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## Oh Well – Fleetwood Mac (song)

At number 22 in BBC Radio 2’s list of the 100 greatest guitar riffs is “Oh Well” by Fleetwood Mac. Although I’m a fan of Fleetwood Mac, I had never heard this song before. It is from 1969 and was recorded when Fleetwood Mac had its original lineup with Peter Green as lead singer and lead guitarist, it is he also who wrote this song. I don’t know any Fleetwood Mac songs before Christie McVie, Lindsey Buckingham and Stevie Nicks joined, so the material with the original lineup is completely unknown to me.

“Oh Well” by Fleetwood Mac is a 1969 song which is at number 22 in BBC Radio 2’s list of the 100 greatest guitar riffs

After saying that, when I heard the opening 10 or so seconds of the song I recognised it straight away. I did not know it was a Fleetwood Mac song, but the opening 30 or so seconds was used at the end of each episode of a BBC Radio 1 documentary called “25 Years of Rock” that was broadcast back in about 1978 or 1979 and which I recorded on cassette tape. I only recorded the episodes from 1964 to 1971, and have only been able to find the cassettes with the 1964 to 1969 episodes, but I have recently transferred these to MP3 files so that I can listen to them on the move. I will say more about that fascinating series in a future blog, but here are the lyrics of “Oh Well”.

I can’t help about the shape I’m in
I can’t sing, I ain’t pretty and my legs are thin
But don’t ask me what I think of you
I might not give the answer that you want me to

Oh well

Now, when I talked to God I knew he’d understand
He said, “Stick by my side and I’ll be your guiding hand
But don’t ask me what I think of you
I might not give the answer that you want me to”

Oh well

Here is a video of this interesting song. Enjoy!

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## State of Alabama residency application

I first saw this about ten or more years ago, but was reminded about it last week so I thought I would share it here.

It is taken from this website here. If there are any questions you feel are missing please add in the comments section…..

+++++++++++++++++++++++++++++++++++++++++++++++
Residency Application for the State of Alabama

Name: ________________ (_) Billy-Bob
(last) (_) Billy-Joe
(_) Billy-Ray
(_) Billy-Sue
(_) Billy-Mae
(_) Billy-Jack
(Check appropriate box)

Age: ____
Sex: ____ M _____ F _____ N/A
Shoe Size: ____ Left ____ Right

Occupation:
(_) Farmer
(_) Mechanic
(_) Hair Dresser
(_) Un-employed

Spouse’s Name:

Relationship with spouse:
(_) Sister
(_) Brother
(_) Aunt
(_) Uncle
(_) Cousin
(_) Mother
(_) Father
(_) Son
(_) Daughter
(_) Pet

Number of children living in household: ___

Number that are yours: ___

Mother’s Name:

Father’s Name: (If not sure, leave blank)

Education: 1 2 3 4 (Circle highest grade completed)

Do you (_)own or (_)rent your mobile home? (Check appropriate box)

___ Total number of vehicles you own
___ Number of vehicles that still crank
___ Number of vehicles in front yard
___ Number of vehicles in back yard
___ Number of vehicles on cement blocks

Firearms you own and where you keep them:
____ truck
____ bedroom
____ bathroom
____ kitchen
____ shed

Model and year of your pickup: _____________ 194_

Do you have a gun rack?
(_) Yes (_) No; please explain:

Newspapers/magazines you subscribe to:
(_) The National Enquirer
(_) The Globe
(_) TV Guide
(_) Soap Opera Digest
(_) Rifle and Shotgun

___ Number of times you’ve seen a UFO
___ Number of times you’ve seen Elvis
___ Number of times you’ve seen Elvis in a UFO

How often do you bathe:
(_)Weekly
(_)Monthly
(_)Not Applicable

Color of teeth:
(_)Yellow
(_)Brownish-Yellow
(_)Brown
(_)Black
(_)N/A

Brand of chewing tobacco you prefer:
(_)Red-Man

How far is your home from a paved road?
(_)1 mile
(_)2 miles
(_)don’t know

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## The 500 Greatest Songs of all Time – no. 29 – “Help” – The Beatles

At number 29 in Rolling Stone Magazine’s 500 greatest songs of all time is “Help” by The Beatles. This John Lennon written song was recorded in 1965, and is the title song of their 5th album, released between “Beatles for Sale” and “Rubber Soul”. There is also a 1965 film by the same name, starring The Beatles in a mad-cap caper in London and the Bahamas.

Lennon wrote this song in what he later described as his “fat Elvis period”, saying that it was indeed a cry for help as he was suffering at the time from depression.

