Answer to Question #122980 in Astronomy | Astrophysics for Thalia

Question

Problem B.1: Temperature of Earth (6 Points)

Our Sun shines bright with a luminosity of 3.828 x 10 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.

litres of heating oil (energy density: 37.3 x 10 J/litre) is this equivalent?

(b) To how many (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 10 km; 1 AU is 1.495 x 10 km.

Answer 1

(a) Calculate the amount of energy arriving on 1 square meter of the Earth:

$$E_0=\frac{L}{A}=\frac{L_S}{4\pi R_S^2}=\frac{3.828⋅10^{26}}{4\pi (1.495⋅10 ^{11})^2}=1360\text{ W/m}^2.$$

This power comes to the earth's cross-section. Multiply by time and get energy:

$$E=E_0\pi R_E^2t,\\ E=1360\cdot\pi\cdot({6370\cdot10^3})^2\cdot(3600\cdot24)=1.5\cdot10^{22}\text{ J}.$$

(b) This is equivalent to the following volume of heating oil:

$$V=\frac{E}{u}=\frac{1.5\cdot10^{22}}{37.3\cdot10^6}=4.02\cdot10^{14}\text{ l}.$$

(c) The energy that the Earth receives is

$$E_E=E_0\pi R_E^2(1-\alpha)=1.21\cdot10^{17}\text{ W}.$$

Apply Stefan-Boltzman law to find the temperature:

$$E_E=\sigma AT^4=\sigma(4\pi R_E^2)T^4,\\\space\\ T=\sqrt[4]{\frac{E_E}{4\sigma\pi R^2_E}}=254\text{ K, or -19°C}.$$

(d) As we know, the average temperature on the Earth is 15°C. The difference is caused by the fact that earth's nucleus provides some heat to the planet as well as activity of living organisms. Also, the value of $\alpha$, which was taken as 0.3, is not a constant value and depends on weather and other factors.