Cecily Zamora
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. This law was formulated by German mathematician and astronomer Johannes Kepler, who lived in Graz, Austria during the early 17th century. Kepler's work built upon the astronomical observations of Tycho Brahe, a 16th-century Danish astronomer. Kepler's first law replaced the previously held belief of circular orbits and epicycles in the heliocentric theory of Nicolaus Copernicus. The elliptical nature of planetary orbits, with the Sun at one of the foci, is a fundamental concept in understanding the motions of the planets in our solar system.
| Characteristics | Values |
|---|---|
| Kepler's First Law | All planets revolve around the Sun in elliptical orbits, with the Sun at one of the two foci. |
| Orbital Shape | Ellipse, resembling a stretched-out circle with the Sun at one focus. |
| Perihelion | Point in the orbit where the planet is closest to the Sun (about 147 million km). |
| Aphelion | Point in the orbit where the planet is farthest from the Sun (about 152 million km). |
| Eccentricity | Parameter in the equation of an ellipse, indicating how stretched-out the orbit is. |
| Kepler's Second Law | The areal velocity of a planet revolving around the Sun remains constant, implying constant angular momentum. |
| Angular Momentum | The rate at which the planet sweeps out areas is proportional to its angular momentum. |
| Kepler's Third Law | The square of the orbital period is directly proportional to the cube of the semi-major axis of the orbit. |
| Derivation | Kepler's Laws are derived from Newton's Law of Universal Gravitation and Laws of Motion. |
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What You'll Learn
The orbit of every planet is an ellipse
Kepler's first law of planetary motion states that "All the planets revolve around the sun in elliptical orbits having the sun at one of the foci". This means that the orbit of every planet is an ellipse, with the Sun situated at one of the ellipse's two foci.
An ellipse is a closed plane curve that resembles a stretched-out circle. The Sun is not at the centre of the ellipse but at one of its foci, with the other focal point having no physical significance for the orbit. The centre of an ellipse is the midpoint of the line segment joining its two focal points. A circle is a specific type of ellipse where both focal points coincide.
The elliptical path of a planet's orbit can be determined using calculus. The general form of an ellipse in polar coordinates is given by the equation:
> r~(1+e*cos(theta))^(-1)
In this equation, r represents the distance from the Sun, and e is the eccentricity, a parameter that characterises the shape of the ellipse. When the eccentricity of an orbit is zero, the orbit is perfectly circular. As e increases towards 1, the ellipse becomes more elongated.
Kepler's first law can be derived from Newton's laws of motion and the universal law of gravitation. The law of conservation of angular momentum also plays a role in verifying this law. Kepler's first law demonstrates that the motion of planets is influenced by the gravitational force of the Sun, resulting in elliptical orbits.
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The Sun is at one of the two foci
Kepler's first law states that "All the planets revolve around the sun in elliptical orbits having the sun at one of the foci". This means that the Sun is positioned at one of the two focal points of the elliptical path that planets follow in their orbit.
The point at which a planet is closest to the Sun is known as perihelion (approximately 147 million kilometres from the Sun), and the point at which the planet is farthest from the Sun is known as aphelion (approximately 152 million kilometres from the Sun). The sum of the distances of any planet from the two foci of an ellipse is always constant.
Kepler's first law can be derived from the work of Brahe, whose dataset revealed that the orbit of planets is elliptical in shape, with the Sun at one focus. This elliptical orbit can be described by the equation:
> r=(1+e.cos(theta))^-1
Here, 'r' represents the general form of an ellipse in polar coordinates, with its origin placed at a focus. The parameter 'e' is the eccentricity of the ellipse. When the eccentricity of a planet's orbit is zero, the orbit is perfectly circular. As 'e' increases, the orbit becomes more elongated and elliptical.
Kepler's laws of planetary motion, which include the first law, describe the motion of planets around the Sun. These laws allow for predictions of planet positions and velocities, as well as the time for a satellite to fall into a planet's surface. Kepler's first law, specifically, reveals that the Sun is at one of the two foci of the elliptical orbits of planets.
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Perihelion and aphelion
Kepler's first law states that "All the planets revolve around the sun in elliptical orbits having the sun at one of the foci". The point at which a planet is closest to the sun is known as perihelion, and the point at which it is farthest from the sun is known as aphelion. The terms perihelion and aphelion come from ancient Greek, where peri means close, apo means far, and helios means the sun.
The Earth reaches perihelion in early January, about two weeks after the December solstice, and reaches aphelion in early July, about two weeks after the June solstice. In 2025, Earth will be about 91,405,993 miles from the Sun at perihelion and 94,502,939 miles from the Sun at aphelion. This means that the Earth is about 4,800,000 km (3,000,000 miles) farther from the Sun in July than in January.
The dates of perihelion and aphelion are not fixed, due to variations in the eccentricity of the Earth's orbit. In 1246, the December solstice occurred simultaneously with the Earth's perihelion. Since then, the dates of perihelion and aphelion have drifted by about one day every 58 years. In the short term, the dates can vary by up to two days from year to year.
