Answer to Question #122772 in Astronomy | Astrophysics for OLJOOO

Question

Our Sun shines bright with a luminosity of 3.828 x 1026 Watt. Her energy is responsible for many processes and the habitable temperatures on the Earth that make our life possible.

(a) Calculate the amount of energy arriving on the Earth in a single day.
(b) To how many litres of heating oil (energy density: 37.3 x 106 J/litre) is this equivalent?
(c) The Earth reflects 30% of this energy: Determine the temperature on Earth’s surface.
(d) What other factors should be considered to get an even more precise temperature estimate?

Note: The Earth’s radius is 6370 km; the Sun’s radius is 696 x 103 km; 1 AU is 1.495 x 108 km.

Answer 1

(a) Let us calculate the solar constant, namely the flux density (see https://en.wikipedia.org/wiki/Solar_constant)

$E_0 = \dfrac{L_{\odot}}{4\pi r^2} = \dfrac{ 3.828 \cdot 10^{26}\,\mathrm{W}}{4\pi \cdot (1.495\cdot10^{11}\,\mathrm{m})^2} = 1.36\cdot10^3\,\mathrm{W/m^2}.$

We should multiply this amount by the area of Earth cross-section and the number of seconds in a day:

$\mathbb{E} = E_0\cdot\pi R^2_{\oplus} \cdot86400\,\mathrm{s} = 1.5\cdot10^{22}\,\mathrm{J}.$

(b) The number of litres will be

$N = \dfrac{\mathbb{E}}{37.3\cdot10^6\,\mathrm{J/l}} = 4\cdot10^{14}\,\mathrm{litres}$ .

(c) The energy absorbed by Earth is $E_0\cdot\pi R_{\oplus}^2\cdot(1-A) = 1.2\cdot10^{17}\,\mathrm{W}.$ All these energy should be re-emitted by Earth. According to the Stefan–Boltzmann law (see https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) this energy is equal to $\sigma 4\pi R_{\oplus}^2 T_{\oplus}^4.$

Therefore, $E_0\cdot\pi R_{\oplus}^2\cdot(1-A) = \sigma 4\pi R_{\oplus}^2 T_{\oplus}^4, \;\; T_{\oplus} = \sqrt[4]{\dfrac{E_0(1-A)}{4\sigma}} = 254\,\mathrm{K} = -18^\circ\,\mathrm{C}.$

(d) We know that the average temperature of Earth is approximately $15^\circ\,\mathrm{C}.$ The inconsistency between two values is mostly due to the greenhouse effect. We should also take into account the human production of energy, the geothermal heat, the internal heat of the Earth (see https://en.wikipedia.org/wiki/Earth%27s_energy_budget). Some amount of solar energy is transformed into biomass by the plants. We should also remember that the albedo of the Earth is not constant: it may be changed due to variations in the area of polar ice caps, during the volcanic eruptions due to the emission of volcanic dust.