Astronomy

Published on January 21st, 2011 | by Carl Mundy

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# Dark matter and our Universe; How galaxies can show us the dark side

In this article, Carl discusses the particular properties of galaxies that show us there must be more in our universe than we can see with our telescopes – there must be another form of matter we cannot see. Welcome to the mysterious world of dark matter…

There’s something in our universe that we cannot see. We can’t point our telescopes up at the sky and observe this stuff because it’s not like normal matter. It’s completely different to the normal matter we are all used to. It’s called dark matter, for obvious reasons, and it is believed to have been key in shaping our universe and the structures within it.

## Evidence for dark matter

Because we cannot see this dark matter, we have to infer its presence from the way other matter is influenced by it. For example, the number of dwarf galaxies predicted to be orbiting the Milky Way is higher than the number observed composed of ordinary matter. This has been a problem for cosmologists for some time now, but a method of detecting those orbiting dwarf galaxies composed mostly from dark matter has been demonstrated which observes tidal disturbances in the gas at the edges of the Milky Way from which the mass and position of the dwarf galaxy can be calculated[1]. This is only one way in which we can infer the existence of dark matter.

Another method is to look at galaxies in the night sky and observe how the velocities of the stars and gas within the galactic disk change as you head out from the centre of the galaxy. Doing this reveals something very interesting indeed.

## Galaxy rotation curves

Observing how the velocity of objects within a galaxy changes with the radial distance $r$from the galactic centre, we can produce a galaxy rotation curve which makes it much easier to understand.

Firstly, let us go back to the basics and calculate what astronomers expected to find when they pointed their telescopes at the galaxies in the night sky. To do this, we assume a galaxy is made entirely from normal matter; the stuff we’re all used to.  We also assume a uniform density $\rho$ such that the density is constant and so not dependent on the radius $r$ as well as assuming the stars and gas have circular orbits.

As our stars, gas and dust have circular orbits, we can equate the centripetal force with the gravitational force and obtain an expression for the velocity of our objects as they orbit in the galaxy.

The mass within a radius $r$ is given by  and the other symbols have their usual meaning. Rearranging this to get the square of the velocity $v^2$ on its own we arrive at:

Now we have to consider how the mass contained within a certain radius depends on that radius. We know that density is mass divided by volume so mass is just the density multiplied by the volume! Thus we get

as our density is constant. From this we know that the mass within a certain radius is proportional to the radius cubed. Using this, we can find how the velocity is dependent on radius by plugging this into our equation for the square of the velocity, arriving at:

so that

So we would expect the velocity to increase linearly with the radius inside the galaxy. But when we reach the visible edge of the galaxy, there is no more mass outside this radius so the mass stays constant and does not depend on the radius any more – the mass within any radius outside the visible edge is just the mass of the galaxy.

So in regime of radii larger than the radius of the visible galaxy, we can see from the above that the velocity is proportional to one over the square root of the radius, $v \propto r^{-1/2}$.

When we look at galaxies and plot our observations against our predictions we find something very interesting indeed. At large radii, the velocities of stars and dust stay constant – they do not depend on the radius at all! This is shown in the figure below.

If the velocity at large radii is constant, then there must be mass at these distances that we cannot observe. The postulation of dark matter stemmed from observations such as this.

## Introducing dark matter

To explain the observations, where velocities at large radii do not show decline but rather stay at a near constant value, we need mass at these distances that scales linearly with radius, or $M(. Seeing as we know mass is the product of density and volume, we can suggest that the density of this invisible matter must scale as one over the radius squared or $\rho \propto 1/r^2$. It is believed that galaxies have a halo of dark matter around them which provides this extra mass to explain the observed velocities of stars and gas.

When asking questions about dark matter, it is easier to answer the question of what we know it isn’t rather than what it is. It is believed to make up about ~25% of the ‘stuff’ in our universe and and we don’t really know anything about it! All we really know is that we see less stuff out there through our telescopes than is needed to explain our universe today; there has to be something else – dark matter.

Research into dark matter is fast paced and full of exciting discoveries that are too complex and too deserving of their own article to be appended to this one.

Things to read up on include…

• gravitational lensing by dark matter
• structure formation
• WIMPs, MACHOs and RAMBOs (yes, these are real!)
• hot,warm and cold dark matter types

## References

[1] S. Chakrabarti, F. Bigiel, Philip Chang, Leo Blitz; Finding Dark Galaxies From Their Tidal Imprints (2011)

Header image courtesy NASA Goddard Space Flight Center.