Source: https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion
Illustration of Kepler's laws with two planetary orbits.The orbits are ellipses, with foci F1 and F2 for Planet 1, and F1 and F3 for Planet 2. The Sun is at F1.The shaded areas A1 and A2 are equal, and are swept out in equal times by Planet 1's orbit.The ratio of Planet 1's orbit time to Planet 2's is .
In astronomy, Kepler's laws of planetary motion give good approximations for the orbits of planets around the Sun. They were published by Johannes Kepler from 1608 to 1621 in three works Astronomia nova, Harmonice Mundi and Epitome Astronomiae Copernicanae. The laws were based on Kepler's concept of solar fibrils adapted to the accurate astronomical data of Tycho Brahe. These laws replaced the circular orbits and epicycles of Copernicus's heliostatic model of the planets with a heliocentric model that described elliptical orbits with planetary velocities that vary accordingly. The three laws state that:[1][2]
The orbit of a planet is an ellipse 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. From this, Kepler inferred that other bodies in the Solar System, 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.
Comparison to Copernicus[edit]
Johannes Kepler's laws improved the model of Copernicus. According to Copernicus:[3][4]
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:[1][2][5]: 53–54
The planetary orbit is not a circle with epicycles, but an ellipse.
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) is constant.
The eccentricity of the orbit of the Earth makes the time from the March equinox 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 of the Earth cuts the orbit into two parts with areas in a 186 to 179 ratio. The areas swept out in an ellipse will differ by twice the area of a triangle with the focus and the two minor axis points as vertices. This triangle has area where is the semi-minor axis length and is the linear eccentricity. Thus the difference in areas is Since the eccentricity is given by (where is the semi-major axis) we have where is the difference in swept areas. Since the area of an ellipse is we have so the eccentricity of the orbit of the Earth is approximately
which still a factor of two off from 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. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22, but this difference (14-15 days) is actually bigger than our difference in inter-equinox times (7 days), which explains our substantial error.
Kepler's laws were developed based on a physical theory of planetary motion in which the Sun emitted magnetic fibrils which pulled the planets into orbits. The fibrils were somewhat elastic allowing non-circular motion driven by the inertia of the planets.[6]: 5
In Astronomia nova (1609), Kepler articulated his first law, showing that Mars' orbit is elliptical,
[7] having found them by analyzing the astronomical observations of Tycho Brahe.[8] 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 of all planets except Mercury.[9] His first law reflected this discovery.
In his Astronomia nova (1609), Kepler did not present his second law in its modern form. He did that only in his Epitome Astronomiae Copernicanae of 1621.[5]: 53
Kepler had two versions of the second law, related in a qualitative sense: the first "distance law" and later the "area law". The distance form was only correct for orbits that were almost circular, but the area form was correct for all elliptical orbit. The "area law" is what became the second law in the set of three. This law had little impact on astronomy because calculations of planetary positions using the law were approximate and time consuming.[6]: 3 [10] 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.[11][12] As the century proceeded it became more widely accepted.[13]
Kepler's third law was published in 1619 in his Harmonice Mundi.[14][8] In 1621, Kepler noted that his third law applies to the four brightest moons of Jupiter.[Nb 1] Godefroy Wendelin, the first well-known astronomer to adopt Kepler's laws, gave a detailed account of the third law in 1652.[6][Nb 2]
Kepler's work had little initial impact. His work was a strong defense of Copernicanism which had fallen out of fashion in part because of opposition by Tycho Brahe. In 1627 Kepler published the Rudolphine Tables containing many accurate astronomical observations accumulated by Brahe. The breadth and accuracy of the tables allowed astronomers to compare Kepler's formula to good quality data. At first these difficult calculations were off putting, but once undertaken more astronomers became convinced of Kepler's approach.[6]: 7
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 on Kepler had been published.[15]
Newton understood 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 of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.[16]
It took nearly two centuries for the current formulation of Kepler's work to take on its settled form. Voltaire's Eléments de la philosophie de Newton (Elements of Newton's Philosophy) of 1738 was the first publication to use the terminology of "laws".[17][18] 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.[19] 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.[20] Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.[18][21]
The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.
