Example Data Table
These examples assume light in vacuum and θ set by the direction.
| Scenario | Source (GHz) | β | θ (deg) | Doppler factor D | Observed (GHz) | Redshift z |
|---|---|---|---|---|---|---|
| Approaching at 0.30c | 1.000 | 0.300 | 0 | 1.3628 | 1.3628 | -0.2663 |
| Receding at 0.30c | 1.000 | 0.300 | 180 | 0.7338 | 0.7338 | 0.3628 |
| Transverse at 0.60c | 1.000 | 0.600 | 90 | 1.2500 | 1.2500 | -0.2000 |
Tip: θ = 90° highlights the time‑dilation part of the shift.
Formula Used
- β = v / c, where c = 299,792,458 m/s.
- γ = 1 / √(1 − β²).
- Doppler factor (light): D = γ(1 + β cosθ)
- Observed frequency: fobs = fsrc · D
- Observed wavelength: λobs = λsrc / D
- Redshift: z = (λobs − λsrc) / λsrc
Angle θ is measured from the velocity direction: 0° toward the observer, 180° away.
How to Use This Calculator
- Select Observed from source to predict the shift, or Velocity from shift to estimate motion.
- Choose Frequency or Wavelength as your input type.
- Pick a direction preset or enter a custom angle between 0° and 180°.
- Enter your source value (and observed value for inverse mode).
- For forward mode, enter speed as β, km/s, or m/s.
- Optionally add uncertainty percentages to see a conservative output range.
- Press Calculate. Use the export buttons to download results.
Relativistic Doppler Effect Calculator Article
Why relativistic corrections matter
Classical Doppler formulas work when speeds are small. At high speeds, time dilation changes the emission rate. Relativistic treatment keeps results physical and consistent with the speed limit.
What the calculator computes
The calculator links a source signal to what an observer measures. It outputs the Doppler factor, the shifted frequency or wavelength, and a redshift z summary. In inverse mode, it estimates β from measured shift.
Choosing frequency or wavelength
Use frequency when your detector counts cycles per second. Use wavelength when you measure spacing between crests. The tool converts common units and reports the equivalent value in base units for checking.
Speed and angle inputs
Speed can be entered as β, meters per second, or kilometers per second. The angle θ describes motion relative to the line of sight. Set θ = 0° for approach and θ = 180° for recession.
Angle and transverse shift
At θ = 90°, the classical shift disappears, yet relativity still predicts a redshift from time dilation. This transverse case is a useful sanity check. If your 90° result looks like a blueshift, review units and input type.
Reading forward-mode results
Forward mode starts from speed and predicts the observed signal. A Doppler factor above one means blueshift for frequency and a shorter observed wavelength. A factor below one means redshift for frequency and a longer observed wavelength.
Using inverse mode safely
Inverse mode solves for β using your source and observed values. The calculator enforces |β| < 1 and warns when inputs imply impossible motion. If you hit a warning, confirm that the observed value matches the chosen direction and angle.
Quick worked example
Imagine a line emitted at 500 nm and observed at 550 nm along θ = 180°. That is a redshift, so the source recedes. Enter wavelengths and inverse mode to estimate β and speed. Swap to θ = 0° to see the opposite trend.
Where it is used
Astronomers compare spectral lines to infer recession and approach. High-speed radar targets and precision timing systems use the same physics. Particle beams and synchrotron sources also depend on relativistic shifts for diagnostics.
Practical accuracy tips
Add uncertainty percentages to see conservative bounds when measurements have tolerances. Keep unit conversions consistent, especially MHz versus GHz and nm versus µm. For small β, results should closely match classical expectations.
Assumptions and limitations
The model assumes special relativity and steady relative velocity during measurement. It does not include gravitational redshift, medium effects, or acceleration within the observation window.