Kepler’s third law states that the square of a planet’s orbital period is proportional to the cube of its semimajor axis. This applies to all celestial bodies in orbit and explains why planets closer to the sun move faster than those farther away. The law can be expressed as (P1)2/(P2)2 = (R1)3/(R2)3 and is demonstrated by comparing the orbits of Mercury and Pluto.
Kepler’s third law of planetary motion states that the square of each planet’s orbital period, represented as P2, is proportional to the cube of each planet’s semimajor axis, R3. A planet’s orbital period is simply the amount of time in years it takes for one complete revolution. A semimajor axis is a property of all ellipses and is the distance from the center of the ellipse to the point on the orbit that is farthest from the center.
The astronomer and mathematician Johannes Kepler (1571-1630) developed his three laws of planetary motion with respect to two objects in orbit, and it makes no difference whether these two objects are stars, planets, comets or asteroids. This is especially true for two relatively massive objects in space. Kepler’s laws changed the way humans studied the motions of celestial bodies.
The following example can be used to demonstrate the properties of each relationship with respect to Kepler’s third law. If P1 represents the orbital period of planet A and R1 represents the semimajor axis of planet A; P2 represents the orbital period of planet B and R2 represents the semimajor axis of planet B; then the ratio of (P1)2/(P2)2, ie the square of each planet’s orbital period, is equal to the ratio of (R1)3/(R2)3, the cube of each planet’s semi-major axis. Thus, as an expression, Kepler’s third law shows that (P1)2/(P2)2 = (R1)3/(R2)3.
Instead of ratios or proportions, Kepler’s third law can be summarized using time and distance. As planets, comets or asteroids approach the Sun, their speed increases; as planets, comets or asteroids move away, their speed decreases. Thus, the increase in velocity of one body is similar to the increase in velocity of another body when both of their distances – their semimajor axes – are taken into account. This is why Mercury, the innermost planet, rotates so fast and Pluto, previously considered the outermost planet, rotates so slowly.
In a real-world example using Mercury and Pluto, notice that the largest numbers are those of Pluto, and remember (P1)2/(P2)2 = (R1)3/(R2)3. In this case, (0.240)2/(249)2 = (0.39)3/(40)3. Therefore, 9.29 x 10-7 = 9.26 x 10-7.
Mercury is always close to the Sun, so its speed is high. Pluto is always far from the Sun, so its speed is slow, but neither object’s speed is constant. Even though Mercury is nearby and Pluto is far away, both have periods during their orbital periods of increasing and decreasing velocity. Regardless of the differences, the square of each planet’s orbital period is proportional to the cube of each planet’s semimajor axis.
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