Gravitational Scalar Potential Values Exterior to the Sun and Planets.

Size: px

Start display at page:

Download "Gravitational Scalar Potential Values Exterior to the Sun and Planets."

Helena Rich
5 years ago
Views:

1 Gravitational Scalar Potential Values Exterior to the Sun and Planets. Chifu Ebenezer Ndikilar, M.Sc., Adam Usman, Ph.D., and Osita Meludu, Ph.D. Physics Department, Gombe State University, PM 7, Gombe, Nigeria. Physics Department, Federal University of Technology, Yola, Adamawa State, Nigeria. ASTRACT In this paper, we compute gravitational scalar potential values along the equator and pole of the Sun and planets. Our computation differs from previous approaches in that the precise shapes (oblate spheroidal) of these astrophysical bodies are used. The values obtained diminish with increasing distance from the massive astrophysical body. A plot of the values agrees satisfactorily with well known experimental facts. (Keywords: scalar potential, oblate spheroid, sun, planets, equator, pole) INTRODUCTION Prior to 95, theoretical gravitational study was restricted almost exclusively to the fields of massive bodies of perfect spherical geometry. Since 95, it has increasingly been recognized that the Earth, Sun, and almost all major astronomical bodies are actually spheroidal in geometry as shown in Table. It has also been confirmed that this geometry has experimentally measurable and physically interesting effects on the motion of test particles in their gravitational fields. These effects exist in Newton s Theory of Universal Gravitation as well as in Einstein s Theory of General Relativity []. It is now well known that satellite orbits around the Earth are governed by not only the simple inverse distance squared gravitational fields due to perfect spherical geometry, but are also governed by second harmonics (pole of order 3) as well as fourth harmonics (pole of order 5) of gravitational scalar potential not due to perfect spherical geometry. Therefore, towards the more precise explanation and prediction of satellite orbits around the Earth, Stern [3] and Garfinkel [4] introduced the method of quadratures for approximating the second harmonics of the gravitational scalar potential of the Earth due to its spheroidal geometry. This method was improved by O Keefe [5]. In 96, Vinti [6] suggested a general mathematical form of the gravitational scalar potential of the spheroidal Earth and how to estimate some of the parameters in it for use in the study of satellite orbits. Recently, [7] the exact and complete Newton s universal gravitational fields interior and exterior to a homogenous oblate spheroidal body were derived. In this article, we use the recent expression [7] for the gravitational scalar potential exterior to a static homogenous oblate spheroid to compute approximate values of the gravitational scalar potential for the oblate spheroidal Sun and planets. Table : Oblateness of odies in the Solar System []. ody Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Oblateness 9x The Pacific Journal of Science and Technology 663 Volume. Number. May 9 (Spring)

2 THEORETICAL ANALYSES The Cartesian coordinates ( x, yz, ) are related to the oblate spheroidal coordinates ( η, φ, ) as in following equations [8]; x= a( η ) ( + ) cosφ () y= a( η ) ( + ) sin φ () z = aη (3) where a is a constant parameter. Now, consider a Cartesian coordinate system fixed at the middle of the oblate spheroidal body as shown Figure. Now, let us consider the Sun and planets to be static homogenous massive oblate spheroids. It is clearly seen from Figure that the x-coordinate point on the surface of the static oblate spheroidal body corresponds to its equatorial radius and its z-coordinate point on the surface correspond to its polar radius. Thus, on the equator and at the surface of the static homogenous oblate spheroidal body ( =, η =, φ = ) ; equation () becomes: ( ) x = a + (4) o where x is the equatorial radius of the body. Also, at the surface along the polar line( =, η = ) ; z a = (5) z where is the polar radius of the body. The polar and equatorial radii of the Sun and planets are well known [9]. Substituting the values of the polar and equatorial radii of the Sun and planets into equations (4) and (5) and solving the equations simultaneously yields the constants and a for the oblate spheroidal bodies in the universe as given in Table. Figure : Static Homogenous Oblate Spheroidal ody. The Pacific Journal of Science and Technology 664 Volume. Number. May 9 (Spring)

