We hear everyday "dark matter" this, "dark energy" that...but what are those things, are they figment of our imagination, just nice explanatory
"device" or a real thing ?
I'll vouch for the second one for now. This does not mean there is no kernel of true or in fact the whole true, but I still don't feel 99% sure ;).
It does not matter what the final picture will be, I think it is a good idea to embrace what we know and build some understanding so we can poke
further in the matter of "dark matter".
The "dark matter" term was invented to explain some anomalies which seemed to imply that the Universal law of gravitation seems to predict
motions in the Solar system fairly well, but once we scale up to the size of Galaxies it worked no more.
Before the middle of 19th century Solar system was composed of 7 planets, Uranus beign the last one. But there was something ikky because Uranus
seemed to perturbe on its orbit, so the question was do we fiddle with the Newtown graviational law or decide there is nothing wrong with it but
instead there is some yet undiscovered planet influencing Uranus. Scientists decided that there is another planet and in 1846 Netuptune was observed
based on the calculations.
Neptune was the "dark matter" of the 19th century.
My point of telling you this little story is to show you that we have been here already i.e. questioning the correctnest of the graviatational law.
We have to first start with some understanding of the law of gravitation and what are its implication for the motion of one object around other.
Let's take for example a planet orbiting a star.
So we have a planet(`M_p`) orbiting around a star (`M_s`), like the image shown on the right.
The planet is moving with velocity `v` and acceleration `a` and the `r` is the distance between the star and the planet.
For the inquiring minds in circular motion acceleration always points toward the center
In real life the planets rotate along ellipse, but as simplification we will pretend that the planet goes around in a circle.
We would make a use of
(1) derived from law of Gravitation i.e. `v^2 = (G*M_s)/r` (
look at the Kepler law derivation below, if you are
interested).
What this relation `v ~ 1/r` is telling us is that the further away the planet is the slower the velocity have to be.
And yes that seems to be the case with our Solar system, if we do the calculations they tend to very well match observation.
Look at the table and the graph below, further away slower the speed.
I didn't saw this before I made the graph, but isn't interesting that Saturn velocity and distance (in A.U.) almost coincide.
A.U. btw is astronomical unit and is equal the distance between the Sun and the Earth i.e ~149 mln km. The other interesting coincidence is that the
Pluto velocity is almost 1/10'th of Mercury ;) and of course the distance is 1/100th.
In general we know that the force acting on the object in our case planet from the Newton second law is :
`F = m*a`
but we know also that the gravitational force that act on planet is :
`F = G*(M_s*m_p)/r^2`
`M_s`  mass of the star
`M_p`  mass of the planet
so :
`F => m_p*a_p = G*(M_s*m_p)/r^2`
when we divide both sides to `m_p`, we get :
`a_p = (G*M_s)/r^2`
We also know that the acceleration for uniform circular motion is :
`a = v^2/r`
so we do again the same trick :
`a_p => v^2/r = (G*M_s)/r^2`
which gives us a solution for the velocity :
(1) `v^2 = (G*M_s)/r`
but we know that T (the period for circular motion) is equal to :
`T = (2*pi*r)/v`
then we rise to second power :
`T^2 = (4*pi^2*r^2)/v^2`
so we substitute reciprocial `v^2` using (1):
`T^2 = (4*pi^2*r^2)*(r/(G*M_s))`
so get finally the Kepler 3rd law :
`G*M_s*T^2 = 4*pi^2*r^3`
G  gravitational consant `6.67 * 10^11`
M  star mass.
T  the period it takes the planet to rotate around the star
r  the distance between the two bodies


I wrote the law in this way because it is easier to remember...look carefully GMT like Greenwich Mean Time ;)

planet 
distance  mln km 
distance  A.U. 
diameter  km 
velocity  km/s 
period  years 
Mercury  57.910  0.387  4,800  47.9  0.241 
Venus  108.200  0.723  12,100  35  0.615 
Earth  149.590  1.000  12,750  29.8  1 
Mars  227.940  1.524  6,800  24.1  1.88 
Jupiter  778.540  5.203  142,800  13.1  11.9 
Saturn  1,433.400  9.523  120,660  9.6  29.5 
Uranus  2,876.670  19.208  51,800  6.8  84 
Neptune  4,503.400  30.087  49,500  5.4  165 
Pluto  5,874.000  39.746  3,300  4.7  248 



