Ch. 11.2 Ellipses

This paragraph introduces the concept of an ellipse as a conic section formed by an angled plane intersecting a cone. It distinguishes an ellipse from a circle by explaining that an ellipse is an elongated circle due to the non-parallel orientation of the intersecting plane. The paragraph also introduces the terms 'focus' and 'foci', explaining that an ellipse has two foci and the constant sum of distances from any point on the ellipse to each focus. It describes the major and minor axes of an ellipse, their lengths, and how they pass through the ellipse's center. The orientation of the ellipse and the axes is also discussed, with examples of how the major axis can be aligned with or rotated relative to the coordinate axes.

The second paragraph delves deeper into the geometry of ellipses, discussing the relationship between the ellipse's foci, vertices, and axes. It explains how the major axis is the longest distance across the ellipse and the minor axis is the shortest, with both axes intersecting at the ellipse's center. The paragraph provides a detailed explanation of how to determine the location of the foci in relation to the major axis and introduces the concept of using the Pythagorean theorem to find the distance 'c' from the center to the foci, given the lengths of the semi-major axis 'a' and the semi-minor axis 'b'. It also touches on the derivation of the ellipse equation based on the constant sum of distances from a point on the ellipse to the foci.

This paragraph focuses on the standard equation of an ellipse and how it can be used to determine the lengths of the semi-major and semi-minor axes, as well as the orientation of the ellipse. It explains that the larger denominator under the squared variables indicates the major axis and its corresponding vertex location. The paragraph provides an example of how to manipulate an equation into the standard form to identify the values of 'a' and 'b', and how to find the foci's location using the values of 'a' and 'b'. It also discusses the eccentricity of an ellipse, which is a measure of how elongated the ellipse is, and how it relates to the distance between the foci and the vertices.

The fourth paragraph applies the concept of ellipses to orbital mechanics, describing how most orbits in space are elliptical, including those of the moon, satellites, and planets. It uses the historical example of the Apollo 11 mission's lunar orbit to illustrate how to derive the equation of an elliptical orbit. The paragraph explains the terms 'perilune' and 'apolune', referring to the closest and furthest points of an orbit from the celestial body, respectively. It provides a step-by-step calculation to find the equation of Apollo 11's orbit around the moon, considering the moon's radius and the spacecraft's distances at perilune and apolune.

This paragraph continues the application of ellipses in orbital mechanics by calculating the specific parameters of Apollo 11's orbit around the moon. It explains how to determine the length of the major axis and the location of the foci using the given distances from the moon's surface at perilune and apolune, as well as the moon's radius. The paragraph demonstrates the process of finding the semi-major axis 'a', the semi-minor axis 'b', and the distance 'c' to the foci by manipulating the standard form of the ellipse equation and applying the Pythagorean theorem.

The final paragraph discusses the concept of eccentricity in the context of Apollo 11's orbit, explaining its significance in determining the shape of the orbit and how circular or elongated it is. It provides a brief analysis of the eccentricity based on the calculated values of 'a' and 'c', and comments on the near-circular nature of the orbit due to the small difference between the squares of 'a' and 'b'. The paragraph concludes with a general observation on the importance of understanding ellipses in various applications, including space orbits.