Resource Lesson
Kepler's Laws
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Remember in our earlier lesson on universal gravitation, that the speed of a satellite in circular orbit is constant. In this lesson we will examine the properties of satellites in elliptical orbits satellites which have variable speeds since their distance from the earth is constantly changing.
These three empirical laws were published by Johannes Kepler (1571-1630). His research was based on the extensive data compiled by Tycho Brahe (1546-1601) on Mars.
Kepler's 1st Law: The Law of Elliptical Orbits
Each planet travels in an elliptical orbit with the sun at one focus.
Here are some properties and vocabulary about ellipses that you should remember.
When the planet is located at point P it is at the
perihelion position
.
R
_{P}
= distance from Sun to P and is called the
perihelion radius
.
When the planet is located at point A it is at the
aphelion position
.
R
_{A}
= distance from Sun to A and is called the
aphelion radius
.
The distance
PA
= R
_{P}
+ R
_{A}
is called the
major axis
which is represented mathematically in formulas as 2a.
The perpendicular bisector of the major axis is called the
minor axis
and its length is represented mathematically in formulas as 2b.
The
center of the ellipse
is where the major and minor axes cross each other.
The distance from the Sun (one of the focii) to the center of the ellipse is called the
interfocal radius
and is represented mathematically in formulas as c. This distance can be calculated with the forumla: c = ½(R
_{A}
-R
_{P}
).
The
average orbital radius
is called R
_{AV}
= a = ½ (R
_{A}
+ R
_{P}
)
R
_{A}
= a + c and R
_{P}
= a - c
Eccentricity
is a measure of how "oval" an ellipse is. It is mathematically expressed as the ratio of e = c / a.
Kepler’s 2nd Law: The Law of Equal Areas
A line from the planet to the sun sweeps out equal areas of space in equal intervals of time.
At the perihelion, the position closest to the sun along the planet’s orbital path, the planet’s speed is maximal.
At the aphelion, the position farthest from the sun along the planet’s orbital path, the planet’s speed is minimal.
Thus, the satellite’s speed is inversely proportional to its average distance from the sun.
v
_{A}
R
_{A}
= v
_{P}
R
_{P}
_{ }
Kepler’s 3rd Law: The Law of Periods
The square of a planet’s orbital period is directly proportional to the cube if its average distance from the sun.
T
^{2}
~ R
_{av}
^{3}
T
^{2}
/ R
_{av}
^{3}
= constant
Since the orbits of the planets in our solar system are EXTREMELY close to being circular in shape (the Earth's eccentricity is 0.0167), we can set the centripetal force equal to the force of universal gravitation and,
F
_{C}
= F
_{G}
m
_{planet}
v
^{2}
/ R = GM
_{Sun}
m
_{planet}
/ R
^{2}
(2 π R / T)
^{2}
= GM
_{Sun}
/ R
4 π
^{2 }
R
^{2}
/ T
^{2}
= GM
_{Sun}
/ R
4 π
^{2 }
R
^{3}
= GM
_{Sun}
T
^{2}
R
^{3}
/ T
^{2}
= GM
_{Sun}
/ 4 π
^{2}
Thus Kepler’s constant for our solar system equals GM
_{Sun}
/ 4 π
^{2}
In general, for any system of satellites, the ratio of T
^{2}
/ R
_{av}
^{3}
equals a constant for that system. Thus, for any group of satellites, the ratio of T
^{2}
/ R
_{av}
^{3}
will be the same. This is one way to determine if two satellites are in orbit about the same central body.
The general rule of thumb used when working with satellites in low Earth orbit is to take their orbital velocity to be 8 km/sec and their orbital period to be 90 minutes. When satellites travel faster than 8 km/sec, then their orbits are no longer circular, but instead become more and more elliptical until their speeds reach 11.2 km/sec. At this speed, called the escape velocity of the Earth, they can no longer orbit the Earth and their orbits become hyperbolic. Achieving this escape velocity does not release them from being members of our solar system, they are just no longer satellites of the Earth.
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