1. Temperatures on planets. To a good approximation, stars and planets emit their energy like
a black body, which means the energy flux (energy per unit area and per unit time) at the solar
surface (radius RS = 7 × 108 m) is given by
F = sSBT
4
, (1)
where sSB = 5.67×10-8 W m-2 K-4
is the Stefan-Boltzmann constant. [Remember, energy per unit
time is power and it is measured in watts (1 J s-1 = 1W).]
(i) Using T = 5778 K for the Sun, compute the energy flux at the solar surface and at a distance
r = 1 AU = 1.5 × 1011 m. (Hint: remember that the energy flux decreases with distance
proportional to 1/r2
.)
(ii) The cross-sectional area of the Earth is pR2
E
, where RE = 6×106 m is the Earth radius. (Don’t
worry about more than just the first or second significant digit.) How much power does the
Earth receive from the Sun? How much more is this than the average consumption in the
world, which is ˜ 1013 W?
(iii) Next, take into account that 30% is reflected (e.g., by clouds and ice). What is now the power
received by the Earth?
(iv) How much power does the Earth emit on average at its surface, assuming an average temperature
T = 300 K? (Hint: use Equation (1) and the fact that the Earth surface area is 4pR2
E
.)
(v) Compare your answers from parts (ii) and (iv). Explain why the Earth appears to release more
power than it receives from the Sun. (Hint: imagine how your body temperature changes if
you wrap yourself up in a blanket.)
(vi) Apply the formula derived in Lecture 9 (page 51 of Lectures 8+9) to estimate the average
temperatures on Venus, Earth, Mars, and Titan, assuming A = 0 for the albedo (nothing is
reflected). (Hint: the distances are 0.7 AU, 1.0 AU, 1.7 AU, and 9.6 AU.) Give the results in
Kelvin, Celsius (TC = T – 273 K), and Fahrenheit (TF = 9 TC/5 + 32), where TC and TF are
the temperatures in Celsius and Fahrenheit, respectively.
(vii) Using the equation from page 48 of the lecture slides, repeat your temperature calculation for
the Earth using 0.3 as a value for the albedo, A. Compare your answer to the value of 300 K
assumed in part (vi) – by what factor is your temperature enhanced/diminished?
(viii) Finally, let’s look at Venus. Compare your calculated temperature for Venus in part (vi) to the
actual surface temperature of 730 K. By what factor is your estimate for the Venusian surface
temperature enhanced/diminished compared to the real-world Venusian temperature?
2. Surprises in 40 K decay and 40Ar production.
(i) A rock sample from the Canadian shield is found to have the following elemental concentrations:
[
40Ar] = 7.63 × 10-7 and [40 K] = 10-6
. If the half-life of 40 K is 1.2 Gyr, how old is this rock
sample? (Hint: you may use the equation on page 21 of Lectures 8+9. Also, remember that
log2 x = ln x/ ln 2.)
1
(ii) When my grand-grand-grand-…-daddy was young, some 3.6 Gyr ago, he placed a rock containing
3 g of 40 K with no (0 g) 40Ar on his bookshelf in a sealed container, for my students to analyze. I
just found it, and so finally it’s your turn: How much 40 K and 40Ar should you find? Remember
that the latter grows like 0.109 × (1 – 2
-t/t1/2 ) for 1 g of initial 40 K; see again page 21 of
Lectures 8+9.
(iii) What are the ratios of [40Ar]/[
40 K] that you found in parts (i) and (ii)? Do you notice anything
peculiar about these values?