Kepler's first law states that:[22]: 3
The orbit of every planet is an ellipse with the sun at one of the two foci.
Kepler's first law placing the Sun at one of the foci of an elliptical orbit
Heliocentric coordinate system (r, θ) for ellipse. Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = rmin and for θ = 180°, r = rmax.
Mathematically, an ellipse can be represented by the formula:
where is the semi-latus rectum, ε is the eccentricity 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.
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
At θ = 90° and at θ = 270° the distance is equal to .
At θ = 180°, aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)
The semi-major axis a is the arithmetic mean between rmin and rmax:
The semi-minor axis b is the geometric mean between rmin and rmax:
The semi-latus rectum p is the harmonic mean between rmin and rmax:
The eccentricity ε is the coefficient of variation between rmin and rmax:
The area of the ellipse is
The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = πr2.
Kepler's second law states that:[22]: 3
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.[23]
The same (blue) area is swept out in a fixed time period. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity.
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.
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 [correct] inverse 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. That force is proportional to velocity is an incorrect tenet of Aristotelian physics, but the errors of assumption in statements #2 and #3 essentially cancel, so that it is approximately true that velocity is inverse to the distance from the sun.
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.[24] 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.
for a small piece of the orbit dx and time to cover it dt.
Thus
Since the final expression is proportional to the total angular momentum , 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[edit]
In a small time the planet sweeps out a small triangle having base line and height and area , so the constant areal velocity is
The area enclosed by the elliptical orbit is . So the period satisfies
and the mean motion of the planet around the Sun
satisfies
And so,
Orbits of planets with varying eccentricities.
Low
High
Planet orbiting the Sun in a circular orbit (e=0.0)
Planet orbiting the Sun in an orbit with e=0.5
Planet orbiting the Sun in an orbit with e=0.2
Planet orbiting the Sun in an orbit with e=0.8
The red ray rotates at a constant angular velocity and with the same orbital time period as the planet, .
S: Sun at the primary focus,
C: Centre of ellipse,
S': The secondary focus.
In each case, the area of all sectors depicted is identical.
Kepler's third law states that:[22]: 3
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.
Or symbolically :
Where is the object's orbital period and is the semi-major axis of its orbit.
This captures the relationship between the distance of planets from the Sun, and their orbital periods.
Kepler enunciated in 1619[14] 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.[25] It was therefore known as the harmonic law.[26] 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.[27]
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:
Then, expressing the angular velocity ω in terms of the orbital period and then rearranging, results in Kepler's third law:
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, , with the semi-major axis, , of the elliptical relative motion of one mass relative to the other, as well as replacing the large mass with . However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is:
where is the mass of the Sun, is the mass of the planet, is the gravitational constant, is the orbital period and is the elliptical semi-major axis, and is the astronomical unit, the average distance from earth to the sun.
The following table shows the data used by Kepler to empirically derive his law:
Data used by Kepler (1618)
Planet
Mean distance to sun (AU)
Period (days)
(10−6 AU3/day2)
Mercury
0.389
87.77
7.64
Venus
0.724
224.70
7.52
Earth
1
365.25
7.50
Mars
1.524
686.95
7.50
Jupiter
5.20
4332.62
7.49
Saturn
9.510
10759.2
7.43
Kepler became aware of John Napier's recent invention of logarithms and log-log graphs before he discovered the pattern.[28]
Upon finding this pattern Kepler wrote:[29]
I first believed I was dreaming... But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance.
