Universal Journal of Geoscience 5(2): 33-39, 2017 http://www.hrpub.org DOI: 10.13189/ujg.2017.050203 Numerical Model for the Orbit of the Earth S. Karna1,*, A. K. Mallik2 1Physics Department, Tri-Chandra M. Campus, Tribhuvan University, Nepal 2Physics Department, Tribhuvan M. Campus, Tribhuvan University, Nepal Copyright©2017 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract In this paper, we constructed a 3-D numerical the main reference for specifying the position of the other model of the Earth-Sun geometry. Our model defines Earth’s celestial objects in our solar system. The perihelion and orbit as an inclined plane of spherically symmetric system. aphelion distances are said to be lie on the same ecliptic We calculated the degree of the tilt of the Earth orbit to the plane, though the shape of the Earth’s orbit is not circular. ecliptic plane by converting from ecliptic frame of reference Since the circular shape appears elliptical when viewed at an to the orbital frame of reference and then we made all the angle, means that for elliptical shape of orbit of Earth around measurement. Initial inputs of our model are aphelion and the Sun, the plane in which Earth orbits must be inclined to perihelion parameters. It is interesting to examine that our some angle with the ecliptic plane. In another word, the plane results obtained from Earth inclined orbit are same that of the Earth’s orbit must be in orbital plane, other than the observed value from Earth’s circular orbit. In other words, ecliptic plane. values of the axial tilt of Earth and Sun, the time taken for the The article is organized as follow: In Section 3, we Sun to move from vernal equinox to autumnal equinox and describe our 3-D numerical model of the Earth-Sun then back to the vernal equinox does not change. Moreover, geometry. In Section 4, we present the results: values of the we were also able to derive mathematical relations for axial tilt of Earth and Sun, the time taken for the Sun to move finding the length of the apparent solar days throughout the from vernal equinox to autumnal equinox and then back to year. On introducing the new types of the length of the day, the vernal equinox. In Section 5, we discuss what we are called Saurya day, the rate of precession of equinox is able to draw from our 3-D numerical model and compare calculated. our results with existing theory. In Section 6, we summarize our results and present our conclusions. Keywords Orbit of the Earth, Earth-Sun Geometry, Equinox, Perihelion, Aphelion, Rate of Precession 2 . Numerical Model The angle of cosine of ratios of mean distance to aphelion 1 . Introduction distance (θ1) and perihelion distance to mean distance (θ2) are as follows: The Solar system consists of the Sun, planets, satellites of -1 o the planets, numerous comets, asteroids, meteoroids and the Cos (1/1.016710) = Cos 10.402169 = Cosθ1⇒ θ1 = o interplanetary medium. Planets and most of their (planet’s) 10.402169 satellites revolve in a nearly circular orbit around the Sun and -1 o and Cos (0.983290/1) = Cos 10.488961 = Cosθ2⇒ θ2 = at the same time, they rotate about their own axis. Earth 10.488961o revolves around the sun in an anti-clockwise direction and or, Mean value, θ= 10.445565o always points into space in the same direction. During the We took the average value of θ1and θ2 for our further revolution period of the Earth around the Sun, the distance calculation. between the Earth and the Sun is not always the same, i.e., the orbit of Earth deviates a bit from the circle. The aphelion 2.1. To Find the Inclination of the Earth’s Orbit (farthest) distance of the Earth is 1.016710 AU, and the perihelion (closest) distance is 0.983290 AU, where 1AU = For Earth-Sun Geometry only, 149,597,870.7 km. The mean distance between the Earth and Consider a fictitious Celestial sphere of radius Rc (= Sun is 1AU [1]. 149,597,870.7 km / Sin10.445565o) = 825,134,101.1 km In our solar system, plane of the Earth's orbit is known as centered on the Sun, with its equator running along the plane the ecliptic plane. The ecliptic coordinate system is used as of the ecliptic as shown in fig. 2. Normal to the ecliptic plane 34 Numerical Model for the Orbit of the Earth defines the ecliptic pole of the Solar System. This Celestial orbital plane is not circular, but deviates a bit from the circle. sphere is rotating uniformly about pole to the ecliptic, so the Thus the projection of this point CE" (lying on the Celestial reference frame fixed on it is an ecliptic frame of reference sphere) in orbital plane as CE, (center of Earth, zso = 0) ES. defines the actual orbit of the Earth around the Sun. The Let us consider another frame of reference OS, called the orbital plane is inclined at an angle λ with the ecliptic plane orbital frame of reference with its origin coincides with ES and the value of λ represents the tilt of the Earth's orbit to the but its coordinate axes inclined at an angle of λ as shown in ecliptic. fig. 1. (This figure is adapted from Mathur (1996) [2]). 