One way to determine the mass of a distant star (or planet) is to observe the motion of its planets (or moons). If exoplanet CSN-2021 is found to have a small moon orbiting at a radius of 7.77 x 10* m with a period of 23.1 days, what is the mass of CSN-2021? (a) 1.90 x 1033 kg (b) 6.97 x 1025 kg (c) 2.54 x 1023 kg (d) 4.49 x 1010 kg 9. If the radius of exoplanet CSN-2021 is 2.97 × 107 m, find the acceleration due to gravity near its surface.
One way to determine the mass of a distant star (or planet) is to observe the motion of its planets (or moons). If exoplanet CSN-2021 is found to have a small moon orbiting at a radius of 7.77 x 10* m with a period of 23.1 days, what is the mass of CSN-2021? (a) 1.90 x 1033 kg (b) 6.97 x 1025 kg (c) 2.54 x 1023 kg (d) 4.49 x 1010 kg 9. If the radius of exoplanet CSN-2021 is 2.97 × 107 m, find the acceleration due to gravity near its surface.
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Chapter1: Units, Trigonometry. And Vectors
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### Problem 9:
One way to determine the mass of a distant star (or planet) is to observe the motion of its planets (or moons). If exoplanet CSN-2021 is found to have a small moon orbiting at a radius of \(7.77 \times 10^8\) m with a period of 23.1 days, what is the mass of CSN-2021?
#### Options:
- (a) \(1.90 \times 10^{33}\) kg
- (b) \(6.97 \times 10^{25}\) kg
- (c) \(2.54 \times 10^{23}\) kg
- (d) \(4.49 \times 10^{10}\) kg
If the radius of exoplanet CSN-2021 is \(2.97 \times 10^7\) m, find the acceleration due to gravity near its surface.
### Explanation:
This problem involves using the gravitational properties and orbital dynamics to estimate the mass of a celestial body. Students need to apply their knowledge of orbital mechanics, specifically the use of Kepler's laws and the gravitational formula, to solve for the exoplanet's mass.
To find the mass of the exoplanet, they might need to use the formula:
\[ M = \frac{4\pi^2r^3}{GT^2} \]
where:
- \( M \) is the mass of the exoplanet,
- \( r \) is the orbital radius,
- \( T \) is the orbital period, and
- \( G \) is the gravitational constant.
Additionally, to find the acceleration due to gravity at the surface of the exoplanet, students can use the formula:
\[ g = \frac{GM}{r^2} \]
where:
- \( g \) is the acceleration due to gravity,
- \( M \) is the mass of the exoplanet,
- \( r \) is the radius of the exoplanet, and
- \( G \) is the gravitational constant.
These calculations help in understanding the gravitational influence of celestial objects and are essential for astrophysical studies and space missions planning.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3524539-73d7-4b1e-9607-006acd5ee568%2F32436496-7907-402e-8fa5-6882590ceb8b%2F9l7gy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Problem 9:
One way to determine the mass of a distant star (or planet) is to observe the motion of its planets (or moons). If exoplanet CSN-2021 is found to have a small moon orbiting at a radius of \(7.77 \times 10^8\) m with a period of 23.1 days, what is the mass of CSN-2021?
#### Options:
- (a) \(1.90 \times 10^{33}\) kg
- (b) \(6.97 \times 10^{25}\) kg
- (c) \(2.54 \times 10^{23}\) kg
- (d) \(4.49 \times 10^{10}\) kg
If the radius of exoplanet CSN-2021 is \(2.97 \times 10^7\) m, find the acceleration due to gravity near its surface.
### Explanation:
This problem involves using the gravitational properties and orbital dynamics to estimate the mass of a celestial body. Students need to apply their knowledge of orbital mechanics, specifically the use of Kepler's laws and the gravitational formula, to solve for the exoplanet's mass.
To find the mass of the exoplanet, they might need to use the formula:
\[ M = \frac{4\pi^2r^3}{GT^2} \]
where:
- \( M \) is the mass of the exoplanet,
- \( r \) is the orbital radius,
- \( T \) is the orbital period, and
- \( G \) is the gravitational constant.
Additionally, to find the acceleration due to gravity at the surface of the exoplanet, students can use the formula:
\[ g = \frac{GM}{r^2} \]
where:
- \( g \) is the acceleration due to gravity,
- \( M \) is the mass of the exoplanet,
- \( r \) is the radius of the exoplanet, and
- \( G \) is the gravitational constant.
These calculations help in understanding the gravitational influence of celestial objects and are essential for astrophysical studies and space missions planning.
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