Physics
Planetary Orbits
Planetary orbits refer to the paths that planets follow as they revolve around a star, such as the sun. These orbits are elliptical in shape, with the star located at one of the foci of the ellipse. The motion of planets in their orbits is governed by gravitational forces, as described by Kepler's laws of planetary motion.
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12 Key excerpts on "Planetary Orbits"
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- 2014(Publication Date)
- University Publications(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 7 Orbit Two bodies of different mass orbiting a common barycenter. The relative sizes and type of orbit are similar to the Pluto–Charon system. In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the solar system. Orbits of planets are typically elliptical. Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesics; though in common practice an approximate force-based theory of universal gravitation based on Kepler's laws of planetary motion is often used instead for ease of calculation. History Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions. Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motions of planets were in fact (at least approximately) elliptical motions. In the geocentric model of the solar system, the celestial spheres model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres ________________________ WORLD TECHNOLOGIES ________________________ or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion.
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- 2014(Publication Date)
- College Publishing House(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 10 Orbit Two bodies of different mass orbiting a common barycenter. The relative sizes and type of orbit are similar to the Pluto–Charon system. In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the solar system. Orbits of planets are typically elliptical. Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesics; though in common practice an approximate force-based theory of universal gravitation based on Kepler's laws of planetary motion is often used instead for ease of calculation. History Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions. Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motions of planets were in fact (at least approximately) elliptical motions. In the geocentric model of the solar system, the celestial spheres model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres ________________________ WORLD TECHNOLOGIES ________________________ or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion.
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- 2014(Publication Date)
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________________________ WORLD TECHNOLOGIES ________________________ Chapter-3 Orbit Two bodies with a slight difference in mass orbiting around a common barycenter. The relative sizes and type of orbit are similar to the Pluto–Charon system. In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the gravitational orbit of a planet around a point in space near a star. Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesics; though in common practice an approximate force-based theory of universal gravitation based on Kepler's laws of planetary motion is often used instead for ease of calculation. History Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions. Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motion of the planets were in fact (at least approximately) elliptical motions. In the geocentric model of the solar system, the celestial spheres model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres ________________________ WORLD TECHNOLOGIES ________________________ or rings, but after measurements of the exact motion of the planets theoretical mechanisms such as the deferent and epicycles were later added. Although it was capable of accurately predicting the planets position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion.
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- 2014(Publication Date)
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________________________ WORLD TECHNOLOGIES ________________________ Chapter 3 Orbit Two bodies with a slight difference in mass orbiting around a common barycenter. The relative sizes and type of orbit are similar to the Pluto–Charon system. In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the gravitational orbit of a planet around a point in space near a star. Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesics; though in common practice an approximate force-based theory of universal gravitation based on Kepler's laws of planetary motion is often used instead for ease of calculation. History Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions. Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motion of the planets were in fact (at least approximately) elliptical motions. In the geocentric model of the solar system, the celestial spheres model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres ________________________ WORLD TECHNOLOGIES ________________________ or rings, but after measurements of the exact motion of the planets theoretical mech-anisms such as the deferent and epicycles were later added. Although it was capable of accurately predicting the planets position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion.
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- 2014(Publication Date)
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________________________ WORLD TECHNOLOGIES ________________________ Chapter 6 Orbit Two bodies of different mass orbiting a common barycenter. The relative sizes and type of orbit are similar to the Pluto–Charon system. In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the solar system. Orbits of planets are typically elliptical. Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesics; though in common practice an approximate force-based theory of universal gravitation based on Kepler's laws of planetary motion is often used instead for ease of calculation. History Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions. Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motions of planets were in fact (at least approximately) elliptical motions. ________________________ WORLD TECHNOLOGIES ________________________ In the geocentric model of the solar system, the celestial spheres model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion.
