Problem 39 - Astronomy: At Play in the Cosmos

Question

A red giant has a temperature of \(3,700 \mathrm{K}\) and luminosity of \(1.8 \times 10^{3} L_{\mathrm{Sun}} .\) What is its radius, in solar radii?

Step-by-Step Solution

Answer

The radius of the red giant is approximately 7.96 solar radii.

Step 1: Understanding the Problem

We need to find the radius of a red giant star in solar radii. We are given the temperature and luminosity of the red giant and need to use the Stefan-Boltzmann Law, which relates these quantities to the radius.

Step 2: Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the luminosity of a star is proportional to its surface area and the fourth power of its temperature. The formula is: \[ L = 4\pi R^2 \sigma T^4 \]where \(L\) is the luminosity, \(R\) is the radius, \(\sigma\) is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \ \mathrm{W/m^2/K^4}\), and \(T\) is the temperature.

Step 3: Rearranging the Formula

We need to solve for the radius \(R\) in terms of known quantities. Rearranging the Stefan-Boltzmann formula gives:\[ R^2 = \frac{L}{4\pi \sigma T^4} \]

Step 4: Substituting Given Values

Substitute the given luminosity \(L = 1.8 \times 10^{3} L_{ ext{Sun}}\) (where \(L_{ ext{Sun}} = 3.828 \times 10^{26} \ \mathrm{W}\)) and temperature \(T = 3700 \ \mathrm{K}\) into the formula. Calculate the radius:\[ L = 1.8 \times 10^{3} \times 3.828 \times 10^{26} \ \mathrm{W} \]\[ R^2 = \frac{1.8 \times 3.828 \times 10^{29}}{4\pi \times 5.67 \times 10^{-8} \times (3700)^4} \]

Step 5: Calculating the Radius

Calculate \(R^2\) from the values:\[ R^2 = \frac{6.8904 \times 10^{29}}{4\pi \times 5.67 \times 10^{-8} \times 1.8741 \times 10^{14}} \]Compute \(R^2\) and then find \(R\) by taking the square root:\[ R^2 \approx 3.0674 \times 10^{19} \rightarrow R \approx 5.54 \times 10^{9} \text{ meters} \]

Step 6: Converting to Solar Radii

Convert the result into solar radii where the solar radius \(R_{ ext{Sun}} = 6.96 \times 10^{8} \ \mathrm{m}\). Thus, calculate:\[ R \approx \frac{5.54 \times 10^{9}}{6.96 \times 10^{8}} \]This yields:\[ R \approx 7.96 R_{ ext{Sun}} \]

Key Concepts

Red Giant Stars

Red giant stars are fascinating celestial objects that represent a late stage in the life cycle of a star. After a star exhausts the hydrogen in its core, it begins to expand. As it does so, it becomes much larger in size, eventually forming a red giant.
At this stage, a red giant's surface temperature decreases, while its brightness (or luminosity) increases dramatically. This increase in size causes the outer layers to cool, giving them a reddish appearance, hence the name "red giant." Some key points about red giants are:

• They have swelled to gigantic sizes, often hundreds of times larger than our Sun.
• Despite cooling on the surface, their core remains extremely hot and dense as it fuses helium into carbon or oxygen.
• They are typically found in older star populations, such as in globular clusters or the halo of our galaxy.

Understanding red giants helps us learn more about stellar evolution and the future of stars like our Sun.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a crucial tool in astrophysics for calculating the luminosity of stars. This law relates a star's luminosity (total energy emitted per unit time) to its temperature and radius. The formula is expressed as:

\[ L = 4\pi R^2 \sigma T^4 \]

In this equation:

• \( L \) is the luminosity of the star.
• \( R \) is the star's radius.
• \( \sigma \) is the Stefan-Boltzmann constant, approximately \( 5.67 \times 10^{-8} \ \mathrm{W/m^2/K^4} \).
• \( T \) is the star's surface temperature in Kelvin.

This law shows that a star's luminosity is proportional to the square of its radius and the fourth power of its temperature. Hence, even a small increase in temperature results in a significant increase in luminosity. The Stefan-Boltzmann Law allows astronomers to infer physical properties of a star by measuring its spectral characteristics.

Luminosity

In astronomy, luminosity is an important measure of a star's intrinsic brightness. It is a measure of the total amount of energy a star emits per unit of time.
Measured in watts, luminosity is often expressed in terms of the Sun's luminosity or \( L_{\mathrm{Sun}} \). For example, if a star has a luminosity of \( 1.8 \times 10^{3} L_{Sun} \), it is 1800 times more luminous than our Sun.

• Star's luminosity depends on both its temperature and size.
• Luminous stars are generally either massive, hot, or both.
• The Stefan-Boltzmann Law, as explained previously, connects luminosity to temperature and radius, making it easier to calculate the other variables if one is known.

Understanding a star's luminosity helps astronomers determine its energy output and its position in the lifecycle of stellar evolution.

Temperature

Temperature plays a key role in a star's properties, including its color and how it radiates energy into space. A star's temperature is measured in Kelvin (K) and can significantly vary during its life cycle.

• Hot stars have high surface temperatures, often appearing bluish, while cooler stars have lower temperatures and appear reddish.
• The temperature affects the star's brightness, or luminosity, through the Stefan-Boltzmann Law, as luminosity increases steeply with temperature.
• The temperature of a red giant star, like the one described in our problem, is typically lower than that of a main-sequence star yet its brightness is enhanced due to its large radius.

Changes in temperature are indicative of a star's evolution, with hotter temperatures generally found in younger or more massive stars.