Suppose our Sun had 10 times its current luminosity. What would be the average blackbody surface temperature of Earth, assuming Earth had the same albedo?

Luminosity is essentially the total amount of energy a star, like our Sun, emits per second. Think of it as the star's power output. This energy is spread out over space, affecting nearby planets and celestial bodies.
In our exercise, the Sun's luminosity is increased to 10 times its current value. This means the Sun now emits 10 times more energy every second. This has a direct impact on the amount of solar radiation reaching Earth, which then influences Earth's temperature.

A blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The concept of blackbody helps us understand how objects radiate energy.
The blackbody temperature is the temperature at which a blackbody emits radiation. When we're talking about planets, we often approximate them as blackbodies to simplify calculations. The Earth's surface temperature, in this ideal model, is what we call the blackbody temperature.
In our scenario, if the Sun's luminosity increases, the Earth's blackbody temperature will also rise. We use this principle to calculate how hot Earth would get with the Sun being 10 times brighter.

The Stefan-Boltzmann Law gives us a direct relationship between the temperature of an object and the power it radiates. The law is expressed as:

\[ P = \text{σ} T^4 \]
where:

• \( P \) is the power radiated per unit area.
• \( \text{σ} \) is the Stefan-Boltzmann constant, approximately \( 5.67 × 10^{-8} W·m^{-2}·K^{-4} \).
• \( T \) is the temperature in Kelvins.

This relationship means that even slight changes in temperature result in significant changes in the power output. In our exercise, to find the new temperature corresponding to increased luminosity, we use this law.

Solar radiation is the energy emitted by the Sun, which reaches Earth and warms it, making life possible. This radiation includes visible light, ultraviolet light, and infrared radiation.
When the Sun's luminosity increases, the solar radiation received by Earth also increases. This leads to a change in Earth's energy balance. In other words, more radiation means more energy, which in turn increases Earth's temperature.
Our exercise involves calculating how Earth's temperature changes if it receives 10 times more solar radiation due to increased solar luminosity.

The equilibrium temperature is the temperature at which the energy a planet receives from its star (like the Sun) is equal to the energy it radiates back into space. For Earth, this balance determines our climate and weather patterns.
When solar luminosity increases, more energy reaches Earth, and thus, the Earth's equilibrium temperature rises. To compute this, we use the proportionate relationship between luminosity and temperature:

• Original Luminosity, \( L \)
• New Luminosity, \( 10L \)
• New Equilibrium Temperature, \( T_{\text{new}} = T \times \root 4\raise 10 \text \right \)
• \(T \) is the temperature before the luminosity change

This mathematical relationship allows us to find the new equilibrium temperature when solar luminosity changes. This process involves the square root and fourth root operations, reflecting the complex relationship between radiation and temperature.