To calculate the temperature of the Earth based on the given parameters, we can use the concept of thermal equilibrium and the Stefan-Boltzmann law. This law relates the temperature of an object to the amount of energy it radiates. Let's break down the steps to find the temperature of the Earth.

Understanding the Problem

We have the following information:

• Surface temperature of the Sun, T_sun = 5500 K
• Radius of the Sun, R_s = 7 x 10¹⁰ cm
• Radius of the Earth, R_e = 6.4 x 10⁶ cm
• Distance between the Earth and the Sun, R = 1.5 x 10¹³ cm

Applying the Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the power radiated by a black body per unit area is proportional to the fourth power of its temperature:

P = σT⁴,

where P is the power, σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m²K⁴), and T is the temperature in Kelvin.

Calculating the Total Power Output of the Sun

The total power output (luminosity) of the Sun can be calculated using its surface area and temperature:

Surface area of the Sun, A_s = 4πR_s².

Substituting the radius of the Sun:

A_s = 4π(7 x 10¹⁰)² cm².

Now, we can find the luminosity (L_s):

L_s = A_s × σ × T_sun⁴.

Calculating the Power Received by the Earth

The power received by the Earth (P_e) can be determined by considering the distance from the Sun:

P_e = L_s / (4πR²),

where R is the distance from the Sun to the Earth.

Finding the Temperature of the Earth

Assuming the Earth is in thermal equilibrium, the power absorbed by the Earth equals the power it radiates:

P_e = σT_e⁴,

where T_e is the temperature of the Earth.

From this, we can rearrange to find T_e:

T_e = (P_e / σ)^1/4.

Putting It All Together

Now, let's calculate each step:

1. Calculate the surface area of the Sun:

A_s = 4π(7 x 10¹⁰)² ≈ 6.16 x 10²¹ cm².

2. Calculate the luminosity of the Sun:

L_s = (6.16 x 10²¹ cm²) × (5.67 x 10^-8 W/m²K⁴) × (5500 K)⁴.

Convert cm² to m² for consistency:

L_s ≈ 3.846 x 10²⁶ W.

3. Calculate the power received by the Earth:

P_e = (3.846 x 10²⁶ W) / (4π(1.5 x 10¹³ cm)²).

P_e ≈ 1.74 x 10¹⁷ W.

4. Finally, calculate the temperature of the Earth:

T_e = (1.74 x 10¹⁷ W / (5.67 x 10^-8 W/m²K⁴))^1/4.

T_e ≈ 255 K.

Final Thoughts

This temperature of approximately 255 K indicates that the Earth, when in thermal equilibrium and absorbing all incoming solar radiation, would have an average surface temperature around -18°C. However, due to the greenhouse effect, the actual average temperature is higher, around 15°C. This example illustrates how energy balance and thermal dynamics play crucial roles in understanding planetary temperatures.