Name: ellipse shape and orbits explained
Uploaded: 2025-11-26T05:44:07.000Z
Duration: 1 min 27 s
Channel: JoVE

Tracing an ellipse with a pencil and a string of fixed length around two fixed pins illustrates a path where the combined distance to the pins remains constant.This defines the geometry of an ellipse, while its shape depends on its eccentricity, e&#8212;the ratio of the focal distance to the semi-major axis length.When e is zero, the ellipse is a perfect circle. As e increases, the ellipse stretches along the major axis.This shape appears in orbital paths such as that of Halley&#8217;s Comet, which has an eccentricity of 0.967 and the Sun at one focus.The major axis of the orbit measures approximately 35.88 astronomical units&#8212;a standard unit of measurement in astronomy.To express Halley&#8217;s orbit in standard form, the ellipse is centered at the origin, with its major axis aligned along the x-axis.Multiplying the semi-major axis by e gives the focal distance.The semi-minor axis length is then calculated using the Pythagorean theorem, which relates it to the semi-major axis and the focal distance, completing the overall shape of the ellipse.These values are substituted into the standard form to model the orbit.

eccentricity of an ellipse

An ellipse is a fundamental conic section defined by the constant sum of distances from any point on its curve to two fixed points, known as the foci. This geometric property can be physically demonstrated using a pencil, string, and two pins. By anchoring the string at both ends and maintaining it taut with a pencil, one can trace the outline of an ellipse.The shape and extent of the ellipse are determined by its eccentricity, e, defined as the ratio of the distance between the center and a focus (c) to the semi-major axis length (a). When e = 0, the ellipse is a circle. As e increases toward 1, the ellipse elongates along its major axis.This property underpins the elliptical nature of celestial orbits. Halley&#8217;s Comet, for instance, travels along a highly eccentric elliptical orbit with e = 0.967. The major axis of its orbit measures approximately 35.88 astronomical units (AU), a common unit in astronomy equivalent to the average distance between Earth and the Sun. This means the semi-major axis, a, is 17.95 AU. The distance from the center of the ellipse to each focus (the focal distance) is given by a &#215; e. For Halley&#8217;s Comet, this results in<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:58.82%;'><img style='aspect-ratio:1027/59;' src='https://cloudfront.jove.com/files/ftp_upload/41036/41036_Equation_1_editor_41036.svg' alt='41036_Equation_1.svg' /></figure>The semi-minor axis b can be computed using the relationship<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:57.91%;'><img style='aspect-ratio:1605/143;' src='https://cloudfront.jove.com/files/ftp_upload/41036/41036_Equation_2_editor_41036.svg' alt='41036_Equation_2.svg' /></figure>Placing the ellipse in standard form, centered at the origin and aligned along the x-axis, the orbital equation becomes:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:56.04%;'><img style='aspect-ratio:731/200;' src='https://cloudfront.jove.com/files/ftp_upload/41036/41036_Equation_3_editor_41036.svg' alt='41036_Equation_3.svg' /></figure>From this model, the perihelion distance (closest approach to the Sun) is a &#8722; c &#8776; 0.59 AU, and the aphelion distance (farthest point from the Sun) is a + c &#8776; 35.29 AU. This substantial variation illustrates Halley&#8217;s orbit&#8217;s pronounced eccentricity and the usefulness of its elliptical geometry in celestial mechanics.

Eccentricity of an Ellipse

An ellipse is a fundamental conic section defined by the constant sum of distances from any point on its curve to two fixed points, known as the foci. ...

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ellipse shape and orbits explained

An ellipse is a key conic section with two foci, or fixed points inside the shape. For any point on the ellipse, the sum of the distances to the two foci stays the same. A pencil, string, and two pins can show this shape in a simple way. If the string is held tight while the pencil moves, it traces an ellipse.

The size and shape of an ellipse depend on its eccentricity, written as e. Eccentricity is the ratio of the distance from the center to a focus, c, to the semi-major axis length, a. When e = 0, the ellipse is a circle. As e gets closer to 1, the ellipse stretches more along its major axis.

This idea helps explain the paths of celestial objects. Elliptical orbits are common in astronomy, and Halley’s Comet is a well-known example. Its orbit has a high eccentricity of 0.967, so it is very elongated. The major axis of the orbit is about 35.88 astronomical units, or AU, which is a unit equal to the average distance between Earth and the Sun.

From that value, the semi-major axis is 17.95 AU. The focal distance is found by multiplying a by e, so c = a × e. The semi-minor axis, b, can also be calculated from the ellipse relationships. When the ellipse is written in standard form, centered at the origin and aligned along the x-axis, it matches the orbital model used for the comet.

That model also gives the comet’s distance from the Sun at different points in its orbit. The perihelion, or closest approach to the Sun, is about 0.59 AU. The aphelion, or farthest point, is about 35.29 AU. These large changes in distance show why Halley’s Comet is a strong example of a highly eccentric ellipse in celestial mechanics.

Ellipse Shape and Orbits Explained

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An ellipse is a key conic section with two foci, or fixed points inside the shape. For any point on the ellipse, the sum of the distances to the two foci stays the same. A pencil, string, and two pins can show this shape in a simple way. If the string is held tight while the pencil moves, it traces an ellipse. The size and shape of an ellipse depend on its eccentricity, written as e. Eccentricity is the ratio of the distance from the center to a focus, c, to the semi-major axis length, a. When...

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key terms and concepts

Learn how ellipse shape changes with eccentricity and how that math models Halley’s Comet, from foci and axes to perihelion and aphelion.

Ellipse Shape and Orbits Explained | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Ellipse Shape and Orbits Explained

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