Dark Matter

Allen E. Martin

Abstract

Math in the news. This application of data analysis discusses recent discoveries in astronomy. The universe itself is changing, and mathematics is a key to discovering and understanding the patterns.

Dark Matter's Existence

For the past 50 years or so there has been a growing body of evidence to indicate that the universe is filled with dark matter. Recent observations have produced compelling evidence that something like 90 to 99 percent of the matter in the universe is not radiating, - at any wavelength! "This dark matter, invisible in the optical, ultraviolet, X-ray, gamma-ray, infrared, and radio regions of the spectrum, is detected by its gravitational attraction on the matter that we can see." [cf. Rubin in Bubbles] Its existence has been deduced from studies of velocities of stars and gas orbiting near the center of galaxies. Furthermore, this "invisible matter" is all around us, permeating every part of the universe that we can see!

Detailed measurements of the gas and stars in several hundred galaxies have been computed and these show that our solar system is not a good model for the universe in general. In our solar system, the orbital velocities of planets near the sun are very high compared to the planets further out. For example, Mercury orbits with a velocity about 10 times as great as that of Pluto. Almost all the mass of our solar system is located at its center (i..e., the Sun) and the planets orbit under the influence of its gravitational attraction. It is a little like when water is swirling in a sink as the sink drains. The water near the hole in the center spins the fastest. The pattern that astronomers are now finding is that stars at the periphery of a galaxy orbit with speeds as great or greater than those close to the center! Seemingly, the only possible way to explain this phenomenon is by postulating the gravitational attraction of enormous amounts of dark matter - matter that we simply cannot "see."

These surprising new findings indicate that the distribution of light that we can see in the sky and through our telescopes does not correctly map the distribution of matter in the universe. Astronomers are having to completely rethink their assumptions and ideas about how matter is distributed throughout the universe. When a galaxy is viewed by telescope, it is brightest near its center. The brightness of the light falls off rapidly the farther away from the center you look. Astronomers had long inferred that the mass of the galaxy was distributed in the same way - much like it is in our own solar system. Dr. Vera Rubin writes, "Thus it was a surprise to learn that in all galaxies studied, orbital velocities remain high, even at the optical edge of the galaxy where there is almost no light."

Some History

By the end of the seventeenth century, Robert Hooke had suggested that the planets were subjected to a gravitational force from the Sun, a force whose intensity decreased as the square of the distance. Newton then recognized that all pairs of objects in the universe have a gravitational attraction for each other that is proportional to the product of their masses and inversely proportional to the square of the distance between them.

QUESTION 1:

Express Newton's law of gravitational attraction algebraically as a law of variation.

QUESTION 2:

If the distance between two objects is increased by a factor of 2, what is the effect on their mutual gravitational attraction?

In our solar system, virtually all of the mass is contained within the Sun. This large central mass exerts enormous gravitational attraction on each of the planets. In fact, it is only the orbital velocity of a planet that keeps it from falling inward and crashing into the Sun. Planets near the Sun must maintain very high orbital velocities to offset this gravitational pull, while those farther out can stay in orbit at slower speeds. Mercury is the planet closest to the Sun, and it orbits with a velocity of 47 km/sec. Pluto is 100 times more distant, and orbits with a velocity of 1/10th as large. Any object at Pluto's distance will orbit with this same velocity - be it a comet, some rocks, or a planet.

QUESTION 3:

Using the modeling assumption that the orbits of the planets are nearly circular, compute the approximate orbital velocity for each planet (in km/sec) - include your results in the table above (column 5)

QUESTION 4:

(a) Sketch and fully label a graph showing

Average Distance from the Sun (Astronomical Units)
vs. # of Earth Days for 1 Orbit

(b) Determine the law of variation that governs this graph.

QUESTION 5:

(a) Sketch and fully label a graph showing

Average Distance from the Sun (in Astronomical Units)

vs. Orbital Velocity (in km/sec)

(b) Determine the law of variation that governs this graph.

As you have seen in your solutions to the previous problems, when (almost) all the mass is located at the center of the rotating system, the orbital velocities get slower and slower the farther out from the center you go. This is a necessary consequence of Newton's gravitational laws. For orbiting masses, the gravitational forces are in exact equilibrium with the centrifugal forces.

where M is all the mass interior to R.

This implies that V varies directly as .

In our solar system, M (essentially) does not vary with R - so we get the relation

.

In fact, as Dr. Rubin writes, this type of variation "is evidence that virtually all of the mass in the solar system is located in the Sun."

However, in careful measurements of hundreds of spiral galaxies, orbital velocities were determined as functions of distance from the center of the galaxy and virtually all the rotation velocities remained high at positions far out from the center of the galaxies. The velocities did not decrease as the distance increased - as one would expect if the mass were distributed like the light from the stars. "The conclusion is inescapable: matter, unlike luminosity, is not concentrated near the center of spiral galaxies. The distribution of light in a galaxy is not at all a guide to the distribution of matter."

We can use the relationship

to help us understand the meaning of all this. Because the orbital velocities of the planets follow the

pattern of variation, we know that virtually all of the mass is located at the Sun. "Conversely, for spiral galaxies, the high velocities at large distances R are convincing evidence that most of the mass is not restricted to the central regions." (See diagram below.) The fact that V does not decrease as R increases means that the quantity

does not decrease as R increases. Therefore M must be increasing! In other words, the mass of a spiral galaxy increases with the radius of the galaxy, and continues to grow well past the limit of the luminous region.

By understanding the meaning and implications of her data, Dr. Vera Rubin and other scientists were able to see that something new was occurring in their data. In trying to explain this data in meaningful ways they were led to the discovery of a radically new understanding of our universe.

Sources and Bibliography

All quotes are from "Weighing the Universe: Dark Matter and Missing Mass" by Vera C. Rubin, Bubbles, Voids and Bumps in Time: The New Cosmology, edited by James Cornell, Harvard-Smithsonian Center for Astrophysics, Cambridge University Press, 1989 (73-104)

The discovery of dark matter and the implications of its existence for astronomy and cosmology are the subject of one of the issues of a NOVA presentation called "The Astronomers". Dr. Vera Rubin is one of the researchers featured in this video. As a companion to this PBS series, St. Martin's Press has published The Astronomers by Donald Goldsmith. Chapter 4 of this book deals with the topic of Dark Matter and Dr. Rubin's contributions to the research.

The data tables used in this article were taken from David Cohen's Algebra & Trigonometry (2nd edition), West Publishing Co., 1989 (See: #70 on p. 27 and #35 on p. 211)

"Dark Matter in Spiral Galaxies" by Vera C. Rubin, Scientific American, June 1983 (96-108) You can find several graphs of the rotation curves for spiral galaxies on page 101 of this article.

"The Rotation of Spiral Galaxies" by Vera C. Rubin, Science 220 (June, 1983)

"Between the Galaxies" by Stephen E. Schneider and Yervant Terzian, American Scientist, Nov./Dec. 1984

"Women and the Stars" by Joseph L. Spradley, The Physics Teacher, Sept. 1990 Very helpful overview of the role of women scientists in astronomy.

"Invisible Galactic Halos" by Karel Vander Lugt, The Physics Teacher, Feb. 1993 Discussion of various models for galaxy dynamics.

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