Space Computation: Computation behind the planet's orbit

Have you ever want to calculate a planet's orbit?

One of the easiest ways to determine the shape of planetary orbits in inverse square force field is to use the rules of calculating gains:

(1)

In the beggining of the coordinates x, y placed center of gravity (eg the Sun). The distance from the beggining is marked as the Pythagorean Theorem is known from:

(2)

The law of universal gravitation can be written as it follows:

(3)

It is the vector form of this law, where

the individual unit vector directed opposite to the radius vector. With this form, we can provide the coordinates of the force:

(4)

We now prove that the equality of two increments means constancy of the difference. We will need it later to determine the equation of the orbit.A simple equation:

(5)

carries important information: the left side is the square of 1 / r, while the right side is opposite r^2. Using the above-mentioned compounds (1), we calculate the growth equation (5). We obtain the following results:

(6)

Using equation (2) we can calculate the increase

for the right side (6):

(7)

Substituting equation (7) into equation (6) and dividing both sides by 2 / r, we obtain the following expression:

(8)

Now we calculate the increase in the following equation:

(9)

The right side of (9) is treated as the product of two factors. Using the above calculated inverse growth (8), we obtain:

(10)

Causing the expression to the common denominator and reducing similar terms, we get:

(11)

The expression in brackets

is the momentum of the velocity vector - twice the areal velocity planet (at constant volume of traffic). Now multiply the identity (11) by the gravitational constant and the mass of the Sun (GM). The term describes the acceleration -GMy/r3, multiplying it by the time we get growth speed increase:

= (-GMy/r³)

. Finally we get the following equation:

(12)

Since the expression in brackets

is constant in time, we can write the product of velocity and increase the expression of the form:

(13)

We can see clearly now that the equality of two increments means that the difference remains constant. This proposal will use now to our main goal - determine the equation of the orbit.We can write the following expression:

(14)

where Ax is the value determined by the initial conditions.Similarly, the y-coordinate we have:

(15)

Now we will make some transformations: multiply equation (14) by x, equation (15) by y, and then add them together:

(16)

Knowing that

we get:

(17)

Using the formula for the angular momentum of the planet:

we obtain the equation of the orbit:

(18)

Raising to the square of both sides of this equation and making the substitutions (2), we get:

(19)

Depending on initial conditions, which determine the constants for a given orbit and the size of J, Ax, Ay, we obtain a circular, elliptical, parabolic, hyperbolic curve.To determine the constant parameters J, Ax, Ay, we choose the simplest position of the body (planet) to the center of gravity (Sun) - in this case, we x-axis coordinate system to the point where the tangent to the path is perpendicular to the radius vector (perihelion - when the point is closest to the center of gravity, Aphelion - when it is just the largest). Then, both y and vx, are at this point equal to zero:

(20)

We received the first permanent Ay = 0R denotes the distance from the Sun at y = 0, and replace with letter v. vx we get the expression defining the constants Ay and J:

(21)

Substituting all the constants determined in equation (18) orbit and dividing the result by GM we get:

(22)

R and V is the size of the initial conditions given to the body moving on the orbit. Now substitute the expression in brackets in equation (22) constant ε:

, then

We now have the equation:

(23)

where ε is known. eccentricity of the ellipse.
For the initial speed equal to the first space velocity

, the eccentricity of the equation disappears and it becomes a simple equation of a circle of radius R, while for lower speed, eccentricity takes values: -1 ≤ ε <0 We see that the radius R, the initial distance is the maximum for a given circuit. If you shoot an body perpendicular to the position vector of velocity less than the first space velocity, the body will be reduced to its distance from the center.

But if we give the body a greater speed, it will initially increase its distance from the center of gravity of the system, on the track of the body depends on the value of ε.

Raising equation (23) to the square we get:

(24)

In the first case we consider ε = 1 (second cosmic speed): get a simple equation of a parabola:

(25)

In turn, for speeds less than the second cosmic velocity we have ε <1Using this assumption, we transform equation (24) to canonical form and divide it by the intercept: