Chapter 18
Mathematical Classification of the Spiral and Ring Galaxy Morphologies

18.1 Introduction

Knowledge of the galaxies and astronomical bodies and their behavior has long been a human striving. In particular, the spiral galaxies have held our attention since their discovery. It has long been our desire to understand and codify their shapes and evolution. This chapter studies the application of the fractional trigonometry to the mathematical definition of the morphology of the spiral and ring galaxies. This chapter is adapted from Lorenzo and Hartley [89] with permission of the ASME:

Perhaps the most fundamental knowledge that we can have about an object or collection of objects is its morphology, that is, its size, shape, and dimension. In most instances, this is the first differentiator between objects. The more accurately or completely this information is known the better we can proceed to more defining issues of kinematics, dynamics, and evolution. This chapter studies the morphology of the spiral and ring galaxies based on the fractional meta-trigonometric spiral functions. There have been numerous classifications proposed and used since that proposed by Hubble [52] in 1936. However, to the knowledge of the authors, there are no classifications of the galaxies that provide a mathematical description of the morphology of the spiral arms. Hopefully, the work to be presented here augments and helps remove some of the subjectivity of the existing morphological classifications.

Van den Bergh [12] has helpfully provided coverage of many galactic classifications including those of Hubble, de Vaucouleurs, and Elmegreen. In an earlier work, Sandage et al. p. 28 [115], references five classification schemes of galaxies that are obtained from direct photography. These are the various modifications and extensions of the Hubble classification, by Pettit [108], Holmberg [53], de Vaucouleurs [29], van den Bergh [13–16], and Morgan [100, 101].

The classification by Hubble [52] was originally based on the so-called “tuning fork diagram” (Figure 18.1). Basically, galaxy morphology is divided into elliptical and spiral shapes with the spirals further divided into normal and barred types, which are further subdivided into classes based on the openness of the spiral arms. A circular type, the S0 class, provided transition between the elliptical and the spiral types. This useful classification method was further refined by de Vaucouleurs to include r and s types, which describe the manner in which the spiral arms emanate from the end of the bar. For the r type, the arms emanate tangential to a ring at the end of the bar, and for the s type, the arms emanate more or less radially from the ends of the bar.

The classifications of van den Bergh and Morgan introduce luminosity effects and along with many later classification schemes provide useful information to the understanding of the nature of the galaxies. Because this study will deal exclusively with purely morphological issues, we will not delve further into these and various other galaxy classifications.

In the last decade or so, computer-based studies identified as Mathematical Morphology (MM) have been forwarded by Candeas et al. [20], Moore et al. [99], Baillard et al. [9], and Aptoula et al. [8]. These studies are computer-based analyses of astronomical images for star/galaxy differentiation and for automatic galaxy classification Baillard et al. [9], and Aptoula et al. [8]. These capabilities are quite important but are also very different from the mathematical classification to be presented here. The study that follows originates from Lorenzo and Hartley [89].

In a 1981 paper, Kennicutt [57] attempted to mathematically describe the shape of galactic spiral arms based on the logarithmic spiral, which is linear in , and the hyperbolic spiral, which is linear in . He concluded: “It quickly became apparent that neither form precisely represents the arms, even the most regular ones.”

The work described here ascribes mathematically defined fractional barred and apparently unbarred (normal) spirals, based on the fractional trigonometric functions, to the description of the galactic spiral arms. That is, we work to analytically define the galaxy spiral arm morphologies.

The classification methods suggested in this study hopefully augment those currently in use or possibly be implemented to work with some of the existing computer identification schemes. A primary objective of this study is to bring to the attention of those members of the astrophysical community dealing with galactic morphology, the existence of the fractional trigonometry and its potential for describing galactic spiral and ring morphologies, and its fundamental mathematical character.

Figure 18.1 The morphological classification of the galaxies as suggested by Edwin Hubble.

Source: Hubble 1936 [52]. Reproduced with permission of Yale University Press.

18.2 Background – Fractional Spirals for Galactic Classification

In Chapters 9 and 15, a variety of fractional spirals associated with the fractional meta-trigonometric functions was presented. In this section, fractional spirals of particular interest relative to galaxy classification are presented, and some important relevant behaviors discussed.

For the initial spiral, we have versus shown in Figure 18.2, here, the effect of the primary order variable q ranges from q = 1.0–1.8, with q = 1.0 defining the unit circle. Clearly, as the variable q increases the growth rate of the spiral increases.

Figure 18.2 Effect of order, q. versus and versus , q = 1.0–1.8 in steps of 0.2, v = 0.0, a = 1.0, .

Note, in Figures 18.2–18.8, as the parameter t increases, the spiral arms diverge either from the origin or the unit circle. Furthermore, t here is the formal parameter of the spirals and does not necessarily represent time since the resulting spirals are considered to be spatial morphologies.

