Which of the following graphs represents the motion of the planet moving about the Sun. T is the periode of revolution and r is the average distance (from centre to centre) between the sun and the planet

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which graph represents the motion of a planet moving about the Sun, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: We know from Kepler's Third Law of Planetary Motion that the square of the period of revolution (T) of a planet is directly proportional to the cube of the average distance (r) from the Sun. This can be mathematically expressed as:
\[ T^2 \propto r^3 \]

2. **Convert to Equation Form**: To express this relationship in a more usable form, we can introduce a constant of proportionality (k):
\[ T^2 = k \cdot r^3 \]

3. **Identify Variables**: In this equation, \(T^2\) is the dependent variable (y-axis), and \(r^3\) is the independent variable (x-axis).

4. **Graph Characteristics**: The equation \(T^2 = k \cdot r^3\) can be rearranged to fit the linear equation format \(y = mx + b\), where:
- \(y = T^2\)
- \(x = r^3\)
- \(m = k\) (the slope)
- \(b = 0\) (the y-intercept)

5. **Conclusion on the Graph**: Since the equation represents a straight line that passes through the origin (0,0), the graph of \(T^2\) versus \(r^3\) will be a straight line with a slope \(k\) and no y-intercept.

6. **Select the Correct Option**: Based on the characteristics of the graph we derived, we need to identify which of the provided options matches this description. The correct graph will be a straight line passing through the origin.

### Final Answer:
The correct answer is **Option C**, which represents a straight line graph passing through the origin, indicating the relationship \(T^2 = k \cdot r^3\).