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Archive for March, 2016

At number 23 in Rolling Stone Magazine’s list of the 100 greatest Beatles songs is “Abbey Road Medley”. This refers to the sequence of songs which merge into each other on the second side of their album Abbey Road. The second side of Abbey Road opens with “Here Comes the Sun”, then “Because”, and this is followed by the medley.

There are 8 songs in the medley, in order they are “You Never Give Me Your Money” (McCartney), “Sun King” (Lennon), “Mean Mr. Mustard” (Lennon), “Polythene Pam” (Lennon), “She Came in Through the Bathroom Window” (McCartney), “Golden Slumbers” (McCartney), “Carry That Weight” (McCartney) and “The End” (McCartney). There is no conceptual link between the songs, rather they just merge into each other, culminating in “The End”, with McCartney’s unforgettable lines “And / in the end / the love you take / Is equal to / the love you make”.

At number 23 in Rolling Stone Magazine's list of the 100 greatest Beatles songs is "Abbey Road Medley".

At number 23 in Rolling Stone Magazine’s list of the 100 greatest Beatles songs is “Abbey Road Medley”.

Most of the songs in the medley do not seem to be available on YouTube, but I have found this video of the last 3, namely “Golden Slumbers”, “Carry That Weight” and “The End”.

Here are the lyrics to each of these songs. First, “Golden Slumbers”

Once there was a way to get back homeward
Once there was a way to get back home
Sleep pretty darling do not cry
And I will sing a lullabye

Golden slumbers fill your eyes
Smiles awake you when you rise
Sleep pretty darling do not cry
And I will sing a lullabye

Once there was a way to get back homeward
Once there was a way to get back home
Sleep pretty darling do not cry
And I will sing a lullabye

Next, “Carry That Weight”

Boy, you’re going to carry that weight,
Carry that weight a long time
Boy, you’re going to carry that weight
Carry that weight a long time

I never give you my pillow
I only send you my invitations
And in the middle of the celebrations
I break down

Boy, you’re going to carry that weight
Carry that weight a long time
Boy, you’re going to carry that weight
Carry that weight a long time

Finally, “The End”

Oh yeah, all right
Are you going to be in my dreams
Tonight?

[Drum solo]

[Guitar solos]

And in the end
The love you take
Is equal to the love
You make

And, here is the video of these three wonderful songs. Enjoy!

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Today I thought I would post this gallery of Beatles photographs which I came across a few months ago. Judging from the appearance of the Beatles, I am guessing they were taken in 1965, during the Rubber Soul and Help! period. One of them appears to be a still from a scene in the movie Help! Enjoy……

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UPDATE: Unfortunately, to reduce my use of the allocated storage space for images, I have had to remove the gallery. This is what the gallery looked like.

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A screen capture of the gallery. I have had to remove the individual images to save space. Sorry!

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Last week I blogged about the most distant galaxy so far found, a galaxy at a spectroscopically measured redshift of z=11.1. When I posted this news item on Twitter (and before I had blogged about it), I made the comment that the surface brightness of a galaxy is constant with distance. As was pointed out to me by numerous colleagues, this is only true for nearby galaxies; something I knew but had forgotten! (I don’t do research in distant galaxies, so that is my excuse for having forgotten this important detail 😉 )

So, let me explain why surface brightness is constant for nearby galaxies but not for more distant ones. First of all, however, I will explain what surface brightness actually is.

The surface brightness of an extended object

When I started my PhD in Cardiff, the research group which I joined was probably best known for work on something called surface brightness selection effects. I won’t go into the details of what this means, but I will explain what the concept of surface brightness means in astronomy.

Apart from stars, pretty much everything else in astronomy is an extended object, which means that it covers an area on the sky. This is particularly true of nebulae, which include emission and reflection nebulae, but also galaxies (which used to be called ‘nebulae’ before it was realised that they were outside of the Milky Way, see my blog here for more about that).

So, for example, the Andromeda galaxy has an apparent visual (V-band) magnitude of 3.44, which should render it easily visible. But it is not, as this magnitude is spread over an area which is about 6 times larger than the Moon. It is the fact that the light is spread out over such a large area which renders the Andromeda galaxy so difficult to see to the naked eye.

How bright (luminous) an object appears to be decreases with distance; it goes as the square of the distance, so an object three times further away appears nine times less luminous. But, the luminosity per unit area, which we call the surface brightness, is constant with distance. Let me explain why. 

As an object moves further away an area of e.g. 1 arc-second squared on the object will encompass more of the object. Let us suppose that some galaxy is initially at 10 Mega parsecs, and 1 square arc-second on the object encompasses an area where there are 10 million stars. The surface brightness we measure will be due to the light from these 10 million stars, as seen from our distance of 10 Mega parsecs away.

If we now move the same galaxy to twice the distance, 20 Mega parsecs, and we assume that the distribution of stars is uniform across the galaxy (a simplification which is not generally true), then the light from each star is reduced by a factor of four. But, now our 1 square arc-second encompasses a larger area of the galaxy. In fact, whereas before we said there were 10 million stars in our one square arc-second, when we move it to twice as far away there will now be 40 million stars in our 1 square arc-second, because the area on the galaxy goes up to four times as large as previously. So, even though the brightness of each star is reduced by a factor of four, we have four times as many stars in our 1 square arc-second, and so the two effects compensate and we have a constant surface brightness with distance.

As I have mentioned before (for example here), astronomers use the strange units of magnitudes to express the brightness/luminosity of objects. In these units, we can express the surface brightness S of an extended object (in units of magnitude per square arc-second) as

S = m + 2.5 log_{10} A \text{ (1) }

where m is the integrated magnitude of the object over an area of A square arc-seconds.

