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Scientists are studying a planet in a faraway galaxy that has the potential to sustain life. It has the same mass 5.97 × 1024 kg as our Earth, and orbits a star with the same mass 1.99 × 1030 kg as our sun. However, its orbit has appreciable eccentricity, so that the distance between this planet and its star varies between 1.20 × 1011 m and 1.80 x 1011 m during its perihelion and aphelion, respectively. (a) Find the semimajor axis and the eccentricity of this planet. What is its period of orbit in Earth days? (b) Find the speeds (both linear and angular) of this planet at its perihelion and aphelion.

Scientists are studying a planet in a faraway galaxy that has the potential to sustain life. It has the same mass 5.97 × 1024 kg as our Earth, and orbits a star with the same mass 1.99 × 1030 kg as our sun. However, its orbit has appreciable eccentricity, so that the distance between this planet and its star varies between 1.20 × 1011 m and 1.80 x 1011 m during its perihelion and aphelion, respectively. (a) Find the semimajor axis and the eccentricity of this planet. What is its period of orbit in Earth days? (b) Find the speeds (both linear and angular) of this planet at its perihelion and aphelion.

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Scientists are studying a planet in a faraway galaxy that has the potential to sustain life. It has the same mass 5.97 × 1024 kg as our Earth, and orbits a star with the same mass 1.99 × 1030 kg as our sun. However, its orbit has appreciable eccentricity, so that the distance between this planet and its star varies between 1.20 x 1011 m and 1.80 x 1011 m during its perihelion and aphelion, respectively. (a) Find the semimajor axis and the eccentricity of this planet. What is its period of orbit in Earth days? (b) Find the speeds (both linear and angular) of this planet at its perihelion and aphelion. (c) Unlike our Earth, whose seasonal cycle is a result of the tilt of Earth's axis, this planet's rotation axis has no tilt (i.e. it is perpendicular to the plane of its orbit), and this planet's seasons depend on its distance from its star. Suppose that the season of summer on this planet occurs when its distance from its star is less than or equal to 1.40 × 1011 m. How many Earth days does the season of summer last on this planet? Hint: You may find the following integral useful: 1 a² – a² dx = ; [xVa² – g² + a° sin=1 a² – x² + a² sin-1 +C 2
Transcribed Image Text:Scientists are studying a planet in a faraway galaxy that has the potential to sustain life. It has the same mass 5.97 × 1024 kg as our Earth, and orbits a star with the same mass 1.99 × 1030 kg as our sun. However, its orbit has appreciable eccentricity, so that the distance between this planet and its star varies between 1.20 x 1011 m and 1.80 x 1011 m during its perihelion and aphelion, respectively. (a) Find the semimajor axis and the eccentricity of this planet. What is its period of orbit in Earth days? (b) Find the speeds (both linear and angular) of this planet at its perihelion and aphelion. (c) Unlike our Earth, whose seasonal cycle is a result of the tilt of Earth's axis, this planet's rotation axis has no tilt (i.e. it is perpendicular to the plane of its orbit), and this planet's seasons depend on its distance from its star. Suppose that the season of summer on this planet occurs when its distance from its star is less than or equal to 1.40 × 1011 m. How many Earth days does the season of summer last on this planet? Hint: You may find the following integral useful: 1 a² – a² dx = ; [xVa² – g² + a° sin=1 a² – x² + a² sin-1 +C 2
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