Introduction: What is dark matter? *

How do we know it exists? (Evidence for Dark Matter) *

Possibilities for dark matter *

Gravitational Lensing/Macrolensing (GL) *

Applications of GL *

Gravitational Microlensing (ML) *

Deviant Microlensing *

EROS, MACHO, and OGLE: The Search for MACHOs *

Conclusion *

Bibliography *

Appendix 1: Glossary *

Appendix 2: Geometrized Units *

More than ninety percent of the mass of the Universe is unaccounted for (Krauss xiv). To explain this "missing mass" problem, scientists believe that most of the Universe is comprised of "dark matter" that neither transmits nor absorbs light. There are many theories that attempt to explain the nature of dark matter, from WIMPs to MACHOs to a failure in Einstein's Theory of Relativity. The exact nature of dark matter is important because the amount of dark matter in existence will determine whether the Universe will collapse in on itself or continue to expand forever.

I have focussed my research on gravitational microlensing, one of the methods that astrophysicists use to learn about dark matter. Microlensing occurs when the light from a distant galaxy is bent by gravity before it reaches the observer. As a result, an observer on Earth can see multiple images of the same galaxy, since the light from the source is "microlensed" by a mass. When no such mass is visible, it is very likely that microlensing is caused by dark matter. The aim of this essay is to determine whether dark matter in the form of MACHOs exists by using microlensing.

Knowledge of dark matter is also important because we can learn more about the structure of the Universe. Although humans have not even left the Solar system yet, we can still point our telescopes to the heavens and observe. Perhaps these observations can prepare us for the mysteries of Space.

In the 1930s, a physicist named Fritz Zwicky collected velocity information available for the Coma cluster. Zwicky measured the speeds of the galaxies in the cluster relative to each other by using red shift. These visible galaxies were moving so fast around the center of the cluster that the gravitational attraction inside the cluster wasn’t strong enough to keep the cluster together. Therefore, there must be some invisible matter inside the cluster that was providing the extra gravitational forces to keep the cluster from breaking apart. He named this invisible mass inside the cluster dunkle Materie, or dark matter. (Krauss 80, Bartusiak 196) In 1936, a colleague, Sinclair Smith, observed the Virgo cluster some 50 million light-years away and reached the same conclusions: "It is possible that the difference [in velocities] represents a great mass of internebular material within the cluster." (Bartusiak 199)

At about the same time, Jan Oort was investigating the stars in the Milky Way. Oort noticed that stars in our galaxy tended to swing back and forth between the plane of our spiral galaxy, and he calculated the mass of the Milky Way required for the stars to be able to accomplish this. He found that the mass of our galaxy was approximately two times the mass of the visible stars and gases. However, this extra mass could simply be stars on the other side of the Milky Way, where Oort's telescope could not reach. (Bartusiak 197)

Stronger evidence for dark matter arrived in the 1970s. Vera Rubin and friends applied Kepler’s Laws to various spiral galaxies. To their surprise, they found that "the rotational velocity of the individual gas clouds remained high at large distances from the center of the galaxy." Since the speeds of stars depends solely on the mass contained inside their orbit, then there must be a "halo" of non-luminous mass outside the orbit of the farthest star. These observations, combined with visible observations, suggest that at least 75% to 80% of the mass in these galaxies are dark. (Krauss 74, Bartusiak 211)

More recently, scientists have used a general theorem in mechanics known as the virial theorem. By measuring the velocities in a system which has reached equilibrium (such as a galaxy cluster) we can calculate the total kinetic energy (E_k = ½mv²). Since the theorem states that the magnitude of the total potential energy is half of that, we can calculate the total mass in the system. (Krauss 81)

However, one must keep in mind the conditions when applying the virial theorem. If a scientist applied the theorem to two galaxies which were merely passing each other, he would think that there was a lot of dark matter in the system. But since their position is not determined by their mutual gravitational attraction, this conclusion is incorrect. A classic example of this fallacy is the Cancer cluster. Before 1984, this cluster appeared to have fifty times as much virial mass than luminous mass. But in that year, new red shift data showed that the Cancer cluster actually consisted of five self-gravitating masses spatially removed from one another. (Krauss 84)

Another concern for the virial theorem is that we need to take average or unbiased values, since the derivation of the theorem relies heavily on statistics. When Jim Peebles applied the virial theorem to the entire visible universe, he took velocity readings from all over the sky and calculated an average. Using this value, the mean mass density of the universe is about ten to thirty times the mass density of luminous matter. (Krauss 85) But since he only took red shifts on visible matter in the sky, and since there appears to be more dark matter in the universe than luminous matter, then this result could be very biased. (Krauss 86)