Figure 18.8 Effect of small parameter changes in , versus with q = 1.06, v = −0.5, , a = 1.0, k = 0, .

In Figure 18.3, another phase plane plot of versus , the value of is varied to show its effect with q = 1.2, v = 0.2. Here, barred spirals result with the rate of divergence of the spiral arms decreasing as increases. These changes effectively start from the end of the bars. That these spirals are barred and are based on this fundamental mathematics should be cause for interest in astrophysical application. Importantly, the Laplace transforms of the fractional meta-trigonometric functions defining fractional differential equations in space may be easily obtained.

Figure 18.3 Effect of , versus and versus , q = 1.2, v = 0.2, a = 1.0, , k = 0.

In Figure 18.4, we use the spiral from Figure 18.3 as the basis of a study of the effect of the secondary order parameter v. In the figure, v is varied from −0.3 to 0.2 in steps of 0.1. The base spiral, v = 0.1, jumps immediately from the origin to the value of 1.0, making it a barred spiral. It is observed that as v is decreased from the value of v = 0.2, the spiral transitions from a hard barred spiral progressively toward a soft barred, or normal type of spiral. That is, the transition to the spiral arms is softened. From Figure 18.4, it can be seen that the choice of the v parameter may control the location of a spiral from being on the SB tine or the SA tine (or anywhere in between) relative to the Hubble tuning fork. The effect of the v variable also applies to the parity-based spiral functions. A family of barred spirals is shown in Figure 18.5 based on variation in the order variable q and enabled by the relationship v = q − 1.

Figure 18.4 Effect of v, versus and versus , .

Figure 18.5 Effect of Order, q, with , versus and versus , q = 1.0–1.75 in steps of 0.25, a = 1.0, .

The parity functions may also be used as the basis for spirals of possible application to galaxy morphology as shown in Figures 18.6–18.8. Figure 18.6 studies versus . Here, it may be seen that increasing increases the divergence rate of the spiral arms while q and v are held constant at , respectively.

Figure 18.6 Effect of , versus and versus , . .

A different set of barred spirals is shown in Figure 18.7 where the parity functions versus form the spiral basis with q, v, and held constant and varied. The divergence of these spirals increases as increases. The acute divergence angle from the end of the bars may remove these spirals from interest in astrophysical application. The following sections are adapted from Lorenzo and Hartley [89] with permission of ASME:

This section has introduced some of the aspects of the fractional trigonometry as it applies to galactic morphology and has shown some of the infinite number of fractional spirals that may be applicable to the classification of spiral galactic morphologies. It is important to note that the fractional trigonometric functions provide, or are elements of, the solutions to linear commensurate, constant-coefficient, fractional differential equations as shown in Chapter 12. This, of course, includes the linear constant-coefficient ordinary differential equations as a subset. Thus, if a meaningful mathematical classification of the spiral (and ring) galaxies based on the fractional trigonometric functions can be created, defining fractional differential equations in space can be inferred, from which there is the potential of deeper analytical understanding of the galactic processes.

Figure 18.7 Effect of , versus and versus , .

18.3 Classification Process

Our intention here is to provide a purely mathematical description of the spiral arm morphology. That is, we exclude any attention to the luminous intensity or the spectral qualities of the galaxy nucleus or spiral arms. The possibility of morphological matching of an image in a particular spectral frequency that better displays the spiral geometry is not precluded. In fact, composite images containing a variety of wavelength information may best describe the morphological features of interest. It is noted that the morphology or arm location may change slightly at different wavelengths. In general, only galaxies with well-defined spiral arms or rings are considered in this initial study.

18.3.1 Symmetry Assumption

For two-armed spiral galaxies, the initial assumption for the mathematical fitting process will be that the spiral arms are (approximately) symmetrical. In the situation where the symmetrical model fits the arms but the center of symmetry does not align with the optical galaxy center, the offset is reported. For the most part, for this initial effort, we avoid galaxies with interacting or tidal effects. Three-armed or multiarmed spirals occasionally occur, they may be treated as n-single arms or if possible with n-symmetric arms. Out of plane or warped disk galaxies are not considered initially. In general, a symmetric form is preferred, but the arms are defined individually if necessary.

18.3.2 Galaxy Image to Spiral or Spiral to Galaxy Image

Two approaches to the morphological matching of mathematical spiral to the galaxy image are possible. First, the image of the galaxy may be manipulated to a “normal” view and then compared with the mathematical spiral.

The second approach does the manipulations with the mathematical spiral. That is, the mathematical spiral is corrected for inclination by correcting either the x- and/or y-axis to achieve the proper morphology, and then rotated in-plane and scaled to match the image spiral. The approach used here is to manipulate the mathematical spiral and superimpose it on the unaltered galaxy image.

