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## Tim Peake is home from space

On Saturday 18 June, as some of you may know, Tim Peake returned from his 6-month stint on the International Space Station (ISS). At the end of January, as a bit of fun, in a blog entitled “Is Tim Peake getting younger or older?”  I worked out whether he was getting younger (due to time dilation in special relativity) or older (due to time running faster due to general relativity). The answer was that the special relativity effect of time slowing down for him was greater than the general relativity effect of time speeding up. But, he would need to stay in space for 100 years to age by 1 second less than if he were on Earth! But, now that he is back on Earth time is running at the same rate for him as for the rest of us. 🙂

Peake held a press conference on Tuesday 21 June, and later that day I was on BBC radio making some comments about his time on the ISS. It was only a short 3-minute interview for the evening news programme (you can listen to it here), but one of the things I was asked was whether Tim Peake’s mission to the ISS had inspired young people (school students).

Astronaut Tim Peake returned to Earth from the International Space Station on 18 June 2016 after a six-month period there.

My answer was that yes, it absolutely had. Peake has captured the public imagination with his trip to the ISS, and has inspired a whole new generation to think about space. As the first person from Britain to go into space at the taxpayers’ expense, he may have had instructions to engage with the public in his time spent there. I don’t know. But, what I do know is that he clearly enjoys communicating science and the wonders of space and the oddities of an astronaut’s life to the public, and has done an excellent job of it.

I just about member the last Apollo mission, Apollo 17, which went to the Moon in December 1971. I’m too young to remember the ones before; even though my mother sat me down in front of the TV to see Neil Armstrong take his historic steps in 1969, I sadly don’t remember it. Seeing astronauts going to the Moon was certainly a factor in igniting my own interest in space and astronomy, but since that time there has been very little to inspire later generations. Going up in the Space Shuttle or going to the ISS are not as exciting as going to the Moon; but thankfully Tim Peake has turned what has become a rather routine activity these days into something very exciting for our younger people.

I don’t know how much it cost the DUK taxpayers to put Peake into space, but I can guarantee you that the money will be recouped dozens of times over. There is no surer way to create wealth than through science and technology, and inspiring a whole new generation of school students into taking an interest in physics, mathematics, engineering and science will, hopefully, see more of them pursue such careers in the future. This can only be a boom for our economy.

## The 100 best Beatles songs – number 17 – Ticket to Ride

At number 17 in Rolling Stone Magazine’s list of the 100 greatest Beatles songs is “Ticket to Ride”. This song was recorded in February 1965 and released as a single in April 1965. The B-side to the single was “Yes It Is”. Unusually for a Beatles single, it was also included on an album, in this case their 1965 album Help!, but this is more because the song is in the movie, and the album was essentially a soundtrack of the movie.

According to the screen-capture of the Rolling Stone blurb below, its writers were Lennon and McCartney. I defer to their greater knowledge, but I have always thought of “Ticket to Ride” as a John Lennon composition. He certainly takes the lead vocal, and its style is very much Lennon rather than a collaboration between him and McCartney. The single got to number 1 in the Disunited Kingdom, the USA, and many other countries.

At number 17 in Rolling Stone Magazine’s list of the 100 greatest Beatles songs is “Ticket to Ride”

Here are the lyrics for this great song. It is one of my favourite Beatles songs from 1965.

I think I’m gonna be sad
I think it’s today, yeah
The girl that’s driving me mad
Is going away

She’s got a ticket to ride
She’s got a ticket to ride
She’s got a ticket to ride
But she don’t care

She said that living with me
Is bringing her down, yeah
For she would never be free
When I was around

She’s got a ticket to ride
She’s got a ticket to ride
She’s got a ticket to ride
But she don’t care

I don’t know why she’s riding so high
She ought to think twice
She ought to do right by me
Before she gets to saying goodbye
She ought to think twice
She ought to do right by me

I think I’m gonna be sad
I think it’s today, yeah
The girl that’s driving me mad
Is going away, yeah

Oh, she’s got a ticket to ride
She’s got a ticket to ride
She’s got a ticket to ride
But she don’t care

I don’t know why she’s riding so high
She ought to think twice
She ought to do right by me
Before she gets to saying goodbye
She ought to think twice
She ought to do right by me

She said that living with me
Is bringing her down, yeah
For she would never be free
When I was around

Ah, she’s got a ticket to ride
She’s got a ticket to ride
She’s got a ticket to ride
But she don’t care

My baby don’t care, my baby don’t care
My baby don’t care, my baby don’t care
My baby don’t care, my baby don’t care (fade out)

Here is a version of “Ticket to Ride” being performed live by the Beatles in 1965 concert at Wembley Stadium, as part of the New Musical Express Poll Winners Concert. Enjoy!