The Tolman effect

Richard Tolman was a mathematical physicist who worked at Caltech, and he is mentioned in my book The Cosmic Microwave Background (click here for more information) for being the first person to show that a blackbody spectrum retains its characteristic shape in an expanding Universe. He also devised a number of tests to determine whether the Universe was expanding or not. One of these (which he suggested in 1930) predicted that the surface brightness of a galaxy would not be constant with distance if the Universe were expanding. I will explain why below.

 

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In 1934, in a book entitled Relativity, Thermodynamics, and Cosmology, Richard Tolman showed that a blackbody spectrum in an expanding Universe would cool due to the expansion of space, but crucially would retain its characteristic shape (this is a screen capture from my book The Cosmic Microwave Background)

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Richard Tolman with Albert Einstein at Caltech in 1932

There are three effects which lead to the surface brightness not being constant with distance in an expanding Universe. These are

  • that photons from a distant galaxy arrive at a different rate compared to in a static Universe
  • the wavelength of each photon from a distant galaxy is different compared to in a static Universe
  • the angular size of the galaxy when the light was emitted was larger, because the galaxy was closer, than its current angular size

Let me explain each of these in more detail.

The rate of arrival of photons

Let us assume that, in a static Universe, a galaxy is emitting a certain number of photons every second in some astronomical passband, such as the B-band filter (see my blog here about astronomical filters). Our detector (a camera) detects, let us say, a million of these photons each second. Of course the galaxy is emitting a lot more photons than this, as we are only intercepting a tiny fraction of the emitted photons with our telescope, and no telescope or detector is 100% efficient. But, at our detector, we record 1 million photons each second through our filter in this static Universe.

Now, let us consider exactly the same galaxy but in an expanding Universe. Because of the expansion, each second which lapses means that the galaxy we are observing is slightly further away than it was the second before. The photons have a further distance to travel as time passes, due to the expansion of the Universe. As a consequence, the number of photons we detect is not going to stay constant, it is going to go down.

The wavelength of each photon

Not only will the number of photons arriving each second go down due to the expansion, but the wavelength of each photon arriving will be longer, due to the cosmological redshift. As space expands each photon’s wavelength gets stretched, this is why we refer to the redshift of distant objects, their light gets shifted to the red end of the spectrum.

This also leads to the light from stars being shifted into a longer wavelength pass-band. This would not matter if the light from galaxies were spread equally across wavelengths, but it is not. So, for example, if a particular galaxy’s spectrum peaks in the blue part of the spectrum in its rest frame, this peak will shift to e.g. the V-band or R-band due to the stretching of the wavelength of each photon, so the light seen in the B-band will be reduced, and hence the surface brightness measured in the B-band will also be reduced.

The luminosity distance d_{L}

The luminosity distance is, as the name implies, a distance determined from the object’s measured (apparent) luminosity. A distance d_{L} can be calculated if its intrinsic luminosity is known. In magnitude units we can write

m - M = 5 log d_{L} - 5
where m is the apparent magnitude, M is the absolute magnitude, and d_{L} is the luminosity distance. Re-arranging this, we can write that the luminosity distance d_{L} is given by

5 log d_{L} = m - M + 5 \rightarrow log d_{L} = \frac{m-M}{5} + 1 \rightarrow \; \boxed{ d_{L} =10^{(m-M)/5 -1} }.

If you prefer to think in terms of luminosity and flux, we can write

F = \frac{ L }{ 4 \pi d_{L}^{2} }
where F is the flux and L is the luminosity. This would then re-arrange to
d_{L} = \sqrt{ \frac{ L }{ 4 \pi F } }

In a static, non-expanding Universe, the transverse comoving distance d_{M} is the same as the luminosity distance, d_{M}=d_{L}. But, in an expanding Universe this is not the case. In an expanding Universe we can write

\boxed{ d_{L} = (1 + z) d_{M} } \text{ (2) }

The angular diameter distance d_{A}

The angular size of an object is simply the angle that the object subtends on the sky. We can measure this. Let us suppose we have a galaxy which is 100 kilo parsecs (100 kpc) across (which is about the size of our Milky Way). If it is at a distance of 10 Mega parsecs then, in a static Universe, its angular size will be measured to be

\theta \text{ (in radians)} = \frac{D_{A}}{d_{A}} = \frac{ 100 \times 10^{3} }{ 10 \times 10^{6} } = 1 \times 10^{-2} \text{ radians} = 0.57^{\circ}

where D_{A} is the actual diameter of the galaxy and d_{A} is its distance. In a static Universe this is all very straightforward, but in an expanding Universe, the measured angular size of the galaxy \theta changes. When the photons left the galaxy it was closer that it is when the light is received, so \theta will be measured to be larger than the angular size the galaxy would currently subtend. As a consequence, \theta is an overestimate of its size, this means that the inferred distance d_{A} is too small. We can write that

\boxed{ d_{A} = \frac{ d_{M} }{ (1 + z) } } \text{ (3) }

where, again, d_{M} is the transverse comoving distance.

Doing the equations

We can write that the surface brightness, which in luminosity units we will write as B, is given by

B = \frac{ F}{ d\omega }
where F is the flux of the object, and d\omega is an element of the solid angle. But, this is the same as luminosity L per unit area, so we can also write
B = \frac{ L }{ d_{L}^{2} } \cdot \frac{ d_{A}^{2} }{ dl^{2} }

In a static Universe d_{M} = d_{L} = d_{A}, but in an expanding Universe (irrespective of the assumed cosmology)
d_{A} = \frac{ d_{M} }{ (1+z) } \text{ (Eq. 3 above) } \rightarrow d_{A}^{2} = \left( \frac{d_{M}}{ (1+z) } \right)^{2}
and
d_{L} = (1+z) d_{M} \text{ (Eq. 2 above) } \rightarrow d_{L}^{2} = (1+z)^{2} d_{M}^{2}
This leads to the our being able to write that
B = \frac{ L }{ dl^{2} } \cdot \frac{ d_{M}^{2} }{ (1+z)^{2} } \cdot \frac{ 1 }{ d_{M}^{2}(1+z)^{2} }
\boxed{ B = \frac{ L }{ dl^{2} } \frac{ 1 }{ (1+z)^{4} } }
So, as we can see, the surface brightness depends on redshift in the sense that it is proportional to (1+z)^{-4}, a very strong dependence on redshift. For small redshift, z << 1, this reduces to what we stated at first, that surface brightness is independent of distance.

