When the distance between two stars decreases by half, the force between them

It is 4 times bigger. Newton's Law of Gravitation says that the gravitational force between two objects is inversely proportional to the square of the distance between them. More specifically F = (G*m1*m2)/(r2). If you cut the distance in half, you have to square that, which is 1/4 so the force is 4 times bigger.