## Why is scale height expressed in terms of 1/e?

Yesterday (Monday 27 June) I wrote a blogpost entitled “What is the scale height of water vapour in the Earth’s atmosphere?“. In that blogpost I said that the scale height of the gas (nitrogen, oxygen) in the Earth’s atmosphere was 7.64km, and that this means that every 7.64km it reduces by a factor of $1/e$, which is a factor of $0.368$.

This means that, at 7.64km, the atmosphere has a thickness (pressure) of  $0.368 (= 36.8\%)$ of its value at sea level. If we go to twice this height, 15.28km, the pressure of the atmosphere is $0.135\% \; (= 0.368 \times 0.368)$ of its value at sea level. At $3 \times 7.64 = 22.92 \text{ km}$ it is $0.368^{3} = 0.05 = 5\%$ of its value at sea level, etc.

The question was asked, why is it a factor of $1/e$ that we quote for the scale height, and not $1/2$, or $1/\text{something else}$? The answer is the way that the formula for the scale height $H$ is derived. It comes about from integrating an infinitesimal change in pressure $dP$ divided by the pressure $P$, that is the integral of $(dP/P)$, and this is what leads to the exponential.

## Deriving the equation for the scale height of the atmosphere

We are going to determine the pressure of the atmosphere as a function of altitude. Pressure is defined as the force per unit area, and for the atmosphere the pressure is due to the weight (not mass) of the overlying atmosphere. This is why pressure goes down with altitude. Let us assume that at a particular height $z$ the atmosphere has a density (mass per unit volume) of $\rho$ and a pressure $P$.

If we have a small slab of volume $dV = Adz$, the mass of this slab will be $dm=\rho dV = A \rho dz$. The weight of any object is given by $dW=gdm$, so the weight of this slab is $Ag\rho dz$. But, pressure is force (weight) per unit area, so

$dP = \frac{dW}{A} = \frac{Ag \rho dz}{A} = g \rho dz \text{ (1)}$

Consider a slab of thickness $dz$ and area $A$, so the volume $dV=Adx$.

Re-arranging Equation (1) we get

$\frac{dP}{dz} = - g \rho \text{ (2)}$

where $g$ is the acceleration due to gravity at that particular altitude. In theory $g$ is a function of $z$, but because the change in $g$ is so small for the changes in $z$ that we will consider, we are going to assume it is constant. The minus sign in the above expression is because $P$ decreases as we increase $z$.

For an ideal gas, we can write

$pdV = NkT$

where $N$ is the number of molecules, $k$ is Boltzmann’s constant, and $T$ is the temperature in Kelvin. If the mass of each molecules is $M$, then the total mass of the slab of gas is $NM$, and so we can say that the density is

$\rho = \frac{NM}{dV} = \frac{NM}{1} \cdot \frac{P}{NkT} = \frac{MP}{kT} \text{ (3)}$

If we combine equations (2) and (3) we get

$\frac{dP}{dz} = -g \frac{MP}{kT} \text{ (4)}$

Re-arranging Equation (4) we get

$\frac{dP}{P} = -\frac{Mg}{kT}dz \text{ (5)}$

Equation (5) is a differential equation, so to solve it we integrate

$\int{ \frac{dP}{P} }= - \frac{Mg}{kT} \int{ dz }$

which becomes

$\ln{P} = -\frac{Mg}{kT} z +C \text{ (6)}$

where $C$ is a constant which we determine by the boundary conditions. $\ln{P}$ is the natural logarithm of the pressure $P$, that is the logarithm to the base $e$. The boundary conditions are that at $z=0, \; P(z)=P_{0}$, the pressure at sea level, so we can write

