Question

Verify the total solar power output of \(4 \times 10^{26} \mathrm{~W}\) based on its surface temperature of \(5,800 \mathrm{~K}\) and radius of \(7 \times 10^{8} \mathrm{~m}\), using Eq. \(1.9 .\)

Short answer

Question: Verify the total solar power output of the Sun given its surface temperature of 5,800 K and radius of \(7 \times 10^8 \mathrm{~m}\), using the Stefan-Boltzmann Law. The given total solar power output is \(4 \times 10^{26} \mathrm{~W}\). Answer: The calculated theoretical power output using the Stefan-Boltzmann Law is approximately \(3.99 \times 10^{26} \mathrm{~W}\), which is very close to the given value of \(4 \times 10^{26} \mathrm{~W}\). This verifies the total solar power output of the Sun based on its surface temperature and radius.

Step-by-step solution

1. Write down the given data

Given: Total solar power output (L) = \(4 \times 10^{26} \mathrm{~W}\) Surface temperature (T) = \(5,800 \mathrm{~K}\) Radius (R) = \(7 \times 10^{8} \mathrm{~m}\)

2. Write down the Stefan-Boltzmann Law (Equation 1.9)

The Stefan-Boltzmann Law is given by: \(L = 4 \pi R^2 \sigma T^4\) Where \(L\) is the luminosity (total power output) of the star, \(R\) is the radius of the star, \(T\) is the surface temperature of the star, and \(\sigma\) is the Stefan-Boltzmann constant, \(\sigma = 5.67 \times 10^{-8} \mathrm{~W~m^{-2}~K^{-4}}\).

3. Plug in the given data and solve for L

Using the given values of \(R\) and \(T\), along with the Stefan-Boltzmann constant, we can now calculate the theoretical power output (L): \(L = 4 \pi (7 \times 10^{8} \mathrm{~m})^2 \cdot (5.67 \times 10^{-8} \mathrm{~W~m^{-2}~K^{-4}}) \cdot (5,800 \mathrm{~K})^4\)

4. Calculate the theoretical power output

Now, compute the value of L: \(L = 4 \pi (7 \times 10^8)^2 \cdot 5.67 \times 10^{-8} \cdot 5,800^4\) \(L \approx 3.99 \times 10^{26} \mathrm{~W}\)

5. Compare the theoretical power output with the given data

The calculated theoretical power output is approximately \(3.99 \times 10^{26} \mathrm{~W}\), which is very close to the given value of \(4 \times 10^{26} \mathrm{~W}\). This verifies the total solar power output of the Sun based on its surface temperature and radius, using Eq. 1.9 (Stefan-Boltzmann Law).

Key concepts

Exploring Solar Power Output

Solar power output is a fascinating topic in astrophysics. It refers to the total energy emitted by the Sun per second in the form of electromagnetic radiation. This total energy, also known as the Sun's luminosity, is incredibly large, around \(4 \times 10^{26} \mathrm{~W}\). Understanding solar power output is crucial because it affects Earth immensely, providing the light and heat essential for life. To accurately determine this solar power, scientists use theoretical calculations based on known physical laws. These calculations help verify the Sun's incredibly high power output and deepen our understanding of solar energy dynamics. Large numbers such as \(4 \times 10^{26}\) can be difficult to grasp. To make it relatable, consider that this power is roughly equivalent to the continuous operation of a trillion trillion lightbulbs, each requiring 100 watts. The ability to calculate and verify these massive outputs, through known scientific laws, fortifies the study of solar physics and offers insights into the functionality of stars.

Mastering Luminosity Calculation with the Stefan-Boltzmann Law

Calculating a star's luminosity, such as that of the Sun, involves using the Stefan-Boltzmann Law. This law mathematically describes how much energy a perfect black body radiates per unit surface area. For stars, it provides a means to compute their total energy output, or luminosity, using measurable properties:

• Radius (\(R\))
• Surface Temperature (\(T\))
• Stefan-Boltzmann Constant (\(\sigma = 5.67 \times 10^{-8} \mathrm{~W~m^{-2}~K^{-4}}\))

The formula is given by \(L = 4 \pi R^2 \sigma T^4\), where \(L\) is the luminosity.Substitute the Sun's known values—its radius of \(7 \times 10^{8} \mathrm{~m}\) and temperature of \(5,800 \mathrm{~K}\). This allows us to compute the solar power output. Through these steps, the theoretical luminosity obtained confirms the empirical data. Calculations like these not only verify observations but also underline the reliability of the Stefan-Boltzmann Law. Understanding how luminosity is calculated thus enhances our capability to study other stars and celestial objects beyond our solar system.

Diving into the Wonders of Solar Physics

Solar physics is a specialized field that examines the Sun's physical properties, dynamic processes, and its pivotal role in the solar system. It encompasses various areas including solar phenomena like sunspots and solar flares, and their effects on space weather. Knowledge gleaned from solar physics is vital as it aids in forecasting space-weather events which can influence satellite operations, communication, and power grids on Earth. By fully grasping elements like solar power output and luminosity, we get insights into the Sun's life cycle and its eventual fate, aiding astrophysical theories. Solar physics doesn't just stop at understanding the Sun but also extends to engaging with the broader field of astrophysics. This includes studying similar stars, providing fundamental data needed in the search for extraterrestrial life, and understanding the origins of our solar system. As an ever-evolving discipline, solar physics continues to provide intriguing discoveries about the Sun's intriguing behaviors and its profound influence on our planet.