— translated from Harmonies of the World by Kepler (1619)
Log–log plot of period T vs. semi-major axis a (average of aphelion and perihelion) of some Solar System orbits (crosses denoting Kepler's values) showing that a3/T2 is constant (green line)
For comparison, here are modern estimates:[citation needed]
Modern data
Planet
Semi-major axis (AU)
Period (days)
(10−6 AU3/day2)
Mercury
0.38710
87.9693
7.496
Venus
0.72333
224.7008
7.496
Earth
1
365.2564
7.496
Mars
1.52366
686.9796
7.495
Jupiter
5.20336
4332.8201
7.504
Saturn
9.53707
10775.599
7.498
Uranus
19.1913
30687.153
7.506
Neptune
30.0690
60190.03
7.504
Planetary acceleration[edit]
Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration 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.[30] Moreover, he does not assign a cause to gravity.[31]
Newton defined the force acting on a planet to be the product of its mass and the acceleration (see Newton's laws of motion). 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:
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[edit]
From the heliocentric point of view consider the vector to the planet where is the distance to the planet and is a unit vector pointing towards the planet.
where is the unit vector whose direction is 90 degrees counterclockwise of , and 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:
So
where the radial acceleration is
and the transversal acceleration is
Kepler's second law says that is constant.
The transversal acceleration is zero:
So the acceleration of a planet obeying Kepler's second law is directed towards the Sun.
The radial acceleration is
Kepler's first law states that the orbit is described by the equation:
Differentiating with respect to time
or
Differentiating once more
The radial acceleration satisfies
Substituting the equation of the ellipse gives
The relation gives the simple final result
This means that the acceleration vector of any planet obeying Kepler's first and second law satisfies the inverse square law
where
is a constant, and is the unit vector pointing from the Sun towards the planet, and is the distance between the planet and the Sun.
Since mean motion where is the period, according to Kepler's third law, 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 or a straight line. (See Kepler orbit.)
Newton's law of gravitation[edit]
By Newton's second law, the gravitational force that acts on the planet is:
where is the mass of the planet and 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, . So
where is the gravitational constant.
The acceleration of Solar System body number i is, according to Newton's laws:
where is the mass of body j, is the distance between body i and body j, 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
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
So in this approximation, the Moon moves around the Earth according to Kepler's laws.
In the three-body case the accelerations are
These accelerations are not those of Kepler orbits, and the three-body problem is complicated. But Keplerian approximation is the basis for perturbation calculations. (See Lunar theory.)
Position as a function of time[edit]
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.
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 n = (2π rad)/P, where P is the period.
Compute the mean anomaly M = nt, where t is the time since perihelion.
Compute the eccentric anomaly E by solving Kepler's equation: where is the eccentricity.
Compute the true anomaly θ by solving the equation:
Compute the heliocentric distance r: where is the semimajor axis.
The position polar coordinates (r,θ) can now be written as a Cartesian vector and the Cartesian velocity vector can then be calculated as , where is the standard gravitational parameter.[32]
The important special case of circular orbit, ε = 0, gives θ = E = M. 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.
Geometric construction for Kepler's calculation of θ. The Sun (located at the focus) is labeled S and the planet P. The auxiliary circle is an aid to calculation. Line xd is perpendicular to the base and through the planet P. The shaded sectors are arranged to have equal areas by positioning of point y.
The Keplerian problem assumes an elliptical orbit 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
The problem is to compute the polar coordinates (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
The sector areas are related by
The circular sector area
The area swept since perihelion,
is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.
where n is the mean motion.
Eccentric anomaly, E[edit]
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.[33] Kepler's solution is to use
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.
Division by a2/2 gives Kepler's equation
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(ry/rx), magnitude of r would be √r · r.
Note from the figure that
so that
Dividing by and inserting from Kepler's first law
to get
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:
Get
Multiplying by 1 + ε gives the result
This is the third step in the connection between time and position in the orbit.
