Dark Matter

This video was produced by DrPhysicsA$^{[1]}$

Suppose we have a very sparse distribution of particles/point-masses surrounding a very dense region of particles where most of the mass is. A familiar example of this is our solar system where the vast majority of the mass is concentrated in the Sun and as you move far away from the Sun the distribution of mass is very sparse and small. If this is the way the mass is distributed, we can approximate and say that the gravitational force $F_g$ acting on any one of the particles in the sparse and mostly empty region is due entirely to the mass $M$ where $M$ can be treated as a single particle/point-mass. For example, when studying our solar system, under this approximation, we can think of the Earth as a speck and all of the other planets, moons, comets and asteroids as other specks which have no gravitational effect on the Earth; under this approximation, we can think of all the mass concentrated at the center making up the Sun as the only lump of mass in the solar system which has a gravitational effect on the Earth and we can treat that entire lump of mass $M_{Sun}$ as a little point-particle. Hence, we can assume that the gravitational force acting on the Earth is $F_{Sun,Earth}=G\frac{M_{Sun}m_{Earth}}{r^2}$. Another good approximation is that the Earth goes around the Sun in a circular orbit (although, in reality, it is an ellipse). Recall that Newton’s second law describe any force $F$ acting on an object (modeled as a particle/point-mass) of mass $m$ as equaling $ma$ where $a$ is the acceleration of the object. In our example, the only force $F$ acting on the Earth is the gravitational force $F_{Sun,Earth}$ due to $M_{Sun}$. Thus, the net force acting on the Earth is just and . Thus, $m_{Earth}a=G\frac{M_{Sun}m_{Earth}}{r^2}$ and $a=G\frac{M_{Sun}}{r^2}$. The acceleration $a$ of any object going in a circular path due to a gravitational force $F_g$ due to a point-mass at the center is $a_c=\frac{v^2}{r}$. Thus,

$$\frac{v^2}{r}=G\frac{M_{Sun}}{r^2}⇒v=\sqrt{G\frac{M_{Sun}}{r}}.$$

(This equation says that the velocity $v$ decreases with increasing $r$.) When we look out through our telescopes at the distribution of stars in our galaxy, we see that the vast majority of the mass is at the center where there are thousands of globular clusters and stars next to each other in a very dense and compact region. The rest of the stars on the outskirts of the galaxy away from the center are very sparsely distributed along the spiral arms. In fact, we have measured that most of the stars and visible matter and mass is at the center (so we know for sure this is where most of the visible matter and mass is). The stars orbit around the center of the galaxy approximately in circular orbits (so we can approximate and say that all the stars have circular orbits). In this example, we can make all the same assumptions that we made about our solar system and go through the same analysis to find that the velocity $v$ of any star at distance $r$ from the center of the galaxy is $v=\sqrt{\frac{MG}{r}}$ where $M$ is the total mass of all the visible matter (mainly stars) at the center. From the way that visible matter is distributed throughout our galaxy and the way that the stars go in roughly circular orbits around the center, we would expect that for stars at greater distances $r$ from the center, they will have smaller velocities $v$. This is actually not the case. In the early 1970s an astronomer named Vera Rubin along with her colleagues measured the rotation rate (that is to say the orbital speed of various stars) in our galaxy to obtain the curve in Figure #. She was awarded the Gold Medal of the Royal Astronomical Society for her pioneering work. We see from this graph that for increasing as we move through the central bulge of the galaxy these measurements agree with our theoretical predictions; however for larger values of that are beyond the central bulge the measured orbital speeds of stars and gas stay roughly constant (in stark contrast to what is actually predicted by theory. We shall see that this suggests that maybe the ordinary, visible masses (stars, planets, etc.) that we see in the night sky (that interacts with light allowing us to see them) isn’t the only kind of mass making up the galaxy.

Let’s come up with a more general model for the mass distribution of the stars in our galaxy. Suppose that the only assumption we make is that the particles are distributed isotropically around a center point. Let’s consider the gravitational force $F_g$ acting on a star at the outermost distances of our galaxy. According to Newton’s theorem, the gravitational force $F_g$ acting on this star of mass $m$ at a distance $R_{Galaxy}(=\text{radius of galaxy})$ from the center is the force $F_g$ due to all the mass $M_{Galaxy}$ enclosed by the sphere of radius $r$. Thus,

$$F_g=G\frac{M(r)m}{r^2}=ma=\frac{mv^2}{r}⇒M(r)=\frac{v^2R_{Galaxy}}{G}.$$

Note that we can measure Newton’s gravitational constant $G$, we can measure the radius $R_{Galaxy}$ of our galaxy, and we can measure the orbital speed $v$ (which is a constant) of each of the stars. When we calculate $M_{Galaxy}=\frac{v^2R_{Galaxy}}{G}$, we obtain a value for $M_{Galaxy}$ which is about 10x bigger than the total mass of all the ordinary matter (stars, planets, etc.) in our galaxy. This suggests that maybe there is some form of invisible matter (called dark matter) distributed throughout the galaxy that does not interact with light.