3 Table : Values for the constants and a for the Static Homogenous Oblate Spheroidal Sun and Planets ody Equatorial radius(x 3 m) Polar Radius(x 3 m) a( m) Sun 7, 699, x 6 Mercury,44, Venus 6,5 6,5 - - Earth 6,378 6, x 5 Mars 3,396 3, x 5 Jupiter 7,49 66, x 7 Saturn 6,7 53, x 7 Uranus 5,56 4, x 6 Neptune 4,76 4, x 6 The gravitational scalar potential exterior to a static homogeneous spheroid [7] is given as: f( η, ) = Q ( i) P( η) + Q ( i) P( η ) (6) where QO and Q are the Legendre functions linearly independent to the Legendre polynomials P and P respectively. and are constants given by: and 4π Gρa = 3Δ = ( ) 9 P i d = (7) 4πGρ a d Δ (8) where and Δ are defined as Δ d ( ) Δ = Q i d = (9) G is the universal gravitational constant. Also, it may be noted [8] that : Q and () t t + = ln t () t + 3 Q () t = ( 3t ) ln t 4 t + () Consequently, it can be shown that the Legendre functions and Q can be expanded to give: QO 3 5 Q ( i) = i and i Q ( i) = Thus, from equations (9) and (3) we have (3) (4) and d d Δ= Q P( i) P( i) Q ( i) O d d = = () Δ = i (5) The Pacific Journal of Science and Technology 665 Volume. Number. May 9 (Spring)

4 Considering only the first two terms in the expansion of Equation (5) (that is approximation 4 of Δ to the order of ) and substituting in equation (7) gives: 4π Gρ a 3 5 ( + ) i Δ (6) Similarly, approximating to the order of and substituting in Equation (8) gives: 5 4πGρa i ( + 3 )( 7+ ) (7) 4 The mean density ( ρ) for various bodies in the universe is given by [9]; the universal gravitational constant is well known to be given as; G = 6.7x Nm kg and with the values of and a computed above (Table ), we get the values for the constants and for the static homogenous oblate spheroidal Sun and planets from Equations (6) and (7) as shown in Table 3. y considering the first two terms of the series expansion of the Legendre functions, Equations (3) and (4); Equation (6) can be written more explicitly as; ( ) ( ) f η, 3 i 3 3 ( 7 5 )( 3η ) i (8) Thus we can write; ( ) ( ) f η, 3 i 3 3 ( 7 5 ) i (9) f η, ) i () ( ) ( ) i ( as the respective expressions for the gravitational scalar potential along the equator and pole exterior to static homogenous oblate spheroidal bodies. Now, with the computation of the constant for the static homogenous oblate spheroidal Sun and planets, we can now evaluate the scalar potential along the equator and the pole at various points (multiples of ) exterior to the static homogenous oblate spheroidal Sun and planets. The results are shown in Tables 4 to and Figures and 3. Table 3: Values of the constants and for the Static Homogenous Oblate Spheroidal Sun and Planets. ody Mean Density, ρ ( kgm 3 ) ( ( Nmkg ) Sun x 3 i 8.938x 7 i Mercury Venus Earth x 8 i.73x 5 i Mars x 8 i x 5 i Jupiter x 9 i.595x 7 i Saturn x 8 i 5.767x 6 i Uranus x 8 i.68x 6 i Neptune x 9 i.785x 6 i The Pacific Journal of Science and Technology 666 Volume. Number. May 9 (Spring)

5 Table 4a: Gravitational Scalar Potential at Various Points along the Equator Exterior to the Static Homogenous Oblate Spheroidal Sun. Radial distance along the equator from center (km) f ( η, ) along the equator ( x (4.5) 7, (483.4),399, (74.56),99, (966.8),799, (7.6) 3,499, (449.) 4,99, (69.64) 4,899, (93.6) 5,599, (73.68) 6,99, (45.) 6,999, Table 4b: Gravitational Scalar Potential at Various Points along the Pole Exterior to the Static Homogenous Oblate Spheroidal Sun. Radial distance along the Pole from center (km) (4.5) 699, f η along the pole( x Nmkg ) (483.4),399, (74.56),99, (966.8),799, (7.6) 3,499, (449.) 4,99, (69.64) 4,899, (93.6) 5,599, (73.68) 6,99, (45.) 6,999, The Pacific Journal of Science and Technology 667 Volume. Number. May 9 (Spring)

