Exercise 1:

  1. What is Te for the Earth? 255 K

  2. Calculate the surface temperature for the earth if the emissivity of the atmosphere is 0.5. \[ \begin{align*} T_s &= T_e\left[\frac{1}{1-\frac{\epsilon_{atmos}}{2}}\right]^{\frac{1}{4}} \\ &= 255\left[\frac{1}{1-\frac{0.5}{2}}\right]^{\frac{1}{4}} \\ &= 255\left[\frac{4}{3}\right]^{\frac{1}{4}} \\ &= 274.015... \\ &\approx 274\ K \end{align*} \]

  3. Using R plot Ts over the full range of possible (atmospheric) emissivity values from \(\epsilon=0\) to \(\epsilon=1\). Insert the plot below.

# Set parameters
effective_temp <- 255
emissivity.atmos = seq(0,1, length=1000)

# Define function for surface temperature
surface_temperature = function(effective_temp, emissivity){

  t.s = effective_temp * (1 / (1 - emissivity / 2))^0.25
  return(t.s)
}

# Plot surface temp over range of emissivity values
plot(emissivity.atmos, surface_temperature(effective_temp, emissivity.atmos), type="l", 
     ylab = "Surface Temperature (K)", xlab = "Atmospheric Emmissivity",
     col = "firebrick", ylim = c(250, 310),
     main = "Surface temperature for 255 K with emissivity values from 0 to 1")
# Add axis ticks
axis(2, at = seq(255, 305, by=10), labels = FALSE, tcl = -0.25)
axis(1, at = seq(0.1, 0.9, by=0.2), labels = FALSE, tcl = -0.25)

# Add grid lines
abline(v = seq(0, 1, by = 0.1), col = "lightgray", lty = 2)
abline(h = seq(250, 310, by = 5), col = "lightgray", lty = 2)

  1. From the graph in part (c) estimate the emissivity of the Earth’s atmosphere given that the average measured value of the surface temperature is 288 K.

For Ts to be 288, \(\epsilon_{atmos}\approx0.8\).

  1. By re-arranging the relevant equation to make \(\epsilon_{atmos}\) the subject, calculate a more accurate measure for the emissivity of the Earth’s atmosphere. Assume the surface temperature of the earth is 288 K.

\[ \begin{align*} T_s &= T_e\left[\frac{1}{1-\frac{\epsilon_{atmos}}{2}}\right]^{\frac{1}{4}} \\ \left[\frac{T_s}{T_e}\right]^4 &= \frac{1}{1-\frac{\epsilon_{atmos}}{2}} \\ 1-\frac{\epsilon_{atmos}}{2} &= \left[\frac{T_e}{T_s}\right]^4 \\ \epsilon_{atmos} &= 2\left(1-\left[\frac{T_e}{T_s}\right]^4\right) \\ \\\ \epsilon_{atmos} &= 2\left(1-\left[\frac{255}{288}\right]^4\right) \\ &=0.770805... \\ &\approx 0.771 \end{align*} \]