Dark matter is a fascinating, frustrating scientific mystery. Astronomers claim that much of it forms halos surrounding galaxies, yet “no one has ever seen this material or been able to study it” [1]. Hence, anticipation accompanied delivery of the “Alpha Magnetic Spectrometer - 02” to the International Space Station in May 2011. Although a goal was to discover this elusive stuff, nothing conclusive has been found [1]. Dark matter supposedly explains gravitational effects, which suggests the merit of analyzing colliding galaxies because gravitational dragging should distort those halos. However, such behavior was not detected [3]; what happened was consistent without the supposed presence of dark matter. A recently-concluded, highly-sensitive and anticipated “Large Underground Xenon” experiment [8] failed to detect a single trace of dark matter. With these and other consistently negative findings, why should we believe this material exists?

Figure 1. Image of Earth’s nearest galaxy, the Andromeda, with trailing stars. Photo credit: Wikimedia Commons.

The search for dark matter relies upon circumstantial evidence. All of the astronomers and astrophysicists I have consulted with agree: The most compelling argument, as described in the press, is that galaxies would spin apart if dark matter did not exist, which is contrary to observations. In other words, dark matter is the difference between a mathematical prediction (mass needed to keep a galaxy from dissipating) and the known amount of mass.

Because dissipation qualifies as dynamics, I investigated it [6-7]. Spoiler alert: Newtonian dynamics do not support the dissipation assertion, nor do they require dark matter and halos. But whether dark matter exists is an empirical issue that mathematics cannot resolve. Something might be found eventually, but what follows compromises, if not eliminates, a standard argument for the existence of massive amounts of this mysterious material.

Computing Mass

Intuition about the computation of mass comes from swinging a weight where “a fast swing plus a weak string can equal a broken window” [7]. To determine the Sun’s mass, \(M_s\), planets are the rotating objects and the string’s strength comes from \(M_s\) and the planet’s distance from the Sun, \(r\). The only unknown in Newton’s two-body equation is \(M_s\). So, with nearly circular orbits (the scalar \(r''\approx 0\)), it follows that

\[M_S = \frac{rv^2}{G}, \tag1 \]

where \(G\) is the gravitational constant and \(v\) is the rotational velocity. To illustrate this with our home planet, \(r=92.96 \enspace \textrm{million miles}\) and \(v= 2\pi r/\textrm{(one year)}\).

\((1)\) also identifies the dissipation concern: if \(M_s\) is much smaller than in \((1)\) for \(r\) and \(v\), the two-body system cannot be stable; i.e., the \(r''\approx 0\) assumption is false. Using the spinning object analogy, a much smaller \(M_s\) value represents a “weak string,” thus encouraging an escaping planet.

Unfortunately, general properties of billion-body Newtonian systems are unknown, so astronomers developed creative approximations to study galactic systems. Intuition for a standard approach comes from star-soup type pictures of galaxies, with bodies pulling others along (as in Figure 1). This star-soup appearance suggests approximating actual \(N\)-body systems with a continuum where, if \(M(r)\) is the galaxy’s mass up to a star at distance \(r\) (from the galaxy’s center) and \(v\) is its rotational velocity [2], then

\[M(r) = \frac{rv^2}{G}. \tag2 \]

Figure 2. Spiral galaxy’s rotational velocity curve (RVC). Image credit: Donald G. Saari.

The \(M(r)\) values depend on \(v\), the rotational velocity of a star at distance \(r\). Here, Vera Rubin’s stunning contributions [4] become relevant. Contrary to expectations, she showed that the rotational velocity curve (RVC) of spiral galaxies behaves as in Figure 2; it increases almost linearly near the galactic center to essentially flatten out with \(v \approx B\).

Substituting \(v \approx B\) into \((2)\) predicts that

\[ M(r) \approx D r, \quad m(r) = M'(r) \approx D, \qquad \textrm{ where } D=\frac{B^2}{G}, \tag3 \]

which significantly exceeds in mass every entity known to exist! Where is the mass? If \((2)\) is correct, and the known \(M(r)\) is significantly smaller than in \((3)\), one of the following must be true:

The galaxy is dissipating (contrary to observations), Newton’s laws are incorrect, or Massive amounts of unobserved matter exist.

The first two options are not palatable, which leaves the third and a search for dark matter – the difference between mathematical predictions and the known mass. The discrepancy is substantial for large \(r\) values, which explains the conjectured halos surrounding galaxies.

Dynamics

Everything depends on whether \((2)\) holds for Newtonian systems of \(N\) discrete bodies (e.g., stars, planets, etc.). It does not! The fact that \((2)\) is identical to the two-body equation \((1)\) offers a clue. Independent of its derivation, the two-body equation \((2)\) approximation is inappropriate for galaxies as it contradicts those pictures of stars pulling other stars along – a dynamic not admitted by \((2)\). But Newton’s equations require strong near-body interactions where faster-moving stars (e.g., body 1 in Figure 3) drag along slower ones (body 2, which then drags body 3, etc.), as in pictures of galaxies. So, a star’s Newtonian rotational velocity is the \(M(r)\) gravitational effect plus dragging terms; these larger \(v\) values exaggerate \(M(r)\) values of \((2)\). As dragging is more pronounced at larger distances, expect this error to incorrectly predict halos.

Figure 3. Center of mass versus near-neighbor attraction. Image credit: Donald G. Saari.

