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## Derivation of Planck’s radiation law – part 1

One of my most popular blogposts is the series I did on the derivation of the Rayleigh-Jeans law, which I posted in three parts (part 1 here, part 2 here and part 3 here). I have had many thousands of hits on this series, but several people have asked me if I can do a similar derivation of the Planck radiation law, which after all is the correct formula/law for blackbody radiation. And so, never one to turn down a reasonable request, here is my go at doing that. I am going to split this up into 2 or 3 parts (we shall see how it goes!), but today in part 1 I am going to give a little bit of historical background to the whole question of deriving a formula/law to explain the shape of the blackbody radiation curve.

## ‘Blackbody’ does not mean black!

When I first came across the term blackbody I assumed that it meant the object had to be black. In fact, nothing could be further from the truth. As Kirchhoff’s radiation laws state

A hot opaque solid, liquid, or gas will produce a continuum spectrum

(which is the spectrum of a blackbody). The key word in this sentence is opaque. The opaqueness of an object is due to the interaction of the photons (particles of light) with the matter in the object, and it is only if they are interacting a great deal (actually in thermal equilibrium) that you will get blackbody radiation. So, examples of objects which radiate like blackbodies are stars, the Cosmic Microwave Background, (which is two reasons why astronomers are so interested in blackbody radiation), a heated canon ball, or even a canon ball at room temperature. Or you and me.

Kirchhoff’s 3 radiation laws, which he derived in the mid-1800s

Stars are hot, and so radiate in the visible part of the spectrum, as would a heated canon ball if it gets up to a few thousand degrees. But, a canon ball at room temperature or you and me (at body temperature) do not emit visible light. But, we are radiating like blackbodies, but in the infrared part of the spectrum. If you’ve ever seen what people look like through a thermal imaging camera you will know that we are aglow with infrared radiation, and it is this which is used by Police for example to find criminals in the dark as the run across fields thinking that they cannot be seen.

The thermal radiation (near infrared) from a person. The differences in temperature are due to the surface of the body having different temperatures in different parts (e.g. the nose is usually the coldest part).

Kirchhoff came up with his radiation laws in the mid-1800s, he began his investigations of continuum radiation in 1859, long before we fully knew the shape (spectrum) of a blackbody.

## Germans derive the complete blackbody spectrum

We actually did not know the complete shape of a blackbody spectrum until the 1890s. And the motivation for experimentally determining it is quite surprising. In the 1880s German industry decided they wanted to develop more efficient lighting than their British and American rivals. And so they set about deriving the complete spectrum of heated objects. In 1887 the German government established a research centre, the Physikalisch-Technische Reichsandstalt (PTR) – the Imperial Institute of Physics and Technology, one of whose aims was to fully determine the spectrum of a blackbody.

PTR was set up on the outskirts of Berlin, on land donated by Werner von Siemens, and it took over a decade to build the entire facility. Its research into the spectrum of blackbodies began in the 1890s, and in 1893 Wilhelm Wien found a simple relationship between the wavelength of the peak of a blackbody and its temperature – a relationship which we now call Wien’s displacement law.

Wien’s displacement law states that the wavelength of the peak, which we will call $\lambda_{peak}$ is simply given by

$\lambda_{peak} = \frac{ 0.0029 }{ T }$

if the temperature $T$ is expressed in Kelvin. This will give the wavelength in metres of the peak of the curve. That is why, in the diagram below, the peak of the blackbody shifts to shorter wavelengths as we go to higher temperatures. Wien’s displacement law explains why, for example, an iron poker changes colour as it gets hotter. When it first starts glowing it is a dull red, but as the temperature increases it becomes more yellow, then white. If we could make it hot enough it would look blue.

