Calculator inputs
Choose a method. The page stays single-column overall, while the calculator uses three columns on large screens, two on smaller screens, and one on mobile.
Example data table
These examples show how different input methods map to common orbital plane types.
| Method | Sample inputs | Inclination | Class | Comment |
|---|---|---|---|---|
| State vectors | r=(7000, 0, 0), v=(0, 7.5, 0) | 0.000° | Equatorial | Angular momentum points along +z. |
| State vectors | r=(7000, 0, 0), v=(0, 0, 7.5) | 90.000° | Polar | Angular momentum lies in the equatorial plane. |
| Angular momentum | h=(12000, 3000, -40000) | ≈162.74° | Inclined Retrograde | Negative hz implies retrograde motion. |
| Launch geometry | Latitude 28.5°, Azimuth 90° | 28.500° | Inclined Prograde | Typical eastward launch from a mid-latitude site. |
Formula used
1) From state vectors
First compute the specific angular momentum vector:
h = r × v
Then compute its magnitude:
|h| = √(hx2 + hy2 + hz2)
Inclination follows from the z-component:
i = cos-1(hz / |h|)
2) From angular momentum vector
If the h-vector is already known, the same angular relation applies directly:
i = cos-1(hz / |h|)
This is efficient when orbital elements or propagator outputs already provide h.
3) From launch geometry
With latitude φ and azimuth A, measured clockwise from north:
cos(i) = cos(φ) × sin(A)
i = cos-1(cos(φ) × sin(A))
How to use this calculator
- Select the input method that matches your data source.
- For state vectors, enter inertial position and velocity components with consistent units.
- For angular momentum mode, enter the three h-vector components directly.
- For launch geometry, enter site latitude and azimuth measured clockwise from north.
- Press Calculate Inclination to place the result above the form.
- Review the classification, motion sense, offsets, and interpretation note.
- Use the CSV button for spreadsheet-friendly output.
- Use the PDF button to save a formatted summary report.
Frequently asked questions
1) What does orbit inclination measure?
Inclination measures the tilt of an orbit relative to the reference equatorial plane. A zero-degree orbit is equatorial, ninety degrees is polar, and angles above ninety degrees are retrograde.
2) Why is angular momentum used to find inclination?
The angular momentum vector is perpendicular to the orbital plane. Its z-component tells you how strongly the plane aligns with the equatorial axis, so it naturally yields inclination.
3) What is the difference between prograde and retrograde?
Prograde orbits have inclination below ninety degrees and move with Earth’s rotation direction. Retrograde orbits exceed ninety degrees and move opposite the planet’s rotation.
4) Can launch latitude limit achievable inclination?
Yes. A launch site cannot directly reach every inclination without plane changes. Latitude and azimuth strongly constrain the initial orbital plane that the rocket can enter.
5) Why does a polar orbit have ninety degrees inclination?
A polar orbit crosses near both poles, so its plane is perpendicular to the equatorial plane. That geometry produces an inclination of approximately ninety degrees.
6) Are the state vector units important?
Yes, but they only need to be consistent. Position and velocity units must match the angular momentum calculation. The final inclination angle is unit-independent after the ratio is formed.
7) Does this calculator include J2 or perturbation effects?
No. It computes the geometric inclination from the supplied vectors or launch geometry. Long-term precession and perturbation modeling require a separate orbital propagation workflow.
8) Can this help check sun-synchronous candidates?
It can flag inclinations near common sun-synchronous bands, but true sun-synchronous behavior also depends on altitude, eccentricity, and Earth’s oblateness-driven nodal precession.