Relativistic Doppler Shift Calculator
Relativistic Doppler Shift Calculator computes wavelength/frequency shift for a moving source, with a spectral plot.
nm
Formulas Used in Relativistic Doppler Shift Calculator
The calculator computes the relativistic Doppler shift for a source moving relative to an observer:
Frequency Shift:
\\[ f_{\text{obs}} = f_0 \frac{\sqrt{1 – \beta^2}}{1 – \beta \cos\theta}, \quad \beta = \frac{v}{c} \\]Wavelength Shift:
\\[ \lambda_{\text{obs}} = \lambda_0 \frac{1 – \beta \cos\theta}{\sqrt{1 – \beta^2}} \\]Redshift:
\\[ z = \frac{\lambda_{\text{obs}}}{\lambda_0} – 1 \\]Where:
- \\(f_{\text{obs}}\\), \\(\lambda_{\text{obs}}\\): Observed frequency (Hz) and wavelength (nm)
- \\(f_0\\), \\(\lambda_0\\): Emitted frequency (Hz) and wavelength (nm)
- \\(v\\): Relative velocity (m/s)
- \\(c\\): Speed of light (\\(2.99792458 \times 10^8 \, \text{m/s}\\))
- \\(\theta\\): Angle between velocity vector and line of sight (degrees)
- \\(\beta = v/c\\): Velocity relative to speed of light
- \\(z\\): Redshift (positive for redshift, negative for blueshift)
Example Calculation
Example: H\\(\alpha\\) line (\\(\lambda_0 = 656.3 \, \text{nm}\\), \\(v = 30000 \, \text{km/s}\\), \\(\theta = 180^\circ\\))
\\[
\beta = \frac{30000 \times 10^3}{2.99792458 \times 10^8} \approx 0.1
\\]
\\[
\lambda_{\text{obs}} = 656.3 \frac{1 + 0.1 \cos(180^\circ)}{\sqrt{1 – 0.1^2}} \approx 726.2 \, \text{nm}
\\]
\\[
f_{\text{obs}} = \frac{2.99792458 \times 10^8}{726.2 \times 10^{-9}} \approx 4.129 \times 10^{14} \, \text{Hz}
\\]
\\[
z = \frac{726.2}{656.3} – 1 \approx 0.107
\\]