FFR - Kepler’s Laws

In this section, we are going to spend all our time dealing with one of the most important Laws that govern the universe. It is known as Kepler’s Three Laws and was proposed by Johannes Kepler in the early 1600s. These laws are very important as they describe the very motion of the planets in our solar system. We deem it important for you to know this as a way to solidify your knowledge of planetary motion and recognize how what you’re learning can be applied to real life principles.

Kepler’s First Law

Also known as the Law of Ellipses, it states that the path (orbits) of the planets around the sun is elliptical in shape with the center of the sun being located at one of the foci. In the previous section, we assumed that the planets orbited that sun in a circular motion. But in reality, they actually follow an elliptical path. The equation of an ellipse is:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ (Equation 3.4.1)

In that equation, $a$ and $b$ are the radii of the ellipse’s x and y-axes, respectively. The larger radius corresponds to the major axis, while the smaller axis corresponds to the minor axis.

An ellipse is a very special shape where the sum of the distances from every point on the elliptical curve to two other points are equal Let’s look at the Gallery 3.4.1 for a better understanding:

We can calculate the location of the foci if we know the radii of the major and minor axes. The equation for calculating the distance of a foci to the center of the ellipse is:

$F=\sqrt{j^2-i^2}$ (Equation 3.4.2)​
Here, $j$ is the radius of the major axis, $i$ is the radius of the minor axes, and $F$ is the distance of a foci to the center of the ellipse.

The foci are always along the major axis of the ellipse. The coordinates of the foci of an ellipse centered at $(\mathrm{0,0})$ and with a major axis along the x-axis are $(F\mathrm{,0})$ and $(-F\mathrm{,0})$

Knowing this equation is very helpful because it is an aid in deriving Kepler’s Second Law.

Kepler’s Second Law

Also known as the Law of Equal Areas, it states that any imaginary line that is drawn from the center of the sun to a planet that lies on the orbital path sweeps the same amount of area in the same period of time. This helps characterize the velocity that the planet has at any given time while orbiting the sun. The green circles represent the planet at different time intervals and the blue and orange triangles are the areas that the planet covers from one instant of time to the next.

Along one elliptical orbit, the planet moves faster near the Sun and slower farther away. Conservation of angular momentum gives constant areal speed; conservation of orbital energy also relates speed to distance.

In equal time intervals, the radius from the Sun to the planet sweeps equal areas. These are orbital sectors bounded by the curved path; drawn triangles only approximate the sector areas when the intervals are small.

Kepler’s Third Law

Kepler’s third law relates orbital period T to semimajor axis a: T²/a³ = 4π²/[G(M+m)] for an isolated Newtonian two-body system. When all orbiting bodies have negligible mass compared with the same central body M, this ratio is approximately the same. For a circular orbit, a equals the radius.

Planets, dwarf planets and other bodies orbiting the same dominant mass approximately share that constant. Satellites of a different central mass have a different constant. The observations support Kepler’s law; a table is not a proof that one constant applies to every orbit.

Table 3.4.1- Kepler’s Third Law

For approximate solar-orbit calculations, express period in Earth years and semimajor axis in au, giving T² ≈ a³. Use 1 year ≈ 3.156 × 10⁷ s. One astronomical unit is exactly 149 597 870 700 m; it is close to Earth’s orbital semimajor axis, not its varying instantaneous distance from the Sun.

Keep this in mind as we will later be able to provide a theoretical explanation to Kepler’s Third Law as we use Newton’s Law to prove it! This is where all the fun begins !

Chapter 3 Equations List

Section 1:

For any interval: v_avg = (x_f − x_i)/Δt (Equation 3.1.1), and a_avg = (v_f − v_i)/Δt (Equation 3.1.2). For constant acceleration in one coordinate: x = x₀ + v₀Δt + ½a(Δt)² (Equation 3.1.3); v² = v₀² + 2a(x − x₀) (Equation 3.1.4); v = v₀ + aΔt (Equation 3.1.5); x = x₀ + ½(v + v₀)Δt (Equation 3.1.6). Here v and a are signed scalar components.

Section 2:

$x_x=x_{\mathrm{0,}x}+{\overrightarrow{v}}_{\mathrm{0,}x}\mathrm{\Delta }t+\frac{1}{2}{\overrightarrow{a}}_x{\left(\mathrm{\Delta }t\right)}^2$ (Equation 3.2.1)​
${\left({\overrightarrow{v}}_x\right)}^2={\left({\overrightarrow{v}}_{\mathrm{0,}x}\right)}^2+2{\overrightarrow{a}}_x\mathrm{\Delta }x$ (Equation 3.2.2)​
${\overrightarrow{v}}_x={\overrightarrow{v}}_{\mathrm{0,}x}+{\overrightarrow{a}}_x\mathrm{\Delta }t$ (Equation 3.2.3)​
$x_x=x_{\mathrm{0,}x}+\frac{1}{2}\left({\overrightarrow{v}}_x+{\overrightarrow{v}}_{\mathrm{0,}x}\right)\mathrm{\Delta }t$ (Equation 3.2.4)

Section 3:

${\overrightarrow{F}}_G=G\frac{m_1{m}_2}{r^2}$ (Equation 3.3.1)​

Section 4:

x²/a² + y²/b² = 1 (Equation 3.4.1). The distance from the ellipse’s center to either focus is F = √(j² − i²) (Equation 3.4.2), where j is the semimajor axis and i is the semiminor axis.

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Chapter 3 Problems