 

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Despite a stutter with about 20 minutes to go, England succeeded in beating France and winning the Grand Slam. Wales thrashed Italy, and Ireland beat a resurgent Scotland in the final weekend of the 2016 6 Nations.

Wales v Italy

Wales needed to make amends for their terrible first half performance against England in Twickenham last week, and in many respects they did. They beat Italy 67-14, running in nine tries and scoring their highest ever points tally in a 6 Nations match. Whereas it took Wales 60 minutes to wake up last week, against Italy they were dominant from the start and showed the kind of attacking flair which fans are so desperate to see in other matches.

Why Wales cannot move the ball wide with such skill and speed against better opposition is something I don’t fully understand. Each time they went wide they looked dangerous, and yet against better opposition Wales play a narrow game and the wingers rarely see the ball. With a daunting three-test tour of New Zealand coming up in June, and a “friendly” against England at Twickenham at the end of May, surely this next few months is the time to try and add some variety to the Warren-ball game plan which has become all too predictable. I fully expect Wales will be taught a rugby-playing lesson by the All Blacks in June, which will once again show the massive gulf between us and the best in the World. So, why not try something different and see if we can move the ball wide as much as possible; we have nothing to lose as we are going to lose anyway!

Ireland v Scotland

Scotland continue their revival by giving Ireland a run for their money in Dublin. Although Ireland won 35-25, Scotland were more than decent opposition and had their chances to have made this match even closer. Until Scotland had a player sin-binned deep into the second half, they genuinely looked like they could win this pulsating match. I am so pleased to see this long-awaited revival of Scottish fortunes, it has been too long since they were a decent team and hopefully they are on their way back to where they were 20-odd years ago.

France v England

You never know where you are with France, and this match proved that cliché as much as any I’ve seen. After looking abject throughout most of this 6 Nations, France put in a performance which rattled England. In fact, if they had not had so many of their line outs stolen, France could have won this match. Going in to the last quarter England were 25-21 up and France looked the better team. But, thanks to some superb line out work by England’s two second rows, les Bleus lost too much vital possession and field position, and England held on to win their first Grand Slam since 2003.

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England held off a second half surge from France to win the Grand Slam for the first time since 2003

Although I was hoping for an upset in Paris, as of course I wanted to see England fail at the final hurdle as they have done 5 times in the last 20 years, I do have to say that they thoroughly deserved to win this year’s 6 Nations. True, Wales could have beaten them if the game had continued another 5 minutes, but had we done so it would not have been a fair reflection of England’s dominance in the match. Yesterday’s match in Paris was, in fact, a much closer affair, despite what the scoreline says. At no point did England look particularly dominant, and a more confident France could well have given the upset so many of us non-English wanted to see.

The summer tours

As has now become customary, the Northern Hemisphere countries will now trek south in June to take on the Southern Hemisphere countries. Wales go to New Zealand, England to Australia and Ireland to South Africa. Scotland go to Japan, which is not the easy tour that it used to be.

I fully expect Wales to lose 3-0 in their test series; which is not being pessimistic, just realistic. England could well give Australia a tough series; they have certainly been the team who have improved most since their terrible display in the World Cup. Ireland have been strange; two decent performances, a reasonable performance against Wales, but two terrible performances against England and France. It will be interesting to see how they get on in South Africa.

Most teams seem to now look upon the period after a World Cup as a time to rebuild, and I sincerely hope that Wales’ rebuilding includes adopting a more adventurous game plan, we certainly need it. We have gone backwards in the last 3 years; it is time for a rethink amongst the Welsh management.

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At number 10 in Rolling Stone Magazine’s list of the 100 greatest songwriters is Stevie Wonder. Stevie Wonder was born Stevland Hardaway Judkins and, when he burst on to the musical scene in 1962, he was initially known as Little Stevie Wonder. He was hailed as a child musical genius, composing his first songs at the age of 11. Of course, the ‘little’ was dropped as he grew older; but in a career that has spanned six decades he will certainly go down as one of the most important singer-songwriters in 20th Century music.

Some of his best known songs are ‘Uptight (Everything’s Alright)’, ‘The Tears Of a Clown’ (a number one hit for Smokey Robinson and the Miracles), ‘Isn’t She Lovely’ (a song to his new-born daughter), ‘For Once in My Life’, ‘Sir Duke’‘Lately’ (which I blogged about here), and ‘Happy Birthday’, the song he wrote in the campaign to have Martin Luther King’s birthday recognised as a public holiday in the USA (which I blogged about here). His 1976 album Songs in the Key of Life is considered one of the greatest albums of all time.

 

At number 10 in Rolling Stone Magazine's list of the 100 greatest songwriters of all time is Stevie Wonder.

At number 10 in Rolling Stone Magazine’s list of the 100 greatest songwriters of all time is Stevie Wonder.

 

The song which I’ve decided to include in this blog is ‘Sir Duke’, which was released as a single in 1977 and is possibly the first Stevie Wonder song I remember hearing on the radio. It comes from his album Songs in the Key of Life, and when released it got to number 1 in the USA and to number 2 in the Disunited Kingdom.

