A New Formula Describing the Scaffold Structure of Spiral Galaxies
  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(phi) = A/log[B
tan(phi/2N)]. This function intrinsically generates a bar in a continuous,
fixed relationship relative to an arm of arbitrary winding sweep. A is simply a
scale parameter while B, together with N, determine 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
1-parameter fit, aside from a scale factor. 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 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.
