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astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
, Kepler's laws of planetary motion, published by
Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...
in 1609 (except the third law, which was fully published in 1619), describe the orbits of planets around the Sun. These laws replaced
circular orbit A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. In this case, not only the distance, but also the speed, angular speed, Potential energy, potential and kinetic energy are constant. T ...
s and epicycles in the heliocentric theory of
Nicolaus Copernicus Nicolaus Copernicus (19 February 1473 – 24 May 1543) was a Renaissance polymath who formulated a mathematical model, model of Celestial spheres#Renaissance, the universe that placed heliocentrism, the Sun rather than Earth at its cen ...
with
elliptical orbit In astrodynamics or celestial mechanics, an elliptical orbit or eccentric orbit is an orbit with an orbital eccentricity, eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. Some or ...
s and explained how planetary velocities vary. The three laws state that: # The orbit of a planet is an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
with the Sun at one of the two foci. # A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. # The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit. The elliptical orbits of planets were indicated by calculations of the orbit of
Mars Mars is the fourth planet from the Sun. It is also known as the "Red Planet", because of its orange-red appearance. Mars is a desert-like rocky planet with a tenuous carbon dioxide () atmosphere. At the average surface level the atmosph ...
. From this, Kepler inferred that other bodies in the
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...
, including those farther away from the Sun, also have elliptical orbits. The second law establishes that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the longer its orbital period.
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own laws of motion and law of universal gravitation. A more precise historical approach is found in '' Astronomia nova'' and '' Epitome Astronomiae Copernicanae''.


Comparison to Copernicus

Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...
's laws improved the model of Copernicus. According to Copernicus: # The planetary orbit is a circle with epicycles. # The Sun is approximately at the center of the orbit. # The speed of the planet in the main orbit is constant. Despite being correct in saying that the planets revolved around the Sun, Copernicus was incorrect in defining their orbits. Introducing physical explanations for movement in space beyond just geometry, Kepler correctly defined the orbit of planets as follows: # The planetary orbit is ''not'' a circle with epicycles, but an ''
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
''. # The Sun is ''not'' at the center but at a '' focal point'' of the elliptical orbit. # Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the '' area speed'' (closely linked historically with the concept of
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
) is constant. The
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
of the
orbit of the Earth Earth orbits the Sun at an astronomical unit, average distance of , or 8.317 light-second, light-minutes, in a retrograde and prograde motion, counterclockwise direction as viewed from above the Northern Hemisphere. One complete orbit takes & ...
makes the time from the
March equinox The March equinox or northward equinox is the equinox on the Earth when the subsolar point appears to leave the Southern Hemisphere and cross the celestial equator, heading northward as seen from Earth. The March equinox is known as the ver ...
to the September equinox, around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the Sun parallel to the
equator The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...
of the Earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately :e \approx \frac \frac \approx 0.015, which is close to the correct value (0.016710218). The accuracy of this calculation requires that the two dates chosen be along the elliptical orbit's minor axis and that the midpoints of each half be along the major axis. As the two dates chosen here are equinoxes, this will be correct when perihelion, the date the Earth is closest to the Sun, falls on a
solstice A solstice is the time when the Sun reaches its most northerly or southerly sun path, excursion relative to the celestial equator on the celestial sphere. Two solstices occur annually, around 20–22 June and 20–22 December. In many countries ...
. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22.


Nomenclature

It took nearly two centuries for the current formulation of Kepler's work to take on its settled form.
Voltaire François-Marie Arouet (; 21 November 169430 May 1778), known by his ''Pen name, nom de plume'' Voltaire (, ; ), was a French Age of Enlightenment, Enlightenment writer, philosopher (''philosophe''), satirist, and historian. Famous for his wit ...
's ''Eléments de la philosophie de Newton'' (''Elements of Newton's Philosophy'') of 1738 was the first publication to use the terminology of "laws". The '' Biographical Encyclopedia of Astronomers'' in its article on Kepler (p. 620) states that the terminology of scientific laws for these discoveries was current at least from the time of Joseph de Lalande. It was the exposition of Robert Small, in ''An account of the astronomical discoveries of Kepler'' (1814) that made up the set of three laws, by adding in the third. Small also claimed, against the history, that these were empirical laws, based on
inductive reasoning Inductive reasoning refers to a variety of method of reasoning, methods of reasoning in which the conclusion of an argument is supported not with deductive certainty, but with some degree of probability. Unlike Deductive reasoning, ''deductive'' ...
. Further, the current usage of "Kepler's second law" is something of a misnomer. Kepler had two versions, related in a qualitative sense: the "distance law" and the "area law". The "area law" is what became the second law in the set of three; but Kepler did himself not privilege it in that way.