The other main piece of evidence for the existence of dark matter is BBN Theory. BBN gives us the average density of matter in the Universe. Thus, we can calculate the total amount of baryonic (normal) matter in existence. Amazingly, the total density of matter calculated by using BBN is very close (within a factor of two) to the mass calculated using the virial theorem. (Krauss 120)

Various hypotheses have been put forward to explain what dark matter could be comprised of. However, all candidates for dark matter can be divided into two categories: baryonic matter and non-baryonic matter.

Dark matter could simply be baryonic matter that we have not found yet. Pools of X-ray emitting gas and clouds of hydrogen between galaxies have been found. Entire disks of gas and dust have been observed orbiting around young and newborn stars, possibly in the process of forming planets and moons. Also, it is likely that there are planets orbiting other stars in the Universe, since we know that our own solar system has many planets. These planets would be difficult to detect since they do not emit enough light and heat. (Bartusiak 222) As telescopes become more refined, it is likely that we will mind more and more dark matter of this form.

However, we do not believe that there are enough planets to account for the amount of dark matter required by BBN Theory and the Virial Theorem. Since our sun makes up more than 99% of the mass in our solar system, we predict that the gravitational effects of planets would be negligible if we assume that all planets orbit stars, and that most solar systems are like our own. (Bartusiak 222)

Dark matter could be composed of larger objects such as Jupiters, Brown dwarfs, and White dwarfs: stars that were unable to become stars due to their (relatively) small masses. These are collectively called MACHOs, which stands for MAssive Compact Halo Objects. (Bartusiak 276) I chose to focus on MACHOs for my essay because it is the easiest to detect from Earth. It also does not venture too far into the realms of particle physics such as the following candidates do.

Dark matter could also be non-baryonic. Neutrinos are the only type of exotic matter that is known to exist. They are created during beta decay. Very recently, scientists from the Super-Kamiokande Collaboration in Takayama, Japan have announced that neutrinos, previously thought to have zero mass, actually have a mass of 0.07 ± 0.04 eV, about one ten millionth of the mass of an electron. Although this is a very small mass, there are around fifty billion neutrinos per electron in the Universe. Thus, there are far more neutrinos in the universe than electrons, and neutrinos could account for a lot of mass. (http://www.phys.hawaii.edu:80/~jgl/nuosc_story.html, Krauss 165) Although this is actually a very good candidate for dark matter, this discovery had not been made when I chose MACHOs as the topic of my essay.

Axions are another prime possibility for dark matter. In interactions such as electromagnetism, gravity, and the weak force, PCT transformations must be obeyed. However, in some systems, only CP symmetries are obeyed, such as the spinning of neutrinos. Fortunately, C and P are only violated at the same time, so PCT still works. In QCD, CP is not a valid symmetry. Because PCT must hold, CP symmetry violation means that T must also be violated, as predicted by ‘t Hooft. Unfortunately, observations show that T violations are very rare, about one in 10⁸. To solve this "strong CP problem", Peccei and Quinn proposed a solution that involved a new type of particle. Named the "axion" by Wilczek, these particles are even harder to detect than neutrinos. Their mass could be anywhere from 10⁵ eV to 10^-8 eV. (Krauss 231, Bartusiak 278)

WIMPs arise from a theory known as SUSY. This theory in turn was created to explain the Higgs boson, which was invented to connect the electromagnetic force and the weak force. SUSY means that problems involving the mass of the Higgs boson relative to larger structures are avoided, since the contribution to the Higgs’ mass by a fermion cancels out with its boson. (Krauss 239) It has already been calculated that WIMPs must have a mass of more than 8 x 10²⁴ grams, the collective mass of five protons. If SUSY is a valid model of the universe, then the more stable fermions such as photinos, Zinos, and Higgsinos could be ideal candidates for dark matter since they would be hard to detect. Neither the Higgs boson nor any of the SUSY counterparts of the known fermions and bosons have been observed. (Bartusiak 276)

The final possibility is that these calculations are incorrect. For example, an error in Einstein's Theory of Relativity or a miscalculation in red shift could very easily account for dark matter. Mordehai Milgrom pointed out that

Because I originally wrote this using MS-Word, I had difficulty translating the diagrams over to HTML. I'm afraid you'll have to get Schneider's book to see what I'm talking about.

