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A New Formula Describing the Scaffold Structure of Spiral Galaxies

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A New Formula Describing the Scaffold Structure of Spiral Galaxies View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Aquila Digital Community The University of Southern Mississippi The Aquila Digital Community Faculty Publications 7-21-2009 A New Formula Describing the Scaffold trS ucture of Spiral Galaxies Harry I. Ringermacher General Electric Global Research Center, [email protected] Lawrence R. Mead University of Southern Mississippi, [email protected] Follow this and additional works at: http://aquila.usm.edu/fac_pubs Part of the Physical Sciences and Mathematics Commons Recommended Citation Ringermacher, H. I., Mead, L. R. (2009). A New Formula Describing the Scaffold Structure of Spiral Galaxies. Monthly Notices of the Royal Astronomical Society, 397(1), 164-171. Available at: http://aquila.usm.edu/fac_pubs/1159 This Article is brought to you for free and open access by The Aquila Digital Community. It has been accepted for inclusion in Faculty Publications by an authorized administrator of The Aquila Digital Community. For more information, please contact [email protected]. Mon. Not. R. Astron. Soc. 397, 164–171 (2009) doi:10.1111/j.1365-2966.2009.14950.x A new formula describing the scaffold structure of spiral galaxies Harry I. Ringermacher1 and Lawrence R. Mead2 1General Electric Global Research Center, Schenectady, NY 12309, USA 2Department of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, MS 39406, USA Accepted 2009 April 20. Received 2009 April 17; in original form 2008 August 14 ABSTRACT Downloaded from We describe a new formula capable of quantitatively characterizing the Hubble sequence of spiral galaxies including grand design and barred spirals. Special shapes such as ring galaxies with inward and outward arms are also described by the analytic continuation of the same formula. The formula is r(φ) = A/log[B tan(φ/2N)]. This function intrinsically generates a bar in a continuous, fixed relationship relative to an arm of arbitrary winding sweep. A is http://mnras.oxfordjournals.org/ simply a scale parameter while B, together with N, determines the spiral pitch. Roughly, greater N results in tighter winding. Greater B results in greater arm sweep and smaller bar/bulge, while smaller B fits larger bar/bulge with a sharper bar/arm junction. Thus B controls the ‘bar/bulge-to-arm’ size, while N controls the tightness much like the Hubble scheme. The formula can be recast in a form dependent only on a unique point of turnover angle of pitch – essentially a one-parameter fit, aside from a scalefactor. The recast formula is remarkable and unique in that a single parameter can define a spiral shape with either constant or variable pitch capable of tightly fitting Hubble types from grand design spirals to late-type large barred at The University of Southern Mississippi on November 30, 2016 galaxies. We compare the correlation of our pitch parameter to Hubble type with that of the traditional logarithmic spiral for 21 well-shaped galaxies. The pitch parameter of our formula produces a very tight correlation with ideal Hubble type suggesting it is a good discriminator compared to logarithmic pitch, which shows poor correlation here similar to previous works. Representative examples of fitted galaxies are shown. Key words: galaxies: fundamental parameters – galaxies: spiral – galaxies: structure. This spiral is usually mathematically characterized by a constant 1 INTRODUCTION angle of pitch (though k may be a function of r as well) allowing The logarithmic spiral has been the traditional choice to describe this parameter to be used to describe galaxy shapes. The pitch, P, the shape of arms in spiral galaxies. Milne (1946) made perhaps is defined from Binney & Tremaine (1987) as the first attempt to derive these shapes from his own theory, but his dφ theory resulted in spiral orbits for stars. Today most astronomers cot(P ) = r(φ) . (2) dr agree that stellar orbits are essentially circular and that the spiral For equation (1), P = tan−1 k is constant. However, it is apparent arms are the result of an evolving pattern, much like a Moire´ pattern when attempting fits that galaxy arms often do not have constant (Lin & Shu 1964; Shu 1992) or a dynamic modal structure (Bertin pitch. This has also been noted by Kennicutt (1981). This is most et al. 1989a,b; Bertin 1993). That is, the stars define a locus of points evident in strongly barred late-type spirals, whereas early types and at a given time among a family of circular orbits. We shall call this grand designs are essentially constant pitch. In this paper, we present locus an isochrone. The simplest such curve that describes galaxies a new formula, differing from any in the standard mathematical