Time Period Of Elliptical Orbit Given Angular Momentum Calculator

Formula Used:

\[ T_e = \frac{2\pi}{[GM.Earth]^2} \times \left( \frac{h_e}{\sqrt{1-e_e^2}} \right)^3 \]

Time Period of Elliptic Orbit:

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1. What is the Time Period of Elliptical Orbit?

The time period of an elliptical orbit is the amount of time a given astronomical object takes to complete one full revolution around another object along an elliptical path. This is a fundamental parameter in orbital mechanics and celestial dynamics.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ T_e = \frac{2\pi}{[GM.Earth]^2} \times \left( \frac{h_e}{\sqrt{1-e_e^2}} \right)^3 \]

Where:

\( T_e \) — Time period of elliptic orbit (seconds)
\( h_e \) — Angular momentum of elliptic orbit (m²/s)
\( e_e \) — Eccentricity of elliptical orbit (unitless)
\( [GM.Earth] \) — Earth's geocentric gravitational constant (3.986004418 × 10¹⁴ m³/s²)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula calculates the orbital period based on the angular momentum and eccentricity of the orbit, using Earth's gravitational parameter.

3. Importance of Time Period Calculation

Details: Accurate calculation of orbital period is crucial for satellite operations, space mission planning, astronomical observations, and understanding celestial mechanics. It helps determine when objects will be in specific positions relative to each other.

4. Using the Calculator

Tips: Enter angular momentum in m²/s and eccentricity as a unitless value between 0 and 1 (0 ≤ e < 1). Both values must be valid positive numbers with eccentricity less than 1.

5. Frequently Asked Questions (FAQ)

Q1: What is angular momentum in orbital mechanics?
A: Angular momentum is a conserved quantity that characterizes the rotational motion of an object in orbit. For elliptical orbits, it remains constant throughout the orbit.

Q2: What does eccentricity represent?
A: Eccentricity measures how elongated an elliptical orbit is. A value of 0 represents a circular orbit, while values closer to 1 indicate more elongated ellipses.

Q3: Why is Earth's GM constant used?
A: The GM constant (gravitational parameter) combines the gravitational constant and Earth's mass, providing a convenient way to calculate orbital characteristics around Earth.

Q4: Can this formula be used for orbits around other celestial bodies?
A: Yes, but you would need to use the appropriate gravitational parameter (GM) for the specific celestial body around which the orbit occurs.

Q5: What are typical time periods for Earth orbits?
A: Low Earth orbits typically have periods of about 90-120 minutes, while geostationary orbits have periods of exactly 24 hours to match Earth's rotation.