The Sun emits energy at a rate of about 3.9 · 1026W. At Earth, this sunshine gives an incident energy flux Ie of about 1.4kWm−2. In this problem, you’ll investigate whether any other planets in our solar systemcould support the sort of water-based life we find on Earth. Consider a planet orbiting at distance d from the Sun (and let de be Earth’s distance). The Sun’s energy flux at distance d is I = Ie(de/d)2, because energy flux decreases as the inverse square of distance. Call the planet’s radius R, and suppose that it absorbs a fraction α of the incident sunlight, reflecting the rest back into space. The planet intercepts a disk of sunlight of area πR2, so it absorbs a total power of πR2αI. Earth’s radius is about 6400 km. The Sun has been shining for a long time, but Earth’s temperature is roughly stable: The planet is in a steady state. For this to happen, the absorbed solar energy must get reradiated back to space as fast as it arrives (see Figure 1.2). Because the rate at which a body radiates heat depends on its temperature, we can find the expected mean temperature of the planet, using the formula radiated heat flux = ασT4. In this formula, σ denotes the number 5.7·10−8Wm−2 K−4 (the “Stefan–Boltzmann constant”). The formula gives the rate of energy loss per unit area of the radiating body (here, the Earth). You needn’t understand the derivation of this formula but make sure you do understand how the units work. a. Using this formula, work out the average temperature at the Earth’s surface and compare your answer to the actual value of 289K. b. Using the formula, work out how far from the Sun a planet the size of Earth may be, as a multiple of de, and still have a mean temperature greater than freezing. c. Using the formula, work out how close to the Sun a planet the size of Earth may be, as a multiple of de, and still have a mean temperature below boiling. d. Optional: If you know the planets’ orbital radii, which ones are then candidates for water-based life, using this rather oversimplified criterion?