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Orbital Speed Formula Calculator

\[ v = \sqrt{\frac{G M}{r}} \]

1. What is the Orbital Speed Formula Calculator?

Definition: This calculator computes the orbital speed (\(v\)) of an object in a circular orbit around a central body, using the formula \( v = \sqrt{\frac{G M}{r}} \), where \(G\) is the gravitational constant, \(M\) is the mass of the central body, and \(r\) is the orbital radius.

Purpose: It is used in astrophysics to determine the speed required for a satellite or celestial body to maintain a circular orbit, applicable in space missions, planetary science, and orbital mechanics.

2. How Does the Calculator Work?

The calculator uses the orbital speed formula:

Formula: \[ v = \sqrt{\frac{G M}{r}} \] where:

  • \(v\): Orbital speed (m/s, km/h, mph, ft/s)
  • \(G\): Gravitational constant (m³/(kg·s²), ft³/(lb·s²))
  • \(M\): Mass of central body (kg, g, lb)
  • \(r\): Radius (m, km, ft, in)

Unit Conversions:

  • Gravitational Constant:
    • 1 m³/(kg·s²) = 1 m³/(kg·s²)
    • 1 ft³/(lb·s²) = 0.068521765856 m³/(kg·s²)
  • Mass of Central Body:
    • 1 kg = 1 kg
    • 1 g = 0.001 kg
    • 1 lb = 0.45359237 kg
  • Radius:
    • 1 m = 1 m
    • 1 km = 1000 m
    • 1 ft = 0.3048 m
    • 1 in = 0.0254 m
  • Orbital Speed (Output):
    • 1 m/s = 1 m/s
    • 1 km/h = \( \frac{1000}{3600} \) m/s \(\approx 0.27777777778 \, \text{m/s}\)
    • 1 mph = 0.44704 m/s
    • 1 ft/s = 0.3048 m/s
The orbital speed is calculated in m/s and can be converted to the selected output unit (m/s, km/h, mph, ft/s). Results greater than 10,000 or less than 0.001 are displayed in scientific notation; otherwise, they are shown with 4 decimal places.

Steps:

  • Enter the gravitational constant (\(G\)), mass of the central body (\(M\)), and radius (\(r\)) with their units (default: \(G = 6.67430 \times 10^{-11} \, \text{m³/(kg·s²)}\), \(M = 5.972 \times 10^{24} \, \text{kg}\), \(r = 6771000 \, \text{m}\)).
  • Convert inputs to SI units (m³/(kg·s²), kg, m).
  • Validate that gravitational constant, mass, and radius are greater than 0.
  • Calculate the orbital speed in m/s using the formula.
  • Convert the orbital speed to the selected output unit.
  • Display the result, using scientific notation if the value is greater than 10,000 or less than 0.001, otherwise rounded to 4 decimal places.

3. Importance of Orbital Speed Calculation

Calculating orbital speed is crucial for:

  • Astrophysics: Determining the speed of satellites, moons, or planets in circular orbits, such as the International Space Station or the Moon around Earth.
  • Space Missions: Designing orbits for spacecraft, ensuring they maintain stable orbits at the correct altitude.
  • Education: Teaching the principles of gravitational forces and circular motion in orbital mechanics.

4. Using the Calculator

Examples:

  • Example 1: Calculate the orbital speed for \(G = 6.67430 \times 10^{-11} \, \text{m³/(kg·s²)}\), \(M = 5.972 \times 10^{24} \, \text{kg}\), \(r = 6771000 \, \text{m}\), output in m/s:
    • Enter \(G = 6.67430 \times 10^{-11} \, \text{m³/(kg·s²)}\), \(M = 5.972 \times 10^{24} \, \text{kg}\), \(r = 6771000 \, \text{m}\).
    • Numerator: \(G M = 6.67430 \times 10^{-11} \times 5.972 \times 10^{24} \approx 3.986 \times 10^{14} \, \text{m³/s²}\).
    • Fraction: \(\frac{G M}{r} = \frac{3.986 \times 10^{14}}{6771000} \approx 58885.7 \, \text{m²/s²}\).
    • Orbital speed: \(v = \sqrt{58885.7} \approx 7673.6 \, \text{m/s}\).
    • Output unit: m/s (no conversion needed).
    • Result: \( \text{Orbital Speed} = 7673.6000 \, \text{m/s} \).
  • Example 2: Calculate the orbital speed for \(G = 3.21693 \times 10^{-12} \, \text{ft³/(lb·s²)}\), \(M = 2.20462 \, \text{lb}\), \(r = 3937.008 \, \text{in}\), output in mph:
    • Enter \(G = 3.21693 \times 10^{-12} \, \text{ft³/(lb·s²)}\), \(M = 2.20462 \, \text{lb}\), \(r = 3937.008 \, \text{in}\).
    • Convert: \(G = 3.21693 \times 10^{-12} \times 0.068521765856 \approx 2.20462 \times 10^{-13} \, \text{m³/(kg·s²)}\), \(M = 2.20462 \times 0.45359237 = 1 \, \text{kg}\), \(r = 3937.008 \times 0.0254 = 100 \, \text{m}\).
    • Numerator: \(G M = 2.20462 \times 10^{-13} \times 1 \approx 2.20462 \times 10^{-13} \, \text{m³/s²}\).
    • Fraction: \(\frac{G M}{r} = \frac{2.20462 \times 10^{-13}}{100} \approx 2.20462 \times 10^{-15} \, \text{m²/s²}\).
    • Orbital speed in m/s: \(v = \sqrt{2.20462 \times 10^{-15}} \approx 4.6953 \times 10^{-8} \, \text{m/s}\).
    • Convert to output unit (mph): \(4.6953 \times 10^{-8} \times \frac{1}{0.44704} \approx 1.0504 \times 10^{-7} \, \text{mph}\).
    • Result: \( \text{Orbital Speed} = 1.0504 \times 10^{-7} \, \text{mph} \).

5. Frequently Asked Questions (FAQ)

Q: What is orbital speed?
A: Orbital speed is the speed required for an object to maintain a circular orbit around a central body, balancing the gravitational force with the centripetal force required for circular motion.

Q: Why must gravitational constant, mass, and radius be greater than zero?
A: These quantities must be greater than zero to represent physical properties: the gravitational constant defines the strength of gravity, the mass of the central body provides the gravitational attraction, and the radius defines the orbit. Zero values would lead to division by zero or be physically meaningless.

Q: What does the radius represent?
A: The radius (\(r\)) is the distance from the center of the central body to the orbiting object, often the sum of the central body’s radius and the altitude of the orbit. For example, for a satellite orbiting Earth, \(r\) is the Earth’s radius plus the satellite’s altitude.

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