History

Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler's third law was published in 1619.Johannes Kepler, ''Harmonices Mundi'' he Harmony of the World(Linz, (Austria): Johann Planck, 1619), book 5, chapter 3
p. 189.
From the bottom of p. 189: ''"Sed res est certissima exactissimaque quod ''proportio qua est inter binorum quorumcunque Planetarum tempora periodica, sit præcise sesquialtera proportionis'' mediarum distantiarum, ... "'' (But it is absolutely certain and exact that the ''proportion between the periodic times of any two planets is precisely the sesquialternate proportion'' .e., the ratio of 3:2of their mean distances, ... ")
An English translation of Kepler's ''Harmonices Mundi'' is available as: Johannes Kepler with E. J. Aiton, A. M. Duncan, and J. V. Field, trans., ''The Harmony of the World'' (Philadelphia, Pennsylvania: American Philosophical Society, 1997); see especiall
p. 411
Kepler had believed in the Copernican model of the Solar System, which called for circular orbits, but he could not reconcile Brahe's highly precise observations with a circular fit to Mars' orbit – Mars coincidentally having the highest
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
of all planets except Mercury. His first law reflected this discovery. In 1621, Kepler noted that his third law applies to the four brightest moons of
Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...
.In 1621, Johannes Kepler noted that Jupiter's moons obey (approximately) his third law in his '' Epitome Astronomiae Copernicanae'' pitome of Copernican Astronomy(Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 4, part 2
pages 554–555
From pp. 554–555: ''" ... plane ut est cum sex planet circa Solem, ... prodit Marius in suo mundo Ioviali ista 3.5.8.13 (vel 14. Galilæo) ... Periodica vero tempora prodit idem Marius ... sunt maiora simplis, minora vero duplis."'' (... just as it is clearly rueamong the six planets around the Sun, so also it is among the four oonsof Jupiter, because around the body of Jupiter any atellitethat can go farther from it, orbits slower, and even that rbit's periodis not in the same proportion, but greater han the distance from Jupiter that is, 3/2 (''sescupla'') of the proportion of each of the distances from Jupiter, which is clearly the very roportionas is used for the six planets above. In his ook''The World of Jupiter'' 'Mundus Jovialis'', 1614 imon Mayr or"Marius" 573–1624presents these distances, from Jupiter, of the four oonsof Jupiter: 3, 5, 8, 13 (or 14 ccording toGalileo) ote: The distances of Jupiter's moons from Jupiter are expressed as multiples of Jupiter's diameter.... Mayr presents their time periods: 1 day 18 1/2 hours, 3 days 13 1/3 hours, 7 days 2 hours, 16 days 18 hours: for all f these datathe proportion is greater than double, thus greater than he proportionof the distances 3, 5, 8, 13 or 14, although less than he proportionof the squares, which double the proportions of the distances, namely 9, 25, 64, 169 or 196, just as power of3/2 is also greater than 1 but less than 2.)
Godefroy Wendelin also made this observation in 1643.Godefroy Wendelin wrote a letter to Giovanni Battista Riccioli about the relationship between the distances of the Jovian moons from Jupiter and the periods of their orbits, showing that the periods and distances conformed to Kepler's third law. See: Joanne Baptista Riccioli, ''Almagestum novum'' ... (Bologna (Bononia), (Italy): Victor Benati, 1651), volume 1
page 492 Scholia III.
In the margin beside the relevant paragraph is printed: ''Vendelini ingeniosa speculatio circa motus & intervalla satellitum Jovis''. (Wendelin's clever speculation about the movement and distances of Jupiter's satellites.) From p. 492: ''"III. Non minus Kepleriana ingeniosa est Vendelini ... & D. 7. 164/1000. pro penextimo, & D. 16. 756/1000. pro extimo."'' (No less clever hanKepler's is the most keen astronomer Wendelin's investigation of the proportion of the periods and distances of Jupiter's satellites, which he had communicated to me with great generosity na very long and very learned letter. So, just as in he case ofthe larger planets, the planets' mean distances from the Sun are respectively in the 3/2 ratio of their periods; so the distances of these minor planets of Jupiter from Jupiter (which are 3, 5, 8, and 14) are respectively in the 3/2 ratio of
heir Inheritance is the practice of receiving private property, titles, debts, entitlements, privileges, rights, and obligations upon the death of an individual. The rules of inheritance differ among societies and have changed over time. Offi ...
periods (which are 1.769 days for the innermost o 3.554 days for the next to the innermost uropa 7.164 days for the next to the outermost anymede and 16.756 days for the outermost allisto.)
The second law, in the "area law" form, was contested by Nicolaus Mercator in a book from 1664, but by 1670 his '' Philosophical Transactions'' were in its favour. As the century proceeded it became more widely accepted. The reception in Germany changed noticeably between 1688, the year in which Newton's '' Principia'' was published and was taken to be basically Copernican, and 1690, by which time work of
Gottfried Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...
on Kepler had been published. Newton was credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law, whereas the other laws do depend on the inverse square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
of planetary motion (the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as
conservation of angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
does via rotational symmetry for the second law.


