Anaya Nolan
Kepler's laws of planetary motion describe how planets orbit the Sun. Kepler's first law states that planets move in elliptical orbits with the Sun as a focus. Kepler's second law can be stated as The areal velocity of a planet revolving around the Sun in an elliptical orbit remains constant, which implies the angular momentum of a planet remains constant. This means that the speed at which the planets move in space continuously changes.
| Characteristics | Values |
|---|---|
| Name | Kepler's Second Law |
| Other Names | Area Law, Law of Areas |
| Date of Publication | 1609 |
| Description | A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time. |
| Formula | The square of the time period of revolution of a planet around the sun in an elliptical orbit is directly proportional to the cube of its semi-major axis |
| Implication | The speed at which the planets move in space continuously changes. |
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What You'll Learn
The Sun is at one focus of the elliptical orbit
Kepler's first law of planetary motion states that all planets move around the Sun in elliptical orbits, with the Sun at one focus of the ellipse. This means that the Sun is located at one of the two points that define an ellipse, called foci. The German astronomer Johannes Kepler derived these laws from his analysis of the observations of the 16th-century Danish astronomer Tycho Brahe.
The ellipse can be thought of as a flattened circle, with its eccentricity measuring how flattened it is. This eccentricity is a number between 0 and 1, with 0 representing a perfect circle and 1 being essentially a flat line. The Earth's orbit, for example, has an eccentricity of 0.0167, making it very close to a perfect circle.
Kepler's first law has important implications for our understanding of the solar system. It implies that the distance between a planet and the Sun is constantly changing as the planet moves along its elliptical orbit. This led to Kepler's second law, which states that the imaginary line joining a planet and the Sun sweeps equal areas of space during equal time intervals as the planet orbits. In other words, planets move faster when they are closer to the Sun (perihelion) and slower when they are farther away (aphelion).
Kepler's laws of planetary motion, including the principle that "The Sun is at one focus of the elliptical orbit," have provided a foundation for subsequent discoveries in astronomy and classical physics. They have helped us understand the motions of the planets in our solar system and have even been applied to the motion of comets.
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Perihelion and aphelion
Kepler's First Law of planetary motion states that all planets orbit the Sun in elliptical paths, with the Sun at one focus of the ellipse. The second focus of the ellipse is empty, with no physical significance as far as classical mechanics is concerned.
The point on a planet's elliptical orbit that is closest to the Sun is called the perihelion, and the point farthest from the Sun is called the aphelion. Due to the changing planet-to-Sun distance as the planet follows the elliptical orbit, the speed of the planet varies. Kepler's Second Law states that the line joining the centres of the Sun and the planet sweeps out equal parts of an area in equal times. This implies that a planet is moving fastest when it is at perihelion and slowest when it is at aphelion.
The distance of the Earth from the Sun at perihelion is about 147 million kilometres, while at aphelion, it is about 152 million kilometres. Thus, the Earth's orbit has very low eccentricity, making it almost a perfect circle.
The concept of perihelion and aphelion also applies to other celestial bodies orbiting the Sun, such as comets. For example, Halley's Comet reaches its perihelion when it is inside the orbit of the Earth, and its aphelion is beyond the orbit of Neptune.
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The speed of planets changes
Kepler's second law of planetary motion states that a planet's speed in space is subject to continuous variation. In other words, the speed at which planets move in space is not constant.
Kepler's second law can be stated as: "The areal velocity of a planet revolving around the sun in an elliptical orbit remains constant, which implies that the angular momentum of a planet remains constant". This means that the angular momentum is constant, and all planetary motions are planar motions, which is a direct consequence of central force.
The law also states that a radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time. In other words, a planet covers the same area of space in the same amount of time, no matter where it is in its orbit. This is because the planet's kinetic energy is not constant in its path. It has more kinetic energy and therefore moves faster when it is closer to the Sun, and it has less kinetic energy and moves slower when it is farther from the Sun.
The point of nearest approach to the Sun is called perihelion, and the point of greatest separation is called aphelion. Thus, a planet is moving fastest when it is at perihelion and slowest when it is at aphelion.
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The angular momentum of a planet is constant
Kepler's second law of planetary motion states that a radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time. This law can also be stated as: "The areal velocity of a planet revolving around the Sun in an elliptical orbit remains constant, which implies the angular momentum of a planet remains constant".
Kepler's second law implies that the angular momentum of a planet is constant. This means that all planetary motions are planar motions, which is a direct consequence of central force. Kepler's second law helps explain that when planets are closer to the Sun, they will travel faster. The speed at which planets move in space continuously changes.
The second law is also referred to as the “area law". It was crucial to Sir Isaac Newton in 1684-85 when he formulated his law of gravitation between the Earth and the Moon and between the Sun and planets. Newton showed that the motion of bodies subject to central gravitational force need not always follow the elliptical orbits specified by the first law of Kepler, but can take paths defined by other, open conic curves.
Carl Runge and Wilhelm Lenz later identified a symmetry principle in the phase space of planetary motion, which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.
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The Sun's mathematical convenience
Kepler's first law of planetary motion states that all planets move around the Sun in elliptical orbits, with the Sun as one focus of the ellipse. The Sun's centre is always located at one focus of the orbital ellipse, with the planet following the ellipse in its orbit. This means that the distance between the planet and the Sun is constantly changing as the planet moves in its orbit.
The second focus of Kepler's first law is simply a mathematical convenience. Kepler's laws derive naturally from vector formulations of Newton's laws, and as such, there is no physical significance to the second focus. The second focus has no unusual properties and its location is not of particular importance.
The first law is one of three laws of planetary motion formulated by Kepler. Kepler's second law states that the imaginary line joining a planet and the Sun sweeps equal areas of space during equal time intervals as the planet orbits. This means that the speed of the planet is constantly changing, with the planet moving fastest when it is closest to the Sun (perihelion) and slowest when it is furthest from the Sun (aphelion). Kepler's third law shows that there is a precise mathematical relationship between a planet's distance from the Sun and the time it takes to revolve around the Sun.
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Frequently asked questions
The second focus of Kepler's first law is that the Sun is located at one of the two foci of the elliptical orbit of a planet.
The second focus highlights that the Sun is not at the center of the ellipse, but rather at one of the two foci, which was a significant departure from the previously held belief that planetary orbits were perfect circles.
The second focus, along with the first focus, contributes to the elliptical shape of the orbit. This means that the distance between the planet and the Sun is constantly changing as the planet moves along its orbit.
Kepler discovered the second focus through meticulous observations of planetary movements, particularly those of Mars. By analyzing the data collected by Tycho Brahe, Kepler realized that the orbit of Mars did not fit a circular model and concluded that the Sun must be at one focus of an ellipse.
