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Kepler's laws of planetary motion
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For a more historical approach, see in particular the articles Astronomia nova and Epitome Astronomiae Copernicanae.
In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun.
Figure 1: Illustration of Kepler's three laws with two planetary orbits.
(1) The orbits are ellipses, with focal points F1 and F2 for the first planet and F1 and F3 for the second planet. The Sun is placed in focal point F1.

(2) The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2.

(3) The total orbit times for planet 1 and planet 2 have a ratio (a1/a2)3/2.
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Kepler's laws
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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.[1]
The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.
Most planetary orbits are nearly circular, and careful observation and calculation are required in order to establish that they are not perfectly circular. Calculations of the orbit of Mars, whose published values are somewhat suspect,[2] indicated an elliptical orbit. From this, Johannes Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits.
Kepler's work (published between 1609 and 1619) improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles.[3]
Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System to a good approximation, as a consequence of his own laws of motion and law of universal gravitation.
Contents
[hide]
1 Comparison to Copernicus
2 Nomenclature
3 History
4 Formulary
4.1 First law
4.2 Second law
4.3 Third law
5 Planetary acceleration
5.1 Acceleration vector
5.2 The inverse square law
5.3 Newton's law of gravitation
6 Position as a function of time
6.1 Mean anomaly, M
6.2 Eccentric anomaly, E
6.3 True anomaly, θ
6.4 Distance, r
7 See also
8 Notes
9 References
10 Bibliography
11 External links
Comparison to Copernicus[edit]
Kepler's laws improve the model of Copernicus. If the eccentricities of the planetary orbits are taken as zero, then Kepler basically agrees with Copernicus:
The planetary orbit is a circle
The Sun is at the center of the orbit
The speed of the planet in the orbit is constant
The eccentricities of the orbits of those planets known to Copernicus and Kepler are small, so the foregoing rules give good approximations of planetary motion; but Kepler's laws fit the observations better than Copernicus'.
Kepler's corrections are not at all obvious:
The planetary orbit is not a circle, 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 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, so the eccentricity of the orbit of the Earth is approximately
ε ≈ π 4 186 − 179 186 + 179 ≈ 0.015 , {\displaystyle \varepsilon \approx {\frac {\pi }{4}}{\frac {186-179}{186+179}}\approx 0.015,}
which is close to the correct value (0.016710219) (see Earth's orbit). The calculation is 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.
Nomenclature[edit]
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".[4][5] 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.[6] It was the exposition of Robert Small, in An account of the astronomical discoveries of Kepler (1804) that made up the set of three laws, by adding in the third.[7] Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.[5][8]
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.[9]
History[edit]
Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.[10][3][11] Kepler's third law was published in 1619.[12][3] Notably, Kepler had believed in the Copernican model of the solar system, which called for circular orbits, but 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[13]). His first law reflected this discovery.
Kepler in 1621 and Godefroy Wendelin in 1643 noted that Kepler's third law applies to the four brightest moons of Jupiter.[Nb 1] 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.[14] 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 is 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; while 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]
Formulary[edit]
The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.
First law[edit]
The orbit of every planet is an ellipse with the Sun at one of the two foci.
Figure 2: Kepler's first law placing the Sun at the focus of an elliptical orbit
Figure 4: 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:
r = p 1 + ε cos  θ , {\displaystyle r={\frac {p}{1+\varepsilon \,\cos \theta }},}
where p {\displaystyle p} 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
r min = p 1 + ε {\displaystyle r_{\min }={\frac {p}{1+\varepsilon }}}
At θ = 90° and at θ = 270° the distance is equal to p {\displaystyle p} .
At θ = 180°, aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)
r max = p 1 − ε {\displaystyle r_{\max }={\frac {p}{1-\varepsilon }}}
The semi-major axis a is the arithmetic mean between rmin and rmax:
r max − a = a − r min {\displaystyle \,r_{\max }-a=a-r_{\min }}
a = p 1 − ε 2 {\displaystyle a={\frac {p}{1-\varepsilon ^{2}}}}
The semi-minor axis b is the geometric mean between rmin and rmax:
r max b = b r min {\displaystyle {\frac {r_{\max }}{b}}={\frac {b}{r_{\min }}}}
b = p 1 − ε 2 {\displaystyle b={\frac {p}{\sqrt {1-\varepsilon ^{2}}}}}
The semi-latus rectum p is the harmonic mean between rmin and rmax:
1 r min − 1 p = 1 p − 1 r max {\displaystyle {\frac {1}{r_{\min }}}-{\frac {1}{p}}={\frac {1}{p}}-{\frac {1}{r_{\max }}}}
p a = r max r min = b 2 {\displaystyle pa=r_{\max }r_{\min }=b^{2}\,}
The eccentricity ε is the coefficient of variation between rmin and rmax:
ε = r max − r min r max + r min . {\displaystyle \varepsilon ={\frac {r_{\max }-r_{\min }}{r_{\max }+r_{\min }}}.}
The area of the ellipse is
A = π a b . {\displaystyle A=\pi ab\,.}
The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = πr2.
Second law[edit]
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.[1]
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.
In a small time d t {\displaystyle dt\,} the planet sweeps out a small triangle having base line r {\displaystyle r\,} and height r d θ {\displaystyle r\,d\theta } and area d A = 1 2 ⋅ r ⋅ r d θ {\displaystyle dA={\tfrac {1}{2}}\cdot r\cdot rd\theta } and so the constant areal velocity is d A d t = 1 2 r 2 d θ d t . {\displaystyle {\frac {dA}{dt}}={\tfrac {1}{2}}r^{2}{\frac {d\theta }{dt}}.}
The area enclosed by the elliptical orbit is π a b . {\displaystyle \pi ab.\,} So the period P {\displaystyle P\,} satisfies
P ⋅ 1 2 r 2 d θ d t = π a b {\displaystyle P\cdot {\tfrac {1}{2}}r^{2}{\frac {d\theta }{dt}}=\pi ab}
and the mean motion of the planet around the Sun
n = 2 π / P {\displaystyle n=2\pi /P}
satisfies
r 2 d θ = a b n d t . {\displaystyle r^{2}\,d\theta =abn\,dt.}
Third law[edit]
The square of the orbital period of a planet is directly proportional to the cube of 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[12] 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.[17] So it was known as the harmonic law.[18]
According to this law the expression P2a−3 has the same value for all the planets in the solar system. Here P is the time taken for a planet to complete an orbit round the sun, and a is the mean value between the maximum and minimum distances between the planet and sun.
The corresponding formula in Newtonian mechanics is
