The star Betelgeuse has a surface temperature of 3000 K and...

The star Betelgeuse has a surface temperature of 3000 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 K).

Key Concepts

Photon Energy

Photon energy is determined by the equation E = h·c/?, where h is Planck’s constant, c is the speed of light, and ? is the wavelength. This concept is essential when calculating the number of photons emitted per second by relating the total radiated power (or energy per unit time) to the energy carried by each photon.

Blackbody Radiation

This concept describes objects that emit electromagnetic radiation in a characteristic spectrum solely determined by their temperature. In astrophysics, stars are often approximated as ideal blackbodies, meaning their emitted radiation follows a specific distribution given by Planck's law, and key parameters like total energy output and peak wavelength depend only on their temperatures.

Wien's Displacement Law

Wien's Displacement Law states that the wavelength at which the emission of a blackbody is maximized is inversely proportional to its temperature. This law is crucial in determining the peak wavelength of radiation from objects like stars, which in turn is used to estimate the energy of individual photons if the star were hypothesized to emit solely at that peak wavelength.

Stefan–Boltzmann Law

The Stefan–Boltzmann Law quantifies the total energy radiated per unit surface area of an ideal blackbody and states that it is proportional to the fourth power of the temperature (?T^4). This law is fundamental in determining the total luminosity of stars and allows for comparison between stars with different temperatures and sizes by integrating over their entire surface area.

Stellar Luminosity and Size Effects

Stellar luminosity not only depends on temperature (through the Stefan–Boltzmann Law) but also on the surface area of the star, which scales with the square of its radius. The comparison of power outputs between different stars thus requires accounting for both their differing temperatures and sizes, emphasizing how a larger surface area can lead to a significantly greater total energy output even at a lower temperature.

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