What is the easiest way to stop a star?

Burning (and fusion) is "unsustainable" by definition because it means to convert an increasing amount of fuel to "energy" plus "waste products" and at some moment, there is no fuel left.

I am not sure whether the word "unsustainable" was used as a joke, a parody of the same nonsensical adjective that is so popular with the low-brow media these days, but I have surely laughed (because it almost sounds like you are proposing to extinguish the Sun to be truly environment-friendly). The thermonuclear reaction in the Sun has been "sustained" for 4.7 billion years and about 7.5 billion years are left before the Sun goes red giant. That's over 10 billion years – many other processes are much less sustainable than that. More importantly, there is nothing wrong about processes' and activities' being "unsustainable". All the processes in the real world are unsustainable and the most pleasant ones are the least sustainable, too.

But back to your specific project.

When it comes to energy, it is possible to blow a star apart without spending energy that exceeds the actual thermonuclear energy stored in the star. Just make a simple calculation for the Sun. Try to divide it to 2 semisuns whose mass is $10^{30}$ kilograms, each. The current distance between the two semisuns is about $700,000$ kilometers, the radius of the Sun. You want to separate them to a distance where the potential energy is small, comparable to that at infinity.

It means that you must "liberate" the semisuns from a potential well. The gravitational potential energy you need to spend is $$ E = \frac{G\cdot M\cdot M}{R} = \frac{6.67\times 10^{-11}\times 10^{60}}{700,000,000} = 10^{41}\,{\rm Joules} $$ That's equivalent to the energy of $10^{24}$ kilograms (the mass of the Moon or so) completely converted to energy via $E=mc^2$, or thermonuclear energy from burning the whole Earth of hydrogen (approximately).

You may force the Sun to do something like the "red giant" transition prematurely and save some hydrogen that is unburned. To do so, you will have to spend the amount of energy corresponding to the Earth completely burned via fusion.

But of course, the counting of the energy which was "favorable" isn't the only problem. To actually tear the Sun apart, you would have to send an object inside the Sun that would survive the rather extreme conditions over there, including 15 million Celsius degrees and 3 billion atmospheres of pressure. Needless to say, no solid can survive these conditions: any object based on atoms we know will inevitably become a plasma. A closely related fact is that ordinary matter based on nuclei and electron doesn't allow for any "higher-pressure" explosion than the thermonuclear one so there's nothing "stronger" that could be sent to the Sun as an explosive to counteract the huge pressure inside the star.

One must get used to the fact that plasma is what becomes out of anything that tries to "intervene" into the Sun – and any intruder would be quickly devoured and the Sun would restore its balance. The only possible loophole is that the amount of this stuff is large. So you may think about colliding two stars which could perhaps tear them apart and stop the fusion. This isn't easy. The energy needed to substantially change the trajectory of another star is very, very large, unless one is lucky that the stars are already going to "nearly collide" which is extremely unlikely.

Physics will not allow you to do such things. You would need a form of matter that is more extreme than the plasma in the Sun, e.g. the neutron matter, but this probably can't be much lighter (and easier to prepare, e.g. when it comes to energy) than the star itself. A black hole could only drill a hole (when fast enough) or consume the Sun (which you don't want).

However, if you allow the Sun to be eaten by a black hole, you will actually get a more efficient and more sustainable source of energy. Well, too sustainable. ;-) A black hole of the mass comparable to the solar mass would have a radius about 3 miles. It would only send roughly one photon of the 3-mile-long wavelength every nanosecond or so in the Hawking radiation and it would only evaporate after $10^{60}$ years or so. It would be so sustainable that no one could possibly observe the energy it is emitting. However, the black hole would ultimately emit all the energy $E=mc^2$ stored in the mass.

If there are powerful civilizations ready to do some "helioengineering", they surely don't suffer from naive and primitive misconceptions about the world such as the word "sustainable" and many other words that are so popular in the mentally retarded movement known as "environmentalism". These civilizations may do many things artificially but they surely realize that the thermonuclear reaction in the stars is a highly efficient and useful way to get the energy from the hydrogen fuel. Even some of us realize that almost all the useful energy that allowed the Earth to evolve and create life and other things came from the Sun.

The Sun may become unsustainable in 7.5 billion years but according to everything we know about Nature, it's the optimum device to provide large enough civilizations – whole planets – with energy.

The most efficient way to save hydrogen for future use by a very advanced civilization is not to try to stop current stars from burning up their hydrogen, but rather to make sure that the star generation rate in the galaxy drops to zero. Basically, give up on current stars as a lost cause and just prevent new stars from forming. This works because the amount of hydrogen in gas clouds in the galaxy is orders of magnitude higher than the amount inside stars.

To do this, the civilization would need to monitor all of the gas clouds in space so they can notice clouds that are getting close to the stage of creating a proto-star. If they can explode a sufficiently energetic bomb near where the expected star would be born, they would be able to increase the pressure and prevent the collapse into a star for a significant period of time. I don't have any calculations of the energy required but it must be very much significantly less than the energy needed to take a significant amount of hydrogen out of the gravitational well of a star that has already started burning hydrogen.

They would have to be very careful that the explosions they create are not too energetic since a bomb that is too big could trigger more star formations if the expanding gas cloud runs into other stationary clouds. This whole scheme would require a lot of monitoring and simulations of all the gas clouds in the galaxy, but I don't think it is impossible from a physics point of view.

Another difficulty would be to monitor stars and to predict supernova explosions since they can also trigger nearby gas cloud to collapse and generate new stars. They would have to either gently move the gas clouds out of the way or come up with someway to prevent the supernova explosion.

The astronomical signature would be a galaxy with no star formation and no supernova explosions over an extended period of time.

