Physics
Third Law of Kepler
The Third Law of Kepler states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. In simpler terms, it describes the relationship between a planet's distance from the sun and the time it takes to complete one orbit. This law is fundamental in understanding the motion of celestial bodies within our solar system.
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8 Key excerpts on "Third Law of Kepler"
- Rudolf Kurth(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
This can also be expressed: the radius vector from the Sun to the planet sweeps out equal areas in equal times. Or, yet again: the areal velocity of the planet (that is the derivative of the area swept out with respect to time) is constant. The inductive generalization made from this to all the planets is known as Kepler's second law. That the three formulations are actually equivalent appears intuitively obvious from a consideration of two positions of Mars so close together that the arc joining them may be regarded, to a sufficient approximation, as the tangent at the first position. The analytical proof may be derived similarly. 2. Kepler had realized, quite early, that a relation must exist between the period, P, and the semi-major axis, a, of the various planets. The periods increase monotonically with the semi-axes. After some failures he found the expected relationship. It is his third law, which may be written: the squares of the periods of two planets are in the same ratio as the cubes of the semi-major axes of their orbits. In symbols we have P* _a* 14 MECHANICS OF THE SOLAR SYSTEM The suffix one denotes a comparison planet. If we choose this to be the Earth, and its orbital semi-axis and period as units, the law is simply P 2 = a 3 . We shall not give a numerical demonstration of this law by means of another table, since the generally accepted values of the semi-major axes (which we have quoted) have all been directly calculated from the sidereal periods by means of Kepler's third law itself. So these values cannot be used as a proof of the law. Data must be used that were obtained by such a method as Kepler's. 3. If the distance, in kilometres, between any two members of the solar system is known at some instant, then all its absolute measurements are known.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
That allows also for highly eccentric orbits (like very long stretched out circles). Bodies with highly eccentric orbits have been identified, among them the comets and many asteroids, discovered after Kepler's time. The dwarf planet Pluto was discovered as late as 1929, the delay mostly due to its small size, far distance, and optical faintness. Heavenly bodies such as comets with parabolic or even hyperbolic orbits are possible under the Newtonian theory and have been observed. To understand the second law let us suppose a planet takes one day to travel from point A to point B . The lines from the Sun to points A and B , together with the planet orbit, will define an (roughly triangular) area. This same area will be covered every day regardless of where in its orbit the planet is. Now as the first law states that the planet follows an ellipse, the planet is at different distances from the Sun at different parts in its orbit. So the planet has to move faster when it is closer to the Sun so that it sweeps an equal area. Kepler's second law is equivalent to the fact that the force perpendicular to the radius vector is zero. The areal velocity is proportional to angular momentum, and so for the same reasons, Kepler's second law is also in effect a statement of the conservation of angular momentum. The third law, published by Kepler in 1619 captures the relationship between the distance of planets from the Sun, and their orbital periods. For example, suppose planet A is 4 ________________________ WORLD TECHNOLOGIES ________________________ times as far from the Sun as planet B. Then planet A must traverse 4 times the distance of Planet B each orbit, and moreover it turns out that planet A travels at half the speed of planet B, in order to maintain equilibrium with the reduced gravitational centripetal force due to being 4 times further from the Sun. In total it takes 4×2=8 times as long for planet A to travel an orbit, in agreement with the law (8 2 =4 3 ).
- Joseph C. Amato, Enrique J. Galvez(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
The law of universal gravita -tion allows us to determine the mass of the Earth, Sun, stars, and all other bodies that possess an orbit -ing satellite (e.g., a moon, planet, or companion star). In recent years, we have used it to discover dark (invisible) matter and hundreds of bodies orbiting stars other than our Sun—the exoplanets envisioned by Giordano Bruno. The law of universal gravitation is the master key that unlocks many of the myster -ies of the universe. 8.2 KEPLER’S LAWS Kepler knew that planets closer to the Sun moved faster than ones farther away. For example, Earth’s aver -age orbital speed is 30 km/s, whereas Venus’ is 35 km/s and Mercury’s is 48 km/s. Moreover, he knew that Mars did not move with constant speed: the closer it was to the Sun, the faster it traveled. He concluded that the planets were propelled by the Sun, not by some power source within the planets themselves (as was supposed by Ptolemy and Copernicus). In 1609, based on his exhaustive analysis of Tycho’s data, Kepler proposed the following two “laws” of planetary motion: 1. The planets move in elliptical orbits, with the Sun at one focus. 2. The line drawn from the Sun to an orbiting planet sweeps out equal areas in equal times. Ten years later, Kepler published his Epitome Astronomiae Copernicanae (summary of Copernican astron -omy), in which he added a third law: 3. The square of a planet’s period T is proportional to the cube of the semimajor axis a of its elliptical orbit. Because of their central importance to the development of phys -ics and astronomy, we will study these laws and their conse -quences with great care. Box 8.1 reviews the mathematics and geometry of the ellipse, while Figure 8.2 illustrates the second law. In the figure, the shaded regions S 1 , S 2 , and S 3 are the areas swept out by a line stretching from the Sun to the orbiting planet during three intervals of time.
