Tidal Force on Satellites Calculator

Estimate the tidal force on a satellite orbiting a planet based on the planet’s mass, orbital distance, and satellite length.

Formulas Used

The tidal force across a satellite is calculated based on the differential gravitational force due to the planet’s mass.

  1. Tidal Force:

    \[ F_{\text{tidal}} = \frac{2 G M m \Delta r}{r^3} \]

    Where:

    • \( G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \) (gravitational constant)
    • \( M \) is the planet’s mass in kg (converted from Earth masses, 1 Earth mass = \( 5.97219 \times 10^{24} \, \text{kg} \))
    • \( m \) is the satellite’s mass (assumed as 1 kg for force per unit mass)
    • \( \Delta r \) is the satellite’s length in meters (radial extent)
    • \( r \) is the orbital distance in meters (converted from km)

    This formula gives the force difference across the satellite’s length due to the planet’s gravity gradient.

Example Calculations

Example 1: Satellite Orbiting Earth

Inputs: Planet Mass = 1 Earth mass, Orbital Distance = 400 km, Satellite Length = 10 m

Calculations:

  • Planet Mass in kg: \[ 1 \times 5.97219 \times 10^{24} = 5.97219 \times 10^{24} \, \text{kg} \]
  • Distance in meters: \[ 400 \times 1000 = 400000 \, \text{m} \]
  • Tidal Force: \[ \frac{2 \times 6.67430 \times 10^{-11} \times 5.97219 \times 10^{24} \times 1 \times 10}{400000^3} \approx 1.24 \times 10^{-7} \, \text{N} \]

Result: Tidal Force: 1.24 × 10^-7 N

Example 2: Satellite Orbiting Jupiter

Inputs: Planet Mass = 317.8 Earth masses, Orbital Distance = 421700 km, Satellite Length = 20 m

Calculations:

  • Planet Mass in kg: \[ 317.8 \times 5.97219 \times 10^{24} \approx 1.898 \times 10^{27} \, \text{kg} \]
  • Distance in meters: \[ 421700 \times 1000 = 421700000 \, \text{m} \]
  • Tidal Force: \[ \frac{2 \times 6.67430 \times 10^{-11} \times 1.898 \times 10^{27} \times 1 \times 20}{421700000^3} \approx 6.76 \times 10^{-8} \, \text{N} \]

Result: Tidal Force: 6.76 × 10^-8 N

Example 3: Satellite in Low Earth Orbit

Inputs: Planet Mass = 1 Earth mass, Orbital Distance = 6378 km, Satellite Length = 5 m

Calculations:

  • Planet Mass in kg: \[ 1 \times 5.97219 \times 10^{24} = 5.97219 \times 10^{24} \, \text{kg} \]
  • Distance in meters: \[ 6378 \times 1000 = 6378000 \, \text{m} \]
  • Tidal Force: \[ \frac{2 \times 6.67430 \times 10^{-11} \times 5.97219 \times 10^{24} \times 1 \times 5}{6378000^3} \approx 1.54 \times 10^{-8} \, \text{N} \]

Result: Tidal Force: 1.54 × 10^-8 N

How to Use the Calculator

Follow these steps to estimate the tidal force on a satellite:

  1. Enter Planet Mass: Input the planet’s mass in Earth masses (e.g., 1 for Earth).
  2. Enter Orbital Distance: Input the distance from the planet’s center to the satellite in kilometers (e.g., 384400 for lunar distance).
  3. Enter Satellite Length: Input the satellite’s length along the radial direction in meters (e.g., 10).
  4. Calculate: Click “Calculate Tidal Force” to see the result.
  5. Interpret Result: The result shows the tidal force in newtons for a 1 kg satellite. If you see “Please fill in all fields,” ensure all inputs are provided and positive.
  6. Share or Embed: Use the share buttons to post results on social media, copy the result, or get an embed code for the calculator.

Note: This calculator assumes a point-mass planet and a simplified tidal force model. Actual tidal forces may vary due to planet oblateness, satellite orbit, or other factors.

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