Interstellar Object 3I/ATLAS:

The above image shows the orbit of interstellar object 3I/ATLAS (discovered by the ATLAS observatory, and the 3rd interstellar object found), possibly a icy body like a comet, passing by the Sun through the solar system. It is on a hyperbolic trajectory, bent closer to the Sun, but it does not enter into an orbit around the Sun. Once it leaves solar system it will never return. It's date of closest approach, perihelion, is October 29, 2025, as seen on the below image. 3I/ATLAS is moving very fast, about twice as fast as the Earth. It's fastest velocity will be at perihelion, almost 247,000 km/hr. At perihelion 3I/ATLAS will be on a trajectory between Earth and Mars. See it's page for more information and links to orrery plots. 

Deriving Orbital Paths from Orbital Elements

The Mathematical Foundation

Every orbit in space follows fundamental conic section equations, derived directly from the orbital elements. Understanding this relationship bridges the gap between the abstract orbital parameters and the actual geometric paths traced by bound elliptical orbits and unbound hyperbolic trajectories of interstellar visitors.

Universal Conic Section Equation

The equation that governs all orbital motion uses a polar coordinate system centered at the focus (where the central body sits):

To plot the actual elliptical path, we convert from polar to Cartesian coordinates in the orbital plane: r = a(1 - e²) / (1 + e·cos(ν))

This single equation describes three types of trajectories:

- **e < 1:** Elliptical orbits (bound systems - planets, moons, comets)

- **e = 1:** Parabolic trajectory (escape velocity)

- **e > 1:** Hyperbolic orbits (unbound systems - interstellar objects)

Where:

- **r** = distance from central body to orbiting object

- **a** = semi-major axis (positive for ellipses, negative for hyperbolas)

- **e** = eccentricity (orbital element) 

- **ν** = true anomaly (orbital element - position angle)

Elliptical Orbits (e < 1) - Bound Systems

Converting to Cartesian Coordinates:

x = r · cos(ν) y = r · sin(ν)

Substituting the ellipse equation:

x = [a(1 - e²) / (1 + e·cos(ν))] · cos(ν) y = [a(1 - e²) / (1 + e·cos(ν))] · sin(ν)

Elliptical Geometric Properties

The orbital elements directly determine the ellipse geometry:

Semi-major axis (a): Defines the size

- Distance from center to farthest point

- Half the longest diameter of the ellipse

Eccentricity (e): Defines the shape

e = 0: Perfect circle

0 < e < 1: Ellipse (most planetary orbits)

e = 1: Parabola (escape trajectory)

e > 1: Hyperbola (interstellar objects)

For elliptical orbits (e < 1):

• Eccentricity measures how "squashed" the ellipse is

• e = 0 is a perfect circle

• As e approaches 1, the ellipse gets more elongated

For hyperbolic orbits (e > 1):

• Eccentricity determines how "open" the hyperbola is

• It controls the deflection angle - how much the object's path bends around the Sun

Specifically:

• e = 1: Parabolic orbit (borderline case)

• e slightly > 1: Very open hyperbola, large deflection (like a hairpin turn)

• e >> 1: Nearly straight path, small deflection (barely affected by the Sun)

Semi-minor axis (b): Calculated from a and e

b = a√(1 - e²)

Focal distance (c): Distance from center to focus

c = a·e

Periapsis and Apoapsis

The closest and farthest points are determined by the orbital elements:

Periapsis distance (closest approach):

r_periapsis = a(1 - e)

Occurs at ν = 0° (true anomaly = 0)

Apoapsis distance (farthest point):

r_apoapsis = a(1 + e)

Occurs at ν = 180° (true anomaly = 180°)

Practical Example: Mars

Using Mars's orbital elements:

- Semi-major axis: a = 1.524 AU

- Eccentricity: e = 0.0934

Calculating key distances:

- Perihelion: r_min = 1.524(1 - 0.0934) = 1.382 AU

- Aphelion: r_max = 1.524(1 + 0.0934) = 1.666 AU

- Semi-minor axis: b = 1.524√(1 - 0.0934²) = 1.517 AU

Distance at any true anomaly:

At ν = 90°: r = 1.524(1 - 0.0934²)/(1 + 0.0934·cos(90°)) = 1.517 AU

Hyperbolic Orbits (e > 1) - Interstellar Objects

When eccentricity exceeds 1, the trajectory becomes a hyperbola - an open, unbound path typical of interstellar visitors passing through our solar system. These objects follow the same universal equation but with fundamentally different geometry.

Hyperbolic Equation Characteristics

For hyperbolic orbits (e > 1):

- Semi-major axis (a) is negative by convention

- The orbit has two asymptotes that the object approaches at infinite distance

- The object makes a single closest approach (perihelion) then escapes forever

The same polar equation applies:

r = |a|(e² - 1) / (1 + e·cos(ν))

Note: For hyperbolas, we use |a| (absolute value) since a < 0 by definition.

Hyperbolic Geometry Properties

Semi-major axis (a): Negative, related to excess velocity

a = -μ / v∞²

Where μ is the gravitational parameter and v∞ is the hyperbolic excess velocity

Semi-minor axis (b): For hyperbolas

b = |a|√(e² - 1)

Focal distance (c): Distance from center to focus

c = |a|·e

Asymptote angle: The angle between asymptotes

α = arccos(-1/e)

Perihelion and Asymptotic Behavior

Perihelion distance (closest approach):

r_perihelion = |a|(e - 1)

Occurs at ν = 0° (true anomaly = 0)

The limits of the hyperbolic orbit plot are defined by the asymptotic true anomaly - the angle at which the hyperbola approaches infinity.