18.3.3 Inclination

The galaxies do not usually present themselves to us in face-on spirals and we must account for the inclination of the plane of the galactic disk relative to the plane that is normal to our line of sight. Two approaches are possible: first published values for the inclination may possibly be used to correct the spiral for comparison to the galaxy image; alternatively, the spiral model may be rotated by some amount (the inclination) and matched to the galaxy image and thereby infer the inclination. This later approach is used in this study.

Consider the galactic plane, that is, the plane in which the disk is located. It can be brought into alignment with the plane normal to the line of sight by two rotations, one around each of the x or y axes of the normal plane. Each rotation is defined by the reciprocal of the cosine of the angle through which each axis of the plane must be rotated. Thus, it is clear that, in general, two datum are required to define or correct for the inclination. In common usage, only a single datum the inclination is quoted. Inferred is the line of nodes about which the inclination is measured or corrected.

The following discussion describes how the issue of galactic inclination is treated in this study. Figure 18.9a shows a face-on fractional spiral defined by and but could also be a spiral based on the parity functions. The result of a rotation of 40° about y₀ and 65° about the x₀ axes is indicated by Figure 18.9b and c, respectively. The result of further rotations of 65° about x₁ and 40° about the y₃ axes are plotted in Figure 18.9d indicating that the order in which the rotations occur is not important. Finally, the spiral in x₄, y₄ plane is rotated in-plane to yield a spiral that may be scaled and translated into position for comparison with the galactic image.

Figure 18.9 Manipulation of a fractional spiral for image comparison. (a) Face-on view of scaled test spiral. (b) Spiral rotated about y₀ axis by 40°. (c) Spiral rotated about x₀ axis by 65°. (d) Spiral rotated about x₁ axis by 65° and spiral rotated about y₃ axis by 40°. (e) In-plane rotation of x₄,y₄ spiral by −15°.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME.

18.3.4 Data Presentation

The basis functions used for mathematical classification in this study are

18.1

where is a spatial parameter and C is a scaling constant. Further for simplicity, we have taken a = 1 and k = 0 limiting consideration to only a subset of the principal functions. In a temporal context, the parameter a is a fractional generalization of the natural frequency; thus, its primary effect is scaling the curves with minor effect on morphology. The scaling may be compensated by the choice of C. Variations in a, however, may allow some improvement in the modeled morphology.

To account for the apparent inclination of the galactic plane, the following corrections are made in the fitting program:

18.2

18.3

where and are in degrees. Then, the coordinate system is rotated degrees in-plane to match the orientation of the galaxy.

The spiral arms of galaxies usually display varying amounts of scattering of the stars defining the spirals. Thus, spirals that bound the scattering are added to main spiral fit when appropriate. In general, any of the fractional spiral parameters may be used as the basis for the bounding spirals; however, in all the cases that follow, only bounds based on variations in C are used. Furthermore, with the bounding spirals, a median or average spiral is always given. It is important to note that a significant number of different parameters or combinations of parameter may be used to determine the bounding spirals. Figure 18.10 illustrates a bounded symmetric spiral pair with an inclined nucleus outline and median spiral. The results in this figure are for illustration only. Estimates of the galactic nucleus outline may be added or not.

Figure 18.10 Sample data display for NGC 1300. This display shows three scaling of the same spiral. Two spirals to enclose the spiral arms along with a median spiral. A nucleus outline is shown displaced from the spiral center.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: Hillary Mathis/NOAO/AURA/NSF. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

In Section 18.4, morphology classifications are listed following the galaxy name. These subjective classifications are taken from NED (NASA/ipac Extragalactic Database) when available and represent the judgments of various investigators listed there. Differences in these judgments usually indicate the need for interpolation between the discrete categories. The axes numerations on the figures that follow are all in pixels as found for the original image.

18.4 Mathematical Classification of Selected Galaxies

In this section, the spirals of various galaxy images are modeled using equation (18.1). The parameters are determined manually by a trial-and-error process. With some experience, a fitting of the quality discussed later may usually be done in a few hours. The galaxies were selected to give some coverage of the Hubble classifications with galaxies having well-defined spirals. With the exception of M51, the galaxies were selected to be free of tidal effects. In the figures that follow, the axes numerations are all in pixels as found in the original image.

18.4.1 NGC 4314 SB(rs)a

This galaxy (Figure 18.11) is highly symmetric and was analyzed on that basis. Furthermore, in addition to the primary arms it appears to have a weak secondary pairs of arms, which are also symmetric and seem to start from the bar. Scale 123.0 pixels/arc-min. The galaxy appears to us truly face-on; thus, common parameters for both primary and secondary arms are . Also, , a = 1, k = 0. Primary arm parameters: q = 1.125, v = −0.1, , C = 210. . Secondary arm parameters: q = 1.08, v = −0.05, , .

