To determine the time interval between the periodic approaches of two satellites moving in circular orbits around the Earth, we can use Kepler's laws of planetary motion. In this scenario, we have two satellites with radii of their orbits given as \( r \) and \( r - r \), where \( r \) is significantly larger than \( r \). Let's break down the problem step by step.

Understanding the Orbits

We have two satellites:

• Satellite 1: Orbit radius \( r = 7000 \) km
• Satellite 2: Orbit radius \( r - r = 7000 \text{ km} - 70 \text{ km} = 6930 \text{ km} \)

Both satellites are in circular orbits around the Earth, which has a mass \( M = 6 \times 10^{24} \) kg.

Calculating Orbital Periods

The orbital period \( T \) of a satellite can be calculated using the formula:

T = 2\pi \sqrt{\frac{r^3}{GM}}

where \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{m}^3/\text{kg} \cdot \text{s}^2 \).

For Satellite 1:

First, we convert the radius from kilometers to meters:

\( r = 7000 \text{ km} = 7000 \times 10^3 \text{ m} = 7 \times 10^6 \text{ m} \)

Now, substituting into the formula:

T_1 = 2\pi \sqrt{\frac{(7 \times 10^6)^3}{6.674 \times 10^{-11} \times 6 \times 10^{24}}}

For Satellite 2:

Similarly, for Satellite 2:

\( r - r = 6930 \text{ km} = 6930 \times 10^3 \text{ m} = 6.93 \times 10^6 \text{ m} \)

Substituting into the formula:

T_2 = 2\pi \sqrt{\frac{(6.93 \times 10^6)^3}{6.674 \times 10^{-11} \times 6 \times 10^{24}}}

Finding the Time Interval Between Approaches

The time interval \( \Delta T \) between the periodic approaches of the two satellites can be found by calculating the difference in their orbital periods:

\Delta T = |T_1 - T_2|

Calculating the Periods

Let's compute \( T_1 \) and \( T_2 \) using the values provided:

• For Satellite 1, \( T_1 \approx 5936 \) seconds (or about 1.65 hours).
• For Satellite 2, \( T_2 \approx 5905 \) seconds (or about 1.64 hours).

Final Calculation

Now, substituting these values into our equation for \( \Delta T \):

\Delta T = |5936 - 5905| \approx 31 \text{ seconds}

This means that the two satellites will periodically approach each other at their minimum distance approximately every 31 seconds. This interval is a result of their differing orbital speeds due to the slight difference in their orbital radii.

In summary, by applying Kepler's laws and understanding the relationship between orbital radius and period, we can effectively determine the time interval between the periodic approaches of the two satellites. This example illustrates the fascinating dynamics of orbital mechanics and how even small changes in radius can lead to measurable differences in satellite behavior.