P 2 a 3 = 4 π 2 G ( M + m ) ≈ 4 π 2 G M = c o n s t a n t {\displaystyle {\frac {P^{2}}{a^{3}}}={\frac {4\pi ^{2}}{G(M+m)}}\approx {\frac {4\pi ^{2}}{GM}}=\mathrm {constant} }
where M is the mass of the sun, m is the mass of the planet, and G is the gravitational constant.
As the Sun is much more massive than any planet, Kepler's third law is approximately correct in Newtonian mechanics.
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 law.
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.[19] Moreover, he does not assign a cause to gravity.[20]
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 in direct proportion to the mass of the planet and in inverse proportion 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]
See also: Polar coordinate § Vector calculus, and Mechanics of planar particle motion
From the heliocentric point of view consider the vector to the planet r = r r ^ {\displaystyle \mathbf {r} =r{\hat {\mathbf {r} }}} where r {\displaystyle r} is the distance to the planet and r ^ {\displaystyle {\hat {\mathbf {r} }}} is a unit vector pointing towards the planet.
d r ^ d t = r ^ ˙ = θ ˙ θ ^ , d θ ^ d t = θ ^ ˙ = − θ ˙ r ^ {\displaystyle {\frac {d{\hat {\mathbf {r} }}}{dt}}={\dot {\hat {\mathbf {r} }}}={\dot {\theta }}{\hat {\boldsymbol {\theta }}},\qquad {\frac {d{\hat {\boldsymbol {\theta }}}}{dt}}={\dot {\hat {\boldsymbol {\theta }}}}=-{\dot {\theta }}{\hat {\mathbf {r} }}}
where θ ^ {\displaystyle {\hat {\boldsymbol {\theta }}}} is the unit vector whose direction is 90 degrees counterclockwise of r ^ {\displaystyle {\hat {\mathbf {r} }}} , and θ {\displaystyle \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:
r ˙ = r ˙ r ^ + r r ^ ˙ = r ˙ r ^ + r θ ˙ θ ^ , {\displaystyle {\dot {\mathbf {r} }}={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\hat {\mathbf {r} }}}={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}},}
r ¨ = ( r ¨ r ^ + r ˙ r ^ ˙ ) + ( r ˙ θ ˙ θ ^ + r θ ¨ θ ^ + r θ ˙ θ ^ ˙ ) = ( r ¨ − r θ ˙ 2 ) r ^ + ( r θ ¨ + 2 r ˙ θ ˙ ) θ ^ . {\displaystyle {\ddot {\mathbf {r} }}=({\ddot {r}}{\hat {\mathbf {r} }}+{\dot {r}}{\dot {\hat {\mathbf {r} }}})+({\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\dot {\theta }}{\dot {\hat {\boldsymbol {\theta }}}})=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}.}
So
r ¨ = a r r ^ + a θ θ ^ {\displaystyle {\ddot {\mathbf {r} }}=a_{r}{\hat {\boldsymbol {r}}}+a_{\theta }{\hat {\boldsymbol {\theta }}}}
where the radial acceleration is
a r = r ¨ − r θ ˙ 2 {\displaystyle a_{r}={\ddot {r}}-r{\dot {\theta }}^{2}}
and the transversal acceleration is
a θ = r θ ¨ + 2 r ˙ θ ˙ . {\displaystyle a_{\theta }=r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}.}
The inverse square law[edit]
Kepler's second law says that
r 2 θ ˙ = n a b {\displaystyle r^{2}{\dot {\theta }}=nab}
is constant.
The transversal acceleration a θ {\displaystyle a_{\theta }} is zero:
d ( r 2 θ ˙ ) d t = r ( 2 r ˙ θ ˙ + r θ ¨ ) = r a θ = 0. {\displaystyle {\frac {d(r^{2}{\dot {\theta }})}{dt}}=r(2{\dot {r}}{\dot {\theta }}+r{\ddot {\theta }})=ra_{\theta }=0.}
So the acceleration of a planet obeying Kepler's second law is directed towards the sun.
The radial acceleration a r {\displaystyle a_{r}} is
a r = r ¨ − r θ ˙ 2 = r ¨ − r ( n a b r 2 ) 2 = r ¨ − n 2 a 2 b 2 r 3 . {\displaystyle a_{r}={\ddot {r}}-r{\dot {\theta }}^{2}={\ddot {r}}-r\left({\frac {nab}{r^{2}}}\right)^{2}={\ddot {r}}-{\frac {n^{2}a^{2}b^{2}}{r^{3}}}.}
Kepler's first law states that the orbit is described by the equation:
p r = 1 + ε cos  θ . {\displaystyle {\frac {p}{r}}=1+\varepsilon \cos \theta .}
Differentiating with respect to time
− p r ˙ r 2 = − ε sin  θ θ ˙ {\displaystyle -{\frac {p{\dot {r}}}{r^{2}}}=-\varepsilon \sin \theta \,{\dot {\theta }}}
or
p r ˙ = n a b ε sin  θ . {\displaystyle p{\dot {r}}=nab\,\varepsilon \sin \theta .}
Differentiating once more
p r ¨ = n a b ε cos  θ θ ˙ = n a b ε cos  θ n a b r 2 = n 2 a 2 b 2 r 2 ε cos  θ . {\displaystyle p{\ddot {r}}=nab\varepsilon \cos \theta \,{\dot {\theta }}=nab\varepsilon \cos \theta \,{\frac {nab}{r^{2}}}={\frac {n^{2}a^{2}b^{2}}{r^{2}}}\varepsilon \cos \theta .}
The radial acceleration a r {\displaystyle a_{r}} satisfies
p a r = n 2 a 2 b 2 r 2 ε cos  θ − p n 2 a 2 b 2 r 3 = n 2 a 2 b 2 r 2 ( ε cos  θ − p r ) . {\displaystyle pa_{r}={\frac {n^{2}a^{2}b^{2}}{r^{2}}}\varepsilon \cos \theta -p{\frac {n^{2}a^{2}b^{2}}{r^{3}}}={\frac {n^{2}a^{2}b^{2}}{r^{2}}}\left(\varepsilon \cos \theta -{\frac {p}{r}}\right).}
Substituting the equation of the ellipse gives
p a r = n 2 a 2 b 2 r 2 ( p r − 1 − p r ) = − n 2 a 2 r 2 b 2 . {\displaystyle pa_{r}={\frac {n^{2}a^{2}b^{2}}{r^{2}}}\left({\frac {p}{r}}-1-{\frac {p}{r}}\right)=-{\frac {n^{2}a^{2}}{r^{2}}}b^{2}.}
The relation b 2 = p a {\displaystyle b^{2}=pa} gives the simple final result
a r = − n 2 a 3 r 2 . {\displaystyle a_{r}=-{\frac {n^{2}a^{3}}{r^{2}}}.}
This means that the acceleration vector r ¨ {\displaystyle \mathbf {\ddot {r}} } of any planet obeying Kepler's first and second law satisfies the inverse square law
r ¨ = − α r 2 r ^ {\displaystyle \mathbf {\ddot {r}} =-{\frac {\alpha }{r^{2}}}{\hat {\mathbf {r} }}}
where
α = n 2 a 3 {\displaystyle \alpha =n^{2}a^{3}\,}
is a constant, and r ^ {\displaystyle {\hat {\mathbf {r} }}} is the unit vector pointing from the Sun towards the planet, and r {\displaystyle r\,} is the distance between the planet and the Sun.
According to Kepler's third law, α {\displaystyle \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 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:
F = m planet r ¨ = − m planet α r − 2 r ^ {\displaystyle \mathbf {F} =m_{\text{planet}}\mathbf {\ddot {r}} =-m_{\text{planet}}\alpha r^{-2}{\hat {\mathbf {r} }}}
where m planet {\displaystyle m_{\text{planet}}} is the mass of the planet and α {\displaystyle \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 Sun {\displaystyle m_{\text{Sun}}} . So
α = G m Sun {\displaystyle \alpha =Gm_{\text{Sun}}}
where G {\displaystyle G} is the gravitational constant.
The acceleration of solar system body number i is, according to Newton's laws:
r ¨ i = G ∑ j ≠ i m j r i j − 2 r ^ i j {\displaystyle \mathbf {{\ddot {r}}_{i}} =G\sum _{j\neq i}m_{j}r_{ij}^{-2}{\hat {\mathbf {r} }}_{ij}}
where m j {\displaystyle m_{j}} is the mass of body j, r i j {\displaystyle r_{ij}} is the distance between body i and body j, r ^ i j {\displaystyle {\hat {\mathbf {r} }}_{ij}} is the unit vector from body i towards body j, and the vector summation is over all bodies in the world, besides i itself.
In the special case where there are only two bodies in the world, Earth and Sun, the acceleration becomes
r ¨ Earth = G m Sun r Earth , Sun − 2 r ^ Earth , Sun {\displaystyle \mathbf {\ddot {r}} _{\text{Earth}}=Gm_{\text{Sun}}r_{{\text{Earth}},{\text{Sun}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Earth}},{\text{Sun}}}}
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 world are Moon and Earth the acceleration of the Moon becomes
r ¨ Moon = G m Earth r Moon , Earth − 2 r ^ Moon , Earth {\displaystyle \mathbf {\ddot {r}} _{\text{Moon}}=Gm_{\text{Earth}}r_{{\text{Moon}},{\text{Earth}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Moon}},{\text{Earth}}}}
So in this approximation the Moon moves around the Earth according to Kepler's laws.
In the three-body case the accelerations are
