Radio Glaxy 4C 73.08

I don't want to publish an astronomy picture just because it's pretty, but I'll make an exception for this.  In any case, 4C 73.08 is a very interesting object.





This Hubble picture of a small cluster of galaxies shows several spiral galaxies; some viewed face on, some viewed edge-on.

The most striking object, however, is the large elliptical galaxy in the upper centre of the image. The supermassive black hole at it's core is ejecting twin jets of material.  These jets of material are not visible optically, however, they emit huge amounts of radio energy.  Hence 4C 73.08's designation as a radio galaxy.

Double Solar Eruption

This double eruption was recorded over a four-hour period by NASA's Solar Dynamics Observatory last Friday:-

 

The video (here) is even more impressive.  The first eruption occurs at the limb, the second from the bright facula on the right-hand side of the picture.

From the "did you know" department: did you know that we actually get more light and energy from the sun during a solar maximum, even though there are more sunspots?  That's because sunspots are often assiciated with these brighter-than-average areas called faculae (singular facula) which more than make up for the reduced light from the sunspots.

Pi to a Million (or a Billion or a Trillion) Decimal Places

This combines two subjects, other than astronomy, in which I'm very interested: mathematics and computing.

A Pi Pie!

Have you ever read about pi (the ratio of the circumference of a circle to it's diameter) being calculated to so many billion decimal places and wondered how they do it?

Well maybe you haven't but I have!

There's two parts to it: obviously you need some sort of formula from which to work out the value of pi, and you also need to be able to perform arithmetic to a million, or a billion, or however many, decimal places.

A Formula!

The first bit is fairly easy: we use the following infinite series expression for the inverse tangent (or acrtangent) which is thought to have been first discovered by the Indian mathematician Madhava of Sangamagrama:-

For any value z this series will give us the angle (in radians) which has that values as it's tangent (i.e. it will give us the angle x such that tan(x)=z).

Now the tan of 45 degrees is one, but in radians 45 degrees is , so if we let z=1 in the above formula we get:-

As with any infinite series, we don't actually carry on adding terms indefinitely - we only use enough terms to give us the accuracy we require. Unfortunately, this formula isn't very useful because the terms get smaller and smaller very slowly. It  would take 500 thousand terms just to get an accuracy of only 5 decimal places.

A formula which converges more quickly was discovered by the English mathematician and astronomer John Machin in 1706:-

While we need to work out the arctan of two numbers, because they are much smaller than one, the terms converge much more quickly.  For example, for arctan(1/5) each term is less than 1/25th of the previous one.

Modern methods to calculate pi to a large number of decimal places use formula which contain even more terms, such as the following:-

This formula was discovered by the Japanese mathematician Yasumasa Kanada who used it in 2002 to calculate pi to over a trillion decimal places.

Notice that the largest value for which we need to calculate the arctangent is 1/49.  Because this number is almost ten times smaller than 1/5 the series will converge much more quickly. 

Trillion Decimal Place Arithmetic

Any computer can do integer arithmetic on integers up to a certain size.  That size depends on the particular architecture of the computer.  Most computer manufacturers these days produce processors that support 64-bit numbers and arithmetic - that's large enough to hold an 18-digit decimal (whole) number.

(Of course they can also do floating point arithmetic, but that's not what we're going to use.)

We use a large number of integers each holding a fixed number (but less than 18) of decimal digits.  For example, if we want to work out pi to a million decimal places then we can use 100,000 integers, each holding 10 digits.

Of course, we need to store more just one of these numbers.  If you look at Machin's formula, or the last formula I gave above, you can see that we need to be able to add and subtract these numbers and to multiply them by "normal" sized numbers such as 4, 12, 32 etc.

Addition is fairly easy: you just add the decimal integers going from right to left.  If the result of adding two integers is greater than 9,999,999,999 we subtract 10,000,000,000 and carry one:-

Subtraction would be done in a similar way. Multiplication by a "normal" sized number is also done from right to left.

The other thing which we need to do is calculate the arctangent of a number to (say) a million decimal places.  However, all formula involve expressions of the form:-

where n is an integer.  This makes things easier; the series for arctangent I showed earlier can be re-written with z replaced by1/n:-

The only new operation we have here is division by a "normal" sized number.  (Update: if we used the origional formula, we would have to multiply two billion decimal place numbers to gether and that would be much less efficient.)  This division is done a bit like multiplication, only proceeding from right to left.  Any remainder from one division operation being carried to the right.

To actually perform this calculation, we would need to keep both the running total and the current term as 1 million digit numbers.

The next term is more efficiently calculated from the previous term than from scratch.  For example, the fourth term above in the equation above: , can be calculated from the third term: ,  by multiplying by 5, dividing by 7 and then dividing again by n squared.

Trivia

 Here is a visual representation of the first four million digits of pi.

Find your date of birth in pi here.

Buy pi related t-shirts here or here! (No, I don't get any commission.)

Update (20/11/12)

• I wrote the program and got it to work (correctly!) up to 100,000 digits.  It took about 2 minutes on my PC - in 1961 this took 8.7 hours on an IBM 7090.   I got bored with it with after that.

• An alternative formula for evaluating pi is the Chudnovsky algorithm:-

• According to the novel Contact by Carl Sagan, after a huge number of digits of pi, they eventually stop being random.  Instead you get a sequence of zeros and ones which, when plotted on a 121 X 121 array form a circle, but only if you calculate pi in base 11!  This is science fiction.  Although they haven't proved it, mathematiciens believe that pi is a normal number.

