1. Kepler’s Laws & Planetary Motion

2. “Laws” of Planetary Motion.
Johannes Kepler
German astronomer (1571 – 1630)
Spent most of his career tediously analyzing huge amounts of observational data (most compiled by Tycho Brahe) on planetary motion (orbit periods, orbit radii, etc.)
Used his analysis to develop
“Laws” of Planetary Motion.
Kepler’s “Laws” are Laws only in the sense that they agree with observation. They are not true theoretical laws, such as Newton’s Laws of Motion & Newton’s Universal Law of Gravitation.

3. Newton’s Universal Gravitation Law
Kepler studied results of other astronomer’s measurements of portions of the Moon, planets, etc.
He found the motion of the Moon and planets could be described by a series of laws
Now called Kepler’s Laws of Planetary Motion
Kepler’s Laws are mathematical rules inferred from the available information about the motion in the solar system
Kepler could not give a scientific explanation or derivation of his laws.
Newton’s Laws of Motion &
Newton’s Universal Gravitation Law
give the explanation

4. Kepler’s “Laws” Kepler’s First Law Kepler’s Second Law
Kepler’s “Laws” are consistent with & are obtainable from Newton’s Laws
Kepler’s First Law
All planets move in elliptical orbits with the Sun at one focus
Kepler’s Second Law
The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals
Kepler’s Third Law
The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit

5. Math Review: Ellipses
The points F1 & F2 are each a focus of the ellipse
Located a distance c from the center
Sum of r1 and r2 is constant
The longest distance through center is The major axis, 2a
a is called the semimajor axis
Shortest distance through center is The minor axis, 2b
b is called the semiminor axis 
Typical Ellipse
The eccentricity is defined as e = (c/a)
For a circle, e = 0
The range of values of the eccentricity for ellipses is 0 < e < 1
The higher the value of e, the longer and thinner the ellipse

6. Ellipses & Planet Orbits
The Sun is at One Focus
Nothing is located at the other focus
The Aphelion is the point farthest away from the Sun
The distance for the aphelion is a + c
For an orbit around the Earth, this point is called the apogee
The Perihelion is the point nearest the Sun
The distance for perihelion is a – c
For an orbit around the Earth, this point is called the perigee

7. All planets move in elliptical orbits with the Sun at one focus
Kepler’s 1st Law
All planets move in elliptical orbits
with the Sun at one focus
A circular orbit is a special case of an elliptical orbit
The eccentricity of a circle is e = 0.
Kepler’s 1st Law can be shown (& was by Newton) to be a direct result of the inverse square nature of the gravitational force. Comes out of N’s 2nd Law + Gravitation Law + Calculus
Elliptic (and circular) orbits are allowed for bound objects
A bound object repeatedly orbits the center
An unbound object would pass by and not return
These objects could have paths that are parabolas
(e = 1) and hyperbolas (e > 1)

8. Kepler’s First Law of Planetary Motion
The Planets Move in Elliptical Orbits
The Sun is at one focus
This was very different from the previous idea that the planets moved in perfect circles with the Sun at the center

9. Orbit Examples
Fig. (a): Mercury’s orbit has the largest eccentricity of the planets. eMercury = 0.21
Note: Pluto’s eccentricity is ePluto = 0.25, but, as of 2006, it is officially no longer classified as a planet!
Fig. (b): Halley’s Comet’s orbit has high eccentricity: eHalley’s comet = 0.97
Remember that nothing is located at the second focus.

10. Kepler’s First Law: Planetary Orbits

11. Kepler’s 2nd Law
The vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals.
Kepler’s 2nd Law can be shown (& was by Newton) to be a direct result of Newton’s Gravitation Law.

12. Kepler’s 2nd Law of Planetary Motion
A line connecting a planet to the sun sweeps out equal areas in equal times as the planet moves around its orbit
If the time required for the planet to sweep out area A1 is equal to the time to sweep out A2, the areas will be equal
The planet’s speed will be slowest when it is farthest from its sun and fastest when it is closest

13. axis a of the elliptical orbit
Kepler’s 3rd Law
The square of the orbital period T of any planet is proportional to the cube of the semimajor
axis a of the elliptical orbit
If the orbit is circular & of radius r, this follows from
Newton’s Universal
Gravitation Law.
This gravitational force supplies a centripetal force for use in
Newton’s 2nd Law
Ks is a constant
Ks is a constant,
which is the same for
all planets.

14. Kepler’s 3rd Law
It can be shown: This also applies to an elliptical orbit, with replacement of r with a, where a is the semimajor axis.
Ks is independent of the planet mass, & is valid for any planet
Note: If an object is orbiting another object, the value of K will depend on the mass of the object being orbited. For example, for the Moon’s orbit around the Earth, KSun is replaced with KEarth, where KEarth is obtained by replacing MSun by MEarth in the above equation.

15. Orbit Examples Low Earth orbit Geosynchronous Orbit
International Space Station, for example
T ~ 90 minutes
r ~ 6.66 x 106 m
Geosynchronous Orbit
T = 1 day
Always above the same position above the Earth
r = 4.2 x 107 m

16. Solar System Data
Kepler’s 3rd Law: The ratio of the square of a planet’s orbital period is proportional to the cube of its mean distance from the Sun.