Blackholes: Accretion Vs Expulsion

[Doum]

i tryed to calculate the final speed a black hole 5 times the mass of the sun and the sun itself would have at impact if the black hole is a thousand AU from the sun and is moving toward it at an initial velocity of 4000 km/s. i think i have the wrong formula. Because i got a speed more then 2 time the speed of light. (((

i suspect it need differential and integer calculus wich i am not good with. i want to know how much time the black hole will take to go trough the sun (Or the sun going trough the black hole) since the black hole is 5 time the sun mass. is it a minute or 5 sec?

i will try to redo some calculus. But i am not sure i can do it. May be Neufer know how to calculate that.

[rstevenson]

Well, if I make two unrealistic assumptions...
1. there is no acceleration at all, so the BH stays at 4000 km/s
2. there is no resistance as it goes through the Sun, so the BH goes right through without slowing
... then at 4000 km/s it will take about 350 seconds, or less than 6 minutes, to pass through the Sun.

I wonder if the no acceleration rule cancels out the no resistance rule? Nah, that would be too easy.

Rob

[Doum]

Rob,

1. No acceleration between 2 body attract by their own gravity. i dont think so. There is an acceleration as they fall toward each other.

May be i write my question badly.

If a black hole is a thousand astronomical unit from the sun and is going toward it, as they both get closer to each other their mutual gravity will attract each other. As they get closer, the strenght of their gravity will increase. So they will gain speed as they get closer. The 4000 km/s speed was the speed the black hole has at a thousand AU from the sun. So, as they fall toward each other they must both be gaining speed toward each other. if not then i am missing something about gravity.

i read somewhere (i dont remember where) that comet fall toward the sun at speed ranging from 70000 km/h to 100000 km/h. That speed is when they are close to the sun and they gain it as they fall toward it. As they start moving away from the sun they loose speed.

So i think the final speed between the black hole and the sun will be a lot higher then 4000 km/s because the black hole is 5 times the strenght gravity of the sun. They both fall toward each other at greater mutual gravity strenght (acceleration) then a comet do with the sun only.

When the black hole pass trough the sun i think it will take a few second to do it. Not minutes. At a 100000km/h (Comet speed) it would take around 14 seconds.
So i think it will take less then 14 seconds for the black hole to pass trough the sun.

i just cant find a formula to do the calculus for the final speed they will have relative to each other. The formula i have seem to work for object that are close to the mass body. As i said i will keep looking or try to find out the final speed.

[BDanielMayfield]

If a 5 solar massed black hole approached to within 50,000 or 100,000 au we should certainly find out if the Oort Cloud was real or not. A certain percentage of them would flare up before being consumed, I would think. Hey ya'll, good news, the Oort Cloud is real. But on the other hand, the world is ending.

Hey Rob, if you don't mind sharing an important little detail in this epic novel your writing, how fast are the BH and the Sun approaching each other at any given point?

Bruce

Edit: This post was in response to Chris' last post.

[BDanielMayfield]

Rob,

1. No acceleration between 2 body attract by their own gravity. i dont think so. There is an acceleration as they fall toward each other.

May be i write my question badly.

If a black hole is a thousand astronomical unit from the sun and is going toward it, as they both get closer to each other their mutual gravity will attract each other. As they get closer, the strenght of their gravity will increase. So they will gain speed as they get closer. The 4000 km/s speed was the speed the black hole has at a thousand AU from the sun. So, as they fall toward each other they must both be gaining speed toward each other. if not then i am missing something about gravity.

Your completely right about gravitational attraction acting on both bodies Doum. But your 4000 km/s initial speed is unrealistically high though, I think. The real Sun orbits the Galactic center at about 220 km/s. The average velocity of the Sun's motion through its stellar neighborhood is only about 20 km/s. Runaway stars (thought to be produced from systems where a star has gone supernova can move at around 100 km/s relative to local movements. Hypervelocity stars shot out from the galactic core are observed at about 1000 km/s.

i just cant find a formula to do the calculus for the final spped they will have relative to each other. The formula i have seem to work for object that are close to the mass body. As i said i will keep looking or try to find out the final speed.

Yeah, I've been tinkering with this problem also. My calculus courses are a distant memory. Calculating the instantaneous accelerations on both bodies is easy. F=Gm₁m₂/r². Calculating the timings and summing up the max velocities are hard. A little help neufer?