eBook - PDF- Richard Fitzpatrick(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
This motion is summed up in three simple laws: 1. The planetary orbits are all ellipses that are confocal with the Sun (i.e., the Sun lies at one of the foci of each ellipse—see Section A.9). 2. The radius vector connecting each planet to the Sun sweeps out equal areas in equal time intervals. 38 39 3.4 Plane polar coordinates 3. The square of the orbital period of each planet is proportional to the cube of its orbital major radii. Let us now see whether we can derive Kepler’s laws from Equation (3.2). 3.3 Conservation laws As we have already seen, gravity is a conservative force. Hence, the gravitational force in Equation (3.1) can be written (see Section 1.4) f = −∇U, (3.3) where the potential energy, U(r), of our planet in the Sun’s gravitational field takes the form U(r) = − G M m r . (3.4) (See Section 2.5.) It follows that the total energy of our planet is a conserved quantity. (See Section 1.4.) In other words, E = 2 2 − G M r (3.5) is constant in time. Here, E is actually the planet’s total energy per unit mass, and v = dr/dt. Gravity is also a central force. Hence, the angular momentum of our planet is a con- served quantity. (See Section 1.5.) In other words, h = r × v, (3.6) which is actually the planet’s angular momentum per unit mass, is constant in time. Assuming that |h| > 0, and taking the scalar product of the preceding equation with r, we obtain h · r = 0. (3.7) This is the equation of a plane that passes through the origin and whose normal is parallel to h. Because h is a constant vector, it always points in the same direction. We therefore conclude that the orbit of our planet is two-dimensional—that is, it is confined to some fixed plane that passes through the origin. Without loss of generality, we can let this plane coincide with the x–y plane.
eBook - PDF- Barbara Ryden, Bradley M. Peterson(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
(3.41) 72 Chapter 3 Orbital Mechanics After dividing by 4a and doing a bit of rearranging, we find r = a(1 − e 2 ) 1 + e cos θ . (3.42) This equation for r as a function of θ is the equation for an ellipse in polar coordinates, with the origin at one focus. This is equivalent in form to equation (3.34), which gives the shape of an orbit if Newton’s law of universal gravitation holds true. Comparison of equations (3.34) and (3.42) tells us that the angular momentum L of a planet’s orbital motion is related to the size and shape of its orbit by the relation L 2 m 2 = GMa(1 − e 2 ). (3.43) Since L = mrv t , this relation can also be written in the form r 2 v 2 t = GMa(1 − e 2 ). (3.44) When a planet is at perihelion, its velocity is entirely tangential (v pe = v t ), and its distance from the Sun is q = a(1 − e). This implies that for a planet at perihelion, v 2 pe a 2 (1 − e) 2 = GMa(1 − e 2 ), (3.45) or v pe = GM a 1 + e 1 − e 1 /2 . (3.46) A similar analysis of the planet’s speed at aphelion, where its velocity is also entirely tangential (v ap = v t ), tells us that v ap = GM a 1 − e 1 + e 1 /2 . (3.47) 3.1.3 Kepler’s Third Law Kepler’s second law (equation 3.21) tells us that the area swept out per unit time by the planet–Sun line is a constant, L/(2m). The area swept out in one orbital period, P , is the area of the ellipse, given by the standard formula A = πab. For one complete orbital period, then, we may write πab P = L 2m . (3.48) By squaring this equation and making the substitution b 2 = a 2 (1 − e 2 ), we have π 2 a 4 (1 − e 2 ) P 2 = L 2 4m 2 . (3.49) 3.1 Deriving Kepler’s Laws 73 Since equation (3.43) gives us a relation among L, a, and e, namely, L 2 m 2 = GMa(1 − e 2 ), (3.50) we can substitute back into equation (3.49) to find π 2 a 4 (1 − e 2 ) P 2 = GMa(1 − e 2 ) 4 , (3.51) or P 2 = 4π 2 GM a 3 , (3.52) which we recognize as Kepler’s third law, P 2 = Ka 3 , with the proportionality constant K ∝ 1 /M.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
Conversely, the closed trajectory is called a subharmonic orbit if k is the inverse of an integer, i.e., if m = 1 in the formula k = m / n . For example, if k = 1/3 (green planet in Figure 5, green orbit in Figure 10), the resulting orbit is called the third subharmonic of the original orbit. Although such orbits are unlikely to occur in nature, they are helpful for illustrating Newton's theorem. ________________________ WORLD TECHNOLOGIES ________________________ Limit of nearly circular orbits In Proposition 45 of his Principia , Newton applies his theorem of revolving orbits to develop a method for finding the force laws that govern the motions of planets. Johannes Kepler had noted that the orbits of most planets and the Moon seemed to be ellipses, and the long axis of those ellipses can determined accurately from astronomical measu-rements. The long axis is defined as the line connecting the positions of minimum and maximum distances to the central point, i.e., the line connecting the two apses. For illustration, the long axis of the planet Mercury is defined as the line through its successive positions of perihelion and aphelion. Over time, the long axis of most orbiting bodies rotates gradually, generally no more than a few degrees per complete revolution, because of gravitational perturbations from other bodies, oblateness in the attracting body, general relativistic effects, and other effects. Newton's method uses this apsidal precession as a sensitive probe of the type of force being applied to the planets. Newton's theorem describes only the effects of adding an inverse-cube central force. However, Newton extends his theorem to an arbitrary central forces F ( r ) by restricting his attention to orbits that are nearly circular, such as ellipses with low orbital eccentricity ( ε ≤ 10%), which is true of seven of the eight planetary orbits in the solar system.
