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Kepler's laws of planetary motion

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Illustration of Kepler's laws with two planetary orbits.
  1. The orbits are ellipses, with fociF1 andF2 for Planet 1, andF1 andF3 for Planet 2. The Sun is atF1.
  2. The shaded areasA1 andA2 are equal, and are swept out in equal times by Planet 1's orbit.
  3. The ratio of Planet 1's orbit time to Planet 2's is(a1/a2)3/2{\textstyle ({a_{1}}/{a_{2}})^{3/2}}.
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Inastronomy,Kepler's laws of planetary motion, published byJohannes Kepler in 1609 (except the third law, which was fully published in 1619), describe the orbits of planets around the Sun. These laws replacedcircular orbits andepicycles in theheliocentric theory ofNicolaus Copernicus withelliptical orbits and explained how planetary velocities vary. The three laws state that:[1][2]

  1. The orbit of a planet is anellipse with the Sun at one of the twofoci.
  2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  3. The square of a planet'sorbital period is proportional to the cube of the length of thesemi-major axis of its orbit.

The elliptical orbits of planets were indicated by calculations of the orbit ofMars. From this, Kepler inferred that other bodies in theSolar 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.

Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his ownlaws of motion andlaw of universal gravitation.

A more precise historical approach is found inAstronomia nova,Harmonice Mundi andEpitome Astronomiae Copernicanae.

Comparison to Copernicus

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Johannes Kepler's laws improved the model ofCopernicus. According to Copernicus:[3][4]

  1. The planetary orbit is a circle withepicycles.
  2. The Sun is approximately at the center of the orbit.
  3. 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 

  1. The planetary orbit isnot a circle with epicycles, but anellipse.
  2. The Sun isnot at the center but at afocal point of the elliptical orbit.
  3. Neither the linear speed nor the angular speed of the planet in the orbit is constant, but thearea speed (closely linked historically with the concept ofangular momentum) is constant.

Theeccentricity of theorbit of the Earth makes the time from theMarch equinox to theSeptember 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 theequator of the Earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately

e≈π4186−179186+179≈0.015,{\displaystyle e\approx {\frac {\pi }{4}}{\frac {186-179}{186+179}}\approx 0.015,}

which is close to the correct value (0.016710218). The accuracy of this calculation requires that the two dates chosen be along the elliptical orbit's minor axis and that the midpoints of each half be along the major axis. As the two dates chosen here are equinoxes, this will be correct whenperihelion, the date the Earth is closest to the Sun, falls on asolstice. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22.

Nomenclature

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It took nearly two centuries for the current formulation of Kepler's work to take on its settled form.Voltaire'sEléments de la philosophie de Newton (Elements of Newton's Philosophy) of 1738 was the first publication to use the terminology of "laws".[6][7] TheBiographical 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 ofJoseph de Lalande.[8] It was the exposition ofRobert Small, inAn account of the astronomical discoveries of Kepler (1814) that made up the set of three laws, by adding in the third.[9] Small also claimed, against the history, that these wereempirical laws, based oninductive reasoning.[7][10]

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.[11]

History

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Kepler published his first two laws about planetary motion in 1609,[12] having found them by analyzing the astronomical observations ofTycho Brahe.[13][14][15][5]: 53  Kepler's third law was published in 1619.[16][14] Kepler had believed in theCopernican 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 highesteccentricity of all planets except Mercury.[17] His first law reflected this discovery.

In 1621, Kepler noted that his third law applies to thefour brightest moons ofJupiter.[Nb 1]Godefroy Wendelin also made this observation in 1643.[Nb 2] The second law, in the "area law" form, was contested byNicolaus Mercator in a book from 1664, but by 1670 hisPhilosophical Transactions were in its favour.[18][19] As the century proceeded it became more widely accepted.[20] The reception in Germany changed noticeably between 1688, the year in which Newton'sPrincipia was published and was taken to be basically Copernican, and 1690, by which time work ofGottfried Leibniz on Kepler had been published.[21]

Newton was credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law, whereas the other laws do depend on the inverse square form of the attraction.Carl Runge andWilhelm Lenz much later identified a symmetry principle in thephase space of planetary motion (theorthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, asconservation of angular momentum does via rotational symmetry for the second law.[22]

Formulary

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

First law

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Kepler's first law states that:

The orbit of every planet is an ellipse with the sun at one of the twofoci.