At number 29 in Rolling Stone Magazine’s ‘500 Greatest Songs of all Time’ is “Help” by The Beatles.

Help, I need somebody
Help, not just anybody
Help, you know I need someone, help

When I was younger so much younger than today
I never needed anybody’s help in any way
But now these days are gone I’m not so self assured
Now I find I’ve changed my mind and opened up the doors

Help me if you can, I’m feeling down
And I do appreciate you being ’round
Help me get my feet back on the ground

And now my life has changed in oh so many ways
My independence seems to vanish in the haze
But every now and then I feel so insecure
I know that I just need you like I’ve never done before

Help me if you can, I’m feeling down
And I do appreciate you being ’round
Help me get my feet back on the ground

When I was younger so much younger than today
I never needed anybody’s help in any way
But now these days are gone, I’m not so self assured
Now I find I’ve changed my mind and opened up the doors

Help me if you can, I’m feeling down
And I do appreciate you being round
Help me, get my feet back on the ground

Here is a video of this song, with some clips from the movie. Enjoy!

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## Derivation of the volume of a sphere – method 2

In this blog here, I derived the expression for the volume of a sphere, $V= \frac{4}{3} \pi r^{3}$ using spherical polar coordinates. There is, however, another way to do it which some people may find easier, and that is to use something called the volume of rotation. Remember, a circle with radius $r$ which is centred on the origin has the equation $x^{2} + y^{2} = r^{2}$, but as I pointed out in my blog on deriving the area of a circle, because we can’t integrate $\sqrt{ \left( r^{2} - x^{2} \right) } \; dx$ we are stuck in trying to use conventional Cartesian coordinates, which is why we had to use polar coordinates instead.

However, if you are not comfortable with spherical polar coordinates to derive the volume of a sphere, there is a trick to get around using them. That is to use the volume of revolution. The idea is quite simple, imagine the strip which has area $dA = y \; dx$ as shown below.

Now, imagine rotating this trip about the x-axis, to produce a disk as shown in the figure below. The surface area of this disk is just the area of a circle with radius $y$, and so its area $dA = \pi y^{2}$. The volume element of the disk is then just $dV = dA \; \cdot dx = \pi y^{2} \; dx$, but as we can write $y^{2} = r^{2} - x^{2}$ this becomes $dV = \pi (r^{2} - x^{2}) \; dx$, which is easy to integrate.

This leads to the total volume $V$ being

$V = \int_{-r}^{+r} (r^{2} - x^{2}) dx = \left[ r^{2}x - \frac{x^{3}}{3} \right]_{-r}^{+r} = (r^{3} - \frac{r^{3}}{3}) - ( -r^{3} + \frac{r^{3}}{3})$

$V = 2r^{3} - \frac{2r^{3}}{3} = \; \boxed{ \; \frac{4}{3} \pi r^{3} }$

just as we had before. QED!

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## Wales crushed in the last 10 minutes

With the score at 15-13 to New Zealand, Welsh full-back Leigh Halfpenny slotted a penalty goal to put Wales 16-15 up, with just 13 minutes left of the match. Welsh fans dared to dream that we could beat NZ for the first time since 1953. Then, with only 11 minutes to go, NZ scored an opportunistic try, and by the time the match had ended they had stretched their lead to a 34-16 win, scoring 19 unanswered points in the last 11 minutes. What heartbreak for Wales; but hats off to New Zealand, this is why they are the best in the World.

This was a nail-biting match, with not an inch given by either side. After a thrilling first half, the score stood at 3-3; in stark contrast to the high scores one sees more and more in rugby these days. Wales had been under pressure for most of the first half, but the Welsh defence was immense. The lacklustre display against Fiji from a week ago was replaced by a level of commitment which was enthralling. Time after time the Welsh players put in the most crunching tackles on their opposition, and New Zealand could not find a way through despite all the possession and territory they were getting.

Early in the second half the try-deadlock was broken, with New Zealand scoring a try from a nice break to the left. Wales fly half Dan Biggar missed a tackle which he really should have nailed, and New Zealand were over. After converting the try they were 10-3 up; but Wales struck back within a few minutes with a lovely try by Rhys Webb, to level the scores at 10-10. Then, Wales were awarded a penalty to go 13-10 in the lead, before New Zealand struck back with another try, which they failed to convert. It was 15-13 to New Zealand before Halfpenny slotted the penalty on 67 minutes to put Wales into the narrowest of leads.