eBook - PDFThe Solar System 2
External Satellites, Small Bodies, Cosmochemistry, Dynamics, Exobiology
- Therese Encrenaz, James Lequeux(Authors)
- 2021(Publication Date)
- Wiley-ISTE(Publisher)
We encounter these laws today when we apply Newton’s laws of dynamics to the gravitational force applied to two bodies (see section 4.2.2). Kepler’s laws correctly describe planetary motion around the Sun because the planets have a negligible mass in comparison to the Sun. FIRST LAW.– The trajectory of a planet is an ellipse with the center of the Sun at one of its foci. An ellipse is a curve in which, for all points, the sum of the distances to two fixed points (the foci) is constant (Figure 4.2). The motion of a planet could be described in the following manner: 𝑟 = (ଵି మ ) (ଵାୡ୭ୱ ) [4.1] a statement in which r is the distance of the planet from the center of the Sun, a is the semi-major axis of the ellipse, and e is its eccentricity. f, the true anomaly, is the angle measured in relation to the point of the shortest heliocentric distance, called perihelion (periapsis). The point of the greatest heliocentric distance is called aphelion (apoapsis in the general case). 208 The Solar System 2 Figure 4.2. Parameters of the ellipse (source: based on (Shields et al. 2016)) SECOND LAW.– While a planet is in motion, the radius vector joining the center of the Sun and the planet describes equal areas during equal amounts of time. The planet slows as it moves away from the Sun and accelerates when it moves toward the Sun. THIRD LAW.– The ratio of the cube of the semi-major axis of the orbit of a planet to the square of its orbital period around the Sun is the same for all of the planets: a 3 /P 2 = C [4.2] where P is the orbital period. If the semi-major axis is expressed in astronomical units (au) and P is expressed in terrestrial years, the constant C is equal to 1. Put another way, for the planets in the Solar System, the period of a planet whose semi-major axis is a au is equal to a 3/2 years. Newton’s law of universal gravitation would later show that this constant C is proportional to the mass of the (star + planet) system (see section 4.2.3).
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- 2014(Publication Date)
- Orange Apple(Publisher)
The apses of an orbit are the points at which the orbiting body is closest or furthest away from the attracting center; for planets orbiting the Sun, the apses correspond to the perihelion (closest) and aphelion (furthest). With the publication of his Principia roughly eighty years later (1687), Isaac Newton provided a physical theory that accounted for all three of Kepler's laws, a theory based on Newton's laws of motion and his law of universal gravitation. In particular, Newton proposed that the gravitational force between any two bodies was a central force F ( r ) that varied as the inverse square of the distance r between them. Arguing from his laws of motion, Newton showed that the orbit of any particle acted upon by one such force is always a conic section, specifically an ellipse if it does not go to infinity. However, this conclusion holds only when two bodies are present (the two-body problem); the motion of three bodies or more acting under their mutual gravitation (the n -body problem) remained unsolved for centuries after Newton, although solutions to a few special cases were discovered. Newton proposed that the orbits of planets about the Sun are largely elliptical because the Sun's gravitation is dominant; to first approximation, the presence of the other planets can be ignored. By analogy, the elliptical orbit of the Moon about the Earth was dominated by the Earth's gravity; to first approximation, the Sun's gravity and those of other bodies of the Solar System can be neglected. However, Newton stated that the gradual apsidal precession of the planetary and lunar orbits was due to the effects of ________________________ WORLD TECHNOLOGIES ________________________ these neglected interactions; in particular, he stated that the precession of the Moon's orbit was due to the perturbing effects of gravitational interactions with the Sun. Newton's theorem of revolving orbits was his first attempt to understand apsidal precession quantitatively.
