3 Kepler's second law

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.

3.1 Proof of Kepler's second law:

Assuming Newton's laws of motion, we can show that Kepler's second law is consistent. By definition, the angular momentum of a point mass with mass and velocity is :

.

where is the position vector of the particle.

Since , we have:

taking the time derivative of both sides:

since the cross product of parallel vectors is 0. We can now say that is constant.

The area swept out by the line joining the planet and the sun, is half the area of the parallelogram formed by and .

Since is constant, the area swept out by is also constant. Q.E.D.

4 Kepler's third law (harmonic law)

The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis.

P² ~ a³

P = object's sidereal period in years

a = object's semimajor axis, in AU

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.

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.)

5 Applicability

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.

6 Application

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:

• c center of auxiliary circle and ellipse
• s sun (at one focus of ellipse);
• p the planet
• z perihelion
• x is the projection of the planet to the auxiliary circle; then
• y is a point on the circle such that

and three angles measured from perihelion:

• true anomaly , the planet as seen from the sun
• eccentric anomaly , x as seen from the center
• mean anomaly , y as seen from the center

Then

giving Kepler's equation

.

To connect E and T, assume then

and

which is ambiguous but useable. A better form follows by some trickery with trigonometric identities:

(So far only laws of geometry have been used.)

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

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]

7 Kepler's understanding of the laws

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:

where:

• P = object's sidereal period
• a = object's semimajor axis
• G = 6.67 × 10⁻¹¹ N · m²/kg² = the gravitational constant
• m₁ = mass of object 1
• m₂ = mass of object 2

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.

8 See also

• Circular motion
• Gravity
• Two-body problem

Celestial mechanics

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