Apsidal Dates

The above image is a screenshot of apsides for the planets, and certain dwarf planets and comets. The apsides are "ideal" in the sense that they are calculated from orbital parameters. In addition, Earth has actual apsides plotted, which are very close to the ideal apsides. Here is the full html file (5 MB). 

The above screenshot shows the Moon's orbit with the next perigee on August 15, 2025, and apogee. You can see the full animation html file (3.3 MBB) here. 

Apsidal dates are the specific calendar dates and times when an orbiting body reaches its closest or farthest point from the object it is orbiting. These two extreme points in an orbit are collectively known as apsides.

☀️ Heliocentric Orbits (Sun-Centered)

In a heliocentric orbit, the central body is the Sun. The apsidal dates correspond to two specific points:

• Perihelion: The date when the object is at its closest point to the Sun. For example, Earth reaches its perihelion in early January each year.

• Aphelion: The date when the object is at its farthest point from the Sun. Earth reaches its aphelion in early July.

This terminology applies to planets, comets, and asteroids orbiting the Sun.

🌍 Geocentric Orbits (Earth-Centered)

In a geocentric orbit, the central body is the Earth. The terms for the apsides change accordingly:

• Perigee: The date when an object, such as the Moon or an artificial satellite, is at its closest point to Earth.

• Apogee: The date when the object is at its farthest point from Earth.

☄️ Hyperbolic Orbits (Unbound)

Hyperbolic orbits are a special case. They are open-ended, unbound trajectories where an object approaches a central body once and then escapes its gravity, never to return.

Because the path is not a closed loop, there is no farthest point of the orbit. Therefore, a hyperbolic trajectory has only one apsidal point:

• Periapsis: The date of the single closest approach to the central body. This term is often used generally, but for hyperbolic solar system visitors like interstellar comets, it specifically refers to their one-time closest pass of the Sun.

Apsides are of major astronomical interest because they represent the extreme points in an orbit, which have significant and predictable effects on a celestial body's speed, energy, physical state, and our ability to observe it.

Orbital Speed and Dynamics 🛰️

According to Kepler's second law of planetary motion, an orbiting body moves fastest when it is at its closest point (periapsis) and slowest when it is at its farthest point (apoapsis). This change in speed is a direct consequence of the conservation of angular momentum. For example, a comet will whip around the Sun at its highest velocity during perihelion and crawl along at its slowest pace when at aphelion in the outer solar system.

Physical and Environmental Effects 🌋

The varying distance at the apsides directly impacts the physical conditions of an orbiting body.

• Solar Energy: For an object orbiting the Sun, the amount of solar radiation it receives changes. Earth receives about 7% more energy at perihelion (in January) than at aphelion (in July), which slightly moderates Northern Hemisphere seasons and intensifies Southern Hemisphere seasons.

• Tidal Forces: Gravitational tidal forces are dramatically stronger at the closest approach. The extreme volcanism on Jupiter's moon Io is a direct result of the immense, flexing tidal forces exerted by Jupiter as Io moves between its perijove (periapsis) and apojove (apoapsis). Similarly, the highest tides on Earth, known as "king tides," occur when the Moon is at or near its perigee.

Observation and Mission Planning 🔭

Apsides are critical for both ground-based observation and space exploration.

• Observation: An object at its closest approach to Earth appears larger and brighter, providing the best opportunity for detailed observation with telescopes.

• Mission Planning: It is far more fuel-efficient to perform engine burns at periapsis. Due to the Oberth effect, a rocket gets more change in kinetic energy from a burn performed at high speed. Therefore, crucial maneuvers like entering orbit around another planet are almost always planned to occur at the point of closest approach.

Tests of Fundamental Physics 🌌

The slow rotation of an orbit's apsides over time, known as apsidal precession, provides a powerful way to test theories of gravity. The most famous example is the precession of Mercury's perihelion. While Newtonian gravity could predict most of this precession, a small, stubborn discrepancy remained unexplained for decades. Albert Einstein's theory of General Relativity precisely predicted this missing amount, providing one of the first and most crucial confirmations of his theory.