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 axisa, semi-minor axisb and semi-latus rectump; center of ellipse and its twofoci marked by large dots. Forθ = 0°,r =rmin and forθ = 180°,r =rmax.

Mathematically, an ellipse can be represented by the formula:

r=p1+εcos⁡θ,{\displaystyle r={\frac {p}{1+\varepsilon \,\cos \theta }},}

wherep{\displaystyle p} is thesemi-latus rectum,ε is theeccentricity of the ellipse,r is the distance from the Sun to the planet, andθ is the angle to the planet's current position from its closest approach, as seen from the Sun. So (r, θ) arepolar coordinates.

For an ellipse 0 < ε < 1 ; in the limiting caseε = 0, the orbit is a circle with the Sun at the centre (i.e. where there is zero eccentricity).

Atθ = 0°,perihelion, the distance is minimum

rmin=p1+ε{\displaystyle r_{\min }={\frac {p}{1+\varepsilon }}}

Atθ = 90° and atθ = 270° the distance is equal top{\displaystyle p}.

Atθ = 180°,aphelion, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)

rmax=p1−ε{\displaystyle r_{\max }={\frac {p}{1-\varepsilon }}}

Thesemi-major axisa is thearithmetic mean betweenrmin andrmax:

a=rmax+rmin2a=p1−ε2{\displaystyle {\begin{aligned}a&={\frac {r_{\max }+r_{\min }}{2}}\\[3pt]a&={\frac {p}{1-\varepsilon ^{2}}}\end{aligned}}}

Thesemi-minor axisb is thegeometric mean betweenrmin andrmax:

b=rmaxrminb=p1−ε2{\displaystyle {\begin{aligned}b&={\sqrt {r_{\max }r_{\min }}}\\[3pt]b&={\frac {p}{\sqrt {1-\varepsilon ^{2}}}}\end{aligned}}}

The semi-latus rectump is theharmonic mean betweenrmin andrmax:

p=(rmax−1+rmin−12)−1pa=rmaxrmin=b2{\displaystyle {\begin{aligned}p&=\left({\frac {r_{\max }^{-1}+r_{\min }^{-1}}{2}}\right)^{-1}\\pa&=r_{\max }r_{\min }=b^{2}\,\end{aligned}}}

The eccentricityε is thecoefficient of variation betweenrmin andrmax:

ε=rmax−rminrmax+rmin.{\displaystyle \varepsilon ={\frac {r_{\max }-r_{\min }}{r_{\max }+r_{\min }}}.}

Thearea of the ellipse is

A=πab.{\displaystyle A=\pi ab\,.}

The special case of a circle isε = 0, resulting inr =p =rmin =rmax =a =b andA =πr2.

Second law

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Kepler's second law states that:

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.

History and proofs

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Kepler notably arrived at this law through assumptions that were either only approximately true or outright false and can be outlined as follows:

  1. Planets are pushed around the Sun by a force from the Sun. This false assumption relies on incorrectAristotelian physics that an object needs to be pushed to maintain motion.
  2. 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.
  3. 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.
  4. 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.
  5. The area swept out is proportional to the overall time. This is also approximately true.
  6. 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 ther anddx vectors, for some short piece of the orbit,dx.

dA=12(r→×dx→)=12(r→×v→dt){\displaystyle dA={\frac {1}{2}}({\vec {r}}\times {\vec {dx}})={\frac {1}{2}}({\vec {r}}\times {\vec {v}}dt)}for a small piece of the orbitdx and time to cover itdt.

