In this paper we calculate the volume, mass, gravitational attraction to the Earth, angular momentum the orbit of the Trojan asteroid (TK<sub>7</sub>) [1]. In this paper, we use classical Newtonian mechanics to analyse some of the physical and orbital properties of the Trojan asteroid, which are still experimentally unknown. The asteroid should remain in Earth’s orbit for the next 100 years. We conclude by providing informed estimates of 2010 TK<sub>7</sub>’s yet unknown physical properties: i.e. mass, volume, gravitational attraction to Earth and angular momentum.
2010 TK<sub>7</sub> Trojan Asteroid Universal Gravitation Horseshoe Orbit Angular Momentum
Newton’s Laws of Motion1. Introduction
TK7 is located in the L4 Lagrangian point, 60˚ ahead of the Earth in orbit of the Sun, and was first observed in October 2010 by NASA’s Wide-field Infrared Survey Explorer (WISE) [2].
From the data received about TK7 from NASA’s WISE satellite we approximate the volume of TK7 as circa 1.1310 × 108 m3. From NASA’s JPL Small Body Database Browser [3], 2010 TK7’s orbital properties have the: aphehelion of 1.19095 AU, perihelion of 0.80955 AU, semi-major axis of 1.00025 AU, eccentricity of 1.9066, an orbital period of 365.394 days, a mean anomaly of 217.384˚, inclination of 20.8656˚, a longitude of ascending node of 96.5426˚ and an argument of perihelion of 45.7551˚. It is currently 80,000,000 km from the Earth. TK7’s dimensions are believed to be 300m in diameter [1].
In Section 2, we will review the current literature on Trojan asteroids, and on Lagrangian points. We will not use the three-body problem [4]1. Instead, in Section 3, we use classical Newtonian mechanics to determine the gravitational attraction of the Trojan to Earth, and we calculate its angular momentum based on its known physical and orbital properties. In Section 4 we provide a synoptic table of the numerical results of this paper. Section 5 concludes the paper discussing its results and possible future uses.
2. Review of Literature on Trojan Asteroids and Lagrangian Points
2010 TK7 was discovered in October 2010 by NASA’s satellite WISE. Once discovered, the asteroid has been followed by the Canada-France-Hawaii Telescope [5]. Figure 1 evaluates the data of orbital properties. These data were published by Connors M. et al. 2011 [1].
The suspicion that Earth had a Trojan asteroid had however been present for some time [1]. 2010 TK7 is the first Trojan asteroid to have been confirmed to be orbiting the Earth. In the past, Trojan asteroids have been found to orbit around many of the gas giants, such as Jupiter or Saturn, however 2010 TK7 is the first known to have begun orbiting the Earth.
Most Trojan Asteroids originate from the Kuiper Belt [6], and follow the orbit of a planet, 60˚ behind, or 60˚ ahead, with the Sun as the pivot. Joseph-Louis Lagrange determined these points as coordinates where Trojan asteroids can be located. Lagrange also worked on the three-body problem [4], where a gravitationally based calculation is made for a system with three celestial bodies instead of two. Since it is impossible to find an exact answer to the three-body problem, we are not exploring this further (Our limitations have been explained by chaos theory [7]). An example of post-Newtonian work on the three-body problem can be found in Fawzy A. Abd ElSalam, et al. (2011) [8]
The Lagrangian points are commonly used to calculate many of the orbital properties of celestial bodies [9].
This paper focuses on celestial planetary mechanics, most of which was developed by Sir Isaac Newton. Other notable examples of modern work on planetary mechanics (although based on a different, Quantum mechanical model) were by Nie Q. X. (2011) [10].
3. Analysis Based on Newtonian Physics3.1. Universal Gravitation
Although the mass of the asteroid TK7 is unknown, we use the average density of Trojan asteroids (2800 kg/m3) [11] to calculate its estimated mass (m) (based on spherical approximations) to be 1.4844 × 108 kg (based on the fact that the radius is 150 m) by
By applying Newton’s Law of Universal Gravitation [12], we obtain the gravitational attraction (F)
3.2. Angular Momentum
The basic law of angular momentum (L) states that L = r × p, r being the distance between the body, and the pivot around which it is orbiting, and p being the linear momentum of the body. We calculate the momentum of TK7 as:
From the data published in “Earth’s Trojan Asteroid” [1], we use the data of the distance between 2010 TK7 and the Earth; 80,000,000 km. Therefore:
Now, by substituting Equations (3) and (4), (giving us p and r) into the equation for angular momentum L = r × p, we obtain:
3.3. Explanation of <xref ref-type="fig" rid="fig1">Figure 1</xref>
Figure 1 [1] shows the orbital path of 2010 TK7 through the inner Solar system. The graph was compiled with data derived from the images taken by the Canada-FranceHawaii Telescope [5], and from NASA’s WISE satellite [2]. Graphical approximation methods suggest that the orbit can be represented by the function:. The horseshoe orbit loosely follows the function of, yet instead of orbiting in a standard fashion, such as other moons or planets, it follows the typical low-eccentricity path of a horseshoe orbiter [13]. The discovery of 2010 SO16 was the first horseshoe orbiter of the Earth. The defining aspect of horseshoe orbiters of planets is their low eccentricity, which in this case is 1.9066 [3].
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. The horseshoe orbit loosely follows the function of
, yet instead of orbiting in a standard fashion, such as other moons or planets, it follows the typical low-eccentricity path of a horseshoe orbiter [