Bruce

[Doum]

mehh just reread myself. gee ...Bruce you are right. i met 4000km/h in the first post above as initial speed. And a comet moving at 100000km/h will take way more then 14 seconds to go trough the sun( more like 14 hours). So Rob you are right, 6 minute is a good figure at 4000 km/s. i made so many mistake.

[rstevenson]

... Yeah, I've been tinkering with this problem also. My calculus courses are a distant memory. Calculating the instantaneous accelerations on both bodies is easy. F=Gm₁m₂/r². Calculating the timings and summing up the max velocities are hard. A little help neufer?

Bruce

The calculus approach is beyond me too, Bruce. What I was going to do as a first stab at the acceleration question is to use that Newtonian equation to get some points on a graph at ever decreasing distances, then extrapolate the graph line to impact. I expect the velocity to be asymptotic at that point (c?) which means it won't reveal anything too accurate, but it will move me in the right direction, I think. (The desire to avoid that multiple recalculation issue is essentially why calculus was invented, as I understand it -- but I've got an iMac and Excel, advantages Sir Isaac didn't have.)

The next intriguing problem is to consider what effect the dense core of the Sun will have. The Sun will leave behind a white dwarf after it blows off its atmosphere at the end of its life, but is there in effect a white dwarf inside the Sun now? I think not; I think the core has yet to evolve to that point, getting denser as it does so. So I'll have to look up its current density and try to work out whether (certainly) and by how much it will slow down the BH.

By the way, I'm assuming the BH is approaching at an initial speed of about 1000 km/s, having been flung our way in a recoil reaction by its formation. (See this Wikipedia article.)

Rob

PS
Yes, doum, I know there's acceleration. I was trying to avoid calculating it by estimating.

[Chris Peterson]

The next intriguing problem is to consider what effect the dense core of the Sun will have. The Sun will leave behind a white dwarf after it blows off its atmosphere at the end of its life, but is there in effect a white dwarf inside the Sun now? I think not; I think the core has yet to evolve to that point, getting denser as it does so. So I'll have to look up its current density and try to work out whether (certainly) and by how much it will slow down the BH.

The Sun's core (which is where fusion occurs) is moderately dense, but not remotely close to the density of a white dwarf (by about four orders of magnitude). It's a plasma fluid.

By the way, I'm assuming the BH is approaching at an initial speed of about 1000 km/s, having been flung our way in a recoil reaction by its formation. (See this Wikipedia article.)

I'd just assume an initial speed of zero and a distance of infinity. That's the general initial condition for considering point sources falling towards each other ballistically. The initial speed will largely not matter, as virtually all the velocity at impact will be from gravitational attraction.

[BDanielMayfield]

By the way, I'm assuming the BH is approaching at an initial speed of about 1000 km/s, having been flung our way in a recoil reaction by its formation. (See this Wikipedia article.)

I'd just assume an initial speed of zero and a distance of infinity. That's the general initial condition for considering point sources falling towards each other ballistically. The initial speed will largely not matter, as virtually all the velocity at impact will be from gravitational attraction.

Rob, the 1000 km/s range works, since it isn't beyond the range of credibility. What's the initial distance?

Chris, say what? My graph paper doesn't extend out to infinity, no matter how large I make the scale. Plus, and seriously, as you know the Forces drop with the inverse square of distance. Zero = Any finite number / infinity².

Bruce

[Chris Peterson]

By the way, I'm assuming the BH is approaching at an initial speed of about 1000 km/s, having been flung our way in a recoil reaction by its formation. (See this Wikipedia article.)

I'd just assume an initial speed of zero and a distance of infinity. That's the general initial condition for considering point sources falling towards each other ballistically. The initial speed will largely not matter, as virtually all the velocity at impact will be from gravitational attraction.

Rob, the 1000 km/s range works, since it isn't beyond the range of credibility. What's the initial distance?

Chris, say what? My graph paper doesn't extend out to infinity, no matter how large I make the scale. :lol2: Plus, and seriously, as you know the Forces drop with the inverse square of distance. Zero = Any finite number / infinity².

It's a limits problem. When two bodies separate at escape velocity, what that means is that they will achieve a relative speed of zero after an infinite amount of time. The problem of two bodies falling toward each other under the influence of gravity is precisely the opposite. If the bodies start at infinity, with no relative speed, they will collide at their escape velocity. In the case of a star and a black hole, this is a large speed- much higher than any realistic initial velocity the pair might have. (Things get complicated when the bodies are close enough for tidal effects to dominate, but for our purposes here we can ignore that.)

