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If the length of the semi-major axis of the elliptical orbit of a planet is doubled, the relation between the new time period of revolution of the planet around the Sun with the original time period will be-

If the length of the semi-major axis of the elliptical orbit of a planet is doubled, the relation between the new time period of revolution of the planet around the Sun with the original time period will be- Correct Answer T<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">2</span><span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">2</span> = 8 × T<span style="position: relative; line-height: 0; vertical-align: baseline; bottom: -0.25em;font-size:10.5px;">1</span><span style="position: relative; line-height: 0; vertical-align: baseline; top: -0.5em;font-size:10.5px;">2</span> 

The correct answer is option 4) i.e. T22 = 8 × T12 

CONCEPT:

  • Kepler's laws of planetary motion: Kepler's laws of planetary motion can be stated as:
    • Kepler's first law: All the planets move around the Sun in elliptical orbits having the Sun at one of the foci. This law is also called the "Law of orbits."
    • Kepler's second law: The radius vector drawn from the Sun to the planet sweeps out equal areas in equal intervals of time. This law is also called the "Law of Equal Areas."
    • Kepler's third law: The square of the time period of revolution of the planet around the Sun in an elliptical orbit is directly proportional to the cube of its semi-major axis. This law is also called the "Law of Periods."

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According to Kepler's third law of planetary motion:

T2 α a3 

where T = Time period of revolution of the planet, and a = length of the semi-major axis of the elliptical orbit of the planet.

CALCULATION:

The original time period(T1) is related to the length of the semi-major axis(a1) as:

T12 = a13

On doubling the length of the semi-major axis, the new time period(T2) will be:

T22 = (2 × a1)3

⇒ T22 = 8 × a13

⇒ T22 = 8 × T12

Hence, on doubling the length of the semi-major axis, the square of the new time period will be 8 times the square of the original time period.

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