In this text, we would like to present one way how to derive keplers laws from the newtons law of universal gravitation and motion. With a bit more involved mathematics than we have presently at our disposal, one can show that the only closed solutions to newtons two body force are elliptical orbits intermediate mechanics for physicists. Planets move around the sun in ellipses, with the sun at one focus. These improved the heliocentric theory of nicolaus copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. Keplers conclusion from this monumental work, are consummated in his three wellknown laws of planetary motion 1. Later, isaac newton, using his universal inversesquare law theory of gravity, was able to show how keplers laws fit into a scientific theory of celestial mechanics. The line connecting the sun to a planet sweeps equal areas in equal times. We need the expression for the normal vector the plane of motion anyway, so we start with. Since the lefthand side is real, it must be that 2r0. They are presented here purely to satisfy curiosity and for your entertainment. Of course, kepler s laws originated from observations of the solar system, but newton s great achievement was to establish that they follow mathematically from his law of universal gravitation and his laws of motion. After stating his 3 laws of motion, the first thing newton proceeded to show was that for any central force, the area swept out per time will be a constant. The geometric locus of points on the plane with the. This ends the proof of keplers first law using polar coordinates.
Keplers laws of planetary motion and his theory of gravitation. One of the highlights of classical mechanics is the mathematical derivation. A central force is a force that is always pointed to a center, as the force of gravity on the earth is always pointed to the sun. The square of the orbital period of a planet is proportional to the cube of the semimajor axis of the ellipse. One justi cation for this approach is that a circle is a special case of an ellipse. This is achieved by applying a simple system identification method using numerical data from the planets orbits in conjunction with the inverse square law for the attractive force between celestial bodies and the concepts of the derivative and differential equation. Observe that we used only the equations 4 and 6 of conservation of angular momentum and of. Keplers second law motion is planar and equal areas are swept out in equal times is an easy consequence of the conservation of angular momentum. The square of the period of a planet is proportional to the cube of its mean distance from the sun. The geometrical way of derivation was discussed in richard feynmann lecture presented on th.
Second law a planet moves in a plane, and the radius vector from the sun to the planet sweeps out equal areas in equal times. Kepler s second law is based on the speed of the object as it orbits. A planet orbits the sun in an ellipse, with the sun at one. Keplers laws of planetary motion are three scientific laws describing the motion of planets. The position vector from the sun to a planet sweeps out area at a constant rate.
It would be a pity to have a course on dynamical astronomy and not at least see a proof of kelpers. Keplers laws math 1 multivariate calculus d joyce, spring 2014 keplers laws of planetary motion. This type of motion is particularly relevant when studying the orbital movement of. A planet moves in a plane in an ellipse with the sun at one focus. Keplers law problems and solutions solved problems in. For simplicity, well consider the motion of the planets in our solar system around the sun, with gravity as the central force. Although he did his work before the invention of calculus, we can more easily develop his theory, as newton did, with multivariate calculus. We can therefore demonstrate that the force of gravity is the cause of keplers laws. The orbit of a planet is an ellipse with the sun at one of its foci. For the special case of an object of mass, m, in a circular orbit around an object of mass. This is easy to show for the simple case of a circular orbit.
Keplers laws of planetary motion 3 perpendicular direction. Laurence department of physics, florida international university, miami, fl 33199. Keplers three laws of planetary motion kepler took the data that brahe had spent his life collecting and used it especially the information on mars to create three laws that apply to any object that is orbiting something else. Derivation of keplers third law for circular orbits we shall derive keplers third law, starting with newtons laws of motion and his universal law of gravitation. Force mass x acceleration and the fact that the centripetal acceleration, a, of a body moving at speed v in a circle of radius r is given by v2r, he inferred that the force on a mass m in a circular orbit must be given by. A detailed look at keplers second law as derived from newtons laws. He became aware of copernicus work at the university of tubingen, where he completed a masters degree. If robs answer is a bit terse for you, see a selfcontained derivation of keplers laws from newtons laws, which assumes less prior knowledge and proceeds in smaller steps. Keplers laws explained using only laws of mechanics and gravity and the calculus, newton could derive keplers three laws.