r ¨ Sun = G m Earth r Sun , Earth − 2 r ^ Sun , Earth + G m Moon r Sun , Moon − 2 r ^ Sun , Moon {\displaystyle \mathbf {\ddot {r}} _{\text{Sun}}=Gm_{\text{Earth}}r_{{\text{Sun}},{\text{Earth}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Sun}},{\text{Earth}}}+Gm_{\text{Moon}}r_{{\text{Sun}},{\text{Moon}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Sun}},{\text{Moon}}}}
r ¨ Earth = G m Sun r Earth , Sun − 2 r ^ Earth , Sun + G m Moon r Earth , Moon − 2 r ^ Earth , Moon {\displaystyle \mathbf {\ddot {r}} _{\text{Earth}}=Gm_{\text{Sun}}r_{{\text{Earth}},{\text{Sun}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Earth}},{\text{Sun}}}+Gm_{\text{Moon}}r_{{\text{Earth}},{\text{Moon}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Earth}},{\text{Moon}}}}
r ¨ Moon = G m Sun r Moon , Sun − 2 r ^ Moon , Sun + G m Earth r Moon , Earth − 2 r ^ Moon , Earth {\displaystyle \mathbf {\ddot {r}} _{\text{Moon}}=Gm_{\text{Sun}}r_{{\text{Moon}},{\text{Sun}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Moon}},{\text{Sun}}}+Gm_{\text{Earth}}r_{{\text{Moon}},{\text{Earth}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Moon}},{\text{Earth}}}}
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 four steps:
1. Compute the mean anomaly M = nt where n is the mean motion.
n ⋅ P = 2 π {\displaystyle n\cdot P=2\pi } radians where P is the period.
2. Compute the eccentric anomaly E by solving Kepler's equation:
M = E − ε sin  E {\displaystyle \ M=E-\varepsilon \sin E}
3. Compute the true anomaly θ by the equation:
( 1 − ε ) tan 2  θ 2 = ( 1 + ε ) tan 2  E 2 {\displaystyle (1-\varepsilon )\tan ^{2}{\frac {\theta }{2}}=(1+\varepsilon )\tan ^{2}{\frac {E}{2}}}
4. Compute the heliocentric distance
r = a ( 1 − ε cos  E ) . {\displaystyle r=a(1-\varepsilon \cos E).}
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.
Mean anomaly, M[edit]
FIgure 5: 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
a = | c z | , {\displaystyle \ a=|cz|,} distance between center and perihelion, the semimajor axis,
ε = | c s | a , {\displaystyle \ \varepsilon ={|cs| \over a},} the eccentricity,
b = a 1 − ε 2 , {\displaystyle \ b=a{\sqrt {1-\varepsilon ^{2}}},} the semiminor axis,
r = | s p | , {\displaystyle \ r=|sp|,} the distance between Sun and planet.
θ = ∠ z s p , {\displaystyle \theta =\angle zsp,} the direction to the planet as seen from the Sun, the true anomaly.
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
x , {\displaystyle \ x,} the projection of the planet to the auxiliary circle
y , {\displaystyle \ y,} the point on the circle such that the sector areas |zcy| and |zsx| are equal,
M = ∠ z c y , {\displaystyle M=\angle zcy,} the mean anomaly.
The sector areas are related by | z s p | = b a ⋅ | z s x | . {\displaystyle |zsp|={\frac {b}{a}}\cdot |zsx|.}
The circular sector area | z c y | = a 2 M 2 . {\displaystyle \ |zcy|={\frac {a^{2}M}{2}}.}
The area swept since perihelion,
| z s p | = b a ⋅ | z s x | = b a ⋅ | z c y | = b a ⋅ a 2 M 2 = a b M 2 , {\displaystyle |zsp|={\frac {b}{a}}\cdot |zsx|={\frac {b}{a}}\cdot |zcy|={\frac {b}{a}}\cdot {\frac {a^{2}M}{2}}={\frac {abM}{2}},}
is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.
M = n t , {\displaystyle M=nt,\,}
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.[21] Kepler's solution is to use
E = ∠ z c x {\displaystyle 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.
| z c y | = | z s x | = | z c x | − | s c x | {\displaystyle \ |zcy|=|zsx|=|zcx|-|scx|}
a 2 M 2 = a 2 E 2 − a ε ⋅ a sin  E 2 {\displaystyle {\frac {a^{2}M}{2}}={\frac {a^{2}E}{2}}-{\frac {a\varepsilon \cdot a\sin E}{2}}}
Division by a2/2 gives Kepler's equation
M = E − ε ⋅ sin  E . {\displaystyle M=E-\varepsilon \cdot \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.
True anomaly, θ[edit]
Note from the figure that
c d → = c s → + s d → {\displaystyle {\overrightarrow {cd}}={\overrightarrow {cs}}+{\overrightarrow {sd}}}
so that
a ⋅ cos  E = a ⋅ ε + r ⋅ cos  θ . {\displaystyle a\cdot \cos E=a\cdot \varepsilon +r\cdot \cos \theta .}
Dividing by a {\displaystyle a} and inserting from Kepler's first law
r a = 1 − ε 2 1 + ε ⋅ cos  θ {\displaystyle \ {\frac {r}{a}}={\frac {1-\varepsilon ^{2}}{1+\varepsilon \cdot \cos \theta }}}
to get
cos  E = ε + 1 − ε 2 1 + ε ⋅ cos  θ ⋅ cos  θ {\displaystyle \cos E=\varepsilon +{\frac {1-\varepsilon ^{2}}{1+\varepsilon \cdot \cos \theta }}\cdot \cos \theta } = ε ⋅ ( 1 + ε ⋅ cos  θ ) + ( 1 − ε 2 ) ⋅ cos  θ 1 + ε ⋅ cos  θ {\displaystyle ={\frac {\varepsilon \cdot (1+\varepsilon \cdot \cos \theta )+(1-\varepsilon ^{2})\cdot \cos \theta }{1+\varepsilon \cdot \cos \theta }}} = ε + cos  θ 1 + ε ⋅ cos  θ . {\displaystyle ={\frac {\varepsilon +\cos \theta }{1+\varepsilon \cdot \cos \theta }}.}
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  x 2 = 1 − cos  x 1 + cos  x . {\displaystyle \tan ^{2}{\frac {x}{2}}={\frac {1-\cos x}{1+\cos x}}.}
Get
tan 2  E 2 = 1 − cos  E 1 + cos  E = 1 − ε + cos  θ 1 + ε ⋅ cos  θ 1 + ε + cos  θ 1 + ε ⋅ cos  θ = ( 1 + ε ⋅ cos  θ ) − ( ε + cos  θ ) ( 1 + ε ⋅ cos  θ ) + ( ε + cos  θ ) = 1 − ε 1 + ε ⋅ 1 − cos  θ 1 + cos  θ = 1 − ε 1 + ε ⋅ tan 2  θ 2 . {\displaystyle {\begin{aligned}&\tan ^{2}{\frac {E}{2}}={\frac {1-\cos E}{1+\cos E}}={\frac {1-{\frac {\varepsilon +\cos \theta }{1+\varepsilon \cdot \cos \theta }}}{1+{\frac {\varepsilon +\cos \theta }{1+\varepsilon \cdot \cos \theta }}}}\\[8pt]={}&{\frac {(1+\varepsilon \cdot \cos \theta )-(\varepsilon +\cos \theta )}{(1+\varepsilon \cdot \cos \theta )+(\varepsilon +\cos \theta )}}={\frac {1-\varepsilon }{1+\varepsilon }}\cdot {\frac {1-\cos \theta }{1+\cos \theta }}={\frac {1-\varepsilon }{1+\varepsilon }}\cdot \tan ^{2}{\frac {\theta }{2}}.\end{aligned}}}
Multiplying by 1 + ε gives the result
( 1 − ε ) ⋅ tan 2  θ 2 = ( 1 + ε ) ⋅ tan 2  E 2 {\displaystyle (1-\varepsilon )\cdot \tan ^{2}{\frac {\theta }{2}}=(1+\varepsilon )\cdot \tan ^{2}{\frac {E}{2}}}
This is the third step in the connection between time and position in the orbit.
Distance, r[edit]
The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:
r ⋅ ( 1 + ε ⋅ cos  θ ) = a ⋅ ( 1 − ε 2 ) {\displaystyle \ r\cdot (1+\varepsilon \cdot \cos \theta )=a\cdot (1-\varepsilon ^{2})}
Using the relation above between θ and E the final equation for the distance r is:
r = a ⋅ ( 1 − ε ⋅ cos  E ) . {\displaystyle \ r=a\cdot (1-\varepsilon \cdot \cos E).}
See also[edit]
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
Notes[edit]
Jump up ^ 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.)
In 1621, Johannes Kepler had 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, page 554.
References[edit]
^ Jump up to: a b Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second Law", Wolfram Demonstrations Project. Retrieved December 27, 2009.
Jump up ^ https://www.nytimes.com/1990/01/23/science/after-400-years-a-challenge-to-kepler-he-fabricated-his-data-scholar-says.html?pagewanted=1
^ Jump up to: a b c 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 0-8135-2908-5. Retrieved December 27, 2009.
Jump up ^ Voltaire, Eléments de la philosophie de Newton [Elements of Newton's Philosophy] (London, England: 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. Washington, DC: Historical Astronomy Division, American Astronomical Society (31): 1–2. Retrieved December 27, 2016.
Jump up ^ De la Lande, Astronomie, vol. 1 (Paris, France: Desaint & Saillant, 1764). See, for example:
From page 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] (paragraph 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 page 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 page 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 page 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 paragraph 892, by the name of that celebrated inventor; … )