[rstevenson]

I'd just assume an initial speed of zero and a distance of infinity. That's the general initial condition for considering point sources falling towards each other ballistically. The initial speed will largely not matter, as virtually all the velocity at impact will be from gravitational attraction.

I'm assuming the unlikely scenario of a perfect north to south pole pass through of the Sun, so I'll have to -- in the best tradition of scifi authors -- wave my hands vaguely in some direction and say the BH was formed out there somewhere. I'm thinking now that I'll set its formation at least a few hundred LYs away, so yes, at that distance there's hardly any difference between zero and 1000 km/s as an initial speed -- but using 1000 km/s keeps a zero out of the equation, which may be handy. Also, the story starts when we notice some disturbance in the Oort Cloud (I'm assuming it's really there) and at that point, the characters eventually figure out, it's going 1000 km/s and that's the point at which they start to calculate the possible consequences.

Thanks for the info about the density of the core. I hadn't looked it up yet. That keeps the pass-through scenario at least likely, pending calculations and modelling.

Rob

[Chris Peterson]

I'd just assume an initial speed of zero and a distance of infinity. That's the general initial condition for considering point sources falling towards each other ballistically. The initial speed will largely not matter, as virtually all the velocity at impact will be from gravitational attraction.

I'm assuming the unlikely scenario of a perfect north to south pole pass through of the Sun, so I'll have to -- in the best tradition of scifi authors -- wave my hands vaguely in some direction and say the BH was formed out there somewhere. I'm thinking now that I'll set its formation at least a few hundred LYs away, so yes, at that distance there's hardly any difference between zero and 1000 km/s as an initial speed -- but using 1000 km/s keeps a zero out of the equation, which may be handy. Also, the story starts when we notice some disturbance in the Oort Cloud (I'm assuming it's really there) and at that point, the characters eventually figure out, it's going 1000 km/s and that's the point at which they start to calculate the possible consequences.

Actually, it's by setting the initial velocity to zero that you make the problem most simple. If I were solving the problem numerically, I'd probably assign an initial speed of zero and an initial distance of a light year. That's going to be close enough to working out the actual minimum collision speed. And yes, I assume a perfect collision where the centers intersect, otherwise it becomes a messy orbit problem dominated at the end by the actual size of the bodies. The best way to solve this problem is to treat both bodies as point sources.

[BDanielMayfield]

By the way, I'm assuming the BH is approaching at an initial speed of about 1000 km/s, having been flung our way in a recoil reaction by its formation. (See this Wikipedia article.)

Rob

That's a great little wiki article Rob Doum, please read it if you haven't and then disregard what I wrote about 4000 km/s being too fast.

Two stellar massed black holes spiral in deep inside a globular cluster. (This is a good source because it could be coming in from whatever direction you wanted.) The holes merge, forming a ten solar massed BH with a recoil velocity of 5,000 km/s! As it zips from the dense stellar core of the cluster its trajectory is altered just a little by close encounters with the stars it passes. It emerges from the cluster with a trajectory that in many millions of years will be very bad news for Rob's world.

A ten solar massed black hole approaching at 5,000 km/s. I think Rob's world is doomed. Full stop.

Bruce

[Chris Peterson]

I'd just assume an initial speed of zero and a distance of infinity. That's the general initial condition for considering point sources falling towards each other ballistically. The initial speed will largely not matter, as virtually all the velocity at impact will be from gravitational attraction.

I'm assuming the unlikely scenario of a perfect north to south pole pass through of the Sun, so I'll have to -- in the best tradition of scifi authors -- wave my hands vaguely in some direction and say the BH was formed out there somewhere. I'm thinking now that I'll set its formation at least a few hundred LYs away, so yes, at that distance there's hardly any difference between zero and 1000 km/s as an initial speed -- but using 1000 km/s keeps a zero out of the equation, which may be handy. Also, the story starts when we notice some disturbance in the Oort Cloud (I'm assuming it's really there) and at that point, the characters eventually figure out, it's going 1000 km/s and that's the point at which they start to calculate the possible consequences.

Actually, it's by setting the initial velocity to zero that you make the problem most simple. If I were solving the problem numerically, I'd probably assign an initial speed of zero and an initial distance of a light year. That's going to be close enough to working out the actual minimum collision speed. And yes, I assume a perfect collision where the centers intersect, otherwise it becomes a messy orbit problem dominated at the end by the actual size of the bodies. The best way to solve this problem is to treat both bodies as point sources.

FWIW, I calculate the relative velocity between a 1 solar mass star and a 5 solar mass black hole, when they are 1 million kilometers apart (close to colliding, but not quite there) to be 1300 km/s (plus any initial velocity). Not as much as I expected. At 100,000 kilometers, so just inside the Sun, the velocity is up to 4000 km/s.

