If a satellites is revolving close to a planet of density `rho` with period `T`, show that the quantity `rho T_(2)` is a universal constant.

Explanation

To show that the quantity \( \rho T^2 \) is a universal constant for a satellite revolving close to a planet of density \( \rho \) with period \( T \), we will follow these steps:

### Step 1: Understand the relationship between gravitational force and circular motion
For a satellite of mass \( m \) revolving around a planet of mass \( M \) at a distance \( r \), the gravitational force provides the necessary centripetal force for circular motion. This can be expressed as:
\[ \frac{GMm}{r^2} = m \omega^2 r \]
where \( G \) is the gravitational constant and \( \omega \) is the angular velocity of the satellite.

### Step 2: Simplify the equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{GM}{r^2} = \omega^2 r \]
Rearranging gives:
\[ \omega^2 = \frac{GM}{r^3} \]

### Step 3: Relate angular velocity to the period
The angular velocity \( \omega \) is related to the period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting this into our equation gives:
\[ \left(\frac{2\pi}{T}\right)^2 = \frac{GM}{r^3} \]
which simplifies to:
\[ \frac{4\pi^2}{T^2} = \frac{GM}{r^3} \]
Rearranging this gives us:
\[ T^2 = \frac{4\pi^2 r^3}{GM} \]

### Step 4: Express mass in terms of density
The mass \( M \) of the planet can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho V = \rho \left(\frac{4}{3} \pi r^3\right) \]
Substituting this expression for \( M \) into our equation for \( T^2 \):
\[ T^2 = \frac{4\pi^2 r^3}{G \left(\rho \frac{4}{3} \pi r^3\right)} \]
This simplifies to:
\[ T^2 = \frac{3\pi}{G\rho} \]

### Step 5: Show that \( \rho T^2 \) is a constant
Now, multiplying both sides by \( \rho \):
\[ \rho T^2 = \rho \cdot \frac{3\pi}{G\rho} = \frac{3\pi}{G} \]
This shows that \( \rho T^2 \) is indeed a constant, specifically \( \frac{3\pi}{G} \), which is a universal constant.

### Conclusion
Thus, we have shown that the quantity \( \rho T^2 \) is a universal constant.

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