The gravitational force between two objects is described by Newton's law of gravitation, which can be expressed as:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the distance between the centers of the two objects.
Let's denote the initial masses as \( m_1 \) and \( m_2 \), and the initial distance as \( r \).
Initially, the gravitational force is:
\[ F_{initial} = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
Now, the first object doubles its mass, so its new mass is \( 2m_1 \). The distance between the objects is also halved, so the new distance is \( \frac{r}{2} \).
The new gravitational force can be calculated as follows:
\[ F_{new} = \frac{G \cdot (2m_1) \cdot m_2}{\left(\frac{r}{2}\right)^2} \]
Now, simplifying the expression for \( F_{new} \):
\[ F_{new} = \frac{G \cdot (2m_1) \cdot m_2}{\frac{r^2}{4}} = \frac{G \cdot (2m_1) \cdot m_2 \cdot 4}{r^2} = \frac{8G \cdot m_1 \cdot m_2}{r^2} \]
Thus, the new force \( F_{new} \) is:
\[ F_{new} = 8 \cdot \frac{G \cdot m_1 \cdot m_2}{r^2} = 8 \cdot F_{initial} \]
This means the gravitational force increases by a factor of 8.
Therefore, the answer is:
Increased by 8 times.