Normal spiral galaxies have exponential disks and logarithmic arms (spirals). They must be interrelated and the relation is suggested to be iso-ratio. Assume that you come off the disk center by such route that the ratio of the star density on your left side to the one on your right side is constant. The route (iso-ratio curve) turns out to be a logarithmic spiral. Such distribution of iso-ratio densities is called a harmonic structure. It can be proved that arms are not harmonic and present (although small) disturbance to the harmonic disks, which explains why elliptical galaxies are clean while spiral galaxies are dusty. Some barred spiral galaxies have elliptical rings (arms). Mathematical study shows that there are very few types of harmonic structures. Most of them are axisymmetric (circularly symmetric). The only non-axisymmetric structure which I can find is the one with elliptical iso-ratio curves. The harmonic structure is handle-shaped while many barred galaxies show a set of symmetric enhancements at the ends of the stellar bar, called ansae or the “handles” of the bar. This suggests that a bar which is the additional harmonic structure to circular disk, consists of several pairs of aligned handles. In far distance from the galaxy center, the bar brightness should not approach the disk one otherwise we could not distinguish between bar and disk in the far distance. It is surprising that my model bar decreases outwards much quicker (cubic-exponential, exp(-r$^3$)) than the exponential disk (exp(-r)). My bar model fits the images of nine galaxies very well. In addition, a similar iso-ratio model fits elliptical galaxies both theoretically and numerically. Because the iso-ratio principle explains and interrelates different observational facts, it must point to some truth.
\section{ Introduction to Galaxy Patterns }
Galaxies are much larger than the solar system. Light travels from the sun to Earth in eight minutes, and travels from one end of the Galaxy to the other end in some hundred thousand years (light travels 300,000 km in a second). Although unimaginably large, thousands of galaxies are taken of images with telescopes. Galaxies demonstrate exceptionally simple and orderly patterns. This shows that the structure of the universe should be simple too.
Independent galaxies present very regular patterns. They are either three-dimensional ellipticals or planar spirals. Ellipticals are very clean while spirals contain dusts which nurture new stars. The life of stars in spiral galaxies is much younger. Images of spirals taken with near-red light show that every spiral galaxy is mainly a disk with its light density decreasing exponentially outwards along the radial direction from the galaxy center (that is, disk center). Therefore, we call them exponential disks. There are other minor or weak structures in spiral galaxies. However, we have only two types of spiral galaxies: the ones with additional bar structures are called barred spirals while the ones without any apparent bar are called normal spirals. Spiral galaxies gain their name by the fact that they present more or less spiral structures, known as arms. The way the arms bend are understood: the angle at each position between the bending direction and the disk radial direction is constant along the arm. A curve which bends in this way is called logarithmic spiral. Therefore, the arms in normal spiral galaxies are called logarithmic spirals.
Now we know that galaxy patterns are exceptionally simple. This poses a question: are the exponential disks and logarithmic arms a coincidence?
Newtonian theory, and other theories are only applicable to the system of one or two bodies. For the system of three or more bodies which have similar masses like stars, these theories have no answer for their patterns!
Exponential disks and logarithmic arms are so elegantly designed that they must be interrelated. The relation is suggested to be the principle of iso-ratio (He, 2005): distribution of stars is harmonic such that star densities on both sides of any curve from a specific orthogonal net keep ratios. The following section proves that the iso-ratio principle interrelates the facts on galaxies and even on mathematics. Therefore, the principle very possibly points to some truth.
\section{ Iso-ratio Principle Interrelates the Facts on Galaxies and Even on Mathematics }
Fact 1: {\it “Iso-ratio curves of exponential disk are logarithmic spirals.”}
Fact 2: {\it “Arm structure is not harmonic and spiral galaxies are dusty.”}
Fact 3: {\it “The only non-axisymmetric harmonic structure which I can find is the one with elliptical iso-ratio curves (bar handle model).”}
Fact 4: {\it “The patterns of galaxy bars are rich and varied, and must themselves consist of several pairs of handles.”}
Fact 5: {\it “The density of the model bar decreases cubic-exponentially (exp(-r$^ 3$)), a consistent result to galaxy images.” }
Fact 6: {\it “The arms near galaxy bars are not logarithmic spirals.”}
Fact 7: {\it “My bar model fits the images of nine galaxies very well.”}
Fact 8: {\it “Fitting my bar model to the images of different galaxies can tell their physical sizes in the universe.”}