Formulary

The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.


First law

Kepler's first law states that:
The orbit of every planet is an ellipse with the sun at one of the two foci.
Mathematically, an ellipse can be represented by the formula: :r = \frac, where p is the semi-latus rectum, ''ε'' is the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
of the ellipse, ''r'' is the distance from the Sun to the planet, and ''θ'' is the angle to the planet's current position from its closest approach, as seen from the Sun. So (''r'', ''θ'') are
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
. For an ellipse 0 < ''ε'' < 1 ; in the limiting case ''ε'' = 0, the orbit is a circle with the Sun at the centre (i.e. where there is zero eccentricity). At ''θ'' = 0°, perihelion, the distance is minimum :r_\min = \frac At ''θ'' = 90° and at ''θ'' = 270° the distance is equal to p. At ''θ'' = 180°, aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°) :r_\max = \frac The semi-major axis ''a'' is the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
between ''r''min and ''r''max: :\begin a &= \frac \\ pt a &= \frac \end The semi-minor axis ''b'' is the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
between ''r''min and ''r''max: :\begin b &= \sqrt \\ pt b &= \frac \end The semi-latus rectum ''p'' is the
harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean ...
between ''r''min and ''r''max: :\begin p &= \left(\frac\right)^ \\ pa &= r_\max r_\min = b^2\, \end The eccentricity ''ε'' is the coefficient of variation between ''r''min and ''r''max: :\varepsilon = \frac. The
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of the ellipse is :A = \pi a b\,. The special case of a circle is ''ε'' = 0, resulting in ''r'' = ''p'' = ''r''min = ''r''max = ''a'' = ''b'' and ''A'' = ''πr''2.


Second law

Kepler's second law states that:
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.Bryant, Jeff; Pavlyk, Oleksandr.
Kepler's Second Law
, ''
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...
''. Retrieved December 27, 2009.
The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation: the planet travels faster when closer to the Sun, then slower when farther from the Sun. Kepler's second law states that the blue sector has constant area.


