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If a hypothetical black hole with the mass of our sun would appear in place of the sun. How long would it take before our entire solar system (everything inside our heliosphere) to be swallowed by this black hole?

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    $\begingroup$ Since this hypothetical black hole would have the exact same mass as the Sun, none of the planets would be gravitationally affected. They would remain in their current orbits. $\endgroup$ – BillDOe Jul 16 '19 at 23:22
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    $\begingroup$ nonetheless it's a clearly-written and interesting question resulting in a valuable and well-received answer, so +1. $\endgroup$ – uhoh Jul 18 '19 at 2:34
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Is a common missconception to think black holes suck everything around them by definition as if the fact they are black holes gave them some kind of different gravitational influence than the one excerted by any other object with a similar mass.

The main difference between the Sun and a solar-mass black hole its the density. The Sun would contain a solar mass in a sphere with 1.3 million km in diameter while the Schwarzschild diameter of a solar-mass black hole would be below 6 km. But the mass is still the same. Take a look to Newton's Universal Law of Gravity:

$F = -G\frac{m_1m_2}{r^2}$

As you can see the gravitational force depends on the masses of the two bodies interacting, $m_1$ and $m_2$ (for example, the Sun and the Earth), but does not depend on the density of the bodies in any way, thus the interaction has the exact same behaviour in both the Sun and the solar-mass black hole scenarios. The planets would continue to circle around the barycenter of the Solar System as if the Sun was there.

Obviously, besides the gravitational influence, a lot of horrible consequences would take place. The absence of the solar light flux would freeze the entire system pretty fast. The first few days of darkness would be fine for Earth (even if air currents, migration patterns and circadian rythms would be totally disrupted). In a matter of months the Earth's surface would start to be sterilized for complex organisms unless moved to geothermal facilities. By the first year life on Earth would have been eradicated as the world oceans become a thick ice crust. After 10 years or so even Earth's atmosphere would start to collapse as air turns liquid and freeze upon arrival to the ground. With some luck the ice shell would still insulate the water below and microbial life could still be there, thriving with the geothermal energy of Earth's interior. The best option for us would be to transfer to Venus since atmospheric greenhouse gases would probably allow for larger thermal inertia and thus the surface would freeze slower than on Earth. In the end, after a few decades, even Venus would be sterilized. The entire Solar System would be a dark cold planetary ghost.

There is a mechanism to allow for the black hole to suck the planets eventually and that is gravitational radiation. We know that orbiting bodies emmit gravitational waves. These waves carry angular momentum away from the system shrinking the sizes of the orbits. We in fact have detected strong gravitational waves coming from inspiraling black holes. Therefore, with enought time, gravitational waves radiated by the planets would put them so close to the black hole as to be "swallowed", but this makes no difference with the Sun scenario we live in, orbits are decaying right now by the same process. But we need to have some orders of magnitude in mind, this process is crazily insignificant, more for small masses like the ones of planets, and even more for the huge orbital periods our planets have.

How long you would have to wait for this to happen? In the case of Earth, it would be swallowed in $10^{13}$ times the history of the universe. Yep, those are not 13 zeros of years but 13 zeros of times the universe age. Mercury, is not swallowed much earlier, in fact only half that time (which is still trillions of times the age of the universe). And for the entire Solar System to be swallowed (considering Neptune the last object of it), you would have to wait 145.000 times that. Not to mention the exponentially increasing waits for Kuiper belt objects and Oort cloud comets.

With these crazy spans of time we also have to think about other strange effects. Hawking radiation would evaporate the solar-mass black hole slowly, meaning that the central mass of the system would start to decrease. This means that planetary orbits would start to get larger (since a planet orbiting a less massive body with the same orbital speed would climb the gravitational dwell by some amount). The solar black hole would evaporate in $10^{58}$ times the age of the universe. Until the planets get free from the influence of an ancient mass that no longer exists in the Solar System. This is a way larger characteristic time period than the one required for orbital decay by gravitational radiation. So the gravitational decaying of the orbits would be the predominant effect even if the wait would be a bit larger (since the mass of the black hole decreases slowly).

So yeah, you can wait $10^{29}$ years until all the planets get swallowed by your black hole. The Sun would swallow them in the same way only that they would touch the Sun's surface even before than the event horizon of your black hole.

Not only that, the Earth is probably going to be swallowed by the Sun way before than in the solar-mass black hole scenario. We know because of stellar evolution that the Sun is going to get larger to the point of reaching our current position in just 6 billion years at max (even if there is some discussion about the solar mass loss in this era enlarging Earth's orbit and making it "safe").

If you want to keep your solar-mass black hole scenario think about those crazy trillions of times the age of the universe wait. Many things are going to happen way before that would disrupt the entire Solar System. It is expected that the orbital perturbations between the planets will eject several planets in a matter of a few billion years. That is a blink of an eye in the wait for them to decay. At these extreme timescales even close flybys of other stars will certainly dirsupt the system if planets don't do it by themselves in the first place.

So my answer is that in real life, turning the Sun into a black hole, would never end in the Solar System been swallowed by it, more like everything disspersed in interstellar space in some hundred of billion years at most.

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