eBook - PDF- Andrew Fraknoi, David Morrison, Sidney C. Wolff(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Chapter Outline 3.1 The Laws of Planetary Motion 3.2 Newton’s Great Synthesis 3.3 Newton’s Universal Law of Gravitation 3.4 Orbits in the Solar System 3.5 Motions of Satellites and Spacecraft 3.6 Gravity with More Than Two Bodies Thinking Ahead How would you find a new planet at the outskirts of our solar system that is too dim to be seen with the unaided eye and is so far away that it moves very slowly among the stars? This was the problem confronting astronomers during the nineteenth century as they tried to pin down a full inventory of our solar system. If we could look down on the solar system from somewhere out in space, interpreting planetary motions would be much simpler. But the fact is, we must observe the positions of all the other planets from our own moving planet. Scientists of the Renaissance did not know the details of Earth’s motions any better than the motions of the other planets. Their problem, as we saw in Observing the Sky: The Birth of Astronomy, was that they had to deduce the nature of all planetary motion using only their earthbound observations of the other planets’ positions in the sky. To solve this complex problem more fully, better observations and better models of the planetary system were needed. Figure 3.1 International Space Station. This space habitat and laboratory orbits Earth once every 90 minutes. (credit: modification of work by NASA) 3 ORBITS AND GRAVITY Chapter 3 Orbits and Gravity 69 3.1 THE LAWS OF PLANETARY MOTION Learning Objectives By the end of this section, you will be able to: Describe how Tycho Brahe and Johannes Kepler contributed to our understanding of how planets move around the Sun Explain Kepler’s three laws of planetary motion At about the time that Galileo was beginning his experiments with falling bodies, the efforts of two other scientists dramatically advanced our understanding of the motions of the planets. These two astronomers were the observer Tycho Brahe and the mathematician Johannes Kepler.
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- 2014(Publication Date)
- Learning Press(Publisher)
They are expressed as equations connecting the coordinates of the planet, and the time variable, with the parameters describing the position, size and shape of the orbit, the socalled orbital elements. Newton's laws of motion are concerned with the motion of objects subject to impressed forces. Newton's law of universal gravitation specifies these forces. Together these laws ________________________ WORLD TECHNOLOGIES ________________________ constitute differential equations satisfied by planetary motions. Solving these equations constitute the n-body problem. The solutions to the two-body problem, where there are only two particles involved, say, the sun and one planet, can be expressed analytically. These solutions include the elliptical Kepler orbits, but motions along other conic section (parabolas, hyperbolas and straight lines) also satisfy Newton's differential equations. The solutions deviate from Kepler's laws in that 1. the focus of the conic section is at the center of mass of the two bodies, rather than at the center of the Sun itself. 2. the period of the orbit depends a little on the mass of the planet. The language of Kepler's laws also applies when the motion of a planet is affected by the attraction from the other planets, as the orbits are described as Kepler orbits with slowly varying orbital elements. And in the case of the two-body problem in general relativity. The derivations below involve the art of solving differential equations. The derivations below use heliocentric polar coordinates, see Figure 4. Kepler's second law is derived first, as the derivation of the first law depends on the derivation of the second law. They can also be formulated and derived using Cartesian coordinates. Acceleration vector From the heliocentric point of view consider the vector to the planet where r is the distance to the planet and the direction is a unit vector.
- Ranjan Vepa(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
The key question is: How close is sufficiently close? That is a question we shall not seek to answer yet, although it is indeed a fundamental one. 2.2.1 Kepler’s Laws The German astronomer Johann Kepler (1571–1630) formulated three empirical laws of planetary motion based on astronomical data provided to him by the Danish astronomer Tycho Brahe in the late 1590s. The laws were published over a period spanning a decade, at about the same time as Galileo was making his landmark astronomical observations. The laws (see, for example, Deutsch [1], Ball and Osborne [2], Prussing and Conway [3], and Bate [4]) are: 28 (i) The orbit of each planet is an ellipse with the Sun at one focus; (ii) The line joining the Sun to the planet sweeps out equal areas in equal lengths of time; and finally, (iii) The squares of the orbital periods of the planets are proportional to the cubes of their mean distances from the Sun. 2.2.2 Keplerian Motion of Two Bodies It is to Sir Isaac Newton (1642–1727), the founder of Newtonian mechanics, the theory of gravitation, and differential calculus, that the undisputed honor of being the origin- ator of the mathematical theory of planetary motion falls. When a particle moves in the vicinity of a celestial body, it experiences a force of attraction, which is directed toward a fixed center of attraction, located at the center of mass of the attracting body. The force is mutual in the sense that the attracting body experiences a force directed toward the center of mass of the particle of the same magnitude. This force is given by Newton’s law of gravitation as, F ¼ G mm 0 r 2 , (2.1) where m 0 = the mass of the attracting body (fixed), G = the universal gravitational constant, and r = the distance between the centers of masses. This attracting force F is always in the negative direction of the direction of r. There is no force acting in the direction of the orbit tangent or θ-direction.