The following calculations and diagrams are completely adapted from Schneider's book, from page 25 onwards. Although I had other sources, none of them went step by step through the calculations to arrive at an understanding of GL; the reader was assumed to already possess the knowledge.

From the General Theory of Relativity, we know that a light ray will change its direction, or be deflected, by a mass. The angle that this light ray is deflected by, known as the Einstein angle, is calculated by the following:

The variables listed above can be found in this diagram:

A "point mass" M is located at a distance D_d from the observer O. The source is at a distance D_s from the observer, and its true angular separation from the lens M is b . The true angular separation is the separation that would be observed in the absence of lensing. The observer will actually see the image at the angular position q = e /D_d. A light ray that passes the lens at a distance e is deflected by a , given by the equation shown above.

This model makes three assumptions. The first assumption is that the entire system is static, or not moving. In other words, we are taking a snapshot of the system at a specific point of time. The second condition is that the mass must be a "point mass" meaning that the mass must be relatively compact. Mathematically, this means that the "impact parameter" e is much larger than the corresponding "Schwarzschild radius", represented by R_s. This radius is represented by

R_s = 2 G M

c²

Substituting into the above equation for the Einstein angle, we get that it is also equal to

a = 2 R_s

e

As a result of this equality, this model is also called the Schwarzschild Lens.

The third assumption is that we are using an "Euclidean background metric" since the diagram shown above has two dimensions. In reality, the distances must be interpreted as "empty cone angular-distances". The system should be pictured as a sphere with O at the center, S on the surface, and M somewhere inside the surface.

Using these assumptions, we create a second model that takes these considerations into effect.

Instead of points on a plane, O, I, and S are now points in 3-space. This new model also has some new features. O is the center of the sphere and the tip of the cone. The line OL is extended to N on the surface of the sphere, and we call line OLN the "optical axis". Angles b and q are both measured from this line. However, L is no longer the point mass; rather, it is known as the "center of lens", and it represents the center of mass of the lensing object. S_s and S_d represent planes that are tangent to the spheres at points S and I respectively. S_o, also known as the source plane, is the plane that contains both b and q .

A new point, I', has been added to the diagram. Both I and I' are on S_d, which is also known as the lens plane. Together, arcs SI'O and SIO represent the asymptotes for the true path of the light ray.

When geometrized units are used, where the constants G and c both equal unity, the equation reduces to

a = 4 M / e (Schutz 286)

For example, the maximum deflection possible for our Sun is when e = R . .

Given M = 1 M . = 1.47 km and R . = 6.96 x 10⁵,

a = 4 x 1.47 km / (6.96 x 10⁵ km) = 8.45 x 10^-6 radians = 1".74 (Schutz 286)

For ground-based telescopes, their resolution is usually around one arc-second, or 1". The HST has resolution of about 0".10.

(http://quest.arc.nasa.gov/hst/about/overview.html)

The result of light from distant objects curving around galaxies has let us see multiple images of the same thing. Depending on the alignment of the observer (O), the lens (L) and the source of the light (S), astronomers can see anywhere from two to an infinite number of images of the same event. On a two-dimensional plane, when S lies on the optical axis, the light from the source can curve from both sides of the lens and reach the observer. (Krauss 91)

Quasar Image 1

Observer Galaxy Quasar

Quasar Image 2

However, when we rotate this picture about the optical axis in three-space, then the observer will see a ring of images. O. Chwolson first published this idea in 1924 in a composition called "Regarding a possible form of fictitious stars". He suggested that "if the foreground and background stars were perfectly aligned, a ring-shaped image of the background star centered on the background star would result." (Schneider 3) These rings would later be called "Einstein rings", despite the fact that Einstein believed that "there is no great chance of observing this phenomenon." He thought that the angular separation of the two images of the background star would be too small to be resolved with optical telescopes then available. In other words, the angle between the two images, q ₁ + q ₂, was too small for telescopes to detect. (Schneider 4) Indeed, it was later determined that the ratio of image separations for GL (and ML) is of the order 10^-6. (Kayser 145) The first Einstein ring, MG 1131+0456, was discovered by MIT using radio telescopes (see above) (http://space.mit.edu/RADIO/mg1131.html). Although GL can be applied to light, visible light is simply one of many forms of electromagnetic waves that stellar phenomena can emit. Others include radio waves, microwaves, IR rays, UV rays, X-rays, and gamma rays.