ThusdAdt=12(r→×v→).{\displaystyle {\frac {dA}{dt}}={\frac {1}{2}}({\vec {r}}\times {\vec {v}}).}

dAdt=1m12(r→×p→).{\displaystyle {\frac {dA}{dt}}={\frac {1}{m}}{\frac {1}{2}}({\vec {r}}\times {\vec {p}}).}

Since the final expression is proportional to the total angular momentum(r→×p→){\displaystyle ({\vec {r}}\times {\vec {p}})}, 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

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In a small timedt{\displaystyle dt} the planet sweeps out a small triangle having base liner{\displaystyle r} and heightrdθ{\displaystyle r\,d\theta } and areadA=12⋅r⋅rdθ{\textstyle dA={\frac {1}{2}}\cdot r\cdot r\,d\theta }, so the constantareal velocity isdAdt=r22dθdt.{\displaystyle {\frac {dA}{dt}}={\frac {r^{2}}{2}}{\frac {d\theta }{dt}}.}

The area enclosed by the elliptical orbit isπab{\displaystyle \pi ab}. So the periodT{\displaystyle T} satisfies

T⋅r22dθdt=πab{\displaystyle T\cdot {\frac {r^{2}}{2}}{\frac {d\theta }{dt}}=\pi ab}

and themean motion of the planet around the Sun

n=2πT{\displaystyle n={\frac {2\pi }{T}}}

satisfies

r2dθ=abndt.{\displaystyle r^{2}\,d\theta =abn\,dt.}

And so,dAdt=abn2=πabT.{\displaystyle {\frac {dA}{dt}}={\frac {abn}{2}}={\frac {\pi ab}{T}}.}

Orbits of planets with varying eccentricities.
LowHigh
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,T=1{\displaystyle T=1}.

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.

Third law

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Kepler's third law states that:

The ratio of the square of an object'sorbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.

By symbol :T2∝a3{\displaystyle T^{2}\propto a^{3}}

WithT{\displaystyle T} is the object'sorbital period anda{\displaystyle a} 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[16] 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 theharmonic 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 thecentripetal force equal to the gravitational force:

mrω2=GmMr2{\displaystyle mr\omega ^{2}=G{\frac {mM}{r^{2}}}}

Then, expressing the angular velocity ω in terms of the orbital periodT{\displaystyle {T}} and then rearranging, results in Kepler's third law:

mr(2πT)2=GmMr2⟹T2=(4π2GM)r3⟹T2∝r3{\displaystyle mr\left({\frac {2\pi }{T}}\right)^{2}=G{\frac {mM}{r^{2}}}\implies T^{2}=\left({\frac {4\pi ^{2}}{GM}}\right)r^{3}\implies T^{2}\propto r^{3}}

A more detailed derivation can be done with general elliptical orbits, instead of circles, as well as orbiting the center of mass, instead of just the large mass. This results in replacing a circular radius,r{\displaystyle r}, with the semi-major axis,a{\displaystyle a}, of the elliptical relative motion of one mass relative to the other, as well as replacing the large massM{\displaystyle M} withM+m{\displaystyle M+m}. However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is:

a3T2=G(M+m)4π2≈GM4π2≈7.496×10−6AU3days2 is constant{\displaystyle {\frac {a^{3}}{T^{2}}}={\frac {G(M+m)}{4\pi ^{2}}}\approx {\frac {GM}{4\pi ^{2}}}\approx 7.496\times 10^{-6}{\frac {{\text{AU}}^{3}}{{\text{days}}^{2}}}{\text{ is constant}}}

whereM{\displaystyle M} is themass of the Sun,m{\displaystyle m} is the mass of the planet,G{\displaystyle G} is thegravitational constant,T{\displaystyle T} is the orbital period anda{\displaystyle a} is the elliptical semi-major axis, andAU{\displaystyle {\text{AU}}} is theastronomical unit, the average distance from earth to the sun.

