How to reduce computer calculation time when solving orbital mechanics task Dr. Donaldas Zanevičius. American Research Institute for Policy Development, American Association of International Researchers Space Technology Research Centre (Lithuania), Chairman of Scientific Council 1. Calculations using classical trigonometric functions Dr. Robert A. Braeunig in the article Rocket and Space Technology. Orbital mechanics. [1] www.braeunig.us/space/orbmech.htm gives many examples of calculations. One of the examples (4.77) is presented as Problem 4.24. Here, like in all other examples, classical geometrictrigonometric mathematical models are used. These are trigonometric functions sinα, cosα, tanα. We present calculation results of the abovementioned mathematical model (4.77). Indicate Write the mathematical model (1) The values of ii, if, Ωi, Ωf are given. We need to calculate the value of γor using (1). As we know, sinα, cosα, tanα can be calculated only using infinite series. (2),(3),(4),(5) (6),(7),(8),(9) (10) Angles are usually given in degrees, minutes and seconds, but in calculations (2)-(9), angles have to be expressed in radians; therefore, they have to be converted to radians using the relation formula. (11),(12),(13),(14) In order to calculate the value of γor (1), we need to calculate fourteen formulas; nine of them are infinite series (2)- (10), what takes quite a lot of computer calculation time. 2. Calculations using h-geometry functions: sph, cph, tph [2],[3]. Here are given the values h11, h12,h21,h22. Then, instead of the expression (1), we can write (15) where (16),(17) (18),(19) (20),(21) Here (15) the value γh is the angle measured in h-parameters. As we can see, calculation in the system of h-parameters requires calculation of seven algebraic expressions (15)-(21). 3. Comparison of calculation complexity and calculation results of the specific task as a given [1] (4.24 problem).
Classical. Given values (22) Using formulas (1)-(14), the calculation program is written for a computer. The result is (23) The same calculated value of γor (20) is also given in [1]. h-geometry models. In order to compare calculation results, we chose h values equivalent to classical trigonometric parameters. We use relation formula (24) From (22) and (24) we get (25) From (!5)-(21), (25) we get (26) As we can see from (23), (26), the calculation results fully correspond. However, computer calculation time using classical method (1)-(14) and using h-geometry formula (!5)- (21),(25) significantly differs.
Literature. 1 Dr. Robert A. Braeunig in the article Rocket and Space Technology. Orbital mechanics. www.braeunig.us/space/orbmech.htm 2 1. Technologies for Calculating Geodetic Coordinates Applying h–Geometry Functions (2) Donaldas Zanevičius Faustas Keršys Magazine GEODESY AND CARTOGRAPHY 201 36(4); 160-163 has published article of Donaldas Zanevičius and Faustas Keršys TECHNOLOGIES FOR CALCULATING GEODETIC COORDINATES APPLYING H–GEOMETRY FUNCTIONS. (The article was written in Lithuanian). This article was cited in various international publications, including Harvard‘s Department of Astronomy, Smithsonian Astrophysical Observatory and NASA Astrophysics Data System (http://adsabs.harvard.edu/abs/2010GeCar..36..160Z ) Article „Technologies for Calculating Geodetic Coordinates Applying h– Geometry Functions (2)“ The article proposes to apply h-geometry functions sph and cph instead of classic geometry functions sin and cos for conversion of coordinates in mathematical models.
As known, numeric values of functions sin and cos can be calculated only when functions are developed to infinite string. h-geometry functions sph and cph have algebraic analytic expressions. It enables system mathematical models to be presented in analytic expressions. This means, that computer calculation times may be shortened 5-10-fold. This should be very important for developers of antiballistic defence systems. Classic geometry functions do not allow to obtain such expressions. In such case, systems’ mathematical models can be calculated only by applying iteration methods. To read the full article, please click here. http://www.kosmose.lt/space-geodesy/ 3. ResearchGate
Dr. Donaldas Zanevičius, Faustas Keršys Geodezinių koordinačių perskaičiavimo technologijos, taikant h – geometrijos funkcijas. Geodesy and Cartography 36(4):160-163 · January 2010 https://www.researchgate.net/publication/254237109_Geodeziniu_koordinaciu_persk aiciavimo_technologijos_taikant_h-geometrijos_funkcijas