1. It is believed that in the Archeaneon (2.5-4 billion years ago) the sun’s radiative output was 30% less than it is today.
(i) What would the temperature of the sun have been at that time?
(ii) At what wavelength would the peak emission from the sun have been?
(iii) Ignoring the effects of the atmosphere, what would the temperature of the Earth have been at this time (i.e. the effective radiative temperature)? Assume that the Earth’s albedo was 0.3, the same as today.
To answer these questions, we need to use the Stefan-Boltzmann Law and Wien's Displacement Law. Here's how you can calculate the answers:
(i) The Stefan-Boltzmann Law states that the total energy radiated by a black body is proportional to the fourth power of its temperature. We can use this law to find the temperature of the sun at that time.
The Stefan-Boltzmann Law: E = σT⁴
Where:
- E is the energy radiated by the sun
- σ is the Stefan-Boltzmann constant (5.67 × 10^−8 W/(m^2K^4))
- T is the temperature of the sun
Since the radiative output was 30% less than it is today, we can say that the energy radiated by the sun at that time was 70% of the current value. So we can write the equation as:
0.7E_today = σT⁴
Rearranging the equation, we can solve for T:
T = (0.7E_today/σ)^(1/4)
You can substitute the value of E_today, which is the current radiative output of the sun, into the equation to find the temperature.
(ii) Wien's Displacement Law states that the wavelength at which a black body emits maximum radiation is inversely proportional to its temperature.
The equation for Wien's Displacement Law is:
λ_max = b/T
Where:
- λ_max is the wavelength at which the peak emission occurs
- b is Wien's constant (2.898 × 10^−3 m·K)
Substituting the value of T (the temperature of the sun) into the equation will give you the peak emission wavelength.
(iii) To find the effective radiative temperature of the Earth at that time, we can use the concept of energy balance. The energy absorbed by the Earth's surface should be equal to the energy it radiates into space.
The equation for this energy balance is:
E_in = E_out
Where:
- E_in is the energy absorbed by the Earth's surface from the sun
- E_out is the energy radiated by the Earth's surface into space
The energy absorbed by the Earth's surface can be calculated using the equation:
E_in = (1 - albedo) * E_sun
Where:
- albedo is the fraction of sunlight reflected by the Earth's surface (given as 0.3)
- E_sun is the current radiative output of the sun
The energy radiated by the Earth's surface can be calculated using the Stefan-Boltzmann Law:
E_out = σ * T_eff^4
Where:
- T_eff is the effective radiative temperature of the Earth
By equating E_in and E_out, you can solve for T_eff, which will give you the temperature of the Earth at that time.
Remember to substitute the appropriate values and units into the equations to get accurate results.