Conic Sections and Orbits

Introduction

In Conic Sections Introduction, it has been shown that the trajectory gravitational bodies make around other, more massive bodies is one of the four conic sections. That is assuming it is a two body problem, bare in mind. This relationship holds due to energy considerations of the massive bodies, but such a derivation is beyond the scope of what is discussed here.

The eccentricity of an orbit is an interesting property of its shape. You can think of it like its 'squishyness'. The orbit is a circle when the eccentricity is zero, think of it as not being squished at all. The orbit is an ellipse when the eccentricity is between zero and one. When it is exactly equal to one, it is a parabola. And finally when the eccentricity is greater than one, it is a hyperbola. Thus showing how this one property bring forth all the possible orbits which are too conic sections.

The Importance of Conic sections in Relation to Orbits

Halley's comet, one of the most prominent sights appearing to human eyes across various points in our history, follows an ellipitical orbit with a relatively high eccentricity. It is because we know of its shape (as well as its velocity), we are able to predict when it will next grace the Earth. If it were a hyperbola for instance, we would see it once and then never again. But because of its ellipitcal nature we repeatedly see it and with the advances of modern astronomy at predictable times.

Conic sections are fundamentally linked to orbits every way, and by examining the mathematical properties of the former we can make real-world observations and predictions about the latter. This doesn't just benefit astrophysicists, but is integral to everyday life. For example in the prediction of lunar cycles, with the aid of the geometry we have been discussing, the exact date of Easter Sunday can be known much further in advance than it could in a past long before Kepler.

Real-world applications are not simply limited to calendar-related affairs but also in certain circumstances to public safety. The prediction of lunar cycles, which we mentioned prior, allows meteorologists to calculate the level of tides. Furthermore in rarer cases dangerous asteroids can potentially be foreseen by astronomers and prepared for if necessary, and NASA tested such capabilities in their 2022 DART mission.