A group of students from the Pennsylvania Governor's School for the Sciences (PGSS) tried to use a light source, lenses and a pinhole to determine what sort of images one can expect from gravitational lensing. The lenses served as gravitational lenses, and the pinhole represented the perspective of one observer. The image was projected on a screen behind the pinhole.

(From http://www-pgss.mcs.cmu.edu/1997/Volume16/physics/GL/GL-III.html)

Although the experiment did not produce any quantitative results, the five students involved did create five pictures on the screen. The variables were the orientation of the lens (45º or 90º) and the location of the pinhole. The angle of the lens represented the orientation of a galaxy relative to a light source, while the pinhole represented the position of the observer. The lenses were actually the feet of pieces of glass stemware.

The apparatus created all of the following pictures on the left, and the HST captured the pictures on the right.

The above pictures were created when the light source, the long lens, and the observer were all along the optical axis. Both exhibit characteristics of Einstein rings.

The above pictures were created when the light source and the medium lens were on the optical axis, but the observer (pinhole) was slightly off-axis. Both exhibit characteristics of twin arc formation.

The above pictures were created when the short lens was 45º to the light source. The pinhole was about half of the vertical axis away from the optical axis. As the lens was angled more, the light rays would bend more. Both exhibit characteristics of broken arcs.

The above pictures were created when the short lens was 45º to the light source. The pinhole was about three-quarters of the vertical axis away from the optical axis. Both exhibit characteristics of non-linear point formation.

The above pictures were created when the light source and the pinhole were on the optical axis, but the short lens was 45º to the light source. Both exhibit characteristics of Einstein crosses.

Although their observations seemed to match that of the HST, the students were not very clear on how their lenses simulated the deflection of light rays in GR. This is arguably the most important part of the experiment. Nevertheless, they did prove that different images can be formed when light rays deflect.

With this understanding of Gravitational lensing, I can now explain the effect that I will focus on for the remainder of this essay. Gravitational lensing is for huge masses such as entire galaxies or black holes. On the other hand, Gravitational Microlensing, more commonly known as simply microlensing, is for lenses in the mass range 10^-3 < M/M _. < 10⁶. (Wambsganss 96) M _. represents the mass of our Sun, which is about 3.33 x 10⁵ times the mass of the Earth. Scientists use this notation very often, since it is more accurate and more useful than conventional mass units. The mass range indicated above includes almost all stars (except for some White dwarfs and Neutron stars) and most black holes.

Because ML lenses are smaller than GL, the deflection angle a will also be very small. This means that the angular separation between the images will also be very small. Whereas GL produces multiple images that are usually distinct, ML simply produces multiple images at the same point. The result is an image that is more intense, or brighter, than normal. (Wambsganss 97)

In microlensing events, a star moves between the observer and the light source. Just as stars on Earth appear to twinkle due to disturbances in the atmosphere, our images of distant light sources will "flicker" when the light rays are deflected by a lens. This causes a "momentary" magnifying effect that last several months or longer (Bartusiak 237). Some real microlensing events are shown in these images taken from the OGLE homepage. Each set of pictures measures 30 arc-seconds by 30 arc-seconds, taken at the minimum and maximum brightness.

When attempting to measure a microlensing event, all of the information relies on the observation of a single parameter: the total time that the event was magnified, known as the timescale t. Deviant microlensing uses what is known as the parallax effect. Tycho Brahe first used this effect in the 1570s to prove that stars were far away. (Bartusiak 25) In 1841, Wilhelm Bessel used the parallax effect and a bit of simple trigonometry to calculated the distance to 61 Cygni. (Bartusiak 51) Deviant microlensing is simply microlensing while allowing for the motion of the Earth around the Sun.

If the Earth were standing still and we observed a microlensing event of another star, the graph of light intensity versus time would be a normal distribution, perfectly symmetric. This is because the speed as the lens heads toward the optical axis is the same as the speed as the lens leaves it. However, since the Earth is orbiting around the Sun, the curves are noticeably asymmetrical. (Bennett 103)

Bennett looked at the light curve of the longest event that they have detected. Using this information along with velocity information calculated through red shifts, we can graph the mass of the lens versus the distance to the lens, with a plot of lens mass as a function of lens distance. By examining the graph, it turns out that the star cannot be a main sequence star; it must be a MACHO in the form of an invisible white dwarf or neutron star. (Bennett 105)

There are a number of projects under way around the world to look for MACHOs using gravitational microlensing. Paczynski in 1986 first proposed the idea to use ML to find MACHOs (Bennett 96). The three main projects under way, EROS, MACHO, and OGLE, all found their first microlensing events in the course of less than a month. By 1996, the total number of ML events discovered had grown to about eighty (Bennett 96).

