Conic section is obtained by the intersection of a plane with a cone. Two such intersections, one for a circle and one for an ellipse are shown in the figure.

Figure 2: Two conic sections representing a circle and an ellipse are shown.

Conic sections

Here, we recount the elementary geometry of an ellipse in order to understand planetary motion. The equation of an ellipse centered at the origin of a rectangular coordinate (0,0) is :

x 2 a 2 + y 2 b 2 = 1 x 2 a 2 + y 2 b 2 = 1

where “a” is semi-major axis and “b” is semi-minor axis as shown in the figure.

Figure 3: Semi major and minor axes of an ellipse

Ellipse

Note that “ F 1 F 1 ” and “ F 2 F 2 ” are two foci of the ellipse.

The eccentricity of a conic section is measure of “how different it is from a circle”. Higher the eccentricity, greater is deviation. The eccentricity (e) of a conic section is defined in terms of “a” and “b” as :

e = 1 − k b 2 a 2 e = 1 − k b 2 a 2

where “k” is 1 for an ellipse, 0 for parabola and -1 for hyperbola. The values of eccentricity for different trajectories are as give here :

Focal points ( F 1 F 1 and F 2 F 2 ) lie on semi major axis at a distance from the origin given by

Figure 4: Focal distances from the center of ellipse

Focal points

f = a e f = a e

The focus of an ellipse is at a distance “ae” from the center on the semi-major axis. Area of the ellipse is "πab".

Semi latus rectum is equal to distance between one of the foci and ellipse as measured along a line perpendicular to the major axis. This is shown in the figure.

Figure 5: Semi latus rectum is perpendicular distance as shown in the figure.

Semi latus rectum

For an ellipse, Semi latus rectum has the expression in terms of “a” and “b” as :

ℓ = b 2 a ℓ = b 2 a

We can also express the same involving eccentricity as :

⇒ ℓ = a 1 − e 2 ⇒ ℓ = a 1 − e 2

Polar coordinates generally suite geometry of ellipse. The figure shows the polar coordinates of a point on the ellipse. It is important to note that one of foci serves as the origin of polar coordinates, whereas the other focus lies on the negative x-axis. For this reason orientation of x-axis is reversed in the figure. The angle is measured anti-clockwise and the equation of ellipse in polar coordinates is :

Figure 6: Second focus lies on negative x-axis.

Equation in polar coordinate

r = ℓ 1 + e cos θ r = ℓ 1 + e cos θ

Substituting expression for semi latus rectum

⇒ r = a 1 − e 2 1 + e cos θ ⇒ r = a 1 − e 2 1 + e cos θ

Perihelion distance

Perihelion position corresponds to minimum distance between Sun and planet. If we consider Sun to be at one focus (say F 1 F 1 ), then perihelion distance is " F 1 A F 1 A " as shown in the figure. We can see that angle θ = 0° for this position.

⇒ r min = a 1 − e 2 1 + e = a 1 − e ⇒ r min = a 1 − e 2 1 + e = a 1 − e

Figure 7: Positions correspond to perihelion and aphelion positions.

Minimum and maximum distance

From the figure also, it is clear that minimum distance is equal to “a - ae= a(1-e)”.

The angular velocity of the planet about Sun is not constant. However, as there is no external torque working on the system, the angular momentum of the system is conserved. Hence, angular momentum of the system is constant unlike angular velocity.

The description of motion in angular coordinates facilitates measurement of angular momentum. In the figure below, linear momentum is shown tangential to the path in the direction of velocity. We resolve the linear momentum along the parallel and perpendicular to radial direction. By definition, the angular momentum is given by :

Figure 8: Angular momentum of the system is constant.

Angular momentum

L = r X p ⊥ L = r X p ⊥

L = r X p ⊥ = r m v ⊥ = r m X ω r = m ω r 2 L = r X p ⊥ = r m v ⊥ = r m X ω r = m ω r 2

Since mass of the planet “m” is constant, it emerges that the term “ ω r 2 ω r 2 ” is constant. It clearly shows that angular velocity (read also linear velocity) increases as linear distance between Sun and Earth decreases and vice versa.

Maximum velocity corresponds to perihelion position and minimum to aphelion position in accordance with maximum and minimum centripetal force at these positions. We can find expressions of minimum and maximum velocities, using conservation laws.

Figure 9: Velocities at these positions are perpendicular to semi major axis.

