Gravitational Wave Signal Strength Calculator

Estimate the strain amplitude of a gravitational wave from a binary black hole merger.

Formulas Used

The strain amplitude \( h \) of a gravitational wave from a binary black hole merger is estimated using a simplified model.

  1. Chirp Mass:

    \[ M_c = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}} \]

    Calculates the chirp mass in solar masses, where \( m_1 \) and \( m_2 \) are the masses of the black holes.

  2. Strain Amplitude:

    \[ h \approx \frac{4 (G M_c)^{5/3} (\pi f)^{2/3}}{c^4 d} \]

    Estimates the dimensionless strain, where \( G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \), \( c = 2.99792458 \times 10^8 \, \text{m/s} \), \( f \) is the orbital frequency in Hz, \( d \) is the distance in meters (converted from Mpc, 1 Mpc = 3.085677581 \times 10^{22} \, \text{m} \)), and \( M_c \) is converted to kg (\( 1 \, M_\odot = 1.98847 \times 10^{30} \, \text{kg} \)).

Example Calculations

Example 1: GW150914-like Event

Inputs: Mass 1 = 36 M☉, Mass 2 = 29 M☉, Distance = 410 Mpc, Frequency = 100 Hz

Calculations:

  • Chirp Mass: \[ \frac{(36 \times 29)^{3/5}}{(36 + 29)^{1/5}} \approx 28.1 \, M_\odot \]
  • Chirp Mass in kg: \[ 28.1 \times 1.98847 \times 10^{30} \approx 5.588 \times 10^{31} \, \text{kg} \]
  • Distance in meters: \[ 410 \times 3.085677581 \times 10^{22} \approx 1.265 \times 10^{25} \, \text{m} \]
  • Strain: \[ \frac{4 \times (6.67430 \times 10^{-11} \times 5.588 \times 10^{31})^{5/3} \times (\pi \times 100)^{2/3}}{(2.99792458 \times 10^8)^4 \times 1.265 \times 10^{25}} \approx 1.04 \times 10^{-21} \]

Result: Strain Amplitude: 1.04 × 10^-21

Example 2: Nearby Binary

Inputs: Mass 1 = 10 M☉, Mass 2 = 10 M☉, Distance = 100 Mpc, Frequency = 50 Hz

Calculations:

  • Chirp Mass: \[ \frac{(10 \times 10)^{3/5}}{(10 + 10)^{1/5}} \approx 8.71 \, M_\odot \]
  • Chirp Mass in kg: \[ 8.71 \times 1.98847 \times 10^{30} \approx 1.732 \times 10^{31} \, \text{kg} \]
  • Distance in meters: \[ 100 \times 3.085677581 \times 10^{22} \approx 3.086 \times 10^{24} \, \text{m} \]
  • Strain: \[ \frac{4 \times (6.67430 \times 10^{-11} \times 1.732 \times 10^{31})^{5/3} \times (\pi \times 50)^{2/3}}{(2.99792458 \times 10^8)^4 \times 3.086 \times 10^{24}} \approx 1.97 \times 10^{-22} \]

Result: Strain Amplitude: 1.97 × 10^-22

Example 3: Distant Massive Binary

Inputs: Mass 1 = 50 M☉, Mass 2 = 50 M☉, Distance = 1000 Mpc, Frequency = 200 Hz

Calculations:

  • Chirp Mass: \[ \frac{(50 \times 50)^{3/5}}{(50 + 50)^{1/5}} \approx 43.55 \, M_\odot \]
  • Chirp Mass in kg: \[ 43.55 \times 1.98847 \times 10^{30} \approx 8.664 \times 10^{31} \, \text{kg} \]
  • Distance in meters: \[ 1000 \times 3.085677581 \times 10^{22} \approx 3.086 \times 10^{25} \, \text{m} \]
  • Strain: \[ \frac{4 \times (6.67430 \times 10^{-11} \times 8.664 \times 10^{31})^{5/3} \times (\pi \times 200)^{2/3}}{(2.99792458 \times 10^8)^4 \times 3.086 \times 10^{25}} \approx 3.59 \times 10^{-21} \]

Result: Strain Amplitude: 3.59 × 10^-21

How to Use the Calculator

Follow these steps to estimate gravitational wave strain amplitude:

  1. Enter Masses: Input the masses of the two black holes in solar masses (e.g., 30 for each).
  2. Enter Distance: Input the distance to the binary system in megaparsecs (e.g., 400).
  3. Enter Frequency: Input the orbital frequency in Hz (e.g., 100 for the inspiral phase).
  4. Calculate: Click “Calculate Strain Amplitude” to see the result.
  5. Interpret Result: The result shows the dimensionless strain amplitude (h). 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 is a simplified model for non-spinning black holes in circular orbits. Actual signals depend on factors like spin, eccentricity, and detector sensitivity.

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