Kepler's 3rd law of planetary motion: [tex]T^2\propto r^3[/tex]
Explanation:
We can relate Newton's universal law of gravitation to Kepler's third law of planetary motion.
In fact, Newton's universal law of gravitation states that the gravitational force between a planet and the Sun is:
[tex]F=G\frac{Mm}{r^2}[/tex]
where
G is the gravitational constant
M is the mass of the Sun
m is the mass of the planet
r is the distance between the Sun and the planet
SInce the motion of the planet around the Sun is approximately circular, we can equate this force to the centripetal force:
[tex]G\frac{Mm}{r^2}=m\frac{v^2}{r}[/tex]
where v is the orbital speed of the planet. Simplifying,
[tex]\frac{GM}{r}=v^2[/tex]
The orbital speed of the planet is equal to the ratio between the length of the orbit ([tex]2\pi r[/tex]) and the orbital period, T:
[tex]v=\frac{2\pi r}{T}[/tex]
So the previous equation becomes
[tex]\frac{GM}{r}=\frac{4\pi^2 r^2}{T^2}[/tex]
Which can be re-arranged as
[tex]T^2 = \frac{4\pi^2}{GM}r^3[/tex]
The term [tex]\frac{4\pi^2}{GM}[/tex] is constant for all planets, so the equation can be rewritten as
[tex]T^2\propto r^3[/tex]
Which is exactly Kepler's third law of planetary motion, which states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit (here we approximated the orbit as a circle, so r representes the average radius of the orbit).
Learn more about Kepler's laws:
brainly.com/question/11168300
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