So far so good, let's pretend now that the planet can orbit inside the star. I know it can't but just bear with me for the sake of argument,
we will need this analogy for the Galaxy primer.
The image on the right shows how such thing may look like.
What is different in this case ? The difference is that only the part of the mass of the star which is enclosed inside the orbit of the planet will have gravitational influence
on the velocity of the planet, the outsidemass part will
cancel out and will have no effect, so if we use formula
(1) again the
conclusion will be that the velocity of the planet should tend to be lower as the orbit get closer and closer toward the center.
Look at the graph above again, do you see the downward slope from Mercury toward the origin of the graph.
What I want to point out is that the more mass there is inside the orbit of an object the faster it will travel and vice versa.
That is nice... law of Gravitation and Kepler third law are very cool tools, just measuring star velocities allows us to calculate the
period of rotation, mass of the objects and so on...
The surprise came when astronomers started to measure the speeds of stars in different galaxies. The planetrelationcurve we just saw
was not obeyed, instead the velocities of the stars were for all intends and purposes the same across the expanse of a galaxy (except those close to the
center).
(Look at the diagram on the right,
case A: solar system,
case B: galaxy).
So the question was then how to explain all this.
One way was to assume that law of Gravitation was not correct on large distances or may be modify it in some way to accommodate for the observed
behaviour (MOND theory tries to do exactly that).
But scientists were reluctant to that, the law worked so well in everyday classical calculation and also on solar system scale that it would
be a total waste. The idea is to simplify the laws of nature, not to make them more complicated if we can. Right ?
What to do ? what to do ?
It was consequence from the Einstein General theory of relativity (not special relativity), which predicted that under big gravitational influence
even light will bend and exhibit effect similar to the well known optical lensing. And so the phenomena was called gravitational lensing.
The good thing is that such effects are observed by astronomers.
Look at the two images. Left one is in the
Abell 2218 galaxy cluster, you can clearly see the lensing.
So if this is true phenomena that happens in the universe, then let's try Einstein proposition.
Below is the equation from General relativity that describes the angle of deflection caused by gravitation :
`alpha = (4*G*M)/(c^2*R)`
where :
 `alpha` is the angle at which light bends
 `R` is the impact parameter (the distance of nearest approach of the lightbeam to the center of mass)
 `G` is the gravitational constant
 `M` the mass the object that bends the light.
 `c` is the speed of light
 and 4 is 4
From the relation `alpha ~ M/R` we can deduce that knowing `alpha` we can calculate the mass of a galaxy or galaxy cluster.
And when the astronomers did the calculations the mass were many times bigger than the one calculated by the normal methods of counting the
average numbers of stars and weighting them or via any other methods...
I'm interested to find details how such calculations are done, so when I have something I will update this document
We can't expect the mass to be concentrated at the center of the galaxy, because if that is the case then the furthest stars speeds will falloff as
the stars are further from the center like in our starplanet example. Remember what we deduced the more mass there is inside the orbit of an object the
faster it will have to orbit around. But we said that the observed mass inside the orbits did not account for the observed velocities.
So the idea then is that most of the mass should spread into some sort of invisible hallo's (dark matter) troughout and outside of the visible part of the galaxy.
Just to clarify in the case of perfect circle and homogenous density (the one we talked above) the gravitational influence will cancel out, but in real world
with the galaxies we observe of course that is not the case.
The distorted image of the distant object can appear in three possible ways depending on the shape of the gravitational lens:
 Ring – the image appears as a partial or complete circle of light known as an Einstein ring. This happens when distant object, lensed galaxy and observer are perfectly aligned.
 Elliptical – where the image gets split into four images and appears as a cross known as an Einstein cross.
 Cluster – where the image appears as a series of bananashaped arcs. Like the Abell2218 shown above
How the astronomers can be sure that the lensed object is the same ?  By measuring the spectrum and twisted shapes of the objects
behind.
By measuring the angle of bending, astronomers can calculate the mass causing the gravitational lens effect (the greater the bend, the more massive the
lens). Using this method, astronomers have confirmed that galaxies indeed have high masses exceeding those measured by luminous matter.
One important thing to mention is that the reason 'dark matter' is invisible is because it does not interact electromagnetically with normal matter (protons, neutrons, electrons) and light (photons), but has a mass i.e. interact gravitationally.
And as we know gravitation is `10^36` weaker than electromagnetism, so you need alot of mass to have visible effect.
So the lesson of the story is that the idea of 'dark matter' preserve the consistency of the Universal law of gravitation.
There is also other theories that try to explain all this, afaik not with such a great success as 'dark matter' idea :
 Modified Newtonian dynamic, aka MOND theory
 Exessive amounts of neutrino i.e. 'dark matter' are neutrinos.
 Wandering brown stars and black holes troughout the galaxy, which are also hardly visible ergo the missing mass.
 Combination of those...
It seems the consensus is on unobserved yet type of particle which is massive but does not interact very well with normal matter.