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) Use the Stefan-Boltzmann formula, and assume an emissivity of 1. You will need the sun's current output.
You can use a simpler method if you know the sun's current surface temperature is 5800 K. You reduce the output 30%, T^4 has to be reduced by a factor 0.7.
(Told/Tnew)^4 = 0.7
Told/Tnew = 0.915
Told = 0.915*5800 = 5300 K
ii) Use the Wien displacement law. If you haven't learned it yet, learn it now. (Lamdamax)*T = 0.29 cm* K
iii) Write a radiation balance equation that says the received radiation from the sun (which is proportional to Tsun^4) equals the radiation emitted by the earth which is proportional to Tearth^4. Solve for Tearth. You will need the solid angle subtended by the sun. You can find online references of how this is done for the sun's actual temperature.
The earth's temperature should also end up being about 0.915 of its present average value (which is about 290K), making it about 265 K.
To answer these questions, we can use the Stefan-Boltzmann law, which relates the temperature of a radiating object to its emitted radiation. The law states that the radiant energy (E) emitted per unit time per unit area by a blackbody is proportional to the fourth power of its absolute temperature (T). Mathematically, it is represented as E = σT^4, where σ is the Stefan-Boltzmann constant.
(i) To find the temperature of the sun during the Archeaneon, we can assume that the sun was a blackbody radiator. We'll use the fact that the sun's radiative output was 30% less than it is today.
Let's take the current solar constant (the amount of solar energy received at the outer atmosphere of Earth) as approximately 1361 Watts per square meter. So, during the Archeaneon, the solar constant would have been 30% less, which is about 0.7 times its current value.
Using the Stefan-Boltzmann law, we can set the emitted radiation equal to the solar constant and solve for the temperature of the sun:
E = σT^4
0.7 * 1361 = σT^4
Now, we can rearrange the equation to solve for T:
T = [(0.7 * 1361) / σ]^(1/4)
By substituting the value of the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 Watts per square meter per Kelvin to four significant figures), we can calculate the temperature of the sun during the Archeaneon.
(ii) The peak emission from the sun corresponds to the wavelength at which it emits the most radiation. According to Wien’s displacement law, the peak wavelength (λmax) is inversely proportional to the temperature of the radiating body. Mathematically, it is represented as λmax = (b / T), where b is Wien's displacement constant, approximately 2.898 x 10^-3 meters per Kelvin.
By substituting the temperature obtained from part (i) into the equation, we can calculate the wavelength at which the peak emission from the sun would have been during the Archeaneon.
(iii) To calculate the effective radiative temperature of the Earth during the Archeaneon, we need to consider the balance between the incoming solar radiation and the outgoing radiation. The effective radiative temperature of the Earth is the equilibrium temperature at which the energy absorbed from the sun is equal to the energy radiated back into space.
Using the Stefan-Boltzmann law, we can equate the incoming solar energy (determined by the solar constant) with the outgoing energy radiated by the Earth. The albedo, which represents the proportion of incoming radiation reflected back into space, is given as 0.3.
Let's assume the Earth is a perfect blackbody radiator, so it absorbs all the incoming radiation. Therefore, the outgoing radiation can be calculated as:
Eout = σT_out^4
To find the effective radiative temperature, we need to solve for T_out:
Ein(1 - albedo) = Eout
(1 - 0.3) * Ein = σT_out^4
By substituting the solar constant (Ein) and the Stefan-Boltzmann constant (σ) into the equation, we can calculate the effective radiative temperature of the Earth during the Archeaneon.
Note: The calculations provided here are based on simplified assumptions and approximations. The actual values and conditions during the Archeaneon may vary.