Table

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The following table shows the data used by Kepler to empirically derive his law:

Data used by Kepler (1618)
PlanetMean distance
to sun (AU)		Period
(days)		R3T2{\textstyle {\frac {R^{3}}{T^{2}}}} (10−6 AU3/day2)
Mercury0.38987.777.64
Venus0.724224.707.52
Earth1365.257.50
Mars1.524686.957.50
Jupiter5.204332.627.49
Saturn9.51010759.27.43

Kepler became aware ofJohn 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 fromHarmonies of the World by Kepler (1619)

Log–log plot of periodT vs. semi-major axisa (average of aphelion and perihelion) of some Solar System orbits (crosses denoting Kepler's values) showing thata3/T2 is constant (green line)

For comparison, here are modern estimates:[citation needed]

Modern data
PlanetSemi-major axis (AU)Period (days)a3T2{\textstyle {\frac {a^{3}}{T^{2}}}} (10−6 AU3/day2)
Mercury0.3871087.96937.496
Venus0.72333224.70087.496
Earth1365.25647.496
Mars1.52366686.97967.495
Jupiter5.203364332.82017.504
Saturn9.5370710775.5997.498
Uranus19.191330687.1537.506
Neptune30.069060190.037.504

Planetary acceleration

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Isaac Newton computed in hisPhilosophiæ Naturalis Principia Mathematica theacceleration of a planet moving according to Kepler's first and second laws.

  1. Thedirection of the acceleration is towards the Sun.
  2. Themagnitude of the acceleration is inversely proportional to the square of the planet's distance from the Sun (theinverse square law).

This implies that the Sun may be the physical cause of the acceleration of planets. However, Newton states in hisPrincipia 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 theforce acting on a planet to be the product of itsmass and the acceleration (seeNewton's laws of motion). So:

  1. Every planet is attracted towards the Sun.
  2. 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, inNewton's law of universal gravitation:

  1. All bodies in the Solar System attract one another.
  2. 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. (Seetwo-body problem.)

Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.

Acceleration vector

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See also:Polar coordinate § Vector calculus

From theheliocentric point of view consider the vector to the planetr=rr^{\displaystyle \mathbf {r} =r{\hat {\mathbf {r} }}} wherer{\displaystyle r} is the distance to the planet andr^{\displaystyle {\hat {\mathbf {r} }}} is aunit vector pointing towards the planet.dr^dt=r^˙=θ˙θ^,dθ^dt=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} }}}

whereθ^{\displaystyle {\hat {\boldsymbol {\theta }}}} is the unit vector whose direction is 90 degrees counterclockwise ofr^{\displaystyle {\hat {\mathbf {r} }}}, andθ{\displaystyle \theta } is the polar angle, and where adot 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^+rr^˙=r˙r^+rθ˙θ^,r¨=(r¨r^+r˙r^˙)+(r˙θ˙θ^+rθ¨θ^+rθ˙θ^˙)=(r¨−rθ˙2)r^+(rθ¨+2r˙θ˙)θ^.{\displaystyle {\begin{aligned}{\dot {\mathbf {r} }}&={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\hat {\mathbf {r} }}}={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}},\\{\ddot {\mathbf {r} }}&=\left({\ddot {r}}{\hat {\mathbf {r} }}+{\dot {r}}{\dot {\hat {\mathbf {r} }}}\right)+\left({\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\dot {\theta }}{\dot {\hat {\boldsymbol {\theta }}}}\right)=\left({\ddot {r}}-r{\dot {\theta }}^{2}\right){\hat {\mathbf {r} }}+\left(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right){\hat {\boldsymbol {\theta }}}.\end{aligned}}}

Sor¨=arr^+aθθ^{\displaystyle {\ddot {\mathbf {r} }}=a_{r}{\hat {\boldsymbol {r}}}+a_{\theta }{\hat {\boldsymbol {\theta }}}}where theradial acceleration isar=r¨−rθ˙2{\displaystyle a_{r}={\ddot {r}}-r{\dot {\theta }}^{2}}and thetransversal acceleration isaθ=rθ¨+2r˙θ˙.{\displaystyle a_{\theta }=r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}.}

Inverse square law

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Kepler's second law says thatr2θ˙=nab{\displaystyle r^{2}{\dot {\theta }}=nab} is constant.

The transversal accelerationaθ{\displaystyle a_{\theta }} is zero:d(r2θ˙)dt=r(2r˙θ˙+rθ¨)=raθ=0.{\displaystyle {\frac {d\left(r^{2}{\dot {\theta }}\right)}{dt}}=r\left(2{\dot {r}}{\dot {\theta }}+r{\ddot {\theta }}\right)=ra_{\theta }=0.}

So the acceleration of a planet obeying Kepler's second law is directed towards the Sun.