Each of these projects has their own homepage on the Internet. EROS, which stands for Experience pour la Recherche d'Objets sombres, "looks for microlensing events induced by brown dwarfs located in our galaxy on stars from the LMC". They are currently based at La Silla, in Chile. They are also trying to set up a second apparatus, EROS II.

(http://www.lal.in2p3.fr/EROS/erosa.html)

OGLE (Optical Gravitational Lensing Experiment)’s main goal is "to search for dark, unseen matter using the microlensing phenomena." They are mainly focussed on the Galactic Bulge, but they are switching to the LMC now. This project uses another telescope located in Chile, the Las Campanas Observatory (LCO). I obtained the majority of my data here, since the observations were easily accessible.

(http://www.astrouw.edu.pl/~udalski/ogle.html)

The third major project currently under way is simply known as MACHO. MACHO's primary aim is "to test the hypothesis that a significant fraction of the dark matter in the halo of the Milky Way is made up of objects like brown dwarfs or planets." They are focussing on the LMC, the SMC, and the Galactic Bulge. They mainly use the Mount Stromlo Observatory in Canberra, Australia.

(http://wwwmacho.anu.edu.au/Project/Overview/status.html)

There are also a host of smaller projects using different telescopes around the world. They go by names such as AGAPE, MEGA, DUO, MOA, GMAN, UKDMC, and PLANET. By the time this paper is submitted, there will probably be even more.

There is definitely dark matter in the form of MACHOs in the Universe. Although we do not yet have enough data to ascertain exactly how much there is, we will almost definitely find out soon, considering how many projects are currently under way that are looking for it. Given that it is possible that dark matter also exists in other forms such as pools of gas, neutrinos, axions, or WIMPs, there might even be enough mass to eventually make the Universe close in on itself, resulting in a "Big Crunch". If we can extrapolate this much information about dark matter while remaining on Earth, imagine what we will find as humans leave the moon's orbit for the first time.

To conclude I shall end with a quote from the first episode of Star Trek: The Next Generation.

"Let's see what's out there."

• Baryonic matter: Matter that consists of protons and neutrons; in other words, normal matter.
• Beta decay: The decay of a neutron into a proton, an electron, and a neutrino.
• BBN (Big Bang Nucleosynthesis) Theory: All of the light elements (with nuclei containing less than seven protons and neutrons) in the Universe was formed during a period that lasted just a few minutes, about 10-15 billion years ago. At that time, the temperature of the matter and the radiation in the Universe was more than one billion degrees Kelvin, one hundred times hotter than the interior of our Sun today. The entire universe consisted of this "soup" of mostly protons, neutrons, electrons, and photons. As time passed and the Universe cooled, some of these combined to form atomic nuclei. These nuclei would later combine with electrons to form the elements of our Periodic Table. BBN has successfully predicted the vast amounts of hydrogen and helium in the Universe. (Krauss 110)
• Blue Shift: When objects are moving towards you, the light from the object is "shifted" towards the blue end of the light spectrum. Almost all of the objects in the sky are red shifted, which provides further evidence that BBN is correct and the Universe is expanding. (See Red Shift)
• GR: See Relativity, General Theory of.
• Higgs boson: A particle thought to carry the mass of other particles. It is the only boson thought to have an angular momentum quantum number of zero. Also known as scalar particle, Higgs scalar.
• HST: Hubble Space Telescope.
• Kepler’s Laws: The orbit of an object depends solely on the distance from the center and the amount of mass inside the orbit (according to Kepler's Laws, established in the 1600's). For example, Mercury's year is just 88 Earth-days long, whereas Pluto's year is equal to 249 Earth years.
• MACHOs: MAssive Compact Halo Objects. Kim Griest, from Berkeley, invented this term in contrast to WIMPs. (See Wimps) (Bartusiak 276)
• Parallax: An apparent displacement in [an object's] position when viewed from widely separated locations. (Bartusiak 25)
• PCT: Parity, Charge conjugation, Time reversal. Represents the three basic symmetries that should be conserved in all our principles of Physics, including both weak and strong interaction. If PCT as a whole did not hold, we would have to create a new physics where probability was not conserved. These symmetries respectively mean that if left were changed into right, particles exchanged with antiparticles, or time were somehow reversed, then the laws of physics must remain the same. (See QCD) (Krauss 227)
• QCD: Quantum Chromodynamics. The theory of strong interactions that describes the interactions of fundamental quarks that combine to make up protons and neutrons. (See PCT) (Krauss 220)
• Red Shift: The main method that scientists use to measure the speeds of distant objects. Since light is a wave, we can measure the wavelength of light. When an object is moving towards you, the wavelength gets shorter; when an object is moving away from you, the wavelength gets longer. If 1/f is the wavelength emitted, the observer will actually observe a wavelength of 1/f (1+V/c), where V is the velocity that the galaxy is moving away from you at, and c is the speed of light. Given that we know the original frequency of the light emitted, we can calculate the velocity of the galaxy. (See Blue Shift)

