Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Kepler's third law states that the period \(T\) of a planet (the time needed to make one complete revolution about the sun) is directly proportional to the \(\frac{3}{2}\) power of its average distance \(d\) from the sun. (a) Express \(T\) as a function of \(d\) by means of a formula that involves a constant of proportionality \(k\). (b) For the planet Earth, \(T=365\) days and \(d=93\) million miles. Find the value of \(k\) in part (a). (c) Estimate the period of Venus if its average distance from the sun is 67 million miles.

Step-by-Step Solution

Answer

(a) \( T = k \cdot d^{\frac{3}{2}} \), (b) \( k \approx 0.4068 \), (c) Venus's period \( \approx 223 \) days.

Step 1: Formulating the relationship

According to Kepler's third law, the period \( T \) is directly proportional to the \( \frac{3}{2} \) power of the distance \( d \). This can be expressed mathematically as \( T = k \cdot d^{\frac{3}{2}} \), where \( k \) is the constant of proportionality.

Step 2: Solving for the constant k

Using Earth's values, where \( T = 365 \) days and \( d = 93 \) million miles, we substitute these into the formula \( T = k \cdot d^{\frac{3}{2}} \). Thus, \( 365 = k \cdot 93^{\frac{3}{2}} \). We solve for \( k \) by dividing both sides by \( 93^{\frac{3}{2}} \), giving \( k = \frac{365}{93^{\frac{3}{2}}} \).

Step 3: Calculating the constant k

First, calculate \( 93^{\frac{3}{2}} \). This involves taking the square root of 93 and then raising the result to the power of 3. Once we find this value, we divide 365 by it to find \( k \). \[ 93^{\frac{3}{2}} = (\sqrt{93})^3 \approx 897.29 \] Thus, \( k \approx \frac{365}{897.29} \approx 0.4068 \).

Step 4: Estimating the period of Venus

We can now use the value of \( k \) to estimate the period of Venus. Venus's average distance from the sun is 67 million miles. Substitute \( d = 67 \) into the formula \( T = k \cdot d^{\frac{3}{2}} \) with \( k \approx 0.4068 \). Calculate \( 67^{\frac{3}{2}} \) and then multiply by \( k \). \[ 67^{\frac{3}{2}} = (\sqrt{67})^3 \approx 548.83 \] Thus, \( T \approx 0.4068 \times 548.83 \approx 223 \) days.

Key Concepts

Period of a Planet

The period of a planet, denoted as \( T \), refers to the time it takes for a planet to complete one full orbit around the sun. This concept is crucial when studying planetary motion. Using Kepler's third law, we understand that this period is not arbitrary but follows a specific rule: it is directly proportional to the \( \frac{3}{2} \) power of its average distance from the sun. This characteristic helps describe the predictable motion of planets within our solar system and allows for calculations of their orbital periods.

Average Distance from the Sun

The average distance from the sun, denoted as \( d \), plays a significant role in determining a planet's orbital period. This average distance is measured in units such as million miles or astronomical units (AU). It directly influences the period of the planet according to Kepler's third law. For instance, Earth has an average distance of about 93 million miles from the sun, which helps determine its 365-day orbital period. By understanding this distance, scientists can predict how long each planet takes to orbit the sun.

Constant of Proportionality

The constant of proportionality, \( k \), in Kepler's third law is crucial for calculating the exact relationship between a planet's period and its distance from the sun. The constant \( k \) ensures that the proportional relation \( T = k \cdot d^{\frac{3}{2}} \) holds true when comparing different planets. By using Earth's data of \( T = 365 \) days and \( d = 93 \) million miles, we can determine that \( k \approx 0.4068 \). This same \( k \) is used when calculating the orbital periods for other planets, like Venus, providing consistency across measurements.

Mathematical Modeling

Mathematical modeling is the method of representing physical phenomena using mathematical expressions. In the context of Kepler's third law, we create a model that uses the formula \( T = k \cdot d^{\frac{3}{2}} \) to depict how a planet's orbital period is related to its distance from the sun. This model allows us to predict unknown periods once we know the distance, such as estimating Venus's period to be approximately 223 days. This process highlights the power of mathematics in turning observational data into predictive insights.