eBook - PDFThe Principia: The Authoritative Translation and Guide
Mathematical Principles of Natural Philosophy
- Sir Isaac Newton(Author)
- 2016(Publication Date)
- University of California Press(Publisher)
4, and the law of areas in phen. 5. While the evidence for phenomena 1—5 is quite convincing, the case for phen. 6, the area law for the moon's orbital motion (reckoned with respect to the center of the earth), is weak, being no more than a correlation of the apparent motion with the apparent diameter. A notable change was introduced in the content of phen. 1 (formerly hyp. 1) in the second edition. Newton now had better data concerning the satellites of Jupiter. Newton also (phen. 2) introduced the satellites of Saturn, whose existence he had not been willing to acknowledge at the time of the first edition. 8 In the second edition a correction was made in phen. 4 of an error in the first edition that had escaped Halley's critical eye as well as Newton's. That is, in discussing Kepler's third law in the first edition (in hyp. 7), Newton had said that the periodic times of the planets and the size of their orbits are the same whether the planets revolve about the earth or about the sun, which is pure nonsense. He obviously meant, as he said in the corrected version in the second edition, that the periodic times and the size of the orbits are the same whether the sun revolves about the earth, or the earth about the sun; that is, they are the same in the Copernican as in the Tychonic or Ricciolian systems, but definitely not the same in the Copernican as in the Ptolemaic system. In stating phen. 4 (formerly hyp. 7), the third law of planetary motion, Newton finally does name Kepler as discoverer. He does not, however, similarly give credit to Kepler in phen. 5 (formerly hyp. 8) for the area law. Newton's Hyp. 3 8.5 Hyp. 3 of the first edition, eliminated in the second, reads: Every body can be transformed into a body of any other kind and successively take on 8.
eBook - PDFGravity
Newtonian, Post-Newtonian, Relativistic
- Eric Poisson, Clifford M. Will(Authors)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
154 Newtonian orbital dynamics The orbital vectors of Eqs. (3.42) and (3.43) are obtained by inserting these results within the relations n = cos f e x + sin f e y and λ = − sin f e x + cos f e y , which are inherited from Eqs. (3.7); the new relations reflect the change of convention regarding the choice of x -direction. 3.3 Perturbed Kepler problem We saw in the preceding section that the motion of two spherical bodies under their mutual gravitational attraction can be solved exactly and completely. As we shall see in Sec. 3.5, the same cannot be said of the three-body problem, and in general, the N -body problem does not admit an exact solution. Many situations of interest, however, involve more than two bodies. The Sun–Earth–Moon system is an extremely pertinent example, and the motion of any planet around the Sun is affected by the massive presence of Jupiter. While these systems cannot be given an exact description, we can nevertheless make progress by appealing to the fact that in many applications, the additional bodies have only a small effect on the orbital motion of a two-body system. In the Sun–Earth–Moon system, the dominant interaction is between the Sun and the Earth, and the gravitational effects of the Moon on Earth’s orbital motion are small. Similarly, while Jupiter is indeed a massive body, it is sufficiently far from the other planets that its gravity has a small effect on them. The approximate analysis of small external influences on a system dominated by an internal interaction is the realm of perturbation theory, and in this section we formulate a perturbation theory for Kepler’s problem. 3.3.1 Perturbing force We return to the two-body problem of Sec. 3.2, but now suppose that the relative acceleration a := a 1 − a 2 between bodies is given by a = − Gm r 2 n + f , (3.47) in which m := m 1 + m 2 , r := r 1 − r 2 , r := | r |, n := r /r , and where f is a perturbing force per unit mass, which may depend on r , v := v 1 − v 2 , and time.
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