Orbital Mechanics Part 2: Orbital Geometry and Properties

Welcome back, dear readers. Today, on AeroIntellect, we are going to continue to dive deeper into the vast world of Orbital Mechanics. Last week, we looked at Kepler's Law, which describes the planetary motion. Today, we are going to talk about the different features and formation of Orbits.

To begin with, an orbit is a curved path described by any body in its revolution around another body. It is formed when an object passes by a planet with a high velocity, which is further attracted to that planet due to gravity, causing the object to be pulled toward the center of the planet.

When the object reaches orbit, one of three things can happen:

•⁠ ⁠If the forces balance (push and pull), a stable orbit is formed.

•⁠ ⁠If the velocity of the object overcomes gravity, an open orbit occurs, allowing the object to fly by.

•⁠ ⁠If gravity overcomes the object's velocity, a collision with the planet occurs.

The first case is commonly used with satellite placement in the Geostationary orbit (GEO, located approximately 35,786km above ground) and Low Earth orbit (LEO, located much closer to Earth's surface, typically ranging from 160 to 2,000km). On the other hand, the second case is quite common when spaceships or satellites want to gain additional velocity while minimizing fuel and energy consumption or orbit transfers. Some examples include Voyager 1 and 2 gaining that extra bit of velocity from conducting fly-bys off Jupiter and Saturn's orbits, as shown in the image.

Geometry of Conic Sections

A conic section is the intersection of a plane and a double right circular cone. The four conic sections shown in the figure are considered orbits. In Newtonian gravity, when one object orbits another, the path of the orbiting object will always be one of these conic sections: Circle, Ellipse, Parabola, and Hyperbola. Each conic section has two foci, and the gravitational center of attraction coincides with one focus for all orbital motion.

Looking back at the Voyager 1 and 2 examples, they conducted a fly-by off Jupiter's orbit, which means they were likely to take the hyperbola pathway to gain that gravity assist and increase their speed.

Classical Orbital Elements

Each orbit of any shape or size will have six elements that can be used to identify it uniquely:

1.⁠ ⁠Eccentricity (e)

Eccentricity, e, is a fixed constant for a conic section and indicates the roundness or flatness of an orbit. It is calculated by dividing the distance between the foci by the length of the central axis.

2.⁠ ⁠Semi-major axis (a)

Kepler's first law of planetary motion states that a planet orbits the sun in an elliptical orbit. Therefore, there will be a major axis (a) and a minor axis (b), as shown in the diagram below.

3.⁠ ⁠Inclination (i)

Inclination is the angle in degrees between the orbital plane and the equatorial plane. In short, it is the tilt of the orbit.

4.⁠ ⁠Longitude of the Ascending Node (Ω)

Longitude of the ascending node is the point at which an orbit intersects a plane of reference, also known as the ecliptic plane, travelling north.

5.⁠ ⁠Argument at Periapsis (ω)

Argument at Periapsis is simply the angle between the body's ascending node and its Periapsis.

6.⁠ ⁠True Anomaly (f)

True Anomaly highlights the position of the object in the orbit. It is the angle between the line that connects the orbiting object to the body and the radius of the Periapsis (the side closer to the main body).

In summary, when one object orbits another, the path of the orbiting object will always be one of these conic sections: Circle, Ellipse, Parabola, and Hyperbola. An orbit will always have these six different elements that are used to define it uniquely: eccentricity and semi-major axis, which represent an orbit's size and shape; inclination and longitude of ascending node, which highlight its orientation in space, and the argument of Periapsis and true Anomaly, which showcase the position of the Periapsis.

Thanks for sticking around till the end. If this sparked your curiosity, I’d love to hear your thoughts in the comments.