In simple terms: yes, the Milky Way's supermassive black hole (SMBH) is at the center of the galaxy, we know approximately where the center is (but not terribly precisely), and we should expect the SMBH to be there.

You can define the Galactic Center using the orbits of stars and gas: what is the (average) center of their orbits? You can also use the spatial distribution of the same (which is probably a bit closer to what you're thinking of as a "center of gravity" or "center of mass"). This is never going to give you an exact answer, and since the galaxy is not perfectly symmetric or unchanging, there will always be some uncertainty and variation.

There's a 2D coordinate definition, which is the point on the sky where Galactic latitude and longitude both = 0. This is intended to approximate the direction toward the GC, but since it was defined back in the 1950s, no one is worried if modern attempts to measure the (3D) center of the galaxy might be slightly different.

As far as we can tell, yes, the SMBH is at the Galactic Center. If a SMBH did form outside the central region of a galaxy, dynamical friction would make it spiral into the center of the galaxy (and damp any radial orbital oscillations as well). This would also cause a SMBH from another (smaller) galaxy that merged with the Milky Way to spiral into the GC and merge with the SMBH there; see my answer to this question.

The SMBH is not the "main player" in our galaxy: it has a mass of about 4 million solar masses, which you can compare with the total stellar mass of about 50 billion solar masses. (And a dark-matter mass of roughly one trillion solar masses.) Nonetheless, there is evidence that SMBHs "co-form", in some sense, with their host galaxies. In particular, the mass of SMBHs is fairly tightly correlated with several properties of the host galaxy (e.g, the mass of the "bulge", or the velocity dispersion of stars in the inner part of the galaxy).

[Edited to fix a typo.]

In simple terms: yes, the Milky Way's supermassive black hole (SMBH) is at the center of the galaxy, we know approximately where the center is (but not terribly precisely), and we should expect the SMBH to be there.

You can define the Galactic Center using the orbits of stars and gas: what is the (average) center of their orbits? You can also use the spatial distribution of the same (which is probably a bit closer to what you're thinking of as a "center of gravity" or "center of mass"). This is never going to give you an exact answer, and since the galaxy is not perfectly symmetric or unchanging, there will always be some uncertainty and variation.

There's a 2D coordinate definition, which is the point on the sky where Galactic latitude and longitude both = 0. This is intended to approximate the direction toward the GC, but since it was defined back in the 1950s, no one is worried if modern attempts to measure the (3D) center of the galaxy might be slightly different.

As far as we can tell, yes, the SMBH is at the Galactic Center. If a SMBH did form outside the central region of a galaxy, dynamical friction would make it spiral into the center of the galaxy (and damp any radial orbital oscillations as well). This would also cause a SMBH from another (smaller) galaxy that merged with the Milky Way to spiral into the GC and merge with the SMBH there; see my answer to this question.

The SMBH is not the "main player" in our galaxy: it has a mass of about 4 million solar masses, which you can compare with the total stellar mass of about 50 billion solar masses. (And a dark-matter mass of roughly one trillion solar masses.) Nonetheless, there is evidence that SMBHs "co-form", in some sense, with their host galaxies. In particular, the mass of SMBHs is fairly tightly correlated with several properties of the host galaxy (e.g, the mass of the "bulge", or the velocity dispersion of stars in the inner part of the galaxy).

[Edited to fix a typo.]

In simple terms: yes, the Milky Way's supermassive black hole (SMBH) is at the center of the galaxy, we know approximately where the center is (but not terribly precisely), and we should expect the SMBH to be there.

You can define the Galactic Center using the orbits of stars and gas: what is the (average) center of their orbits? You can also use the spatial distribution of the same (which is probably a bit closer to what you're thinking of as a "center of gravity" or "center of mass"). This is never going to give you an exact answer, and since the galaxy is not perfectly symmetric or unchanging, there will always be some uncertainty and variation.

There's a 2D coordinate definition, with is the point on the sky where Galactic latitude and longitude both = 0. This is intended to approximate the direction toward the GC, but since it was defined back in the 1950s, no one is worried if modern attempts to measure the (3D) center of the galaxy might be slightly different.

As far as we can tell, yes, the SMBH is at the Galactic Center. If a SMBH did form outside the central region of a galaxy, dynamical friction would make it spiral into the center of the galaxy (and damp any radial orbital oscillations as well). This would also cause a SMBH from another (smaller) galaxy that merged with the Milky Way to spiral into the GC and merge with the SMBH there; see my answer to this question.

The SMBH is not the "main player" in our galaxy: it has a mass of about 4 million solar masses, which you can compare with the total stellar mass of about 50 billion solar masses. (And a dark-matter mass of roughly one trillion solar masses.) Nonetheless, there is evidence that SMBHs "co-form", in some sense, with their host galaxies. In particular, the mass of SMBHs is fairly tightly correlated with several properties of the host galaxy (e.g, the mass of the "bulge", or the velocity dispersion of stars in the inner part of the galaxy).

In simple terms: yes, the Milky Way's supermassive black hole (SMBH) is at the center of the galaxy, we know approximately where the center is (but not terribly precisely), and we should expect the SMBH to be there.

You can define the Galactic Center using the orbits of stars and gas: what is the (average) center of their orbits? You can also use the spatial distribution of the same (which is probably a bit closer to what you're thinking of as a "center of gravity" or "center of mass"). This is never going to give you an exact answer, and since the galaxy is not perfectly symmetric or unchanging, there will always be some uncertainty and variation.

There's a 2D coordinate definition, which is the point on the sky where Galactic latitude and longitude both = 0. This is intended to approximate the direction toward the GC, but since it was defined back in the 1950s, no one is worried if modern attempts to measure the (3D) center of the galaxy might be slightly different.

As far as we can tell, yes, the SMBH is at the Galactic Center. If a SMBH did form outside the central region of a galaxy, dynamical friction would make it spiral into the center of the galaxy (and damp any radial orbital oscillations as well). This would also cause a SMBH from another (smaller) galaxy that merged with the Milky Way to spiral into the GC and merge with the SMBH there; see my answer to this question.

The SMBH is not the "main player" in our galaxy: it has a mass of about 4 million solar masses, which you can compare with the total stellar mass of about 50 billion solar masses. (And a dark-matter mass of roughly one trillion solar masses.) Nonetheless, there is evidence that SMBHs "co-form", in some sense, with their host galaxies. In particular, the mass of SMBHs is fairly tightly correlated with several properties of the host galaxy (e.g, the mass of the "bulge", or the velocity dispersion of stars in the inner part of the galaxy).

[Edited to fix a typo.]