To solve this problem, we can use a simplified energy balance model for Earth's temperature, assuming a one-layer atmosphere. The key concept here is that the energy received from the Sun is balanced by the energy Earth radiates back to space. Let's break it down:

Step 1: Define the energy balance equation.

The Earth's energy balance can be written as:

$(1 - \alpha) \frac{S}{4} = \sigma T^4$

Where:

• $\alpha$ is Earth's albedo (initially 0.3).
• $S$ is the solar constant (1,360 W/m²).
• $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4$).
• $T$ is Earth's surface temperature.

The left side represents the energy absorbed by Earth (with albedo $\alpha$), and the right side is the energy Earth radiates (as a blackbody, following Stefan-Boltzmann law).

Step 2: Calculate the initial temperature of Earth.

Using the given values ($\alpha = 0.3$, $S = 1360 \, \text{W/m}^2$):

$(1 - 0.3) \frac{1360}{4} = \sigma T^4$

$0.7 \times 340 = \sigma T^4$

$238 = 5.67 \times 10^{-8} T^4$

Now, solve for $T$:

$T^4 = \frac{238}{5.67 \times 10^{-8}} = 4.197 \times 10^9$

$T = \left( 4.197 \times 10^9 \right)^{1/4} \approx 255 \, \text{K}$

So the initial surface temperature is approximately 255 K.

Step 3: Reduce the temperature by 1 K.

If we want to reduce Earth's temperature by 1 K, the new temperature would be $T_{\text{new}} = 254 \, \text{K}$.

Step 4: Calculate the new albedo.

Now, we reverse the process. We want to find the new albedo, $\alpha_{\text{new}}$, that corresponds to a surface temperature of 254 K.

The energy balance equation is:

$(1 - \alpha_{\text{new}}) \frac{S}{4} = \sigma T_{\text{new}}^4$

Substitute the new temperature:

$(1 - \alpha_{\text{new}}) \times 340 = 5.67 \times 10^{-8} \times 254^4$

Calculate $254^4$:

$254^4 = 4.142 \times 10^9$

Now:

$5.67 \times 10^{-8} \times 4.142 \times 10^9 = 235$

Thus:

$(1 - \alpha_{\text{new}}) \times 340 = 235$

Solve for $(1 - \alpha_{\text{new}})$:

$1 - \alpha_{\text{new}} = \frac{235}{340} = 0.691$

So the new albedo is:

$\alpha_{\text{new}} = 1 - 0.691 = 0.309$

Step 5: Conclusion

To reduce Earth's surface temperature by 1 K, the albedo would need to increase from 0.30 to approximately 0.309. This means an increase of 0.009 in albedo.

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Would you like more details on any of the steps? Here are some related questions you might explore:

1. How does Earth's current albedo compare to that of other planets in the solar system?
2. What are other methods to influence Earth's albedo, besides space mirrors?
3. How does the greenhouse effect impact the calculation of surface temperature?
4. What are the implications of increasing albedo on global ecosystems?
5. How do clouds contribute to the Earth's albedo?

Tip: Albedo is highly variable across Earth's surface—snow-covered regions, deserts, and oceans all reflect sunlight differently. Understanding these variations can help improve models of climate change.