Time Since Periapsis In Hyperbolic Orbit Given Hyperbolic Eccentric Anomaly Calculator

Formula Used:

\[ Time\ Since\ Periapsis = \frac{h_h^3}{[GM.Earth]^2 \times (e_h^2 - 1)^{3/2}} \times (e_h \times \sinh(F) - F) \]

Time Since Periapsis:

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1. What is Time Since Periapsis in Hyperbolic Orbit?

Time Since Periapsis in Hyperbolic Orbit is a measure of the duration that has passed since an object in a hyperbolic trajectory passed through its closest point to the central body, known as periapsis. This parameter is crucial for predicting the object's future position and velocity in its hyperbolic path.

2. How Does the Calculator Work?

The calculator uses the hyperbolic orbit equation:

\[ t = \frac{h_h^3}{[GM.Earth]^2 \times (e_h^2 - 1)^{3/2}} \times (e_h \times \sinh(F) - F) \]

Where:

\( t \) — Time since periapsis (seconds)
\( h_h \) — Angular momentum of hyperbolic orbit (m²/s)
\( [GM.Earth] \) — Earth's geocentric gravitational constant (3.986004418 × 10¹⁴ m³/s²)
\( e_h \) — Eccentricity of hyperbolic orbit
\( F \) — Eccentric anomaly in hyperbolic orbit (radians)
\( \sinh \) — Hyperbolic sine function

Explanation: This equation relates the time elapsed since periapsis passage to the orbital parameters and hyperbolic eccentric anomaly through Kepler's equation for hyperbolic orbits.

3. Importance of Time Since Periapsis Calculation

Details: Accurate calculation of time since periapsis is essential for mission planning, trajectory prediction, and orbital mechanics analysis of hyperbolic orbits, particularly for interplanetary missions and spacecraft entering or leaving planetary spheres of influence.

4. Using the Calculator

Tips: Enter angular momentum in m²/s, eccentricity (must be greater than 1 for hyperbolic orbits), and eccentric anomaly in radians. All values must be valid positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What distinguishes hyperbolic orbits from elliptical orbits?
A: Hyperbolic orbits have eccentricity greater than 1, meaning the object has excess orbital energy and will escape the central body's gravitational influence, unlike elliptical orbits where the object remains bound.

Q2: How is hyperbolic eccentric anomaly different from true anomaly?
A: Hyperbolic eccentric anomaly (F) is an angular parameter used in the hyperbolic form of Kepler's equation, while true anomaly is the actual angle between periapsis and the object's current position as seen from the central body.

Q3: What are typical applications of this calculation?
A: This calculation is used in interplanetary mission planning, spacecraft escape trajectories, and analysis of objects entering or leaving planetary systems on hyperbolic paths.

Q4: How does the hyperbolic sine function differ from the regular sine function?
A: The hyperbolic sine (sinh) is defined as (eˣ - e⁻ˣ)/2 and describes hyperbolas, while the regular sine function describes circular motion and is defined through the unit circle.

Q5: Can this formula be used for other celestial bodies?
A: Yes, by replacing [GM.Earth] with the appropriate gravitational parameter for other celestial bodies (e.g., [GM.Sun] for heliocentric orbits).