The derivation of Kepler s first law, what are Kepler s first and second laws?

Proof of Kepler's first law.

The gravitational pull of the planet on the Sun is f=-(gmm r)r°

First of all, it is proved that the planets must move in the same plane, and there is Newton's second law: f=m(dv dt).

Moment r f=-(gmm r)r° r°=0.i.e. r (dv dt) = 0.

d(r×v)/dt=×v+r×dv/dt=0。

integral, rv=h (constant vector).

The above equation shows that the planetary radius vector r is always orthogonal to the constant vector h, so the planets must be moving in the same plane.

In order to derive the trajectory of the planetary motion, the plane polar coordinate direction in the figure is used.

Take the Sun at rest as the pole o and the planet position as (r, ) in the plane.

In polar coordinates, the physical quantities related to the motion of the planets are as follows:

Radial r=r·r° ; velocity v=dr dt=(dr dt)·r°+r· (d dt)·

r° is the radial unit vector, and ° is the radial vertical unit vector.

dr dt is the radial velocity component, r· (d dt) is the transverse velocity component.

The velocity size satisfies v =(dr dt) +r· (d dt))

Momentum mv=m(dr dt)+m( r·( d dt))

Angular momentum l=r mv=m r (d dt) (r°

get l=m r d dt).

The gravitational pull of the sun on the planet points to point o, so the moment of point o m=0, according to the angular momentum theorem, the angular momentum is conserved. l is constant.

The mechanical energy of the solar planetary system is conserved, and if the total energy of the system is e, then.

e=½mv²-gmm/r

Because dt=l mv dr dt= (l mv) (dr d) is substituted into the above equation.

l²/m²r²r²)(dr/dα)²l²/m²r=2e/m+2gm/r

The above two types are multiplied by m l, and you get.

dr²/dα²r²r²+1/r²=2me/l²+2mm²/l²r

To simplify the formula, let =1 rthen dr d = -r (d d ).

Then the equation becomes (dr d) 2gm m l = 2me l

The above equation is the derivative. Note that e and l are constants. Get.

2（dr/dα）（d²r/dα²）2ρ(dρ/dα)

gmm/l²(dρ/dα)=0

Kepler's first law was derived from mathematical derivation.

It is derived from the experience and theories of previous people, and then through actual observations.

Precise calculations were demonstrated.

Kepler's Law.

The contents and formulas of Kepler's three laws are as follows:

Kepler's First Law (Law of Orbit): Each planet orbits the Sun in an elliptical orbit, and the Sun is in a focal point of the ellipse.

Kepler's second law (area law): A straight line from the Sun to the planets sweeps the same area in equal time. It is expressed by the formula: sab=scd=sek.

The third law (periodic law): the square of the period of the revolutions of the planets around the Sun is proportional to the cube of the semi-major axis of their elliptical orbits. Formula: (r 3) (t 2) k (where k gm (4 2)).

Details:

In 1609, Kepler published two laws of planetary motion, one is Kepler's first law, also known as the orbital law, which states that the orbits of all planets around the sun are elliptical, and the sun is at a focal point of the ellipse.

Kepler's second law, also known as the law of area, is that for any planet, its line with the sun sweeps an equal area at the same time.

It is expressed by the formula: sab=scd=sek.

In 1619, Kepler discovered a third law, Kepler's third law, also known as the periodic law, which states that all planets have an equal ratio of the cubic of the semi-major axis of their orbits to the quadratic of the period of revolution.

The above content refers to: Encyclopedia-Kepler's law.

Kepler's first law, also known as the law of ellipse, the law of orbit, and the law of planets.

Each planet orbits the Sun in its own elliptical orbit, and the Sun is at one of the focal points of the ellipse. Kepler's first law was developed by the German astronomer Johannes Kepler.

Proposed. Before this law, it was believed that the orbits of celestial bodies were: "perfectly circular".

Kepler's law of the first spike is also known as the law of the ellipse, the law of orbit, and the law of the planets.

First Law: Each planet orbits the Sun in an elliptical orbit, and the Sun is at one focal point of the ellipse.

Kepler's first law is that the orbit of the planets around the Sun is elliptical, and the Sun is located at a focal point of this elliptical orbit.

What was the trajectory of the planets like before Kepler? At the time when geocentrism was popular, we all now know that geocentrism itself is wrong, and the earth is not the center of the universe.

If we want to verify that the trajectory of the planet is an ellipse, then we need to find the equation of the trajectory, that is, the orbit equation, and compare it with the equation of the ellipse to determine whether it is consistent. But to know the equation of the ellipse, which coordinate system is more appropriate? Considering that the ellipse is a regular closed curve, it is more appropriate to use a polar coordinate system.