The blackbody spectra for three different temperatures, and the Rayleigh-Jeans law, which was behind the term “the UV catastrophe”

By 1898, after a decade of experimental development, the PTR had developed a blackbody which reached temperatures of 1500 Celsius, and two experimentalists working there Enrst Pringsheim and Otto Lummer (an appropriate name for someone working on luminosity!!) were able to show that the blackbody curve reached a peak and then dropped back down again in intensity, as shown in the curves above. However, this pair and others working at the PTR were pushing the limits of technology of the time, particularly in trying to measure the intensity of the radiation in the infrared part of the spectrum. By 1900 Lummer and Pringsheim had shown beyond reasonable doubt that Wien’s ad-hoc law for blackbody radiation did not work in the infrared. Heinrich Rubens and Ferdinand Kurlbaum built a blackbody that could range in temperature from 200 to 1500 Celsius, and were able to accurately measure for the first time the intensity of the radiation into the infrared. This showed that the spectrum was as shown above, so now Max Planck knew what shape curve he had to find a formula (and hopefully a theory) to fit.

In part 2 next week, I will explain how he went about doing that.

## Why do we have leap seconds?

At midnight on the night of Monday the 30th of June, an extra second was added to our clocks. A so-called leap second. Did you enjoy it? Me too 🙂 I got so much more done….. But, why do we have leap seconds?

In this blog here, I explained the difference between how long the Earth takes to rotate $360^{\circ}$ (the sidereal day) and how long it takes for the Sun to appear to go once around the Earth (the mean solar day). We set the length of our day, 24 hours, by the solar day. If there are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, then there should be $24 \times 60 \times 60 = 86,400 \text{ seconds}$ in a solar day. But, there aren’t! The Earth’s rotation is not consistent, that is if we measure the length of a mean solar day, it is not consistently 86,400 seconds. This difference is why we need leap seconds.

A leap second was added at midnight on the 30th of June. It was the first leap second to be added since 2012.

But, how do we accurately measure the mean solar day (the average time the Sun appears to take to go once around the sky) , and what is causing the length of the mean solar day to change?

## How do we define a second?

When the second was first defined, it was defined so that there were 86,400 seconds in a mean solar day. But, since the 1950s, we have a very accurate method qof measuring time, atomic clocks. Using these incredibly accurate time pieces (the most accurate atomic clocks will be correct to 1 second over some tens of thousands of years) we have been able to see that the mean solar day varies. It varies in two ways, there is a gradual lengthening, but there are also random changes which can be either the Earth speeding up or slowing down its rotation.

## How do we measure the Earth’s rotation so accurately

In order to measure the Earth’s rotation accurately we use the sidereal day, which is roughly four minutes shorter than the mean solar day. By definition, the sidereal day is the time it takes for a star to cross through a local meridian a second time. But, actually, stars in our Galaxy are not good for this as they are moving relative to our Sun. So, in fact, we use quasars, which are active galactic nuclei in the very distant Universe; and use radio telescopes to pinpoint their position.

## The gradual slowing down of the Earth’s rotation

There is a gradual and unrelenting slowing down of the Earth’s rotation, which may or may not be greater than the random changes I am going to discuss below. This gradual slowing down is due to the Moon, or more specifically to the Moon’s tidal effects on the Earth. As you know, the Moon produces two high tides a day, and this bulge rotates as the Earth rotates. But, the Moon moves around the Earth much more slowly (a month), so the Moon pulls back on the bulge of the Earth, slowing it down. To conserve angular momentum, the Earth slowing down means the Moon moves further away from the Earth, about 3cm further away each year.

## The random fluctuations in the Earth’s rotation

In addition to the unrelenting slowing down of the Earth’s rotation due to the Moon, there are also random changes in the Earth’s rotation. These can be due to all manner of things, including volcanoes and atmospheric pressure. These random fluctuations can either speed up or slow down the Earth’s rotation.

We have been having leap seconds since the 1970s when atomic clocks became accurate enough to measure the tiny changes in our planet’s rotation. Since them we have added a leap second when it is decided that we need it, typically but not quite once a year. However, having that extra second at the end of June can cause glitches with computers, and so there are discussions to remove the leap second and replace it with something larger on a less frequent basis.

