The Red Supergiant Betelgeuse. The star Betelgeuse has a...

The Red Supergiant Betelgeuse. The star Betelgeuse has a surface temperature of 3000 $\mathrm{K}$ and is 600 times the diameter of our sun. (If our sun were that large, we would be inside it!) Assume that it radiates like an ideal blackbody. (a) If Betelgeuse were to radiate all of its energy at the peak-intensity wavelength, how many photons per second would it radiate? (b) Find the ratio of the power radiated by Betelgeuse to the power radiated by our sun (at 5800 $\mathrm{K} )$ .

Key Concepts

Luminosity Scaling

Luminosity scaling is an application of the Stefan-Boltzmann Law, which considers that a star’s total power output (luminosity) is proportional to its surface area (which scales with the radius squared) multiplied by the fourth power of its temperature. This concept is pivotal for understanding how variations in a star’s size and temperature affect its overall brightness and energy output compared to other stars.

Blackbody Radiation

This concept refers to the idealized emission of electromagnetic radiation by an object that absorbs all incident radiation. The radiation spectrum of a blackbody is fully determined by its temperature and is described by Planck’s law, which is fundamental for estimating both the distribution of energy across wavelengths and the total energy output of astronomical objects such as stars.

Wien's Displacement Law

Wien’s Displacement Law establishes the relationship between the temperature of a blackbody and the wavelength at which it emits radiation most intensely. This law provides the means to determine the peak wavelength of radiation for any given temperature, which is essential for estimating the energy per photon at that wavelength.

Photon Energy

Photon energy is derived from the relationship E = hc/?, where h is Planck’s constant, c is the speed of light, and ? is the wavelength. This concept is critical when calculating the number of photons emitted, as it allows one to convert the radiated energy at the peak wavelength into a photon count by determining the energy carried by an individual photon.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law relates the total power radiated per unit area of a blackbody to the fourth power of its temperature. In astrophysics, this law is used to compute the luminosity of a star by taking into account its surface area and temperature. This provides a means to compare the radiated power of stars with differing temperatures and sizes.

Transcript

00:01 In this question we are looking at a red super giant star.

00:04 We're given that it's temperature of its surface, it's 3 ,000 kelvin's and the diameter is 600 times of that of our sun.

00:17 We want to find what is the number of photons it radiates, assuming that it just emits at the peak intensity wavelength.

00:29 Well, what we will first need to do is to find what is the total energy that is being emitted per second.

00:38 So this is the power.

00:40 We will need to use the stefan boltzmann's law.

00:43 When the power is equal to sigma times a, the surface area, apply by the temperature to the power 4.

01:01 So we know that the stephen boltzman's constant, the surface area would be taking 4 pi times r square but this r is actually 600 times the radius of our sun so we use our radius 6 .96 times 10 power 8 this is the radius of the sun multiply this square then the then the temperature is 3 ,000 kelvin's right so 3 ,000 kelvin to the power 4 this is the power irradiated, what we need to find next is the energy of each photon.

01:53 This equals to hc over lambda.

01:59 And the lambda, we can find what is the peak wavelength based on our wain's displacement law.

02:08 Wain's displacement law basically states that the pink wavelength is equal to some constant divided by the temperature.

02:16 So this is just hc type c temperature divided by the constant, which is 2 .9, stand power minus 3.

02:35 So the temperature over here is 3 ,000 kelvin's...

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