History and proofs

Kepler notably arrived at this law through assumptions that were either only approximately true or outright false and can be outlined as follows: # Planets are pushed around the Sun by a force from the Sun. This false assumption relies on incorrect Aristotelian physics that an object needs to be pushed to maintain motion. # The propelling force from the Sun is inversely proportional to the distance from the Sun. Kepler reasoned this, believing that gravity spreading in three dimensions would be a waste, since the planets inhabited a plane. Thus, an inverse instead of the orrectinverse square law. # Because Kepler believed that force would be proportional to velocity, it followed from statements #1 and #2 that velocity would be inverse to the distance from the sun. This is also an incorrect tenet of Aristotelian physics. # Since velocity is inverse to time, the distance from the sun would be proportional to the time to cover a small piece of the orbit. This is approximately true for elliptical orbits. # The area swept out is proportional to the overall time. This is also approximately true. # The orbits of a planet are circular (Kepler discovered his second law before his first law, which contradicts this). Nevertheless, the result of the second law is exactly true, as it is logically equivalent to the conservation of angular momentum, which is true for any body experiencing a radially symmetric force. A correct proof can be shown through this. Since the cross product of two vectors gives the area of a parallelogram possessing sides of those vectors, the triangular area dA swept out in a short period of time is given by half the cross product of the ''r'' and ''dx'' vectors, for some short piece of the orbit, ''dx''. dA = \frac (\vec \times \vec) = \frac (\vec \times \vec dt) for a small piece of the orbit ''dx'' and time to cover it ''dt''. Thus \frac = \frac (\vec \times \vec). \frac = \frac \frac (\vec \times \vec). Since the final expression is proportional to the total angular momentum (\vec \times \vec), Kepler's equal area law will hold for any system that conserves angular momentum. Since any radial force will produce no torque on the planet's motion, angular momentum will be conserved.


In terms of elliptical parameters

In a small time dt the planet sweeps out a small triangle having base line r and height r \, d\theta and area dA = \frac \cdot r \cdot r \, d\theta, so the constant areal velocity is \frac = \frac \frac. The area enclosed by the elliptical orbit is \pi ab. So the period T satisfies :T \cdot \frac \frac = \pi ab and the mean motion of the planet around the Sun :n = \frac satisfies :r^2\,d\theta = abn\,dt. And so, \frac = \frac = \frac.


Third law

Kepler's third law states that:
The ratio of the square of an object's orbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.
This captures the relationship between the distance of planets from the Sun, and their orbital periods. Kepler enunciated in 1619 this third law in a laborious attempt to determine what he viewed as the " music of the spheres" according to precise laws, and express it in terms of musical notation. It was therefore known as the ''harmonic law''. The original form of this law (referring to not the semi-major axis, but rather a "mean distance") holds true only for planets with small eccentricities near zero. Using Newton's law of gravitation (published 1687), this relation can be found in the case of a circular orbit by setting the centripetal force equal to the gravitational force: : mr\omega^2 = G\frac Then, expressing the angular velocity ω in terms of the orbital period and then rearranging, results in Kepler's third law: : mr\left(\frac\right)^2 = G\frac \implies T^2 = \left(\frac \right)r^3 \implies T^2 \propto r^3 A more detailed derivation can be done with general elliptical orbits, instead of circles, as well as orbiting the center of mass, instead of just the large mass. This results in replacing a circular radius, r, with the semi-major axis, a, of the elliptical relative motion of one mass relative to the other, as well as replacing the large mass M with M + m. However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is: :\frac = \frac \approx \frac \approx 7.496 \times 10^ \frac \text where M is the mass of the Sun, m is the mass of the planet, G is the gravitational constant, T is the orbital period and a is the elliptical semi-major axis, and \text is the
astronomical unit The astronomical unit (symbol: au or AU) is a unit of length defined to be exactly equal to . Historically, the astronomical unit was conceived as the average Earth-Sun distance (the average of Earth's aphelion and perihelion), before its m ...
, the average distance from earth to the sun.