6 Table 5a: Gravitational Scalar Potential at Various Points along the Equator Exterior to the Static Homogenous Oblate Spheroidal Earth. Radial distance along the equator from center (km) f ( η, ) along the equator 7 ( x (.) 6, (4.), (36.3) 9, (48.4) 5, (6.5) 3, (7.6) 38, (84.7) 44, (96.8) 5, (8.9) 57, (.) 63, Table 5b: Gravitational Scalar Potential at Various Points along the Pole Exterior to the Static Homogenous Oblate Spheroidal Earth. Radial distance along the pole from center (km) f η along the pole( (.) 6, (4.), (36.3) 9, (48.4) 5, (6.5) 3, (7.6) 38, (84.7) 44, (96.8) 5, (8.9) 57, (.) 63, x 7 The Pacific Journal of Science and Technology 668 Volume. Number. May 9 (Spring)

7 Table 6: Gravitational Scalar Potential at Various Points along the Equator and Pole Exterior to the Static Homogenous Oblate Spheroidal Mars. f ( η, ) along the equator 7 ( x f η along the pole( x (9.7) (8.34) (7.5) (36.68) (45.85) (55.) (64.9) (73.36) (8.53) (9.7) Table 7: Gravitational Scalar Potential at Various Points along the Equator and Pole Exterior to the Static Homogenous Oblate Spheroidal Jupiter. f ( η, ) along the equator f η along the pole( x 9 ( x (.64) (5.8) (7.9) (.56) (3.) (5.84) (8.48) (.) (3.76) (6.4) The Pacific Journal of Science and Technology 669 Volume. Number. May 9 (Spring)

8 Table 8: Gravitational Scalar Potential at Various Points along the Equator and Pole Exterior to the Static Homogenous Oblate Spheroidal Saturn. f ( η, ) along the equator 8 ( x f η along the pole( x (.97) (3.94) (5.9) (7.88) (9.85) (.8) (3.79) (5.76) (7.73) (9.7) Table 9: Gravitational Scalar Potential at Various Points along the Equator and Pole Exterior to the Static Homogenous Oblate Spheroidal Uranus. f ( η, ) along the equator f η along the pole( x 8 ( x (3.99) (7.98) (.97) (5.96) (9.95) (3.94) (7.93) (3.9) (35.9) (39.9) The Pacific Journal of Science and Technology 67 Volume. Number. May 9 (Spring)

9 Table : Gravitational Scalar Potential at Various Points along the Equator and Pole Exterior to the Static Homogenous Oblate Spheroidal Neptune. f ( η, ) along the equator 8 ( x f η along the pole( x (4.3) (8.6) (.9) (7.) (.5) (5.8) (3.) (34.4) (38.7) (43.) Scalar Potential(x Nmkg - ) Radial distance along the Equator(km) Figure a: Gravitational Scalar Potential at Various Points along the Equator Exterior to the Static Homogenous Oblate Spheroidal Sun. Gravitational Scalar Potential (x 7 Nmkg - ) Radial distance along Equator(km) Figure 3a: Gravitational Scalar Potential at Various Points along the Equator Exterior to the Static Homogenous Oblate Spheroidal Earth. Scalar Potential (x Nmkg - ) Radial distance along Pole(km) Figure b: Gravitational Scalar Potential at Various Points along the Pole to the Static Homogenous Oblate Spheroidal Sun. Gravitational Scalar Potential (x 7 Nmkg - ) Radial distance along Pole(km) Figure 3b: Gravitational Scalar Potential at Various Points along the Pole Exterior to the Static Homogenous Oblate Spheroidal Earth. The Pacific Journal of Science and Technology 67 Volume. Number. May 9 (Spring)