Angular Momentum

Accepting Newton’s equations requires accepting that \((2)\) exaggerates \(M(r)\) values. But are the incorrect values “close enough” for practical purposes? To explore this concern, I applied \((2)\) to analytic billion-body solutions [6] with properties similar to those motivating \(2)\). My solutions with \(m(r) \le \frac1r\) have \(M(r) \le \ln(r)\). But the values of \((2)\) exceed \(M(r) \approx D r\) (see \((3)\)), proving that \((2)\) can yield exponentially exaggerated predictions! No solution has halos, yet \((2)\) incorrectly predicts massive ones! These egregious errors manifest differences between systems of discrete bodies and approximations; actual systems involve dragging effects, while continuum approximations ignore this crucial dynamic.

To exploit Rubin’s seminal contributions, the rotational velocities suggest using the system’s angular momentum \(\sum_{j=1}^N m_j {\mathbf r_j} \times \mathbf v_j = \mathbf c\) (\(\mathbf{c}\) is a constant of integration). In the simpler coplanar setting (for exposition), this is

\[\sum_{j=1}^N m_j r_j v_j = c, \tag4\]

where \(r_j\) is the \(\mathbf{r}_j\) length and, here, \(v_j\) is the rotational velocity.

There exists \(S(t)\) [5] where the \(j^{th}\) body's contribution to the angular momentum about the axis of rotation is \(Sr_j\); i.e., \(c = \sum m_j r_j(S r_j)\). Should each body’s rotational velocity be \(Sr_j\), these rotational terms would represent a rigid body rotation, which does not happen. Instead, a star’s rotation relative to the system is \(\omega_j=v_j - S r_j\).

To depict \(\omega_j\), start with the angular momentum line (AML) \(y=Sr\), which identifies a body’s contribution to the system’s angular momentum (see Figure 4). In Figure 4, \(\omega_j\) is the vertical difference between the RVC and the AML. If \(\omega_j>0\) (the RVC is above the AML), the star is a “leader” and rotates faster than the system. If \(\omega_j<0\) (the RVC is below the AML), the star is a “laggard” and falls behind the system’s rotation. These differences are related with the following interesting equality (substitute \(v_j= S r_j + \omega_j\) into \((4)\):

\[\sum_{\textrm{Leader}} m_jr_j \omega_j =\sum_{\textrm{Laggard}} m_jr_j |\omega_j|. \tag5 \]

As \(\omega_j\) values define AML-RVC differences, \(W=\sum m_j \omega_j^2\) measures the AML positioning. If, for instance, \(W=0\), then \(\omega_j=0\) forces the RVC to lie on the AML. Similarly, small \(W\) values require the RVC to be “close” to the AML. Estimates on \(W\) values involve standard galactic stability assumptions: the galaxy’s moments of inertia and kinetic energy are nearly constant. With two-body systems, these conditions require that the velocity emphasize rotations. The same conclusion [5, 7] holds for \(N\)-body systems; they require a small \(W\) value, forcing the RVC to be close (defined by \(W\)) to the AML.

Figure 4. Effects of angular momentum on mass. Image credit: Donald G. Saari.

And so, with small \(m_j\omega_j^2\) values, large masses must have small \(\omega_j\) values, forcing the RVC to approach the AML. This is precisely what we observe! Near the galactic center with its heavier masses, the RVC is nearly a straight line (see Figure 2). So, dynamics explain this initial, linear growth where the RVC hugs the AML (see Figure 4).

For large distances, Figure 4 requires \(\omega_j \approx S r\); i.e., \(m_j\omega_j^2\) nearly equals \(S^2m_jr^2\). Thus, small \(m_j\omega_j^2\) values require \(m(r) \le A/{r^2}\), which is compatible with known mass levels but significantly contradicts \(m(r)\approx D\) assertions in \((3)\) and the existence of massive halos. Indeed, \(m(r) \approx D\), and small \(W\) values require \(\omega_j\le \frac A{r}\), so the RVC approaches the linear AML, contradicting observations. Similarly with \((5)\), as the large laggard region distances have \(|\omega_j|\) values asymptotically approaching \(S r_j\), the summation eventually has \(\sum_{\textrm{Laggard}} m_j r_j^2\) sizes; \((5)\) indicates that \(m(r) \le A/{r^2}\).

Thus, Newtonian dynamics dismiss \((2)\) and \((3)\). One reason for this is that two-body approximations of Newtonian systems in \((2)\) use the RVC height (see Figure 4). The variable for actual Newtonian systems is the RVC-AML difference; for example, analysis of \(N\)-body systems differs significantly from that of its parts or approximations in \((2)\). This casts doubt about a standard argument claiming massive amounts of dark matter.

References

[1] Alpha Magnetic Spectrometer. (2016). NASA. Retrieved from https://ams.nasa.gov.

[2] Binney, J., & Tremaine, S. (2008). Galactic Dynamics (2nd ed.). Princeton, NJ: Princeton University Press.

[3] Harvey, D., Massey, R., Kitsching, T., Taylor, A., & Tittley, E. (2015, March 27).The nongravitational interactions of dark matter in colliding galaxy clusters. Science, 347, 1462-1465.

[4] Rubin, V. (1993). Galaxy dynamics and the mass density of the universe. Proc. Natl. Acad. Sci., 90, 4814-4821.

[5] Saari, D.G. (2005). Collisions, Rings, and Other Newtonian N-Body Problems. Providence, RI: American Math Society.

[6] Saari, D.G. (2015, May). N-body solutions and computing galactic masses. The Astronomical Journal, 149, 174-180.

[7] Saari, D.G. (2015, May). Mathematics and the ‘Dark Matter’ puzzle. American Math Monthly, 122, 407-423.

[8] World’s most sensitive dark matter detector completes search. (2016, July 22). Science Explorer.

Donald G. Saari is a distinguished professor and director of the Institute for Mathematical Behavioral Sciences at the University of California, Irvine. His research interests range from the Newtonian N-body problem to voting theory and evolutionary properties of the social and behavioral sciences.