Table

The following table shows the data used by Kepler to empirically derive his law: Kepler became aware of
John Napier John Napier of Merchiston ( ; Latinisation of names, Latinized as Ioannes Neper; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8 ...
's recent invention of logarithms and log-log graphs before he discovered the pattern. Upon finding this pattern Kepler wrote: For comparison, here are modern estimates:


Planetary acceleration

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
computed in his '' Philosophiæ Naturalis Principia Mathematica'' the
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
of a planet moving according to Kepler's first and second laws. # The ''direction'' of the acceleration is towards the Sun. # The ''magnitude'' of the acceleration is inversely proportional to the square of the planet's distance from the Sun (the ''inverse square law''). This implies that the Sun may be the physical cause of the acceleration of planets. However, Newton states in his ''Principia'' that he considers forces from a mathematical point of view, not a physical, thereby taking an instrumentalist view. Moreover, he does not assign a cause to gravity. Newton defined the
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
acting on a planet to be the product of its
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
and the acceleration (see
Newton's laws of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
). So: # Every planet is attracted towards the Sun. # The force acting on a planet is directly proportional to the mass of the planet and is inversely proportional to the square of its distance from the Sun. The Sun plays an unsymmetrical part, which is unjustified. So he assumed, in
Newton's law of universal gravitation Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is Proportionality (mathematics)#Direct proportionality, proportional to the product ...
: # All bodies in the Solar System attract one another. # The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them. As the planets have small masses compared to that of the Sun, the orbits conform approximately to Kepler's laws. Newton's model improves upon Kepler's model, and fits actual observations more accurately. (See two-body problem.) Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.


Acceleration vector

From the heliocentric point of view consider the vector to the planet \mathbf = r\hat where r is the distance to the planet and \hat is a
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
pointing towards the planet. \frac = \dot = \dot\hat,\qquad \frac = \dot = -\dot\hat where \hat is the unit vector whose direction is 90 degrees counterclockwise of \hat, and \theta is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time. Differentiate the position vector twice to obtain the velocity vector and the acceleration vector: \begin \dot &= \dot\hat + r\dot = \dot\hat + r\dot\hat, \\ \ddot &= \left(\ddot\hat + \dot\dot \right) + \left(\dot\dot \hat + r\ddot\hat + r\dot\dot \right) = \left(\ddot - r\dot^2\right)\hat + \left(r\ddot + 2\dot\dot\right)\hat. \end So \ddot = a_r \hat+a_\theta\hat where the radial acceleration is a_r = \ddot - r\dot^2 and the transversal acceleration is a_\theta = r\ddot + 2\dot\dot.


Inverse square law

Kepler's second law says that r^2\dot = nab is constant. The transversal acceleration a_\theta is zero: \frac = r\left(2\dot\dot + r\right) = ra_\theta = 0. So the acceleration of a planet obeying Kepler's second law is directed towards the Sun. The radial acceleration a_\text is a_\text = \ddot - r\dot^2 = \ddot - r\left(\frac\right)^2 = \ddot - \frac. Kepler's first law states that the orbit is described by the equation: \frac = 1 + \varepsilon\cos(\theta). Differentiating with respect to time -\frac = -\varepsilon\sin(\theta)\,\dot or p\dot = nab\,\varepsilon\sin(\theta). Differentiating once more p\ddot = nab\varepsilon\cos(\theta)\, \dot = nab\varepsilon\cos(\theta)\, \frac = \frac\varepsilon\cos(\theta). The radial acceleration a_\text satisfies pa_\text = \frac\varepsilon\cos(\theta) - p\frac = \frac\left(\varepsilon\cos(\theta) - \frac\right). Substituting the equation of the ellipse gives pa_\text = \frac\left(\frac - 1 - \frac\right) = -\fracb^2. The relation b^2 = pa gives the simple final result a_\text = -\frac. This means that the acceleration vector \mathbf of any planet obeying Kepler's first and second law satisfies the inverse square law \mathbf = -\frac\hat where \alpha = n^2 a^3 is a constant, and \hat is the unit vector pointing from the Sun towards the planet, and r\, is the distance between the planet and the Sun. Since mean motion n=\frac where T is the period, according to Kepler's third law, \alpha has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire Solar System. The inverse square law is a differential equation. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is a hyperbola or
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
or a straight line. (See Kepler orbit.)