The radial accelerationar{\displaystyle a_{\text{r}}} isar=r¨−rθ˙2=r¨−r(nabr2)2=r¨−n2a2b2r3.{\displaystyle a_{\text{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:pr=1+εcos⁡(θ).{\displaystyle {\frac {p}{r}}=1+\varepsilon \cos(\theta ).}

Differentiating with respect to time−pr˙r2=−εsin⁡(θ)θ˙{\displaystyle -{\frac {p{\dot {r}}}{r^{2}}}=-\varepsilon \sin(\theta )\,{\dot {\theta }}}orpr˙=nabεsin⁡(θ).{\displaystyle p{\dot {r}}=nab\,\varepsilon \sin(\theta ).}

Differentiating once morepr¨=nabεcos⁡(θ)θ˙=nabεcos⁡(θ)nabr2=n2a2b2r2εcos⁡(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 accelerationar{\displaystyle a_{\text{r}}} satisfiespar=n2a2b2r2εcos⁡(θ)−pn2a2b2r3=n2a2b2r2(εcos⁡(θ)−pr).{\displaystyle pa_{\text{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 givespar=n2a2b2r2(pr−1−pr)=−n2a2r2b2.{\displaystyle pa_{\text{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 relationb2=pa{\displaystyle b^{2}=pa} gives the simple final resultar=−n2a3r2.{\displaystyle a_{\text{r}}=-{\frac {n^{2}a^{3}}{r^{2}}}.}

This means that the acceleration vectorr¨{\displaystyle \mathbf {\ddot {r}} } of any planet obeying Kepler's first and second law satisfies theinverse square lawr¨=−αr2r^{\displaystyle \mathbf {\ddot {r}} =-{\frac {\alpha }{r^{2}}}{\hat {\mathbf {r} }}}whereα=n2a3{\displaystyle \alpha =n^{2}a^{3}}is a constant, andr^{\displaystyle {\hat {\mathbf {r} }}} is the unit vector pointing from the Sun towards the planet, andr{\displaystyle r\,} is the distance between the planet and the Sun.

Since mean motionn=2πT{\displaystyle n={\frac {2\pi }{T}}} whereT{\displaystyle T} is the period, 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 adifferential equation. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is ahyperbola orparabola or astraight line. (SeeKepler orbit.)

Newton's law of gravitation

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ByNewton's second law, the gravitational force that acts on the planet is:F=mplanetr¨=−mplanetαr−2r^{\displaystyle \mathbf {F} =m_{\text{planet}}\mathbf {\ddot {r}} =-m_{\text{planet}}\alpha r^{-2}{\hat {\mathbf {r} }}}

wheremplanet{\displaystyle m_{\text{planet}}} is the mass of the planet andα{\displaystyle \alpha } has the same value for all planets in the Solar System. According toNewton'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,mSun{\displaystyle m_{\text{Sun}}}. Soα=GmSun{\displaystyle \alpha =Gm_{\text{Sun}}}whereG{\displaystyle G} is thegravitational constant.

The acceleration of Solar System body numberi is, according to Newton's laws:r¨i=G∑j≠imjrij−2r^ij{\displaystyle \mathbf {\ddot {r}} _{i}=G\sum _{j\neq i}m_{j}r_{ij}^{-2}{\hat {\mathbf {r} }}_{ij}}wheremj{\displaystyle m_{j}} is the mass of bodyj,rij{\displaystyle r_{ij}} is the distance between bodyi and bodyj,r^ij{\displaystyle {\hat {\mathbf {r} }}_{ij}} is the unit vector from bodyi towards bodyj, and the vector summation is over all bodies in the Solar System, besidesi itself.

