Universidad Central de Venezuela From the SelectedWorks of Jorge A Franco September, 8 Estimation of Planetary Orbits via Vectorial Relativity (VTR) Jorge A Franco, Universidad Central de Venezuela Available at: https://works.bepress.com/jorge_franco/17/
Journal of Vectorial Relativity JVR 3 (8) 4 33-41 Estimation of Planetary Orbits via Vectorial Relativity (VTR) E Valdebenito 1 ABSTRACT: The equation (approximate) for the Movement of Planets about a Massive Body, according to the theory VTR, is: du (1 6 )u u 1 y, they r d, where are parameters. Applying the previous model we obtain the paths of several planets of the Solar System. In addition we obtain the precession for every planet using the formula given by the Vectorial Relativity and we compare it with the values of the precession given by the GTR ( General Theory of the Relativity ). KEYWORDS: Path of Planets, VTR. INDEX I. Introduction II. Equation of the Path of a Planet III. Path of some Planets of the Solar System IV. Precession of Planets V. Conclusion References I. INTRODUCTION In [1] J. A. Franco presents the following exact equation for the Movement of a Planet about a Massive Body. u GM v v d u m r v dr p p u d L p p p dv p p p (1) 1 Independent Researcher, Santiago de Chile, Chile September 16th, 8
E Valdebenito: Estimation of Planetary Orbits via Vectorial Relativity (VTR) September 16th, 8 where u 1 r M : mass massive body (Mass of the Sun) L : Angular Momentum m : mass planet v : tangential velocity G : Constant of the Universal Gravitation p : momentum The mass m of the planet is considered to be a variable. The equation (1) is solved in approximate form using all the results and definitions of the vectorial Theory of the relativity [1] [6]. (a) The new definition of relativistic mass is used [6], M m 1 v c 3 () thats corrects that given by Einstein s in 195, M m 1 v c 1 (3) (b) The new definition of Relativistic Kinetic Energy is used []. K m( v c ) m ( v c ) (4) (c) The new definition of tangential velocity is used [3]. m 1 1m m v v GM r r m (5) For details to see references. The (approximate) final equation for movement of planets is: where: du (1 6 )u d (6) JVR 4 (9) 1 68-94 Journal of Vectorial Relativity 34
E Valdebenito: Estimation of Planetary Orbits via Vectorial Relativity (VTR) September 16th, 8 M GM GM 6 L c r (7) M G M Lc c : velocity of the light M : mass of the planet. (In rest). r : minimal distance of the planet to the Sun We will apply the model (6) to obtain the equation of the path of several planets of our solar system. II. EQUATION OF THE PATH OF A PLANET The solution of the differential equation (6) is: u( ) u cos( 16 ) 16 16 (8) where u 1. r The equation for the path is: 1h r 1 1 1cos A rh (9) where: h 16 A 16 (1) III. PATHS OF SOME PLANETS OF THE SOLAR SYSTEM We will obtain the equation of the path of the following planets of the solar system: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto. JVR 4 (9) 1 68-94 Journal of Vectorial Relativity 35
E Valdebenito: Estimation of Planetary Orbits via Vectorial Relativity (VTR) September 16th, 8 III.1. Data The general used data is: Constant of the Universal Gravitation: 11 17 G 6.6766 1 [ Nm / kg ] 6.6766 1 [ Nkm / kg ] Velocity of the ligth: 8 5 c.99 1 [ m / s ].99 1 [ km / s ] Mass of the Sun: 3 Msol 1.9889 1 [ kg ] In the Table 1 they present the information for every planet. PLANET MASS (kg) PERIHELION (km) ORBITAL MAXIMUM VELOCITY (km/s) MERCURY 3.4 1 46 58.98 VENUS 4 4.8685 1 7 1.7476 1 35.6 EARTH 4 5.9737 1 6 147.9 1 3.9 MARS 3 6.4185 1 66 6.5 JUPITER 7 1.8987 1 74746 13.7 SATURN 6 5.6851 1 1349467 1.18 URANUS 5 8.6849 1 73556 7.11 NEPTUNE 6 1.44 1 445963 5.5 PLUTO 1.3 1 44368 6.1 Table 1. Mass, Perihelion and Orbital Maximum Velocity. Orbital Maximum Velocity: Orbital maximum Velocity (in the Perihelion), in kilometers per second. Perihelion: Minimal distance of the planet to the Sun. JVR 4 (9) 1 68-94 Journal of Vectorial Relativity 36
