Reference: https://catalina.lpl.arizona.edu/faq/what-are-orbital-elements

🛰️ Transforming JPL Horizons Ephemerides: 

From Reference Frames to Refined Orbits

The Challenge: JPL Horizons provides satellite ephemerides in various reference frames—sometimes the ecliptic plane, sometimes a planet's equatorial plane, and for distant satellites, often relative to the Laplace plane. To create accurate visualizations and perform orbital mechanics calculations, we need to transform these diverse coordinate systems into a unified reference frame.

(Orbital elements image credit: Cosmos Also a great resource!)

Astronomers determine orbital parameters by taking several observations of an object's position in the sky over time and then using mathematical methods, grounded in the laws of gravity, to find the unique orbit that best fits those data points.

The Raw Data: Observation 🔭

The process begins with collecting observational data. The most fundamental measurement is astrometry, which is the precise measurement of an object's position in the sky against the background of "fixed" stars.

• What is measured? For each observation, astronomers record two key things:

  1. Position: The object's location on the celestial sphere, given as two angles: Right Ascension (RA) (the celestial equivalent of longitude) and Declination (Dec) (the celestial equivalent of latitude).

  2. Time: The exact time of the observation.

• How many observations? To define a unique orbit, a minimum of three high-quality observations taken at different times is required. In modern astronomy, dozens or even hundreds of observations are used to refine the orbit with incredible precision.

The Calculation: From Points to a Path 🛰️

You can't see the orbit itself; you only see points where the object was. The job of the orbital mechanic is to find the one elliptical (or hyperbolic) path that connects those dots according to the law of universal gravitation.

The Classical Approach (Gauss's Method)

The foundational method was developed by Carl Friedrich Gauss in the early 1800s to find the orbit of the newly discovered asteroid Ceres. While computationally intensive by hand, it's a brilliant mathematical process that essentially:

1. Uses the three observed sky positions (RA/Dec) and the times between them.

2. Applies the laws of Keplerian motion and Newtonian gravity.

3. Solves a series of complex geometric and algebraic equations to determine the object's 3D position vector (r) and 3D velocity vector (v) in space at the time of one of the observations.

Once you have the object's exact position and velocity at a single moment in time, you can directly calculate all six of the orbital parameters.

The Modern Approach (Iterative Orbit Determination)

Today, with computers, astronomers use a more powerful iterative process that can handle hundreds of observations:

1. Initial Guess: An initial, approximate orbit is calculated from the first few observations.

2. Prediction: A computer model predicts where the object should have appeared in the sky for every single observation time, based on this guess orbit.

3. Comparison: The predicted positions are compared to the actual observed positions. The differences are called residuals.

4. Refinement: Using a mathematical technique called least-squares fitting, the computer slightly adjusts the six orbital parameters of the guess orbit to minimize the sum of the squares of the residuals.

5. Iteration: Steps 2-4 are repeated, with each iteration producing a more refined orbit that better fits all the available data. This continues until the orbit converges on a final solution that is statistically the best possible fit.

The Result: The Six Orbital Elements 🌍

The end result of this process is the set of six Keplerian orbital elements that you have been visualizing. These elements completely describe the orbit and the object's position on it.

• a (Semi-major axis): Determines the orbit's overall size.

• e (Eccentricity): Determines the orbit's shape (0 for a circle, >1 for a hyperbola like 3I/ATLAS).

• i (Inclination): The tilt of the orbital plane relative to Earth's orbit (the ecliptic). This is what makes Halley's Comet's orbit retrograde (i > 90°).

• Ω (Longitude of the Ascending Node): The "swivel" of the tilted orbit, defining where it crosses the ecliptic plane.

• ω (Argument of Periapsis): The orientation of the ellipse within its own tilted plane, defining where the closest point (periapsis) is located.

• M (Mean Anomaly at Epoch): The object's exact position along its orbital path at a specific, known time (the "epoch").

🔭 Understanding Orbital Elements

The six classical orbital elements define a celestial object's orbit in 3D space:

🔵 Semi-major Axis (a)

Defines the size of the orbit - half the longest diameter of the ellipse. For satellites, typically given in kilometers; for heliocentric objects in AU.

🟡 Eccentricity (e)

Describes the shape: e = 0 (perfect circle), 0 < e < 1 (ellipse), e = 1 (parabola), e > 1 (hyperbola). 

Most planets and planetary satellites have low eccentricity orbits, like Earth. Comets often have high eccentricities but still have elliptical orbits. Interstellar objects that approach but do not orbit the sun have hyperbolic trajectories. 

Conic Sections 

• Circle (e=0): 

A perfectly circular orbit.

• Ellipse (0 < e < 1): 

The most common type of orbit for planets, moons, and satellites. The central body is at one of the two foci of the ellipse.

• Parabola (e=1): 

This is an "escape orbit." An object with this trajectory has just enough energy to escape the gravitational pull of the central body and will never return.

• Hyperbola (e>1): 

This is also an escape orbit, but with more energy than a parabolic orbit. The object is moving too fast to be captured and will continue on its path into space.

🔴 Inclination (i)

Angle between the orbital plane and reference plane (0-180°). Values > 90° indicate retrograde motion.

Actual retrograde motion is when an object physically moves or spins in the opposite direction of the main flow of the system. 🔄

• Retrograde Orbit: Many irregular moons, like Neptune's Triton, orbit their planet in the opposite direction of the planet's rotation. This often suggests the moon was a captured object rather than forming with the planet.

