New Theorem and Formula for Circle Arc Length Calculations with Trigonometric Approach Application in Astrophysics

: The circle and sphere have been studied since the ancient Egyptians and Greeks before the Common Era (BCE). The recent scientific renaissance has also used them in different fields. It is also mentioned in the Prophet Mohamed`s Holy Quran. This article introduces a new Theorem (S. El-Mongy`s Theorem) as an empirical formula to correlate the constant (e) with circle and sphere . It states that “the arc length is correlated as a direct function in { (e π r s A )}, whatever the central angle ( ϴ ) and radius (r). The factor s A is ( ϴ /10 ϕ ). The formula can also be written as; A L = {(0.0174533185 r ϴ )}. Where the value 0.0174533185 is a constant called Sayed`s number (I s ) and equals ( eπ/10ϕ ). This factor is very close to value ( π/180 = 0.0174532925) with ~1.5x10 -4 % difference. The formula was applied for calculation the arc length (A L ) of circles of different radii and angles. The results of this formula were validated and verified for very wide range; from 0.5 cm to 4.4x10 23 km (46.5x10 9 light-years; ly) and compared with the old published arc length formula results. The difference is from 0.000% to 0.002% only. The formula was also used as trigonometric functions of circular orbits for calculation the distances between the Earth and Sun, Moon, planets, stars and EH-M87 Black hole with relatively small error; the difference is from 0.26% to maximum ~ 2.27%. The error was 0.29% for ~54 x 10 6 ly distance to the M87 black hole. The S. El-Mongy formula may open the door for further scientific and engineering applications.


Introduction
The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon and Sun. The circle was first observed in one of Pharos papyrus (1). The philosopher and mathematician Thales (624-546 BCE), who transferred the science and geometry from the Egyptians to Greeks has a theorem. The word circle in Greek, means "hoop" or "ring" (1,2,3). A circle is a plane figure bounded by one curved line and all the bounding line is equal as given by Euclid, Elements Book, 300 BCE (1,3).
The circle is 360° all the way around. The circle may also be defined as a special kind of ellipse in which the two foci are coincident (zero eccentricity) (1,3). Both circles and spheres are circular. They are 2D figure (plane) and 3d object (space) respectively. The sphere is a dual geometrical surface, fully symmetrical, resulting from a circle rotation around one of its diameters. The sphere and the circle arc, sector and segment are given in (Fig.  1).
The three major established circle theorems and concepts indicate that the (i) circle arc length is directly proportional to its radius. (ii) If central angles (ϴ) congruent, then arcs congruent. (iii) The chord lengths dividing the circumference of a circle into equal number of segments.
The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus (1,3).
Our galaxy revolves on its own axis which is its center with the result that the Sun revolves around the same center in a circular orbit. Recently, 1917, Shapley estimated the distance between the Sun and the center of our galaxy at 10 kilo-parsecs, i.e., 10x10 17 km. The orbital movement of the Sun was already referred to by the Prophet Mohamed`s Quran (4,5).

II.
Theoretical Aspects of the Formula: II.A. S. El-mongy`s Theorem: "The circle arc length is a function in (e π r s A )".
A new empirical formula for calculation of the circle arc length (AL) was introduced in this article. It correlated the circle circumstance with other mathematical constants. The old formula for arc length can be given as follows (2,3): The circle circumstance = 2 π r (1) Where, (r) is the circle radius and ϴ is the central angle in degree. The (π) is the irrational constant equal to (22/7) or 3.14159265 (π is also called Archimedes constant (3, 6). The arc length can also be calculated by the other known formula; (AL = r. ϴ). Where, ϴ is in Radian.
The proposed empirical formula was practically found as mentioned and is mathematically given and expressed as follows:

Arc length (AL) = {(e π r s A )} (3)
This empirical formula is called S. El-Mongy`s theorem. The factor (s A ) represents the ratio {(ϴ/ 10ϕ)}. Where, ϴ is the central angle in degree and ϕ corresponding 48.929 0 . This formula was applied tested and validated for any circular orbit whatever the radius length and ϴ.
The constant e is the base of the natural logarithm (Euler`s number) and equal to 2.7182818 (7). This formula number 3 can also be rewritten to be;

