Department of Physics, Carnegie Mellon, Fall ’13, Deserno/Franklin
Homework 18, due in recitation on Thursday, December 5th
The weight of sunlight
In this problem we’re going to combine some astrophysics question with a bit of relativity—using the most
famous of all physics equations in the process.
According to Einstein’s special theory of relativity, energy is equivalent to mass: E = mc2 . In this
equation, m is the mass of of the thing we’re contemplating, and c is the speed of light; if the mass is
measured in kilogram and the speed of light is measured in meters per second, the answer will come out
in joule, because kg m2 /s2 = J.
In this problem we apply this equation to the energy which the Earth receives from the sun. We know
the power of sunlight (recall our discussion in the context of solar cells?), and we know the circular area
which the Earth faces towards the sun (the Earth’s radius is approximately 6366 km, courtesy of Google).
What would be the total relativistic mass m = E/c2 of all the sunlight hitting the Earth in one day, if we
were to take Einstein’s equation and convert that sunlight’s energy into an equivalent mass?
Let’s find out!
1. What’s the cross-sectional area which the Earth displays towards the sun at any given point in time?
2. What’s the total energy which hits this area per second? And per day?
3. What’s the mass that is equivalent to this energy per day?
4. If all that energy were absorbed by the Earth, by how much would the mass of the Earth increase in
one day? How does this compare to the average amount of about 40 tons of extraterrestrial material
(mostly cosmic dust) which also hits the Earth every day?
5. In fact, our Earth does not gain that mass from solar radiation per day, but that is not because the
equation E = mc2 would not apply to light. The reason is a very different one. Which one?
Hint: recall what we learned in the climate chapter about the total energy balance of the Earth!