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I was recently asked the following question:

Which of the Kepler Laws change if the Universal Constant Of Gravitation is changed? And why?

I list the 3 laws of Kepler below:

  1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.

  2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times.

  3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit.

Intuitively I don't expect first and the third laws to change.
Writing the equation of Time period even if the G changes, the square of the period is still proportional to cube of semi major axis.
So I don't expect any of the laws to change.

However the answer provided to me by some professor was that the second and third laws both would change slightly.

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    $\begingroup$ You are indeed correct. All that changing $G$ would do is multiply gravitational force by some constant. The first two laws only depend on the force law being an inverse-square law, and the third law only makes a statement of proportionality. $\endgroup$ – probably_someone Feb 8 '18 at 23:25
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    $\begingroup$ @probably_someone Isn't the second law just conservation of angular momentum, since the rate of area sweeping is ${\bf r} \times {\bf v}$--so it holds for any central potential. $\endgroup$ – JEB Feb 8 '18 at 23:41
  • $\begingroup$ @JEB Oops, yeah, you're right. $\endgroup$ – probably_someone Feb 8 '18 at 23:43
  • $\begingroup$ I've added the homework-and-exercises tag. Please use this in the future for homework problems. $\endgroup$ – Ben Crowell Feb 9 '18 at 0:45
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If you change the gravitational constant $G$, all three of Kepler's laws will still be valid. The first and the third law only depend on the $1/r^2$ dependence of Newtons gravitational force law, and the second law, which corresponds to the conservation of angular momentum, even holds for any central force law.

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