How much time dilation do you get at those masses though?

I’m having more trouble visualizing how accretion disks would work for a binary black hole. Because the light is coming from the disks, not the black holes. So those are what are actually pulsing/girating.

Kepler’s laws should still provide a pretty good estimate, at least until black holes get much closer. I did a quick back of the envelope calculation, and looks like they’ll be roughly 14k astronomical units, or 0.22 light years apart.

In Newtonian gravity, the relation between the orbital period T and the semimajor axis a of the orbital ellipse is a^3 / T^2 = GM / 4π^2, where M is the reduced mass of the system (in this case, with 99% of the mass being in one of the two black holes, it's simply the mass of the heavier one).

Plugging 12 years and 18e9 solar masses gives about 2e12 kilometers, or roughly a fifth of a lightyear. This also means the smaller black hole is zipping around the bigger one at around 6% of the speed of light, which is low enough that the Newtonian approximation is probably reasonable accurate (at least to give a rough idea of how large the distances must be).

  Feature               Primary Black Hole         Secondary Black Hole
  --------------------  -------------------------  ----------------------------
  Mass                  18 billion Solar Masses    150 million Solar Masses
  Schwarzschild Radius  ~356 AU                    ~3.0 AU
  Orbital Speed         ~4,200 km/s (1.4% c)       ~502,000 km/s (16.7% c)

  Feature               Value
  --------------------  ---------------------------------
  Orbital Period        12 years
  Average Separation    ~13,800 AU (~0.22 light-years)

So the "smaller" SMBH is punching through the larger one's disk at 16.7% of c twice every 12 years. But it's losing energy to gravitational waves so quickly that they'll probably merge in around 10,000 years [1]

[1] https://archive.is/Ccy5M

edit: just noticed the issue with 502,000 km/s being more than c... recalculating before I can't edit!

edit2: this is gemini's 2nd attempt, and seems more realistic, sorry don't have time to deep dive to verify...

  Feature               Primary Black Hole         Secondary Black Hole
  --------------------  -------------------------  ----------------------------
  Mass                  18 billion Solar Masses    150 million Solar Masses
  Schwarzschild Radius  ~356 AU                    ~3.0 AU
  Orbital Speed         ~282 km/s (0.09% c)        ~33,833 km/s (11.3% c)

maybe this:

One more flare happened since then, in 2022, but because of instrumental limitations, it was caught only at a prestage (M. J. Valtonen et al. 2023; M. J. Valtonen 2024). At the same time, more flares were discovered in historical photographic plate studies so that only eight of the expected 26 flares remain unconfirmed (R. Hudec et al. 2013). All the unconfirmed ones are due to lack of known photographs at the expected epochs.

https://iopscience.iop.org/article/10.3847/1538-4357/ae057e