Newton's law of gravitation

By Newton's second law, the gravitational force that acts on the planet is: \mathbf = m_\text \mathbf = - m_\text \alpha r^ \hat where m_\text is the mass of the planet and \alpha has the same value for all planets in the Solar System. According to Newton's third law, the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun, m_\text. So \alpha = Gm_\text where G is the gravitational constant. The acceleration of Solar System body number ''i'' is, according to Newton's laws: \mathbf_i = G\sum_ m_j r_^ \hat_ where m_j is the mass of body ''j'', r_ is the distance between body ''i'' and body ''j'', \hat_ is the unit vector from body ''i'' towards body ''j'', and the vector summation is over all bodies in the Solar System, besides ''i'' itself. In the special case where there are only two bodies in the Solar System, Earth and Sun, the acceleration becomes \mathbf_\text = Gm_\text r_^ \hat_ which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws. If the two bodies in the Solar System are Moon and Earth the acceleration of the Moon becomes \mathbf_\text = Gm_\text r_^ \hat_ So in this approximation, the Moon moves around the Earth according to Kepler's laws. In the three-body case the accelerations are \begin \mathbf_\text &= Gm_\text r_^ \hat_ + Gm_\text r_^ \hat_ \\ \mathbf_\text &= Gm_\text r_^ \hat_ + Gm_\text r_^ \hat_ \\ \mathbf_\text &= Gm_\text r_^ \hat_ + Gm_\text r_^ \hat_ \end These accelerations are not those of Kepler orbits, and the
three-body problem In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses orbiting each other in space and then calculate their subsequent trajectories using Newton' ...
is complicated. But Keplerian approximation is the basis for perturbation calculations. (See Lunar theory.)


Position as a function of time

Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a transcendental equation called
Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his ''Epitome of ...
. The procedure for calculating the heliocentric polar coordinates (''r'',''θ'') of a planet as a function of the time ''t'' since perihelion, is the following five steps: # Compute the mean motion , where ''P'' is the period. # Compute the mean anomaly , where ''t'' is the time since perihelion. # Compute the eccentric anomaly ''E'' by solving Kepler's equation: M = E - \varepsilon\sin E , where \varepsilon is the eccentricity. # Compute the true anomaly ''θ'' by solving the equation: (1 - \varepsilon) \tan^2 \frac = (1 + \varepsilon)\tan^2\frac # Compute the heliocentric distance ''r'': r = a(1 - \varepsilon\cos E) , where a is the semimajor axis. The position polar coordinates (''r'',''θ'') can now be written as a Cartesian vector \mathbf = r \left\langle \cos, \sin\right\rangle and the Cartesian velocity vector can then be calculated as \mathbf = \frac \left\langle -\sin, \sqrt \cos\right\rangle, where \mu is the standard gravitational parameter. The important special case of circular orbit, ''ε'' = 0, gives . Because the uniform circular motion was considered to be ''normal'', a deviation from this motion was considered an anomaly. The proof of this procedure is shown below.


Mean anomaly, ''M''

The Keplerian problem assumes an
elliptical orbit In astrodynamics or celestial mechanics, an elliptical orbit or eccentric orbit is an orbit with an orbital eccentricity, eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. Some or ...
and the four points: * ''s'' the Sun (at one focus of ellipse); * ''z'' the perihelion * ''c'' the center of the ellipse * ''p'' the planet and * a = , cz, , distance between center and perihelion, the semimajor axis, * \varepsilon = , the eccentricity, * b = a\sqrt, the semiminor axis, * r = , sp, , the distance between Sun and planet. * \theta = \angle zsp, the direction to the planet as seen from the Sun, the true anomaly. The problem is to compute the
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
(''r'',''θ'') of the planet from the time since perihelion, ''t''. It is solved in steps. Kepler considered the circle with the major axis as a diameter, and *x, the projection of the planet to the auxiliary circle *y, the point on the circle such that the sector areas , ''zcy'', and , ''zsx'', are equal, *M = \angle zcy, the mean anomaly. The sector areas are related by , zsp, = \frac \cdot , zsx, . The circular sector area , zcy, = \frac2. The area swept since perihelion, , zsp, = \frac \cdot, zsx, = \frac \cdot , zcy, = \frac \cdot \frac = \frac, is by Kepler's second law proportional to time since perihelion. So the mean anomaly, ''M'', is proportional to time since perihelion, ''t''. M = nt, where ''n'' is the mean motion.


