Which of the following graphs between the square of the time period and cube of the distance of the planet from the sun is correct ?

Options

A.
B.
C.
D.

Answer

To solve the problem regarding the relationship between the square of the time period (T²) and the cube of the distance (R³) of a planet from the Sun, we can follow these steps:

### Step 1: Understand Kepler's Third Law
Kepler's Third Law states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (R) of its orbit. Mathematically, this can be expressed as:
\[ T^2 \propto R^3 \]
This means that:
\[ T^2 = k \cdot R^3 \]
where \( k \) is a constant of proportionality.

### Step 2: Rearranging the Equation
From the above relationship, we can rearrange it to show that:
\[ T^2 = kR^3 \]
This indicates that if we plot \( T^2 \) on the y-axis and \( R^3 \) on the x-axis, the relationship will be linear.

### Step 3: Graph Characteristics
Since the equation \( T^2 = kR^3 \) is linear, the graph will be a straight line. The line will pass through the origin (0,0) because when \( R = 0 \), \( T^2 \) will also be 0.

### Step 4: Identify the Correct Graph
Given the options for the graph, we need to identify which one represents a straight line passing through the origin. This will be the graph that correctly represents the relationship between \( T^2 \) and \( R^3 \).

### Conclusion
Based on our analysis, the correct graph that represents the relationship between the square of the time period and the cube of the distance of the planet from the Sun is option C.

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