He used a geometrical argument similar to the following. In astronomy, kepler s laws of planetary motion are three scientific laws describing the motion of planets around the sun. Derivation of keplers laws from newtons laws keplers laws ki, kii, and kiii can be derived from newtons laws using calculus which newton also invented. Keplers laws the german astronomer kepler discovered three fundamental laws governing planetary motion. His second law is that equal areas of the position vector from the sun to the planet are swept out in equal times. At that time he developed these laws, there was not yet a developed theory of gravity capable of explaining why the planets moved as they were observed to. Keplers laws describe the motion of objects in the presence of a central inverse square force. General astronomykeplers laws wikibooks, open books. We present here a calculusbased derivation of kepler s laws. Keplers laws of planetary motion, stated with a generality that covers comets as well as planets, are as follows. Unlike keplers first and second laws that describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The intent is to make this crowning achievement of newtonian mechanics easily accessible to students in introductory physics courses.
Planetary motion 3 initsmotionaroundthesunisl mrv nwheremisthemassoftheplanet,equation1. K2 period squared proportional to radius cubed this proof is easy for the special case of a circular orbit of radius 4, where the planets speed is also constant at every point. By knowing some very basic formula we can derive the equation for keplers 3rd law. In sections 28 we present newtons derivation of keplers laws from the inverse square law for gravity, which only uses basic calculus. Plenty of literature deals with the relation between keplers and newtons laws. Kepler discovered them, but newton understood them. A selfcontained derivation of keplers laws from newtons. Newtons law of motion is derived from keplers laws of planetary motion. Keplers third law sometimes referred to as the law of harmonies compares the orbital period and radius of orbit of a planet to those of other planets. Kohout and lamar layton goddard space flight center 1. Laws of planetary motion johannes kepler 15711630 kepler johannes kepler came from a poor protestant family in germany.
Gravitation from his laws of motion coupled with keplers third law of planetary motion from his second law. A selfcontained derivation of keplers laws from newtons laws. Although keplers math was essentially wrong, the three laws he came up with were correct. A modern newtonian derivation of keplers second law requires the concept of an orbiting bodys angular momentum l r x p m r x v where m is the bodys mass, r is its position vector and p its linear momentum mv, where v is its velocity. Kepler 15711630 developed three laws of planetary motion. This is achieved by applying a simple system identification method using numerical data from the planets orbits in. A simple derivation of keplers laws without solving. A planet, mass m, orbits the sun, mass m, in a circle of radius r and a period t.
The orbit of the planet is a conic ellipse, parabola, or hyperbola with the sun at one focus. We can derive keplers third law by starting with newtons laws of motion and the universal law of gravitation. Keplers second law by studying the danish astronomer tycho brahes data about the motion of the planets, kepler formulated three empirical laws. Keplers 3rd law is often called the harmonic law, and states that, for each planet orbitting the sun, its sidereal period squared divided by the cube of the semimajor axis of the orbit is a constant. Derivation of newtons law of motion from keplers laws of. The derivation of keplers laws of planetary motion from. Also we will see how much information about gravity we can get from keplers.
Among other things, keplers laws allow one to predict the position and velocity of the planets at any given time, the time for a satellite to collapse into the surface of. Proceeding like newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain keplers laws. The distance the planet travels in one orbit is the circumference 2. In this note, i am going sketch the derivationof all three of keplers laws from classical newtonian mechanics. The point is to demonstrate that the force of gravity is the cause for keplers laws although we will only derive the third one. As the earth moves away from the sun, it will move slower and slower. Start with keplers 2nd law, da dt l 2m 1 since the rhs is constant, the total area swept out in an orbit is a l 2m p 2. The equations of planetary motion and their solution by. Satellite s in elliptical orbit about the earth f figure 1 shows a satellite s is in an elliptical orbit of period t about the earth f where t is the time. The bonus is that i have done the complete proof using.
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