Eccentric anomaly, ''E''

When the mean anomaly ''M'' is computed, the goal is to compute the true anomaly ''θ''. The function ''θ'' = ''f''(''M'') is, however, not elementary. Kepler's solution is to use E = \angle zcx, ''x'' as seen from the centre, the eccentric anomaly as an intermediate variable, and first compute ''E'' as a function of ''M'' by solving Kepler's equation below, and then compute the true anomaly ''θ'' from the eccentric anomaly ''E''. Here are the details. \begin , zcy, &= , zsx, = , zcx, - , scx, \\ with , scx, &= \frac \\ \frac &= \frac2 - \frac \end Division by ''a''2/2 gives
Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his ''Epitome of ...
M = E - \varepsilon\sin E. This equation gives ''M'' as a function of ''E''. Determining ''E'' for a given ''M'' is the inverse problem. Iterative numerical algorithms are commonly used. Having computed the eccentric anomaly ''E'', the next step is to calculate the true anomaly ''θ''. But note: Cartesian position coordinates with reference to the center of ellipse are (''a'' cos ''E'', ''b'' sin ''E'') With reference to the Sun (with coordinates (''c'',0) = (''ae'',0) ), ''r'' = (''a'' cos ''E'' – ''ae'', ''b'' sin ''E'') True anomaly would be arctan(''r''''y''/''r''''x''), magnitude of ''r'' would be .


True anomaly, ''θ''

Note from the figure that , cd, = , cs, + , sd, so that a\cos E = a \varepsilon + r\cos\theta. Dividing by a and inserting from Kepler's first law \frac = \frac to get \cos E = \varepsilon + \frac \cos\theta = \frac = \frac. The result is a usable relationship between the eccentric anomaly ''E'' and the true anomaly ''θ''. A computationally more convenient form follows by substituting into the trigonometric identity: \tan^2\frac = \frac. Get \begin \tan^2\frac &= \frac = \frac \\ pt &= \frac = \frac \cdot \frac = \frac \tan^2\frac. \end Multiplying by 1 + ''ε'' gives the result (1 - \varepsilon)\tan^2\frac = (1 + \varepsilon)\tan^2\frac This is the third step in the connection between time and position in the orbit.


Distance, ''r''

The fourth step is to compute the heliocentric distance ''r'' from the true anomaly ''θ'' by Kepler's first law: r(1 + \varepsilon\cos\theta) = a\left(1 - \varepsilon^2\right) Using the relation above between ''θ'' and ''E'' the final equation for the distance ''r'' is: r = a(1 - \varepsilon\cos E).


See also

* Circular motion * Free-fall time *
Gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
* Kepler orbit * Kepler problem *
Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his ''Epitome of ...
* Laplace–Runge–Lenz vector * Specific relative angular momentum, relatively easy derivation of Kepler's laws starting with conservation of angular momentum


Explanatory notes


References


General bibliography

* Kepler's life is summarized on pp. 523–627 and Book Five of his ''magnum opus'', '' Harmonice Mundi'' (''harmonies of the world''), is reprinted on: * A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example: . * *


External links


The Feynman Lectures on Physics - Kepler's laws
* Crowell, Benjamin,
Light and Matter
', an online book that gives a proof of the first law without the use of calculus (see section 15.7) * David McNamara and Gianfranco Vidali,
Kepler's Second Law – Java Interactive Tutorial
, an interactive Java applet that aids in the understanding of Kepler's second law. * Cain, Gay (May 10, 2010), ''Astronomy Cast'',
Ep. 189: Johannes Kepler and His Laws of Planetary Motion
* University of Tennessee's Dept. Physics & Astronomy: Astronomy 161,

* Solar System Simulator

*

in ''From Stargazers to Starships'' by David P. Stern (10 October 2016) * by Jens Puhle (Dec 27, 2023) – a video explaining and visualizing Kepler's three laws of planetary motion {{DEFAULTSORT:Kepler's Laws Of Planetary Motion 1609 in science 1619 in science Copernican Revolution Eponymous laws of physics Equations of astronomy Johannes Kepler Orbits