What is the temperature of a planet with a reflectivity of 98.4 % at a distance of 1 AU?  



To calculate the **equilibrium temperature** of a planet, we assume that the planet is a perfect blackbody (absorbing and re-radiating energy) with some reflectivity, also known as **albedo**. 

Given:  
- **Reflectivity (albedo)** = \( A = 0.984 \) (98.4%)  
- **Distance from the Sun** = \( d = 1 \, AU \)  
- **Solar constant** at 1 AU = \( S = 1361 \, \text{W/m}^2 \)  

---

### Step-by-Step Calculation:

The formula for the equilibrium temperature \(T\) of a planet is:  
\[
T = \left( \frac{(1 - A) \cdot S}{4 \cdot \sigma} \right)^{1/4}
\]
where:  
- \( A \) is the albedo (reflectivity)  
- \( S \) is the solar constant (energy flux from the Sun at the planet's orbit)  
- \( \sigma = 5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4 \) is the Stefan-Boltzmann constant  

---

### Substituting Values:

1. \( (1 - A) = 1 - 0.984 = 0.016 \)  
2. Solar energy absorbed = \( 0.016 \times 1361 = 21.78 \, \text{W/m}^2 \)  
3. Now, divide by 4:  
   \[
   \frac{21.78}{4} = 5.445 \, \text{W/m}^2
   \]
4. Applying the Stefan-Boltzmann constant:  
   \[
   T = \left( \frac{5.445}{5.67 \times 10^{-8}} \right)^{1/4}
   \]
   \[
   T = \left( 9.6 \times 10^7 \right)^{1/4} 
   \]
5. Taking the fourth root:  
   \[
   T \approx 253.2 \, \text{K}
   \]

---

### Final Answer:
The equilibrium temperature of the planet is approximately **253 K** (or **-20°C**).

---

Do you need more details or have any questions?

Here are five related questions:
1. How does albedo affect the temperature of a planet?
2. What is the significance of the Stefan-Boltzmann constant?
3. How does distance from the Sun influence planetary temperature?
4. How would greenhouse gases change the result of this calculation?
5. What would the temperature be if the planet’s albedo were 0%?

**Tip:** A planet with a high albedo reflects most incoming radiation, which can significantly reduce its surface temperature!

What is the temperature of a planet with a reflectivity of 98.4% at a distance of 1 AU?

Discover how to calculate the equilibrium temperature of a planet with 98.4% reflectivity (albedo) at a distance of 1 AU. Using the Stefan-Boltzmann law and radiative energy balance, the temperature is found to be approximately 253 K (-20°C). Learn step-by-step calculations and explore key concepts like albedo and the solar constant.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation