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.
(i) To determine the temperature of the sun during the Archeaneon, we can use the Stefan-Boltzmann law, which states that the power radiated by a black body (like the sun) is proportional to its temperature raised to the fourth power. The equation is given by:
P = σT^4
where P is the power radiated, σ is the Stefan-Boltzmann constant (approximately equal to 5.67 x 10^-8 Wm^-2K^-4), and T is the temperature.
Since it is mentioned that the sun's radiative output was 30% less than it is today, we can assume that the power radiated (P) during the Archeaneon is 70% of the current value.
Therefore, we can set up the following equation:
0.7P = σT^4
Solving for T, we get:
T = (0.7P/σ)^0.25
To find the temperature, we need to know the current power radiated by the sun. According to reference data, the current solar constant (average power flux received at the Earth's orbit) is approximately 1361 W/m^2.
So, substituting this value into the equation, we find:
T = (0.7 * 1361 / σ)^0.25
Calculating this equation will give us the temperature of the sun during the Archeaneon.
(ii) The peak emission wavelength (λmax) from a black body is given by Wien's displacement law:
λmax = b / T
where λmax is the wavelength, b is Wien's displacement constant (approximately equal to 2.898 x 10^-3 mK), and T is the temperature.
Using the temperature obtained in part (i), we can calculate the peak emission wavelength from the sun during the Archeaneon.
(iii) To determine the temperature of the Earth at that time (effective radiative temperature), we can use the Stefan-Boltzmann law again. However, we need to consider the albedo of the Earth as well.
The Earth's effective radiative temperature can be calculated using the following equation:
Teff = (√(L(1 - A))/(4εσ))^0.25
where Teff is the effective radiative temperature, L is the solar constant, A is the albedo, ε is the emissivity of the Earth (assumed to be 1), and σ is the Stefan-Boltzmann constant.
Substituting the values for L (current solar constant) and A (0.3), we can calculate the effective radiative temperature of the Earth during the Archeaneon.