Orbital Alignment, what is K?

[Masklein]

I'm using a unit circle to track the paths of two bodies around a point. I've found a formula that can work for my purposes at: http://mathpages.com/home/kmath161/kmath161.htm .

My question is about k in this formula:



The integers that k can equal, are they arbitrary? In other words does a person just guess at what number to enter that will yield the whole number t for alignment?


My ultimate goal is to determine which two bodies may or may not have aligned in the past based on a given current t where they are possibly aligned. But I need to understand the workings of this formula further before I can work on that problem.

Thank you for your time.

 

[mmm4444bot]

My question is about k in this formula:

View attachment 2175

The integers that k can equal, are they arbitrary? In other words does a person just guess at what number to enter that will yield the whole number t for alignment?

The hands will align at any time t coming out of that formula, as long as k is an Integer (any Integer).

To model the statement that the expression t(ω₁-ω₂) must be a multiple of 2π, we write:

t(ω₁-ω₂) = k*2π, where k is an Integer

The formula comes from dividing both sides of this equation by ω₁-ω_2.

Are you thinking that t will also be an Integer? I'm wondering about your reference to Whole numbers for t. :cool:

 

[Masklein]

My thought on t being a whole number relates to this statement from the link:

The positions of the two hands at any time t are w₁ t and w₂ t, and they are aligned if and only if (w₁-w₂) t is an integer multiple of 2p.

Since the final formula I'm using has (ω₁-ω₂) on the other side with k2π, I thought that whole numbers would mean alignment and fractions would indicate progress toward alignment.

 

[Masklein]

I thought about this more and realized that what I am trying to formulate is not alignment. So this formula doesn't help me in that regard. Though I am still glad to have found it and am learning as a result.

I'm certain of this because my thinking of what the numbers I was using represent was wrong. I was thinking something like this:

Given:

_{One object at (5}π)/3
A second object at (7π)/3

These positions are also the distance they move in 1 second.

How long until they arrive at (35π)/3

There is actually a very easy way to find this, it is the upper limit of 35/6. Which is 6. However for the orbital alignment formula, I had some things wrong. (5π)/3 arrives at (35π)/3 in 6 seconds. While (7π)/3 arrives at (35π)/3 in 5 seconds. So it isn't specifically alignment that I am looking for. I realize there are other easier ways to calculate this and I am using them to check my work. This is also why I was expecting t above to be a whole number, because I was looking for t = 6 ( which is k = 2 ). Though I realized a number of different answers would also lead to whole numbers. So I obviously had something wrong in my thinking.

I would still like to explore ways to describe orbital movement. My reason is that my next question to answer will be: Which two and/or more objects originating along the π/3 or (5π)/3 axis will arrive at (35π)/3?