Kepler's laws

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Graphic representation of Kepler's laws. The Sun is located in one of the spotlights. In equal times, the areas sweeped by the planet are the same. Therefore, the planet will move faster near the Sun.

Kepler's laws were enunciated by Johannes Kepler to describe mathematically the motion of the planets in their orbits around the Sun.

First Law (1609): All planets move around the Sun describing elliptical orbits. The Sun is located in one of the hot spots of the ellipse.
Second Law (1609): The vector radio that unites a planet and the Sun runs equal areas in equal times.; The law of the areas is equivalent to the constancy of the angular moment, that is, when the planet is farther away from the Sun (felio) its speed is less than when it is closer to the Sun (perihelium).; The razor and the perihellio are the only two points of the orbit in which the radio vector and the speed are perpendicular. So only in those 2 points the module of the angular moment $L$ can be calculated directly as the product of the mass of the planet by its speed and distance to the center of the Sun.; ${displaystyle L=mcdot r_{a}cdot v_{a}=mcdot r_{p}cdot v_{p},}$; At any other point in the orbit other than the Afelio or the Perihelio the calculation of the angular moment is more complicated, since the speed is not perpendicular to the vector radio, the vector product must be used; ${displaystyle mathbf {L} =mcdot mathbf {r} times mathbf {v} ,}$
Third Law (1619): For any planet, the square of its orbital period is directly proportional to the cube of the length of the larger semieje of its elliptical orbit.; ${displaystyle {frac {T^{2}}{a^{3}}}=C={text{constante}}}$; Where, T It's him. orbital period (time it takes to turn around the Sun), athe average distance of the planet with the Sun and C the constant proportionality.; These laws apply to other astronomical bodies that are in mutual gravitational influence, such as the system formed by the Earth and the sun.

Newton's formulation of Kepler's third law

Before Kepler's laws were written, there were other scientists such as Claudius Ptolemy, Nicolaus Copernicus and Tycho Brahe whose main contributions to the advancement of science were in having achieved very precise measurements of the positions of the planets and stars. Kepler, who was a disciple of Tycho Brahe, took advantage of all these measurements to be able to formulate his third law.

Kepler was able to describe the motion of the planets. He used the mathematical knowledge of his time to find relationships between the data of astronomical observations obtained by Tycho Brahe and with them he managed to compose a heliocentric model of the universe. He began working with the traditional model of the cosmos, proposing eccentric trajectories and movements in epicycles, but he found that the observational data placed him outside the scheme established by Copernicus, which led him to conclude that the planets did not describe a circular orbit around of the Sun. He tried other forms for the orbits and found that the planets describe elliptical orbits, which have the Sun at one of their foci. Analyzing Brahe's data, Kepler also discovered that the speed of the planets is not constant, but that the radius vector that joins the Sun (located at one of the foci of the elliptical path) with a given planet describes equal areas in equal times. Consequently, the speed of the planets is greater when they are close to the Sun (perihelion) than when they move through the most distant areas (aphelion). This gives rise to Kepler's three Laws of planetary motion.

Kepler's laws represent a kinematic description of the solar system.

First Law of Kepler: All planets move around the Sun following elliptical orbits. The Sun is in one of the hot spots of the ellipse.

Second Law of Kepler: Planets move with constant areolar speed. I mean, the vector position r of each planet with respect to the Sun sweeps equal areas in equal times.

It can be shown that the angular momentum is constant which leads us to the following conclusions:

The orbits are flat and stable.

They always travel in the same sense.

The force that moves the planets is central.

Third Law of Kepler: It is fulfilled that for all planets, the reason between the period of revolution squarely and the greater semieje of the ellipse to the cube remains constant. This is:

{displaystyle {frac {T^{2}}{a^{3}}}=C}

Illustration of the relationship between orbital radio and orbital period.

Newton's study of Kepler's laws led to his formulation of the law of universal gravitation.

Newton's mathematical formulation of Kepler's third law for circular orbits is:

The gravitational force creates the necessary centripetal acceleration for circular motion of radius a:

{displaystyle {frac {GMm}{a^{2}}}=momega ^{2}a}

remembering the expression that relates the angular velocity and the period of revolution:

{displaystyle omega ={frac {2pi }{T}}}

from which it can be deduced that the square of the time of a complete orbit or period is:

{displaystyle T^{2}={frac {4pi ^{2}}{GM}}a^{3}}

and clearing:

{displaystyle {frac {T^{2}}{a^{3}}}={frac {4pi ^{2}}{GM}}=C}

where $C$ It's Kepler's constant, TIt's the orbital period, a the major semieje of the orbit, M is the mass of the central body and Ga constant called Universal Gravitation Constant whose value marks the intensity of the gravitational interaction and the system of units to be used for the other variables of this expression. This expression is valid for both circular and elliptical orbits.