$\ln{ P_{0}} = 0 + C \rightarrow C = \ln{ P_{0} }$

Putting this back in Equation (6) we have

$\ln{P} = -\frac{Mgz}{kT} + \ln{ P_{0} } \rightarrow \ln{P} - \ln{ P_{0} } = -\frac{Mgz}{kT} \rightarrow \ln{ \left( \frac{ P }{ P_{0} } \right) } = -\frac{Mgz}{kT} \text{ (7)}$

We get rid of the logarithm in Equation (7) by taking the exponent, so it becomes

$\frac{ P }{ P_{0} } = e^{ -\frac{Mgz}{kT} } \text{ (8)}$

Finally, we define the scale height $H$ as $H = \frac{ kT }{Mg}$ so we have
$\boxed{ \frac{P}{P_{0}} = e^{-\frac{z}{H}} \text{ or } P=P_{0}e^{-\frac{z}{H}} \text{ (9)} }$

As we can see, the pressure varies with altitude in the sense that the ratio of pressure at any altitude $P$ to its value at sea level $P_{0}$ is given by an exponent; the negative sign in the exponent tells us that pressure will decrease with increasing altitude.

## The variation of pressure with altitude

If we plot Equation (9) we get the following (with a value of $H=7.64 \text{ km}$)

The variation of pressure with altitude assuming a scale height of 7.64km

This shows the exponential drop off of atmospheric pressure with altitude, as given in Equation (9) above. We can, however, plot the pressure (y-axis) on a logarithmic scale. We take Equation (9) and write

$\ln{ \frac{P}{P_{0} } } = - \frac{z}{H} \rightarrow \ln{P} - \ln{P_{0}} = -\frac{z}{H}$

which we can re-arrange to give

$\boxed{ \ln{P} = -\frac{1}{H}z + \ln{P_{0}} \text{ (10)} }$

This is the equation of a straight line (c.f. $y=mx+c$), so the intercept of our straight line is $\ln{P_{0}}$ and our gradient is $-(1/H)$. It is because the integration of our expression $dP/P$ (Equation (5) above) produces an exponential that the scale height $H$ is expressed as the altitude one needs to ascend for the pressure to drop by a factor of $1/e$ and not, e.g. 1/2.

If we plot the pressure as a function of altitude with the pressure (on the y-axis) plotted on a logarithmic scale, we get a straight line. The equation of this line is $\ln{P} = - \frac{1}{H}z + \ln{P_{0}}$ The gradient of the line is $-1/H$, where $H$ is the scale height of the atmosphere. So, on this linear-log plot, if we increase the altitude by $H$, the natural log of the pressure will drop by 1.

## What is the scale height of water vapour in the Earth’s atmosphere?

Someone recently asked me what was the scale height of water vapour in the Earth’s atmosphere, so I decided to see if I could find out. The scale height of water vapour is particularly important for infrared, sub-millimetre, millimetre and microwave astronomy, as it is the water vapour in the Earth’s atmosphere which prevents large fractions of these parts of the electromagnetic spectrum from reaching the ground. This is why we can, for example, only study the Cosmic Microwave Background from space or from a few particularly dry places on Earth such as Antarctica and the Atacama desert in Chile.

## What does the term ‘scale height’ mean?

First of all, let me explain what the term “scale height” means. It is the altitude by which one needs to go up for the quantity of something (water, nitrogen, oxygen, carbon dioxide) to go down by a factor of $1/e$, where $e$ is the base of natural logarithms, and $e=2.71828.....$. The scale height, usually written as $H$, is dependent upon the temperature of the gas, the mass of the molecules, and the gravity of the planet. We can write that

$H = \frac{ kT }{ Mg } \text{ (1)}$

where $k$ is Boltzmann’s constant, $T$ is the temperature (in Kelvin), $M$ is the  mass of the molecule and $g$ is the value of the acceleration due to gravity. If we were to plot the atmospheric pressure as a function of altitude we find that it follows an exponential, this is because of the differential equation which produces Equation (1) above (I will go into the mathematics of how Equation (1) is derived in a separate blog).

In the case of air, which is some 80% nitrogen molecules and 20% oxygen molecules, the scale height has been well determined and is $7.64 \text{ kilometres}$ (or, to put it another way, it drops by a half every $5.6 \text{ km}$). So, if one were at an altitude of $5.6 \text{ km}$, half of the atmosphere would be below you. Go up another $5.6 \text{ km}$ and it drops by a half again, so at $11.2 \text{ km}$ 75% of the atmosphere is below you.