## Crash landing on Mercury

A few weeks ago, NASA’s Mercury MESSENGER space probe crash landed on the surface of the planet. This was not a mistake, scientists had deliberately sent it hurtling towards the surface of mysterious Mercury. It brought to an end a highly successful mission to learn more about the smallest planet in the inner solar system.

NASA’s Mercury MESSENGER space probe crashed into the surface of the planet on the 30th of April

MESSENGER (MErcury Surface, Space ENvironment, GEochemistry, and Ranging) was launched by NASA in August of 2004 and arrived at Mercury in April 2011. You might be wondering why it took so long to get to Mercury, which is much closer to us than e.g. Jupiter. The reason is that the space probe could not fly directly to Mercury, otherwise it would have just whizzed straight past. Instead it had to go on a circuitous route so that when it arrived at Mercury it was moving slowly enough to be able to go into orbit about the planet. During this flight it flew past Earth once and past Venus twice. These fly-bys, as well as being used to slow down a space probe (in this case, usually they are used to speed them up), are also used to test the instruments.

The path that Mercury MESSENGER took to get to the planet, and the dates

During the four years that MESSENGER has been orbiting Mercury it has obtained a wealth of data. It would take me too long to describe all of its findings, but some highlights are

Mercury has a magnetic field
Discovery of water in craters
Discovery of volcanism
Discovery of organic compounds
Discovery of unusually high concentrations of calcium and magnesium

As is often the case with gathering more information than we have ever previously gathered, we now have more questions about Mercury than we have answers. How can such a slowly rotating planet (it rotates once every 58.6 Earth days) produce a magnetic field? Scientists are now going to have to wait a while to find out more about Mercury, the European Space Agency (ESA) plan to launch BepiColombo in January 2017, it will arrive at Mercury in January 2024.

## Philae lands on Comet 67P, but is it stable?

Yesterday (Wednesday the 12th of November) the European Space Agency (ESA) successfully landed a tiny space-probe called Philae on to the surface of a comet. This is a remarkable achievement, and one which has clearly captured the imagination of the public and was the top story on the news here in the Disunited Kingdom yesterday. I was on the BBC talking about this just before 8am, with Philae detaching from its mother ship Rosetta just over half an hour later, at 08:35 UTC.

This remarkable photograph was taken of Philae by Rosetta’s cameras as Philae descended towards the surface of the comet. You can see in the photograph that Philae has deployed its landing legs, but unfortunately just before its release from Rosetta it was realised that a thruster on top of the washing-machine sized lander was not working. They decided to drop it anyway.

A remarkable photograph of Philae taken by its mother-ship Rosetta as it drops towards the comet. The photograph clearly shows that Philae has deployed its landing legs.

Philae descended very slowly towards Comet 67P. A common misconception seems to be that the comet does not have any gravity, but this is not true. It does, but its gravity is very very weak. Philae has a mass of 100kg, but on the surface of the comet this would feel more like a few grams. If you were on the surface of Comet 67P and you were to jump gently upwards, you would probably not return back to the surface, so weak is its gravity. This photograph was taken by Philae as it slowly descended towards the surface.

A photograph of Comet 67P as Philae descended towards its surface. This was taken from about 3km away.

ESA finally reported that Philae had landed soon after 16:00 UTC, and here is a screen capture of the robot’s Twitter feed (@Philae2014). As you can see from the announcement of its landing, it also says that the harpoons did not fire. These were designed to secure the robot to the surface.

A screen capture from Philae’s Twitter feed

As of my writing this on Thursday morning, we still do not know how securely Philae is on the surface of the comet. The latest reports are saying that Philae bounced a few times, but that it is now stationary on the surface. My understanding is that ESA are now trying to decide whether to fire the harpoons to secure it better to the surface. The danger of doing this is by firing them Philae will be sent in the opposite direction, ie. away from the surface, due to Newton’s 3rd law (to every action there is an equal and opposite reaction). If the harpoons manage to get a grip on the comet then firing them is not a risk, as then the motors can winch Philae back to the surface. But, if they fail to grip, Philae may float away from the surface not to return.

