Mathematically Describing Planetary Orbits in Two Dimensions

This article describes use of Mathcad mathematical package to solve problem of the motion of two, three and four material points under the influence of gravitational forces on the planar motion and in three-dimensional space. The limits of accuracy of numerical methods for solving ordinary differential equations are discussed. Usual concept of Kepler hours with uneven movement arrows in planet. Projectile motion may be thought of as an example of motion in space-that is to say, of three-dimensional motion rather than motion along a line, or one-dimensional motion. In a suitably defined system of Cartesian coordinates, the position of the projectile at any instant may be specified by giving the values of its three coordinates, x(t), y(t), and z(t). Citation: Ramanakumar V (2018) Mathematically Describing Planetary Orbits in Two Dimensions. J Appl Computat Math 7: 414. doi: 10.4172/21689679.1000414


Introduction
One of the biggest applications of mathematics is in the field of astrophysics, which is the study of the mechanical motion and behavior of celestial bodies [1]. Using math, we can describe any celestial phenomena using numerical quantities which represent certain physical characteristics [2][3][4][5].
In this theoretical paper, mathematically describe the orbits of natural satellites and planets, and use this to calculate certain characteristics of the orbits of the planets within our solar system [6][7][8][9]. Once done this using mathematical method, calculated values to literature values already determined. The properties of a planetary orbit plan on investigating are: At first glance, the motion of planetary bodies may seem complex and hard to understand [10]. However, believe Math is a black and white subject that simplifies certain situations to numbers and equations. In this report, it attempt to simplify planetary revolutions through mathematical concepts.
Before proceeding with this investigation, it is important to introduce and layout the basic variables and quantities that will be used in this exploration [6]. These quantities are included in the diagram below: • a: This represents the semi-major axis, which is the largest distance between the center of the ellipse and the curve of the orbit.
• b: This represents the semi-minor axis, which is the shortest distance between the center of the ellipse and the curve of the orbit.
• c: This is the distance between the center of the orbit and the focus of the orbit, which is where the Sun would be in a planetary orbit.

e:
Although not included in the diagram, this quantity is known as eccentricity, which is the measure of the deviation of the shape of an orbit from that of a perfect circle (a measure of how elliptical an orbit is). The higher the eccentricity of an orbit, the more elliptical it is. It can be calculated using the simple formula: c e a = (for an orbit into parametric 1).
After some research from wolframalpha.com and windows2universe.org, Tabulated these values for each planet. All values are given in astronomical units.

Cartesian Form of an Orbit
Given the fact that any planetary orbit is an ellipse, the Cartesian form will follow the basic form of an elliptical curve: Where h is the horizontal displacement of the curve, k is the vertical displacement of the curve a is the radius of the curve in the x direction, and b is the radius of the curve in the y direction.
Generally, in a planet's orbital curve, there is no displacement in the y direction; the focus only shifts away from the origin in the x direction. Therefore, the value of k in the above equation for a planetary orbit becomes 0. Now, we need to plug in the variables introduced in the previous section. Since c is the distance between the focus, which represents the origin O, and the center of the curve, this can be denoted as the horizontal displacement of the curve, so we can replace h with c. The variable a represents the semi-major axis, or the radius in the x direction, and will take the position of the variable a in the general form. The variable b represents the semi-minor axis, or the radius in the y direction, and will replace the variable b in the general form equation.
With the given replacements and adjustments, we get the general Cartesian form for the elliptical orbit of a planetary body:

Parametric Form of an Orbit
The parametric form of an equation expresses both variables in a function, x and y, in terms of another distinct variable. In this paper, we will refer to this variable as t (not to be confused with the variable for time).
We can convert our Cartesian equation for an orbit into parametric form in a few simple steps:

Polar Form
Polar form equations are in terms of r, the distance from the origin, and θ, the angle between the line that connects a point on the curve and the origin and the positive x-axis.
It attempted to directly convert the Cartesian equation to the polar equivalent, but was quite unsuccessful. It decided to manipulate some key characteristics of an ellipse to derive the polar form. The two characteristics of an ellipse to keep in mind are as follows: 1. In an ellipse, with two foci (equidistant from the center/origin), the sum of the lines F 1 P and F 2 P are always equal to 2a, where P is any point on the ellipse (Figures 1 and 2).