## What is the scale height of water vapour in the Earth’s atmosphere?

Determining the scale height of water vapour in the Earth’s atmosphere is, I have discovered, essentially impossible. Or, to put it better, it is a meaningless figure. This is because it varies too much. It depends on temperature, so even in a given place it can vary quite a bit. So, instead, we talk of precipitable water vapour (PWV) at a particular place (both location and altitude). PWV is the equivalent height of a column of water if we were to take all the water vapour in the atmosphere above a particular location and it were to precipitate as rain.

The Mount Abu Infrared Observatory in India, for example, is at an altitude of 1,680 metres, and quotes a PWV of 1-2mm in winter. The PWV would be higher in summer, as water sinks in the atmosphere when it is cold. For Kitt Peak in Arizona, which is at an altitude of 2,090 metres, the PWV varies between about 15mm and 25mm. This is why very little infrared astronomy is done at Kitt Peak. For Mauna Kea in Hawaii, which is at an altitude of just over 4,000 metres, it varies between 0.5mm and 2mm. This is why there are a number of infrared, sub-mm and millimetre wave telescopes there.

At the South Pole, which is at an altitude of 2,835 metres, the PWV is measured to be between 0.25 and 0.4 in the middle of the Austral winter (June/July/August). Why is this so much lower than Mauna Kea, even though it it is at a lower altitude? It is because it is so much colder.

The average Precipitable Water Vapour at the South Pole averaged over a 50-year period from 1961 to 2010. Even in the Austral summer it is low, but in the Austral winter (June/July/August) it drops to as low as 0.25 to 0.35mm, one of the lowest values found anywhere on Earth.

High in the Atacama desert, on the Llano de Chajnantor (the Chajnantor plateau), which is at an altitude of 5,000 metres and where ALMA and other millimetre and microwave telescopes are being located, the PWV is typically about 1mm, and drops to as low as 0.25mm some 25% of the time (see e.g. this website). This is why Antarctica and the Atacama desert (in particular the Chajnantor plateau) have become places to study the Cosmic Microwave Background from the Earth’s surface; we need exceptionally dry air for the microwaves to reach the ground.

## Summary

To summarise, it is meaningless to talk about a scale height for water vapour in the Earth’s atmosphere, as the vertical distribution of water vapour not only varies from location to location, but varies at a given location. So, instead, we talk about Precipitable Water Vapour (PWV); the lower this number the drier the air is above our location. To be able to do infrared, sub-millimetre, millimetre and microwave astronomy we need the PWV to be as low as possible, the best sites (Antarctica and the Atacama) get as low as 0.25mm and are usually below 1mm. The exceptionally dry air above Antarctica and the Atacama desert enable us to study the Cosmic Microwave Background from the ground, something we usually have to do from space.

## Wales are still in the 2016 Euros!

We may have voted to leave the EU, a disappointment from which it will take me a long time to recover, but Wales are still in the 2016 Euros. On Saturday evening we beat Northern Ireland 1-0 to advance to the Quarter Finals. The dream is still alive, and Wales, who have never been in the Euros before, are now in the last eight. In the Quarter Finals we will meet Belgium, who were in the same qualifying group as us. We beat them in our home game, and held them to a draw in the away game. So, we should not be afraid of them.

Wales beat Northern Ireland 1-0 on Saturday to advance to the Quarter Finals. We play Belgium in Lille on Friday at 20:00 BST (19:00 GMT)

Belgium advanced to the Quarter Finals with an emphatic 4-0 win over Hungary, and certainly they looked very good. The Wales win over Northern Ireland was much more scrappy, but although Northern Ireland are not the most attacking of teams they are also a very difficult team to break down, as we discovered.

Wales are already in dreamland, but it is far from inconceivable that we can beat Belgium. If we do, we would play either Portugal or Poland in the semi-finals. But, let’s not get ahead of ourselves. One game at a time. We first need to beat Belgium, and we showed in qualifying for the Euros that we can do that.