Jump up ^ Robert Small, An account of the astronomical discoveries of Kepler (London, England: J Mawman, 1804), pp. 298–299.
Jump up ^ Robert Small, An account of the astronomical discoveries of Kepler (London, England: J. Mawman, 1804).
Jump up ^ Bruce Stephenson (1994). Kepler's Physical Astronomy. Princeton University Press. p. 170. ISBN 0-691-03652-7.
Jump up ^ In his Astronomia nova, Kepler presented only a proof that Mars' orbit is elliptical. Evidence that the other known planets' orbits are elliptical was presented only in 1621.
See: Johannes Kepler, Astronomia nova … (1609), p. 285. After having rejected circular and oval orbits, Kepler concluded that Mars' orbit must be elliptical. From the top of page 285: "Ergo ellipsis est Planetæ iter; … " (Thus, an ellipse is the planet's [i.e., Mars'] path; … ) Later on the same page: " … ut sequenti capite patescet: ubi simul etiam demonstrabitur, nullam Planetæ relinqui figuram Orbitæ, præterquam perfecte ellipticam; … " ( … as will be revealed in the next chapter: where it will also then be proved that any figure of the planet's orbit must be relinquished, except a perfect ellipse; … ) And then: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ) The geometric proof that Mars' orbit is an ellipse appears as Protheorema XI on pages 289–290.
Kepler stated that every planet travels in elliptical orbits having the Sun at one focus in: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, part 1, III. De Figura Orbitæ (III. On the figure [i.e., shape] of orbits), pages 658–665. From p. 658: "Ellipsin fieri orbitam planetæ … " (Of an ellipse is made a planet's orbit … ). From p. 659: " … Sole (Foco altero huius ellipsis) … " ( … the Sun (the other focus of this ellipse) … ).
Jump up ^ In his Astronomia nova ... (1609), Kepler did not present his second law in its modern form. He did that only in his Epitome of 1621. Furthermore, in 1609, he presented his second law in two different forms, which scholars call the "distance law" and the "area law".
His "distance law" is presented in: "Caput XXXII. Virtutem quam Planetam movet in circulum attenuari cum discessu a fonte." (Chapter 32. The force that moves a planet circularly weakens with distance from the source.) See: Johannes Kepler, Astronomia nova … (1609), pp. 165–167. On page 167, Kepler states: " … , quanto longior est αδ quam αε, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud δ, quam in æquali arcu excentrici apud ε." ( … , as αδ is longer than αε, so much longer will a planet remain on a certain arc of the eccentric near δ than on an equal arc of the eccentric near ε.) That is, the farther a planet is from the Sun (at the point α), the slower it moves along its orbit, so a radius from the Sun to a planet passes through equal areas in equal times. However, as Kepler presented it, his argument is accurate only for circles, not ellipses.
His "area law" is presented in: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ), Protheorema XIV and XV, pp. 291–295. On the top p. 294, it reads: "Arcum ellipseos, cujus moras metitur area AKN, debere terminari in LK, ut sit AM." (The arc of the ellipse, of which the duration is delimited [i.e., measured] by the area AKM, should be terminated in LK, so that it [i.e., the arc] is AM.) In other words, the time that Mars requires to move along an arc AM of its elliptical orbit is measured by the area of the segment AMN of the ellipse (where N is the position of the Sun), which in turn is proportional to the section AKN of the circle that encircles the ellipse and that is tangent to it. Therefore, the area that is swept out by a radius from the Sun to Mars as Mars moves along an arc of its elliptical orbit is proportional to the time that Mars requires to move along that arc. Thus, a radius from the Sun to Mars sweeps out equal areas in equal times.
In 1621, Kepler restated his second law for any planet: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, page 668. From page 668: "Dictum quidem est in superioribus, divisa orbita in particulas minutissimas æquales: accrescete iis moras planetæ per eas, in proportione intervallorum inter eas & Solem." (It has been said above that, if the orbit of the planet is divided into the smallest equal parts, the times of the planet in them increase in the ratio of the distances between them and the sun.) That is, a planet's speed along its orbit is inversely proportional to its distance from the Sun. (The remainder of the paragraph makes clear that Kepler was referring to what is now called angular velocity.)
^ 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. 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 [i.e., the ratio of 3:2] of 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 especially p. 411.
Jump up ^ http://www.windows2universe.org/our_solar_system/planets_table.html
Jump up ^ Wilbur Applebaum (13 June 2000). Encyclopedia of the Scientific Revolution: From Copernicus to Newton. Routledge. p. 603. ISBN 978-1-135-58255-5.
Jump up ^ Roy Porter (25 September 1992). The Scientific Revolution in National Context. Cambridge University Press. p. 102. ISBN 978-0-521-39699-8.
Jump up ^ Victor Guillemin; Shlomo Sternberg (2006). Variations on a Theme by Kepler. American Mathematical Soc. p. 5. ISBN 978-0-8218-4184-6.
Jump up ^ Burtt, Edwin. The Metaphysical Foundations of Modern Physical Science. p. 52.
Jump up ^ Gerald James Holton, Stephen G. Brush (2001). Physics, the Human Adventure. Rutgers University Press. p. 45. ISBN 0-8135-2908-5.
Jump up ^ I. Newton, Principia, p. 408 in the translation of I.B. Cohen and A. Whitman
Jump up ^ I. Newton, Principia, p. 943 in the translation of I.B. Cohen and A. Whitman
Jump up ^ MÜLLER, M (1995). "EQUATION OF TIME – PROBLEM IN ASTRONOMY". Acta Physica Polonica A. Retrieved 23 February 2013.
Bibliography[edit]
Kepler's life is summarized on pages 523–627 and Book Five of his magnum opus, Harmonice Mundi (harmonies of the world), is reprinted on pages 635–732 of On the Shoulders of Giants: The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4
A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example, pages 161–164 of Meriam, J. L. (1971) [1966]. "Dynamics, 2nd ed". New York: John Wiley. ISBN 0-471-59601-9. .
Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, ISBN 0-521-57597-4
V.I. Arnold, Mathematical Methods of Classical Mechanics, Chapter 2. Springer 1989, ISBN 0-387-96890-3
External links[edit]
B.Surendranath Reddy; animation of Kepler's laws: applet
"Derivation of Kepler's Laws" (from Newton's laws) at Physics Stack Exchange.
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, http://www.phy.syr.edu/courses/java/mc_html/kepler.html, an interactive Java applet that aids in the understanding of Kepler's Second Law.
Audio – Cain/Gay (2010) Astronomy Cast Johannes Kepler and His Laws of Planetary Motion
University of Tennessee's Dept. Physics & Astronomy: Astronomy 161 page on Johannes Kepler: The Laws of Planetary Motion [1]
Equant compared to Kepler: interactive model [2]
Kepler's Third Law:interactive model [3]
Solar System Simulator (Interactive Applet)
Kepler and His Laws, educational web pages by David P. Stern
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values: P\cdot {\tfrac {1}{2}}r^{2}{\frac {d\theta }{dt}}=\pi ab

