Halley s comet orbits the Sun with a period of 76.2 years. a) Find the semi-major axis of the...

Question:

Halley s comet orbits the Sun with a period of 76.2 years.

a) Find the semi-major axis of the orbit of Halley 's comet in astronomical units (1 AU is equal to the semi-major axis of the Earth s orbit).

Kepler's law of gravitation:

According to Kepler's third law of gravitation, the square of the time period of the rotation of the body is directly proportional to the cube of the semi-major axis of the rotation.

Answer and Explanation:

Given data

Time period of the comet {eq}T = 76.2 \ years \\ T = 76.2 \times (365\times 24 \times 3600) \ s\\ T = 2.403 \times 10^{9} \ s {/eq}

Now, the time period is given by

{eq}T^{2} = \dfrac{4\pi^{2} R^{3}}{GM} \\ (2.403 \times 10^{9}) = \dfrac{4\pi^{2} R^{3}}{(6.67 \times 10^{-11})(5.98 \times 10^{24})} \\ R = 3.878 \times 10^{10} \ m \\ R = (3.878 \times 10^{10} \ m )\left(\dfrac{6.684 \times 10^{-12} \ AU}{1 \ m }\right) \\ R = 0.259 \ AU {/eq}

where

  • M is the mass of the earth

Learn more about this topic:

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Elliptical Orbits: Periods & Speeds

from Physics: High School

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