Final answer:
To find the Sun's radius, we use the intensity of solar radiation at Earth's orbit (1370 W/m²) and the Stefan-Boltzmann law. The total power radiated by the Sun is determined and compared to the power emitted based on the temperature and emissivity of the Sun to estimate its radius.
Step-by-step explanation:
The student is asking about determining the power of radiation from the Sun and calculating the Sun's radius using its temperature and the concept that the Sun emits energy as a perfect black body (Stefan-Boltzmann law).
The intensity of solar radiation at Earth's orbit is given as 1370 W/m².
To find the total power P radiated by the Sun, this intensity is multiplied by the surface area of a sphere with a radius equal to the orbit of the Earth (the distance from the Earth to the Sun). The radius of this sphere is 1.496 × 10¹¹ m.
Using the formula for the surface area of a sphere (4πr²) and the solar constant (the intensity of solar radiation), we can calculate the total power output of the Sun:
P = Intensity × Area = 1370 W/m² × 4π(1.496 × 10¹¹ m)² = 3.82 × 10²¶ W.
To find the radius of the Sun, we use the Stefan-Boltzmann law in the form P = σAeT⁴, where σ is the Stefan-Boltzmann constant, A is the surface area of the Sun, e is the emissivity (which is given as 1 for the Sun), and T is the temperature of the Sun.
Given that the temperature of the Sun is 5780 K and substituting the total power P we've just calculated, we can solve for A and then find the radius of the Sun by rearranging the area of a sphere formula A = 4πr².
Therefore, by using the given solar radiation intensity at Earth's orbit and the temperature of the Sun, we can estimate the radius of the Sun.