According to Kepler's first law, any planet in the solar system describes an elliptical orbit for which the Sun is located at one of the focal points of the ellipse. The point on the ellipse (representing the orbit of the planet) closest to the Sun is called perihelion, and aphelion is the most distant point. The objective of this question is to estimate the maximum speed of the planet Mercury which is reached at its perihelion. The orbit of the planet Mercury around the Sun is represented in the Cartesian plane (IR2), with the Sun (r+ c)² y? located at the origin, by the ellipse of equation: where a = 5.791 x 10^7 km, b = 5.667 x 10^7 km and c = 1.191 x 10^7 km. The Perihelion and the aphelion of the planet Mercury are therefore respectively located at the point (a - c, 0) and at the point (-a - c, 0), with a - c = 4.600 x 10^7 km and -a -c = -6.982 x 10^7 km. Mercure Soleil aphélie =(-a-c, 0) périhélie = (a-c, 0) a) Using the formula for calculating an arc length, determine the distance traveled by the planet Mercury for a period of complete revolution (around of the Sun). b) Knowing that the period of revolution of Mercury lasts 88 days, determine the average speed of this planet (in km / s). c) Using defined integrals, determine the total area St of the ellipse generated by the orbit of Mercury.
According to Kepler's first law, any planet in the solar system describes an elliptical orbit for which the Sun is located at one of the focal points of the ellipse. The point on the ellipse (representing the orbit of the planet) closest to the Sun is called perihelion, and aphelion is the most distant point. The objective of this question is to estimate the maximum speed of the planet Mercury which is reached at its perihelion. The orbit of the planet Mercury around the Sun is represented in the Cartesian plane (IR2), with the Sun (r+ c)² y? located at the origin, by the ellipse of equation: where a = 5.791 x 10^7 km, b = 5.667 x 10^7 km and c = 1.191 x 10^7 km. The Perihelion and the aphelion of the planet Mercury are therefore respectively located at the point (a - c, 0) and at the point (-a - c, 0), with a - c = 4.600 x 10^7 km and -a -c = -6.982 x 10^7 km. Mercure Soleil aphélie =(-a-c, 0) périhélie = (a-c, 0) a) Using the formula for calculating an arc length, determine the distance traveled by the planet Mercury for a period of complete revolution (around of the Sun). b) Knowing that the period of revolution of Mercury lasts 88 days, determine the average speed of this planet (in km / s). c) Using defined integrals, determine the total area St of the ellipse generated by the orbit of Mercury.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Hello I need to solve part a, b and c. Thank you
Transcribed Image Text:According to Kepler's first law, any planet in the solar system describes an elliptical orbit for which the
Sun is located at one of the focal points of the ellipse. The point on the ellipse (representing the orbit of
the planet) closest to the Sun is called perihelion, and aphelion is the most distant point. The objective of
this question is to estimate the maximum speed of the planet Mercury which is reached at its perihelion.
The orbit of the planet Mercury around the Sun is represented in the Cartesian plane (IR2), with the Sun
(r+ c)²
y?
located at the origin, by the ellipse of equation:
where a = 5.791 x 10^7 km, b = 5.667 x 10^7 km and c = 1.191 x 10^7 km. The Perihelion and
the aphelion of the planet Mercury are therefore respectively located at the point (a - c, 0)
and at the point (-a - c, 0), with a - c = 4.600 x 10^7 km and -a -c = -6.982 x 10^7 km.
Mercure
Soleil
aphélie =(-a-c, 0)
périhélie = (a-c, 0)
a) Using the formula for calculating an arc length, determine the distance traveled by the planet
Mercury for a period of complete revolution (around of the Sun).
b) Knowing that the period of revolution of Mercury lasts 88 days, determine the average speed of this
planet (in km / s).
c) Using defined integrals, determine the total area St of the ellipse generated by the orbit of Mercury.
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