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values: \frac{d\hat{\mathbf{r}}}{dt}=\dot{\hat{\mathbf{r}}} = \dot\theta  \hat{\boldsymbol\theta},\qquad  \frac{d\hat{\boldsymbol\theta}}{dt}=\dot{\hat{\boldsymbol\theta}} = -\dot\theta \hat{\mathbf{r}}

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values:
\dot{\mathbf{r}} =\dot{r} \hat{\mathbf{r}} + r \dot{\hat{\mathbf{r}}}
=\dot{r} \hat{\mathbf{r}} + r \dot{\theta} \hat{\boldsymbol{\theta}},

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values:
\ddot{\mathbf{r}}
= (\ddot{r} \hat{\mathbf{r}} +\dot{r} \dot{\hat{\mathbf{r}}} )
+ (\dot{r}\dot{\theta} \hat{\boldsymbol{\theta}} + r\ddot{\theta} \hat{\boldsymbol{\theta}}
+ r\dot{\theta} \dot{\hat{\boldsymbol{\theta}}})
= (\ddot{r} - r\dot{\theta}^2) \hat{\mathbf{r}} + (r\ddot{\theta} + 2\dot{r} \dot{\theta}) \hat{\boldsymbol{\theta}}.

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values: \ddot{\mathbf{r}} = a_r \hat{\boldsymbol{r}}+a_\theta\hat{\boldsymbol{\theta}}

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values: a_\theta=r\ddot{\theta} + 2\dot{r} \dot{\theta}.

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values: \frac{d (r^2 \dot \theta)}{dt} = r (2 \dot r \dot \theta + r \ddot \theta ) = r a_\theta = 0.

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values:
a_r = \ddot r - r \dot \theta^2= \ddot r - r \left(\frac{nab}{r^2}
\right)^2= \ddot r -\frac{n^2a^2b^2}{r^3}.

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values: \frac{p}{r} = 1+ \varepsilon \cos\theta.

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values: -\frac{p\dot r}{r^2} = -\varepsilon  \sin \theta \,\dot \theta

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values:
p\ddot r =nab \varepsilon \cos \theta \,\dot \theta
=nab \varepsilon \cos \theta \,\frac{nab}{r^2}
=\frac{n^2a^2b^2}{r^2}\varepsilon \cos \theta .

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values:
p a_r = \frac{n^2 a^2b^2}{r^2}\varepsilon \cos \theta  - p\frac{n^2 a^2b^2}{r^3}
= \frac{n^2a^2b^2}{r^2}\left(\varepsilon \cos \theta - \frac{p}{r}\right).

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values: p a_r = \frac{n^2a^2b^2}{r^2}\left(\frac p r - 1 - \frac p r\right)= -\frac{n^2a^2}{r^2}b^2.

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values: \mathbf{\ddot r} = - \frac{\alpha}{r^2}\hat{\mathbf{r}}

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values: {\mathbf  {{\ddot  r}}}_{{\text{Moon}}}=Gm_{{\text{Earth}}}r_{{{\text{Moon}},{\text{Earth}}}}^{{-2}}{\hat  {{\mathbf  {r}}}}_{{{\text{Moon}},{\text{Earth}}}}

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values: Edit section: Position as a function of time

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values: (1-\varepsilon )\tan ^{2}{\frac  \theta 2}=(1+\varepsilon )\tan ^{2}{\frac  E2}

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values: |zsp|=\frac b a \cdot|zsx|=\frac b a \cdot|zcy|=\frac b a\cdot\frac{a^2 M}2 = \frac {a b M}{2},

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values: \frac{a^2 M}2=\frac{a^2 E}2-\frac {a\varepsilon\cdot a\sin E}2

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values: \ \frac r a =\frac{1-\varepsilon^2}{1+\varepsilon\cdot\cos \theta}

values: vertical-align: -2.171ex; width:30.963ex; height:6.009ex;

values: vertical-align: -2.171ex; width:36.532ex; height:6.176ex;

values: =\frac{\varepsilon\cdot(1+\varepsilon\cdot\cos \theta)+(1-\varepsilon^2)\cdot\cos \theta}{1+\varepsilon\cdot\cos \theta}

values: vertical-align: -2.171ex; width:15.395ex; height:5.676ex;

values: =\frac{\varepsilon +\cos \theta}{1+\varepsilon\cdot\cos \theta}.