In the 1930’s astronomers measured the orbital speeds of galaxies in galaxy clusters. These galaxies had roughly circular orbits around the galaxy cluster’s center of mass (or gravity?). An astrophysicist named Fritz Zwicky measured the orbital speeds of a few dozen galaxies in an enormous galaxy cluster called the Coma cluster. For any one of these galaxies orbiting (in a circle) around the Coma cluster’s center of mass with a radius of orbit $r$, astronomers can measure the mass $M_{OM}$ (from the luminosity of each galaxy) of all the mass enclosed in the sphere of radius $r$ surrounding the center of mass. Using the equation $v=\sqrt{\frac{M_{OM}G}{r}}$, we can calculate what the orbital speed of that galaxy “should” be if $M_{OM}$ was the only mass there. Zwicky measured that all of these galaxies had speeds greater than the escape speed given by $v_{esc}=\sqrt{\frac{2M_{OM}G}{r}}$. Subsequent experiments in the decades following Zwicky’s work got analogous results for other galaxy clusters.

Based on the measured mass of all the ordinary matter in our galaxy (which we call $M_{OM}$), the gravitational force $F_{OM,m}=G\frac{M_{OM}m}{R^2_{Galaxy}}$ exerted on a star located at the distant outskirts of our galaxy (where $r=R_{Galaxy}$) will be too small to keep it in orbit based on its measured orbital speed $v$ (where, because $v$ does not fall off as $\frac{1}{\sqrt{r}}$, $v$ is very big at $R_{Galaxy}$). The formula $v=\sqrt{\frac{MG}{r}}$ applies to all objects with a circular orbit around some center; since the stars really do go around the center of our galaxy in circular orbits, the formula $v=\sqrt{\frac{MG}{r}}$ must apply. Given that the orbital speed $v$ of all the stars in our galaxy really is constant (as has been measured), it means that $\frac{M}{r}$ must be constant:

$$v^2=G\frac{M}{r}⇒\frac{M}{r}=k⇒M(r)=kr⇒M(r)∝r.$$

There have been two proposed resolutions which modify our models to agree with experiment. The first is that Newton’s Law of Gravity must be modified over very large distances so that the force $F_g$ doesn’t diminish as rapidly as $\frac{1}{r^2}$. The second hypothesis which would reconcile our models with observation is the claim that the mass $M_{OM}$ due to all the ordinary matter in the galaxy (stars, planets, etc.) isn’t all the mass in the galaxy. This hypothesis is the claim that there is another kind of matter (which is given the name dark matter) which permeates the galaxy that doesn’t interact with light, but which has mass $M_{DM}$. By Einstein’s mass-energy equivalence principle, it must also have energy $E_{DM}$ and energy density $ρ_{DM}$. Since dark matter has energy density $ρ_{DM}=T^{00}$, it acts as a source of gravity and curves space and time according to the EFE’s. Therefore, the path of a beam of light passing by a distribution of dark matter will be deflected from that of a straight line by an angle $α$. Using General Relativity, we can derive a relationship between the angle of deflection $α$ and the distribution of mass $M_{DM}$. If dark matter could somehow be “isolated” from a galaxy, we could use this method/technique of gravitational lensing to prove or disprove the existence of dark matter.

Shortly after the Big Bang, the cosmic web was formed—a colossal structure made of mostly dark matter which spans across the entire observable universe. Astronomers think that the cosmic web played an important role in the formation of galaxies and galaxy clusters. Some theories suggest that the filaments/strands of the cosmic web intersect where galaxy clusters happen to be. Astronomers using the Hubble Space Telescope analyzed the deflection of galaxy light passing by an enormous filament extending across 60 million light-years from the galaxy cluster MACS J0717; from the deflection of light, they were able to determine where the filament is and what its distribution of mass looked like. Using additional data astronomers were able to construct a three-dimensional model of the filamentary structure. (Click on this link for more details: https://www.youtube.com/watch?v=GjO0zdXqCU0)

We can also use this technique to determine the mass of one of the largest structures in the Universe, namely a supercluster of galaxies. By determining the mass of superclusters, we can then estimate the average mass and energy density of the Universe; it might then seem like a good idea to use the EFE’s to determine the curvature of the entire Universe (not just local regions of it). The problem with this approach is that we are measuring the gravitational effects (namely, the deflection angle of light rays) caused by mostly visible objects (namely galaxy superclusters) in order to determine the mass; but on cosmological scales there are great voids in the Universe which are totally dark. If there is dark matter hidden in these voids, our calculation of total curvature of the universe would be incorrect since it only accounted for the masses of galaxy superclusters and not whatever dark matter might be hidden away in these voids.

This article is licensed under a CC BY-NC-SA 4.0 license.

References

1. DrPhysicsA. "Dark Matter and Galaxy Rotation". Online video clip. YouTube. YouTube, 14 July 2014. Web. 21 May 2017.

2. Leonard Susskind. "Dark matter and allocation of energy density." http://theoreticalminimum.com

3. Wikipedia contributors. "Dark matter." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 17 May. 2017. Web. 18 May. 2017.