- Osamu Morita(Author)
- 2022(Publication Date)
- CRC Press(Publisher)
Chapter 11 Orbital Motion of PlanetsDOI: 10.1201/9781003310068-11Johannes Kepler found three laws of the orbital motion of planets based on the enormous observational data of Tycho Brahe. Isaac Newton established the basis of classical mechanics and found the law of universal gravitation in which Kepler's three laws played a very important role. In this chapter, we will review the process classical mechanics was established and prove Kepler's three laws exactly. Further, we will discuss the universal gravitation exerted by bodies of finite extent, and the oceanic tides and tidal effects on the Earth–Moon system. We will discuss the general orbits due to a central force. As an application of the orbital motion, Rutherford scattering is discussed precisely.11.1 The Law of Universal Gravitation
Newton considered that an apple falls to the ground owing to some attractive force exerted by the Earth and the same force exerts on the Moon. Orbital motion of the Moon around the Earth is the same that the Moon is continuously falling to the Earth. Newton calculated the falling distance of the Moon in 1 s, from which he obtained the acceleration due to gravity at the center of the Moon. He found that the ratio of the magnitude of the acceleration due to gravity at the center of the Moon to that of the Earth's surface is almost equal to the square ratio of the Earth's radius to the distance between the Earth and the Moon. We will review the calculation performed by Newton. Here we will use the following physical quantities,r = 3.84 ×: the distance from the Earth to the Moon,10 8ma = 6.37 ×: the radius of the Earth,10 6mT = 2.36 ×: the orbital period of the Moon.10 6sAt first, we will calculate the distance that the Moon falls to the Earth in 1 s. Let the center of the Earth be O, the position of the Moon at time t(s) be A and the position of the Moon att + 1(s) be B. Suppose that the Moon would move to point C att + 1(s) if it traveled at the constant velocity (Fig. 11.1
- Joseph C. Amato, Enrique J. Galvez(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
Inset: telescopic image of the four moons. Their orbits are in a plane nearly perpendicular to the page. (Image courtesy of Chanan Greenberg, www.greenhawkobservatory.com.) 286 Physics from Planet Earth - An Introduction to Mechanics From Equation 8.4 the force exerted by the Sun on a planet of mass m P is F m r Planet P Sun = 4 1 2 2 π κ , where r is the distance between the two bodies. If the inverse-square force is universal, then the force on the Sun, due to the planet, must obey an equation of the same form: F M r Sun Sun P = 4 1 2 2 π κ , where κ P depends only on the properties of the planet. By Newton’s third law (action–reaction), F Sun = F Planet , so m P / κ Sun = M Sun / κ P , or m P / M Sun = κ Sun / κ P . The simplest way to satisfy this relation is to require that m P ∝ 1/ κ P and M Sun ∝ 1/ κ Sun , with the same proportionality constant in each case. Using this result, F F M m r Sun P Sun P = ∝ 2 or F F G M m r Sun P Sun P = = 2 , where the constant of proportionality G is called the gravitational constant , and must be determined experimentally. More generally, the force between any two bodies of mass m 1 and m 2 is F G m m r 12 1 2 2 = , (8.7) where r is the distance between the bodies. The direction of the force on body 1 is toward body 2, and vice versa. This is Newton’s law of universal gravitation , applicable to bodies whose dimensions (e.g., radii) are small compared to the distance r between them or to bodies (such as stars and planets) that are spherically symmetric. Equation 8.7 is one of the most important and far-reaching equations in physics. It is “that mighty principle under the influence of which every star, planet, and satellite in the universe pursues its allotted course.” * For a body of mass m in a circular orbit about a central body of mass M , the force on m is F m v r m r T G mM r T r GM = = = = 2 2 2 2 2 3 2 4 4 π π or .
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