E Valdebenito: Estimation of Planetary Orbits via Vectorial Relativity (VTR) September 16th, 8 III.. Path of Planets Using the equation (9) and the previous data we obtain: Path planet Mercury: r 55463697.55874 1.57 cos.9999 Path planet Venus: r 18196.79951 1.6838 cos.9999 Path planet Earth: r 149571381.6816777 1.1686 cos.9999 Path planet Mars: r 585899.453161 1.931 cos.9999 Path planet Jupiter: r 7786397.3183 1.565 cos.9999 Path planet Saturn: r 1413696.8886483 1.5375 cos.9999 Path planet Uranus: r 854685.1416756 1.4 cos.9999 Path planet Neptune: r 45331941.966 1.165 cos.9999 Path planet Pluto: JVR 4 (9) 1 68-94 Journal of Vectorial Relativity 37
E Valdebenito: Estimation of Planetary Orbits via Vectorial Relativity (VTR) September 16th, 8 r 5519337837.5176615 1.439 cos.9999 According to the equations, the path of the planets are elliptical. In the Figure 1 they present the graphs of the orbits of the planets: Mercury, Venus, Earth, Mars.. 1 8 Tierra 1. 1 8 Venus. 1 8 1. 1 8 1. 1 8. 1 8 Mercurio 1. 1 8 Marte. 1 8 Figura1. Orbits planets Mercury, Venus, Earth, Mars In the Figure they present the graphs of the orbits of the planets: Jupiter, Saturn, Uranus, Neptune, Pluto. JVR 4 (9) 1 68-94 Journal of Vectorial Relativity 38
E Valdebenito: Estimation of Planetary Orbits via Vectorial Relativity (VTR) September 16th, 8 4. 1 9. 1 9 Urano Saturno 6. 1 9 4. 1 9. 1 9. 1 9 4. 1 9 Jupiter. 1 9 Neptuno Pluton 4. 1 9 Figura. Orbits planets Jupiter, Saturn, Uranus, Neptune and Pluto. IV. PRECESSION OF PLANETS We obtain the precession of the planets of the Solar System using the following formula given by the VTR (Equation 35 [1]): M. GM 6.. (11) L. c IV.1. Data JVR 4 (9) 1 68-94 Journal of Vectorial Relativity 39
E Valdebenito: Estimation of Planetary Orbits via Vectorial Relativity (VTR) September 16th, 8 We obtain the values of the precession using the previous information and those of the Table 3. PLANETS Returns every 1 years MERCURY 414.9378 VENUS 16.616 EARTH 1. MARS 53.1915 JUPITER 8.4317 SATURN 3.3944 URANUS 1.193 NEPTUNE.668 PLUTO.43 Tabla 3. Orbits per century. In the Table 4 appear the obtained results: PLANETS VTR (arc sec) GTR (arc sec) MERCURY 43.139 4.9195 VENUS 8.665 8.6186 EARTH 3.855 3.8345 MARS 1.358 1.35 JUPITER.64.63 SATURN.137.137 URANUS.4.4 NEPTUNE.77.8 PLUTO.4.4 Table 4. Precession per century VTR y GTR. V. CONCLUSION In the frame of the theory of the vectorial relativity (VTR) an estimation of the orbits is obtained for the following planets of the Solar System: Mercury, Venus, Earth, Mars, Jupiter, JVR 4 (9) 1 68-94 Journal of Vectorial Relativity 4
E Valdebenito: Estimation of Planetary Orbits via Vectorial Relativity (VTR) September 16th, 8 Saturn, Uranus, Neptune, Pluto. The values of the orbital precession obtained with the VTR and GTR are similar. REFERENCES [1] J A Franco R. First Solutions to Gravitation and Orbital Precession under Vectorial Relativity. Published by JVR on March 3th 8. JVR 3 (8) 1 1-13. [] J A Franco R. Energy in Vectorial Relativity, E m.c². Published by JVR on November 16th 6. JVR 1 (6) 1 1-7. [3] J G Quintero D y J A Franco R. Gravitational Forces in Vectorial Relativity. Published by JVR on March 16th 7. JVR (7) 1 33-4. [4] J G Quintero D y J A Franco R. Precession in Vectorial Relativity. Published by JVR on March 16th 7. JVR (7) 1 53-6. [5] J A Franco R. Vectorial Lorentz Transformation. Published by EJTP on February 5th 6. EJTP 9 (6) 35-64. [6] J G Quintero D y J A Franco R. Mass in Vectorial Relativity. Published by JVR on November 16th 6. JVR 1(6) 1 33-4. JVR 4 (9) 1 68-94 Journal of Vectorial Relativity 41