# Why is Sun's energy entropy low on Earth?

Warning I don't have a physics background, having said that I was recently looking into the sun's radiation energy entropy and I had a couple of questions.

So it is said that we can utilize the energy from the sun on Earth because of its low entropy. The entropy according to the second law of thermodynamics can be estimated as: $$S=\dfrac{E}{T_{sun\_surface}}$$ [1]

Now given that the energy received from the sun is exponentially proportional (inverse square law) to the distance from the sun, does it mean that the closer we get to the sun the higher the entropy of the energy of the sun would be? (because the E would get bigger and bigger)

Is this because the earth would heat up faster and faster the closer we get to the sun? what is some intuitive explanation of why this is expected to happen?

Asking because normally I don't see people giving distance as one of the major factors when they explain why the sun has low entropy, eg [2]

Edit: ok, so distance doesn't seem to matter. After reading [3] I would summarize the following based on my understanding: because the energy we receive from the sun is emitted by a much hotter surface (sun's) compared to the objects that receive it here on earth (eg 300K), as a whole is expected to be lower entropy (interesting note: the spectral radiance of the sun is set to maximize entropy given its temperature/energy given its similarity to blackbody radiation). since the sun's radiation displays similar characteristics to blackbody radiation and its surface temperature is high, its spectral radiance is more concentrated in the visible range. This combined with the fact that smaller wavelength radiation has lower entropy allows us to have more usable energy in the visible and near visible range, ie higher energy[4]. Some energies in the longer wavelengths may or may not be usable due to either low entropy or low intensity through the exergy definition

The amount of radiant power that passes through this area is called the solar constant and is equal to 1373 Joules/second (Lide 2004-5: 14-2). In the absence of the earth's atmosphere, the entropy of this sunlight would equal this energy divided by the temperature of the sun's surface, known from spectroscopy to equal 5780 K. The result would give the entropy of this amount of sunlight as 0.238 J/K every second

In contrast, the Sun's temperature is enormous—around 5500°C, which makes the denominator of the effective entropy term S=E/T quite small. Thus, it's not the energy of the sunlight that's particularly useful—it's its low entropy

• Where did you get the relation $S=E/T_\text{sun}$? Commented Jul 19 at 7:20
• Could you please cite the sources (and possibly even quote the relevant statements)? It is sometime shard to understand what is meant in popular explanations, and even harder when one hears this explanation second-hand from a non-physicist. No offense intended. Commented Jul 19 at 7:58
• This may be useful (and links therein) physics.stackexchange.com/q/796434/226902 physics.stackexchange.com/q/693008/226902 Commented Jul 19 at 7:59
• The max. amount of energy that one can extract from the sun should be given roughly by a Carnot process between the temperature of the sun's surface (which is pretty close to a black body) and the temperature of space (i.e. the temperature of the CMB). I don't see how this would depend on the distance of Earth to the sun. At most the Carnot machine would have to be much, much larger in area far from the sun than it would have to be if placed close to the sun. Am I missing something? Commented Jul 19 at 10:36
• The source [1] confuses me. If two systems are in detailed balance, or thermodynamical equilibrium, entropy does NOT increase. So the calculation with the sun radiatiating and our earth's surface having constant temperature does not seem to work out in such a way. But then again, it also states our earth is (and this is true) a non-equilibrium system. Commented Jul 19 at 10:36

It is said that we can utilize the energy from the sun on earth because it's low entropy

Actually, we can utilise energy from the Sun because the temperature of the Sun is higher than the temperature of outer space i.e. the Sun is a hot area in an otherwise cold sky. This allows us to capture a given amount of energy $$E_1$$ from the sun, extract some work $$W$$ from this energy, and radiate the remaining waste energy $$E_1-W$$ into space without breaking the second law of thermodynamics.

Does it mean that the closer we get to the sun the higher the entropy of the energy of the sun would be ?

The entropy per unit amount of energy, which is $$S/E$$, remains the same because this is equal to $$\frac 1 {T_{sun}}$$. The entropy per unit area per unit time increases because the energy received per unit are per unit time increases following the inverse square law (basically because at a closer distance the amount of energy emitted by the sun in all directions stays the same but it is spread over a smaller area).

Is this because earth would heat up faster and faster the closer we get to the sun ?

Not really. The relationship between the surface temperature of a planet and its distance from the sun is complex, and depends on factors such as the depth and composition of the planet's atmosphere and its day length. The average surface temperature of Mercury, for example, is actually lower than the average surface temperature of Venus.

• this answer seems to contradict your first point with regards to having as only requirement a cold sink. it seems to suggest that distance is also an important factor for entropy physics.stackexchange.com/q/796490 Commented Jul 19 at 11:13
• @MarinosEftichiou The entropy per unit amount of energy, which is $S/E$, remains the same - it does not depend on distance. And it is this entropy that determines the maximum proportion of energy received from the sun that can be converted into work. Commented Jul 19 at 11:27
• so based on this, you would say that entropy per energy remains the same regardless of distance and the linked stackexchange answer is potentially a more elaborate explanation why even with the varying energy flux resulting from different distances which could result in changes in entropy, those entropy changes balance out with the points mentioned in (2) and (3). Note, in the referenced answer in the first comment, i think he implies that S1=S2+S3 but i cant comment to verify. I assumed that S2+S3>S1 Commented Jul 19 at 13:00

Near the sun the temperature of the thermal energy of the photon gas is quite high, and the amount of thermal energy of the photon gas is quite high, which is why the entropy of the photon gas is quite low.

Far away from the sun the temperature of the thermal energy of the photon gas is quite low, and the amount of thermal energy of the photon gas is quite low, which is why the entropy of the photon gas is quite low.

The temperature of photon gas is measured by a thermometer that is at rest relative to the gas, obviously.

Hmm ... in case you guys don't understand, I will say it in another way:

Near the sun a photon-gas pressure meter reads a small value. Far away from the sun, a photon-gas pressure meter reads a very small value. According to Bernoulli's equation pressure energy becomes kinetic energy when the photon gas flows in space.