2.
If the first diagram is altered slightly, so that point P is on the semi-minor axis, both F 1 P and F 2 P are equal in length at a, and 2 right angle triangles are formed between the center/origin, both foci and point P. Using Pythagoras theorem, we can deduce that b 2 =a 2 -c 2 ( Figure 3). Now, if we were to set F 1 as the origin and let F 2 some point on the positive x-axis denoted by (2c, 0), we get the in Figure 4.
We take w to be a vector connecting O and P. The vector connecting F 2 and P is represented by w-2ci (the i is added to give direction to the magnitude 2c). The sum of the magnitude of both vectors is equal to 2a, according to the first statement made above: Since the magnitude of w is equal to r (recall that r is the distance between the origin and the curve), we get: Now we need to split the w-2ci vector into x and y components. We can denote the magnitude of the y component simply as y, but for the x component, we need to take into account 2ci, as this is in the x direction (Table 1). Therefore, the magnitude of the x component is x-2c: The magnitude of the second vector is nothing but the square root of the sum of the squares of the x magnitude and y magnitude: Remember that in polar equations, x=γ cos θ and y=γ sin θ. Plugging these into the original equation gives:

Length of the Curve of an Orbit
Now that we have the equations of the curve, we can calculate the length of this curve, which represents the length of each planet's revolution. But first, let us derive the formula for length of a curve. If we have a graph of a function f(x), then the distance between any two distinct points on the curve is given by the following formula:

Note:
The unit of the axes of the above graphs are Astronomical Units (AU) If we were to bring these two points infinitely close to each other along the curve, the segment on the curve would be very close to a straight line, andΔx→0 and Δy→0. Due to this change, the above equation becomes: Plugging this into the parametric equation formula, we get the general formula for the length of a polar curve: With these three formulas, we can give the general equation for the length of a planetary elliptical orbit. Using Cartesian form: To get dy dx , we implicitly differentiate our general Cartesian equation for a planetary orbit: Plugging this into the length of a curve formula gives us: In order to substitute for y 2 , we need to find y 2 in terms of x: Then, we need to simplify the denominator of In order to further simplify, you need to differentiate r with respect to θ using the quotient rule: We can input this simplification into our length of a polar curve formula. Integrating with the limits [0, 2π], we get: We can also calculate the volume covered by a planet in a single orbit by multiplying the area of a circle with the same radius as the planet into the length formulas derived above, similar to how a prism's volume is calculated by multiplying the cross sectional area and the length of the solid. Using this formula gives the volume as either:

Mercury's Orbit
To see whether my derivations were accurate, it plugged in already calculated values for the constants a, b, and c and compared the result with already calculated values found on reliable sources on the internet.
My first test was on the orbit of the planet Mercury, the first planet in the solar system which happens to have the largest value of e, eccentricity.
After in depth research I found that the approximated, widely accepted values for a, b are as follows: • a: 0.3870 astronomical units (a unit of distance used in astronomy which is equal to the average distance between the Earth and Sun) or 5.8 × 10 7 km • b: 0.3788 astronomical units or 5.7 × 10 7 km.
We can calculate the total distance covered by a planet in a single revolution based off of the length of a curve formula given above. We can choose to use either the Cartesian or the parametric form. Since the parametric form is slightly simpler to handle, let's use that equation. Plugging in the above values gives us:

Area Covered by the Elliptical Orbit of a Planet
If we visualize the elliptical orbit as a graph on a Cartesian plane, we can calculate the area covered by the curve, or in the words, the planet,  x c a ab  abπ As expected, the above formula is the formula for the area of any ellipse, where a is the maximum radius and b is the minimum radius (Table 4).

Conclusion
This proof has become monumental for its ingenuity and its immense intellectual motion of Kepler concept. Therefore, the author considered it worthwhile to attempt to reach Mathcad conclusion via an alternate route, so that the beauty of the journey along this path be exposed to its full extent. Subsequently, based on the law of universal gravitation and his other great discovery, the concept of astrophysics, one-dimensional motion proved theoretically that planets do indeed prescribe elliptic trajectories around the sun. The position and velocity vectors at a particular instant of time discovered generating an  ephemeris (right ascension and declination as a function of time) from the orbital elements in the real Two-dimensional situation.