What would be the most energy-efficient way (using known physics) to blow apart a star or otherwise prevent or greatly slow the rate at which it performs fusion?

Spin it until fusion stops. Do this using the sun's own energy.

To accomplish this, I will have to ask you to envision something like a Dyson Sphere, but the primary function of the matter encircling the star will be mirrors. I will be making statements about the force balances and basic physics, how this could be actually done in practice is out of the scope of this answer.

I propose that mirrors would at some distance from the sun, and the focal length of these mirrors would be equal to this distance. The idea will be to reflect the sun's light back onto the sun in a way that makes it spin faster. We would like to redirect all the light nearly tangentially to the edge of the sun, but we can't do this because of entropic limits. Remember, nothing can be focus the sun's rays to heat something hotter than the surface of the sun. This is why I select the parameter of $R=f$.

Focal length equal to distance, per Wikipedia

With this type of mirror we could focus the sun directly back onto the sun. We will assume sufficient distance from the sun to treat it as a simple circle. To make the sun spin faster, we will redirect the image right of the center, so the image will have a center a distance of $d$ to the right of the object. In order to get the optimal location to direct the reflection, we will

• Integrate the distance from the axis of rotation from the lower bound of the image circle to the upper bound of the object circle
• Integrate this value between the two intersections of the circles
• Find the greatest value of this moment integral over all valid values of $d$

I actually did that calculation. I obtained $d=0.836 R$. To summarize, the proposal is to refocus the light back onto the sun so that it looks like a Venn diagram.

By doing this, we trash a lot of the radiation. I calculate the average radius of interaction to be $0.418 R$, but if we dilute that number by the number of photons lost, we will get a multiplier of $0.202 R$ to convert the photon's momentum to the average torque exerted. I believe this is a fundamental, entropic limit, and if I'm wrong, it's at least conservative.

At this point we're still not finished. That's because this mirror isn't balanced. If it always reflected light in this way, it would not have a stable orbit since the gravitational force can only act radially and there is a tangential component to the photon's force on the mirror. One could compensate for this quite simply by holding a flat mirror at a $45^{\circ}$ angle to the sun's radiation, directing them in the other tangential direction. The torque from that balancing mirror, however, would depend on the distance from the sun. Because of that, we can't numerically correct for it here. If the distance of this satellite from the sun was large compared to the sun's radius then the loss from this balancing mirror would be negligible.

For simplicity I'll assume all the radiation from the sun is photons (it's only about 98%). The power output of the sun is:

$$P = 3.846 \times 10^{26} W$$

The photonic momentum (which is totally isotropic normally) can be found from $E=pc$ applied to the above power output. This gives the total momentum of photons emitted per unit time, or in other words, the isotropic photonic force.

$$F = P/c = 1.282 \times 10^{18} N$$

The moment of inertia of the sun can be found from the forumula for the moment of inertia for a solid ball.

$$m = 1.989 \times 10^{30} kg$$

$$I = \frac{2}{5} m R^2 = 3.848 \times 10^{47} m^2 kg$$

Using the methods I've laid out here, I can calculate the torque. This is assuming that all photons from the sun are used as efficiently as possible.

$$ \tau = F \bar{R} = (1.282 \times 10^{18} N) ( 0.202 R) = 1.801 \times 10^{26} N m $$

How fast would it need to spin in order to stop fusion, and then how much to break it apart? This is a difficult question to answer. However, one thing we can say is that if you have enough energy to completely disociate everything in the sun gravitationally you have enough energy to do both of the tasks of stopping fusion and breaking it apart less spectacularly. The energy to fully spread out the sun's mass over all space can be calculated. This is similar to a recent question, in short, the full dissociation energy is the half of the potential integrated over the entire volume. If I've done this right, the full disociation energy is:

$$ E_{diss} = \frac{3 G M^2}{8 R} = 1.423 \times 10^{41} J$$

It is also difficult to estimate the time needed for the available torque to dissociate the sun. But let's do a limit case where the sun doesn't deform due to the increased rotation. In that case we can seek an angular velocity that is equivelant to the above energy of dissociation, then ask how long it would take the available torque to accelerate it to that point.

$$ E_{diss} = \frac{1}{2} I \omega^2$$

$$ \omega = 0.00086 \frac{rad}{s}$$

This seems small, but consider that this would be the state of rotating once every 2 hours. Now, how long would the given torque take to get it to this state?

$$ I \omega = \Delta t \tau $$

$$ \Delta t = 58.23 \text{ billion years} $$

Now, if someone started spinning the sun with the sun's own power, eventually the fusion would stop, but the radiation wouldn't stop right away. In order to see if the stored energy is sufficient to break the sun apart, we'll consider the thermal energy stored in the core alone. The core's temperature is about 15,000,000 kelvin. This region goes out to around 0.25 solar radii. The average kinetic energy of a helium nuclei in the sun's core would then come from $E_k = 3/2 k T$, coming out to

$$ E_k = 3.106 \times 10^{-16} J $$

The average molecular mass in the sun is about 1.67 http://web.njit.edu/~gary/321/Lecture7.html I can use this to find the number of nuclei in the core of the sun. I can then combine that with the previous value for energy per nuclei to find the total stored kinetic energy in the sun.

$$ E = E_k N = E_k m (0.25)^3 / (1.67 amu) = E_k (1.12 \times 10^{55} \text{particles} ) = 3.481 \times 10^{39} J $$

This is the total thermal energy stored in the sun's core. Roughly. Divide this by the normal power output to get a $s$ valued number for the sun's power.

$$ E / P = 9.05 \times 10^{12} s = 6.9 \text{million years} $$

We conclude that the stored energy of the sun is insufficient to fully break it apart by about 4 orders of magnitude. The described method would still be viable to stop the fusion reaction.