Solar luminosity is estimated to have been 30% lower than today at the time when the solar system formed, 4.6 billion years ago.
(a) If Earth's albedo was the same as it is now (A=0.3), what would have been its effective radiating temperature Te at that time?
(b) If the magnitude of the greenhouse effect had also remained unchanged (Delta Tg = 33K), what would Earth's average surface temperature have been?
I highly recommend this reference for assistance:
http://www.atmos.washington.edu/2001Q1/211/notes_for_011001_lecture.html
(a) Earth's effective radiation temperature now is 255 K. (See above reference)
If the solar luminosity were 30% of what it is now, the effective radiating temperature would be lower by a factor (0.3)^(1/4) = 0.74, because of the Stefan-Boltzmann T^4 law.
That would make the effective radiating temperature 189 K.
(b) Sorry, but I don't understand the delta Tg concept. The reference quoted above has some discussion of the greenhouse effect, which may be helpful to you.
To calculate the effective radiating temperature (Te) at that time when the solar system formed, we can use the Stefan-Boltzmann law:
(a) The Stefan-Boltzmann law states that the power radiated by a blackbody is proportional to the fourth power of its temperature. The equation is given by:
I = σ * Te^4
Where I is the solar constant, σ is the Stefan-Boltzmann constant (5.67 × 10^-8 W m^-2 K^-4), and Te is the effective radiating temperature.
Given that the solar luminosity was 30% lower than today, we can calculate the reduced solar constant (Ir) using the following relationship:
Ir = (1 - 0.3) * I
The current solar constant is approximately 1361 W m^-2.
Substituting the values into the equation, we can solve for Te:
Ir = σ * Te^4
Thus,
(1 - 0.3) * I = σ * Te^4
Te^4 = (0.7 / σ) * Ir
Te = (0.7 / σ)^(1/4) * Ir^(1/4)
Calculating this value will give us the effective radiating temperature (Te).
(b) To calculate Earth's average surface temperature at that time, we need to consider the greenhouse effect. The greenhouse effect is the process by which certain gases in the Earth's atmosphere trap heat and raise the temperature at the surface.
The equation to calculate the average surface temperature with the greenhouse effect is:
Ts = Te + ΔTg
Where Ts is the average surface temperature, Te is the effective radiating temperature, and ΔTg is the magnitude of the greenhouse effect.
Given that ΔTg remained unchanged, we can substitute the values in and solve for Ts.
To calculate the effective radiating temperature (Te) of Earth 4.6 billion years ago when the solar luminosity was 30% lower and the albedo (A) was the same as it is now (A=0.3), follow these steps:
(a) Calculate the solar constant (St) at that time:
The solar constant represents the amount of solar energy received at the outer atmosphere of Earth. It is given by:
St = L / (4 * π * R²)
where L is the solar luminosity and R is the average distance between the Earth and the Sun.
In this case, the solar luminosity is estimated to have been 30% lower than today. If we know the current solar luminosity (L0), we can calculate the solar luminosity at that time (L):
L = L0 * (1 - 0.3)
Substituting the value of L, you can calculate the solar constant at that time.
(b) Use the Stefan-Boltzmann Law to calculate the effective radiating temperature (Te):
The Stefan-Boltzmann Law relates the temperature of a radiating body to the rate at which it radiates energy:
L = 4 * σ * π * R² * Te⁴
where σ is the Stefan-Boltzmann constant.
Rearrange the equation to solve for Te:
Te = ⁴√(L / (4 * σ * π * R²))
Substitute the values of L and R, which were calculated in step (a), to find the effective radiating temperature (Te) at that time.
To calculate Earth's average surface temperature when the magnitude of the greenhouse effect (ΔTg) remained unchanged, follow these steps:
(c) Calculate the adjusted effective radiating temperature (Te_adj):
Te_adj = Te - ΔTg
(d) Calculate the average surface temperature (Ts):
The average surface temperature is best approximated by the surface temperature at the blackbody effective radiation temperature (Te_adj).
Use the Stefan-Boltzmann Law again to calculate Ts:
Ts = (Te_adj)⁴
Substitute the value of Te_adj to find the average surface temperature of Earth 4.6 billion years ago with an unchanged greenhouse effect.
By following these steps, you will be able to calculate the effective radiating temperature (Te) at that time and the average surface temperature of Earth with an unchanged greenhouse effect. Note that these calculations are based on certain assumptions and simplified models, and actual values may vary.