 

Music is a world within itself
With a language we all understand
With an equal opportunity
For all to sing, dance and clap their hands
But just because a record has a groove
Don’t make it in the groove
But you can tell right away at letter A
When the people start to move

They can feel it all over
They can feel it all over people
They can feel it all over
They can feel it all over people

Music knows it is and always will
Be one of the things that life just won’t quit
But here are some of music’s pioneers
That time will not allow us to forget
For there’s Basie, Miller, Satchmo
And the king of all Sir Duke
And with a voice like Ella’s ringing out
There’s no way the band can lose

You can feel it all over
You can feel it all over people
You can feel it all over
You can feel it all over people

You can feel it all over
You can feel it all over people
You can feel it all over
You can feel it all over people

You can feel it all over
You can feel it all over people
You can feel it all over
You can feel it all over people

You can feel it all over
You can feel it all over people
You can feel it all over
I can feel it all over-all over now people

Can’t you feel it all over
Come on let’s feel it all over people
You can feel it all over
Everybody-all over people

 

Here is a video of this great song. Enjoy!

Which is your favourite Stevie Wonder song?

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Just over a week ago, on March 3, the news broke that the Hubble Space Telescope has found the most distant galaxy seen to date, GN-z11. At a spectroscopically measured redshift of z=11.1, it breaks the previous record of z=10.7 (which is a redshift currently based on the photometric redshift method, rather than spectroscopy). The previous record for a galaxy which had been spectroscopically measured was z=8.68, so this new discovery breaks that record by some margin.

Here is an image of the galaxy. It is superimposed on an image of a survey called GOODS North to show the part of the sky where the galaxy was found.

 

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An image of the most distant galaxy seen so far, at a redshift of z=11.1; meaning we are seeing it as it was when the Universe was only some 400 million years old.

Here is a screen capture of the summary of the Space Telescope Science Institute’s press release, you can find the actual press release here.

 

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A screen capture of the summary of the press release announcing the discovery of a galaxy at a redshift of z=11.1, which has been measured by spectroscopy rather than based on the photometric redshift technique.

You can see in the text of this press release summary that the galaxy is referred to as “surprisingly bright”, and below I show the beginning of the preprint of the paper announcing the result (dated March 3). You can read the preprint for yourself by following this link. Again, the galaxy is referred to as “remarkably luminous” in the preprint. Today I just wanted to present this exciting story, but next week I will explain more about a “schoolboy error” (or undergraduate error 😉 ) I made in discussing the brightness of this galaxy with colleagues.

 

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A screen capture of the preprint submitted to Astrophysical Journal announcing the discovery of a galaxy at a redshift of z=11.1

But, more on my schoolboy undergraduate error next week; as I say today I just want to present the story. At a redshift of 11.09 +0.08 and -0.12 (pretty small errors), this galaxy is being seen when the Universe was only some 400 million years old, or to put it another way we are seeing it some 13.3 billion years ago! Truly remarkable.

 

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At number 24 in Rolling Stone Magazine’s list of the 100 greatest Beatles songs is their 1968 song “Happiness Is a Warm Gun”. This John Lennon composition appears as the 8th (and last) track on the first side of the double album The Beatles (usually known as The White Album). According to Lennon, the song was partly inspired (see below) by the cover of a gun magazine which producer George Martin showed him.

Apparently the song is a fusion of three separate songs. The first was a series of visual images which Lennon had when on an acid trip with Derek Taylor, Neil Aspinall and Pete Shotton (a childhood friend from Liverpool). Lennon wanted to describe a girl who was really sharp, so Taylor suggested the line “she’s not a girl who misses much”. The second song  fragment was some lines about Yoko, whom Lennon would often call “mother”. Finally, the third fragment of song was based on his seeing the cover of American Rifleman, the magazine of the National Rifle Association (NRA), where a single feature by a write named Warren W. Herlihy was entitled “Happiness Is A Warm Gun”.

“Happiness Is a Warm Gun” was recorded in September of 1968, and appeared when The White Album was released in November of the same year. It is, apparently, both Paul McCartney’s favourite song on the album, and was also George Harrison’s favourite. Because of supposed sexual references, the song was originally banned from being played on BBC radio.

At number 24 in Rolling Stone Magazine's list of the 100 greatest Beatles songs is "Happiness Is a Warm Gun".

At number 24 in Rolling Stone Magazine’s list of the 100 greatest Beatles songs is “Happiness Is a Warm Gun”.

She’s not a girl who misses much
Do do do do do do do do, oh yeah
She’s well acquainted with the touch of the velvet hand
Like a lizard on a window pane
The man in the crowd with the multicoloured mirrors
On his hobnail boots
Lying with his eyes while his hands are busy
Working overtime
A soap impression of his wife which he ate
And donated to the National Trust

Down
I need a fix ’cause I’m going down
Down to the bits that I left uptown
I need a fix ’cause I’m going down

Mother Superior jump the gun
Mother Superior jump the gun
Mother Superior jump the gun
Mother Superior jump the gun
Mother Superior jump the gun
Mother Superior jump the gun

Happiness is a warm gun (Happiness bang, bang, shoot, shoot)
Happiness is a warm gun, mama (Happiness bang, bang, shoot, shoot)
When I hold you in my arms (Oo-oo oh yeah)
And I feel my finger on your trigger (Oo-oo oh yeah)
I know nobody can do me no harm (Oo-oo oh yeah)

Because happiness is a warm gun, mama (Happiness bang, bang, shoot, shoot)
Happiness is a warm gun, yes it is (Happiness bang, bang, shoot, shoot)
Happiness is a warm, yes it is, gun (Happiness bang, bang, shoot, shoot)
Well, don’t you know that happiness is a warm gun, mama? (Happiness is a warm gun, yeah)

Unfortunately I have not been able to find a video of this song anywhere. If anyone is aware of one please let me know in the comments section below, but it looks like all copies have been removed due to copyright infringement. But, as of Christmas Eve 2015, The Beatles’ music is now available on various streaming services, so if you have not heard this fantastic song you can now do so via one of those.

