Problem 46 - College Algebra

Question

For the following exercises, use Kepler's Law, which states that the time, \(T\) , required for a planet to orbit the Sun varies directly with the cube of the mean distance, \(a\) , that the planet is from the Sun. Using the Earth's time of 1 year and mean distance of 93 million miles, find the equation relating \(T\) and \(a\) .

Step-by-Step Solution

Answer

The equation is \(T = \frac{1}{804357} \cdot a^3\).

Step 1: Understand Kepler's Law of Periods

Kepler's Laws of Planetary Motion include the Law of Periods, which states that the square of the period of orbit (T) of a planet is proportional to the cube of the semi-major axis of its orbit (a), not exactly the time T directly. However, for this exercise, the statement simplifies it to a direct proportion: \(T \propto a^3\). This implies a relation: \(T = k \cdot a^3\) where \(k\) is the constant of proportionality.

Step 2: Utilize Known Values to Find Constant

We use the Earth's known values to find the constant \(k\). We know that for Earth, \(T = 1\) year and \(a = 93\) million miles. Substitute these into the equation: \(1 = k \cdot 93^3\). This will let us solve for \(k\).

Step 3: Calculate the Constant \(k\)

Calculate \(93^3\) which is \(93 \times 93 \times 93 = 804357\). Then, we substitute \(93^3\) and \(T = 1\) into our equation to find \(k\): \(1 = k \cdot 804357\). So, \(k = \frac{1}{804357}\).

Step 4: Write the Final Equation

Now that we have \(k\), substitute \(k\) back into the original relation to write the final equation: \(T = \frac{1}{804357} \cdot a^3\). This equation relates the time period \(T\) to the distance \(a\) for any planet using the Earth's orbit as a reference.

Key Concepts

Planetary Motion

The concept of planetary motion describes how planets orbit a central celestial body, like the Sun, following predictable paths. According to Johannes Kepler, planets move in elliptical orbits with the Sun at one focus. This motion is governed by the laws of physics and is predictable over time.
Kepler's Laws of Planetary Motion consist of three laws that describe this behavior in detail:

• The Law of Ellipses states that planets move in ellipses, not perfect circles.
• The Law of Equal Areas describes how a line segment joining a planet and the Sun sweeps out equal areas in equal times.
• The Law of Harmonies relates the orbital period to the size of the orbit.

By understanding these laws, we can accurately predict the location and motion of any planet at any given time. This forms the foundation of celestial mechanics.

Orbital Period

The orbital period refers to the time it takes for a planet to complete one full orbit around the Sun. It's typically measured in Earth years for convenience. The period can vary widely among the planets in our solar system. For instance, the Earth takes exactly one year to orbit the Sun, giving us our seasonal cycle.
Kepler's Second Law, or the Law of Equal Areas, supports the concept that the speed at which a planet orbits can vary but will achieve a full orbit in a predictable timeframe. This law implies that planets move faster when they are closer to the Sun and slower when they are further away. Consequently, understanding the orbital period helps in determining other critical factors like velocity and position over time.

Proportional Relationship

A proportional relationship is one where two quantities increase or decrease at the same rate. Kepler's Third Law demonstrates this by providing a mathematical connection between the time a planet takes to orbit the Sun (its orbital period) and the size of its orbit (specifically the semi-major axis). This relationship is expressed as:

• The square of the orbital period (T) is proportional to the cube of the semi-major axis (a): \[ T^2 \propto a^3 \].

This means, as the mean distance from the Sun increases, the time to complete an orbit also increases. So if you know the semi-major axis, you can predict the orbital period and vice versa. This proportionality is fundamental in astronomy for calculating celestial dynamics.

Semi-Major Axis

The semi-major axis is one of the most crucial aspects of an ellipse, which describes a planet's orbit. It is half of the longest diameter of the ellipse, effectively representing the average distance of the planet from the Sun over the course of its orbit.
In Kepler's Third Law, as mentioned earlier, the semi-major axis helps define the orbital period.

• It gives a clear geometric representation of the size and scale of a planet's orbit.

Importantly, the semi-major axis is not affected by the eccentricity of the orbit; whether the elliptic path is nearly circular or highly elongated, the semi-major axis remains consistent in length. Thus, understanding this component helps astronomers calculate the orbital characteristics of any celestial body following an elliptical path around a focal point.