Image 1 Caption: A cartoon of a typical galaxy rotation curve plot. The "naively expected (A) and observed (B) star velocity as a function of distance from the barycenter of the galaxy." (Somewhat edited.)
Features:
The observed rotation curve B tells us that there is dark matter that we see NO trace of except through its gravitational effects.
However, the ratio of baryonic matter to dark matter in galaxies seems vary various ranging from 1/10??? to 1/30 to less (Dekel et al. 2019, Figure 1).
Baryonic matter in the intergalactic medium (IGM) is ∼ 9 more abundant than baryonic matter in galaxies (see pie_chart_cosmic_energy.html). So most baryonic matter by far is NOT in galaxies. The intergalactic medium (IGM) is mainly primordial cosmic composition (fiducial values by mass fraction: 0.75 H, 0.25 He-4, 0.001 D, 0.0001 He-3, 10**(-9) Li-7). Of course, there is lots of dark matter in intergalactic space, but that is usually NOT counted as part of the intergalactic medium (IGM).
Because of its dominance in galaxies, dark matter is the main determinant of the barycenters of the galaxies.
Moreover, baryonic matter is hard to hide in all respects. If the dark matter were baryonic matter, we believe we would have detected by some means by now.
In fact, we do NOT know what dark matter is. It is thought most likely to be an exotic fundamental particle that we have NOT detected yet, except, as aforesaid, by its gravitational effects. This exotic fundamental particle (i.e., the dark matter particle) apparently only interacts very weakly with itself and baryonic matter, except through gravity.
Another theory is that primordial black holes (PBHs) (i.e., those formed before Big Bang nucleosynthesis era (cosmic time ∼ 10--1200 s ≅ 0.17--20 m)) make up the dark matter.
Hopefully, we will discover what dark matter is in the 2020s.
We can analyze the behavior qualitatively using a very simplified picture. There is a general formula for circular orbit orbital velocity (which is a uniform circular motion) that follows from Newtonian physics (specifically Newton's 3rd law of motion and Newton's law of universal gravitation) for a test particle in a spherically symmetric mass distribution:
v = sqrt[GM(r)/r] ,
where the gravitational constant G = 6.67430(15)*10**(-11) (MKS units), r is radius from the center of the distribution, M(r) is the interior mass (i.e., mass interior to a sphere of radius r), and the mass exterior to radius r has absolutely NO effect as proven by the shell theorem. We consider special cases of formula:
v = sqrt[GM(R)/r] ∝ 1/sqrt(r) ,
where all the mass is confined to within R and r > R. This is the Keplerian orbit case which holds for planets, moons, etc.. This case would hold approximately for the outer part of galaxies if the baryonic matter were all the matter. This is case for curve A in Image 1.
v = sqrt[GCr/r]=sqrt[GC] ,
where C is a constant. In this case M(r)=Cr ∝ r and there is a cancellation of r's and v stays constant with r. This is the case for the plateau region of curve B in Image 1 and approximately the case of real galaxies showing that there is a lot of dark matter. It usually extends out as far as we can find observable tracers to measure orbital velocity from. Dark matter halos extend well beyond the baryonic matter which sits in their gravitational wells.
v = sqrt[G(4π/3)r**3*ρ/r]=sqrt[qrt[G(4π/3)ρ]*r ∝ r ,
where ρ is uniform density (i.e., constant density). For this case the velocity increasely linearly with r. This is very approximately the case for inside planets---if test particles were free to orbit there. The result implies that the gravitational force in planets increases linearly from their centers to their surface. On the size scale of a kiloparsec, galaxies at least to order of magnitude seem constant density in the central regions, and so this this velocity cases suggests the orbital velocities should go toward zero near the centers of galaxies. This behavior is seen for both curves A and B in Image 1.
In fact, from analyzing galaxy rotation curves, the density profiles of galaxies can be determined to a degree. However, the outer regions of the dark matter halos are hard to determine because we run out of tracers for the orbital velocities. Also deviations from spherical symmetry in dark matter halos are also hard to detect.
To illustrate: for Keplerian orbits around SMBHs, we have the following fiducial value formulae:
v = (2.0739 ... km/s)*sqrt[M/(10**6*M_☉)/(r/ 1 kpc)] v = (65.58 ... km/s)*sqrt[M/(10**6*M_☉)/(r/ 1 pc)] v = (65.58 ... km/s)*sqrt[M/(10**9*M_☉)/(r/ 1 kpc)] v = (2073.9 ... km/s)*sqrt[M/(10**9*M_☉)/(r/ 1 pc)]
As one can see on the size scale of 1 kpc (which is about where the plateau orbital velocity starts in Image 1), even a very large SMBH of mass 10**9 M_☉ gives velocities (∼ 65 km/s) that are still relatively small compared to the 200 km/s of the plateau orbital velocity. However, it is possible in some galaxies with very massive central SMBHs (i.e., ∼> 10**9 M_☉) that galaxy rotation curves do stay very high (∼> 200 km/s) in the central region of the galaxies.
The formulae for the gravitational field are:
g = G(4π/3)ρ*r**3/r**2 = G(4π/3)ρ*r = (GM/R**2)(r/R) for inside, g = GM/R**2 at the surface, g = GM/r**2 for outside,
where the gravitational constant G = 6.67430(15)*10**(-11) (MKS units), ρ is the uniform density, and M is the total mass of the sphere. The formulae can be derived from the shell theorem. The gravitational field points radially inward.
Note that the gravitational field goes to the zero at r = 0. Although, stars, planets, etc. are NOT uniform density spheres, they also have gravitational fields going to the zero at their centers. This can be proven by the shell theorem.
Also note that outside of the sphere, the gravitational field obeys an inverse-square law as for a point mass. Thus, though this is NOT clear in Image 3, the gravitational field outside the sphere only goes to zero as →∞. This outside behavior also holds for any spherically symmetric mass distribution (as can be proven by the shell theorem), and so holds for stars, planets, etc.