Here is the link to the song on Spotify

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The 4th weekend of the 2016 6 Nations has left England still on course for the Grand Slam, and already crowned 2016 champions with the final weekend still to come. In a thorough demolition of Wales at Twickenham they showed that they are much improved from the side that Wales beat there just 5 months ago. England saw off a late surge from Wales to hold on to win 25-21. In the first match of the weekend Ireland thrashed Italy 58-15 to register their first win of the Championships, and in Sunday’s match Scotland beat France 29-18, the first time they have beaten Les Bleus in 10 years!

Ireland v Italy

It is difficult to understand what has happened to Ireland in the last twelve months. To only be registering their first victory of the 2016 6 Nations in its fourth weekend is very strange for the country which has won the Championships the last two years running. Whether they have been traumatised by their heavy defeat to Argentina in the quarter finals of the World Cup, or whether it is something else, I have no idea.

Finally they put in a display which is worthy of them, running in nine tries in Dublin to wallop Italy 58-15. I did not see the game, but am pleased to see Ireland back in form.

England v Wales

This was, for any Welsh person, the big match of the weekend. The build-up during the week had been relentless, as I blogged about on Friday. But, when the game started Wales were nowhere to be seen. We were largely absent for the first 60 minutes of this match, going down 19-0 and 25-7 before we rallied in the last 7 minutes to make the final score a more respectable 25-21.

I really don’t know what happened to Wales in the first half. It is true that England were good, in fact they were considerably better than Wales. But, Wales did not look like they were interested; they looked lethargic, slow, lacking commitment and as if they did not have the will to play a decent game. We missed so many tackles in the first half that I could not believe what I was seeing, 19 missed tackles in the first half, more than we typically miss in a whole championship! The total number of missed tackles by the end of the match was 27. Crazy for a team who pride themselves on their defence. We also kept getting turned over at the breakdown, it was truly shocking. At  half time it was 16-0, and Wales were lucky to have zero.

Whatever Gatland and Shaun Edwards said at half time to the hapless Welsh players clearly had some impact, because they came out in the second half a different team. After Farrel put England ahead 19-0, Dan Biggar charged a kick down to score under the posts. Suddenly it was 19-7. Then, two more penalties to England put them 25-7 ahead, and the brief hope of a Welsh revival seemed to be extinguished.

With the score at 25-7 and only 7 minutes left on the clock the Welsh team came to life. We scored two tries in quick succession to make it 25-21, and England looked rattled. Wales could have snatched it with a late break down the touchline by George North. On replays it is clear to me that the assistant referee made a mistake, North got the ball away before he was in touch. Whether Wales would have scored a try is debatable, as the English player and man of the match Maro Itoje looked ready to nail Rhys Webb who had caught North’s pass. But, who knows? It is frustrating from a Welsh viewpoint that a bad call should deny us a possible last-gasp victory; but I must repeat that England were the better team on the day and deserved to win.

Obviously I wanted Wales to snatch victory from the jaws of defeat, but had they done so it would not have been a fair result as, for 60-plus minutes of this game, we were thoroughly outplayed. England deserved their victory, and Wales need to get to the bottom of how they can put in such an abject performance for 40-plus minutes of a match with as much riding on it as this one had. It is difficult to tell how good England were in the first half, as Wales were clearly below par. But, there is no doubt that England are a much better team than they were in the World Cup, and this is clearly due to Eddie Jones, their new coach.

 

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Wales’ first half display was abject, I have rarely seen such a lacklustre performance by a Welsh team since Gatland took over in 2008

Scotland v France

With Scotland beating France 29-18, this was the surprise result of the weekend. Not only have Scotland not beaten France in 10 years, but until their victory over Italy two weeks ago, they had gone some ten 6 Nations matches without a win. Now they have two back-to-back wins! By beating France they have ensured that England have won the 2016 6 Nations, irrespective of what happens in the final weekend.

Final Weekend Preview

Next weekend will be the fifth and final weekend of the 2016 6 Nations. The main question is can England go to France and win there to secure the first Grand Slam since 2003? They should do, as not only are they a much better team under Eddie Jones, but France have been pretty useless in this championships. However, of any team in the 6 Nations, France are the most unpredictable, so England cannot be complacent. England have fallen several times at the last hurdle in the last 5-6 years, but I am sure Eddie Jones will ensure that this does not happen this time.

Scotland go to Dublin on a two-match winning streak, and it is going to be very interesting to see how well a confident Scotland can do against an Ireland who are still below par and lacking confidence. I am pretty neutral when it comes to Ireland v Scotland, but I would like to see Scotland do well and lift themselves from the whipping boys they have become in the last several years.

Wales take on Italy at home for their final game. It should be a comfortable win, but the Welsh fans will want to see some expansive rugby and plenty of tries. We go on tour to New Zealand in the summer, and if we play as we have done in this 6 Nations we are going to get thumped by the All Blacks. If we beat Italy we will finish second in the table, but this should not mask the serious work that Wales need to do to improve, we have gone backwards since the World Cup. 

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Today I thought I would suspend my usual Friday post of the countdown of the 100 greatest songwriters as determined by Rolling Stone Magazine and post, instead, a poem by one of my favourite Welsh-language poets – Waldo Williams. The poem I have chosen has been in the news a bit this week as BBC Wales have used an English translation of it in their trailer for tomorrow’s (Saturday’s) big rugby showdown between England and Wales.

As anyone who knows anything about Wales will tell you, we are big on rugby. It has become our religion. We get pretty excited about any rugby international, but when it is against England (the old enemy), and by beating England we can both scupper their chances of a Grand Slam and put us in a position to win the 6 Nations Championship, then the excitement goes into overdrive.

But, more about the rugby later in this blogpost, first Waldo Williams and the poem.

Who was Waldo Williams?