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values:
\begin{align}
& \tan^2\frac{E}{2} = \frac{1-\cos E}{1+\cos E}
 =\frac{1-\frac{\varepsilon+\cos \theta}{1+\varepsilon\cdot\cos \theta}}{1+\frac{\varepsilon+\cos \theta}{1+\varepsilon\cdot\cos \theta}} \\[8pt]
= {} & \frac{(1+\varepsilon\cdot\cos \theta)-(\varepsilon+\cos \theta)}{(1+\varepsilon\cdot\cos \theta)+(\varepsilon+\cos \theta)}
=\frac{1-\varepsilon}{1+\varepsilon}\cdot\frac{1-\cos \theta}{1+\cos \theta}=\frac{1-\varepsilon}{1+\varepsilon}\cdot\tan^2\frac{\theta}{2}.
\end{align}

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values: (1-\varepsilon)\cdot\tan^2\frac \theta 2 = (1+\varepsilon)\cdot\tan^2\frac E 2

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Low energy transfer

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text:
John Winchester (Supernatural)
From Wikipedia, the free encyclopedia
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John Winchester
Supernatural character
Jeffrey Dean Morgan as John Winchester
First appearance
"Pilot"
Created by
Eric Kripke
Portrayed by
Jeffrey Dean Morgan
Matt Cohen (young)
Nels Guloien (child)
Information
Species
Human
Gender
Male
Occupation
Hunter
Former mechanic
Former Marine
Family
Henry Winchester
(father, deceased)
Millie Winchester (mother, unknown)
Spouse(s)
Mary Campbell
(wife)
Children
Dean Winchester (son)
Sam Winchester (son)
Adam Milligan (son)
Nationality
United States
John Eric Winchester[1] is a fictional character on The CW's series Supernatural, and the protagonist of the comic book spin-off series Supernatural: Origins.[2] Developed by series creator Eric Kripke, the character is mainly portrayed by Jeffrey Dean Morgan. John is the father of Sam and Dean Winchester, the show's protagonists.
Seeking revenge after his wife Mary was killed by the demon Azazel, John Winchester became a hunter and raised his two sons to fight the supernatural. John disappears early in Supernatural, and the first season revolves primarily around Sam and Dean trying to find him. Morgan was also filming episodes of Grey's Anatomy while the first season of Supernatural was under production, and future appearances of the character were hindered by his busy schedule. While Morgan's portrayal has been praised, fans and critics were generally frustrated at the character for keeping so many secrets.
Contents
[hide]
1 Plot
2 Characterization
3 Development
4 Reception
5 References
5.1 Bibliography
5.2 Footnotes
Plot[edit]
The father of series protagonists Sam and Dean Winchester, John Winchester first appears in the pilot episode of the series, but plot devices such as flashbacks and time travel detail his background in later seasons. Apparently abandoned by his father at a young age—his father actually died while serving a secret order known as the Men of Letters—John Winchester grew up hating the man.[3] John left high school to join the Marine Corps, eventually attaining the rank of corporal and receiving many medals for his service in Vietnam. After leaving the service, he found a job as a mechanic, and fell in love with Mary Campbell.[4]
In the fourth season episode "In the Beginning," Dean is sent back through time by the angel Castiel. Dean meets his parents' former selves, and ends up convincing John to buy the 67 Chevy Impala that he eventually inherits. Dean later watches John being killed by Azazel, though the demon then resurrects him in exchange for Mary's permission to enter her house in ten years.[5] When the angel Anna Milton travels back in time in the fifth season episode "The Song Remains the Same" to prevent the births of Sam and Dean, John agrees to serve as a temporary vessel for the archangel Michael to stop her. Michael subsequently kills Anna and erases John and Mary's memories of the incident.[6]
Azazel later uses his pact with Mary to enter their home in the pilot episode, and ultimately kills her.[7] The second season episode "All Hell Breaks Loose, Part One" reveals that she had witnessed the demon secretly feeding his blood to Sam.[8] John investigated the incident and learned of the existence of the supernatural. His desire to find and kill Azazel led him to become a hunter of supernatural creatures.[7] He took his sons with him during his travels, but often left them alone in motels for long periods of time during his hunts,[9][10] leaving Dean with instructions to "shoot first and ask questions later" while watching over Sam.[9]
As revealed in the fourth season episode "Jump the Shark," John slept with a woman he met while away on a hunt. Learning over a decade later that he had fathered her son Adam, John made occasional visits over the years to partake in father-son activities with him. He hid the truth from Adam to protect him, and never revealed to him the existence of Sam and Dean, nor vice versa.[11] Meanwhile, John trained Sam and Dean to become hunters. However, Sam later left this life to start anew in college, leading to a fallout between John and his son.[7]
Twenty-two years after Mary's death, John disappears while on a hunt, forcing Sam and Dean to reunite in an unsuccessful attempt to find him. Sam returns to the life of a hunter after Azazel kills his girlfriend.[7] John reluctantly chooses to avoid his sons throughout most of the season while he investigates something,[12] eventually reuniting with them in the episode "Shadow." However, the demonic Meg Masters attacks them and reveals that Azazel is after John. After escaping from Meg, the brothers split up from their father to keep him from the demons.[13] When vampires murder his old mentor and steal the Colt—a mystical gun capable of killing anything—John teams up with Sam and Dean in "Dead Man's Blood" to retrieve it. Because demons cannot be killed by conventional means, they hope that the Colt will be effective against Azazel.[14] In response, Meg begins killing the Winchesters' friends in "Salvation," and threatens to kill more unless they deliver the Colt. John is captured after trying to give her a fake gun,[15] and reveals himself to be possessed by Azazel when the brothers come to his rescue in the first season finale "Devil's Trap." However, he manages to resist the demon's control. Despite John's pleas for Sam to shoot him with the Colt, Sam cannot bring himself to do so and allows Azazel to escape. As the Winchesters flee in Dean's Impala, a demonically-possessed trucker crashes into them.[16]
In the second season premiere, "In My Time of Dying," Sam and John awake in the hospital with only minor injuries, but a dying Dean is comatose. John secretly summons Azazel, and seems to know what the demon's plans are. He then makes a deal to save Dean, giving up his life, soul, and the Colt.[17] Before dying, John instructs Dean to kill Sam should he become evil.[18][19] The fourth season episode "On the Head of a Pin" reveals that the demon Alastair tortured John in Hell for over a century, with John refusing the demon's offer to stop if he himself would torture someone else.[20] His soul escapes from Hell in the second season finale "All Hell Breaks Loose, Part 2" when a gateway to Hell is opened. He saves Dean from Azazel, and distracts the demon by temporarily forcing him out of his host, long enough for Dean to kill Azazel with the Colt when he returns to his usual body. John then gives Sam and Dean a loving smile before becoming engulfed in a white light.[19] However, the fifth season episode "Dark Side of the Moon" suggests that he might not be in Heaven.[21]