The try on 69 minutes which put NZ back in front was cruel on Wales, the All Black outside half Beauden Barrett chipped ahead and the awkward bounce caught Halfpenny going in the wrong direction, and the Kiwi was in under the posts. A few minutes later Mike Phillips had a clearance kick charged down and all of a sudden Wales were 29-16 behind, when only 5 minutes before they had been 16-15 up. Another unconverted try for NZ in the last few minutes left a final score of 34-16, but the scoreline didn’t indicate what a tight match it had been until the last 10 minutes.

So, heartbreak once again for Wales, and it is now 61 years and counting since we last beat New Zealand. It is difficult to see what else Wales could have done, they put in a huge effort but were undone in the last 10 minutes by a better team. NZ have the ability to absorb pressure, and then just move up another gear at the business end of the second half, and this is what they did on Saturday. In my opinion, Wales’ play didn’t significantly worsen in those last 10 minutes, although some substitutions did disrupt things. But, for me, it was more that NZ just applied that killer instinct that they have and moved to a level of play that was too good for Wales.

We now have to pick ourselves up for our final game of this Autumn series – the Springboks next Saturday. If Wales can put in a display like we did against NZ, and go into the last 10 minutes with a larger lead, I don’t see any reason why we cannot record our second ever victory over South Africa. We shall see!

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## Loving You – Minnie Riperton (song)

Today I thought I would share this song by Minnie Riperton – ‘Loving You’. I remember this song very vividly from my childhood, it was released in January 1975, and it is my understanding that it has one of the highest notes ever recorded on a pop song. It reached number 2 in the Disunited Kingdom, and number 1 in the US.

Here are the lyrics.

Lovin’ you is easy cause you’re beautiful
Makin’ love with you is all I wanna do
Lovin’ you is more than just a dream come true
And everything that I do is out of lovin’ you
La la la la la la la… do do do do do

No one else can make me feel
The colors that you bring
Stay with me while we grow old
And we will live each day in springtime
Cause lovin’ you has made my life so beautiful
And every day my life is filled with lovin’ you

Lovin’ you I see your soul come shinin’ through
And every time that we oooooh
I’m more in love with you
La la la la la la la… do do do do do

And here is a YouTube video of her performing the song (I’m not sure why she has a bush growing out of the side of her head, but this was the 1970s! 😛 ). Enjoy!

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## Birth of a new Saturn moon?

This story caught my attention back in April, the possible creation of a new Saturn moon on the outskirts of its famous ring system (I’m not sure why it has taken me so long to blog about it, probably because in April I was working frantically finishing up my book on the Cosmic Microwave Background). The particles in Saturn’s rings are basically small lumps of rocky water-ice, and so are pretty good at sticking together if they collide. What we think is happening is that this little moon is being created by the collisions of the particles in the ring system, which is slowly building up a new moon. Quite spectacular really, to see the possible birth of a new moon!

A story about the birth of a “new” moon about Saturn.

We don’t fully know the origin of Saturn’s rings, there are two competing theories. One is that a larger moon which would have formed at the same time as Saturn (i.e. 4.6 billion years ago) and at some time in the past came too close to Saturn and was torn apart by Saturn’s gravitational tidal forces. The point at which an extended body will be torn apart by tidal forces is called the Roche limit, and once an object is closer than this the gravitational forces on e.g. the near and the far side are too different and the object gets torn apart. The size of Roche limit depends on the planet (so would be different for the Earth and Saturn), but also on the size of the object. So, the Roche limit for a larger moon will be different (at a larger distance) than for a smaller object.

The other theory is that they formed from debris left over from Saturn’s formation, just as the asteroid belt is debris left over from the formation of the Solar System. The problem with this theory is that the rings are thought to be just a few hundred million years old, whereas if they are left over debris we would expect them to be as old as Saturn. As of writing this, we are not sure which of these two competing theories is correct. Or, maybe there is another explanation entirely!

Galileo looked at Saturn through his telescope back in 1610, but was not able to make out the rings as such. He saw that Saturn had what looked like “ears”, but it was the Dutch astronomer Christian Huygens who was the first to describe them as rings. They are a complex system, and it would take several blogs to describe them, but suffice it to say that in the time that we have been sending space probes past or to Saturn (e.g. Voyager, Pioneer, and now Cassini), we have been learning more and more about how complex a system the rings are.

Saturn is not the only planet with a ring system, it may surprise you to hear that Jupiter, Uranus and Neptune also have ring systems, but none is as spectacular as Saturn’s. Here is an amazing photograph taken by Cassini of Saturn and her ring system, looking back towards the Sun. It also shows several of Saturn’s moons, Mars, Venus and even the Earth!!

