There have been suggestions that the value of the gravitational constant `G` becomes smaller when considered over very large time period (in billions of years) in the future. If that happens, for our earth

Options

A. nothing will change.
B. we will become hotter after billions of years.
C. we will be going around but not strictly in closed orbits.
D. after sufficiently long time we will leave the solar system.

Explanation

To analyze the implications of a decreasing gravitational constant \( G \) over billions of years on Earth, we can break down the problem step by step.

### Step-by-Step Solution:

1. **Understanding the Gravitational Force**:
The gravitational force \( F \) between two masses (Earth and Sun) is given by Newton's law of gravitation:
\[ F = \frac{G \cdot M_1 \cdot M_2}{R^2} \]
where \( M_1 \) is the mass of the Earth, \( M_2 \) is the mass of the Sun, \( R \) is the distance between the centers of the two masses, and \( G \) is the gravitational constant.

2. **Assumption of Decreasing \( G \)**:
If \( G \) decreases over a long time period (billions of years), it implies that the gravitational force \( F \) will also decrease since \( F \) is directly proportional to \( G \).

3. **Effect on Gravitational Force**:
As \( G \) decreases, the gravitational force \( F \) acting between the Earth and the Sun will decrease:
\[ F \propto G \quad \Rightarrow \quad F \text{ decreases as } G \text{ decreases.} \]

4. **Implications of Decreasing Force**:
A decrease in gravitational force means that the attraction between the Earth and the Sun weakens. Since the gravitational force is what keeps the Earth in orbit around the Sun, a reduction in this force will affect the orbital dynamics.

5. **Change in Orbital Radius**:
If the gravitational force \( F \) decreases, the Earth will not be held as tightly in its orbit. This leads to an increase in the orbital radius \( R \) because:
\[ F = \frac{G \cdot M_1 \cdot M_2}{R^2} \quad \Rightarrow \quad \text{If } F \text{ decreases, } R \text{ must increase.} \]

6. **Long-Term Consequence**:
As \( R \) increases, the Earth will eventually move away from its current orbit. If this trend continues over billions of years, it could lead to the Earth leaving the solar system entirely.

7. **Conclusion**:
Therefore, if the gravitational constant \( G \) continues to decrease over a long time, the Earth could potentially drift away from the Sun and leave the solar system.

### Final Answer:
If the gravitational constant \( G \) decreases over billions of years, the Earth could leave the solar system due to the weakening gravitational attraction between the Earth and the Sun.