In the special case where there are only two bodies in the Solar System, Earth and Sun, the acceleration becomesr¨Earth=GmSunrEarth,Sun−2r^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 Solar System are Moon and Earth the acceleration of the Moon becomesr¨Moon=GmEarthrMoon,Earth−2r^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 arer¨Sun=GmEarthrSun,Earth−2r^Sun,Earth+GmMoonrSun,Moon−2r^Sun,Moonr¨Earth=GmSunrEarth,Sun−2r^Earth,Sun+GmMoonrEarth,Moon−2r^Earth,Moonr¨Moon=GmSunrMoon,Sun−2r^Moon,Sun+GmEarthrMoon,Earth−2r^Moon,Earth{\displaystyle {\begin{aligned}\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}}}\\\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}}}\\\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}}}\end{aligned}}}

These accelerations are not those of Kepler orbits, and thethree-body problem is complicated. But Keplerian approximation is the basis forperturbation calculations. (SeeLunar theory.)

Position as a function of time

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Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of atranscendental equation calledKepler's equation.

The procedure for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of the timet sinceperihelion, is the following five steps:

  1. Compute themean motionn = (2π rad)/P, whereP is the period.
  2. Compute themean anomalyM =nt, wheret is the time since perihelion.
  3. Compute theeccentric anomalyE by solving Kepler's equation:M=E−εsin⁡E,{\displaystyle M=E-\varepsilon \sin E,} whereε{\displaystyle \varepsilon } is the eccentricity.
  4. Compute thetrue anomalyθ by solving the equation:(1−ε)tan2⁡θ2=(1+ε)tan2⁡E2{\displaystyle (1-\varepsilon )\tan ^{2}{\frac {\theta }{2}}=(1+\varepsilon )\tan ^{2}{\frac {E}{2}}}
  5. Compute the heliocentric distancer:r=a(1−εcos⁡E),{\displaystyle r=a(1-\varepsilon \cos E),} wherea{\displaystyle a} is the semimajor axis.

The position polar coordinates (r,θ) can now be written as a Cartesian vectorp=r⟨cos⁡θ,sin⁡θ⟩{\displaystyle \mathbf {p} =r\left\langle \cos {\theta },\sin {\theta }\right\rangle } and the Cartesian velocity vector can then be calculated asv=μar⟨−sin⁡E,1−ε2cos⁡E⟩{\displaystyle \mathbf {v} ={\frac {\sqrt {\mu a}}{r}}\left\langle -\sin {E},{\sqrt {1-\varepsilon ^{2}}}\cos {E}\right\rangle }, whereμ{\displaystyle \mu } is thestandard gravitational parameter.[32]

The important special case of circular orbit,ε = 0, givesθ =E =M. Because the uniform circular motion was considered to benormal, a deviation from this motion was considered an anomaly.

The proof of this procedure is shown below.

Mean anomaly,M

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Main article:Mean anomaly
Geometric construction for Kepler's calculation of θ. The Sun (located at the focus) is labeledS and the planetP. The auxiliary circle is an aid to calculation. Linexd is perpendicular to the base and through the planetP. The shaded sectors are arranged to have equal areas by positioning of pointy.

The Keplerian problem assumes anelliptical orbit and the four points:

  • s the Sun (at one focus of ellipse);
  • z theperihelion
  • c the center of the ellipse
  • p the planet

and

The problem is to compute thepolar 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|zsp|=ba⋅|zsx|.{\displaystyle |zsp|={\frac {b}{a}}\cdot |zsx|.}

Thecircular sector area|zcy|=a2M2.{\displaystyle |zcy|={\frac {a^{2}M}{2}}.}

The area swept since perihelion,|zsp|=ba⋅|zsx|=ba⋅|zcy|=ba⋅a2M2=abM2,{\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=nt,{\displaystyle M=nt,}wheren is themean motion.

Eccentric anomaly,E

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When the mean anomalyM is computed, the goal is to compute the true anomalyθ. The functionθ = f(M) is, however, not elementary.[33] Kepler's solution is to useE=∠zcx,{\displaystyle E=\angle zcx,}x as seen from the centre, theeccentric anomalyas an intermediate variable, and first computeE as a function ofM by solving Kepler's equation below, and then compute the true anomalyθ from the eccentric anomalyE. Here are the details.|zcy|=|zsx|=|zcx|−|scx|with|scx|=|cs|.|dx|2a2M2=a2E2−aε⋅asin⁡E2{\displaystyle {\begin{aligned}|zcy|&=|zsx|=|zcx|-|scx|\\with|scx|&={\frac {|cs|.|dx|}{2}}\\{\frac {a^{2}M}{2}}&={\frac {a^{2}E}{2}}-{\frac {a\varepsilon \cdot a\sin E}{2}}\end{aligned}}}

Division bya2/2 givesKepler's equationM=E−εsin⁡E.{\displaystyle M=E-\varepsilon \sin E.}

This equation givesM as a function ofE. DeterminingE for a givenM is the inverse problem. Iterative numerical algorithms are commonly used.

