Weighing Black Holes

Let B be any astronomical object of unknown mass M. Suppose it's far from our galaxy--and invisible at any wave length. How to estimate M?

Here's a simple case: Suppose there's a small visible object b orbiting B, and that the line of sight from us to b makes a reasonable angle with the plane of b's orbit. Now examine b's light with a spectrograph. It will show the same spectral lines changing between two limit positions. (This is how the fact that b is orbiting something will show up--it's too far away to inspect visually.)

Evidently, these measurements give the rate of rotation of b. But they will also give the speed v of b, because light from b is Doppler-shifted to higher frequencies when it is coming toward us and to lower frequences--half an orbit later--when it's going away from us.

So the problem is to express M in terms of and v.

To keep things simple, suppose b is in approximately circular orbit around B, at radius r. Though r is not directly measurable, we can get it from v=r.

Now apply Newton's motion law F=ma combined with his gravitation law F=Mm/r² (both as scalar equations) to the motion of b. In convenient units,

   Mm/r²=F=ma=m²r.

Hence M=²r³= v³/.

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