Video Transcript
A satellite orbiting Earth is initially in Earth’s shadow. As the satellite moves around Earth, it passes into a part of the orbit where it is exposed to sunlight, as shown in the diagram. The satellite reaches thermal equilibrium before it passes back into Earth’s shadow. Which pair of the following sketch graphs most correctly shows how both the infrared radiation emitted by the satellite and the satellite’s temperature change while it travels through sunlit space?
So here we see two sets of sketch graphs. One has to do with the infrared radiation emitted by the satellite, and the other set has to do with the change in satellite temperature over time as it moves through sunlit space. For our answer, we’ll identify a pair of graphs, one graph from the first set and one from the second, that correctly describe these phenomena.
The physical situation we’re thinking about is a satellite orbiting the Earth. We’re told that initially the satellite is in a position so that it’s shielded from sunlight. But as it moves through its orbit, the satellite eventually is exposed to sunlight, and it’s exposed to sunlight all through this part of its orbit. That’s the section of its orbit that we’re considering. During this interval of time, we’re thinking about two things: first, how the infrared radiation emitted by the satellite changes, and second, how the temperature of the satellite changes.
Importantly, we’re told that, while the satellite is in the sun, it reaches thermal equilibrium. This means that the rate at which energy is transferred to the satellite from the sunlight it receives and the rate at which energy is transferred away from the satellite by infrared radiation at some point become equal to one another. This is when the satellite reaches thermal equilibrium. If we think about the satellite’s temperature throughout its orbit, we can imagine that at this moment, just as it emerges into the sunlight, it’s at a temperature minimum. At this point in its orbit, the satellite has not been exposed to sunlight and its warming effects for as long as possible, given its orbit.
But then as it moves through the sunlit portion of its orbit, the satellite’s temperature will increase. This is because of the sunlight that it absorbs, specifically the infrared radiation from the sun. If we think then about which of these four sketches correctly represents the change in temperature of the satellite over time, since we’re letting the initial time be the moment where the satellite crosses back into the sunlit portion of its orbit, we know that over the time interval of interest, the satellite’s temperature will begin to rise.
This means we can eliminate answer option (D) from consideration. That option shows the temperature of the satellite that remains constant, but we know this doesn’t happen. Considering the remaining three graphs, we recall that at some point in the sunlit portion of its orbit, the satellite, we’re told, reaches thermal equilibrium. As we’ve said, that involves a balance in the energy transferred to and the energy transferred from the satellite. When those transfer rates are equal, that means the temperature of the satellite will level out; it will reach a constant value.
If we look at graph (G), we see that this indicates a temperature which seems to be increasing at a faster and faster rate over time. Meanwhile, graph (H) shows a temperature that increases at a steady rate with time. But because our satellite does reach thermal equilibrium, we expect that graph (E) will accurately represent the change in temperature of the satellite. Note that this graph does show the temperature flattening out with time, reaching a constant value. This is the sign that the satellite is reaching thermal equilibrium.
We’ll choose graph (E) then to represent the satellite’s change in temperature over time during the sunlit portion of its orbit.
Now let’s think about how the satellite emits infrared radiation over the same part of its orbit. The infrared radiation given off by the satellite actually follows the temperature of the satellite fairly closely. When the temperature of the satellite is at a relative low point, as it is here, the satellite will not transfer much energy to its surroundings through infrared radiation. But then, as the satellite’s temperature goes up and up and up, this increasingly hotter object will transfer more and more energy away through infrared radiation.
There’s actually a connection between the energy radiated away from the satellite and the satellite’s temperature. The rate of energy emission through infrared radiation increases as temperature gets higher. But then since the temperature of the satellite reaches a steady value, as we’ve seen, due to the fact that it reaches thermal equilibrium, this means that the rate at which the satellite emits infrared radiation also reaches a level or constant value. This is required for thermal equilibrium. And we know that that equilibrium is achieved.
Just like for temperature — which starts at a low value, increases quickly, but then levels out — we expect the infrared radiation emitted by the satellite to follow the same progression. We choose graph (C) so that our final answer is that the pair of graphs most accurately representing on the one hand the infrared radiation emitted by the satellite and on the other its change in temperature over time are graphs (C) and (E).
