Basics of Orbits Parameters and Elements

Table of Contents

Orbital elements are the parameters required to uniquely identify a specific orbit. The path of satellite revolving around the earth is known as orbit. This path can be represented with mathematical notations. A satellite orbit is always in a plane around the heaviest body.

In the 17th century, Johannes Kepler and Isaac Newton discovered the basic physical laws governing orbits; in the 20th century, Albert Einstein’s general theory of relativity supplied a more exact description.

An orbit is completely described by six geometric properties called its elements, from them the future positions of the planet can be calculated. This system has six degrees of freedom. Orbital elements are the parameters, which are helpful for describing the orbital motion of satellites.

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Basic Orbital Parameters and Elements

There are six parameters to describe the orbit. These six parameters are called the Keplerian elements or orbital elements.
Three of the parameters will used to describe how the plane looks like and the position of the satellite on the ellipse.
The other three parameters will used to describe how that plane is oriented in the celestial inertial reference frame and where the satellite is in that plane.

There are six orbital elements define the orbit of earth satellites. Therefore, it is easy to distinguish one satellite from other satellites based on the values of orbital elements. They are

Semi major axis
Eccentricity
Mean anomaly
Argument of perigee
Inclination
Right ascension of ascending node

Semi-Major and Semi-Minor Axis

The heaviest body in the Keplerian orbit (seen as a star), is in one of the focal points of the orbiting satellite.
The length of Semi-major axis defines the size of satellite’s orbit. It is half of the major axis. This runs from the center through a focus to the edge of the ellipse. So, it is the radius of an orbit at the orbit’s two most distant points. If circular orbit is considered as a special case, then the length of semi-major axis will be equal to radius of that circular orbit.
The point where the satellite is the closest to the central body is called the periapsis, with the length to the central body usually denoted as $r_\text{p}$ .
The point where the satellite is the farthest away is called the apoapsis, and has the associated length $r_\text{a}$ .
The semi-major axis is on the line segment between the periapsis and the apoapsis, and it is half of the distance between them, that is $a = (r_\text{p} + r_\text{a})/2$ .

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Eccentricity

The value of Eccentricity (e) fixes the shape of satellite’s orbit. This parameter indicates the deviation of the orbit’s shape from a perfect circle.
If the lengths of semi major axis and semi minor axis of an elliptical orbit are a & b, then the mathematical expression for eccentricity (e) will be,

e=√a2−b2 /a

The value of eccentricity of a circular orbit is zero, since both a & b are equal. Whereas, the value of eccentricity of an elliptical orbit lies between zero and one.

Mean anomaly

Mean anomaly (M) gives the average value of the angular position of the satellite in its orbit with reference to perigee.
At perigee, the Mean Anomaly is zero, it increases to 180 degrees at apogee, then back to perigee at 360 degrees.
If the orbit is circular, then Mean anomaly gives the angular position of the satellite in the orbit. But, if the orbit is elliptical, then calculation of exact position is very difficult.

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Argument of perigee

Satellite orbit cuts the equatorial plane at two points. First point is called as descending node, where the satellite passes from the northern hemisphere to the southern hemisphere.
Second point is called as ascending node, where the satellite passes from the southern hemisphere to the northern hemisphere.
The argument of perigee, ω , is the angle measured from the ascending node (or line of nodes) to the perigee in the counter clockwise sense. 0º < < 360º ω . It determines the orientation of the orbit inside its plane.
Argument of perigee (ω) is the angle between ascending node and perigee. If both perigee and ascending node are existing at same point, then the argument of perigee will be zero degrees.
Argument of perigee is measured in the orbital plane at earth’s center in the direction of satellite motion.

Inclination

The angle between orbital plane and earth’s equatorial plane is known as inclination (i).
The inclination, i , is the angular distance of the orbital plane with respect to a reference plane (the reference plane is the equatorial or ecliptic plane). Normally stated in degrees 0º<180º<i . It is measured at the ascending node with direction being east to north.

Based on the angle of inclination, orbits ae divided into four types.

Equatorial orbit − Angle of inclination is either zero degrees or 180 degrees.
Polar orbit − Angle of inclination is 90 degrees.
Prograde orbit − Angle of inclination lies between zero and 90 degrees.
Retrograde orbit − Angle of inclination lies between 90 and 180 degrees.

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Right ascension of ascending node

Right Ascension of ascending node (Ω) is the angle between line of Aries and ascending node towards east direction in equatorial plane. Aries is also called as vernal and equinox.
Satellite’s ground track is the path on the surface of the Earth, which lies exactly below its orbit.
The ground track of a satellite can take a number of different forms depending on the values of the orbital elements.

Orbital Velocity

Orbital velocity of satellite is the velocity at which, the satellite revolves around earth. Satellite doesn’t deviate from its orbit and moves with certain velocity in that orbit, when both Centripetal and Centrifugal forces are balance each other.

Alternative parameter,

Object	Elements used
Major planet	e, a, i, Ω, ϖ, L₀
Comet	e, q, i, Ω, ω, T₀
Asteroid	e, a, i, Ω, ω, M₀
Two-line elements	e, i, Ω, ω, n, M₀

Orbital Parameters of Earth

As viewed from above the Northern Hemisphere Earth orbits the Sun is at an average distance of 149.60 million km in a counter clockwise direction.
Orbit takes 365.256 days (1 sidereal year) to complete one rotation, during which time the Earth has traveled 940 million km (584 million mi).
Earth’s orbit is an ellipse with the Earth-Sun barycenter as one focus and a current eccentricity of 0.0167.
The center of the orbit is relatively close to the center of the sun, hence the eccentricity value is close to zero.
The radius of the earth’s orbit around the sun (assumed to be circular) is 1.50×108 1.50 × 10 8 km, and the earth travels around this orbit in 365 days.

There are three known parameters that describe the Earth orbit around the sun,

(1) eccentricity

(2) axial tilt (or obliquity)

(3) time of perihelion (or precession)

Eccentricity

Eccentricity describes the degree of variation of the Earth’s orbit around the Sun from circular to more elliptical.
Eccentricity has two main periodicities, one cycle with an average of ~100,000 years and a longer cycle with a periodicity of ~413,000 years.

Axial tilt (or obliquity)

Axial tilt or Obliquity describes the tilt of the Earth’s axis in relation to its orbital plane, which ranges from 22.1–24.5 degrees with a periodicity of ~41,000 years.

Time of perihelion (or precession)

Time of perihelion or Precession describes the motion of the Earth’s axis of rotation, which does not point towards a fixed direction in the sky through time.
The axis of rotation describes a clockwise circle in space, like the spinning of a wobbling top, with a periodicity of 19,000–23,000 years

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