## Can Wales advance to the Quarter Finals?

After the heartbreak of the Brexit vote yesterday, at least I can look forward to some exciting sport today. It is not often that, with a big rugby match on the same day as a football match, that I would choose to blog more about the football, but today is  not an ordinary day. Later today, at 17:00 BST (16:00 GMT), Wales take on Northern Ireland in the round  of 16 of the 2016 Euros. As I have mentioned in previous blogposts, Wales have never been in the Euros before, so getting to the Quarter Finals would be a wonderful achievement.

Before then, in rugby Wales take on New Zealand in the final test of their three test tour. I really don’t have too much to say about that match; we are expected to lose and the main aim will be to stay within 15-20 points of New Zealand. Such is the gulf between the standard of the two teams. If we can give New Zealand a good match, we can come home thinking the tour has been a qualified success.

More interestingly, at 11am BST Australia v England kicks off in the third test of England’s tour down under. England are 2-0 up in the series, and are going for a white wash. It would be a remarkable achievement should they do it. Not only will they humiliate Australia, but it will really set down a marker that England are on their way back to being as good as anyone else in world rugby, a position they have not enjoyed since 2003. I am going to be watching that match with more interest than the NZ v Wales match earlier.

But, the real highlight for most Welsh sporting fans today is our 2016 Euros match against Northern Ireland. After our incredible display against Russia, the Welsh team must be full of confidence. We tore Russia apart in a 3-0 victory, not only ensuring our advancement to the knock-out rounds, but also we finished top of Group B after England failed to beat Slovakia.

Wales take on Northern Ireland later today in the 2016 Euros. Kick off is at 17:00 BST (16:00 GMT). Neither team has been this far in the Euros before, for Wales it is our first time ever in the Euros.

Certainly expectation is higher of our beating Northern Ireland in the football than it is of beating New Zealand in the rugby. The Welsh football team are playing full of confidence, and if we can play anything like as well as we did against Russia I cannot see us not advancing to the Quarter Finals. Fingers crossed.

## A sad day for Wales, and the world

I am writing this on Friday (24 June), the day that the result of the referendum to stay or leave the European Union (EU) was announced. I assume everyone reading this knows the result, the citizens of the (Dis)united Kingdom have voted by 51.9% to 48.1% to leave the EU. To say that I am shocked and disappointed would be an understatement. And, I am also ashamed. I am ashamed that my country, Wales, voted by 52.5% to leave. That is a higher percentage than the DUK average. I am ashamed to be Welsh at this moment.

Scotland, not surprisingly, voted to stay, in fact 62% of those voting in Scotland want to stay in the EU. Nicola Sturgeon, the First Minister of Scotland, has already said that a fresh referendum for Scottish indpendence is “highly likely”, as she feels it is totally wrong for Scotland to be forced out of the EU against its will. And, I agree with her. I only wish I could say the same for Wales, but we actually voted to leave.

As anyone who reads my blog (all two of you) will know, I am a massive rugby fan. Tomorrow, Wales will take on New Zealand in the 3rd and final test of their summer tour. We have not beaten New Zealand since 1953. Also, later tomorrow, our football team take on Northern Ireland in the 2016 Euros; if we win we will get to the Quarter Finals.

I would love us to beat NZ for the first time in 63 years, and for us to advance to the Quarter Finals of the 2016 Euros. But, I would willingly give up all of this to have had Wales mirror Scotland and have voted to stay in the EU. I have always thought of my small country as outward looking and inclusive, but it seems I was wrong. A majority want to turn their backs on our European neighbours. I would bet my mortgage that Wales will regret this decision in 5-10 years’ time and wish they had voted differently.

By 2020, I predict, Scotland will be back in the EU as an independent country; whilst Wales becomes an increasingly economically poor western part of the rump which is left of the (Dis)united Kingdom. With Scotland independent, the London government will be perpetually a Conservative one, and do the Welsh people honestly think people like Boris Johnson (the most likely person to become Britain’s next Prime Minster) or Michael Gove give a damn about the poverty blighting the South Wales valleys? The poverty that Maggie Thatcher set in motion when she dismantled the coal industry in the 1980s? They probably don’t even know where Wales is.

I have just seen this on Twitter, and so thought I would add it. Although I’m a little too young to be a baby boomer, my generation voted overwhelmingly to “leave” too. “Sorry” doesn’t seem adequate……

I am sad, I am angry, I am shocked. But, most of all I am ashamed. And envious of Scotland, a beacon of sanity in a sea of madness…….