I feel a bit of a connection with Waldo Williams as he was born in Haverfordwest where I grew up. Then, at 7 years of age, he moved with his family to Mynachlog Ddu in the Preseli mountains, a place where some of my ancestors on my paternal grandfather’s side of the family also lived. He spoke only English before he moved to Mynachlog Ddu; his father was a Welsh speaker but his mother spoke only English. As Mynachlog Ddu was (and still is) a Welsh-speaking community he quickly became fluent in Welsh; but apparently always spoke to his sister in English as that is the language in which they had started their relationship.

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Waldo Williams (1904-1971) was a Welsh poet, anti-war campaigner and political activist who grew up in Mynachlog Ddu, Pembrokeshire

After graduating in English from the University College of of Wales, Aberystwyth (now Aberystwyth University) he became a teacher, and went on to become headmaster of the local school in Maenclochog (near Mynachlog Ddu). He became a Quaker in the 1950s, and during the Korean War he refused to pay his taxes as a protest against the war. For this refusal, he was sent to prison several times.

As a teenager I  had a poster of one of Waldo’s poems on my bedroom wall, a beautiful poem called Cofio, which I will have to blog about in the future. I also included two lines from his poem Preseli at the beginning of my PhD thesis back in 1992. These lines are

Mur fy mebyd, Foel Drigarn, Carn Gyfrwy, Tal Fynydd

Wrth fy nghefn ym mhob annibyniaeth barn

which I translated as

The Wall of my youth, Bare Three Cairns, Saddle Cairn, Tall Mountain,

Behind me in all my independence of opinion

(Foel Drigarn, Carn Gyfrwy and Tal Fynydd are three mountains one can see from Mynachlog Ddu). The same words are on the memorial stone to Waldo, which stands overlooking these three mountains of his youth. I quoted these lines at the start of my Thesis as it summed up, for me, what growing up in the rugged countryside of Pembrokeshire engenders in its people; an independence of opinion and a preparedness to choose the path less followed.

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The memorial stone to Waldo, which stands overlooking the three mountains mentioned in the lines of his poem

Pa Beth yw Dyn?

Pa Beth yw Dyn? was published in Waldo’s only book of poetry, Dail Pren (The Leaves of the Tree), which came out in 1956.

Beth yw byw? Cael neuadd fawr
Rhwng cyfyng furiau
Beth yw adnabod? Cael un gwraidd
Dan y canghennau.

Beth yw credu? Gwarchod tref
Nes dyfod derbyn.
Beth yw maddau? Cael ffordd trwy’r drain
At ochr hen elyn.

Beth yw canu? Cael o’r creu
Ei hen athrylith.
Beth yw gweithio ond gwneud cân
O’r coed a’r gwenith?

Beth yw trefnu teyrnas? Crefft
Sydd eto’n cropian
A’i harfogi? Rhoi’r cyllyll
Yn llaw’r baban.

Beth yw bod yn genedl? Dawn
Yn nwfn y galon.
Beth yw gwladgarwch? Cadw ty
Mewn cwmwl tystion.

Beth yw’r byd i’r nerthol mawr?
Cylch yn treiglo.
Beth yw’r byd i blant y llawr?
Crud yn siglo.

Dr. Rowan Williams, the former Archbishop of Canterbury, has done a translation of this poem, and it is his translation which is used in the BBC Wales trailer for tomorrow’s match. His translation reads

What is living? The broad hall found
between narrow walls.
What is acknowledging? Finding the one root
under the branches’ tangle.

What is believing? Watching at home
till the time arrives for welcome.
What is forgiving? Pushing your way through thorns
to stand alongside your old enemy.

What is singing? The ancient gifted breath
drawn in creating.
What is labour but making songs
from the wood and the wheat?

What is it to govern kingdoms? A skill
still crawling on all fours.
And arming kingdoms? A knife placed
in a baby’s fist.

What is it to be a people? A gift
lodged in the heart’s deep folds.
What is love of country? Keeping house
among a cloud of witnesses.

What is the world to the wealthy and strong? A wheel,
turning and turning.
What is the world to earth’s little ones? A cradle,
rocking and rocking.

This is an alternative translation by Tony Conran

To live, what is it? It’s having
A great hall between cramped walls.
To know another, what’s that? Having
The same root under the branches

To believe, what is it? Guarding a town
Until acceptance comes.
Forgiveness, what’s that? A way through thorns
To an old enemy’s side.

Singing, what is that? The ancient
Genius of the creation.
What’s work but making a song
Of the trees and the wheat?

To rule a kingdom, what’s that? A craft
That is crawling still.
And to arm it? You put a knife
In a baby’s hand.

Being a nation, what is it? A gift
In the depths of the heart.
Patriotism, what’s that? Keeping house
In a cloud of witnesses.

What’s the world to the strong?
Hoop a-rolling.
To the children of earth, what is it?
A cradle rocking.

The England v Wales BBC Trailer

Now, finally, tomorrow’s (Saturday’s) big rugby match between England and Wales. It is the fourth weekend of the 2016 6 Nations, and as things stand England and Wales are the only two undefeated sides. England have 3 wins from 3, and Wales have 2 wins and a draw from 3. The winner at Twickenham tomorrow is almost certainly going to win the 2016 Championship, so the stakes could not be higher.

Wales and England have played each other 127 times. Remarkably, both sides are incredibly even; England have won 58 times and Wales have won 57 times, with 12 matches drawn. Wales have beaten England more times since 2008, and the last time we played (at Twickenham) was when we helped dump England out of the  World Cup.

Wales v England results since 2008
Year Venue Competition Score Winner
2015 Twickenham 2015 Rugby World Cup 25-28 Wales
2015 Cardiff 2015 6 Nations 16-21 England
2014 Twickenham 2014 6 Nations 29-18 England
2013 Cardiff 2013 6 Nations 30-3 Wales
2012 Twickenham 2012 6 Nations 12-19 Wales
2011 Cardiff 2011 World Cup Warm Up Match 19-9 Wales
2011 Twickenham 2011 World Cup Warm Up Match 23-19 England
2011 Cardiff 2011 6 Nations 19-26 England
2010 Twickenham 2010 6 Nations 30-17 England
2009 Cardiff 2009 6 Nations 23-15 Wales
2008 Twickenham 2008 6 Nations 19-26 Wales

As this table shows, since 2008 Wales and England have played 11 times. Wales have won 6 times, England have won 5 times, and there have been no draws. It couldn’t be much closer!