[Doum]

Mehh i try using excel and make 2 table of 2760 lines for the black hole (surface 10 km) and 2360s line for the sun ( surface at 0,7 million Km). Each line is a distance between the sun and the black hole that decrease as they get closer to impact. i diminish the distance between line as the 2 bodys get closer because gravity get stronger so do the speed . i calculate the force at each of these distance for the black hole and for the sun. At the end i got a final speed for the sun falling the that blackhole of 470171 m/s for an initial speed at 1000AU of 4000 Km/s (1111m/s). And for the black hole going toward the sun i got 182890 m/s.(the table are for a falling object). So, the 2 of them going toward each other give a total collision speed of 653060 m/s or 653,06Km/s. So according to my calculus the black hole will go trought the sun for around 0,595 hour or 35,7 minutes.

Now looking at Chris calculus i am way far from his result.(1300Km/s for a million Km. While i have half of that for a 1000UA fall)

Rob you are right. The difference for an initial speed of 4000 Km/s i not that important. i only got 653Km/s. Will it add that same speed if it have 4000Km/s at a 1000 AU instead of the 1.11 Km/s i use?

Now i am not sure i want to see what i do wrong compare to Chris calculus. im tired of trying to do that. there is fun and there is fun. May be later.

[Doum]

OK i verified my calcul ( For what its Worth.). i found some error and also i redo it with a 4000Km/s initial speed for the black hole. i also add 7000 different distance to do the calcul. i have a final total speed of 4986355,487 m/s.
So the black hole go trough the sun in 5,34 minutes. The acceleration between the sun and the black hole before the encounter seem to add near 20% to the 4000Km/s initial speed.

i did the calculus (no excel did it ) from a 1000 UA from the sun untill it get to 0,7 million Km (or 7 x10E8 meter) from the sun ( wich i assume is near the sun surface).

Data use:
Sun diameter= around 1.4 million KM
1000 AU= 150x10E12 meters
Black hole mass= 9.95x10E30 (5 times the sun mass)
Sun mass= 1.19x10E30
G= 6.674x10E-11
Black hole initial speed = 4000km/s

Dunno if any of this is good at all. My result is far from Chris calculus from a 1 million KM distance from the sun.

Ohh and good luck Rob for your story.

P.S. if i try to use F= G* M1 M2/D square i get amazing speed. Way more then the speed of light. so i did it for the sun mass only and then for the black hole mass only and add the 2 final speed i get. Cant do more. But it was annoyingly fun.

[BDanielMayfield]

OK i verified my calcul ( For what its Worth.). i found some error and also i redo it with a 4000Km/s initial speed for the black hole. i also add 7000 different distance to do the calcul. i have a final total speed of 4986355,487 m/s.
So the black hole go trough the sun in 5,34 minutes. The acceleration between the sun and the black hole before the encounter seem to add near 20% to the 4000Km/s initial speed.

i did the calculus (no excel did it ) from a 1000 UA from the sun untill it get to 0,7 million Km (or 7 x10E8 meter) from the sun ( wich i assume is near the sun surface).

Data use:
Sun diameter= around 1.4 million KM
1000 AU= 150x10E12 meters
Black hole mass= 9.95x10E30 (5 times the sun mass)
Sun mass= 1.19x10E30
G= 6.674x10E-11
Black hole initial speed = 4000km/s

Dunno if any of this is good at all. My result is far from Chris calculus from a 1 million KM distance from the sun.

Ohh and good luck Rob for your story.

P.S. if i try to use F= G* M1 M2/D square i get amazing speed. Way more then the speed of light. so i did it for the sun mass only and then for the black hole mass only and add the 2 final speed i get. Cant do more. But it was annoyingly fun.

This is both fun and annoying. You have an error in the Sun mass figure. Just round it up to 2x10E30 kg. (1x10E31 kg then for the BH) You know this, but be careful about mixing up units like km and meters.

As a check, what Force in Newtons do you get at 1000 AU? (You MUST use F=Gm₁m₂/d² because, in this universe, it's the LAW.)

I come up with 5.932 x 10^22 Newtons. I think this is right, anyone please correct me if not.

Do we agree? If not, what do you come up with?
Next step, what acceleration does each body see at T₀?

