Dragonhead Cyclone in the Falkland Islands has Double Cote's Spiral

Introduction

Cyclone

A cyclone is a large air mass that rotates around a strong center of low atmospheric pressure, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere as viewed from above (opposite to an anticyclone). A subtropical cyclone is a weather system that has some characteristics of a tropical cyclone and some characteristics of an extratropical cyclone [1-4, 27,29]. They can form between the equator and the 50th parallel. These storms usually have a radius of maximum winds that is larger than what is observed in purely tropical systems, and their maximum sustained winds have not been observed to exceed about 32 m/s (64 knots) [1, 5-8, 26, 27]. Subtropical cyclones sometimes become true tropical cyclones, and likewise, tropical cyclones occasionally become subtropical storms.

Subtropical cyclones in the Atlantic basin are classified by their maximum sustained surface winds: Subtropical depressions have surface winds less than 18 m/s (35 knots), while subtropical storms have surface winds greater than or equal to 18 m/s. Tropical cyclones are compact, circular storms, generally some 320 km (200 miles) in diameter, whose winds swirl around a central region of low atmospheric pressure [8-21, 26-29, 39]. The winds are driven by this low-pressure core and by the rotation of the Earth, which deflects the path of the wind through a phenomenon known as the Coriolis force. As a result, tropical cyclones rotate in a counterclockwise (or cyclonic) direction in the Northern Hemisphere and in a clockwise (or anticyclonic) direction in the Southern Hemisphere [8-21, 26-29, 39].

NGC 5247 Spiral Galaxy

Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work The Realm of the Nebulae and, as such, form part of the Hubble sequence [40-43]. Most spiral galaxies consist of a flat, rotating disk containing stars, gas and dust, and a central concentration of stars known as the bulge. These are often surrounded by a much fainter halo of stars, many of which reside in globular clusters [54]. Spiral galaxies are named by their spiral structures that extend from the center into the galactic disc. The spiral arms are sites of ongoing star formation and are brighter than the surrounding disc because of the young, hot OB stars that inhabit them [42, 43, 47, 54].

                  Figure 1: HAWK-I image of NGC 5247 Galaxy

                                          Source: [55]

The Figure (1) shows NGC 5247, a grand design barred spiral galaxy, located 60–70 million light-years away. The galaxy lies face-on towards Earth, thus providing an excellent view of its pinwheel structure and multiple arms. It is in the zodiacal constellation of Virgo (the Maiden). The image was made in infrared light with the HAWK-I camera on ESO’s Very Large Telescope at Paranal Observatory in Chile. HAWK-I is one of the most powerful infrared imagers in the world, and this is one of the sharpest and most detailed pictures of this galaxy ever taken from Earth. The filters used were Y (shown here in blue), J (in light blue), H (in green), and K (in red). The field of view of the image is about 6.4 arc minutes [55].

NGC 5247 is a face-on unbarred spiral galaxy located some 60 million light years away in the constellation Virgo. It is a member of the Virgo II Groups, a series of galaxies and galaxy clusters strung out from the southern edge of the Virgo Super cluster. This grand design spiral galaxy displays no indications of distortion caused by interaction with other galaxies. It has two spiral arms that bifurcate after wrapping halfway around the nucleus. The disk is estimated to be 4.9 ± 2.0 kly (1.5 ± 0.6 kpc) in thickness and it is inclined by roughly 28° to the line of sight [56].

Cote’s Spiral

In physics and in the mathematics of plane curves [29, 30, 31, 32, 33, 34, 35, 36], a Cote's spiral (also written Cotes' spiral and Cotes spiral) is one of a family of spirals classified by Roger Cotes.

Figure 2: Shapes of Curves Called Cote's Spiral Gives the Solution to The Central Orbit Problem

                                                 Source: [58,59].

Development

After an analysis of the different types of spirals, it was concluded that the form that came closest the spiral, Figure (1, 2, 4-7), is a double Cotes's Spiral, for the case μ< h2 , Figure (2) [26, 29-36]. It was determined the mathematical equation of the shape of the DC, in the shape of a spiral called Cotes's Spiral", for the case μ< h2 , Figure (2) [24-26].

A spiral that gives the solution to the central orbit problem under a radial force law

rÃÂ?? = -μ [r] - 3rÃÂ??                           (1)

where µ is a positive constant. There are three solution regimes,

r = Asec (kθ + ϵ)                    (2)

where ð??ð??² � 1 —  ^μ , when μ � â??² ,

where A and ÃÂ?? are constants, and h is the specific angular momentum [26, 37, 38].

Analyzing the shape of the Spiral shape called "Cotes’s Spiral." for μ< h2 [24, 25], it appears that adding two constants to Equation (2) makes the necessary adjustments for the Isobaric ones. In the case of DC, the spiral that gives the solution to a radial force law is given by Equation (1).

An adjustment in Equation (2) is necessary to obtain the graph of Figure (7). Then, adding the constants B≠0 and C where for μ< h2 .

 = BAsec (kθ + ϵ) + C                         (3)