(http://www.physics.uiuc.edu/~white/darkmatter/dopplershift.html)

• Relativity, General Theory of (1916): In Einstein’s General Theory of Relativity, he taught that the force of gravity is due to the curvature of space-time. For example, assume that space occupied just two dimensions, like a huge rubber mat. Objects could sit on this rubber mat, and their mass would cause them to "sink" into the mat. In this way, if a marble were fired next to a bowling ball sitting on this mat, it would follow the "curvature" of the mat and curve towards the bowling ball. When we go back to stars and planets existing in three-dimensional space, we can compare this "mat" to the fourth dimension we call space-time. Gravity is actually the result of objects following the space-time curvature. (Bartusiak 48)
• Relativity, Special Theory of (1905): Albert Einstein’s Special Theory of Relativity is based on the idea that the speed of light is constant, regardless of an observer's state of motion. But to let two observers, one stationary and one moving, to measure the same speed of light, we must let either their measurements of time to change, or their measurements of distance to change (since speed is time divided by distance). In fact, both time and distance become relative quantities with respect to the speed of light. (Krauss 21)
• Scientists
• Brahe, Tycho: Danish astronomer who did not believe in the Copernican view of the solar system.
• Einstein, Albert: German scientist famous for his ideas on special/general relativity, quantum mechanics, Brownian motion, and others.
• Newton, Sir Isaac: English scientist who discovered calculus, gravity, and the laws of motion.
• Oort, Jan: Dutch astronomer who demonstrated in 1920s that the Milky Way rotates about its center. Pioneered radio astronomy and thought of the Oort cloud, a reservoir of comets just outside our solar system. (Bartusiak 197)
• Peebles, Jim: Professor at Princeton University who uses statistics to determine the structure of the universe.
• Rubin, Vera: American scientist who specializes in measuring the speeds of spiral galaxies.
• ‘t Hooft, Gerard: Dutch physicist specializing in quantum theories.
• Zwicky, Fritz: Swiss national educated at Caltech who first conceived of the supernova, the implosion of a neutron star, with Walter Baade. (Bartusiak 191)
• SR: See Relativity, Special Theory of.
• Supersymmetry (SUSY): This theory states that every boson (force particle) is paired with a fermion (matter particle). Conventionally, the fermion is obtained from the boson by adding the suffix -ino to the boson. Thus, the photon has the photino; the W boson, the Wino; the Z particle, the Zino; and the graviton, the gravitino. The boson could be obtained by adding the prefix s- to the fermion: selectrons, sneutrinos, and squarks would exist. (Bartusiak 274)
• Virial Theorem: For a stable, self-gravitating, spherical distribution of equal mass objects (stars, galaxies, etc) the total kinetic energy is equal to negative one-half of the total potential energy: KE = -½PE. (By convention, PE is negative and KE is positive.)

http://astrosun.tn.cornell.edu/courses/astro201/vt.htm

• WIMPs: Weakly Interacting Massive Particles. (See Supersymmetry, MACHOs)

In the chapter detailing Gravitational Lensing, I switched to geometrized units. These units are used in SR to make the equations simpler. In normal SI units, the speed of light, c, is approximately 2.998 x 10⁸ m s^-1 and the gravitational constant G is 6.67 x 10^-11 m³ kg^-1 s^-2. In geometrized units, both c and G are equal to simply one. What these conversions mean is that both seconds and kilograms have been converted to meters. This means that almost all measurements can simply be done in meters!

Here is a table with some basic fundamental constants expressed in SI and geometrized units:

(Schutz 198)