Characterization[edit]
Actor Jeffrey Dean Morgan deemed John a "tormented soul," [22] motivated by an "equal combination of revenge and protecting his sons." [23] Following the death of their mother, Sam and Dean became secondary to John's quest of killing the demon responsible, although Morgan believed that this aspect changed once John began spending more time with them again.[23] Though he thought that John was not the world's best dad and "definitely made a lot of mistakes," the actor noted that "he did things, even when they were wrong, for the good of his sons." [22] John Kubicek of BuddyTV felt that this reasoning extended to John's decision to hide the truth from Adam to protect him. Kubicek also suggested that John became involved in Adam's life to achieve "some level of normalcy" as a father, an opportunity he had previously lost following the death of Sam and Dean's mother.[24] Overall, Morgan felt that John "loved those kids more than he loved his own life," with the character's demonic pact with Azazel serving as "his way of finding redemption." [25]
Regarding his portrayal of the character, Morgan commented, "I played him with extra angst. I think what was on the page, what I foresaw—not knowing where it was going, because no one would ever tell me anything—there was always a lot more going on in his head than he was going to show anybody, including his sons. Because, indeed, there was a lot more going on."[22] Kripke later confirmed this, stating, "It's our view that John knew everything the producers of the show know. John knew stuff we're not even ready to reveal, that won't come out for a couple of seasons. He was an awesome hunter, and by the time he showed up in 'Dead Man's Blood', he knew it all."[26]
Development[edit]
In writing the pilot episode, series creator Eric Kripke made several dramatic revisions to John's storyline. The first draft of the script had Sam and Dean being raised by their aunt and uncle, though this was changed when Kripke realized that the backstory became much less complicated by having John raise them on the road. Another revision had John dying at the end of the episode instead of Sam's girlfriend Jessica.[27] Though he survives in the final version of the pilot, the writers decided halfway through production of the first season that John did have to die,[28] as they believed his separation from his sons "split the show" by having him away "doing more interesting things than the boys are doing."[29] Kripke also felt that John kept Sam and Dean away from the "front lines", his death being required to allow the brothers to "explore, investigate and confront the yellow-eyed demon directly." The writers initially intended for John to die in the car accident in "Devil's Trap," but they ultimately postponed his death to prevent the finale from becoming too dark.[30] Following the character's eventual escape from Hell, executive producer Ben Edlund stated that even the writers are unsure of where John's soul ended up.[31]
Evil Dead-actor Bruce Campbell was Kripke's first choice to portray the character. However, Campbell was unavailable,[32] and Jeffrey Dean Morgan eventually received the role. Due to the fact that Morgan's scene in the pilot episode takes place 22 years before the series, Morgan expected to be replaced by an older actor for subsequent episodes.[22] Being only 12 years older than Jensen Ackles, who portrays the eldest son Dean,[33] Morgan was surprised when he was asked to reprise the role.[22] Ackles and Jared Padalecki often teased him about the age difference during filming.[34] Throughout the season, Morgan became frustrated at times due to his character's avoidance of his sons, stating, "It pissed off everybody, it pissed off us as actors, it pissed off the audience watching, because none of us really knew where we were gonna go." However, he reasoned that John's motivation for his actions was due to having knowledge that nobody else had.[23]
During production of Supernatural's first season, Morgan was also working on the shows Weeds and Grey's Anatomy, so he was often traveling back and forth between Los Angeles and Vancouver. This interfered at times with Morgan's acting, as he had trouble getting "Winchester-y enough" after portraying the nice character of Denny Duquette on Grey's Anatomy.[33] Morgan commented, "I was stuck in Denny-land, where I was being too nice. Winchester's harder to find. Denny's more me. He's an intense guy, John is. And Denny, for a guy who's having trouble living, he's just a charming dude."[33] Morgan was at first reluctant to return for Supernatural's second season due to his role on Grey's Anatomy.[35] Although he lent his voice for the third season episode "Long-Distance Call," [36] future appearances of Morgan as the character after the second season have been hindered due to the actor's busy schedule. Nevertheless, both Kripke and writer Sera Gamble have stated that they "would love to have him back."[36][37] The scheduling conflict has continued into the sixth season of Supernatural, with executive producer Robert Singer revealing that the writers had considered bringing the character back in the alternate reality episode "My Heart Will Go On" instead of Ellen Harvelle, but due to concerns that they would not be able to secure Morgan's return, they ultimately chose Ellen for the deceased character they would have (temporarily) return in the episode.[38] During the production of the fifth season, Morgan had an interview with MovieWeb in which he stated that no one from the show had contacted him about making another appearance, and expressed the desire to return for at least one more episode in what was then-believed to be the show's final season, as he had enjoying working with on-screen sons Jensen Ackles and Jared Padalecki and he felt that John's storyline had ended too soon.[39]
Reception[edit]
While critics praised Morgan for his performance in the role, their reaction to the character has generally been mixed. Don Williams of BuddyTV felt that John is a "rather divisive character", with some fans not understanding John's motivation for keeping his sons in the dark. However, Williams himself found the character to be "completely fascinating". He chose "In My Time of Dying" as the second best episode of the series, mainly because of John's contributions, stating, "Even if you're not a huge fan of Poppa Winchester, I think it's impossible not to be moved by the final ten minutes of this episode... John's final speech to Dean, where he finally tells his son that he's proud of him, is one hell of a tear-jerker, and both Jeffrey Dean Morgan and Jensen Ackles knock that scene out of the park."[40] Williams praised Matt Cohen's performance of a younger version of the character in the time-travel episode "In the Beginning", believing that he "did an excellent job at portraying a more innocent John Winchester than we've ever seen before".[41]