The Moon's orbit around the Earth is also elliptical. The point in the Moon's orbit that is closest to Earth is called the perigee, and the farthest point is the apogee. These terms are sometimes used interchangeably with the Earth's perihelion and aphelion.
The terms aphelion and perihelion are relevant not only to Earth but also to other planets, comets, and bodies orbiting the Sun. For example, the planet Mars has an even more elliptical orbit than Earth. The perihelion and aphelion of a planet can be related to its kinetic energy, with the planet having more kinetic energy near perihelion and less near aphelion, implying greater speed at perihelion and lower speed at aphelion.
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Derivation from the Inverse-Square Law
Kepler's laws describe the motion of objects in the presence of a central inverse square force. Kepler's first law states that "All the planets revolve around the sun in elliptical orbits having the sun at one of the foci". The point at which the planet is closest to the sun is known as perihelion, and the point farthest from the sun is known as aphelion.
The inverse-square law, as it relates to Kepler's laws, refers to the relationship between the force acting on a planet and its distance from the Sun. According to Newton's Law of Universal Gravitation, the force of gravity between two objects is inversely proportional to the square of the distance between them. In the context of planetary motion, this means that the force acting on a planet is directly proportional to the mass of the planet and inversely proportional to the square of its distance from the Sun. This is expressed by the equation:
> F = GMm/r^2
Where:
- F is the force
- G is the gravitational constant
- M is the mass of the Sun
- M is the mass of the planet
- R is the distance between the Sun and the planet
By applying this inverse-square relationship and using calculus, we can derive Kepler's first law. Kepler analysed the dataset of Tycho Brahe, and from this, he determined that the orbit of a planet is an ellipse with the Sun at one of its foci. This is visualised by plotting the orbit for various values of eccentricity, which is a parameter in the equation for an ellipse:
> r = (1 + e*cos(theta))^-1
Here, e represents the eccentricity, which determines the shape of the orbit. When e is 0, the orbit becomes a perfect circle. As e increases towards 1, the orbit becomes more elongated and elliptical.
In summary, Kepler's first law can be derived from the inverse-square law by recognising that the force acting on a planet varies with the inverse square of its distance from the Sun. This relationship, combined with empirical data analysis, allowed Kepler to conclude that planetary orbits are elliptical with the Sun at one focus.
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Derivation from Newton's Laws
Kepler's laws of planetary motion, published by Johannes Kepler in 1609, describe the orbits of planets around the Sun. Kepler's first law is proof that the orbit is an ellipse if the gravitational force is inverse square.
Newton's Second Law, F = ma, can be used to derive Kepler's First Law. The force is GMm/r^2 in a radial inward direction. The angular momentum L is constant, L = mr^2w, so we can get rid of w in the equation. This equation can be integrated using two tricks.
The first trick is to change the variable r to its inverse, u = 1/r. The second trick is to use the constancy of angular momentum to change the variable t to q. This results in the standard (r, q) equation of an ellipse of semi-major axis a and eccentricity e, with the origin at one focus.
The time it takes a planet to make one complete orbit around the Sun (one planet year) is related to the semi-major axis of its elliptic orbit. The area of an ellipse is pab, and the rate of sweeping out of the area is L/2m, so the time T for a complete orbit is given by the equation.
The top point B of the semi-minor axis of the ellipse must be exactly a distance from F1. Using Pythagoras' theorem, we can show that the distance from F2 is b. Therefore, the distance from the focus to either side of the ellipse is a and b, which is the standard form of an ellipse.
Newton derived Kepler's laws as pieces of his orbital mechanics. Newton's Principia does not use calculus, but it does make extensive use of the infinitesimal techniques he used to develop calculus. Newton computed in his Principia the acceleration of a planet moving according to Kepler's first and second laws. The direction of the acceleration is towards the Sun, and the magnitude of the acceleration is inversely proportional to the square of the planet's distance from the Sun. This implies that the Sun may be the physical cause of the acceleration of planets.
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Frequently asked questions
Kepler's First Law states that all planets orbit the Sun in elliptical paths, with the Sun at one of the foci.
Kepler's First Law can be derived from the Inverse-Square Law, which states that the gravitational force is inverse square. This means that the orbit must be an ellipse with the Sun at one focus.
Key terms include "ellipse," "focus," "perihelion," and "aphelion." An ellipse is a stretched-out circle with two foci, and Kepler's First Law states that the Sun is at one of these foci. Perihelion is the point in a planet's orbit where it is closest to the Sun, while aphelion is the point farthest from the Sun.
Kepler's First Law was derived from empirical data collected by Tycho Brahe and analysed by Johannes Kepler. Kepler's Laws describe the motion of objects in the presence of a central inverse square force, such as the Sun's gravity.
Kepler's First Law has several implications, including the fact that planets are closer to the Sun at perihelion and farther from the Sun at aphelion. This results in planets travelling faster at perihelion and slower at aphelion.
