Orbits!

OK, this is a really interesting post for me. Just stick with it, OK?

In math, there are two types of orbits:
1) An elliptical path traced by an object due to gravitational pull towards another object. AKA the orbit that you know.
2) A repetition of a function infinitely upon itself. (this term might be outdated- I found the lesson in a precalculus book at my college-aged cousins' house)

For orbit #1, an orbit is elliptical due to two factors:
1) Gravity is imperfect.
2) the way that an object enters (or forms in) the gravitational area.
Anyway, the point is that the object that it orbits around is a focus of that ellipse. The foci are the two points that determine the ellipse-- the sum of the distances from any point on the ellipse to each of the foci is constant.

If you think about it, the Sun isn't at the center of the solar system–– it's at one of the foci.

Here's a good problem about it.

Denying the laws of probability and physics itself, let's say the planets in our solar system are perfectly lined up to one side of the sun. The center of the ellipse is approximately 1.5 million km from the center of the sun. If Earth is at its closest point to the sun (148 million km), and the center of gravity of the solar system remains where it is no matter where the planets travel (which is a lie), what is the furthest point of Earth's orbit from the sun?

In the ellipse above, A and B are the foci, D is the center, and C and E are points on the major and minor axes respectively. 

Anyhow, in this case, AD is equal to 1.5 million kilometers and EA is equal to 148 million kilometers. The sun is at A and Earth is at E. 

Since the furthest point is basically the point on the other side of point B, the formula for the farthest point would be:

( (EA) + (AD) ) * 2 - (EA)

Which is about 151 million km, or an accurate-ish 93.8 million miles.

Another thing you might want to know about ellipses is that in the above example (pretend the scale is off), ED=AC=(AD)^2 + (CD)^2.

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Soooo, now to the other orbit.

Function orbits are similar to orbits: they go around and around a particular value. Now, to explain them.

Imagine a function f(x) for any value x1. Now imagine f(f(x1)). Then f(f(f(x1))). Continue this pattern until you are 106 years old. The orbit of f(x) for that value of x1 would be the infinite sequence 

x, f(x), f(f(x)).... on until you are 106. Or really, infinity years old. The last (or technically last) term of that sequence would be called the limit of the orbit, because the sequence would get infinitely close to the limit but not to the limit.

The limits are also called 'fixed points', because if x1 is equal to its limit, it goes automatically to the limit. Therefore, the limit becomes 'fixed', as its position never changes after the first term.

If you think about it, the fixed points are all on the line y=x, as shown for f(x)= sqrt(x):

In fact, this is a good function to show you something about.

For f(x)=sqrt(x):

For the x1 values of 0<x1<1, what is the limit?

For the x1 values of x>1, what is the limit?

As you can see, both limits to the domain containing non-fixed points are 1. So, 1 is an attracting limit while 0 is repelling.

There are a couple important uses of orbits.

What is the value of 1+(2/(1+(2/(1+(2/...

This is the limit of the orbit 1+(2/x). Calculating the value of the limit would be:

1+(2/x) = x

x^2 - x - 2 = 0

x= 2, -1

Since all of the signs are positive, the root must be positive 2.

*Note*- The above problem could be thought of as the limit of (2/(1+x)) plus one.

The easier way to explain population is through orbits. Population prediction is really important for people in government, and they use dynamic systems to answer their questions. Here's an example of a model:

f(x+1)= 4 * f(x) * (1 - f(x))

where f(x) is a decimal less than one representing the current population. This function is true because of four things:

1)–– (1-f(x)) is a representation of death, especially by starvation or sickness. This would be true because as population goes up, resources get used quicker and lead to a severe lack of supplies.

2)–– f(x) is a representation of birth, which also increases as the population increases. 

3)–– 4 is needed to multiply because 0.5 is lost twice when the other terms are multiplied.

4)–– The function is recursive because we're trying to find the population next year, not any general year.

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Have fun pondering over what I'm trying to explain!

Stay coolio,

John