Actually, this last expression is only an approximation of the more general expression that can be deduced with all rigor from Newton's Laws and which is:

{displaystyle {frac {T^{2}}{a^{3}}} (M+m)={frac {4pi ^{2}}{G}}}

Where $M$ is the mass of the central body, $m$ the star that revolves around him and $a$ would be the major semieje regarding the mass center of the system. As in the Solar System the mass of the Sun is far superior to that of any planet, ${displaystyle mll M}$ , simplified expression is obtained from the most general by ${displaystyle M+msimeq M}$

Mathematical deduction of Kepler's laws from Newton's laws

The proof of Kepler's first and second laws is based on Newton's laws and the universal law of gravitation.

Polar coordinate reference system

Proof of Kepler's second law

Enunciado mathematician
Sean. ${displaystyle {[t_{1},t_{2}]}}$ , ${displaystyle {[t_{3},t_{4}]}}$ two time intervals such that ${displaystyle {t_{2}-t_{1}=t_{4}-t_{3}}}$ and be ${displaystyle A_{theta _{1},theta _{2}}={1 over 2}int limits _{theta _{1}}^{theta _{2}}r^{2}(theta)dtheta }$ , ${displaystyle forall theta _{1},theta _{2}}$ .
So, ${displaystyle {A(t_{2}-t_{1})=A(t_{4}-t_{3})}}$ .

First, a polar coordinate reference system is set:

{textstyle {overrightarrow {x}}(t)=r(t)cdot (cos(theta (t)),sen(theta (t)))}

{displaystyle {overrightarrow {u_{r}}}(t)=(cos(theta (t)),sen(theta (t)))}

{displaystyle {overrightarrow {u_{theta }}}(t)=(-sen(theta (t)),cos(theta (t)))}

where ${displaystyle {overrightarrow {x}}(t)}$ denotes the position of the body with mass $m$ in the instant $t$ ; body with mass $M$ is still and at the origin; and ${displaystyle {overrightarrow {u_{r}}}}$ and ${displaystyle {overrightarrow {u_{theta }}}}$ are unitary vectors in radial and circumferential directions, respectively; and ${displaystyle theta (t)}$ is the shaped angle ${displaystyle r(t)}$ with the polar axis (reference from which the polar angle is measured).

${displaystyle {overrightarrow {u_{r}}}}$ and ${displaystyle {overrightarrow {u_{theta }}}}$ satisfy the following properties:

{displaystyle {d{overrightarrow {u_{r}}} over dtheta }={overrightarrow {u_{theta }}}}

;

{displaystyle quad {d{overrightarrow {u_{r}}} over dt}={overrightarrow {u_{r}}}'={overrightarrow {u_{theta }}}theta '}

;

{displaystyle quad {overrightarrow {u_{r}}}bot {overrightarrow {u_{theta }}}}

The force ${displaystyle {overrightarrow {F}}}$ on the body of mass $m$ is broken down into: ${displaystyle {overrightarrow {F}}=F_{r}{overrightarrow {u_{r}}}+F_{theta }{overrightarrow {u_{theta }}}}$ . Plus, like ${displaystyle {overrightarrow {F}}}$ It's a central force, ${displaystyle F_{theta }equiv 0}$ .

Therefore, applying Newton's second law,

{displaystyle F_{r}{overrightarrow {u_{r}}}={overrightarrow {F}}=m{overrightarrow {x}}''(t)}

{displaystyle [1]}

The speed of the planet is the derivative of the position:

{displaystyle x'(t)=r'(t){overrightarrow {u_{r}}}+r(t)theta '(t){overrightarrow {u_{theta }}}}

{displaystyle [2]}

and its acceleration is the derivative of velocity:

{textstyle {begin{aligned}x''(t)&=r''(t){overrightarrow {u_{r}}}+r'(t)theta '(t){overrightarrow {u_{theta }}}+r'(t)theta '(t){overrightarrow {u_{theta }}}+r(t)theta ''(t){overrightarrow {u_{theta }}}-r(t)theta '(t)theta '(t){overrightarrow {u_{r}}}\&=(r''(t)-r(t)(theta '(t))^{2}){overrightarrow {u_{r}}}+(2r'(t)theta '(t)+r(t)theta ''(t)){overrightarrow {u_{theta }}}.\end{aligned}}}

{displaystyle [3]}

Using ${displaystyle [1]}$ and ${displaystyle [3]}$ :

{displaystyle {begin{cases}F_{r}=[r''(t)-r(t)(theta '(t))^{2}]m\0=2r'(t)theta '(t)+r(t)theta ''(t)end{cases}}}

${displaystyle {begin{matrix}[4]\{[5]}end{matrix}}}$

Multiplying by $r$ on both sides ${displaystyle [5]}$ :

{displaystyle 0=2r(t)r'(t)theta '(t)+r^{2}(t)theta ''(t)=(r^{2}(t))'theta '(t)+r^{2}(t)theta ''(t)=(r^{2}(t)theta '(t))'}

So ${displaystyle r^{2}(t)theta '(t)=c}$ (constant). ${displaystyle [6]}$

Representation of the area of the sweeping sector between the angles

{displaystyle theta _{1}}

and

{displaystyle theta _{2}}

On the other hand, ${displaystyle A_{theta _{1},theta _{2}}}$ the area of the sweeping sector between the angles ${displaystyle theta _{1}}$ and ${displaystyle theta _{2}}$ :

{displaystyle A_{theta _{1},theta _{2}}={1 over 2}textstyle int limits _{theta _{1}}^{theta _{2}}displaystyle r^{2}(theta)dtheta }

Let's take it. ${displaystyle theta _{1}=0}$ for simplicity and denote ${displaystyle A_{theta _{1},theta _{2}}=A}$ .

