Who would win? One Sun sized Earth or 1.3 million Earth sized Suns?
The short answer is neither, they both die.
Why is the Sun sized Earth Dead?
First off the Sun sized Earth is dead, no question. Why? Well first we have to make the assumption that the Earth simply ADDS mass to be as large as the Sun. So how much mass is being added to the Earth? If we assume the Earth is just a sphere we know that the Earth has a radius of 6,371 kilometres with a mass of 5.972 * 1024 kg. The Sun, on the other hand, has a radius of 696,340 kilometres, more than 100x the radius of Earth. Because the mass of an object is proportional to it's volume and the volume of a sphere is proportional the cube of the radius we can calculate that the new mass of the earth is 7.798 * 1030 kg!
That's bloody heavy, on the surface of our new Earth-Sun gravity is 1.073 km/s, this is more than 40x the surface gravity on Jupiter and 4x the surface gravity of our Sun. A house cat would weigh as much as a grand piano on the surface! So that's great, because a lot of pressure and gravitational force means a lot of energy and because our Earth weighs MORE than the Sun you'd expect it would turn into a Sun right?
The problem is the Earth is made of very different things than our Sun. You see the Sun gets its energy from the fusion of hydrogen into helium and this fusion releases a tremendous amount of energy which causes the Sun to expand against the gravitational force the Sun is experiencing. The problem is Earth's core is not made of hydrogen but it's made of iron which cannot be fused to create energy. Iron is the most nuclear stable element in the universe and so changing it into something else such as via fusion (or even fission) actually costs energy instead of gives energy like in the case of hydrogen (or uranium for fission). The outer layers of the Earth have less iron and have elements such as silicon and oxygen that CAN fuse but these requires a tremendous amount of pressure to fuse and even our Earth-Sun is not massive enough to fuse it. The hydrogen found within a planet is simply too sparse to cause any kind of real fusion.
So what would our Earth-Sun become? Well it wouldn't remain a planet, it is simply beyond the size and mass to function as a rocky planet. The tremendous amount of pressure the core would experience would be far beyond what the core would maintain and so the planet would immediately begin to collapse in on itself. The collapse would be equivalent to a stellar supernova as a tremendous amount of energy is released in the violent collapse, dispersing a huge amount of mass outwards. Our Earth-Sun is not QUITE heavy enough to form into more exotic stellar objects like a black hole or neutron star, instead the core collapses to become extremely dense degenerate matter. Degenerate matter is not held up by the electromagnetic force like most matter is, instead it's held up by Quantum Mechanics via a principle called the Pauli Exclusion Principle. In a nutshell matter (specifically fermions like electrons, protons and neutrons) don't like to be in the same quantum state as each other but gravity is trying to force them into the same state by compressing them a tonne and so Pauli Exclusion fights against it in what's called degenerency pressure. The degenernecy pressure is more than the gravitational energy for our Earth-Sun so the core stops from collapsing completely into nuclear matter (IE: Into a neutron star).
Degenerency matter however is extremely dense at about 10000 kg/cm3, this is approximately two million times denser than Earth is. This means our Sun sized Earth quickly becomes much smaller with a radius of just 1776 km according to this website. That's shockingly close to the size of the moon! The surface gravity of the Earth-Sun is now an astounding 165,001 km/s! A house cat on this Earth-Sun would weigh as much as the Titanic! The Earth-Sun is now officially what is called a white dwarf, usually this is the result of a dead star but the Earth-Sun fits the bill quite nicely. The heat of all the energy from gravity and degenerency pressure causes the Earth-Sun to heat up to well above 30,000 C which causes it to shine brilliantly. The Earth-Sun is essentially dead, in that it's not gaining any energy and only losing energy for the rest of it's life, however it's losing energy due to thermal radiation outwards which is fairly inefficient for massive objects so you'd expect the Earth-Sun to shine for a few billion years, maybe much, much longer!
Why are the Earth sized Suns Dead?
There are two possible answers to this question. The first is a boring one because if we just simply take the components of a star and put them in a mass sized earth we essentially just get a big cloud of gas that behaves pretty inertly. Gravity would heat up the gas and cause the formation of what's called a gas planet (essentially a small gas giant) but there wouldn't be enough energy for fusion and so nothing interesting would happen. Pretty boring. So let's make it more interesting and instead we say what if we got the core of an actively fusing star and then suddenly teleport the core to only have the mass of the earth while the fusion was still ongoing? And what if we repeated that 1.3 million times?
As said before the Sun produces a certain amount of energy from fusion which fights against the energy due to gravity. The sun fuses 620 million tonnes of hydrogen every second, fusion carries away 0.7% of the mass of mass into energy (the rest forming into helium). Using Einstein's famous E = mc2 equation we get that the Sun produces 5.58*1025 Joules of energy per second. That's the equivalent of one hundred million of the most powerful thermonuclear bombs going off at once, every second. But unfortunately we have to keep in mind the premise of the question, we're REDUCING the size of the Sun to the size of the Earth so we're gonna reduce the Sun's mass. Using the same proportion as we calculated before we know that the Sun is about 331,000 times heavier than the Earth so we're gonna reduce the energy to that proportion. So for one Sun-Earth we release about 1.683*1020 joules of energy with the fusion stopping immediately because the Sun-Earth is not massive enough to keep fusion going. This is now only 100 times more powerful than the most powerful thermonuclear bomb so it's not nearly as impressive despite the fact we're causing a tremendous amount of destruction. All in all sadly the fate of our Sun-Earth even in this scenario doesn't change much. Essentially a large chunk of gas would be blown away by the explosion but ultimately the explosion probably wouldn't be powerful enough to blow away all the gas and so the gas will eventually condense again after millions of years and form a slightly smaller gas planet. All in all it's a little lame, but we need to remember we have 1.3 million of these little bombs...
So who would win?
Well as I said before they both kinda die, but maybe the question of who would win is less about who survives and more about how spectacularly they die. So a better question is who produces the most energy?
We'll start with our Earth sized Suns, we calculated before that one Earth sized Sun produces 1.683*1020 Joules of Energy upon it's death but we have 1.3 million of them giving a total of a massive 2.188 * 1026 joules of energy or over ONE BILLION of the most powerful thermonuclear weapons humanity has ever created all going off at once. That's a very big bomb!
But this, unfortunately, absolutely pales in comparison to the bomb we created with our Sun sized Earth. As I said before the collapse of our Sun sized Earth releases as much energy as a small supernova. To calculate the energy released we notice that the energy comes from the difference in the gravitational binding energy. The equation for gravitational binding energy is given as:
We have to calculate how much energy was released by comparing the gravitational binding energy from when the Earth-Sun was the size of the Sun to when it was the size of a white dwarf (the moon). The initial gravitational binding energy is -1.748 *1043 Joules, that's a LOT but we have deduct it from the gravitational binding energy of our white dwarf which is even more massive at -6.855 * 1045 Joules! This means the total amount energy we outputted is a massive 6.837 * 1045 Joules! That's 30 quintillion times more powerful than our 1.3 million Earth sized Suns could ever hope to achieve. How many of the most powerful thermonuclear bombs is it? It's 1,000,000,000,000,000,000,000,000,000 many thermonuclear bombs going off.
So both stars die, but the Sun sized Earth dies much more spectacularly than the 1.3 million Earth sized Suns, and now we know!
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