Figure 18.11 Classification for NGC 4314. See text for fit parameters. Solid curve is primary arm and dashed is secondary arm.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: David W. Hogg, Michael R. Blanton, and the Sloan Digital Sky Survey Collaboration. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.2 NGC 1365 SBb/SBc/SB(s)b/SBb(s)

NGC 1365 (Figure 18.12) is a dramatic barred spiral at a distance of 60 million light-years. In this infrared image, the bar appears to be horizontal; however, the fitting of the well-defined spiral arms seems to indicate a phantom bar (dark gray dashed lines) at approximately −45°. The implications of this are not clear, it may indicate a discontinuous evolutionary process. The analysis considers the arms as symmetric, and the results show good containment by close bounding fractional spirals. Common parameters: q = 1.23. v = −0.20, , , . Primary arm parameters: . Phantom arm parameters: .

Figure 18.12 Classification for NGC 1365. See text for fit parameters. Dashed curve is primary arm and phantom arm extends primary to origin.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: ESO/P. Grosbol. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.3 M95 SB(r)b/SBb(r)/SBa/SBb

M95 (NGC3351) (Figure 18.13) is a barred spiral with the bar sitting at about 50° to the x-axis in the image. The fitting parameters are q = 1.07, v = 0.0, , , , , , and , . The fitting only applies to the bar and the inner part of the spiral arms. The nucleus shown has a radius . The scale is approximately 214 light-years/pixel.

Figure 18.13 Classification for M 95. See text for fit parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit & Copyright: Adam Block, Mt. Lemmon SkyCenter, University of Arizona. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.4 NGC 2997 Sc/SAB(rs)c/Sc(s)

The spiral arms of NGC 2997 (Figure 18.14) were assumed to be symmetric for the fractional spirals used to fit the image. This galaxy is an excellent example of a normal spiral. The modeling parameters are q = 1.19, v = 0.19, , , , , . The radius of the nucleus is . The scale is approximately 88.2 pixels/arc-min.

Figure 18.14 Classification for NGC 2997. See text for fit parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: ESO. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.5 NGC 4622 (R′)SA(r)a pec/Sb

The galaxy NGC 4622 (Figure 18.15) appears to be rotating clockwise while the spiral arms give the inference of counterclockwise rotation. The classification of this galaxy assumed a symmetric form for its two spiral arms. While the arms could have been modeled slightly better individually, the symmetric assumption yielded good results. The galaxy appears to us face-on. The modeling parameters for Figure 18.15 are q = 1.07, v = 0.07, , , , , and . The nucleus radius was arbitrarily set at 130 pixels. A size calibration was not available for the image.

Figure 18.15 Classification for NGC4622. See text for parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image credit: NASA and The Hubble Heritage Team (STScI/AURA), Acknowledgement: Dr. Ron Buta (U. Alabama) and Tarsh Freeman (Bevill State Community College). Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.6 M 66 or NGC 3627 SAB(s)b/Sb(s)/Sb

M 66 or NGC 3627 is modeled here as a symmetric barred spiral, although significant asymmetry may be seen in the control image. The galaxy is presented to us at an inclination of about 65°. In this image from the European Space Organization, “North is towards upper left, West towards upper right.” The modeling parameters for Figure 18.16 are q = 1.28, v = 0.0, , , , and . The geometric center is shifted from the optical center by 10 pixels to the left in x and 20 pixel down in the y direction. The nucleus radius was pixels. A size calibration was not available for the image.

Figure 18.16 Classification for M 66 or NGC 3627. See text for fit parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: ESO/P. Barthel. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.7 NGC 4535 SAB(s)c/SB(s)c/Sc/SBc

Initially, the galaxy NGC 4535 was modeled as a symmetric barred spiral. This is shown in Figure 18.17a where the fit parameters are; q = 1.14, v = 0, a = 1.0, , , , , , and . However, the arms appeared to be tangent to the nucleus, type s, so a second model attempted to capture that feature and the result is shown in Figure 18.17b where the fit parameters are; q = 1.26, v = 0.26, a = 1.0, , , , and . Note the considerable differences in the parameters of the two models, even the apparent inclination parameter, , changes from 20° to 40°.

Figure 18.17 (a) Classification for NGC 4535. See text for fit parameters for first model. (b) Classification for NGC 4535. See text for fit parameters for second model.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: ESO. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.8 NGC 1300 SBc/SBb(s)/SB(rs)bc

NGC 1300 (Figure 18.18) is often described as the prototypical barred spiral galaxy. The modeling done here assumes symmetry of the spiral arms although it is clear that the left-most arm fit could be improved if modeled individually. The modeling parameters for Figure 18.18 are q = 1.145, v = −0.05, , , , and . The nucleus radius was . The galaxy size has been estimated at 85,000 light-years from which the calibration may be estimated at 65.4 light-years/pixel. It is interesting to note that various investigators Lindblad et al. [66] and Peterson and Huntley [107] estimated the inclination at 35°, 35.5°, 40°, 44°, 48°, 49.3°, and 53°. The value of , found here appears to validate the largest estimate and indicates the galaxy is far from face-on. It should also be noted that the optical center of the galaxy indicated by the white axis was not used as the geometrical center but a shift of 10 pixels to the short (dark gray) y-axis provided a much better fit. This represents a distance of 654 light-years!

