Calculator Inputs
Formula Used
This calculator uses the Newtonian analogue of geodesic deviation for a small separation vector near a central gravitating mass. The gravitational potential is Φ(r) = -GM / r.
The local tidal equations are:
d²ξr / dt² = (2GM / r³) ξrd²ξt / dt² = -(GM / r³) ξt
Define ω = √(GM / r³) and α = √2 · ω.
The exact local solutions used here are:
ξr(t) = ξr0 cosh(αt) + (vr0 / α) sinh(αt)ξt(t) = ξt0 cos(ωt) + (vt0 / ω) sin(ωt)
Relative accelerations follow directly from the tidal coefficients:
ar = (2GM / r³) ξrat = -(GM / r³) ξt
This approach is useful for weak-field intuition, orbital mechanics, and small-offset motion studies. It is a Newtonian tidal approximation, not a full relativistic spacetime solver.
How to Use This Calculator
- Enter the central mass and pick a mass unit.
- Enter the reference radius from the mass center.
- Provide initial radial and tangential separations.
- Add any initial relative velocities for both directions.
- Choose an observation time and chart resolution.
- Press calculate to show results above the form.
- Review the graph, summary table, and stretch ratio.
- Download CSV or PDF for reporting or later comparison.
Example Data Table
Example case: Earth mass, 6771 km reference radius, 10 m radial offset, 10 m tangential offset, zero initial relative velocities, 60 seconds.
| Example Metric | Value | Unit |
|---|---|---|
| Angular frequency | 0.001133 | rad/s |
| Radial tidal coefficient | 2.568098e-6 | 1/s² |
| Tangential tidal coefficient | -1.284049e-6 | 1/s² |
| Final radial separation | 10.046261 | m |
| Final tangential separation | 9.976896 | m |
| Final total separation | 14.158595 | m |
| Final relative speed | 0.001725 | m/s |
| Stretch ratio | 1.001164 | ratio |
Frequently Asked Questions
1. What does this calculator estimate?
It estimates how a small separation between nearby particles changes over time near a central mass. It reports radial stretching, tangential motion, relative velocity, relative acceleration, and total separation using a Newtonian tidal model.
2. Is this an exact relativity calculator?
No. It is a Newtonian analogue of geodesic deviation. The page is useful for intuition, weak-field estimates, and small-offset orbital behavior, but it does not solve the full general relativistic geodesic deviation equation.
3. Why does radial separation often increase?
In a central gravity field, a point slightly farther outward feels a weaker pull than a point slightly inward. That difference stretches the pair along the radial direction, producing a positive radial tidal coefficient.
4. Why can tangential separation oscillate?
The local tangential equation behaves like a restoring mode under the Newtonian tidal approximation. That makes sideways separation evolve with sine and cosine terms, which often look oscillatory over the chosen time window.
5. Which units can I use?
You can choose multiple units for mass, radius, separation, velocity, and time. The calculator converts everything internally into SI units, so mixed input units stay consistent during the computation.
6. When are these results most reliable?
They are most reliable when the separation is small compared with the main radius and the field does not vary much across the offset. Large separations need a more complete dynamical model.
7. What happens if initial relative velocity is not zero?
The initial relative velocity directly enters the exact local solution. Positive or negative starting velocity can amplify, reduce, or reverse the later offset trend, depending on direction and observation time.
8. Can I export the results?
Yes. After calculation, download the results as CSV for spreadsheet work or as PDF for sharing. The export tools capture the computed values and the visible summary content.