Problem 63 - Algebra Form and Function

Question

One of Kepler's three laws of planetary motion states that the square of the period, \(P\), of a body orbiting the sun is proportional to the cube of its average distance, \(d,\) from the sun. The earth has a period of 365 days and its distance from the sun is approximately 93,000,000 miles. (a) Find a formula that gives \(P\) as a function of \(d\). (b) The planet Mars has an average distance from the sun of 142,000,000 miles. What is the period in earth days for Mars?

Step-by-Step Solution

Answer

Answer: The period of Mars is approximately 687 Earth days.

Step 1: Understand the relation between period and distance

According to Kepler's third law, the square of the period (\(P^2\)) is proportional to the cube of the distance (\(d^3\)). That is, there is a constant \(k\) such that \(P^2 = k d^3\).

Step 2: Find the constant of proportionality

We are given the period and distance of Earth from the sun: \(P_{\text{earth}} = 365\text{ days}\) and \(d_{\text{earth}} = 93,000,000\text{ miles}\). We can substitute these values into the equation to find the constant of proportionality \(k\): \begin{align*} P_{\text{earth}}^2 &= kd_{\text{earth}}^3 \\ (365)^2 &= k (93,000,000)^3 \end{align*} Now solve for \(k\): $$ k = \frac{(365)^2}{(93,000,000)^3} $$

Step 3: Find the function relating P and d

Now that we have the constant \(k\), we can write the function that gives the period \(P\) as a function of the distance \(d\) by plugging \(k\) back into the equation \(P^2 = kd^3\): $$ P(d) = \sqrt{\frac{(365)^2}{(93,000,000)^3} d^3} $$ Now we have a formula for the period as a function of the distance.

Step 4: Find the period of Mars

We are given that the average distance of Mars from the sun is \(d_{\text{mars}} = 142,000,000\text{ miles}\). To find the period of Mars, substitute \(d_{\text{mars}}\) into the function \(P(d)\): \begin{align*} P_{\text{mars}} &= \sqrt{\frac{(365)^2}{(93,000,000)^3} (142,000,000)^3} \\ &\approx 687 \text{ days} \end{align*} So, the period of Mars is approximately 687 Earth days.

Key Concepts

Period of Orbit

In astronomy, the **period of orbit** refers to the time it takes for a celestial object, such as a planet or a moon, to complete a full orbit around a more massive object like the Sun. This period is crucial as it helps scientists understand the dynamics and stability of planetary systems. For example, Earth has a period of 365 days, which we recognize as a year. But how do we derive this from Kepler's laws? Kepler's Third Law explains the relationship between the period and the distance from the central star. Specifically, it states that the square of the period (\(P^2\)) is proportional to the cube of the average distance from the star (\(d^3\)).

This means:

• If a planet is farther from the Sun, it will have a longer period of orbit compared to planets closer to the Sun.
• The relationship is not linear but rather a cubic function, which helps explain the dramatic increase in orbital periods with distance.

Understanding this relationship helps predict whether planets might have longer or shorter years than Earth, based only on their distance from the Sun.

Distance from Sun

The **distance from the Sun** plays a crucial role in determining a planet's period of orbit. In Kepler's Third Law, this distance is critical as it forms the base of the cubic relationship with the period of orbit. For Earth, this average distance is approximately 93 million miles. How does this compare with other planets? For example:

• Mars has a greater average distance from the Sun at about 142 million miles. Hence, as per Kepler's law, Mars has a longer orbital period compared to Earth.
• Planets like Mercury, with shorter distances, complete their orbits much faster.

It is vital to remember that only the average distance is considered here because orbits are not perfect circles but ellipses. The average gives a more accurate approximation of the planets' full orbital characteristics.

When comparing different planets, the increased distance means not just a longer orbit, but usually a slower speed around the sun as well. This dual effect increases the period of orbit dramatically compared to inner planets.

Constant of Proportionality

The **constant of proportionality** in Kepler's Third Law links the period of a planet to its distance from the Sun. This constant remains the same for all planets orbiting around the same central body, such as the Sun.Mathematically, it is represented in the equation:\[P^2 = k \cdot d^3\]Here, \(k\) is the constant of proportionality.
To find \(k\), we use Earth's known orbit details:

• Period \(P_{\text{earth}} = 365\) days
• Distance \(d_{\text{earth}} = 93,000,000\) miles

Substituting these into the equation allows us to solve for \(k\): \[k = \frac{(365)^2}{(93,000,000)^3}\]Once we have \(k\), it can be used universally for any other planet in the Solar System. So, when calculating the period for Mars or any other planet, \(k\) helps maintain the accuracy of these calculations. Note that any change in \(d\) directly affects \(P\) due to this constancy relationship across different celestial bodies reflected against their central orbit."