Problem 37 - Astronomy: At Play in the Cosmos

Question

A star's temperature is 3 times as high as the Sun's, and its luminosity is 48 times that of the Sun. What is the ratio of the star's radius to the Sun's radius?

Step-by-Step Solution

Answer

The ratio of the star's radius to the Sun's radius is \( \frac{4\sqrt{3}}{9} \).

Step 1: Understanding the Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the luminosity \( L \) of a star is proportional to its surface area \( A \) and the fourth power of its temperature \( T \): \[L =  1sigma A T^4\] where \( 1sigma \) is the Stefan-Boltzmann constant. For a spherical star, \( A = 4pi R^2 \), where \( R \) is the radius of the star.

Step 2: Relating Luminosities and Temperatures of the Star and the Sun

Let \( L_s \) and \( T_s \) be the Sun's luminosity and temperature, respectively, and \( L \) and \( T \) be the star's. Given: - \( T = 3T_s \)- \( L = 48L_s \) Using the relationship from the Stefan-Boltzmann Law: \[L = 4pi R^2 1sigma (3T_s)^4= 81L_s \frac{R^2}{R_s^2}\]

Step 3: Setting Up the Equation for Radius Ratios

Since the given star's luminosity \( L = 48L_s \), and from the relationship developed in Step 2, we equate:\[48L_s = 81L_s \frac{R^2}{R_s^2}\] From this, solve for \( \frac{R^2}{R_s^2} \):\[\frac{R^2}{R_s^2} = \frac{48}{81}\] Simplifying the right side gives:\[\frac{16}{27}\]

Step 4: Calculating the Radius Ratio

Take the square root of both sides to solve for the ratio of the radii:\[\frac{R}{R_s} = \sqrt{\frac{16}{27}} = \frac{4}{\sqrt{27}} = \frac{4}{3\sqrt{3}}\] Rationalize the denominator to get the final form:\[\frac{R}{R_s} = \frac{4\sqrt{3}}{9}\]

Key Concepts

Star Luminosity

Star luminosity is a measure of the total amount of energy emitted by a star per unit of time. It is an intrinsic property of the star and provides insight into its energy output. Luminosity is often compared to that of the Sun, as the Sun serves as a reference point being our closest star. The formula for luminosity according to the Stefan-Boltzmann Law is given by:\[ L = \sigma A T^4 \]where:

• \( L \) is the star's luminosity.
• \( \sigma \) is the Stefan-Boltzmann constant.
• \( A \) is the surface area of the star.
• \( T \) is the temperature of the star.

In our example, understanding these components allows us to compare a star's luminosity with that of the Sun by considering their surface areas and temperatures. Comparing luminosity gives clues about other stellar properties, such as temperature and size.

Star Temperature

The temperature of a star is a crucial factor that influences its luminosity and color. Typically, the surface temperature of a star determines how luminous it will appear from a given distance. According to the Stefan-Boltzmann Law:\[ L \propto T^4 \]This means that even a small change in temperature can drastically affect the star's brightness. The exercise states that the star's temperature is three times that of the Sun's. This increase in temperature results in a huge increase in luminosity, owing to the fourth power dependence in the formula. If the temperature of the Sun is used as a baseline, understanding how such multiplication affects energy output helps in visualizing how much more powerful or radiant a star is compared to our own Sun.

Radius Ratio

The radius ratio provides information about the comparative sizes of a star and the Sun, based on their luminosities and temperatures. By examining the Stefan-Boltzmann Law equation:\[ L = 4\pi R^2 \sigma T^4 \]and using the given conditions that the star’s luminosity is 48 times that of the Sun and its temperature is three times greater, we can establish a relationship between the radii:\[ \frac{R^2}{R_s^2} = \frac{48}{81} \]Solving this equation involves taking the square root to find:\[ \frac{R}{R_s} = \frac{4\sqrt{3}}{9} \]This ratio reveals how smaller a hotter star can be while maintaining a high luminosity, demonstrating the interplay of star size and temperature in producing light and explaining why some smaller stars are more luminous than much larger ones.

Spherical Star Area

The concept of a star's surface area is essential for calculating its luminosity. For a spherical star, the surface area is given by the formula:\[ A = 4\pi R^2 \]where \( R \) is the radius of the star. This equation highlights how the star’s size impacts its energy emission. Larger radii equate to larger surface areas. Since luminosity depends directly on surface area, changes in radius have a profound effect on how much energy the star emits. In our problem, understanding that this star's increased temperature offset its smaller surface area relative to the Sun is key. By being smaller yet hotter, it can still exhibit significantly higher luminosity, as described by the Stefan-Boltzmann Law. This concept exemplifies how size and temperature balance to influence a star’s overall radiant output.