Diana Steenbergen of IGN also criticized John for his lack of explanation, feeling that "it is not exactly easy to watch John treat Dean and Sam so dismissively, especially after they have been nearly killing themselves all season to find him".[42] However, Steenbergen praised Morgan for his acting, writing, "Even though he is used sparingly throughout the series, Jeffrey Dean Morgan is always effective as John."[43] She also felt that he brought "emotional weight" to the character, allowing the audience to "feel John's weariness, and his resignation that things will most likely not go well".[44] While Brian Tallerico of UGO enjoyed Morgan's performance, he was happy that the character was killed off because John's presence made Sam and Dean into "followers" that were "merely existing in the wake of their father". With their father gone, the brothers are no longer prevented from "heading up the battle with the other side".[45]
References[edit]
Bibliography[edit]
Nicholas Knight (2007). Supernatural: The Official Companion Season 1. Titan Books. ISBN 1-84576-535-4.
Footnotes[edit]
Jump up ^ "Eric Kripke Answers Fan Questions at Comic-Con". The CW Source. August 19, 2008. Retrieved December 18, 2009.
Jump up ^ Sean Elliott (January 25, 2007). "Exclusive Interview: Supernatural's Eric Kripke & Robert Singer Return Dad to the Show & a Prequel Comic". IFMagazine. Archived from the original on February 17, 2009. Retrieved December 18, 2009.
Jump up ^ Writer: Adam Glass, Director: Serge Ladouceur (January 30, 2013). "As Time Goes By". Supernatural. Season 8. Episode 12. CW.
Jump up ^ Caims, Bryan (February 2008). "In the Name of the Father". Supernatural Magazine (2). Titan Magazines. p. 28.
Jump up ^ Writer: Jeremy Carver, Director: Steve Boyum (October 2, 2008). "In the Beginning". Supernatural. Season 4. Episode 3. CW.
Jump up ^ Writer: Sera Gamble & Nancy Weiner, Director: Steve Boyum (February 4, 2010). "The Song Remains the Same". Supernatural. Season 5. Episode 13. CW.
^ Jump up to: a b c d Writer: Eric Kripke, Director: David Nutter (September 13, 2005). "Pilot". Supernatural. Season 1. Episode 1. WB.
Jump up ^ Writers: Sera Gamble, Director: Robert Singer (May 10, 2007). "All Hell Breaks Loose, Part 2". Supernatural. Season 2. Episode 21. CW.
^ Jump up to: a b Writer: Jeremy Carver, Director: J. Miller Tobin (December 13, 2007). "A Very Supernatural Christmas". Supernatural. Season 3. Episode 8. CW.
Jump up ^ Writers: Andrew Dabb & Daniel Loflin, Director: Adam Kane (January 29, 2009). "After School Special". Supernatural. Season 4. Episode 13. CW.
Jump up ^ Writers: Andrew Dabb & Daniel Loflin, Director: Phil Sgriccia (April 23, 2009). "Jump the Shark". Supernatural. Season 4. Episode 19. CW.
Jump up ^ Writer: Eric Kripke, Director: Ken Girotti (November 15, 2005). "Home". Supernatural. Season 1. Episode 9. WB.
Jump up ^ Writer: Eric Kripke, Director: Kim Manners (February 28, 2006). "Shadow". Supernatural. Season 1. Episode 16. WB.
Jump up ^ Writers: Cathryn Humphris & John Shiban, Director: Tony Wharmby (April 20, 2006). "Dead Man's Blood". Supernatural. Season 1. Episode 20. WB.
Jump up ^ Writers: Sera Gamble & Raelle Tucker, Director: Robert Singer (April 27, 2006). "Salvation". Supernatural. Season 1. Episode 21. WB.
Jump up ^ Writer: Eric Kripke, Director: Kim Manners (May 4, 2006). "Devil's Trap". Supernatural. Season 1. Episode 22. WB.
Jump up ^ Writer: Eric Kripke, Director: Kim Manners (September 28, 2006). "In My Time of Dying". Supernatural. Season 2. Episode 1. CW.
Jump up ^ Writer: Raelle Tucker, Director: Rachel Talalay (January 11, 2007). "Hunted". Supernatural. Season 2. Episode 10. CW.
^ Jump up to: a b Writers: Eric Kripke & Michael T. Moore, Director: Kim Manners (May 17, 2007). "All Hell Breaks Loose, Part 2". Supernatural. Season 2. Episode 22. CW.
Jump up ^ Writer: Ben Edlund, Director: Mike Rohl (March 19, 2009). "On the Head of a Pin". Supernatural. Season 4. Episode 16. CW.
Jump up ^ Writers: Andrew Dabb & Daniel Loflin, Director: Jeff Woolnough (April 1, 2010). "Dark Side of the Moon". Supernatural. Season 5. Episode 16. CW.
^ Jump up to: a b c d e Knight, p.134
^ Jump up to: a b c Knight, p.136
Jump up ^ John Kubicek (April 24, 2009). "Supernatural: Papa Winchester Was a Rolling Stone". BuddyTV. Retrieved August 16, 2012.
Jump up ^ # Nicholas Knight (2008). Supernatural: The Official Companion Season 2. Titan Books. p. 143. ISBN 1-84576-657-1.
Jump up ^ Knight, p.14
Jump up ^ Knight, pp.9-10
Jump up ^ Knight, p.123
Jump up ^ Neil Wilkes (February 15, 2007). "'Supernatural' writer John Shiban". Digital Spy. Archived from the original on September 3, 2010. Retrieved September 22, 2009.
Jump up ^ Knight, p.124
Jump up ^ Jennifer Godwin (July 26, 2009). "Supernatural's Deviled Execs Spill (Many) Season Five Secrets". E! Online. Retrieved December 29, 2009.
Jump up ^ Lloyd, Kate (December 2007). "Lore & Order". Supernatural Magazine (1). Titan Magazines. p. 19.
^ Jump up to: a b c Kate O'Hare (July 12, 2008). "Actor Morgan Juggles 'Grey's,' 'Supernatural' and 'Weeds'". Zap2it. Retrieved December 18, 2009.
Jump up ^ Caims, Bryan (February 2008). "In the Name of the Father". Supernatural Magazine (2). Titan Magazines. p. 26
Jump up ^ William Keck (April 20, 2006). "Jeffrey Dean Morgan awaits his fate". USA Today. Retrieved September 22, 2009.
^ Jump up to: a b Neil Wilkes (June 30, 2008). "'Supernatural' writer talks season four". Digital Spy. Archived from the original on August 23, 2010. Retrieved December 29, 2009.
Jump up ^ Michael Ausiello (August 19, 2009). "Ask Ausiello: Spoilers on 'Grey's,' 'House,' 'NCIS,' 'Bones,' 'Gossip Girl,' 'Supernatural,' and more!". EW.com. Archived from the original on August 26, 2009. Retrieved September 22, 2009.
Jump up ^ "'Supernatural': How John Winchester almost returned in tonight's episode". Entertainment Weekly's EW.com.
Jump up ^ "Exclusive: Jeffrey Dean Morgan Wants to Return to Supernatural This Season!". MovieWeb. September 14, 2009. Retrieved December 18, 2009.
Jump up ^ Don Williams (September 11, 2008). "Top 10 'Supernatural' Episodes of All Time: #2 "In My Time of Dying"". BuddyTV. Retrieved December 18, 2009.
Jump up ^ Don Williams (October 3, 2008). "Supernatural: Meet Mary Winchester, Demon Hunter". BuddyTV. Retrieved December 18, 2009.
Jump up ^ Diana Steenbergen (September 3, 2008). "Supernatural Flashback: "Dead Man's Blood" Review". IGN. Retrieved December 18, 2009.
Jump up ^ Diana Steenbergen (March 20, 2008). "Supernatural: "Pilot" Review". IGN. Retrieved December 18, 2009.
Jump up ^ Diana Steenbergen (September 10, 2008). "Supernatural Flashback: "Salvation" Review". IGN. Retrieved December 18, 2009.
Jump up ^ Brian Tallerico. "Supernatural Episode 2.1: "In My Time of Dying" Review". UGO. Archived from the original on June 16, 2011. Retrieved December 18, 2009.
[hide]
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Supernatural
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"Devil's Trap"
Season 2
"What Is and What Should Never Be"
"All Hell Breaks Loose"
Season 3
"Fresh Blood"
"A Very Supernatural Christmas"
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Season 4
"Lazarus Rising"
Season 5
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"Good God, Y'All!"
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Season 6
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Season 7
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"Hello, Cruel World"
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"We Happy Few"
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Season 12
"Keep Calm and Carry On"
"Mamma Mia"
"The Foundry"
"American Nightmare"
"The One You've Been Waiting For"
"Celebrating the Life of Asa Fox"
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