So you say the max temp on earth by this method is the sun's temperature. Why wouldn't this be enough?

It may, but source says "You will need: Means for focusing a good few percent of the Sun's energy output directly on the Earth.", then posits using mirrors to do this. Mirrors could get Earth to something approaching 5800 kelvin, but that is nowhere near "a few good percent of the sun's energy output."

How much energy is a good few percent?

The sun has surface area of 6 x 10^18 m^2. Earth has surface area of 5 x 10^14 m^2. Earth can not be bombarded by more than the ratio of these to numbers with solar radiation using only mirrors. <<1 percent.

I see what you mean, but it seems like a self-correcting problem. Focus enough of the sun's energy on Earth, and it will be shrouded with a rapidly expanding ball of hot plasma, at which point its surface area would be much greater. Now you can focus even more of the sun's energy on it without conjuring Maxwell's demons.

I don’t think that is true. Energy can be focused to a smaller volume with a higher temp and you can keep adding energy. This is how laser fusion works.

Author doesn't speak of lasers. Author says to use mirrors. The sun takes up about 1/2 degree width as seen from Earth. This limits the focus you can get using mirrors. (see 2nd law of thermodynamics) You could put solar panels on every surface of a Dyson swarm, convert to laser light, and aim it all at Earth, but that is not what author proposes.

The degrees dont matter and you dont need to convert to laser light. You can have a dyson sphere of mirrors to reflect the entire energy output of the sun.

As you say, The sun has surface area of 6 x 10^18 m^2. Earth has surface area of 5 x 10^14 m^2 for a ratio of 10^4.

IF you reflect all of this energy to the earth, the incoming energy per meter on the earth will be 10^4 higher than the energy per meter leaving the surface of the sun.

The surface of the sun is 5,700K and can use the Stefan–Boltzmann law to describe how hot the earth would have to be to radiate the same amount of energy. Earth would need 57,000K to match the the same output.

You can't reflect all the energy of the sun onto the earth with a mirror or a lens. Put a one million mile wide source 93 million miles from the earth. The photons from the left side of the sun will reflect off the mirror about 1/2 degree from those from the right side of the sun. Get more than about 1 million miles from earth, and the size of the reflection of sunlight off the mirror is already larger than the earth, i.e. many of those photons will miss the earth entirely. Optics obeys the 2nd law of thermodynamics like all other heat transfer mechanisms. The target must be cooler than the source.

One of the main reasons why optics obeys the 2nd law of thermodynamics is because as the temperature of the earth approaches that of the sun, the earth starts emitting radiation approaching the same intensity, and the net influx of energy goes down towards zero.

That temperature should still be enough to make a fair dent though.

I think that helps. You’re saying that because the sun is not a Point Source and the rays aren’t parallel.

Following that logic, couldn’t you still put a dyson mirror around thesis and earth raising the sun and earth temp above 5700? This would raise the temp of both above 5700.

Interesting idea. I think if you put perfectly reflective mirrors inside a Dyson sphere larger in radius than Earth's orbit, the photons would bounce around until they hit the Sun, Earth, the Moon, Venus or Mercury. It seems before too long, (a few reflections), the radiation would be uniform from all directions throughout the volume of the sphere. The mirror would get as warm as all the planets, and would fail at some not too extreme temperature. Unless I'm missing something.

I Im not sure that the mirror has to be at the same temperature as a heat source. The area of a dyson sphere at earths orbit would be 2.810^23 m^2 The area of the sun is 610^18 m^2.

This is where my understanding of optics and reflections breaks down. How much heat/kinetic energy is transferred to a mirror when it reflects a photon?

From a heat transfer perspective. the dyson sphere has 100,000X the surface area of the sun to radiate heat and 1/100,000 the photon flux hitting it's surface. Based on this I would expect it to be cooler.

2nd law says heat flows from hot to cold. Mirrors are passive, temperature of target, (Earth), must be lower than surface temperature of source (Sun).

As a second point, even if mirros can't make the earth hotter than the sun, Why isnt the suns temperature enough to vaporize the the earth?

The sun is a plasma bound together by gravity and earth does not have enough gravity to retain it's mass?