Eccentricity Formula
Eccentricity Formula - Related Videos
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FAQs
The expressions e = c/a and e = √(1 − b²/a²) describe the same geometric quantity from different measurements. The first uses the center-to-focus distance, while the second uses the semimajor and semiminor axes. This equivalence allows eccentricity to be found either from focal geometry or from the ellipse’s measured dimensions.
With a held constant, decreasing b makes the term b²/a² smaller, so the value of e increases. Geometrically, the ellipse becomes more elongated as its semiminor axis shrinks relative to its semimajor axis. Increasing b has the opposite effect, bringing the shape closer to the circular case represented by e = 0.
In orbital mechanics, the value of e classifies the trajectory: an ellipse has eccentricity below 1, a parabola has e = 1, and a hyperbola has e greater than 1. This classification extends the formula beyond closed elliptical orbits and provides a compact way to identify the geometric type of an orbit.
Eccentricity also indicates how strongly an orbit departs from circular symmetry. A smaller value corresponds to a less elongated ellipse, whereas a value closer to 1 signals greater elongation. Because orbital distance changes along a noncircular path, this single dimensionless parameter helps connect conic geometry with the changing separation between an orbiting body and its focus.
First identify the semimajor axis a and semiminor axis b, ensuring that a is the larger axis. Substitute them into e = √(1 − b²/a²), then evaluate the square root. The result is dimensionless, so the same units may be used for both axes without conversion, provided their ratio is formed consistently.
Use e = c/a when the distance from the ellipse’s center to a focus and the semimajor axis are available. Use e = √(1 − b²/a²) when the two axis lengths are easier to measure. Choosing the form that matches the available geometric data simplifies calculations while preserving the same eccentricity.