Figure 18.18 Classification for NGC 1300. See text for fit parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: Hillary Mathis/NOAO/AURA/NSF. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.9 Hoag's Object

Hoag's Object (Figure 18.19) is a ring galaxy that is about 100,000 light-years in diameter, giving an approximate calibration of 100 light-years/pixel. Initial symmetric (two arm) modeling showed satisfactory results, however, a single-arm barred spiral was found to be superior. The spiral parameters (Figure 18.19) are q = 1.015, v = −0.011, , , , , , and . The nucleus radius as shown is , but could easily be considered to be a larger value. Because there is no evidence of a bar between the nucleus and outer ring, the initiation angle of the spiral is somewhat arbitrary. Thus, at an angle of −85° a phantom semi-bar with marks the assumed starting point of the spiral. The very low value of q = 1.015 indicates the closeness to a circular geometry, at q = 1.0. Thus, in this mathematical classification approach the ring galaxies are clearly viewed as extremely weak barred galaxies.

Figure 18.19 Classification for Hoag's Object. See text for fit parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: NASA, R. Lucas (STSc/AURA). Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.10 M 51 Sa + Sc

The image (Figure 18.20) shows M 51 (NGC 5194) and its nearby neighbor NGC 5195. By modeling the two arms as a symmetric pair, based on the right-most arm, we can see the effect of the gravitational attraction of NGC 5195 on the spiral arms of NGC 5194 as the difference between the model and the physical location of the arms in the upper portion of the image. The spiral parameters of Figure 18.20 are q = 1.19, v = −0.19, , , , , , , and . The galaxy diameter is approximately 65,000 light-years giving a scale factor of approximately 65 light-years/pixel. The nucleus radius (not outlined) is approximately . The geometric center of the galaxy is displaced downward from the optical center by 30 pixels or 1950 light-years. This is marked by the green subaxis. The in-plane (projected) deflection of the upper arm is approximately 100 pixels (6500 light-years) at the tip.

Figure 18.20 Classification for M 51. See text for fit parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: T.A. Rector and Monica Ramirez/NOAO/AURA/NSF. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.11 AM 0644-741 Sc/Strongly peculiar/

The galaxy AM_0644-741, a Lindsay-Shapley Ring Galaxy, is considered to be a ring collisional galaxy. Because of the regularity of the ring, a mathematical fitting was attempted, and is shown in Figure 18.21a. The ring is fit quite well as a single spiral arm with a very low growth rate, that is, q = 1.018. The geometrical center is shown by the white subaxis, which is shifted along the x-axis by 408 pixels from the optical center of the image. Since the uncropped image is 260,000 light-years wide the calibration is 65.8 light-years/pixel. Thus, the displacement appears to be approximately 27,000 light-years. The spiral parameter values for the fitting are q = 1.018, v = 0.018, , , , , , pixels from galaxy optical center, and . The low value of q appears to indicate that this is a type S0 galaxy. There is a faint string of stars near the nucleus not modeled previously, the parameter values for this segment (Figure 18.21b) were found to be; q = 1.018, v = −0.3820. , , , , , and . The two segments appear to match smoothly at the interface. Notice both segments have the same inclinations , and same value of q.

Figure 18.21 Classification for AM 0644-741. See text for fit parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: NASA, ESA, and The Hubble Heritage Team (AURA/STScI) Acknowledgement: J. Higdon (Cornell U.) and I. Jordan (STScI). Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.12 ESO 269-G57 (R')SAB(r)ab/Sa(r)

Two models are applied for the galaxy ESO 269-G57, that is, the symmetric outer arms and the inner ring are modeled separately. The spiral arm parameter values for the fitting are q = 1.130, v = 0.130, , , , , , and . The inner ring was well fit by a single-arm spiral with parameter values of q = 1.016, v = 0.016, , , , , , and . The nucleus radius is 17 pixels. The cropped image of the galaxy (Figure 18.22) is approximately 4.06 × 3.55 arc-min, and the galaxy width was given as 200,000 light-years yielding a calibration of approximately 250 light-years/pixel. The galaxy optical center is displaced upward from the geometrical center by about 10 pixels, 2500 light-years. Interestingly, the inclination of the spiral arm and the inner ring differ by , possibly indicating a warping of the disk were it possible to view from the side.

