Calculator Inputs
Formula Used
Semi-minor axis: b = a√(1 − e²)
Linear eccentricity: c = ae
Semi-latus rectum: p = a(1 − e²)
Periapsis distance: q = a(1 − e)
Apoapsis distance: Q = a(1 + e)
Unit vector: u = (cos θ, sin θ)
Normal vector: v = (−sin θ, cos θ)
Periapsis vertex: C + au
Apoapsis vertex: C − au
Co-vertices: C ± bv
Kepler radius: r = a(1 − e²) / (1 + e cos ν)
Optional period: T = 2π√(a³ / μ)
How to Use This Calculator
Enter the semi-major axis and eccentricity first.
Select whether your reference point is the primary focus or center.
Enter the reference coordinates.
Add the periapsis axis angle in degrees.
Add a true anomaly when you need an orbit point.
Enter μ only when period and mean motion are required.
Press Calculate to show the result below the header.
Use CSV or PDF buttons to save the same result.
Example Data Table
| Case | a | e | Reference | Angle | True Anomaly | Expected Use |
|---|---|---|---|---|---|---|
| Simple orbit | 10 | 0.4 | Focus at 0, 0 | 25 | 60 | Basic Kepler vertex check |
| Centered ellipse | 12 | 0.25 | Center at 2, 3 | 45 | 120 | Rotated coordinate layout |
| Near circular | 8 | 0.05 | Focus at 1, 1 | 0 | 30 | Low eccentricity comparison |
Understanding Kepler Ellipse Vertices
Kepler orbits use an ellipse with the attracting body at one focus. This detail matters. The center is not the same point as the focus when eccentricity is greater than zero. A vertex is an end point of the major axis. The periapsis vertex is the closest point to the focus. The apoapsis vertex is the farthest point.
Why Vertices Matter
Vertices describe the widest path of the orbit. They also show the nearest and farthest orbital distances. These distances help with simple mission sketches, astronomy lessons, geometry checks, and simulation inputs. When an ellipse is rotated, the coordinates are less obvious. The calculator handles that rotation and returns usable points.
Using Kepler Data
The main inputs are semi major axis, eccentricity, reference point, and axis angle. The semi major axis sets half the long diameter. Eccentricity sets how stretched the ellipse is. A value near zero gives a circle. A value close to one gives a narrow orbit. The angle points toward periapsis. This lets the tool place the vertices in any plane.
Coordinate Meaning
If the reference point is a focus, the tool treats it as the primary body. It then shifts the ellipse center backward by c equals a times e. If the reference point is the center, it builds both foci around that center. The same formulas then return periapsis, apoapsis, co vertices, radius at true anomaly, area, and perimeter.
Practical Use
Use consistent units. If distance is in kilometers, coordinates are also kilometers. If a gravitational parameter is entered, it must match the same distance and time units. The period result is optional. It is useful only when the parameter is correct. Always review the eccentricity range. Closed Kepler ellipses need eccentricity from zero up to, but not including, one. The downloadable files help save results for homework, reports, and quick comparisons.
Limits and Checks
The tool is for closed elliptical paths. It does not model parabolic or hyperbolic escape paths. It also does not replace numerical orbit propagation. Use it for geometry, first estimates, clear diagrams, and final reports. Compare important work with trusted sources every time. Small input changes can move rotated vertices noticeably, so keep enough decimal places.
FAQs
What are ellipse vertices in a Kepler orbit?
They are the two endpoints of the major axis. In orbital terms, they represent periapsis and apoapsis when the focus is the attracting body.
What does eccentricity control?
Eccentricity controls how stretched the ellipse is. Zero gives a circle. Values closer to one create a longer and narrower ellipse.
Should I enter the focus or the center?
Use focus when your data starts from the attracting body. Use center when your geometry problem gives the ellipse center directly.
What is the periapsis axis angle?
It is the rotation angle of the major axis. The calculator uses it to place the vertices and foci on rotated coordinates.
What is true anomaly?
True anomaly is the angle of an orbiting point measured from periapsis. It helps find a position on the ellipse.
Why is μ optional?
The gravitational parameter is only needed for period and mean motion. Coordinate and vertex results do not require it.
Can this handle hyperbolic paths?
No. This calculator is designed for closed ellipses. Eccentricity must be at least zero and less than one.
What units should I use?
Use one consistent unit system. If the semi-major axis is in kilometers, all coordinate results are also in kilometers.