How the orrery handles apsides:

1. **Accurate apsidal date calculations**:

   - Dynamic perihelion/apohelion date computation: Calculates actual dates when objects reach their closest and farthest points from the Sun based on their current orbital position

   - Support for all orbit types: Handles elliptical orbits (planets, asteroids), hyperbolic orbits (some comets), and satellite orbits (moons with perigee/apogee)

2. **Legend integration**:

   - Apsidal markers now appear in the plot legend for easy identification

3. **Intelligent date display**:

   - Shows calculated future dates for perihelion/apohelion, with special handling for hyperbolic orbits that may never return

### Dynamic Apsidal Marker System

1. **Real-time orbital calculations**:

   - Automatically calculates and displays when objects reach perihelion/aphelion

   - Works for all orbiting bodies: planets, asteroids, comets, and spacecraft

   - Intelligent handling of elliptical and hyperbolic orbits

2. **Dual marker system**:

   - **Ideal apsidal points**: Calculated from orbital elements showing theoretical perihelion/aphelion positions

   - **Actual apsidal markers**: Fetched from JPL Horizons for specific dates, showing true positions

   - Visual validation: Compare ideal vs actual positions to verify orbital accuracy

3. **Enhanced information display**:

   - Hover over markers to see predicted dates and distances

   - Actual measured distances displayed at true apsidal dates

   - Automatic terminology adjustment (perihelion/aphelion for solar orbits, perigee/apogee for moons)

**Apsidal Markers:**

- Plot any object with an orbit (planets, asteroids, comets)

- Look for the square markers indicating perihelion/apohelion

- **Ideal markers**: Show theoretical apsidal points based on orbital elements

- **Actual markers**: Display true positions on specific dates with measured distances

- Hover over markers to see:

  - Calculated or actual dates when the object reaches these points

  - Precise distances in AU and kilometers

  - Visual confirmation that ideal orbits match real ephemeris data

- Observe special handling for hyperbolic objects that may show "Past perihelion"

- Compare ideal vs actual positions to build confidence in the orbital calculations

**Apsidal Marker Validation**: The software now provides visual proof of its accuracy by displaying both theoretical and actual apsidal points. Users can see that the ideal orbital calculations (based on Keplerian mechanics) align precisely with actual JPL Horizons ephemeris data, building confidence in the visualizations when actual data is unavailable or for performance reasons.

**ORBITAL DYNAMICS CALCULATOR**

**`apsidal_markers.py`** ✅ 

- **Core functionality**: Calculates accurate dates for perihelion, apohelion, perigee, and apogee based on current orbital positions and Keplerian mechanics.

- **Key Features**:

  - True anomaly calculation: Determines an object's current position in its orbit from 3D coordinates.

  - Kepler's equation solver: Converts between true, eccentric, and mean anomalies for time calculations.

  - Dual marker system: Generates both ideal (calculated) and actual (measured) apsidal markers.

  - Multi-orbit support: Handles elliptical orbits (e < 1) and hyperbolic trajectories (e ≥ 1).

  - Automated marker generation: Creates properly formatted Plotly markers with hover text and date information.

  - Satellite terminology: Automatically uses perigee/apogee for moon orbits instead of perihelion/apohelion.

  - Supports elliptical and hyperbolic orbits.

**ORBITAL MECHANICS MODULE**

**`idealized_orbits.py`** ✅ 

-   **Core functionality**: Contains the orbital elements (like semi-major axis, eccentricity, and inclination) for a wide range of solar system bodies, including planets, moons, and asteroids. It provides the foundational data for calculating and plotting idealized elliptical and hyperbolic orbits. It also handles the complex transformations required to correctly orient satellite orbits around their parent planets. Includes refined transformation logic for the satellite systems of planets with significant axial tilts, such as Mars, Saturn, and Uranus.

- Provides ideal orbital calculations from JPL Horizons ephemeris orbital elements.

- NAIF ID system integration for consistent object identification.

- Contains orbital element definitions for accurate elliptical and hyperbolic trajectory modeling for all supported JPL Horizons objects, including planets, moons, and comets.

-   **Recent enhancements**: Integration with apsidal_markers.py for accurate perihelion/apohelion date calculations based on current orbital positions.