2.2. Calculating λ (Tilt of the Earth's Orbit to the Ecliptic Plane) Let the Cartesian Coordinates of the point CE" are xse, yse, zse in frame ES and xso, yso, zso in frame OS. Then the transformation equations from ES to OS is xso = xsecosλ + zsesinλ (1) yso = yse (2) zso = - xsesinλ + zsecosλ (3) Let the spherical polar coordinates of the point CE" are specified by (Rc, θse, Φse) with respect to the axes xse, yse, zse in frame ES and CE"(Rc, θso, Φso) with respect to xso, yso, zso in frame OS. Here, θse and θso is the co-latitude i.e. the angle between CE"S' and zse- axis in frame ES and CE"S' and zso-axis in frame OS respectively. Φse and Φso is the longitudinal or azimuthal angle. Figure 1. Representation of the point CE" in ecliptic and orbital frame of reference The longitudinal or azimuthal angle (Φse) is measured by starting the revolution of the Earth from the point, where the o Let any point CE" lies at the co-latitude of 10.445565 on Earth is at a farthest distance to the Sun. the surface of the Celestial sphere, and from CE", CE"CE' Substituting the value of xse for aphelion distance and perpendicular to xseyse- plane is drawn, then the radial perihelion distance in the transformation equation and distance from the center of the celestial sphere (S') in its solving it, we get, λ = 0.176506 degree. celestial equatorial plane (S'CE') is Here, the value of λ represents the degree of tilt of the Earth's orbit to the ecliptic plane. Thus the deviation in the R o = R Sin θ = 149,597,870.7 km. 10.445565 c path of the Earth from the circle indicates that the Earth's This Celestial Sphere rotates uniformly, then the point CE' orbital plane is inclined at an angle of 0.176506 degree with rotates on the equatorial plane (or the ecliptic plane also) of the ecliptic plane. the celestial sphere (zse = 0) uniformly at the distance of 149,597,870.7 km. from the center. Here C'E is taken as the 2.3. Distance between Earth and Sun in Orbital Plane imaginary center of the Earth of radius 6371 km., with its equator running along the equator of the Celestial Sphere, For the given value of Rc = 825,134,101.1 km., θse o and the path followed by C'E in equatorial plane is the =10.445565 , zse = Rccosθse = 811,459,649 km., λ = o imaginary orbit of the Earth around the Sun. 0.176506 the value of xse, yse, zse, xso, zso, and Rso (the Again, from CE", CE"CE perpendicular to xsoyso- plane is distance between Earth and Sun in orbital plane) can be drawn. Then, the radial distance from the center of the calculated as follows in Table: 1; Celestial sphere in orbital plane is S'CE. The path of CE in Universal Journal of Geoscience 5(2): 33-39, 2017 35 Table1. Distance between Earth and Sun in orbital plane at different azimuthal angle (Φse). 2 2 1/2 Φse xse = R sinθsecosΦse yse = R sinθsesinΦse = yso xso = xsecosλ+ zsesinλ zso = - xsesinλ + zsecosλ Rso = (xso + yso ) 0o 149,597,870.7 0 152,096,948.3 810,994,946.5 152,096,948.3 30o 129,555,556.4 74,798,935.35 132,054,729.1 811,056,688.9 151,767,362.1 60o 74,798,935.35 129,555,556.4 77,298,367.9 811,225,372.5 150,863,116.3 90o 0 149,597,870.7 2,499,787.4 811,455,798.6 149,618,755.0 120o -74,798,935.35 129,555,556.4 -72,298,793.0 811,686,224.6 148,363,599.5 150o -129,555,556.4 74,798,935.35 -127,055,154.2 811,854,908.2 147,437,759.5 180o -149,597,870.7 0 -147,097,373.4 811,916,650.7 147,097,373.4 210o -129,555,556.4 -74,798,935.35 -127,055,154.2 811,854,908.2 147,437,759.5 240o -74,798,935.35 -129,555,556.4 -72,298,793.0 811,686,224.6 148,363,599.5 270o 0 -149,597,870.7 2,499,787.4 811,455,798.6 149,618,755.0 300o 74,798,935.35 -129,555,556.4 77,298,367.9 811,225,372.5 150,863,116.3 330o 129,555,556.4 -74,798,935.35 132,054,729.1 811,056,688.9 151,767,362.1 360o 149,597,870.7 0 152,096,948.3 810,994,946.5 152,096,948.3 o The distance between the Earth and the Sun (Rso) at any yso at Φse= 90.957464 and at this point xso = 0 ).