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Two stars, both of which behave like ideal blackbodies, radiate the same total energy per second. The cooler one has a surface temperature T and a diameter 3.0 times that of the hotter star. (a) What is the temperature of the hotter star in terms of T ? (b) What is the ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star?

Short Answer

Expert verified
(a) T1=T3; (b) Ratio is 13.

Step by step solution

01

Understand the Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the power radiated per unit area from a blackbody is proportional to the fourth power of its temperature: P=σAT4.Where σ is the Stefan-Boltzmann constant, A is the surface area, and T is the temperature.
02

Express the Power for Each Star

Let T1 and T2=T be temperatures of the hotter and cooler stars, respectively. The radius of the cooler star is 3 times the radius of the hotter star, so the surface area is A2=9A1. Thus, the power emitted from each star can be given as follows:- For the hotter star: P1=σA1T14- For the cooler star: P2=σA2T24.Since A2=9A1, then P2=σ×9A1×T4.
03

Equate the Powers

Since both stars radiate the same total energy per second, we have P1=P2. Substitute the expressions for P1 and P2:σA1T14=σ×9A1×T4.Cancel σA1 from both sides to simplify:T14=9T4.
04

Solve for the Temperature of the Hotter Star

Taking the fourth root on both sides of the equation T14=9T4, we get:T1=T×91/4.Simplify this to find:T1=T×31/2. This means T1=T3.So the temperature of the hotter star in terms of T is T3.
05

Understand Wien’s Displacement Law

Wien's Displacement Law states that the peak wavelength λmax is inversely proportional to the temperature:λmax=bT, where b is Wien's displacement constant.
06

Calculate the Ratio of Peak Wavelengths

Let λ1 and λ2 be the peak-intensity wavelengths for the hotter and cooler stars, respectively. From Wien’s Law, we have:λ1=bT1 and λ2=bT.Thus, the ratio of the wavelengths is:λ1λ2=TT1=TT3=13.The ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star is 13.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stefan-Boltzmann Law
The Stefan-Boltzmann Law is a fundamental principle in the study of blackbody radiation. It describes how an ideal blackbody emits energy. According to this law, the power emitted by a blackbody per unit area is directly proportional to the fourth power of its temperature. This can be mathematically represented as:
P=σAT4
where:
  • P is the power emitted,
  • σ is the Stefan-Boltzmann constant, approximately 5.67×108 W/m2K4,
  • A is the surface area of the blackbody, and
  • T is the temperature in Kelvin.
For two stars emitting the same power, the law dictates that a larger star must have a lower temperature if their radii are different. This is because the power output is also a function of the surface area. In our exercise, the cooler star, which is larger, must emit more from a larger area to match the same total power as the smaller hot star.
Wien's Displacement Law
Wien's Displacement Law provides a critical insight into understanding the behavior of blackbody radiation in terms of temperature and wavelength. This law states that the peak wavelength of the emitted radiation from a blackbody is inversely proportional to its temperature. The formula for Wien's Displacement Law is:
λmax=bT
where:
  • λmax is the peak wavelength,
  • b is Wien's displacement constant, approximately 2.898×103 m K, and
  • T is the temperature in Kelvin.
In essence, as the temperature of a blackbody increases, the wavelength at which it emits most prominently shifts toward shorter wavelengths. Hence, hotter stars will have peak emissions in shorter wavelengths, meaning they appear bluer. Conversely, cooler stars will peak at longer wavelengths, often displaying a reddish color. This principle helps astronomers determine stellar temperatures based on their color.
Peak-intensity wavelength ratio
The concept of the peak-intensity wavelength ratio is derived from the application of Wien's Displacement Law, especially useful when comparing two blackbodies. From the exercise, we deal with two stars at different temperatures. Using Wien's Law, we can express the peak-intensity wavelengths for both stars given their temperatures.
For the hotter star:λ1=bT1
For the cooler star:λ2=bT
This allows us to calculate the ratio of their peak wavelengths:
λ1λ2=TT1
From our exercise, with the temperature of the hotter star being T3, the ratio simplifies to:13
This tells us that the peak wavelength of the hotter star is shorter than that of the cooler star, precisely by the factor 3. Such ratios are crucial in astrophysics, as they enable the understanding of the relative temperatures of celestial bodies through observational data.

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