Whether ESA gets Philae secured to the surface or not, this landing has still been a remarkable success. Many of the instruments on Philae are working and taking data, and of course we also need to remember that the Philae landing was just a part of a larger mission for Rosetta. Rosetta will stay in orbit about Comet 67P and observe and study it up close as it passes at its closest to the Sun, and is scheduled to continue sending us data until December 2015.

So, whether Philae does manage to get a firm grip on the comet or not, let us remember that this is one of the greatest achievements in our history of space exploration – to land a robotic probe onto the surface of a comet which is hurtling through space some 500 million km away from Earth. Well done ESA!

## The Origins of the BBC

I am slightly embarrassed to say that, mainly due to pressure from my wife, I watch Downton Abbey. Whereas it seems to be the highlight of my wife’s viewing week, I find it largely tedious and boring. But, every now and again something in it sparks my interest, and in the episode on Sunday the 5th of October it was the plot-line about the house getting a “wireless” (radio). The head of the house, the crusty Earl of Grantham (played by Hugh Bonneville), is dead against getting a wireless, until he is told that King George V will be making a live broadcast on the radio to open the Wembley Empire Exhibition.

My wife had told me that this current series (series 5) starts in 1924, and I vaguely remembered that sometime in the last few years the British Broadcasting Corporation (BBC) had celebrated its 90th anniversary, so I decided to read more about the origins of the BBC. For quite a while I have been planning to blog about the origins of radio, as the first ever radio transmission over water occurred just a few miles from Cardiff; but that is a much bigger task and something for which I don’t currently have the time.

It surprised me to read that the BBC was not the first organisation to broadcast radio in the Disunited Kingdom. In June 1920, Lord Northcliffe, the then owner of the Daily Mail newspaper, organised the live radio transmission of a concert from the Marconi factory in Chelmsford. However, the Government quickly moved to ban future such broadcasts, believing that radio should be confined to military and other important government use, and not for entertainment.

However, there was such a public backlash to this decision that, by 1922, the Government had to make a U-turn. It decided to give radio broadcasting rights to one sole operator, namely a newly created company which they decided should be called the “British Broadcasting Company Ltd.” and which would be headed up by a dour Scottish calvinist by the name of John Reith. Its founding principles, as laid down by Reith, were “to inform, educate and entertain”, principles which have guided it to this day.

John Reith was the first head of the BBC.

The BBC started daily broadcasts from London on the 14th of November 1922, and it has been broadcasting daily ever since. Initially these broadcasts were made from studios in The Strand, in the west end of London. They then moved their studios to nearby Savoy Hill, but in 1928 construction was begun in Portland Place on what would become known as “Broadcasting House”.

Broadcasting House opened in 1932, and has been the headquarters of the BBC ever since.

The first transmissions from Broadcasting House took place on the 15th of March 1932. By this time the British Broadcasting Company Ltd. had changed its name to the “British Broadcasting Corporation”, a change which happened in 1927. Until 1936 the BBC only broadcast radio transmissions, and that on two stations, the BBC National Programme and the BBC Regional Programme. The Regional Programme was created to accommodate local broadcasting, which quickly sprung up after the BBC’s initial broadcasts from London in November of 1922.

The local stations up until 1924.

With the outbreak of World War 2, the BBC decided to change its two stations to the “BBC Light Programme” and the “BBC Home Service”. After the end of the war, these two stations were joined by the “BBC Third Programme”, which broadcast mainly classical music.

