A planet is revolving around a star with a time period of 2 days. If radius of star trippled keeping same. Planet will revolve with these changed parameters. Select correct open (s) :

Options

A. with time period of 6 days
B. with `(1)/(sqrt(3))rd` orbital speed of initial speed
C. with potential energy `(1)/(sqrt(3))` times of its initial potential energy
D. None of these

Answer

To solve the problem, let's analyze the situation step by step.

### Given:
- A planet revolves around a star with a time period \( T = 2 \) days.
- The radius of the star is tripled, but the mass of the star remains the same.

### Step 1: Understand the relationship between time period, radius, and mass
The time period \( T \) of a planet revolving around a star can be described by Kepler's third law of planetary motion, which states:

\[ T^2 \propto r^3 \]

where \( T \) is the time period and \( r \) is the orbital radius (distance from the star).

### Step 2: Analyze the effect of changing the radius of the star
In this case, the radius of the star is tripled, but it is important to note that the distance between the planet and the star (the orbital radius) does not change. The mass of the star also remains constant.

### Step 3: Determine the time period after the change
Since the orbital radius \( r \) remains unchanged, we can conclude that the time period \( T \) will also remain unchanged. Therefore, the new time period \( T' \) is still:

\[ T' = T = 2 \text{ days} \]

### Step 4: Analyze the orbital velocity
The orbital velocity \( v \) of the planet can be expressed as:

\[ v = \sqrt{\frac{GM}{r}} \]

where \( G \) is the gravitational constant, \( M \) is the mass of the star, and \( r \) is the orbital radius. Since neither \( G \), \( M \), nor \( r \) has changed, the orbital velocity will also remain constant.

### Step 5: Analyze the potential energy
The gravitational potential energy \( U \) between the planet and the star is given by:

\[ U = -\frac{GMm}{r} \]

where \( m \) is the mass of the planet. Since \( G \), \( M \), \( m \), and \( r \) remain unchanged, the potential energy will also remain constant.

### Conclusion
- The time period remains the same at 2 days.
- The orbital velocity remains constant.
- The potential energy remains constant.

### Final Answer
None of the options provided in the question are correct based on the analysis. The correct conclusions are:
- Time period: remains the same (2 days).
- Orbital velocity: remains constant.
- Potential energy: remains constant.