Kepler’s Laws & 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.
• 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
Kepler’s “Laws”
• 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
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
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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
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
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)
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
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.
Kepler’s First Law: Planetary Orbits
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.
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
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.
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.
Orbit Examples
• Low Earth 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
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.