Gravitational Wave Strain Calculator
Gravitational Wave Strain Calculator computes strain from a binary system, with a waveform plot.
Formulas Used in Gravitational Wave Strain Calculator
The calculator computes the dimensionless strain for a binary system:
Reduced Mass:
\\[ \mu = \frac{m_1 m_2}{m_1 + m_2} \\]Total Mass:
\\[ M = m_1 + m_2 \\]Strain:
\\[ h \approx \frac{4 G^2 \mu M}{c^4 r} \left( \frac{2\pi}{P} \right)^{2/3} \\]Waveform:
\\[ h(t) \approx h \sin(2\pi f t), \quad f = \frac{2}{P} \\]Where:
- \\(m_1, m_2\\): Masses of the binary objects (kg)
- \\(\mu\\): Reduced mass (kg)
- \\(M\\): Total mass (kg)
- \\(r\\): Distance to observer (m)
- \\(P\\): Orbital period (s)
- \\(f\\): Gravitational wave frequency (Hz)
- \\(G\\): Gravitational constant (\\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\\))
- \\(c\\): Speed of light (\\(2.99792458 \times 10^8 \, \text{m/s}\\))
- \\(h\\): Dimensionless strain
Example Calculation
Example: Binary Black Holes (\\(m_1 = m_2 = 1.989 \times 10^{31} \, \text{kg}, r = 10 \, \text{Mpc}, P = 0.01 \, \text{s}\\))
\\[
\mu = \frac{(1.989 \times 10^{31}) \times (1.989 \times 10^{31})}{1.989 \times 10^{31} + 1.989 \times 10^{31}} \approx 9.945 \times 10^{30} \, \text{kg}
\\]
\\[
M = 1.989 \times 10^{31} + 1.989 \times 10^{31} = 3.978 \times 10^{31} \, \text{kg}
\\]
\\[
h \approx \frac{4 \times (6.67430 \times 10^{-11})^2 \times 9.945 \times 10^{30} \times 3.978 \times 10^{31}}{(2.99792458 \times 10^8)^4 \times (10 \times 3.086 \times 10^{22})} \left( \frac{2\pi}{0.01} \right)^{2/3} \approx 1.06 \times 10^{-21}
\\]