Question

The amount of light emitted by a hot region is proportional to the temperature to the fourth power $\left(L \propto T^{4}\right) .$ If a sunspot is at a temperature of $4500 \mathrm{K}$, how much less luminosity does it generate than if it were at $6000 \mathrm{K}$ ?

Name: Problem 4, Chapter 51: Pathways to Astronomy
Uploaded: 2021-09-21T17:40:03-08:00
Duration: 2 min 9 s
Channel: Vishal Gupta
Description: The amount of light emitted by a hot region is proportional to the temperature to the fourth power (L ∝T^4) . If a sunspot is at a temperature of 4500 K, how much less luminosity does it generate than if it were at 6000 K ?

   The amount of light emitted by a hot region is proportional to the temperature to the fourth power $\left(L \propto T^{4}\right) .$ If a sunspot is at a temperature of $4500 \mathrm{K}$, how much less luminosity does it generate than if it were at $6000 \mathrm{K}$ ?

Pathways to Astronomy

Stephen E. Schneider 2nd Edition

Chapter 51, Problem 4

Instant Answer

Solved by Expert Vishal Gupta

09/21/2021

Step 1

Since $L \propto T^4$, we have: $\frac{L_{4500}}{L_{6000}} = \frac{T_{4500}^4}{T_{6000}^4} = \frac{4500^4}{6000^4}$ Now, we can calculate the value of this ratio: $\frac{4500^4}{6000^4} = \frac{4.100625 \times 10^{12}}{1.296 \times 10^{16}} = 0.3164$ So, the Show more…

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The amount of light emitted by a hot region is proportional to the temperature to the fourth power $\left(L \propto T^{4}\right) .$ If a sunspot is at a temperature of $4500 \mathrm{K}$, how much less luminosity does it generate than if it were at $6000 \mathrm{K}$ ?

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Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law links the power radiated per unit area of a blackbody to the fourth power of its temperature. This key principle in astrophysics explains how the energy output (or luminosity) of stars and other hot objects is highly sensitive to temperature changes.

Temperature Dependence of Luminosity

Luminosity is heavily reliant on temperature due to its proportional relationship, often expressed as T^4 in the context of blackbody radiation. This means that even small variations in temperature can result in significant differences in luminosity.

Power Law Relationships

Power law relationships describe how one quantity varies as a fixed power of another, as seen in the T^4 dependence in radiative processes. Such relationships are common in physics and other sciences, providing insights into scaling behaviors and system dynamics.

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