# Calculating when the center points of two orbiting bodies will be at the same circumferential position

I'm working on a game idea that includes objects orbiting around a common center point. I would like to have some way of knowing (mathematically) when two given objects/bodies will be at the same circumferential position (in radians) in their respective orbits.

What is known:

1. The movement of objects is 2 dimensional

2. All objects follow a circular orbital path around a common center point.

3. The distance of the orbiting object from the center point is known, but may vary from object to object.

4. The time it takes for any given object to complete one full revolution is known in seconds. This will also vary from object to object.

5. The starting point of any given object is known in radians. This will also vary from object to object.

6. All objects begin moving/orbiting at the same time.

7. All objects are circles and their positions on the coordinate plane are dictated by their center points.

8. All objects will continue their orbits indefinitely.

9. No external forces are acting on the objects.

10. Let's assume we are only comparing two objects at a time.

Example:

Object 1 is at a radius of 180 pts from the common center point and is located at 0.5 radians. It completes one full revolution in 3 seconds.

Object 2 is at a radius of 220 pts and is located at 0 radians. It completes one revolution in 5 seconds.

The objects are orbiting in the same direction.

What am I trying to figure out?

How can I, for example, determine the first 5 points when/where the two orbiting objects will cross paths. That is, each time a single straight line can be drawn from the common center point through the center points of both objects.

If any additional information is needed to calculate or if this isn't as straight forward as I'm imagining it to be, just let me know and I will do my best to provide additional info. Thanks ahead of time, really appreciate any help you guys can provide. :)

I am not sure I understand your question, but if it is as straightforward as it seems, then you might calculate the alignement time as follows. If the period of object 1 is $$T_1$$ and the period of object 2 is $$T_ 2$$, the time $$t_n$$ at which they are aligned is:

$$2 \pi \frac{t_n}{T_ 1} = 2 \pi \frac{t_n}{T_ 2} + 2 n \pi$$ $$t_n = n\frac{T_1\, T_2}{T_2-T_1}$$

where n is any natural number.

• What would the value n represent? – SquidInSuperMario Mar 31 at 22:56
• Nevermind, I understand now. Thank you for your help, really appreciate it! – SquidInSuperMario Mar 31 at 23:14

Let $$T_j$$ and $$T_k$$ be the different periods of objects $$j$$ and $$k$$, with $$T_k > T_j$$.

If the objects are at different angles $$\phi_j$$ and $$\phi_k$$ at time $$t=0$$, they are aligned when: $$\frac{t}{T_j} - \frac{t}{T_k} - \frac{\theta}{2 \pi} = n \in \mathbb{Z}$$ with $$\theta = \phi_j - \phi_k$$. Note that there may be candidates for the solution for negative values of $$n$$.

The times $$t$$ at which $$j$$ and $$k$$ are aligned will be given by $$t = \frac{T_j T_k}{T_k - T_j} \left(n + \frac{\theta}{2 \pi} \right)$$

(While I was writing this, I notice that Filipe Miguel posted a similar answer.)