By the 1960s both musical tastes and the radio broadcasting landscape had changed considerably, and there was increasing demand for more variety on the BBC. In particular, the BBC broadcast very little “rock ‘n’ roll” music on its Light Programme in the mid 1960s, and so teenagers had to go elsewhere to hear the music of The Beatles, The Rolling Stones, The Kinks etc.

This teenager demand led to a number of so-called “pirate radio stations” exploiting a loop-hole in the broadcasting legislation by broadcasting from ships anchored a few miles off the coast. The most famous of these stations was probably Radio Caroline, and a thoroughly entertaining picture of this pirate station is given in the movie “The Boat that Rocked”, which I really enjoyed when I saw it.

Despite most members of the Government abhorring “popular music” (rock-n-roll), it was clear that the BBC had to make changes. It closed the loophole in the law, leading to the pirate radio stations being forced to close, and in 1967, the BBC re-jigged its TV and radio provisions. To satisfy the desire for rock-n-roll, Radio 1 was created. Radio 2 essentially took over the output of the former BBC Light Programme, with light music of the likes of Bing Crosby, Nat King Cole etc. The BBC Third Programme became Radio 3, and the BBC Home Service became Radio 4.

## The 2014 Nobel Prize for Physics

The 2014 Nobel Prize in Physics has been won by three Japanese physicists who developed the blue LED (light emitting diode). This is important, as red LEDs were developed in 1962 and green LEDs in the early 1970s, but white light could not be produced by LEDs until a blue LED had been developed. Several large companies tried, but failed, but these three academics succeeded through persistence and ingenuity.

Anyone who has bought a torch (flashlight) in the last few years will know that they almost all now use LEDs. In fact, they use blue LEDs, the kind developed by this Japanese trio, as the blue photons are energetic enough to excite phosphorous which then fluoresces and produces white light.

Computer displays, more and more TVs, and of course smartphones also use LEDs, but in this case they combine red, green and blue LEDs to produce the colour display we see. Smartphones and tablet devices which run on batteries would just not be possible without this technology, as the LEDs used in their screens require significantly less power than any other current technology. Similarly, in lighting applications, be it torches or lighting a room, LEDs use significantly less power than the old incandescent lights and even less than the more recent fluorescent lights to create the same amount of light.

It has been calculated that, by using LEDs, the World-wide use of energy for lighting can be reduced significantly. Currently it is estimated that some 20% of all electricity use is for lighting, but by switching to LED lighting this can be reduced to about 4%.

It is interesting to look at the Nobel Prize in Physics over the last few years to see how it sometimes honours fundamental physics (or astronomy), and other times it honours more applied physics such as this year.

• 2009 – awarded for the development of fibre optic cables, which are used for e.g. transmitting telephone and internet traffic with a much higher efficiency than the old copper-wire technology
• 2010 – the discovery of “wonder material” graphene
• 2011 – the discovery of the acceleration of the expansion of the Universe
• 2012 – work on quantum optics, which promises amongst other things super-fast computers
• 2013 – the discovery of the Higgs boson

## Ground control to Major Tim

Last week, in a big media event at London’s Science Museum, it was announced that Major Tim Peake would become Britain’s first “government backed” astronaut. He is scheduled to go to the International Space Station in late 2015.

He will not be the first British person in space, that was Helen Sharman who, in 1991, went into space through a joint venture between a number of private UK companies and the Russian government. There are also several British born astronauts who have gone into space through NASA, but to do so they have had to become US citizens. Michael Foale is the most well-travelled British born astronaut.

In my opinion it is wonderful news that the Disunited Kingdom’s government is now officially getting involved in sending humans into space. Although we learn a great deal about the Universe beyond Earth when we send space probes, there is nothing which captures the public imagination more than sending humans into space. One only had to see the press coverage this announcement received to see the proof of that, and today’s teenagers will hopefully be inspired to take up a career in science or engineering through being inspired by people like Tim Peake.

Of course the title of my blog is a play on the lines in David Bowie’s Space Oddity, so it also gives me a good excuse to add a YouTube clip of that wonderful song.