• Retrograde Rotation: Some planets have a retrograde spin. Venus rotates backward so slowly that its day is longer than its year. Uranus is tilted on its side so extremely that its rotation is also considered retrograde. These unusual rotations are likely the result of massive impacts early in the solar system's history.

🟢 Longitude of Ascending Node (Ω)

Where the orbit crosses the reference plane going "upward" (0-360°). Defines the orientation of the orbital plane.

In orbital mechanics, a node is a point where an orbit crosses a reference plane.

Think of the reference plane as a flat, imaginary disk in space, like the ecliptic plane (the plane of Earth's orbit around the Sun). A satellite's orbit is usually tilted with respect to this plane. The two points where the satellite's orbital path punches through this disk are the nodes.

There are two kinds of nodes that define an orbit:

• Ascending Node (☊): 

This is the point where the orbiting object passes from south to north through the reference plane. It's the "upward" crossing. 📈

• Descending Node (☋): 

This is the point where the object passes from north to south through the reference plane. It's the "downward" crossing. 📉

The Line of Nodes

The line of nodes is the straight line that connects the ascending and descending nodes. ↔️ This line passes through the central body being orbited (like the Sun or a planet).

The orientation of this line is a crucial orbital element known as the longitude of the ascending node. It helps define the three-dimensional orientation of the orbit in space.

The longitude of the ascending node (Ω) is about orientation in space, and determining it requires a reference system. Here's how it works:

The Reference System

1. Reference Plane: Usually the ecliptic (Earth's orbital plane) or the equator

2. Reference Direction: The First Point of Aries (♈) - also called the vernal equinox direction (0 degrees in Pisces)

What It Measures

The longitude of the ascending node is the angle measured:

• From: The First Point of Aries (♈)

• To: The ascending node (where the orbit crosses the reference plane going northward)

• Direction: Measured eastward (counterclockwise when viewed from north)

• Range: 0° to 360°

The First Point of Aries (Reference Direction)

This is where the Sun appears to cross Earth's equator at the March equinox. It's defined by:

• The intersection of Earth's equatorial plane and the ecliptic plane

• The direction toward the Sun at the moment of the March equinox

• In the constellation Pisces (historically was in Aries, hence the name)

Why This Reference?

1. Observable: The equinoxes are directly observable astronomical events

2. Stable: While it precesses slowly (~26,000 year cycle), it's predictable

3. Universal: All astronomers can determine the same reference point

4. Historical: Been used since ancient Greek astronomy

Different Reference Frames

• Heliocentric ecliptic: Ω measured from ♈ in the ecliptic plane

• Geocentric equatorial: Ω measured from ♈ in Earth's equatorial plane

• Planet-centric: For moons, measured in the planet's equatorial plane

Practical Determination

Modern astronomers determine Ω by:

1. Observing the object's position over time

2. Calculating the orbital plane from multiple observations

3. Finding where this plane intersects the reference plane

4. Measuring the angle from ♈ to this intersection point

This is why different reference frames (like we discussed with Earth's elements) can give different Ω values - they're measuring from the same direction (♈) but in different planes!

The above image is a screenshot of the html plot showing Mercury's orbit in the perifocal plane and in the ecliptic plane, showing the final transformation. It also shows the reference direction towards the Vernal Equinox. 

Defining "North" and "South" 

The concepts of "north" and "south" are determined by the angular momentum of the system, which establishes a primary axis.

• North: If you curl the fingers of your right hand in the direction of the primary body's rotation (or the orbit's direction), your thumb points to the celestial north pole of that reference plane. This "north" is perpendicular to the plane itself.

• South: This is simply the opposite direction of north, pointing away from the celestial south pole of the plane.

For the ecliptic plane, "north" points perpendicular to the plane of Earth's orbit around the Sun, in the direction defined by the orbit's counter-clockwise motion. For a planetary equatorial plane, "north" is aligned with that planet's rotational north pole.

Defining "East" and "West"

"East" and "west" refer to the direction of motion within the reference plane, not directions in 3D space.

• East: This is the standard, or prograde, direction of motion. It's the direction that the planets orbit the Sun (counter-clockwise when viewed from above the Sun's north pole).

• West: This is the opposite, or retrograde, direction of motion.

This is why an ascending node is the point where a satellite crosses the plane moving from south to north, and a descending node is where it crosses from north to south.

🟣 Argument of Periapsis (or argument of perihelion) (ω)

Angle from ascending node to periapsis (closest approach) within the orbital plane (0-360°).

🟠 Mean Anomaly (M) or True Anomaly (ν)

Position of the satellite along its orbit at a specific time. Mean anomaly increases uniformly with time.

The mean anomaly is a parameter that describes the position of an orbiting body along its elliptical path. It represents the fraction of the orbital period that has elapsed since the body last passed its point of closest approach to the central body (the periapsis), expressed as an angle.

Imagine a fictitious object orbiting in a perfect circle with the same orbital period as the actual satellite. The mean anomaly is the angle that this fictitious object would have moved through since passing the periapsis point. It increases uniformly with time, from 0 to 360 degrees over one full orbit.

The mean anomaly (M) is calculated as:

M=n(t−tp​)

where:

• n is the mean motion (the average angular speed of the satellite).

• t is the current time.

• tp​ is the time of periapsis passage.