A Somewhat Complicated Explanation of Kepler’s Laws 75

As a quick but important side note,

in mathematics, the eccentricity does not

necessarily represent only the degree of

ellipticality. We have a true circle or an

ellipse only when the eccentricity (normally

represented by e) is greater than or equal

to 0 and less than 1. For other values, we

will have a parabola or hyperbola.

Eccentricity (e) = 0 True circle

0 < Eccentricity (e) < 1 Ellipse

Eccentricity (e) = 1 Parabola

1 < Eccentricity (e) Hyperbola

Second Law: A Line Joining a Planet and the Sun Sweeps

Out Equal Areas During Equal Intervals of Time

Or in other words, a planet moving along an elliptical orbit will move faster when it is closer

to the Sun and slower when it is farther away from the Sun. If we illustrate this, the areas

of the shaded portions in the following figure are all the same.

This is the same as the conservation of angular momentum in Newtonian mechanics.

Although it is a little difficult to prove mathematically, it is intuitively similar to the rotation of

a figure skater—a skater who begins to turn with her arms extended will turn faster as she

brings her arms closer to her body.

e = 1

e = 2

e = 0

e = 0.5

The relationship between eccentricity and

geometric shape

Perihelion

Sun

Aphelion

Orbit of a planet according to Kepler’s Second Law

76 Chapter 1 Is Earth the Center ofthe Universe?

We can also consider a situation in which we attach a weight to a string and swing

it around in a circle. When the string is longer, the weight will be harder to rotate, and its

speed will be slower.

Conceptually, the following explanation may be easier to understand. When no external

force acts on a body, uniform linear motion will continue, due to the law of conservation of

momentum. In the diagram here, the body will move as follows: P

1

→ P

2

→ P

3

, where the

length from P

1

to P

2

is the same as the length from P

2

to P

3

.

However, a planet does not move with linear motion, since it is affected by the gravity

of the Sun at point S. Although a planet is continuously pulled toward the Sun and moves

with circular motion, to make the explanation easier to understand here, we’ll assume that

gravity continuously pulls the planet toward the Sun when the planet moves from P

2

to P

3

.

The planet’s motion will be pulled to the left by the force of gravity f to arrive at position

P

3

′ (composition of the force of gravity f and the force that tries to continue uniform linear

motion). Since momentum does not change, the lengths of P

2

P

3

and P

2

P

3

′ are the same.

Now, if we compare the triangles that were formed, since P

1

P

2

= P

2

P

3

due to uniform

linear motion, SP

1

P

2

and SP

2

P

3

will be triangles with equal length sides and equal

heights, so their areas will also be equal.

Next, since SP

2

P

3

and SP

2

P

3

′ share base SP

2

and have the same height (since f is

the force of gravity at P

2

, the arrow used in the composition of forces will also be parallel to

SP

2

), and their areas will be equal. In other words, the following holds true:

SP

2

P

3

= SP

2

P

3

′

Since this relationship holds regardless of the positions of the Sun and planet, a line

joining a planet and the Sun sweeps out equal areas during equal intervals of time.

Sun

Planet

S

P

1

P

2

P

3

P

3

′

f

f

Orbit of a planet according to Kepler’s Second Law

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