Kepler's Third Law

The third law, the Law of Harmonies, relates the period of a planet's orbit, T, to the length of its average radius. It states that the square of the period of the orbit T is proportional to the cube of the radius r, and that every planet around the same Sun has the same constant of proportionality.


The proof begins with dA/dt = l/(2 m), the expression to represent the area swept out by the radius of the planet's orbit.
Multiply both sides by dt to get dA = [l/(2 m)] dt.
This can be integrated to get an expression relating the total area of the orbit to the period of the orbit:
A = [l/(2 m)] T.
Square both sides and solve for T^2:
T^2 = 4 (m2/l2) A^2.

The area A of an ellipse is pi a b, where a is the length of the semimajor axis and b the length of the semiminor axis. Thus our expression of T^2 becomes:
T^2 = 4 pi^2 (m2/l2) a^2 b^2.

b is related to a and c, the distance between the focus and the center, by a2 = b2 + c2, 

so b^2 = a^2 - c^2: T2 = 4 pi^2 (m^2/l^2) a^2 (a^2 - c^2).

Since c = a e, we find that
T^2 = 4 pi^2 (m^2/l^2) a^2 (a^2 - a^2 e^2)
T^2 = 4 pi^2 (m^2/l^2) a^2 [a^2 (1 - e^2)]
T^2 = 4 pi^2 (m^2/l^2) a^4 (1 - e^2).

Then, a property of ellipse geometry is used which indicates that e p = a (1 - e2).

Factor out a (1 - e^2) from our expression of T^2 and replace it with e p .

T^2 = 4 pi^2 (m^2/l^2) a^3 [a (1 - e^2)]
T^2 = 4 pi^2 (m^2/l^2) a^3 (e p).

T2 is proportional to a3. Kepler's first law shows that e p = l2/(G m2 M). Substitute this into our expression for T^2 to find that T^2 = 4 pi^2 (m^2/l^2) l^2/(G m^2 M) a^3. The m2 and l2 cancel, and we are left with T^2 = (4 pi^2)/(GM) a^3. (4 pi^2)/(GM) is the constant of proportionality. It depends on G (which represents the universal constant of gravitational force) and M, the mass of the sun around which the planets orbit. As a result, the square of the period of a planet is proportional to the cube of the length of the semimajor axis (radius), and this rule applies to all planets and satellites with complete elliptical orbits.

The above proof is adapted from:

"Francis, Erik M. "Kepler's Laws: Kepler's Third Law." Kepler's Laws: Kepler's Third Law. Alcyone Systems, 2015. Web. 22 Apr. 2015. <http://www.alcyone.com/max/physics/kepler/6.html>."

"The Apple, the Moon, and the Inverse Square Law." The Physics Classroom. N.p., n.d. Web. 22 Apr. 2015. <http://www.physicsclassroom.com/class/circles/Lesson-3/The-Apple,-the-Moon,-and-the-Inverse-Square-Law>.

Proof of Second and Third Laws

 

from Youtube user mathymadison

Kepler's Second Law

Kepler's Law of Equal Areas states the radius of a planet's orbit (as drawn between itself and the Sun) will cover equal areas of space in equal amounts of time.

The blue section maintains the same area.

Depending on where the planet is in its orbit, its radius and its angular velocity is subject to change due to the gravitational pull between it and the Sun.

The planet will travel at a higher velocity as it approaches the Sun, causing it to cover more distance over time, and it will travel more slowly when it is farther away from the Sun, causing it to cover less distance over time.

In the time dt, the planet clears a triangle having the base r and height r dƟ with an area  dA=1/2 * r *rd(Ɵ).
From this, the  constant areal velocity can be written as dA/dt=1/2 r^2 dƟ/dt.
The area enclosed by the elliptical orbit is πab.
So the period (P) works with P * 1/2 r^2 dƟ/dt = πab.
The average motion of the planet as it orbits the Sun is  n= 2π/P and it works with r^2 dƟ= abn dt.



Fowler, Michael. "Deriving Kepler’s Laws from the Inverse-Square Law." Galileo. University of Virginia, n.d. Web. 22 Apr. 2015. <http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/KeplersLaws.htm>.

"Kepler's Laws of Planetary Motion." Wikipedia. Wikimedia Foundation, n.d. Web. 22 Apr. 2015. <http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#The_inverse_square_law>.

Newton’s Proof of Kepler’s Second Law (n.d.): 1-2. Department of Mathematics. University of Kansas. Web. 22 Apr. 2015. <http://www.math.ku.edu/~lerner/m500f09/NewtonKepler.pdf>.

Kepler's First Law

To put it briefly, The Law of Ellipses states that the orbit of a planet traces an elliptical path. All ellipses have two focii, and in this case, one of them is the Sun.

image from UStudy Initiative

     He came to this conclusion after studying Tycho Brahe's observations on Mars. Conventional thought was that all planetary orbits were circular. Kepler's circular model caused the satellite to move too fast at aphelion and perihelion and then move too slowly at the sides.

     As a way to fix this problem, Kepler chose to alter the shape of the orbit. He was able to find the right information from Brahe's observations to support his mathematical calculations and prove that the orbit was elliptical. He extended this theory to the orbits of the other planets and gave evidence to support his assertions after doings thousands of pages of arithmetic.

     Kepler did not yet have calculus to prove this, so he worked through it another way.
He did this by first locating the equant point, the place in the middle of the two focii from which all of the planets appear to have the same angular velocity (rate of change of the angle of displacement of an object in its orbit). It is expressed in this manner:

Sir Isaac Newton later would return to Kepler's laws to apply them to calculus, applying inverse square law to help explain his work.

Mathematically formulated: Intensity ∝ 1/(distance ^2)

α

Reference https://www.physicsforums.com/threads/what-does-this-symbol-mean.225342/

α

Reference https://www.physicsforums.com/threads/what-does-this-symbol-mean.225342/

--and it can actually be used in many applications of physics, including gravity.

image from Physics Classroom

"The Apple, the Moon, and the Inverse Square Law." The Physics Classroom. N.p., n.d. Web. 22 Apr. 2015. <http://www.physicsclassroom.com/class/circles/Lesson-3/The-Apple,-the-Moon,-and-the-Inverse-Square-Law>.

Hwang, Jenny. "Kepler and the First Law of Planetary Motion." Kepler and the First Law of Planetary Motion. Berkeley University, n.d. Web. 22 Apr. 2015. <https://math.berkeley.edu/~robin/Kepler/>.

Knisley, Jeff. "The Inverse Square Law." Vectors. Dept. of Math., East Tennessee State University, 2001. Web. 22 Apr. 2015. <http://math.etsu.edu/multicalc/prealpha/Capstone/intro.htm>.

Nave, Carl R. "Kepler's Laws." Kepler's Laws. Hyperphysics, Georgia State University, 2000. Web. 22 Apr. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html>