or is the logarithmic spiral and has been used by many (Lin & Shu astronomical literature, which is capable of describing all spiral 1964; Roberts, Roberts & Shu 1975; Kennicutt 1981; Kennicutt & shapes, constant pitch or variable, in an elegant way. Hodge 1982; Elmegreen & Elmegreen 1987; Ortiz & LePine´ 1993; Seigar & James 1998a,b, 2002; Block & Puerari 1999; Seigar et al. 2006; Vallee´ 2002) in their morphological descriptions: 2 NEW FORMULA kφ r(φ) = r0e . (1) Our formula derives from an examination of equations found in the non-Euclidean geometry of negatively curved spaces. This hy- perbolic geometry was first discovered and published by Bolyai (1832) and independently by Lobachevsky. Their work is dis- E-mail: [email protected] cussed in Coxeter (1998). The central formula describing multiple C 2009 The Authors. Journal compilation C 2009 RAS The scaffold structure of spiral galaxies 165 parallels measures ‘the angle of parallelism’ (Coxeter 1998) be- 3 GALAXY FITS tween a given line and ‘parallel’ lines through a given point not on the line – the violation of Euclid’s fifth postulate. The an- 3.1 Down- and up-projection gle of parallelism, known as Lobachevsky’s formula, is given by φ(x) = 2tan−1(e−x ). The Gudermannian function is closely related Galaxy shapes in the sky are projections with respect to a north– and is given by φ(x) = 2tan−1(ex ). The latter function directly re- south, east–west coordinate system which we simply define as ori- lates circular to hyperbolic functions. We have found a new function ented along Y-andX-axes, respectively, on a graph facing us. Two closely related to the above that describes the shapes of spiral galax- angles, namely position angle (PA) and inclination angle (I), are ies remarkably well. This formula is given in radial form, where r−1 necessary to down-project a shape from a ‘sky plane’ to a ‘graph replaces x in the Gudermannian and scaling degrees of freedom are plane’. By the previous definition, the two planes are actually one added: and the same. The end result of a two-angle down-projection, PA followed by I, is a correct but oriented graph shape at a third angle, = A r(φ) φ . (3) γ . In this case, the third angle is the orientation, γ , in the plane log B tan 2N with respect to ‘Y’ or north–south. We recognize the shape in any Downloaded from This function intrinsically generates a bar in a continuous, fixed direction so it is not important. relationship relative to an arm of arbitrary winding sweep. Though This procedure is, however, not arbitrarily reversible. If one cre- in some instances, observations show gaps between the bar and ates a theoretic shape function to compare to an observed galaxy arms (e.g. Seigar & James 1998b), nevertheless, arms begin where and simply starts with the major axis aligned along ‘Y’, then up- bars end so that a continuous bar–arm formula serves as a galactic projects using the known I followed by PA (reversing the order and fiducial for fitting. This is particularly evident in NGC 1365 of our sign of angle), the shape would, in general, be incorrect and we http://mnras.oxfordjournals.org/ galaxy selection and will be described later. A is simply a scale would require a third Euler rotation, γ . Alternatively, we could ap- parameter for the entire structure while B, together with a new ply the ‘final orientation’, γ , determined from down-projection as parameter N, determines the spiral pitch. The ‘winding number’, N, the first rotation about Z, and then apply I followed by PA and find need not be an integer. Unlike the logarithmic spiral, this spiral does the correct sky shape. Equivalently, one could replace φ in formula not have constant pitch but has precisely the pitch variation found (2) by φ − γ and achieve the same effect. It is clear from either in galaxies. The use of this formula assumes that all galaxies have view that the third angle is necessary for up-projection otherwise a ‘bars’ albeit hidden within a bulge consistent with recent findings. serious shape error could result. The necessity for a third angle is Roughly, greater N results in tighter winding. Greater B results most obvious in cases where a galaxy shape is not equiaxial in its in greater arm sweep and smaller bar/bulge, while smaller B fits plane. There are then two unique axes in the sky plane, the major at The University of Southern Mississippi on November 30, 2016 larger bar/bulge with a sharper bar/arm junction. Thus, B controls axis as viewed and the intrinsic long axis, and thus the need for a the ‘bulge-to-arm’ size, while N controls the tightness much like third angle to reconcile them. Circularly symmetric tightly wound the Hubble scheme. Fig. 1 shows several examples of these spirals. spirals and face-on galaxies do not require a third angle, but many We divide the examples according to N-value.
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