Kepler's first law, also known as the elliptic law, the law of orbit: each planet orbits the Sun in its own elliptical orbit, and the Sun is in a tung oak focal point of the ellipse.

Kepler's statement in The Harmony of the Universe states that each planet orbits the Sun in its own elliptical orbit, and the Sun is in a focal point of the ellipse.

Kepler's first law was proposed by the German astronomer Johannes Kepler, who published two laws on planetary motion in his scientific journal New Astronomy in 1609, and in 1618, discovered the third law. Before this law was disturbed by the imaginary chakra, it was believed that the orbits of celestial bodies were "perfectly circular".

In astronomy and physics, Kepler's laws posed great challenges to the Aristotelians and Ptolemaics. Kepler asserted that the earth is constantly moving; The orbits of the planets are not circular, but elliptical; The speed at which the stars rotate is unequal. These arguments greatly shook the astronomy and physics of the time.

After almost a century of research, physicists have finally been able to explain principles using physical theories. Newton applied his second law and the law of universal gravitation to mathematically rigorously prove Kepler's law and give an understanding of its physical significance. Thus, Kepler's three laws of planetary motion changed the entire astronomy, completely destroyed Ptolemy's complex cosmic system, perfected and simplified Copernicus's heliocentric theory, and he became a key figure in the scientific revolution of the seventeenth century.

Kepler's first law, also known as the law of ellipse, the law of orbit, and the law of planets.

Each planet orbits the Sun in its own elliptical orbit, and the Sun is at one of the focal points of the ellipse. Kepler's first law was proposed by the German astronomer Johannes Liqu Kepler. Prior to this law, it was believed that the orbits of celestial bodies in defeat were:

Perfectly rounded".

1. Kepler's first law, also known as elliptic law: each planet orbits the sun in its own elliptical orbit, and the sun is in a focal point of the ellipse.

2. Kepler's second law, also known as the law of area: in equal time, the area swept by the line between the sun and the moving planets is equal. This law actually reveals the conservation of angular momentum around the Sun.

It is expressed by the formula k=a3 t2.

3. Kepler's third law, also known as the law of harmony: the square of the orbital period of each planet around the Sun is proportional to the cube of the semi-major axis of their elliptical orbits.

It is not difficult to derive from this law: the gravitational force between the planet and the sun is inversely proportional to the square of the radius. This is an important basis for Newton's law of universal induction.

Here, a is the semi-major axis of the planet's orbit, t is the planet's orbital period, and k is a constant.

Proof of Kepler's first law Let the masses of the sun and planets be m and m respectively, and the plane poles are taken as the standard system, and the positions of the planets are described by (r, ). As shown in the figure, the planet position vector is a vertical unit vector. The gravitational pull of the planet on the Sun is f=-(gmm r)r° First of all, it is proved that the planetary remnants must move in the same plane, and there is Newton's second law:

f=m(dv dt) moment r f=-(gmm r)r° r°=0i.e. r (dv dt) = 0. d(r×v)/dt=×v+r×dv/dt=0。

integral, r v=h (constant vector) The above equation shows that the planetary radius vector r is always orthogonal to the constant vector h, so the planets must be moving in the same plane. In order to obtain the trajectory of the planetary motion, the direction of the plane polar coordinates in the figure is adopted, the stationary sun is taken as the pole o, and the planetary position is (r, ) In the plane polar coordinates, the physical quantities related to the planetary motion are as follows: Radial r=r·r° ; velocity v=dr dt=(dr dt)·r°+r· (d dt)· r° is the radial unit vector, and ° is the radial vertical unit vector.

dr dt is the radial velocity component, r· (d dt) is the transverse speed or measurement component of the speed size satisfies v = (dr dt) +r· (d dt)) Momentum mv=m(dr dt)+m( r·( d dt)) angular momentum l=r mv=m r (d dt) (r° get l=m r d dt) The solar gravitational force experienced by the planet points to point o, so the moment for point o m=0, from the angular momentum theorem, the angular momentum is conserved. l is a constant The mechanical energy of the solar planetary system is conserved, and if the total vertical energy of the system is e, then e= mv -gmm r because dt=l mv dr dt= (l mv) (dr d ) is substituted into the above equation (l m r r) (dr d ) l m r = 2e m+2gm r The above two formulas are multiplied m l , and dr d r r +1 r =2me l +2mm l r In order to simplify the formula, Req =1 rThen dr d = -r (d d ) then the equation becomes (dr d ) 2gm m l =2me l and the above equation is the derivative of .

Note that e and l are constants. 2(dr d )(d r d )2 (d d ) gmm l (d d )=0 is mathematically derived, and Kepler's first law is derived.