Elliptical Orbital Speed Calculator

Elliptical Orbital Speed Equation:

\[ v = \sqrt{GM \left( \frac{2}{r} - \frac{1}{a} \right)} \]

Gravitational Constant (G):

m³/kg s²

Central Body Mass (M):

Distance (r):

Semi-major Axis (a):

Orbital Speed (v):

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1. What is the Elliptical Orbital Speed Equation?

The Elliptical Orbital Speed Equation calculates the speed of an object in an elliptical orbit around a central body. It is derived from the vis-viva equation and provides the orbital speed at any point in the orbit based on the distance from the central body and the semi-major axis of the orbit.

2. How Does the Calculator Work?

The calculator uses the elliptical orbital speed equation:

\[ v = \sqrt{GM \left( \frac{2}{r} - \frac{1}{a} \right)} \]

Where:

\( v \) — Orbital speed (m/s)
\( G \) — Gravitational constant (6.67430 × 10⁻¹¹ m³/kg s²)
\( M \) — Mass of the central body (kg)
\( r \) — Distance from the central body (m)
\( a \) — Semi-major axis of the orbit (m)

Explanation: The equation calculates the orbital speed at any point in an elliptical orbit, accounting for both the current distance from the central body and the overall size of the orbit.

3. Importance of Orbital Speed Calculation

Details: Accurate orbital speed calculation is crucial for spacecraft trajectory planning, satellite operations, and understanding celestial mechanics. It helps determine the required velocity for orbital insertion, transfers, and maintenance.

4. Using the Calculator

Tips: Enter the gravitational constant (typically 6.67430e-11), the mass of the central body in kilograms, the current distance from the central body in meters, and the semi-major axis of the orbit in meters. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between circular and elliptical orbital speed?
A: In a circular orbit, the speed is constant, while in an elliptical orbit, the speed varies depending on the distance from the central body, being fastest at periapsis and slowest at apoapsis.

Q2: What are typical orbital speeds for Earth satellites?
A: Low Earth orbit satellites typically travel at about 7.8 km/s, while geostationary satellites travel at about 3.1 km/s.

Q3: How does mass affect orbital speed?
A: For a given distance and semi-major axis, orbital speed increases with the square root of the central body's mass.

Q4: What happens when r equals a?
A: When r = a, the equation simplifies to the circular orbit speed formula: v = √(GM/r).

Q5: Can this equation be used for parabolic or hyperbolic trajectories?
A: No, this equation is specifically for elliptical (and circular) orbits. For parabolic or hyperbolic trajectories, different equations apply.