A Polar Ring Galaxy

A Pierson's Puppeteer

When I was younger, much younger, I particularly enjoyed the novels of Larry Niven.  They fizzed and sparkled with strange and novel ideas: Klemperer rosettes, aliens like the Kzin, Pierson's Puppeteers (see above) and Bandersnatchi, humans bred for luck and, of course, the Ring World:-

 

 

One of the reasons I'm so fascinated by astronomy is that, in the last few years, we have discovered many strange and exciting new things that far out-do Larry Niven's vivid imagination.  Ringworld was a construct the size of a Solar System.  Below is a picture of a ring the size of a galaxy:-

NGC 660 - A Polar Ring Galaxy

NGC 660 is a rare polar ring galaxy - only a handful have been discovered.  In the above picture the galaxy is aligned roughly NW to SW and the ring is approximately horizontal.  Dark dust lanes can be seen in the host galaxy.

The host galaxy is a lenticular galaxy.  A lenticular galaxy is one that's in transition between a spiral galaxy and an elliptical galaxy.  The difference between spiral galaxies and elliptical galaxies is that spiral galaxies contain the gas and dust necessary for star formation to take place, and so contain many young blue stars.

Elliptical galaxies have either lost, or used up, all their gas and dust, so the formation of new stars has stopped and the galaxy contains mainly contains old red stars,

Polar ring galaxies are believed to be formed either by a piercing merger between two galaxies aligned roughly at right angles, or when one galaxy tidally strips material from a passing gas-rich spiral and strews it into a ring.  Numerous star formation areas can be seen in the ring which also contains numerous blue and red supergiant stars.

Studies of the rotational speed of the ring indicate that NGC 660 is embedded in a huge cloud of dark matter.

Dark Matter

Early in the 20th centuary, astronomers discovered that many of the patches of light they saw in the sky, were actually huge collections of stars, like the Milky Way, seen from a great distance. These are, of course, what we now call galaxies.

M31 - The Andromeda Galaxy

When astronomers started measuring the velocities of stars in external galaxies, like Andromeda, they found that they were moving far too fast, given the amount of matter they could see in the form of stars, gas and dust.

Similarly, when it was realised that galaxies were grouped together in clusters, there didn't seem to be enough matter in these cluster to hold them together.  Galaxies and clusters of galaxies should be flying apart!

The answer is, of course, that there is much more matter present than we can see - dark matter. The existence of dark matter has been confirmed using weak gravitational lensing.  The following 3D map of dark matter was created using from a Hubble Space Telecope survey called COSMOS using gravitational lensing techniques.

3D Map of Dark Matter

One of the big question in modern astronomy is what form does this dark matter take?  One idea is that it's made up of some as yet undiscovered sub-atomic particle that interacts weakly, if at all, with normal matter.  These are termed WIMPs (Weakly Interacting Massive Particles).

The other alternative is called MACHOs (Massive Compact Halo Objects). The most likely candidates for MACHOs are brown dwarfs.  Brown dwarfs are failed stars - object that are  not quite massive enough for nuclear fusion to take place.  The mass of a brown dwarf is between thirteen and seventy-five times the mass of Jupiter.

Since their only source of energy is their own gravitational contaction, brown dwarfs may glow a dull brown (hence the name) and may be visible under certain circumstances.

And now I'm beginning to get to the point.  In my previous post I mentioned that astronomers had discovered a stream of stars between the Large and Small Magallenic Clouds (the LMC and SMC).

What they were actually looking for was MACHOS.  They were hoping to detect MACHOS using microlensing.  During a microlensing event, a nearby object passes in front of a more distant star. The gravity of the closer object bends light from the star like a lens, magnifying it and causing it to brighten.

What they hoped to see was MACHOS within the Milky Way microlensing stars in the LMC.  The number of microlensing events seen, however, was not enough to account for dark matter but was higher than expected.

Computer simulations showed that the most likely explanation for the observed microlensing events was an unseen population of stars removed by the LMC from its companion, the SMC. Foreground stars in the LMC are gravitationally lensing the trail of removed stars located behind the LMC from our point of view.

Supersymmetry

Since MACHOs have been rules out, what about WIMPS?  The Large Hadron Collider wasn't just built to find the Higgs Boson.  (Before I carry on, the Higgs boson is not a likely candidate for dark matter - it disintegrates into lots of smaller particles almost instantly.)

One of the other purposes of the LHC was to investigate the theory of Supersymmetry (SUSY).  According to Supersymmetry, for every known particle there is much heavier superpartner.  A stable supersummetry particle (e.g. the superpartner of the neutron is called the neutralino) would be a good candidate for a WIMP.

Unfortunately, researchers at the LHC have dealt a blow to the theory of supersymmetry.  They have been looking for the decay of a particle called the Bs meson into two muons.  However, this particular decay was seen only three times out of a billion Bs meson decays.  If supersymmetry was correct, this decay would bee seen many more times.

There are of course other candidates for WIMP particles, e.g. axions.  More exitingly however, this could be a clue point us towards to a new theory (or theories) that might one day supplant both quantum mechanics and general relativity.

Milky Way Rising

The picture shows the Milky Way rising over the hosizon at the European Southern Observatory, located in the Atacama Desert region of Chile.

The two patches of light on the left-hand side of the pictures are the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC), which are smaller satellite galaxies of the Milky Way. They are only visible from the Southern hemisphere and cannot be seen from Europe or Norh America.

Astronomers believe that the LMC stripped stars from the SMC when the two collided about 300 million years ago. They have detected a thin stream of stars that still connects the two dwarf galaxies.