Hopefully, with Wales having beaten England the last time they played, and it having been at Twickenham, Wales will have the edge tomorrow. I cannot wait for the match. And, to get you in the mood, here is the BBC Wales trailer, with Rowan Williams’ translation of Pa Beth Yw Dyn? read by Welsh actress Erin Richards…..

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Erin Richards reading Waldo Williams’ poem Pa Beth Yw Dyn? (What is Man?), as translated by Rowan Williams

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On Tuesday of last week I blogged about the H.E.S.S. telescopes in Namibia which detect Cherenkov radiation produced by high energy rays (cosmic rays or gamma rays) entering the Earth’s atmosphere from outer space. These high energy rays can hit atoms in the Earth’s atmosphere; it is these collisions which can give rise to Cherenkov radiation. Today I am going to give more detail about Cherenkov radiation; this is based on my lecture notes from teaching High Energy Astrophysics at Cardiff University, a Masters level astrophysics course.

Who was Cherenkov and why does he have a type of radiation named after him?

Cherenkov radiation is named in honour of Pavel Alekseyevich Cherenkov, a Soviet scientist who won the 1958 Nobel Prize for his experimental observation of this type of radiation.

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Figure 1: Pavel Alekseyevich Cherenkov (1904-1990), who won the 1958 Nobel Prize in Physics for his theoretical prediction of what we now call Cherenkov radiation

Cherenkov worked at the Lebedev Physical Institute, which is based in Moscow. His observation of Cherenkov radiation was made in 1934, and 24 years later, in 1958, he won the Nobel Prize for this work. He shared the prize with fellow Russian-scientists Igor Tamm and Ilya Frank who worked out the theory of the radiation. Cherenkov was also awarded two Stalin prizes, the first in 1946 and the second in 1952.

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Figure 2: The characteristic blue light produced by Cherenkov radiation in a nuclear reactor

What Cherenkov observed in 1934 was a blue light being give off by a bottle of water which was being subjected to bombardment by radioactivity (what we call irradiated). By the 1950s, with the advent of nuclear reactors, this blue glow had become a common sight – an example is shown in Figure 2. It shows some nuclear fuel rods submerged in water (the moderator), and  the interaction of the high-energy radiation from the fuel rods with electrons in the water causes the water to glow blue – this is Cherenkov radiation.

The details of Cherenkov radiation

Cherenkov radiation is only one type of interaction between either high-energy particles, or high-energy radiation, and matter. Other such interactions include when a high-energy nucleon (proton or neutron) interacts with an electron, or a high-energy electron interacts with an electron, or a high-energy electron interacts with a gamma-ray. But, in this blogpost I am going to concentrate on Cherenkov radiation.

Cherenkov radiation is produced when a charged particle travels faster than the speed of light in that medium. As nothing can travel faster than the speed of light in a vacuum, Cherenkov radiation can only happen in a medium such as air, water or glass. Many of you are probably familiar with the idea of the refractive index n of a medium. It crops up for example in Snell’s law, which allows us to calculate the angle through which radiation deviates when it travels from one medium to another. Snell’s law states that

\frac{ n_{2} }{ n_{1} } = \frac{ sin \theta_{1} }{ sin \theta_{2} }

where n_{1} \text{ and } n_{2} are the refractive indices of the two media and \theta_{1} \text{ and } \theta_{2} are the angles between the direction of the ray and the normal, as shown in Figure 3.

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Figure 3: Snell’s law gives the relationship between the angles between the direction of the rays and the refractive indices n of the two media

Snell’s law is usually taught in A-level physics, but what is not taught is the relationship between the refractive index and the speed of light. It is a very simple relationship; if n_{1} is the refractive index in medium 1, we can simply write

n_{1} = \frac{ c }{ v_{1} }

where v_{1} is the speed of light in that medium (and c is, of course, the speed of light in a vacuum). We should note that, in general, the refractive index n is a function of wavelength; this means that red light and blue light travel at different speeds in e.g. glass (or water). This is why these media disperse (split up) light and you see the colours of the rainbow when light passes through a prism, the blue and red lights are refracted different amounts in the glass because they travel at different speeds.

This page gives a list of refractive indices for different media. For pure water at 20 Celsius, n=1.3325, whereas for heavy water (deuterium oxide, each hydrogen atom is replaced with deuterium which has a neutron in the nucleus in addition to the proton present in hydrogen), the refractive index according to this page is n=1.328, so slightly less than for pure water. Heavy water is used as a moderator in nuclear reactors as the additional neutron helps slow down the fast neutrons from the nuclear reactions more effectively than pure water.

Assuming we are dealing with heavy water, this means that the speed of light in heavy water is v = 3 \times 10^{8} / 1.328 = 2.259 \times 10^{8} metres per second. If a charged particle travels faster than this speed in heavy water, it will emit Cherenkov radiation. A cone of constructive interference will form at an angle \theta to the direction of travel of the charged particle.

If v is the speed of the particle in the medium and v_{1} is the speed of light in that medium (where v_{1} = c/n), we can write that

\boxed{ cos \theta = \frac{ v_{1} }{ v } = \frac{ c }{ nv } } \; \; \text{(1)}

Clearly, as cos \theta can never be greater than 1, v (the speed of the particle) cannot be less than c/n. This is the threshold, Cherenkov radiation will only happen if v > c/n.

