Kepler's laws of planetary motion Guide, Meaning , Facts, Information and Description
Johannes Kepler's primary contribution to astronomy/astrophysics were the three laws of planetary motion. Kepler derived these laws, in part, by studying the observations of Brahe. Isaac Newton would later design his laws of motion and universal gravitation and verify that Kepler's laws could be derived from them. The generic term for an orbiting object is "satellite".
There is no object at the other focus of a planet's orbit. The semimajor axis, a, is the average distance between the planet and its star.
Newton proposed that "every object in the universe attracts every other object along a line of the centers of the objects proportinal to each objects mass, and inversely proportional to the square of the distance between the objects."
We begin with Newton's law F=ma:
Kepler's laws of planetary motion
Kepler's first law
The orbit of a planet about a star is an ellipse with the star at one focus. Proof of Kepler's first law
Here we express F as the product of its magnitude and its direction. Recall that in polar coordinates:
In component form we have:
Now consider the angular momentum:
This differential equation has the general solution:
- .
A line joining a planet and its star sweeps out equal areas during equal intervals of time.
This is also known as the law of equal areas. Suppose a planet takes 1 day to travel from points A to B. During this time, an imaginary line, from the Sun to the planet, will sweep out a roughly triangular area. This same amount of area will be swept every day.
As a planet travels in its elliptical orbit, its distance from the Sun will vary. As an equal area is swept during any period of time and since the distance from a planet to its orbiting star varies, one can conclude that in order for the area being swept to remain constant, a planet must vary in velocity. Planets move fastest when at perihelion and slowest when at aphelion.
This law was developed, in part, from the observations of Brahe that indicated that the velocity of planets was not constant.
This law corresponds to the angular momentum conservation law in the given situation.
By definition, the angular momentum of a point mass with mass and velocity is :
Since , we have:
The area swept out by the line joining the planet and the sun, is half the area of the parallelogram formed by and .
The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis.
See the actual figures: attributes of major planets.
Newton would modify this third law, noting that the period is also affected by the orbiting body's mass, however typically the central body is so much more massive that the orbiting body's mass may be ignored. (See below.)
The laws are applicable whenever a comparatively light object revolves around a much heavier one because of gravitational attraction. It is assumed that the gravitational effect of the lighter object on the heavier one is negligible. An example is the case of a satellite revolving around Earth.
Assume an orbit with semimajor axis a, semiminor axis b, and eccentricity ε. To convert the laws into predictions, Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points:
Then
Note that is the area swept since perihelion; by the second law, that is proportional to time since perihelion. But we defined and so M is also proportional to time since perihelion—this is why it was introduced.
We now have a connection between time and position in the orbit. The catch is that Kepler's equation cannot be rearranged to isolate E; going in the time-to-position direction requires an iteration (such as Newton's method) or an approximate expression, such as
Kepler did not understand why his laws were correct; it was Isaac Newton who discovered the answer to this more than fifty years later. Newton, understanding that his third law of motion was related to Kepler's third law of planetary motion, devised the following:
Kepler's second law
Proof of Kepler's second law:
where is the position vector of the particle.
taking the time derivative of both sides:
since the cross product of parallel vectors is 0. We can now say that is constant.
Since is constant, the area swept out by is also constant. Q.E.D.Kepler's third law (harmonic law)
Thus, not only does the length of the orbit increase with distance, also the orbital speed decreases, so that the increase of the sidereal period is more than proportional.Applicability
Application
and three angles measured from perihelion:
giving Kepler's equation
To connect E and T, assume then
which is ambiguous but useable. A better form follows by some trickery with trigonometric identities:
(So far only laws of geometry have been used.)
via the Lagrange reversion theorem. For the small ε typical of the planets (except Pluto) such series are quite accurate with only a few terms; one could even develop a series computing T directly from M.[1]Kepler's understanding of the laws
where:
Astronomers doing celestial mechanics often use units of years, AU, G=1, and solar masses, and with m2<<m1, this reduces to Kepler's form. SI units may also be used directly in this formula.