At school we have spent a lot of time learning the equations of circumferences, ellipses, parables, and hyperboles, and many times, we asked ourselves: “When will I ever use these… what do you call them? Uh, right, conic sections.”
Conic sections are found in many real-word situation, one of them is the calculation of celestial body orbits.
In 2017 they helped us understand where a strange asteroid named Oumuamua came from, so let’s see how.
Conic sections are the curves obtained by sectioning a cone with a plane.
To visualize ‘conics’ there is a very simple way: just project the light of a flashlight onto a wall.
The light created by the flashlight has the shape of a cone and the wall represents the plane that intersects this cone.
If we hold the flashlight perpendicular to the wall, we get a circle (top left image).
If we slightly turn the flashlight while continuing to project the whole beam onto the wall, we obtain ellipses (top right image).
When the part of the beam that is farthest from the wall is parallel to the wall, the light forms a parabola (bottom left image).
If we further turn the flashlight outward, we obtain a hyperbole (bottom right image).
The Orbits of Celestial Bodies
A body attracted by the gravitational field of another body with a much larger mass moves along trajectories that are conical sections.
But what’s the reason why, sometimes, the orbit is an ellipse, as with planets, while other times it’s a different conic?
It’s not easy to explain this without going into technical details, to keep things simple we can say that the type of conic depends on how much energy the body has. For example:
1. If the body does not have enough kinetic energy to escape the gravitational field of the heavier body, then it will follow a circumference or an ellipse.
2. If the body has enough kinetic energy to completely escape the gravitational field of the heavier body moving further and further away, then it will follow a hyperbola.
3. If the body has a precise energy which represents the critical value between the first and second scenario above, then it will follow a parabola, moving further and further away, but with its speed tending towards zero.
Here are some examples of these three cases.
Planet motion belongs to case 1. Planets do not travel at a fast-enough speed to escape the gravitational field of the Sun, so they follow elliptical orbits. The same consideration applies to artificial satellites that orbit around the Earth, so they also follow circular or elliptical orbits.
Space probes performing a flyby of a planet follow hyperbolic orbits because they have enough energy to escape the planet’s gravitational field, so their path belongs to case 2.
Non-periodic comets are examples of case 3. They are comets that originate at the most extreme boundaries of the solar system, complete a single orbit around the Sun and then return just as far from where they started following a parabolic trajectory.
The Oumuamua Comet
In October 2017 an asteroid with an anomalous orbit was sighted. After a month of observations, astronomers found it didn’t follow a common elliptical orbit around the Sun but it was traveling on a hyperbolic orbit.
The object was initially considered an asteroid, then in June 2018 more detailed analysis clarified it was a comet.
It was the first time that anyone had observed a comet with this kind of orbit and the conclusion was clear: the comet came from the outside of our solar system.
This 230-meter-long piece of rock is making an incredible journey inside our galaxy, visiting different star systems from time to time.
Having been discovered by an astronomical observatory located in the Hawaiian archipelago, the comet has been named “Oumuamua” which in the local language, roughly translated, means “first distant messenger”.
Oumuamua is now moving away from our solar system at great speed and who knows how many other planetary systems it will visit in its wanderings through our galaxy.
And here they are again. The good old conics that made us struggle at school, yet which appear every time we use a flashlight, or we see the beam of a street light against a wall. They help us understand the world around us and draw conclusions which, however bizarre, represent the logical consequence of our observations.
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NB: Regarding the three examples of conics (planets, long-period comets, and the artificial satellite flyby), the trajectories are very close but not exactly equal to ellipses, parabolas, and hyperbolae. The main reason is that each body doesn’t only interact with the heavier body that is orbiting around, but also with all the other planets and celestial objects in the solar system!