Calculating Time Interval of Two Earth Satellites in Circular Orbits

[NEILS BOHR]

Homework Statement

Two satellites of the Earth move in a common plane along circular orbits , the radii being r and r-\Deltar ( \Deltar << r ). What is the time interval b/w their periodic approaches to each other over the min. distance . Take M to be the mass of the Earth
M = 6 * 10 ²⁴ kg , r = 7000 km , \Deltar = 70 km ).

Homework Equations

The Attempt at a Solution

i m unable to understand the quesn...

 

[gneill]

Which part of the question do you not understand?
Which parts do you understand?

 

[NEILS BOHR]

i mean what's the meaning of by time interval b/w their periodic approaches to each other over the min. distance??

uptill now i m doing this :

T = 2\pi^1/2( r³ / GM)

so dT = 2 pi / root of GM * 3/2 r^1/2 dr

so i have found dT / T and hence dT...

what to do next??

 

[gneill]

i mean what's the meaning of by time interval b/w their periodic approaches to each other over the min. distance??

uptill now i m doing this :

T = 2\pi^1/2( r³ / GM)

so dT = 2 pi / root of GM * 3/2 r^1/2 dr

so i have found dT / T and hence dT...

what to do next??

Perhaps you were going for the formula for the period of a circular orbit?

T = \frac{2 \pi}{\sqrt{G M}}r^{3/2}

Bodies in circular orbits with different radii will have different periods. Inner ones have shorter periods than the outer ones. In the present case, this means that the satellite with the smaller orbit will periodically "lap" (pass) the outer one.

The problem is asking for you to compute that time period. I must admit that the phrase, "over the min. distance" is a bit vague. It could be that they want you to divide the period by the distance between the satellites when they're at closest approach. On the other hand, it could imply something a bit more devious -- what if the directions of the orbits are not the same (one going clockwise, the other counterclockwise)? Then the distance traveled along the orbit from meeting to meeting would be minimized.

Which ever way it turns out, the concept in play is what is known as the synodic period. A web search will turn up some adequate material.

 

[NEILS BOHR]

hmmm
still confused with what the quesn is asking exactly??