Having computed the eccentric anomalyE, 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 ofr would be√r · r.

True anomaly,θ

[edit]

Note from the figure that|cd|=|cs|+|sd|{\displaystyle |cd|=|cs|+|sd|}so thatacos⁡E=aε+rcos⁡θ.{\displaystyle a\cos E=a\varepsilon +r\cos \theta .}

Dividing bya{\displaystyle a} and inserting from Kepler's first lawra=1−ε21+εcos⁡θ{\displaystyle {\frac {r}{a}}={\frac {1-\varepsilon ^{2}}{1+\varepsilon \cos \theta }}}to getcos⁡E=ε+1−ε21+εcos⁡θcos⁡θ=ε(1+εcos⁡θ)+(1−ε2)cos⁡θ1+εcos⁡θ=ε+cos⁡θ1+εcos⁡θ.{\displaystyle \cos E=\varepsilon +{\frac {1-\varepsilon ^{2}}{1+\varepsilon \cos \theta }}\cos \theta ={\frac {\varepsilon (1+\varepsilon \cos \theta )+\left(1-\varepsilon ^{2}\right)\cos \theta }{1+\varepsilon \cos \theta }}={\frac {\varepsilon +\cos \theta }{1+\varepsilon \cos \theta }}.}

The result is a usable relationship between the eccentric anomalyE and the true anomaly θ.

A computationally more convenient form follows by substituting into thetrigonometric identity:tan2⁡x2=1−cos⁡x1+cos⁡x.{\displaystyle \tan ^{2}{\frac {x}{2}}={\frac {1-\cos x}{1+\cos x}}.}

Gettan2⁡E2=1−cos⁡E1+cos⁡E=1−ε+cos⁡θ1+εcos⁡θ1+ε+cos⁡θ1+εcos⁡θ=(1+εcos⁡θ)−(ε+cos⁡θ)(1+εcos⁡θ)+(ε+cos⁡θ)=1−ε1+ε⋅1−cos⁡θ1+cos⁡θ=1−ε1+εtan2⁡θ2.{\displaystyle {\begin{aligned}\tan ^{2}{\frac {E}{2}}&={\frac {1-\cos E}{1+\cos E}}={\frac {1-{\frac {\varepsilon +\cos \theta }{1+\varepsilon \cos \theta }}}{1+{\frac {\varepsilon +\cos \theta }{1+\varepsilon \cos \theta }}}}\\[8pt]&={\frac {(1+\varepsilon \cos \theta )-(\varepsilon +\cos \theta )}{(1+\varepsilon \cos \theta )+(\varepsilon +\cos \theta )}}={\frac {1-\varepsilon }{1+\varepsilon }}\cdot {\frac {1-\cos \theta }{1+\cos \theta }}={\frac {1-\varepsilon }{1+\varepsilon }}\tan ^{2}{\frac {\theta }{2}}.\end{aligned}}}

Multiplying by 1 + ε gives the result(1−ε)tan2⁡θ2=(1+ε)tan2⁡E2{\displaystyle (1-\varepsilon )\tan ^{2}{\frac {\theta }{2}}=(1+\varepsilon )\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 distancer from the true anomalyθ by Kepler's first law:r(1+εcos⁡θ)=a(1−ε2){\displaystyle r(1+\varepsilon \cos \theta )=a\left(1-\varepsilon ^{2}\right)}

Using the relation above betweenθ andE the final equation for the distancer is:r=a(1−εcos⁡E).{\displaystyle r=a(1-\varepsilon \cos E).}

See also

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Explanatory notes

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  1. ^In 1621, Johannes Kepler noted that Jupiter's moons obey (approximately) his third law in hisEpitome 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.)
  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]).)