III.
Results and discussion:

III.A. Validation of S.El-Mongy`s Theorem: The arc length is a function in (e π r s A ).
The arc length can be calculated using our abovementioned empirical formula (eq.3). It was validated for different circle radii (0.5 cm -4.4x10 23 km) and central angles ϴ (0.0005 0 -360 0 ). The precise results compared with the old published formula are given in Table 1.  (8). ***Radius of a galactic gaseous halo (9.461x10 21 km) around our Milky Way galaxy (9). It should be mentioned that the correlation between e to power π was only given by Euler`s identity; (e iπ = -1) (10).
It can be observed that the results arc length calculation achieved by El-Mongy`s formula are almost identical with these obtained by the old published formula. Our formula was accurately and sharply validated using circles radii (r) from 0.5 cm to 4.4x10 23 km (46.5x10 9 ly). The highest difference does not exceed 0.002%.

III.B. Applications of S. El-Mongy`s Theorem and Formula in Astrophysics:
The small-angle approximation is a useful simplification of the basic trigonometric functions which is approximately true in the limit where the angle approaches zero (11,12,13,14,15). Figure 2 shows clarification how to calculate the distance of object (e.g. Sun) as seen from Earth based on small angle approximation (5,14,15). For example, the arc that runs through the Sun's diameter has an angle (angle subtended by an object) and an arc length equal to actual diameter (the arc-length formula). This is presented for a right triangle with the trigonometric relationship as follows: Where, D>>d, Tan ϴ ≈ ϴ and angular diameter (ϴ) is in radian. For conversion, it is Figure 2: Angular diameter and Arc length approach for distance calculation (15). By using our formula; eq.3: AL={(e π r s A )}, that can be rewritten to be:

D = (AL x ϕ x 10) / (e π ϴ)
This formula was also and simply used for astronomical distances (D) calculation. It can also be written as follows;

III.B.1 Validation and verification of the formula:
Validation of this formula was also carried out by calculating the distance (D) between the Earth and Sun, Moon, planets, Stars and M87 Black hole in the center of our Milky Way galaxy (5,16). The Table 2 shows the calculated results compared with the literature values. Where, the arc length (AL) equals to diameter (the arc-length approach) and ϴ is the sighting angle in degree (angular diameter) from the Earth (14,15).  (16). Note: The distance between Earth and the Sun orbit is an average 149597870 km. ** One of the nearest stars to our Earth is Alpha Centauri at 4.3 light years (ly) away (40.68 x10 9 km away from the Earth). ***The distance between the Sun and our Galaxy center (10 5 ly circular) equal 9.5x10 17 km.
The relatively small difference (0.26% to maximum ~ 2.27%) with the reference values is mainly due to the variation of these values (e.g. angular diameter and distances) given in different references and literature (14,15).
The formula was also validated by calculating the radius of some planets and stars. The result between the certified published values and the proposed formula are given in the following Table 3. It should be mentioned that the small angle approximation is useful in many areas of engineering and physics, including mechanics, electromagnetics, optics, cartography, astronomy, computer science (14,15). S. El-Mongy formula is currently developed to be used in nuclear physics fields.

Conclusion
It can be observed that S.El-Mongy`s theorem, is a new empirical formula for calculating the circle arc length as a function in (eπrs A ). This formula can also be given as; AL = {(0.0174533185 r ϴ)}. Where, the value 0.0174533185 is called Sayed`s number (Is) and equal to (eπ/10ϕ). The difference between the calculated arc lengths using the old well-known formula and our formula does not exceed 0.002% whatever the central angle and circular radius.
The distances between the Earth and sun, moon, planets, stars and EH-M87 black hole were precisely calculated using the S. El-Mongy`s formula with differences from 0.26 to ~ 2.27%. The formula was also used for radius calculations of some planets and stars.
Finally, it should be stated that the our new formula that correlates (eπrs A ) with circle arc length and its applications may dramatically leads to new and developed concept in many scientific fields. Previously, the study of the circle has helped inspire the development of many fields; geometry, astronomy, physics and calculus.

Conflicts of Interest
The Author declares that there is no conflict of interests with any other author.