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I was reading this article that mentions a blackhole as having a gravitational sphere of 4,000 light-years.

I'd not heard of the term (gravitational sphere) before so looked it up, and it looks like a simple definition is the same as sphere of influence and according to Wikipedia is

A sphere of influence (SOI) in astrodynamics and astronomy is the spherical region (actually is an oblate sphere) around a celestial body where the primary gravitational influence on an orbiting object is that body.

From the above, I take it to mean that an object could be tiny, but providing there are no objects anywhere near with another object with a bigger gravitational pull near them, then its sphere could be larger than a more massive object. i.e. I can't directly look at a gravitational sphere of 4000 light-years and think "that must be a huge object".

Is my understanding correct?

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up vote 2 down vote accepted

No you are not exactly right. There is a difference bethween SOI for celestial bodies (ie planets) and that of a black hole (BH). The SOI for a BH is usually used when trying to determine the mass of a supermassive BH at the center of a galaxy. The definition, "The region around a supermassive black hole in which the gravitational potential of the black hole dominates the gravitational potential of the host galaxy" (This definition is also taken from Wikipedia though since I couldn't find it in my Cosmology book), is similar to the one you've stated above but there is nothing comparable in size in the surrounding area and the mass of the BH is strongly correlated to the size of the SOI.

The radii of the SOI is defined by

$r_h = \frac{G M_{BH}}{\sigma^2}$

where $\sigma$ is the stellar velocity dispersion, $G$ is the gravitational constant and $M_{BH}$ the mass of the BH. As you can see the radii of the SOI is dependent on the mass of the BH in this equation.

The spheroid bulge of stars around the BH can be measured by observations in the near infrared spectrum and is closely related to the ROI. When the radii is combined with the galaxys effective stellar velocity dispersion it is then possible to determine the mass of the central BH.

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