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Sunday, 4 November 2012

Kepler's Laws of Planetary Motion

Last time, we looked at the basics of getting into orbit, namely the speed you would need, and the amount of fuel. And while I made mention that the next post in this series would look at the different orbits and launch site considerations, I decided to insert this brief coverage of Kepler's three laws of planetary motion. When I was drafting the post about the different types of orbits, these laws kept coming back to me and so I think we should have a look at them in order to better understand how orbits work.

The first commonly held misconception is that orbits are circular. Thanks to the tireless work of astronomer Johannes Kepler, and others, we know this not to be the case, most of the time.

A tireless astronomer, Kepler watched the skies and made careful notes as to what he saw, night after night. Eventually, he noticed some patterns and his documentation of them eventually became Kepler's laws of planetary motion.

The first law is that all naturally occurring orbits, whether it concern planets, moons, or asteroids, all are elliptical, and thus, the distance between the orbiting and orbital body increases and decreases over time.

I've included a picture below to help illustrate this. Trust me, it was the best one I could find and while I could make one for you, it would appear more or less the same.

So, let's take a look at this. Earth's orbit is elliptical as it goes around the Sun. At one time of the year, the Earth is the closest it can get to the Sun, called the perihelion, and later in the year, the Earth is at its farthest point, known as the aphelion.

One quick thing to point out, look at the dates. Some people have the misconception that the distance from Earth to the Sun determines the season, when in fact, you'll notice that it's summer in the northern half of the Earth when it's farthest from the Sun. It's actually the tilt of the Earth which causes the seasons.

Now, these terms and conditions also apply to an object in orbit around the Earth, case in point, the Moon. So, if you were interested in getting into orbit around the Earth, you should probably consider this.

The second law, is that a line joining the two bodies in orbit, sweeps out equal areas in equal times. Once again, a picture is included below:
This one was a little confusing to me at first, so allow me to explain. The purple segments above, taken by looking at the distance the Earth has moved in its orbit after a month, are supposed to be equal in area. The better way of explaining this is that the closer the two bodies are, the faster the one in orbit moves around the other.

Imagine a figure skater, with his/her arms out. When that skater spins, they move at a certain speed. When the skater brings their arms closer, he or she spins much faster. It's the same with orbiting bodies. As the orbiting object gets closer, it moves much faster in its orbit. The Russians made expert use of this, which we'll see in my next post about orbit types.

As you might imagine, Kepler's third law was that the period of the orbit, the time it takes to go around, is related to the total length of the ellipse.

So why are these laws important? Well, first off, because they work, consistently, with any body in orbit. That's one of the strength of physics, and is in fact its underlying goal; to better understand and model the universe.

If you're considering putting a satellite in orbit, you should know these laws and how they might affect your mission. The fact that all naturally occurring orbits are ellipses and not circles means that a lot of your math and predictions might be off. In fact, when I did the calculations last time, I assumed the orbit was a circle. It's not a large margin of error, especially for the casual reader, but as our standards and computing power increase, and we need to know exact orbital location to the micro or even nanosecond, well, we should start by knowing the correct shape of our orbit. You can have a nearly circular orbit but since it is not the natural default, it requires more energy to keep it that way.

The second law is important because if your orbit is more elliptical, it means your satellite will spend more time in the part of its orbit which places it farther away.

Consider a satellite meant to look at the Arctic ice, to determine how large its coverage is and to watch for changes over time. A satellite like that should really spend as much time as possible over the Arctic and as close as possible to the Earth to get the best results. But, if the orbit is too elliptical, we know that it will spend the least amount of time there. It will slingshot around the Earth and spend more of its time farther away.

The third law is important for the same reasons, but I like to think about how it relates to exoplanets. You might have heard news stories about how astronomers keep finding planets in other solar systems. You might wonder how they know so much about planets they can barely see. I mean, sure, we have some excellent telescopes, but how do we know how far a planet is from its sun? Most people cannot guess distances very well. But, knowing Kepler's third law means if we know how long it takes a planet to go around its sun, we can determine how far it is from said sun.

And that concludes our look at Kepler's three laws of planetary motion. Next time, we'll look at inclination, launch sites, and the types/uses of different orbits.

Thank you for the readership so far; I understand that this post may have been less exciting than the one prior, and didn't have nearly enough hand-drawn pictures or calculations, but there's more of that to come, so stay tuned!

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