Looking at the generate_hyperbolic_orbit_points function:

python

Example (see above images): for 3I/ATLAS with e ≈ 6.3:

• theta_inf = arccos(-1/6.3) ≈ 1.73 radians ≈ 99°

• So the plot shows from about -99° to +99° of true anomaly

Asymptotic distance: As ν approaches ±α, r approaches infinity

True anomaly range: For hyperbolas, ν is limited to:

-α ≤ ν ≤ +α, where α = arccos(-1/e)

Real Examples: Interstellar Visitors

'Oumuamua (1I/2017 U1) - First Confirmed Interstellar Object:

- Semi-major axis: a = -1.280 AU (negative for hyperbola)

- Eccentricity: e = 1.201 (hyperbolic)

- Perihelion: r_p = |a|(e-1) = 1.280(1.201-1) = 0.257 AU

- Asymptote angle: α = arccos(-1/1.201) = 146.4°

- True anomaly range: -146.4° ≤ ν ≤ +146.4°

2I/Borisov - Interstellar Comet:

- Semi-major axis: a = -2.015 AU

- Eccentricity: e = 3.357

- Perihelion: r_p = 2.015(3.357-1) = 4.75 AU

- Asymptote angle: α = arccos(-1/3.357) = 107.3°

- Much more extreme hyperbolic trajectory than 'Oumuamua

3I/2022 ATLAS (projected values):

- Semi-major axis: a ≈ -2.9 AU  

- Eccentricity: e ≈ 2.8

- Perihelion: r_p ≈ 2.9(2.8-1) = 5.22 AU

- Another highly hyperbolic interstellar trajectory

- See image above showing the hyperbolic orbit and perihelion as an open square symbol. 3I/ATLAS is moving more than twice as fast as Earth at its current location. See the second image showing 3I/ATLAS at perihelion moving almost 2.5 times as fast as Earth, and a year later it will be already past the orbit of Sedna! 

Calculating Hyperbolic Positions

Distance at any true anomaly (within valid range):

For 'Oumuamua at ν = 90°:

r = 1.280(1.201² - 1) / (1 + 1.201·cos(90°)) r = 1.280(0.4424) / (1 + 0) = 0.566 AU

Cartesian coordinates in orbital plane:

x = r · cos(ν) y = r · sin(ν)

Hyperbolic Orbital Period and Timing

Unlike elliptical orbits, hyperbolic objects don't have periods—they pass through once and never return. Instead, we use time since perihelion passage:

Mean anomaly for hyperbolas:

M = n(t - T)

Where:

- n = √(μ/|a|³) (mean motion)

- t = current time

- T = time of perihelion passage

Eccentric anomaly (E) relationship:

M = e·sinh(E) - E

True anomaly from eccentric anomaly:

tan(ν/2) = √((e+1)/(e-1)) · tanh(E/2)

Observational Implications

Visibility window: Interstellar objects are only observable for a limited time around perihelion passage

Trajectory asymmetry: The inbound and outbound paths are mirror images, but observations typically occur during one phase

High velocities: These objects move much faster than typical solar system bodies, requiring rapid follow-up observations

Detection challenges: Most interstellar objects are discovered when already leaving the inner solar system

Universal 3D Transformation

Both elliptical and hyperbolic equations give us the orbit in the orbital plane. To place either trajectory correctly in 3D space, we apply the same three rotational transformations you've detailed:

1. Rotation 1: Longitude of ascending node (Ω) around Z-axis

2. Rotation 2: Inclination (i) around X-axis  

3. Rotation 3: Argument of periapsis (ω) around Z-axis

The mathematical sequence (identical for elliptical and hyperbolic orbits):

1. Calculate (x,y) in orbital plane using conic equation

2. Apply rotation matrices for Ω, i, and ω

3. Transform from reference frame to target coordinate system

Key insight: The 3D transformation process is identical whether the object is Mars in an elliptical orbit or 'Oumuamua on a hyperbolic trajectory. Only the conic equation parameters change.

Special Cases and Validations

Circular orbits (e ≈ 0):

The equation simplifies to r ≈ a (constant radius)

Parabolic escape (e = 1):

The limiting case between bound and unbound motion

r = a/(1 + cos(ν)) with infinite aphelion

Highly eccentric ellipses (e approaching 1):

Comet-like orbits with dramatic distance variations

Moderate hyperbolas (e slightly > 1):

Objects with small excess velocities, like some long-period comets that became unbound

Extreme hyperbolas (e >> 1):

Fast-moving interstellar objects like 2I/Borisov

Validation checks:

For ellipses: r_periapsis + r_apoapsis = 2a ✓

For hyperbolas: r_perihelion = |a|(e-1) ✓

Integration with Your Orbital Mechanics Pipeline

This conic section derivation fits perfectly into your transformation process:

1. Start with orbital elements (from JPL Horizons - works for any object)

2. Generate conic path using the universal polar equation

3. Apply 3D rotations (your detailed rotation sequence)

4. Transform reference frames (equatorial → ecliptic)

5. Apply time corrections (precession, perturbations)

The mathematical elegance is that the same equations work for every object in space—from Phobos orbiting Mars to 'Oumuamua visiting from interstellar space—with only the orbital element values and eccentricity determining whether you get a closed ellipse or open hyperbola.