By the fundamental theorem of the calculation, ${displaystyle {dA over dtheta }={r^{2}(theta) over 2}}$ . Like $theta$ is function of $t$ For ${displaystyle [6]}$ :

{displaystyle A'(t)={dA(theta (t)) over dtheta }theta '(t)={r^{2}(theta (t)) over 2}theta '(t)=c}

(constant).

So, ${displaystyle A(t)=ct+k}$ For some constant $k$ .

You get: ${displaystyle A(t)=ct}$ .

Applying this to two time intervals of equal length, ${displaystyle [t_{1},t_{2}]}$ and ${displaystyle [t_{3},t_{4}]}$ :

{displaystyle A(t_{2}-t_{1})=(t_{2}-t_{1})c=(t_{4}-t_{3})c=A(t_{4}-t_{3})}

. ■

Proof of Kepler's First Law

Enunciado mathematician
The trajectory of the mass body m It's a conical.

Assuming that $F_r$ comply with the universal law of gravitation:

{displaystyle F_{r}={-{GmM over r^{2}(t)}}}

{displaystyle [7]}

Applying Newton's second law and the law of universal gravitation:

{displaystyle {overrightarrow {x''}}(t)=-{Bigl (}G{M over r^{2}(t)}{Bigr)}{overrightarrow {u_{r}}}}

{displaystyle [8]}

Just like ${displaystyle [4]}$ and ${displaystyle [7]}$ :

{displaystyle -{GM over r^{2}(t)}=r''(t)-r(t)(theta '(t))^{2}}

{displaystyle [9]}

Clearing of the equation ${displaystyle [6]}$ is obtained: ${displaystyle theta '(t)={c over r^{2}(t)}}$ For c constant. ${displaystyle [10]}$

You can rewrite the equation ${displaystyle [9]}$ using ${displaystyle [10]}$ like:

{displaystyle -{GM over r^{2}(t)}=r''(t)-{c^{2} over r^{3}(t)}}

{displaystyle [11]}

Making change ${displaystyle z(t)={1 over r(t)}}$ and drifting twice, we get the following:

{displaystyle r'(t)=-{1 over z^{2}(t)}z'(t)=-{1 over z^{2}(t)}{dz(t) over dtheta (t)}theta '(t)=-c{dz(t) over dtheta (t)}}

{displaystyle r''(t)=-c{d over dt}{biggl (}{dz(t) over dtheta (t)}{Biggr)}=-c{d^{2}z(t) over dtheta ^{2}(t)}theta '(t)}

Using ${displaystyle [10]}$ :

{displaystyle r''(t)=-c^{2}z^{2}(t){d^{2}z(t) over dtheta ^{2}(t)}}

Replacement ${displaystyle r''(t)}$ on the right side ${displaystyle [11]}$ :

{displaystyle r''(t)-{c^{2} over r^{3}(t)}=-c^{2}z^{2}(t){d^{2}z(t) over dtheta ^{2}(t)}-{c^{2} over r^{3}(t)}=-c^{2}z^{2}(t){d^{2}z(t) over dtheta ^{2}(t)}-c^{2}z^{3}(t)}

and on the left side applying the change again ${displaystyle z(t)={1 over r(t)}}$ :

{displaystyle -{GM over r^{2}(t)}=-GMz^{2}(t)}

In this way, the equation ${displaystyle [9]}$ you can write as:

{displaystyle z+{d^{2}z over dtheta ^{2}}={GM over c^{2}}}

This differential equation has the only solutions:

{displaystyle z(theta)=Acos(theta -alpha)+{GM over c^{2}}}

Where

A

and

alpha

They're constant.

Choosing the polar axis so ${displaystyle alpha =0}$ :

{displaystyle r(theta)={1 over Acos(theta)+{GM over c^{2}}}={{c^{2} over GM} over Bcos(theta)+1}}

Where

{displaystyle B={Ac^{2} over GM}}

Making changes ${displaystyle e=B}$ and ${displaystyle p={1 over A}}$ , you get the equation of a conic with focus on the origin:

{displaystyle r(theta)={ep over ecos(theta)+1}}

Where

e

is eccentricity and

p

is the distance from the focus to the guideline.

According to the value of $e$ , this conic can be an ellipse, a hyperbola or a parable.

In this case, if the trajectory of the celestial bodies is covered, the only possible case is that it is an ellipse, $<math alttext="{displaystyle 0<e0.e.1{displaystyle 0 ingredient1} <img alt="{displaystyle 0<e$ . ■

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Kepler's laws

Newton's formulation of Kepler's third law

Mathematical deduction of Kepler's laws from Newton's laws

Proof of Kepler's second law

Proof of Kepler's First Law

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