Figure 18.22 Classification for ESO 269-G57. See text for fit parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: ESO, Data: Henri Boffin. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.13 NGC 1313 SBc/SB(s)d/SB(s)d

NGC 1313 has been called the Topsy-Turvy galaxy because of its very active state. This is reflected in the large value of q in the fitting parameters. The galaxy in Figure 18.23 has been fit with symmetric spirals with the parameters; q = 1.57, v = 0.31, , , , , , and . The apparent optical center is marked by the white axes. The geometrical center for the spiral fit was found to be 52 pixels above that and is marked by the dark gray subaxis. The size of the galaxy has been given as 50,000 light-years, giving a calibration of about 83 light-years/pixel.

Figure 18.23 Classification for NGC 1313. See text for fit parameters.

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME. Image Credit: Henri Boffin (ESO), FORS1, 8.2-meter VLT, ESO. Please see www.wiley.com/go/Lorenzo/Fractional_Trigonometry for a color version of this figure.

18.4.14 Carbon Star 3068

This star was found to have a faint spiral structure of great regularity [20]. It was of some interest to determine the relevance of the fractional spirals to this structure. The results are presented in Section 18.7.

18.5 Analysis

Continuing from Ref. [89] with permission of ASME:

A summary of the galaxy morphologies based on the fittings of the fractional trigonometric spirals is given in Table 18.1. Also contained in the table are the Hubble classifications of various investigators as given in the NASA Extragalactic Database, NED. Furthermore, the table also contains the parameter , which is plotted against Hubble class in Figure 18.24. The Hubble class is assumed to be linear in the plot. The q parameter was chosen because of the strong effect on spiral growth rate that is seen in Figures 18.2 and 18.6. Note that NGC 1313 has not been included in the plots that follow. This galaxy does not follow our initial requirement of well-defined spiral arms. Also, Hoag's Object, a ring galaxy, has been plotted as type S0.

A roughly linear correlation of q with Hubble class is seen in Figure 18.24. Because the number of galaxies studied is small, the scatter appears to be somewhat large. In particular, the single datum at Hubble class b appears to be high and the lower datum point at Hubble class c appears to be low. There are at least two possible explanations here: first, as can be seen from the table, the Hubble classifications by various investigators are substantially not self-consistent and can easily slide horizontally as indicated by the horizontal bars. Furthermore, as can be seen in Figure 18.4, and possibly , also influence the spiral growth rate and their effects are not included in the q parameter. It is believed by the authors that a larger sample of galaxies will give a stronger correlation. This, however, is beyond the scope of this initial investigation.

de Vaucouleurs [28, 29] expanded the Hubble classification by adding the r–s class. The r type indicates the spiral arms exit tangent to an external ring at the termination of a bar while the s type the arms exit the end of the bar. In Figure 18.25, q − 1 is plotted as a function of v for galaxies in which is near unity. The galaxy number as listed in Table 18.1, and the r–s type are given next to each data point. Importantly, all points falling on or near the v = q − 1 line are of the r type. This, of course, is expected from the results shown in Figure 18.6. Furthermore, as v moves away from that line toward v = 0 or negative v, the results become increasingly r–s to s types. Thus, we see that the value of v relative to q − 1 dictates the nature of the exit of the spiral arms and falls directly out of the fitting process. Interestingly, in this limited study, no value of v was found that was more negative than the v = 1 − q line (shown dashed). This may indicate that the v = 1 − q line gives pure s types and the space between these lines are the mixed types.

Table 18.1 Summary of galaxy modeling data and Hubble classifications as found on the NASA NED database

#	Galaxy	q	v			Hubble Class.	r/s
1	NGC 4314	1.125	−0.1	0.10	0.90	SB(rs)a	r s
2	NGC 1365	1.23	−0.2	1	0	SBb/SBc/SB(s)b/SBb(s)	s
3	M 95	1.07	0.0	0.45	0.541	SB(r)b/SBb(r)/SBa/SBb	r
4	NGC 4622	1.07	0.07	1	0	(R′)SA(r)a pec/Sb	r
5	NGC 2997	1.19	0.19	1	0	Sc/SAB(rs)c/Sc(s)	r s/s
6	M 66	1.28	0	1	0	SAB(s)b/Sb(s)/Sb	s
7	NGC 4535	1.14	0	1	0	SABc/SBc/Sc	s
	NGC 4535	1.26	0.26	0.8	0.2	–	–
8	NGC 1300	1.145	0.05	1	0	SBc/SBb(s)/SB(rs)bc	s/r s
9	Hoag's Ob.	1.015	−0.011	1	0	–	–
10	M 51	1.19	0.19	1	0	Sa + Sc?	–
11	AM 0644-741	1.018	0.018	1	0	Sc/Strongly peculiar/	–
12	ESO 269-G57	1.130	0.130	1	0	(R′)SAB(r)ab/Sa(r)	r
13	NGC 1313	1.57	0.31	0.98	0.02	SBc/SB(s)d/SB(s)d	s

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME.

