Kepler's second law of planetary motion, also known as the area law, refers tosolar systemThe connecting line (vector diameter) between the middle sun and the moving planet sweeps an equal area in an equal time.[1]
This law was discovered by German astronomer Johannes KeplerKepler's lawone of.It was first published in the New Celestial Literature published in 1609, which also pointed out that the law was also applicable to other central motionsphere systems Medium.
Kepler's second law is a more accurate description of the orbits of planetsCopernicusOfHeliocentric theoryProvides strong evidence and providesNewtonLaterUniversal gravitationThe proof provides evidence and, together with the other two Kepler's laws, lays the foundation of classical astronomy.
Johannes Kepler's original statement in New Astronomy:In the same time, the area swept by the line between the sun and the moving planet is equal.[2]
It is expressed as:The line between the central celestial body and the surrounding celestial body (called vector diameter)[3]Sweep the same area in the same time.That is:
Kepler's second law
Where rmIs the position vector leading from the center of mass of the central celestial body to planet m, and vm is the velocity of planet m,
Is the angle between the planet speed v and the vector diameter r,
Is the rotation angle of m around the center of mass.
Formula derivation
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For the two body problem, as shown in Figure 1, the masses of the two bodies are recorded as m1 and m2, the distance between the two bodies is r, and the position vector from the two bodies to the origin is recorded as r1 and r2 in turn. Then, the gravitational action between the two bodies is:
Subtract the above formula, and record
The simplified equation is:
Both sides at the same time
, due to
, get
。
By integrating it, we can get:
, i.e
。
For a single object m2
The area element swept by m2 object in dt time is:
Figure 1 Formula derivation
The above equation can be proved by integrating.[4]
For m2, the angular velocity is:
It can be seen from the format that,This law is essentially a two body problem with a central force fieldangular momentumconservation。
For a planet in a large gravitational field, such as MercuryPerihelion precessionAt this time, Kepler's second law needsGeneral relativityAmend.Specifically:
In 1915, according to the general theory of relativity, Einstein regarded the motion of the planet around the sun as its motion in the solar gravitational field. Because of the curvature of the surrounding space caused by the mass of the sun, the perihelion precession of the planet every revolution is:
Where a is the long half axis of the planetary orbit, c is the speed of light, and e is the orbitEccentricity, T is the revolution period.For Mercury, calculate ε=43 ″/100 years.[5]
Explanation of laws
1. Kepler's second law is not only applicable to gravitational environment, but also applicable to all two body problems of central force field.[4]
2、
The constant values in are generally different for different celestial systems;For different celestial bodies,
The values of are generally different.
3. In the two body problem, two celestial bodies move around the common center of mass.If one of the celestial bodies is regarded as static, you can useReduced quality。
For m2 object, the ratio of its area velocity before and after using reduced mass simplification is
[6]
4. It can be found from this law that the Earth and other planets are fast at perihelion and slow at aphelion[7]
Further analysis
remember
，
reference resourcesKepler's first lawFor an elliptical orbit, the equation of motion is
Referring to the polar coordinate equation of the ellipse, we can get
[4]
Development History
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Danish astronomerTycho Brahe After his death, he left more than 20 years of observational data and a precise catalog.Tycho put forward a theory between geocentric theory and heliocentric theory, which was introduced into China in the 17th century and had a significant impact.Without an astronomical telescope, Tycho has observed the orientation of celestial bodies for decades. With amazing perseverance and patience, he has accumulated a large number of accurate materials. Kepler's discovery was obtained through induction and analysis of these materials.
Kepler believes that through careful mathematical analysis of Tycho's records, we can determine which planetary motion theory is correct: Copernicus heliocentric theory, ancientPtolemy Geocentric theoryOr the third theory proposed by Tycho himself.But after years of careful calculation and research, he found that these three theories were not consistent with Tycho's catalog and observation data.
Johannes Kepler
When Johannes Kepler could not explain Tycho's observation data with the existing theory of planetary motion, he resolutely gave up the idea that planets move in a uniform circle and tried to explain it with other geometric figures. After four years of hard thinking, that is, in 1609, he found that the ellipse was completely suitable for the requirements here and could make the same accurate explanation,So we got“Kepler's first law”: Mars edgeellipseOrbit around the sun, the sun is in one of the two focal points.
When Kepler continued his research, the "treacherous and multiterminal" Mars deceived him again.It turns out that Kepler and his predecessors studied planetary motion as constant velocity.He worked hard with this method for one year, but still could not get the result.Later, he found that the speed of Mars is uneven. When it is closer to the sun, it moves faster (perihelion), and when it is far from the sun, it moves slower (apohelion).
After Kepler found this problem, after precise and hard calculation, he found that the speed of the planet running in an elliptical orbit is not constant, but the area swept by the line between the planet and the sun is equal in the same time.This is the second law of planetary motion, also known as the "area law".[8]
These two laws were published in the New Astronomy (also known as On the Movement of Mars) published in 1609, which also pointed out that the two laws were also applicable to the movement of other planets and the moon.
application area
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Kepler's second law connects the coordinates and time of planets with orbital parameters, either in geometric language or in equations.It effectively solves the explanation of the laws of celestial body movement.In the study of the movement of celestial bodies, the effective combination of Newton's mechanics and Kepler's three laws can be used to predict the orbit, speed and rotation period of celestial bodies, so as to predict the position of celestial bodies in space at a certain time, and can be applied to celestial body detection, satellite launch and other fields.[9]
Law influence
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Once Kepler's law was established,This roundThe system collapsed completely, the movement of celestial bodies was no longer irregular, and Kepler's law became the "law" of the sky world.Later scholars honored Kepler as "the legislator of the sky".[9]
First, Kepler's law has an important influence on scientific thought.Its extremely brave spirit of creation and questioning inspired later scholars to be brave in innovation and questioning.
Secondly, the second law of Kepler and the first law of Kepler completely destroyed Ptolemy's epicycle system, liberated Copernicus system from the shackles of epicycle, and brought it full integrity and preciseness.From then on, no need to resort to any current round andEccentric circleThe motion of the planet can be calculated simply and accurately.[9]
Third, Kepler's law, including Kepler's second law, makes people's understanding of planetary motion clear.It proves that the planet world is a symmetrical and computable system.The sun is at one of the focal points of each planet's orbit.The orbital period of planets is determined by the distance between each planet and the sun, and has nothing to do with the mass.[9]
Fourth, Kepler's second law strongly proved the heliocentric theory, further overturned the creationism, carried forward the scientific spirit, and promoted the development of the times.It laid the foundation for Newton's proposition of universal gravitation and provided a powerful argument.
