Keplers laws and a hypothetical 1/r^3 dependence

[standardflop]

Hello,
i've been asked a hypothetical question about Keplers three laws: What if the gravitational force was proprotional to 1/r^3 instead of 1/r^2? And for one of the laws it apparently "isent easy to decide". My thoughts:

keplers 1. : my first thought was that this law was the "not easy to decide"
keplers 2. : must still be valid, because the vector product r x F will always be zero, and hence dL/dt = 0 (conservation of momentum for any central force)
keplers 3 : a planets period would change?

Is this correct? And can somebody help me clearify matters?

 

[member 553083]

Thanks in advance!Kepler's first law: This law states that the orbits of planets around the Sun are ellipses, with the Sun at one focus. If the gravitational force was proportional to 1/r^3, then the orbits would still be ellipses, but the shape of the ellipse may change.Kepler's second law: This law states that a planet sweeps out equal areas in equal times along its orbit. Since the gravitational force is determined by the ratio of the masses of the two objects and the square of the distance between them, a change in the force to 1/r^3 would not affect this law.Kepler's third law: This law states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. If the gravitational force was proportional to 1/r^3, then the orbital period would change, as the force would influence the acceleration of the planet along its orbit.