How Long To 'FreeFall' Into TON 618 From ISCO

[Daharen]

So I'm trying to help someone get an idea of the 'scale' of some of the most massive black holes, but, I'm at best a laymen when it comes to this stuff. Because ISCO for a non-rotating black hole is just 3 times it's radius, and I can calculate apparent arc radius from ISCO, I could tell them how large it would appear if our solar system orbited the black hole at the ISCO, but then I was asked how long it would take to 'fall' into the black hole if we suddenly stopped orbiting (A very similar question to the classic how long it would take to fall into the sun if the Earth stopped orbiting, but on a very different scale).

Now... I imagine this gets quite complicated quite fast since at this point we're not just dealing with simple Newtonian Mechanics, but probably a lot of relativistic stuff too.. However, if we can disregard relativity for the sake of this conversation, how long would it take for a planet that was orbiting TON 618 at about 3900 AU from it's Event Horizon, and suddenly stopped orbiting and went into free fall, to reach the event horizon, which is about 1300 AU from the singularity, given a mass of 66 Billion Solar Masses?

If this is beyond the reasonable scope of this forum I understand, I tried really hard to work with calculators to find the answers, but couldn't account for the change in acceleration over time (Integral Calculus is definitely not my strong point).

 

[stefan r]

So I'm trying to help someone get an idea of the 'scale' of some of the most massive black holes, but, I'm at best a laymen when it comes to this stuff. Because ISCO for a non-rotating black hole is just 3 times it's radius, and I can calculate apparent arc radius from ISCO, I could tell them how large it would appear if our solar system orbited the black hole at the ISCO, but then I was asked how long it would take to 'fall' into the black hole if we suddenly stopped orbiting (A very similar question to the classic how long it would take to fall into the sun if the Earth stopped orbiting, but on a very different scale).

Now... I imagine this gets quite complicated quite fast since at this point we're not just dealing with simple Newtonian Mechanics, but probably a lot of relativistic stuff too.. However, if we can disregard relativity for the sake of this conversation, how long would it take for a planet that was orbiting TON 618 at about 3900 AU from it's Event Horizon, and suddenly stopped orbiting and went into free fall, to reach the event horizon, which is about 1300 AU from the singularity, given a mass of 66 Billion Solar Masses?

If this is beyond the reasonable scope of this forum I understand, I tried really hard to work with calculators to find the answers, but couldn't account for the change in acceleration over time (Integral Calculus is definitely not my strong point).

If you disregard relativity you can plug it into the Kepler's equations. Falling straight in is equivalent to a very narrow ellipse so R/2 is semi-major axis.

You can probably disregard m. M is the hole mass 6.6x 10¹⁰, G is constant but convert M into kilogram. R is just distance but should match units used for G.

 

[Daharen]

Thank you, I did work it out eventually using this, I weirdly came up with 'roughly' the exact same amount of time it would take for Earth to free-fall into the sun for the solar system to free-fall into the black hole from its ISCO... Is this just a coincidence, or is there some mechanical reason that it would work out this way? FYI, the math works out to just a bit under three months of free fall before we enter the event horizon. Turns out that the solar system would begin to experience spaghettification even with the largest black hole, but individual bodies like the Earth could actually remain entact for a 'short while' beyond the event horizon (Ignoring the photon sphere, holographic principle, or any other ideas that would prohibit the possibility of passing the event horizon at all entact). Acceleration has to be 'stopped' at some point because it exceeds light speed if you don't account for relativity, so obviously this answer is grossly inaccurate at some point, but still a fascinating thought experiment.

I would love to see the answer for this problem incorporating relativity to the full extent possible, but I wouldn't even know where to begin. In either case, thank you for providing me with the formula it was extremely helpful. Sorry about the delay in my reply.

 

[stevebd1]

You might find these lecture notes of use-
https://studylib.net/doc/10899930/physics-161--black-holes--lecture-10--27-jan-2010-10