Bruce

[BDanielMayfield]

I'm assuming the unlikely scenario of a perfect north to south pole pass through of the Sun, so I'll have to -- in the best tradition of scifi authors -- wave my hands vaguely in some direction and say the BH was formed out there somewhere. I'm thinking now that I'll set its formation at least a few hundred LYs away, so yes, at that distance there's hardly any difference between zero and 1000 km/s as an initial speed -- but using 1000 km/s keeps a zero out of the equation, which may be handy. Also, the story starts when we notice some disturbance in the Oort Cloud (I'm assuming it's really there) and at that point, the characters eventually figure out, it's going 1000 km/s and that's the point at which they start to calculate the possible consequences.

Thanks for the info about the density of the core. I hadn't looked it up yet. That keeps the pass-through scenario at least likely, pending calculations and modelling.

Rob

Your story is yours, naturally, so you have license to frame it any which way you please, but why limit this to a perfect solar poleward pass? Perfect angles and nice round numbers are great for simulations of what might occur, but they detract from story value by adding to the implausibly factor. I think simulations will show that it wouldn't matter what angle this impact comes from, the Sun is going down the drain. What you call "the best tradition of scifi authors" can be a weakness. There's way too little science in most sci-fi. The more realistic a story is, the better.

Bruce

[Chris Peterson]

Your story is yours, naturally, so you have license to frame it any which way you please, but why limit this to a perfect solar poleward pass?

Only, I think, because the discussion started around the interesting scenario of an actual collision between the Sun and a black hole, and the possibility of the Sun surviving mostly intact. It's easy enough to demonstrate that all of the realistic cases of a black hole passing anywhere the Solar System will fatally disrupt planetary orbits. So then it just degenerates to a story about the end of the world, and maybe the story of some heroic attempt to save humans. Old story, done to death. More interesting then, is there a special case of a black hole passing through that might actually be survivable?

[BDanielMayfield]

Your story is yours, naturally, so you have license to frame it any which way you please, but why limit this to a perfect solar poleward pass?

Only, I think, because the discussion started around the interesting scenario of an actual collision between the Sun and a black hole, and the possibility of the Sun surviving mostly intact. It's easy enough to demonstrate that all of the realistic cases of a black hole passing anywhere the Solar System will fatally disrupt planetary orbits. So then it just degenerates to a story about the end of the world, and maybe the story of some heroic attempt to save humans. Old story, done to death. More interesting then, is there a special case of a black hole passing through that might actually be survivable?

Ok, then this comes down to what happens to the Sun and it's planets. The planets go haywire first, since the BH is pulling on them with 5 times the sun's pull, adjusted by distance of course. As the BH approaches the Sun it will be stretched out by tidial forces. The solar wind will be drawn in and the surface of the sun near the line of contact will be pulled up and compressed, heating it before it falls out of sight into the hole. Acceleration of the BH will continue as it penetrates into the sun while more and more solar plasma falls into it as it moves into ever denser material. Meanwhile all points on and inside the Sun are falling down the hole's steep gravity well, shrinking the Sun and causing it to flare up as a nova. A point of maximum acceleration will be reached deep inside the shrinking Sun and then the acceleration will rapidly fall off to zero as the BH reaches the Sun's center. While it's passing through it is literally eating the heart out of the Sun, swallowing the densest, most massive material in its core. Solar fusion is snuffed out (at least briefly) increasing the freefall of Solar material since there is no pressure in the core while the BH is passing through it. A substantial fraction of the Sun's mass will have been transferred to the BH by the time it reaches the center of the Sun. The destruction will continue, ...

Bruce

[Chris Peterson]

Ok, then this comes down to what happens to the Sun and it's planets. The planets go haywire first, since the BH is pulling on them with 5 times the sun's pull, adjusted by distance of course.

Probably true.

As the BH approaches the Sun it will be stretched out by tidial forces. The solar wind will be drawn in and the surface of the sun near the line of contact will be pulled up and compressed, heating it before it falls out of sight into the hole.

Agreed, but this won't happen until the BH is very near the Sun. For most of the time, tidal forces will be small and the actual effect will be to pull the Sun (as a sphere) towards it.

Acceleration of the BH will continue as it penetrates into the sun while more and more solar plasma falls into it as it moves into ever denser material. Meanwhile all points on and inside the Sun are falling down the hole's steep gravity well, shrinking the Sun and causing it to flare up as a nova. A point of maximum acceleration will be reached deep inside the shrinking Sun and then the acceleration will rapidly fall off to zero as the BH reaches the Sun's center. While it's passing through it is literally eating the heart out of the Sun, swallowing the densest, most massive material in its core. Solar fusion is snuffed out (at least briefly) increasing the freefall of Solar material since there is no pressure in the core while the BH is passing through it. A substantial fraction of the Sun's mass will have been transferred to the BH by the time it reaches the center of the Sun.