References

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  1. ^ab"Kepler's Laws".hyperphysics.phy-astr.gsu.edu.Archived from the original on 2022-12-13. Retrieved2022-12-13.
  2. ^ab"Orbits and Kepler's Laws".NASA Solar System Exploration. 26 June 2008.Archived from the original on 2022-12-13. Retrieved2022-12-13.
  3. ^"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. Retrieved2022-12-13.
  4. ^"Nicolaus Copernicus".history.com. 9 November 2009.Archived from the original on 2022-12-13. Retrieved2022-12-13.
  5. ^abGingerich, 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. Retrieved27 July 2023.
  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 ...)
  7. ^abWilson, Curtis (May 1994)."Kepler's Laws, So-Called"(PDF).HAD News (31):1–2. RetrievedDecember 27, 2016.
  8. ^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; ... )
  9. ^Robert Small,An account of the astronomical discoveries of Kepler (London: J Mawman, 1804),pp. 298–299.
  10. ^Robert Small,An account of the astronomical discoveries of Kepler (London: J. Mawman, 1804).
  11. ^Bruce Stephenson (1994).Kepler's Physical Astronomy. Princeton University Press. p. 170.ISBN 978-0-691-03652-6.
  12. ^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.
  13. ^In hisAstronomia 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) ... ).
  14. ^abHolton, 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. RetrievedDecember 27, 2009.
  15. ^In hisAstronomia nova ... (1609), Kepler did not present his second law in its modern form. He did that only in hisEpitome 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.)
  16. ^abJohannes 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 quodproportio qua est inter binorum quorumcunque Planetarum tempora periodica, sit præcise sesquialtera proportionis mediarum distantiarum, ... " (But it is absolutely certain and exact that theproportion 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'sHarmonices Mundi is available as: Johannes Kepler with E. J. Aiton, A. M. Duncan, andJ. V. Field, trans.,The Harmony of the World (Philadelphia, Pennsylvania: American Philosophical Society, 1997); see especiallyp. 411.
  17. ^National Earth Science Teachers Association (9 October 2008)."Data Table for Planets and Dwarf Planets".Windows to the Universe. Retrieved2 August 2018.
  18. ^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.
  19. ^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])
  20. ^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.
  21. ^Roy Porter (1992).The Scientific Revolution in National Context. Cambridge University Press. p. 102.ISBN 978-0-521-39699-8.
  22. ^Victor Guillemin; Shlomo Sternberg (2006).Variations on a Theme by Kepler. American Mathematical Soc. p. 5.ISBN 978-0-8218-4184-6.
  23. ^Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second LawArchived 2019-09-11 at theWayback Machine",Wolfram Demonstrations Project. Retrieved December 27, 2009.
  24. ^Holton, Gerald; Brush, Stephen (2001).Brush and Holton - Physics: the Human Adventure. Princeton University Press. pp. 42–43.ISBN 978-0-8135-2908-0.
  25. ^Burtt, Edwin.The Metaphysical Foundations of Modern Physical Science. p. 52.
  26. ^Gerald James Holton, Stephen G. Brush (2001).Physics, the Human Adventure. Rutgers University Press. p. 45.ISBN 978-0-8135-2908-0.
  27. ^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.
  28. ^Caspar, Max (1993).Kepler. New York: Dover. p. 304.ISBN 978-0-486-67605-0.
  29. ^Caspar, Max (1993).Kepler. New York: Dover. p. 286.ISBN 978-0-486-67605-0.
  30. ^I. Newton,Principia, p. 408 in the translation of I.B. Cohen and A. Whitman
  31. ^I. Newton,Principia, p. 943 in the translation of I.B. Cohen and A. Whitman
  32. ^Schwarz, René."Memorandum № 1: Keplerian Orbit Elements → Cartesian State Vectors"(PDF).Archived(PDF) from the original on 5 January 2019. Retrieved4 May 2018.
  33. ^Müller, M (1995)."Equation of Time – Problem in Astronomy". Acta Physica Polonica A.Archived from the original on 14 November 2017. Retrieved23 February 2013.

General bibliography

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