The Simplest Gravitational Evidence for Dark Matter

The nature of dark matter

Dark matter is one of the major unresolved questions in modern physics. It cannot be observed directly through ordinary electromagnetic radiation, yet its gravitational influence is evident in the motion of stars and galaxies, gravitational lensing, galaxy clusters, and the large-scale structure of the Universe.

Among these observations, galactic rotation provides one of the simplest ways to understand the problem. The argument requires only basic Newtonian gravity: if the motion of an object is known, that motion can be used to infer the mass responsible for the gravitational field.

Consider a star orbiting a galaxy at a distance r from its centre. For an approximately circular orbit, its orbital velocity is given by: 

where G is the gravitational constant and M is the mass enclosed within the star's orbit. Rearranging,

This relationship provides a direct connection between the observed motion of stars and the gravitational mass of a galaxy. The same gravitational reasoning can also be expressed through escape velocity,

In both cases, the important point is the same: gravitational motion provides information about mass, including mass that cannot be seen directly.

The problem with galactic rotation

If the visible matter of a galaxy contained essentially all of its mass, the orbital velocity of stars should generally decrease with increasing distance from the galactic centre. Once the enclosed mass becomes approximately constant,

Thus, the outer regions of a galaxy should exhibit progressively lower orbital velocities.

Observations of spiral galaxies, however, show a different behaviour. Their rotation curves often remain approximately flat over substantial radial distances. Stars and gas in the outer regions continue to orbit at velocities that are higher than expected from the distribution of visible matter alone. This creates a mass discrepancy. The gravitational mass inferred from the observed motion is greater than the mass accounted for by the visible components:

The conventional interpretation is that galaxies contain an additional, non-luminous component known as dark matter.

Figure 1. Galactic rotation curve of the Andromeda Galaxy (M31) comparing predicted velocities based on visible matter with observed velocities (Rubin & Ford, 1970).

The Milky Way as a case study

The Milky Way provides an especially useful example because its stellar motions can be studied in considerable detail. The European Space Agency's Gaia mission has provided extensive astrometric data that allow the kinematics of our Galaxy to be investigated with much greater precision.

Cautun et al. (2020) used the Galactic rotation curve from Gaia DR2, together with other observational constraints, to construct a physically motivated model of the Milky Way's mass distribution. Their preferred model includes the stellar and gaseous components of the Galaxy embedded within a dark-matter halo.

Their best-fitting model gives a dark-matter halo mass of:

and a total halo mass of:

The corresponding stellar mass is:

Here,  denotes the solar mass. Using: 

  

the central value of the inferred dark-matter halo mass corresponds to approximately:

with the quoted uncertainty corresponding to approximately:

This number should be interpreted carefully. It is not a direct measurement or weighing of dark-matter particles. Rather, it is the dark-matter mass inferred from a gravitational model that reproduces the observed dynamics of the Milky Way. The quantity M_200 represents the mass within a conventional halo boundary defined by an average density of 200 times the critical density of the Universe.

The mass distribution tells an even clearer story

The significance of the result becomes more apparent when the mass distribution is considered as a function of distance from the Galactic centre.

Cautun et al. provide an enclosed mass profile for the Milky Way, separating the contributions from stars, baryonic matter, dark matter, and total mass. Their model shows that the inferred dark-matter contribution becomes increasingly important with increasing Galactic radius.

Figure 2. Enclosed mass profiles of the Milky Way as a function of distance from the Galactic centre. The solid black line shows the enclosed baryonic mass; the coloured lines show the enclosed dark-matter mass for haloes of different masses and concentrations, before (dotted) and after (solid) baryonic contraction. Based on Cautun et al. (2020), Figure 5.

At relatively small radii, the stellar component makes a substantial contribution to the enclosed mass. At larger radii, the dark-matter contribution becomes progressively more significant. This provides the physical basis for describing the Milky Way as a visible stellar disc embedded within a much more extended dark-matter halo.

The model also gives a local dark-matter density at the Solar position of: 

or equivalently:

The local density and the total halo mass describe different aspects of the same inferred component: one characterizes the dark-matter density near the Solar neighbourhood, while the other represents the integrated mass of the Galactic halo within the adopted halo boundary.

What does this actually demonstrate?

It is important not to overstate what the rotation curve establishes.

The motion of stars does not constitute a direct detection of a dark-matter particle. Instead, it demonstrates that the observed gravitational field cannot be adequately accounted for by the visible matter alone under the standard gravitational interpretation.

Dark matter is the conventional explanation for this additional gravitating mass. However, alternative explanations based on modified theories of gravity have also been investigated. Consequently, the most precise statement is that galactic dynamics provide strong evidence for an additional unseen gravitating component, conventionally identified with dark matter.

The significance of dark matter does not rest on galactic rotation alone. Independent evidence comes from gravitational lensing, the dynamics of galaxy clusters, the cosmic microwave background, and the observed formation and distribution of large-scale cosmic structures. The convergence of these different observations is what makes the dark-matter interpretation particularly important in modern cosmology.

References

Rubin, V. C., & Ford, W. K., Jr. (1970). Rotation of the Andromeda Nebula from a spectroscopic survey of emission regions. The Astrophysical Journal, 159, 379–403. https://doi.org/10.1086/150317

Cautun, M., Benítez-Llambay, A., Deason, A. J., Frenk, C. S., Fattahi, A., Gómez, F. A., Grand, R. J. J., Oman, K. A., Navarro, J. F., & Simpson, C. M. (2020). The Milky Way total mass profile as inferred from Gaia DR2. Monthly Notices of the Royal Astronomical Society, 494(3), 4291–4313. https://doi.org/10.1093/mnras/staa1017.