Keplers Laws

By Kepler's third law, days

Step 1: Definition.
Geo-stationary satellite appears fixed relative to Earth.
Step 2: Condition.
Same angular velocity and direction as Earth’s rotation.

Step 1: For circular motion, (GMm)/(r²) = (mv²)/(r) ⟹ v² = (GM)/(r)
Step 2: Time period: T = (2π r)/(v) ⟹ T² = (4π² r³)/(GM)
Step 3: Comparing with T² = Kr³: K = (4π²)/(GM) ⟹ GMK = 4π²

Areal velocity & angular momentum of a planet are related by equation , where is the mass of planet. Since in planetary motion is constant , hence is also constant.

Step 1: For circular motion of a planet: (mv²)/(r) = (GMm)/(r²)
Step 2: Simplifying: v² = (GM)/(r)
Step 3: Time period of revolution: T = (2π r)/(v) ⟹ T² = (4π² r³)/(GM)
Step 4: Comparing with Kepler’s law T² = Kr³: K = (4π²)/(GM) ⟹ GK = 4π²

Step 1: Understanding the Concept
For a satellite in a circular orbit, the total mechanical energy is a function of the orbital radius. When a satellite loses energy due to resistive forces (like atmospheric drag), it "falls" into a lower orbit with a smaller radius.

Step 2: Key Formula or Approach
The total mechanical energy ( ) of a satellite of mass orbiting a planet of mass at a radius is: From this, we see that .

Step 3: Detailed Explanation
1. Let the initial radius be and the initial energy be . 2. The new radius is . 3. Using the proportionality: 4. Substitute the values:

Step 4: Final Answer
The new total mechanical energy of the satellite is . Note that since is negative, is a "more negative" value, representing a loss of energy.