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?

 

[Nate810]

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.

 

[kyle8921]

Hello,

Your thoughts are correct. If the gravitational force were proportional to 1/r^3 instead of 1/r^2, it would have a significant impact on Kepler's laws. Let's break it down and see how each law would be affected:

1. Kepler's First Law states that planets move in elliptical orbits with the Sun at one focus. If the gravitational force were 1/r^3, it would result in a different shape of the orbit. In fact, the shape would no longer be an ellipse but instead a shape called a "lemniscate" which looks like a figure eight. This is because the force would not decrease as quickly with distance, causing the planet to move in a more elongated orbit.

2. Kepler's Second Law states that a line joining a planet and the Sun sweeps out equal areas in equal times. This law would still hold true because, as you mentioned, the vector product r x F would still be zero, meaning that the angular momentum of the planet would still be conserved.

3. Kepler's Third Law states that the square of a planet's orbital period is proportional to the cube of its semi-major axis. If the gravitational force were 1/r^3, this law would also be affected. The orbital period would no longer be proportional to the cube of the semi-major axis, but instead, it would be proportional to the square root of the semi-major axis. This means that planets farther from the Sun would have longer orbital periods than they do under the 1/r^2 force.

In summary, a hypothetical 1/r^3 dependence of the gravitational force would have significant implications for Kepler's laws. It would result in different shaped orbits, but the conservation of angular momentum would still hold true. The most significant change would be in Kepler's Third Law, where the relationship between a planet's orbital period and its distance from the Sun would be altered. I hope this helps clarify the matter for you.