Would isolating the Solar System prevent its heat death?

[Feynstein100]

I have a thought experiment in mind. Crudely speaking, the second law of thermodynamics implies that there is only a finite amount of change possible in the universe. Once this limit is reached, no more change can occur. The key thing here though is reversibility. If changes were reversible, then the entropy change would be zero and we could stay at the same entropy level indefinitely.
Considering only the Solar System, the sun is burning up its mass and converting it to energy, which then spreads out across the Milky Way and the rest of the universe. Again, according to the second law, once energy gets spread out, it cannot be concentrated again without expending more energy. So what if we stopped the energy from spreading out? Let's say we have some kind of Dyson sphere but instead of enclosing just the Sun, it encloses the entire Solar System. This hypothetical Dyson sphere is made out of ideal mirrors that reflect all of the Sun's energy without absorbing any of it, i.e. now all of the Sun's energy comes back to it and doesn't get spread out. What would happen in this theoretical scenario?
Would this cause the entire Solar System to be in a reversible state? The sun emits energy. This energy gets reflected back to the sun, and then re-emitted by the sun again, closing the loop. Of course, fusion would no longer occur but that's kind of the point. The sun is no longer burning itself up, and yet still shining.
Although, any closed system should eventually reach an equilibrium temperature across its entirety. So will that happen in this case too? Will the entire Solar System reach some equilibrium temperature and then not change anymore? Which, ironically, is heat death. I'm not sure what would happen. My mental simulation of this is kinda stuck lol. Could anybody help me out here?

 

[Drakkith]

If the Sun is still fusing: the solar system and everything in it continues to heat up. The rocky planets eventually become fully molten again, while the gas giants puff up and become superhot balls of gas. The Sun's photosphere heats up because of all the reflected radiation and its spectrum slowly shifts more and more towards the UV end. The inability to get rid of energy and heat causes the Sun's interior to heat up. Fusion occurs more rapidly in the core, which increases the energy generated, which raises the internal temp, which just further adds to the whole process, becoming a massive positive feedback loop.

Eventually, perhaps before the planets turn into superheated plasma from the ever increasing temperature from the Sun and the reflected radiation, the Sun either 1.) explodes from runaway fusion from the ever increasing temperature, 2.) puffs up into a giant sphere of plasma that engulfs most of the solar system and never shrinks, 3.) completes its multi-billion-year lifecycle in record time and we are left with a solar system that just consists of a variety of 'dead' objects sitting at thousands or millions of degrees (probably the latter, but maybe even hotter).

Keep in mind that the Sun contains about $10^{44}$ joules of potential fusion energy. This is about 500x more than its gravitational binding energy, which is the amount of energy it would take to disassemble it and pull all of its pieces very far away. This is a stupid amount of energy. If none of this energy can escape the solar system, the entire system would heat up beyond my ability to accurately calculate.

Of course, fusion would no longer occur but that's kind of the point.

Why wouldn't fusion still occur?

 

[Drakkith]

Although, any closed system should eventually reach an equilibrium temperature across its entirety. So will that happen in this case too? Will the entire Solar System reach some equilibrium temperature and then not change anymore?

Yes. Ultimately, whatever actually happens to the Sun is irrelevant. Eventually the entire solar system would be at the same temperature, whatever that happens to be in this hypothetical scenario.

 

[DaveC426913]

The sun emits energy. This energy gets reflected back to the sun, and then re-emitted by the sun again, closing the loop.

This is the flaw, as others may have pointed out. It is not reversible.

The reflected energy does not return to the sun, and it cannot "re-emit" it. Instead, the reflected heat will be distributed within the sphere.

Will the entire Solar System reach some equilibrium temperature and then not change anymore?

This.