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Imagine that hypothetically the black hole in the center of the milky way gradually increased in mass by for example 50% every year. That is exponential increase in mass.

Which visual effects would we see on for example Alpha Centauri, stars more distant and on M31? Red or blueshift? Lensing?

How about linear increase in mass?

Are there software I can use to visualize it as seen from earth?

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Sound like a great question for what-if.xkcd.com. –  this Jan 16 at 18:45
    
BTW: It is an assumption that there is a black hole in the center of our galaxy –  Ahmed Hamdy Jan 21 at 12:14
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this is not really an answer to your question, but to the assumption you made.

The BH in the galactic center cannot grow that much. There are two problems. First, there is not enough food close by. Any potential food (gas clouds, stars, and dark matter) will have some non-zero angular momentum preventing it from coming close enough. Loosing this angular momentum is difficult (it cannot be radiated away like energy), the only way is to exchange it with other objects either via impact (of gas clouds) or gravitational interactions with other objects (not the BH).

The second problem is that a feeding BH is surrounded by an accretion disc of hot gas. In that disc, angular momentum is slowly transported outwards and mass inwards via viscosity (that viscosity most likely originates from turbulent magnetic fields that become unstable -- the magneto-rotational instability). This process inevitably heats the accretion disc to very high temperatures ($10^{6-9}$K) such that it emits a wind of raditation and particles (similar to the Solar wind, but much much stronger). If the BH feeds too much, this wind becomes so strong that it pushes away any further infalling material (potential food).

This second process limits the growth of any BH to double in mass in no less than about $10^6$ years, I think (I'm not too sure--if you want a precise value, consult the literature), even if the first problem was no issue.

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Yep, but i am more interested in the optical effects for us, not about how the black hole would grow. Perhaps you could imagine it receiving intense gamma rays directed at its center would increase its mass without us being able to see where the increase of mass came from. I don't really care how, but about the optics. –  frodeborli Jan 18 at 13:45
    
It is the first process, which ultimately matters. Exponential growth would give you arbitrary masses sooner or later. –  Alexey Bobrick Jan 19 at 1:01
    
@AlexeyBobrick Nope. The fact that these beasts grow to $10^{6-9}$ Solar masses shows that somehow they find a way to overcome the first problem. But eventually the feedback form the hole is so strong that it drives all the gas out of its host galaxy. This is thought to occur when all the energy of the wind is dumped into the galactic ISM, i.e. when the swept-up ISM cannot cool efficiently (if it can, only the wind momentum drives the outflow). Most efficient cooling is due to inverse Compton scattering off photons from the AGN itself (the shocked gas is much hotter than the accretion disc). –  Walter Jan 19 at 19:52
    
@Water, well, yes, but anyway Eddington limit (or more complex analagous processes as the ones you describe) does not stop accretion. The accretion could even be long and very sub-eddington, but eventually you will run into the problem of not having enough 'accretable' mass. –  Alexey Bobrick Jan 19 at 20:16
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The question wasn't about the problem of getting the required mass. –  this Jan 20 at 10:17
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