This I'm not inclined to agree with. From first contact, the passage time of the BH through the Sun is 600 seconds. That's not very long. Just how much material can something with so little surface area collect in that time? Just how much can the Sun's shape be changed in that time? My thinking is that there will be almost no mass transfer at all, and all of the significant effects will be due to the increased gravity. But what those effects might actually be I really have no idea.

[neufer]

Click to play embedded YouTube video.

<<"Black Hole Sun" is a song by US rock band Soundgarden. Written by frontman Chris Cornell, the song was released in 1994 as the third single from the band's fourth studio album Superunknown (1994). Cornell stated, "It's just sort of a surreal dreamscape, a weird, play-with-the-title kind of song."He also said that "lyrically it's probably the closest to me just playing with words for words' sake, of anything I've written. I guess it worked for a lot of people who heard it, but I have no idea how you'd begin to take that one literally." In another interview he elaborated further, stating, "I was just sucked in by the music and I was painting a picture with the lyrics. There was no real idea to get across." Commenting upon how the song was misinterpreted as being positive, Cornell said, "No one seems to get this, but 'Black Hole Sun' is sad. But because the melody is really pretty, everyone thinks it's almost chipper, which is ridiculous.">>

[Doum]

Bruce,

This is both fun and annoying. You have an error in the Sun mass figure. Just round it up to 2x10E30 kg. (1x10E31 kg then for the BH) You know this, but be careful about mixing up units like km and meters.

As a check, what Force in Newtons do you get at 1000 AU? (You MUST use F=Gm1m2/d2 because, in this universe, it's the LAW.)

- yes i do get the same force.

-Tought more about it. i remake the excel calculus. Now i have a final speed at 700 000 km from the sun center(surface) of 4039,5 km/s.
wich is not much more then the initial speed of 4000km/s. Even a third Inside the sun i get 4061 km/s.

Ohhh and thank for the sun mass.

Next step, what acceleration does each body see at T0?

-I got 4,92E-09 m/ s2 for the acceleration at t0.

[BDanielMayfield]

Bruce,

This is both fun and annoying. You have an error in the Sun mass figure. Just round it up to 2x10E30 kg. (1x10E31 kg then for the BH) You know this, but be careful about mixing up units like km and meters.

As a check, what Force in Newtons do you get at 1000 AU? (You MUST use F=Gm1m2/d2 because, in this universe, it's the LAW.)

- yes i do get the same force.

-Tought more about it. i remake the excel calculus. Now i have a final speed at 700 000 km from the sun center(surface) of 4039,5 km/s.
wich is not much more then the initial speed of 4000km/s. Even a third Inside the sun i get 4061 km/s.

Ohhh and thank for the sun mass.

Next step, what acceleration does each body see at T0?

-I got 4,92E-09 m/ s2 for the acceleration at t0.

Ok then. We are getting there Doum. 4.92E-9 m/s² is close to what I get for the BH's acceleration. Are we using the same masses? (Sun=2E30, BH=1E31)

For the BH I get A=F/M = 5.932E22/1E31 = 5.93E-9.

But remember the old, for every action there is an equal and opposite reaction rule? (The Sun sees five times as much acceleration as the BH!)

Bruce

[Chris Peterson]

Ok then. We are getting there Doum. 4.92E-9 m/s² is close to what I get for the BH's acceleration. Are we using the same masses? (Sun=2E30, BH=1E31)

For the BH I get A=F/M = 5.932E22/1E31 = 5.93E-9.

But remember the old, for every action there is an equal and opposite reaction rule? (The Sun sees five times as much acceleration as the BH!)

FWIW, as I hinted earlier, this problem is easy if treated as the reverse of the escape velocity calculation (where the relative velocity at infinity is zero). Indeed, it uses the same equation as the one for escape velocity, with just one slight difference- we normally calculate escape velocity as applied to a massless point moving away from a massive body. In this case, both bodies have significant mass, so we need to consider their sum.

The velocity of two massive point sources at some distance r, drawn together by self-gravity from a starting distance of infinity, is just

v = √(2G(m₁+m₂)/r)

To this you can add any initial velocity you wish to assume, but at the least this will provide the minimum possible speed at any distance.

I based my calculations on the mass of the Sun as 2e30 kg and the mass of the black hole at 1e31 kg.