The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:
Using the relation above between θ and E the final equation for the distance r is:
Circular motion
Free-fall time
Gravity
Kepler orbit
Kepler problem
Kepler's equation
Laplace–Runge–Lenz vector
Specific relative angular momentum, relatively easy derivation of Kepler's laws starting with conservation of angular momentum
^ In 1621, Johannes Kepler noted that Jupiter's moons obey (approximately) his third law in his Epitome Astronomiae Copernicanae [Epitome 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 [true] among the six planets around the Sun, so also it is among the four [moons] of Jupiter, because around the body of Jupiter any [satellite] that can go farther from it, orbits slower, and even that [orbit's period] is not in the same proportion, but greater [than the distance from Jupiter]; that is, 3/2 (sescupla) of the proportion of each of the distances from Jupiter, which is clearly the very [proportion] as is used for the six planets above. In his [book] The World of Jupiter [Mundus Jovialis, 1614], [Simon Mayr or] "Marius" [1573–1624] presents these distances, from Jupiter, of the four [moons] of Jupiter: 3, 5, 8, 13 (or 14 [according to] Galileo) [Note: 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 [of these data] the proportion is greater than double, thus greater than [the proportion] of the distances 3, 5, 8, 13 or 14, although less than [the proportion] of the squares, which double the proportions of the distances, namely 9, 25, 64, 169 or 196, just as [a power of] 3/2 is also greater than 1 but less than 2.)
^ 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 [than] Kepler'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 [in] a very long and very learned letter. So, just as in [the case of] the 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 [their] periods (which are 1.769 days for the innermost [Io], 3.554 days for the next to the innermost [Europa], 7.164 days for the next to the outermost [Ganymede], and 16.756 days for the outermost [Callisto]).)
^ Jump up to: a b "Kepler's Laws". hyperphysics.phy-astr.gsu.edu. Archived from the original on 2022-12-13. Retrieved 2022-12-13.
^ Jump up to: a b "Orbits and Kepler's Laws". NASA Solar System Exploration. 26 June 2008. Archived from the original on 2022-12-13. Retrieved 2022-12-13.
^ "Planetary Motion: The History of an Idea That Launched the Scientific Revolution". earthobservatory.nasa.gov. 2009-07-07. Archived from the original on 2022-12-13. Retrieved 2022-12-13.
^ "Nicolaus Copernicus". history.com. 9 November 2009. Archived from the original on 2022-12-13. Retrieved 2022-12-13.
^ Jump up to: a b Gingerich, Owen (2011). "The great Martian catastrophe and how Kepler fixed it" (PDF). Physics Today. 64 (9): 50–54. Bibcode:2011PhT....64i..50G. doi:10.1063/PT.3.1259. Retrieved 27 July 2023.
^ Jump up to: a b c d Russell, J. L. (June 1964). "Kepler's Laws of Planetary Motion: 1609–1666". The British Journal for the History of Science. 2 (1): 1–24. doi:10.1017/S0007087400001813. ISSN 1474-001X.
^ Astronomia nova Aitiologitis, seu Physica Coelestis tradita Commentariis de Motibus stellae Martis ex observationibus G.V. Tychnonis.Prague 1609; Engl. tr. W.H. Donahue, Cambridge 1992.
^ Jump up to: a b Holton, Gerald James; Brush, Stephen G. (2001). Physics, the Human Adventure: From Copernicus to Einstein and Beyond (3rd paperback ed.). Piscataway, NJ: Rutgers University Press. pp. 40–41. ISBN 978-0-8135-2908-0. Retrieved December 27, 2009.
^ National Earth Science Teachers Association (9 October 2008). "Data Table for Planets and Dwarf Planets". Windows to the Universe. Retrieved 2 August 2018.
^ Bruce Stephenson (1994). Kepler's Physical Astronomy. Princeton University Press. p. 170. ISBN 978-0-691-03652-6.
^ Mercator, Nicolaus (1664). Nicolai Mercatoris Hypothesis astronomica nova, et consensus eius cum observationibus [Nicolaus Mercator's new astronomical hypothesis, and its agreement with observations] (in Latin). London, England: Leybourn.