Figure 18.24 Galaxy order parameter q versus Hubble class.

Source: Lorenzo and Hartley 2011 [89]. Adapted with permission of ASME.

Figure 18.25 Galaxy order parameter q − 1 versus v for galaxies with and .

Source: Lorenzo and Hartley 2011 [89]. Reproduced with permission of ASME.

The following question arises: are there theoretical limits on the values of v for spiral galaxies of the , and k = 0 type? This was studied numerically, applying the program used for the mathematical classification study. The results are shown in Figure 18.26. The upper limit results in a hard barred spiral similar to that seen in Figure 18.5 when v = 0.1. When v is larger than q − 1, the program yields a nonspiral result; thus, it appears that v cannot be larger than q − 1.

Figure 18.26 Limits for v parameter, for galaxies with , , and .

A search was then conducted at various values of q to probe the lower limits for v. At any given value of q − 1, the value of v was reduced until the spiral arms interfered with each other. These spirals were similar to that seen in Figure 9.42. Analytically, and possibly physically, of course v could be reduced further; however, the results would not be representative of the spirals of this class. This process yielded the resulting circular markers in Figure 18.26. Surprisingly, this is closely approximated by the linear condition . Note the lines from Figure 18.25 are repeated in Figure 18.26, but not the data.

The results of Figure 18.26 raise the further speculative question; if the spiral arms have a significant material content, does the interference limit possibly indicate a collapse of the spiral structure of a galaxy? On the other hand, it is more widely accepted that the spiral arms are the result of the hypothesized density waves. If this is the case, then, for galaxies of the , type, we should expect wave interference affecting the wave amplitude.

18.5.1 Fractional Differential Equations

Continuing from Ref. [89] with permission of ASME:

The spiral fitting values for the galaxies may now be used to infer fractional-order differential equations describing the geometry. From the forms, we have used

18.4

and the Laplace transform equations (9.66) and (9.69); the following fractional differential equations are inferred:

18.5

18.6

where a = 1, , and . We see that the left-hand sides of both of these simultaneous fractional differential equations are identical in form. The right-hand side filtering, however, changes for the x and y components. A common spatial synchronizing impulse drives both equations.

In these equations, the independent variable is the spatial parameter defining the resulting fractional spiral. This parameter, however, does not define the physical length along the spiral which must be calculated from the x and y position variables defining the spirals.

18.5.2 Alternate Classification Basis

In consideration of alternatives to the use of the fractional versus as the basis of galactic morphology, we note the following. In the classical trigonometry, the sin(t) versus the cos(t) provides the basis of oscillation of the harmonic oscillator. Of all the meta-trigonometric functions, we note that most accurately generalizes the cos(t), because of its even and real basis, that is from Chapter 9:

9.55

Similarly, the most accurately generalizes the sin(t), because of its odd and imaginary basis, also from Chapter 9

9.58

Because of these features, galactic morphology modeling-based versus is worthy of future consideration. Here, we take an initial look at a possible parameter set for this phase plane combination. Figure 18.27 is phase plane of versus and versus , which is highlighted. The spatial t parameter ranges from 0 to 5. For this plot, we have set , and with q = 1.14 and v = 0.14; this results in a barred spiral with a very hard corner. We have set v = 0.1399 to show the transition from the origin to the beginning of the spiral, which would otherwise not display. The various combinations of and produce a set of spirals, which may very well match galaxy spiral morphology. In Figure 18.28, the parameter set is identical except that the v parameter is set to v = −0.400. Comparison of the two figures reveals a startling difference in the transition from the origin to spiral arms. In fact, the spiral appears to be a normal one. Furthermore, the arms of the spirals in Figure 18.28 has been rotated backward relative to those of Figure 18.27. Clearly, this combination of fractional meta-functions is of interest for this task. Other combinations of functions may also have application.

Figure 18.27 Effect of and , Hard barred spiral. versus and versus , , .

Figure 18.28 Effect of and . Normal spiral. versus and versus , , .

18.6 Discussion

Current methods of classification of galaxy morphology are subjective comparisons to idealized galactic shapes established originally by Hubble in 1936 and later extended by de Vaucouleurs to include the effect of the manner in which the spiral arms exit barred spirals, the r–s property. This chapter defines a classification system that uses the fractional trigonometric sine and cosine functions to describe the spiral and ring galaxies on a mathematical basis. The foundation for this basis is quite fundamental, being ultimately sourced by the fractional generalization of the exponential function in the form of the R-function. The significance of this is the connection back to the solutions of fractional differential equations as seen in Chapters 12 and 2. Importantly, the mathematical basis of this classification method may allow further studies of arm evolution and kinematics.

