge dominated galaxies, such as NGC 7814, will probably be characterized by dimension functions which aretypical of the spherically-symmetric structures analyzed in paper I: D slowly decreasing from the Newtonian D ≈ D ≈ . − . D ≈ D ≈ D ≈ < D (cid:46) − ∆ is replaced by a fractional version ( − ∆) s with s ∈ [1 , / s = 1 the Newtonian limit is recovered, while s = 3 / Fractional Newtonian Gravity (FNG) in a second paper on the subject [10].As already remarked in papers I-II and in Sect. I of this work, the main difference between FNG and our NFDGis that the former is a MOND-like (non-local) fractional theory, i.e., fractional operators are used to determine the6
NFDG ( MOND ) NFDG ( SPARC ) Line of Unity - - - - - - - - - - ( g bar ) [ m s - ] l og ( g ob s )[ m s - ] FIG. 6. NFDG log-log plots for NGC 3741. The NFDG (MOND) curve (blue-solid) represents the general MOND-RARrelation, while the NFDG (SPARC) curve (green-solid) represents the particular case of NGC 3741, since it is based directlyon the SPARC data for this galaxy. Also shown: Newtonian behavior-Line of Unity (black-dashed diagonal line). gravitational potential and field, while the latter employs field equations of integer order (and thus local). The“fractional flavor” of NFDG comes instead from the non-integer dimension D ≤ (cid:96) = π (cid:112) GM/a forFNG, l = (cid:112) GM/a for NFDG) which ensures dimensional correctness of all the equations.The gravitational potentials obtained by the two models also present some similarities with regard to their generalforms, but they differ in the mathematical details [1]. Still, they show similarities when plotted for equivalent valuesof the fractional dimension, so it is possible that these two models might be practically equivalent, when applied togalactic structures. In the second FNG paper [10] Giusti et al., apply their model to the gravitational potential forKuzmin disks and also consider the possibility of turning FNG into a variable order theory, by assuming s = s ( x /(cid:96) ),i.e., considering the fractional order as a function of the field point. This is equivalent to our NFDG assumption of avariable fractional dimension D = D ( w ), depending on the rescaled field position w ≡ r /l .It will be interesting to compare the detailed NFDG dimension/rotation curves presented in this paper for the threegalaxies that we have analyzed, with similar FNG fits, if they will become available. In particular, after fitting therotational velocity data for a certain galaxy (e.g., NGC 6503), if the two models will yield the same variable dimension7function (i.e., if the FNG s = s ( x /(cid:96) ) function will be found to be equivalent to the NFDG D = D ( R )), then thetwo models could also be considered equivalent and the case for a “fractional” explanation of MOND would becomestronger. IV. CONCLUSION
In this work, we continued our study of NFDG, a fractional-dimension gravity model which might provide a possibleexplanation of the MOND theory and the related RAR. As in our papers I-II, we assumed that Newtonian gravitymight act on a metric space of variable dimension D ≤
3, when applied to galactic scales, and used NFDG to modelthree rotationally supported galaxies: NGC 7814, NGC 6503, and NGC 3741.For these three cases, we have shown that NFDG can accurately fit the rotation data, as it was done by MOND-RARmodels, but these results are explained in terms of dimension functions D ( R ) which are peculiar to each of the galaxiesbeing studied. In particular, a bulge-dominated galaxy such as NGC 7814 is characterized by a dimension functiontypical of spherically symmetric structures (2 < D ≤ D ≈ < D (cid:46) D ≈
2, is more typical of disk-dominatedgalaxies, while it is achieved only asymptotically by bulge and gas-dominated galaxies.Further work will be needed to check this general interpretation of the results. Detailed fitting of several othergalaxies in the SPARC database will need to be performed in upcoming work to show how the three different com-ponents (bulge, disk, and gas) play different roles in determining the dimension function for each case. Additionally,other structures for which MOND is less effective, such as globular clusters or similar, will need to be studied to seeif their dynamical behavior can still be explained by NFDG without any use of DM.A relativistic version of NFDG, which can be tentatively called Relativistic Fractional-Dimension Gravity (RFDG)also needs to be introduced, by expanding General Relativity to metric spaces with fractional dimension. We willleave these and other topics to future work on the subject.
ACKNOWLEDGMENTS
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