^ Mercator, Nic. (25 March 1670). "Some considerations of Mr. Nic. Mercator, concerning the geometrick and direct method of signior Cassini for finding the apogees, excentricities, and anomalies of the planets; ...". Philosophical Transactions of the Royal Society of London (in Latin). 5 (57): 1168–1175. doi:10.1098/rstl.1670.0018. Mercator criticized Cassini's method of finding, from three observations, an orbit's line of apsides. Cassini had assumed (wrongly) that planets move uniformly along their elliptical orbits. From p. 1174: "Sed cum id Observationibus nequaquam congruere animadverteret, mutavit sententiam, & lineam veri motus Planetæ æqualibus temporibus æquales areas Ellipticas verrere professus est: ... " (But when he noticed that it didn't agree at all with observations, he changed his thinking, and he declared that a line [from the Sun to a planet, denoting] a planet's true motion, sweeps out equal areas of an ellipse in equal periods of time: ... [which is the "area" form of Kepler's second law])
^ Wilbur Applebaum (2000). Encyclopedia of the Scientific Revolution: From Copernicus to Newton. Routledge. p. 603. Bibcode:2000esrc.book.....A. ISBN 978-1-135-58255-5.
^ Jump up to: a b Johannes Kepler, Harmonices Mundi [The Harmony of the World] (Linz, (Austria): Johann Planck, 1619), book 5, chapter 3, p. 189.
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 especially p. 411.
^ Roy Porter (1992). The Scientific Revolution in National Context. Cambridge University Press. p. 102. ISBN 978-0-521-39699-8.
^ Victor Guillemin; Shlomo Sternberg (2006). Variations on a Theme by Kepler. American Mathematical Soc. p. 5. ISBN 978-0-8218-4184-6.
^ Voltaire, Eléments de la philosophie de Newton [Elements of Newton's Philosophy] (London: 1738). See, for example:
From p. 162: "Par une des grandes loix de Kepler, toute Planete décrit des aires égales en temp égaux : par une autre loi non-moins sûre, chaque Planete fait sa révolution autour du Soleil en telle sort, que si, sa moyenne distance au Soleil est 10. prenez le cube de ce nombre, ce qui sera 1000., & le tems de la révolution de cette Planete autour du Soleil sera proportionné à la racine quarrée de ce nombre 1000." (By one of the great laws of Kepler, each planet describes equal areas in equal times; by another law no less certain, each planet makes its revolution around the sun in such a way that if its mean distance from the sun is 10, take the cube of that number, which will be 1000, and the time of the revolution of that planet around the sun will be proportional to the square root of that number 1000.)
From p. 205: "Il est donc prouvé par la loi de Kepler & par celle de Neuton, que chaque Planete gravite vers le Soleil, ..." (It is thus proved by the law of Kepler and by that of Newton, that each planet revolves around the sun ...)
^ Jump up to: a b Wilson, Curtis (May 1994). "Kepler's Laws, So-Called" (PDF). HAD News (31): 1–2. Retrieved December 27, 2016.
^ De la Lande, Astronomie, vol. 1 (Paris: Desaint & Saillant, 1764). See, for example:
From p. 390: "... mais suivant la fameuse loi de Kepler, qui sera expliquée dans le Livre suivant (892), le rapport des temps périodiques est toujours plus grand que celui des distances, une planete cinq fois plus éloignée du soleil, emploie à faire sa révolution douze fois plus de temps ou environ; ..." (... but according to the famous law of Kepler, which will be explained in the following book [i.e., chapter] (para. 892), the ratio of the periods is always greater than that of the distances [so that, for example,] a planet five times farther from the sun, requires about twelve times or so more time to make its revolution [around the sun] ...)
From p. 429: "Les Quarrés des Temps périodiques sont comme les Cubes des Distances. 892. La plus fameuse loi du mouvement des planetes découverte par Kepler, est celle du repport qu'il y a entre les grandeurs de leurs orbites, & le temps qu'elles emploient à les parcourir; ..." (The squares of the periods are as the cubes of the distances. 892. The most famous law of the movement of the planets discovered by Kepler is that of the relation between the sizes of their orbits and the times that the [planets] require to traverse them; ...)
