Thanksgiving Kepler's Laws Essay
Deriving Kepler’s Laws
Tanner Morrison
November 16, 2012
Abstract
Johannes Kepler, a world renowned mathematician and astronomer, formulated three of today’s most influential laws of physics. These laws describe planetary motion around the sun. Deriving these laws (excluding Kepler’s First Law) will stress the concept of planetary motion, as well as provide a clear understanding of how these laws became relevant.
1
Kepler’s First Law
Kepler’s First Law states: The orbit of every planet is an ellipse with the Sun at one of the two foci.
2
Kepler’s Second Law
Kepler’s Second Law states: A line joining a planet and the Sun sweeps out equal areas during equal time intervals. In more simpler terms, the rate at
which …show more content…
The fundamental theorem of calculus states that the integral of the derivative is equal to the integrand,
T
T
dA =
0
h
2
dt
0
2
by simplifying we get the area of the planetary motion h T
2
A=
(7)
recall that A = πab, inputting this into our area equation we get πab =
h
T
2
Solving for the period (T), we get
2πab
h
T=
By squaring this period we get,
4 π 2 a 2 b2 h2 T2 =
(8)
2
Recall the directrix of an ellipse is (d = h ) and the eccentricity of an ellipse is c c
(e = GM ). Multiplying these together and simplifying we get ed =
h2 e h2 = eGM GM
(9)
Also recall that the square of half of the major axis of an ellipse is a2 = and the square of half of the minor axis is b2 =
√
Consider
√ a2 =
e2 d2
(1 −
e2 ) 2
e2 d 2
(1−e2 ) .
=a=
e2 d2
(1−e2 )2
Solving for a
ed
1 − e2
2
b a b2 e2 d2 (1 − e2 )
=
= ed a (1 − e2 ) ed
(10)
Equating equations (9) and (10) yields h2 b2
=
GM a Simplifying this we get h2 = recalling T 2 =
4π 2 a2 b2
,
h2
b2 GM a (11)
inserting the new found h we get
T2 =
4π 2 a2 b2 a
4π 2
Tanner Morrison
November 16, 2012
Abstract
Johannes Kepler, a world renowned mathematician and astronomer, formulated three of today’s most influential laws of physics. These laws describe planetary motion around the sun. Deriving these laws (excluding Kepler’s First Law) will stress the concept of planetary motion, as well as provide a clear understanding of how these laws became relevant.
1
Kepler’s First Law
Kepler’s First Law states: The orbit of every planet is an ellipse with the Sun at one of the two foci.
2
Kepler’s Second Law
Kepler’s Second Law states: A line joining a planet and the Sun sweeps out equal areas during equal time intervals. In more simpler terms, the rate at
which …show more content…
The fundamental theorem of calculus states that the integral of the derivative is equal to the integrand,
T
T
dA =
0
h
2
dt
0
2
by simplifying we get the area of the planetary motion h T
2
A=
(7)
recall that A = πab, inputting this into our area equation we get πab =
h
T
2
Solving for the period (T), we get
2πab
h
T=
By squaring this period we get,
4 π 2 a 2 b2 h2 T2 =
(8)
2
Recall the directrix of an ellipse is (d = h ) and the eccentricity of an ellipse is c c
(e = GM ). Multiplying these together and simplifying we get ed =
h2 e h2 = eGM GM
(9)
Also recall that the square of half of the major axis of an ellipse is a2 = and the square of half of the minor axis is b2 =
√
Consider
√ a2 =
e2 d2
(1 −
e2 ) 2
e2 d 2
(1−e2 ) .
=a=
e2 d2
(1−e2 )2
Solving for a
ed
1 − e2
2
b a b2 e2 d2 (1 − e2 )
=
= ed a (1 − e2 ) ed
(10)
Equating equations (9) and (10) yields h2 b2
=
GM a Simplifying this we get h2 = recalling T 2 =
4π 2 a2 b2
,
h2
b2 GM a (11)
inserting the new found h we get
T2 =
4π 2 a2 b2 a
4π 2