If it takes a planet [tex]$2.8 \times 10^8 s$[/tex] to orbit a star with a mass of [tex]$6.2 \times 10^{30} kg$[/tex], what is the average distance between the planet and the star?
A. [tex]$1.43 \times 10^9 m$[/tex]
B. [tex]$9.36 \times 10^{11} m$[/tex]
C. [tex]$5.42 \times 10^{13} m$[/tex]
D. [tex]$9.06 \times 10^{17} m$[/tex]
Answer (2)
Use Kepler's Third Law: T 2 = GM 4 π 2 r 3 .
Rearrange the formula to solve for r : r = 3 4 π 2 GM T 2 .
Substitute the given values: G = 6.674 × 1 0 − 11 N ( m / k g ) 2 , M = 6.2 × 1 0 30 k g , and T = 2.8 × 1 0 8 s .
Calculate r ≈ 9.36 × 1 0 11 m . The final answer is 9.36 × 1 0 11 m .
Explanation
Understanding the Problem We are given the period of a planet's orbit around a star and the mass of the star. We need to find the average distance between the planet and the star. We can use Kepler's Third Law to solve this problem. Kepler's Third Law relates the period of an orbit to the average distance between the two objects and the mass of the central object.
Stating Kepler's Third Law Kepler's Third Law states that T 2 = GM 4 π 2 r 3 , where:
T is the period of the orbit,
G is the gravitational constant ( 6.674 × 1 0 − 11 N m 2 / k g 2 ),
M is the mass of the star,
r is the average distance between the planet and the star.
Solving for the Average Distance We need to solve for r . Rearranging the formula, we get: r 3 = 4 π 2 GM T 2 r = 3 4 π 2 GM T 2
Plugging in the Values Now, we plug in the given values:
G = 6.674 × 1 0 − 11 N m 2 / k g 2
M = 6.2 × 1 0 30 k g
T = 2.8 × 1 0 8 s So, r = 3 4 π 2 ( 6.674 × 1 0 − 11 ) ( 6.2 × 1 0 30 ) ( 2.8 × 1 0 8 ) 2
Calculating the Result Calculating the value of r , we get: r = 3 4 π 2 ( 6.674 × 1 0 − 11 ) ( 6.2 × 1 0 30 ) ( 2.8 × 1 0 8 ) 2 ≈ 9.3665 × 1 0 11 m
Final Answer Therefore, the average distance between the planet and the star is approximately 9.3665 × 1 0 11 m .
Examples
Understanding orbital mechanics is crucial in space exploration. For instance, when planning a mission to Mars, scientists use Kepler's laws to calculate the precise trajectories and distances involved. By knowing the mass of the Sun and Mars' orbital period, they can determine the spacecraft's required velocity and fuel consumption to ensure a successful journey. This ensures efficient and accurate navigation through space, optimizing mission resources and timelines.
Using Kepler's Third Law, we calculated the average distance between the planet and the star to be approximately 9.36 × 1 0 11 m , which corresponds to option B. The calculation used the period of the orbit and the mass of the star to find this distance. Therefore, the correct choice is B.
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