Could an hypothetical hidden black hole companion to the sun be revealed by proper motion data?

If the Sun were in a binary star system with a black hole companion, the Sun would be orbiting the black hole, and this should be reflected in a change in the observed proper motion of nearby stars.

Are the current measurements of proper motion, old enough, precise and extensive enough to detect such a change?

Have any calculations been performed to estimate the function of mass and distance that would place a black hole companion (or a passing black hole in the vicinity) below the detectable threshold of our current data?

A black hole at a distance of 10,000 AU (comparable to the separation of Proxima from Alpha Centauri) could mean that the sun was in an orbit lasting longer than 200,000 years.

So in two hundred years, it would have moved a third of a degree around its orbit, so I did a rough calculation of how much the sun's velocity would have changed, and it was about 8 m/s. That is well below the uncertainties in stars' relative velocities.

(Assume circular motion, the sun would have a velocity of about 1.4km/s relative to the black hole, and an acceleration of v^2/r or 1.3e-9 m/s^2. over 200 years, (and now assuming linear motion) that is about 8 m/s)

But This change in velocity would be towards the black hole, so if this had occurred we would see a systematic change in the radial velocity of stars along a particular axis. Radial velocities are known to a much higher accuracy than relative velocity. It should be possible to detect a systematic change of 8m/s, or even a change of 1m/s over 25 years, using more recent and more accurate data. But I'm not aware of any attempt to do this.

The key to this is not to depend on proper motion, which is not accurate enough, but only radial motion which can be determined spectroscopically to great accuracy. However you'd need to have data from a range of stars, as you would be looking for a statistical effect by averaging a number of stars.

I've made several assumptions about the nature of the black hole. Clearly a very large, or very close black hole can be discounted. Detecting a smaller or more distant black hole would be more challenging and depend on a statistical result.

• At 10.000 AU such a black hole would be right inside the Oort cloud. I would assume that if such a black hole existed one would model the Oort cloud very differently. I wonder whether the observations that let to our current model of the Oort cloud could be used to disprove the existence of such a black hole. Commented Dec 3, 2023 at 10:49
• Probably, I've not considered any other means of observing a slowly moving, nearby, 10M black hole. But you're right, it would mess up the Oort cloud a treat. Commented Dec 3, 2023 at 10:59