The study was based on a small selection of galaxies intended to give coverage of the Hubble classification scheme. The galaxies were also chosen based on the quality of the definition of the spiral arms. The classification basis used only the fractional trigonometric sine and cosine functions with no cross coupling and with constant growth rate, q. Furthermore, to simplify the manual fitting process the variable a, a spatial fractional generalization of the natural frequency was set to one. The matches of the analytical spirals to the physical spirals were considered to be quite good over the ranges shown thus providing mathematical descriptions over significant segments of the spiral arms. However, in some cases, the extension of the analytical model beyond that shown resulted in deviations from the spiral arms. It appears reasonable to claim that the method allows the classification of spiral galaxy morphology to be set on an analytical basis.

18.6.1 Benefits

This study allows the spiral arms to be mathematically defined, a fact that greatly refines the classification process and presents the possibility of detailed correlation studies involving the spiral parameters. The following benefits were found from this mathematical approach.

1. 1. The process yielded mathematical descriptions of the spiral arms, which are superior to subjective descriptions. The mathematical descriptions allow mathematical correlations, projections, and inferences not possible with the subjective classification. It is important to note that the fractional trigonometric-based classifications, and the fractional differential equations based on them, serve to model both the normal and barred spirals, as well as the r–s variants of the spiral initiation and the morphology of the ring galaxies with just a few parameters.
2. 2. Using the fractional trigonometric functions with programs to account for the inclinations of the galactic planes, good to excellent agreement was found between the actual galaxy morphology and the mathematical spiral morphology.
3. 3. A single fractional trigonometric formulation was found to unify descriptions of both barred and normal spirals, and both r and s spiral-exits. Ring morphologies are also unified for most cases.
4. 4. For most cases, the mathematical spiral q parameter related well to galaxy spiral growth rate, or in Hubble terms, to the openness of the spiral arms. The mathematical spiral v parameter correlated to the barred versus normal galactic spiral property.
5. 5. Plots of v versus q − 1 were found to predict limiting bounds for the r, s behavior, thus relating the galactic spiral arm exit-behavior to the galactic spiral growth rate.
6. 6. The method provides a spiral based method to determine inclination of the galactic plane. A total inclination comparable to contemporary quoted values may be determined by vector considerations of the and computationally inferred inclinations.
7. 7. For NGC 1313, ESO 269-G57, AM 0644-741, M 66, and NGC 1300, major deviations of the optical center of the galaxy from the galaxy geometric center were found based on the mathematical symmetry of the spiral arms. Some of these offsets were many light-years! An indication of galaxy warping was seen for ESO 269-G57.
8. 8. In the case of M 51, the method provides the potential for computing the tidal driven deflection of the upper areas of the spiral arms from an inferred undisturbed position, based on a symmetry assumption for the spiral arms.
9. 9. In this mathematical classification system, the ring galaxies may be interpreted as one-armed barred galaxies (with bars not visible).
10. 10. Finally, availability of the fractional trigonometry-based mathematical classification of the galactic spiral morphology has allowed the easy computation of the value of v for any q, which achieves interference of the galactic spiral arms. This limiting equation may indicate the conditions for transition of the spiral galaxies into elliptical galaxies. This limiting equation may allow validation of the density wave hypothesis relative to material arms.

We note that models based on the fractional trigonometric functions identified by the classification can be considered to be the solution of the spatial part of a fractional wave-diffusion field equation. The development of such an equation in cylindrical form is an important challenge; see Chapter 20.

It should be clear that other subsets and/or additions of the fractional trigonometric spirals, perhaps using the parity functions, may provide superior matches of the mapping of the functions to the galactic spiral arms.

18.7 Appendix: Carbon Star

As mentioned earlier a very regular spiral was found around Carbon Star AFGL 3068. It was of some interest to determine whether this spiral could be fit with the fractional meta-trigonometric spirals. This is done in this appendix.

18.7.1 Carbon Star AFGL 3068 (IRAS 23166 + 1655)

The pattern of mass ejection from this star has been considered to be “one of the most perfect geometrical forms created in space [102].” Thus, there is interest to know if the fractional spirals also fit the form seen here. The star spiral (Figure 18.29) has been fit with a single spiral with the parameters: q = 1.035, v = 0.035, , , , , , and three segments for , , and . These segments together with the dashed lines create a single spiral that matches the observed spiral quite well. A paper by Morris et al. [102] discusses the large mass ejection rates, outflow velocity, and other aspects associated with the spiral.

Figure 18.29 Classification for Carbon Star 3068. See text for fit parameters. (a) Original image (b) magnified view, (c) with analytical fit.

Source: ESA/NASA & R. Sahai. Public domain.