Maximum and minimum velocities

Let “ r 1 r 1 ” and “ r 2 r 2 ” be the minimum and maximum distances, then :

r 1 = a 1 − e r 1 = a 1 − e

r 2 = a 1 + e r 2 = a 1 + e

We see that velocities at these positions are perpendicular to semi major axis. Applying conservation of angular momentum,

L = r 1 m v 1 = r 2 m v 2 L = r 1 m v 1 = r 2 m v 2

r 1 v 1 = r 2 v 2 r 1 v 1 = r 2 v 2

Applying conservation of energy, we have :

1 2 m v 1 2 − G M m r 1 = 1 2 m v 2 2 − G M m r 2 1 2 m v 1 2 − G M m r 1 = 1 2 m v 2 2 − G M m r 2

Substituting for “ v 2 v 2 ”, “ r 1 r 1 ” and “ r 2 r 2 ”, we have :

v 1 = v max = { G M a 1 + e 1 − e v 1 = v max = { G M a 1 + e 1 − e

v 2 = v min = { G M a 1 − e 1 + e v 2 = v min = { G M a 1 − e 1 + e

As no external force is working on the system and there is no non-conservative force, the mechanical energy of the system is conserved. We have derived expression of linear velocities at perihelion and aphelion positions in the previous section. We can, therefore, find out energy of “Sun-planet” system by determining the same at either of these positions.

Let us consider mechanical energy at perihelion position. Here,

E = 1 2 m v 1 2 − G M m r 1 E = 1 2 m v 1 2 − G M m r 1

Substituting for velocity and minimum distance, we have :

⇒ E = m G M 1 + e 2 a 1 + e − G M m a 1 − e ⇒ E = m G M 1 + e 2 a 1 + e − G M m a 1 − e

⇒ E = m G M a 1 − e 1 + e 2 − 1 ⇒ E = m G M a 1 − e 1 + e 2 − 1

⇒ E = m G M a 1 − e e − 1 2 ⇒ E = m G M a 1 − e e − 1 2

⇒ E = − G M m 2 a ⇒ E = − G M m 2 a

We see that expression of energy is similar to that of circular trajectory about a center with the exception that semi major axis “a” replaces the radius of circle.

Johannes Kepler analyzed Tycho Brahe’s data and proposed three basic laws that govern planetary motion of solar system. The importance of his laws lies in the fact that he gave these laws long before Newton’s laws of motion and gravitation. The brilliance of the Kepler’s laws is remarkable as his laws are consistent with Newton’s laws and conservation laws.

Kepler proposed three laws for planetary motion. First law tells about the nature of orbit. Second law tells about the speed of the planet. Third law tells about time period of revolution.

Law of velocities

Law of velocities is a statement of comparative velocities of planets at different positions along the elliptical path.

Definition 2: Law of velocities

The line joining a planet and Sun sweeps equal area in equal times in the planet’s orbit.

This law states that the speed of the planet is not constant as generally might have been conjectured from uniform circular motion. Rather it varies along its path. A given area drawn to the focus is wider when it is closer to Sun. From the figure, it is clear that planet covers smaller arc length when it is away and a larger arc length when it is closer for a given orbital area drawn from the position of Sun. It means that speed of the planet is greater at positions closer to the Sun and smaller at positions away from the Sun.

Figure 10: Speed of the planet is greater at positions closer to the Sun and smaller at positions away from the Sun.

Equal area swept in equal time

Further on close examination, we find that Kepler’s second law, as a matter of fact, is an statement of the conservations of angular momentum. In order to prove this, let us consider a small orbital area as shown in the figure.

Figure 11: Time rate of area is statement of conservation of angular momentum.

Area swept

Δ A = 1 2 X Base X Height = 1 2 r Δ θ r = 1 2 r 2 Δ θ Δ A = 1 2 X Base X Height = 1 2 r Δ θ r = 1 2 r 2 Δ θ

For infinitesimally small area, the “area speed” of the planet (the time rate at which it sweeps orbital area drawn from the Sun) is :

⇒ đ A đ t = 1 2 r 2 đ θ đ t = 1 2 r 2 ω ⇒ đ A đ t = 1 2 r 2 đ θ đ t = 1 2 r 2 ω

Now, to see the connection of this quantity with angular momentum, let us write the equation of angular momentum :

L = r X p ⊥ = r m v ⊥ = r m X ω r = m ω r 2 L = r X p ⊥ = r m v ⊥ = r m X ω r = m ω r 2

⇒ ω r 2 = L m ⇒ ω r 2 = L m

Substituting this expression in the equation of area – speed, we have :

⇒ đ A đ θ = L 2 m ⇒ đ A đ θ = L 2 m

As no external torque is assumed to exist on the “Sun-planet” system, its angular momentum is conserved. Hence, parameters on the right hand side of the equation i.e. “L” and “m” are constants. This yields that area-speed is constant as proposed by Kepler.

We can interpret the above result other way round also. Kepler’s law says that area-speed of a planet is constant. His observation is based on measured data by Tycho Brahe. It implies that angular momentum of the system remains constant. This means that no external torque applies on the “Sun – planet” system.