Question

Suppose planet a is exactly twice the size (in surface area) of planet

Answer 1

For the first equation, let's approach this using the general formula for the heat transfer via radiation:

Q = σA(T₁⁴ - T₂⁴)
where 
Q is the heat transfer rate
σ is the Stephen-Boltzmann constant
A is the surface area
T is temperature

Thus, the greater the surface area, the greater the rate of heat transfer through radiation. Thus, the planet with twice the surface area would emit the most radiation.

For the second question, the Wiens Law is

λmax = 0.0029 m·K / T
Substituting T = 1,500 K,
λmax = 0.0029 m·K / (1,500 K)
λmax = 1.933×10⁻⁶ m or 1.933μm