From p. 430: "Les Aires sont proportionnelles au Temps. 895. Cette loi générale du mouvement des planetes devenue si importante dans l'Astronomie, sçavior, que les aires sont proportionnelles au temps, est encore une des découvertes de Kepler; ..." (Areas are proportional to times. 895. This general law of the movement of the planets [which has] become so important in astronomy, namely, that areas are proportional to times, is one of Kepler's discoveries; ...)
From p. 435: "On a appellé cette loi des aires proportionnelles aux temps, Loi de Kepler, aussi bien que celle de l'article 892, du nome de ce célebre Inventeur; ..." (One called this law of areas proportional to times (the law of Kepler) as well as that of para. 892, by the name of that celebrated inventor; ... )
^ Robert Small, An account of the astronomical discoveries of Kepler (London: J Mawman, 1804), pp. 298–299.
^ Robert Small, An account of the astronomical discoveries of Kepler (London: J. Mawman, 1804).
^ Jump up to: a b c Murray, Carl D.; Dermott, S. F. (1999). Solar system dynamics. Cambridge; New York: Cambridge University Press. ISBN 978-0-521-57295-8.
^ Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second Law Archived 2019-09-11 at the Wayback Machine", Wolfram Demonstrations Project. Retrieved December 27, 2009.
^ Holton, Gerald; Brush, Stephen (2001). Brush and Holton - Physics: the Human Adventure. Princeton University Press. pp. 42–43. ISBN 978-0-8135-2908-0.
^ Burtt, Edwin. The Metaphysical Foundations of Modern Physical Science. p. 52.
^ Gerald James Holton, Stephen G. Brush (2001). Physics, the Human Adventure. Rutgers University Press. p. 45. ISBN 978-0-8135-2908-0.
^ Vijaya, G. K. (2019). "Original form of Kepler's Third Law and its misapplication in Propositions XXXII-XXXVII in Newton's Principia (Book I)". Heliyon. 5 (2) e01274. Bibcode:2019Heliy...501274V. doi:10.1016/j.heliyon.2019.e01274. PMC 6395789. PMID 30886926.
^ Caspar, Max (1993). Kepler. New York: Dover. p. 304. ISBN 978-0-486-67605-0.
^ Caspar, Max (1993). Kepler. New York: Dover. p. 286. ISBN 978-0-486-67605-0.
^ I. Newton, Principia, p. 408 in the translation of I.B. Cohen and A. Whitman
^ I. Newton, Principia, p. 943 in the translation of I.B. Cohen and A. Whitman
^ Schwarz, René. "Memorandum № 1: Keplerian Orbit Elements → Cartesian State Vectors" (PDF). Archived (PDF) from the original on 5 January 2019. Retrieved 4 May 2018.
^ Müller, M (1995). "Equation of Time – Problem in Astronomy". Acta Physica Polonica A. Archived from the original on 14 November 2017. Retrieved 23 February 2013.
General bibliography[edit]
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: Hawking, Stephen, ed. (2002). On the shoulders of giants: the great works of physics and astronomy. Philadelphia: Running Press. pp. 635–732. ISBN 978-0-7624-1348-5.
A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example: Meriam, J. L. (1971) [1966]. Dynamics (2nd ed.). New York: Wiley. pp. 161–164. ISBN 978-0-471-59601-1..
Murray, Carl D.; Dermott, S. F. (1999). Solar system dynamics. Cambridge; New York: Cambridge University Press. ISBN 978-0-521-57295-8.
Arnolʹd, V. I. (1997). "Chapter 2: Investigation of the Equations of Motion". Mathematical methods of classical mechanics. Graduate texts in mathematics (2nd ed.). New York: Springer Publishing. ISBN 978-0-387-96890-2.
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, "Johannes Kepler: The Laws of Planetary Motion"
Solar System Simulator (Interactive Applet) Archived 2018-12-13 at the Wayback Machine
"Kepler and His Laws" in From Stargazers to Starships by David P. Stern (10 October 2016)
"Kepler's Three Laws of Planetary Motion" on YouTube by Jens Puhle (Dec 27, 2023) – a video explaining and visualizing Kepler's three laws of planetary motion
Archived URL: https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion
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