Calculator
Example Data Table
These sample cases illustrate how exhaust speed and mass ratio strongly affect near-light mission performance.
| Scenario | β₀ | βₑ | m₀ (kg) | m₁ (kg) | Mass Ratio | Approx. βf | Gamma γf |
|---|---|---|---|---|---|---|---|
| Deep Probe A | 0.000000 | 0.100000 | 50,000 | 20,000 | 2.500000 | 0.091033 | 1.004170 |
| Fusion Craft B | 0.050000 | 0.300000 | 120,000 | 30,000 | 4.000000 | 0.424435 | 1.104600 |
| Extreme Mission C | 0.100000 | 0.700000 | 200,000 | 20,000 | 10.000000 | 0.943078 | 3.010560 |
Formula Used
η = atanh(β) = 0.5 × ln((1 + β) / (1 - β))
ηf = η₀ + βₑ × ln(m₀ / m₁)
βf = tanh(ηf)
γ = 1 / √(1 - β²)
γβ
Coordinate time = D / βf
Proper time = (D / βf) / γf
ΔK = (γf - γ₀) × m₁ × c²
This implementation uses an idealized Ackeret-style relativistic rocket model. It is excellent for fast comparison, but it does not replace a full mission simulation.
How to Use This Calculator
- Select whether you want to solve for final velocity or required mass ratio.
- Enter the initial velocity fraction and exhaust velocity fraction as values between 0 and 1.
- Provide the starting mass of the spacecraft.
- Enter final mass if solving forward, or target velocity if solving backward.
- Add payload mass if you want payload fraction outputs.
- Enter mission distance in light-years to estimate travel time.
- Press the calculate button to display results above the form.
- Use the CSV or PDF buttons to export the generated mission summary.
FAQs
1) What does this calculator solve?
It estimates final relativistic speed, gamma, mass ratio, propellant mass, payload fractions, kinetic energy change, and travel time for idealized near-light rocket missions.
2) Why use rapidity instead of ordinary delta-v?
Rapidity adds cleanly under relativistic motion. That makes it more reliable than classical delta-v when the spacecraft approaches a significant fraction of light speed.
3) What is exhaust velocity fraction βₑ?
βₑ is the effective exhaust speed divided by light speed. Higher values dramatically improve achievable velocity and reduce the mass ratio needed for the same mission.
4) Does the tool include gravity losses or staging?
No. It assumes an ideal one-dimensional rocket in free space with constant effective exhaust speed. Real missions need added margins and more detailed modeling.
5) What does gamma represent?
Gamma is the Lorentz factor. It shows how strongly time dilation and relativistic energy rise as the rocket’s velocity approaches light speed.
6) How is travel time estimated?
Travel time uses the final computed cruise speed across the full entered distance. It does not separately model acceleration, coasting, or braking segments.
7) Why can the required mass ratio become huge?
Near light speed, each extra gain in rapidity becomes expensive. Even efficient exhaust still demands enormous propellant mass for modest additional speed.
8) Can I use this for interstellar concept studies?
Yes. It is useful for early feasibility checks, quick comparisons, and educational analysis of extreme missions before building a more detailed trajectory model.