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Figure 4: In Cherenkov radiation, a wavefront forms at an angle \theta to the direction of travel of the charged particle. In this figure, the particle is travelling at a velocity v horizontally; the radiation in the medium travels at a velocity v_{1}

 

This cone of light is very analogous to the cone of sound which forms when an object travels faster than the speed of sound in that medium. In Figure 5 we show an aeroplane travelling at less than the speed of sound (on the left), at the speed of sound (in the middle) and in excess of the speed of sound (on the right). The cone of sound produced at speeds greater than the speed of sound has the same relationship for the angle as with Cherenkov radiation, it is given by the ratio of the speed of the object to the speed of the waves in that medium.

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Figure 5: The sound waves created as an aeroplane travels at different speeds. When it exceeds the speed of sound (figure on the right), a cone of sound is formed around the aeroplane, the angle depends on the ratio of the aeroplane to the speed of sound, just like the angle of the cone of light in Cherenkov radiation

Another way to illustrate the radiation given off in Cherenkov radiation is shown in figure 6. It is just a different way of showing the same thing as shown in Figure 4, but may be clearer to some of you. In Figure 6 the wavefronts are shown after a time t, where the particle will have travelled a distance vt.

c/n in Figure 6 is the same as v_{1} in Figure 4, the speed of the radiation in the medium. \beta is defined as v/c which means that \beta c = v, the speed of the particle in the medium. Figure 6 shows nicely why a cone of light develops which shines forwards at an angle of \theta to the direction of travel of the charged particle.

Cherenkov.svg

Figure 6: This is a different way of illustrating the emission of Cherenkov radiation.

 

The energy emitted per unit length travelled by an electron per angular frequency interval d\omega is given by the Frank-Tamm formula (see here)

\boxed{ \frac{ dE }{ dx d\omega } = \frac{ e^{2}}{ 4 \pi } \mu(\omega) \omega \left( 1 - \frac{ c^{2} }{ v^{2} n^{2}(\omega) } \right) }

where v is the speed of the particle, e is the charge on an electron, \mu(\omega) is the magnetic permeability of the medium (which may be frequency dependent), n(\omega) is the refractive index of the medium (which may be frequency dependent) and c/n(\omega) is the speed of light in the medium. Frank and Tamm are the two other physicists who shared the 1958 Nobel with Cherenkov.

As we saw for Equation (1), which gave the angle of the wavefronts to the direction of the charged particle, the specific intensity will go to zero if the speed of the particle v is equal to c/n_{\nu}, and becomes negative if v becomes less than c/n_{\nu}.

Rather than show the energy, it is more usual to calculate the number of photons emitted per particle length per frequency interval or wavelength interval. We are going to show it per wavelength interval d\lambda. We start with Equation (2), and first we change from angular frequency \omega to temporal frequency \nu. To do this we remember that \omega = 2 \pi \nu, and so d \omega = 2 \pi d\nu. This gives us

\frac{ dE }{ dx d\nu } = \frac{ e^{2}}{ 4 \pi } \mu(\nu) 2 \pi \nu \left( 1 - \frac{ c^{2} }{ v^{2} n^{2}(\nu) } \right) \cdot 2 \pi

which simplifies to

\frac{ dE }{ dx d\nu } = \frac{ \pi e^{2}}{ 1 } \mu(\nu) \nu \left( 1 - \frac{ c^{2} }{ v^{2} n^{2}(\nu) } \right)

We are now going to assume that the permeability \mu(\nu) is unity. To go from the energy to the number of photons we remember that the energy of each photon is h \nu, where h is Planck’s constant, so dE = N h \nu, where N is the number of photons. So, we can write

\frac{ d^{2}N }{ dx d\nu } = \frac{ \pi e^{2} }{ h \nu } \nu \left( 1 - \frac{ c^{2} }{ v^{2} n^{2}(\nu) } \right)

We are now going to express this per wavelength interval. Remember c = \lambda \nu so \nu = c/\lambda and d\nu = - (c/\lambda^{2})d\lambda. We can ignore the minus sign, this is just telling us that wavelength decreases as frequency increases, so then we can write

\frac{ d^{2}N }{ dx d\lambda } = \frac{ \pi e^{2} }{ h } \frac{ \lambda }{ c } \frac{ c }{ \lambda }\left( 1 - \frac{ c^{2} }{ v^{2} n^{2}(\lambda) } \right) \frac{ c }{ \lambda^{2} }

which simplifies to

\frac{ d^{2}N }{ dx d\lambda } = \frac{ \pi e^{2} }{ hc \lambda^{2} } \left( 1 - \frac{ c^{2} }{ v^{2} n^{2}(\lambda) } \right)

Finally, we multiply by 4 \pi, the solid angle in a sphere, to give us our final expression

\boxed{ \frac{ d^{2}N }{dxd\lambda} = \frac{ 4 \pi^{2} e^{2} }{ h c \lambda^{2} } \left( 1 - \frac{ c^{2} }{ v^{2} n_{\lambda}^{2} } \right) } \; \; (3)

In Figure 7 I use Equation (3) to show the numbers for two different particle speeds, (a) v=2.8 \times 10^{8} m/s and (b) 2.3 \times 10^{8} m/s. I have illustrated the visual part of the spectrum by the shaded box.

 

Figure 7: the number of photons emitted in Cherenkov radiation per unit length per wavelength interval for two different particle velocities – v=2.8\times 10^{8} m/s (green) and (b) v=2.3 \times 10^{8} m/s (blue), both for heavy water with n_{\lambda}=1.328. The shaded area represents the visible part of the spectrum.

 

As can be seen from Figure 7, (a) the number of photons is more when the charged particles are moving quicker and (b) the number of photons increases with shorter wavelength, which is why Cherenkov radiation has its characteristic blue appearance. As the number goes as 1/\lambda^{2}, the number of blue photons is roughly four times as many as the number of red photons. In fact, the intensity is even higher in the ultraviolet, but of course we cannot see this light with our eyes.

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