How cold would it be on Earth if the sun were not there?

Answer

A very nice question, with a lot of physics involved!

But first a little warning: if the Sun had never been there, the Earth wouldn’t be there either. Because planets like the Earth arise as a by-product of the formation of stars like the Sun. But from what we observe in other planetary systems, we have learned that in many of those systems planets can play billiards among themselves, whereby the mutual attraction can cause planets to be flung out of the system. It is highly unlikely that this will ever happen to us, but at the same time it is likely that ejected planets such as Earth are indeed among the stars.

How much such an isolated planet radiates depends on two things: the energy it receives from its surroundings, and the energy it still retains within itself.

The energy that planets can absorb in the tenuous regions between stars contains several components. There is initially a radiation everywhere that is the remnant of the big bang that started our universe; that ensures that there is a minimum temperature of about 3 Kelvin everywhere in the universe, three degrees above the absolute zero of the temperature, which is at -273 degrees Celsius; for example, there is a minimum temperature of -270 degrees Celsius for the ‘planet’. If the planet stays in our galaxy, it gets a similar amount of heat from the stars. If you add both forms of energy, you hardly get above those 3 degrees from there (be careful: it is not the case that if you add two amounts of radiation of 3 Kelvin each, you get 6 Kelvin; twice more radiation, means an increase in temperature of only 20 percent). A third component of energy that the planet can capture – and actually slightly larger than the previous two – comes from cosmic rays; these are high-energy particles floating around in the interstellar region, originating from various explosions that happen in the universe. All together you barely bring the planet above 5 Kelvin (-268 degrees Celsius).

For a planet like Earth, most of the radiated energy would come from its own accumulated energy. It contains two approximately equivalent components. First of all, there is the heat that was built up during the formation process of the planet, which has still not disappeared from the Earth. Planets are formed by the amalgamation of many small ‘planets’. As the planet grows in size, those “planets” fall at an increasing rate, and the energy from that incursion heats up the planet’s interior considerably. When the incursion process stops, the planet gradually radiates that energy, but it takes billions of years, and the Earth is still at it. The second form of energy that heats up the Earth’s interior is the result of the radioactivity of some elements in the Earth. When those elements disintegrate, it warms up the interior. That process is still ongoing: the decay time of some elements such as uranium and thorium is on the order of billions of years, so energy is still being released; moreover, it is again the case that that energy needs a lot of time to get out. These two processes together ensure that the Earth still dissipates heat to the outside. If there were only those processes (so no Sun), then that would ensure a surface temperature of about 40 Kelvin, ie -230 degrees Celsius.

The Earth’s own energy is therefore much greater than what it would receive from its environment, and the answer to your question is therefore ‘of the order of -230 degrees Celsius’.

The result thus depends on the amount of internal energy of the Earth and on the rate at which the Earth loses that energy. The amount of energy depends on the volume of the Earth, which is proportional to the cube of the radius. The energy loss depends on the size of the area through which that energy is lost, and that area depends on the radius squared. Dividing the contents of the ‘tank’ by the ‘consumption’ says something about how long the process can take. So here that ratio is divided by ‘radius to the third’ divided by ‘radius squared’, so that is proportional to the radius. The larger a planet, the longer it can maintain and radiate its internal energy. That it is not yet done for the Earth, we see from the fact that volcanoes still erupt! But on a smaller planet like Mars, where we still see many extinct volcanoes, we notice that there has been hardly any volcanic activity for a long time. A smaller planet thus has hardly any internal energy left, and would therefore be much colder if it had to reside only in the interstellar center.