10 REMARKS AND CONCLUSION It is deduced from equation (8) that: ( η) lim f, = () This shows that the scalar potential diminishes as one moves away from the massive oblate spheroid. We can conveniently conclude from our computations (see Tables 4 to and Figures and 3) above that the theoretical values of the gravitational scalar potential exterior to the static homogenous oblate spheroidal Sun and planets decreases sharply in magnitude as we move away from the surface of the Sun and planets along the equator and the pole. The exponential decrease is similar to what is obtained when the Sun and planets are considered to be static homogenous spheres. Our computations agree satisfactorily with the experimental fact that the gravitational scalar potential exterior to any regularly shaped object has maximum magnitude on the surface of the body and decreases to zero at infinity. The immediate consequence of the results obtained above is that the exact shape of the planets and Sun was used to obtain the gravitational scalar potential on the surface at the pole and equator. Thus, instead of using the values obtained by considering the Sun and planets as homogenous spheres, our experimentally convenient values obtained can now be used. The door is now open for the computation of values for various gravitational phenomena exterior to the static homogenous oblate spheroidal Sun and planets along the equator and pole. Some of these phenomena include gravitational length contraction and time dilation. For more accurate results, additional terms of the series expansion of the Legendre polynomials can be included. Also, our analysis in this article can be expanded to evaluate the values of scalar potential at any point exterior to the oblate spheroid. The results obtained in this article can be compounded with the correction to Scalar potential of these massive bodies due to their rotation []. Recently, Ioannis and Michael [] proposed the Sagnac interferometric technique as a way of detecting corrections to the Newton s gravitational scalar potential exterior to an oblate spheroid. If this technique is developed in the near future, the theoretical study in this article will be confirmed experimentally. REFERENCES. Howusu, S.X.K and E.F. Musongong. 5. Newton s Equations of Motion in the Gravitational Field of an Oblate Mass. Galilean Electrodynamics. 6(5):97-.. Solar System Data oldata3.html 3. Stern, T.E Theory of Satellite Orbits. Astronomical Journal. 6(96). 4. Garfinkel, Problem of Quadratures. Astronomical Journal. 63(88). 5. O Keefe Improved Methods on Quadratures. Science. 8(568). 6. Vinti, J.P. 96. New Approach in the Theory of Satellite Orbits. Physics Review Letters. 3():8. 7. Howusu, S.X.K. 5. Gravitational Fields of Spheroidal odies-extension of Gravitational Fields of Spherical odies. Galilean Electrodynamics. 6(5): Arfken, G Mathematical Methods for Physicists, 5th edition. Academic Press: New York, NY Redmond, W.A. 8. The Solar System. Microsoft Student Encarta. Microsoft Corporation: Redmond, WA.. Ioannis, I.H. and Micheal, H. 8. Detection of the Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer. Progress in Physics. 3:3-8 AOUT THE AUTHORS Chifu E. Ndikilar is currently an Assistant Lecturer in the Department of Physics, Gombe State University, Nigeria. He is currently working towards his doctoral degree in General Relativity applied and applicable to gravitational phenomena. Adam Usman is currently a Senior Lecturer in the Department of Physics, Federal University, of Technology Yola, (FUTY), Nigeria. His research interests include Quantum Optics, Non Linear The Pacific Journal of Science and Technology 67 Volume. Number. May 9 (Spring)

11 Optics, Relativistic Mechanics, and Applied Physics. Osita C. Meludu is currently a Senior Lecturer in the Department of Physics, Federal University, of Technology Yola, (FUTY), Nigeria. His research interests are Applied Geophysics, Health Physics, and Relativistic Mechanics SUGGESTED CITATION Chifu, E.N, A. Usman, and O.C. Meludu. 9. Gravitational Scalar Potential Values Exterior to the Sun and Planets. Pacific Journal of Science and Technology. (): Pacific Journal of Science and Technology The Pacific Journal of Science and Technology 673 Volume. Number. May 9 (Spring)

Gravitational Field of a Stationary Homogeneous Oblate Spheroidal Massive Body

International Journal of Theoretical Mathematical Physics 08, 8(3): 6-70 DOI: 0.593/j.ijtmp.080803.0 Gravitational Field of a Stationary Homogeneous Oblate Spheroidal Massive Body Nura Yakubu,*, S. X.