Two Satellites in Parallel Orbits

Homework Statement

Two earth satellites are in parallel orbits with radii 6700 km and 6701 km. One day they pass each other, 1 km apart, along a line radially outward from the earth. How long will it be until they are again 1 km apart?

s=r*θ

The Attempt at a Solution

I have an idea that when the two satellites are again 1km apart, they should have the same radian measure. Therefore, if $s_{1}=r_{1}θ$ and $s_{2}=r_{2}θ$, then $\frac{s_{1}}{r_{1}}=\frac{s_{2}}{r_{2}}$. But this is just a proportion and I can't find a way to the time required.

At the moment, I have an idea of substituting a value for s2 and then find s1. Then I compare the time taken by each satellite to cover the required distance, if they two values of time do match, then it will be the final answer...

Andrew Mason
Homework Helper

Homework Statement

Two earth satellites are in parallel orbits with radii 6700 km and 6701 km. One day they pass each other, 1 km apart, along a line radially outward from the earth. How long will it be until they are again 1 km apart?

s=r*θ

The Attempt at a Solution

I have an idea that when the two satellites are again 1km apart, they should have the same radian measure. Therefore, if $s_{1}=r_{1}θ$ and $s_{2}=r_{2}θ$, then $\frac{s_{1}}{r_{1}}=\frac{s_{2}}{r_{2}}$. But this is just a proportion and I can't find a way to the time required.

At the moment, I have an idea of substituting a value for s2 and then find s1. Then I compare the time taken by each satellite to cover the required distance, if they two values of time do match, then it will be the final answer...
Can you find the orbital period for each satellite (the time it takes to complete one complete orbit)?

AM

@Andrew Mason

Yes, I have found the orbital period for both using the equation $T^2=\frac{4\pi^2r^3}{GM}$, specifically T1≈5456.053s and T2≈5457.274s. Yet, I still cannot see how this helps...

Think about the rendez-vous situation in terms of the orbital angle. Let's say at the initial rendez-vous the angle was zero for both satellites. What will it be at the next one?

Andrew Mason
Homework Helper
@Andrew Mason

Yes, I have found the orbital period for both using the equation $T^2=\frac{4\pi^2r^3}{GM}$, specifically T1≈5456.053s and T2≈5457.274s. Yet, I still cannot see how this helps...
Can you now express the angle between their respective radial vectors as a function of time? (hint: find the difference in angular velocity and relate that to the angle between their respective radial vectors).

AM

@voko and Andrew Mason

Thank you very much.
I can easily find angular velocity $ω=\frac{2\pi}{T}$. Also, since this is uniform circular motion, I then think that $θ=ωt$, so the angle between radial vectors as a function of time is: $\Deltaθ=t(ω_{1}-ω_{2})$. But $\Deltaθ=0$ only when t=0 (initial condition). Please tell me if I am missing something...

Thank you!

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But $\Deltaθ=0$ only when t=0 (initial condition).

What about ## \Delta \theta = 2 \pi n ##, where ## n ## is integer?

@voko

Yes!!! Why didn't I think about that??? Thanks a lot, I got the correct answer! Now I just let n=1 and then find t. :)