This amazing photograph, taken by NASA’s Cassini space probe, shows the rings back-lit by the Sun. The photograph also shows Mars, Venus and the Earth!

The Cassini probe has been an overwhelming success. Launched in 1997, it arrived at Saturn in 2004, and some of you may remember it sent a little probe called Huygens which landed on the surface of Titan in January 2005. For the last ten years it has been orbiting Saturn and some of its moons, and has made numerous discoveries in doing so. It is scheduled to remain in operation until 2017. Here are some remarkable images from the probe, together with a model of Enceladus based on the observations that Cassini has made of it.

A Cassini image of Enceladus, showing plumes of water coming from its pole.

One of the surprises of the Cassini mission has been the moon Enceladus, which shows geysers of water shooting out from its poles. Because of the presence of water, Enceladus has become a prime candidate to look for life beyond our Earth.

A Cassini image showing the rings nearly edge-on, with the moon Dione in the background.

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## The 500 Greatest Songs of all Time – no. 30 – “I Walk the Line” – Johnny Cash

As I’ve now finished counting down the 500 greatest albums according to Rolling Stone Magazine (see, for example, number 1 here), I thought I would next take a look at their 500 greatest songs. I am not going to count down the entire 500! Just the top 30, but for those of you interested the top 100 from 100 to 31 are

• 100 – Gnarls Barkley – ”Crazy” (2006)
• 99 – Credence Clearwater Revival – ”Fortunate Son” (1969)
• 98 – Al Green – ”Love and Happiness” (1972)
• 97 – Chuck Berry – ”Roll Over Beethoven” (1956)
• 96 – Jerry Lee Lewis – ”Great Balls of Fire” (1957)
• 95 – Carl Perkins – ”Blue Suede Shoes” (1956)
• 94 – Little Richard – ”Good Golly, Miss Molly” (1958)
• 93 – U2 – ”I Still Haven’t Found What I’m Looking For” (1987)
• 92 – Ramones – ”Blitzkrieg Bop” (1976)
• 91 – Elvis Presley – ”Suspicious Minds” (1969)
• 90 – The Five Satins – ”In the Still of the Night” (1956)
• 89 – The Mamas and the Papas – ”California Dreamin” (1965)
• 88 – The Temptations – ”My Girl” (1965)
• 87 – Johnny Cash – ”Ring of Fire” (1963)
• 86 – Bruce Springsteen – ”Thunder Road” (1975)
• 85 – Patsy Cline – “Crazy” (1961)
• 84 – The Police – ”Every Breath You Take” (1983)
• 83 – The Beatles – ”Norwegian Wood (This Bird Has Flown)“ (1965)
• 82 – Fats Domino – ”Blueberry Hill” (1956)
• 81 – Marvin Gaye – ”I Heard it Through the Grapevine” (1968)
• 80 – The Kinks – ”You Really Got Me” (1964)
• 79 – The Byrds – ”Mr. Tambourine Man” (1965)
• 78 – James Brown – ”I Got You (I Feel Good)“ (1965)
• 77 – Elvis Presley – ”Mystery Train” (1955)
• 76 – The Beatles – ”Strawberry Fields Forever” (1967)
• 75 – Led Zeppelin – “Whole Lotta Love” (1969)
• 74 – Eddie Cochran – “Summertime Blues” (1958)
• 73 – Stevie Wonder – “Superstition” (1972)
• 72 – The Beach Boys – “California Girls” (1965)
• 71 – James Brown – “Papa’s Got A Brand New Bag” (1966)
• 70 – Dionne Warwick – “Walk On By” (1964)
• 69 – Roy Orbison – “Cryin'” (1961)
• 68 – Bob Dylan – “Tangled Up in Blue” (1975)
• 67 – Elvis Presley – “Jailhouse Rock” (1957)
• 66 – Bob Marley and the Wailers – “Redemption Song” (1980)
• 65 – Cream – “Sunshine of Your Love” (1968)
• 64 – The Beatles – “She Loves You” (1963)
• 63 – Buffalo Springfield – “For What It’s Worth” (1967)
• 62 – Bo Diddley – “Bo Diddley” (1955)
• 61 – Jerry Lee Lewis – “Whole Lotta Shakin’ Going On” (1957)
• 60 – Al Green – “Let’s Stay Together” (1971)
• 59 – Bob Dylan – “The Times They Are-A-Changin'” (1964)
• 58 – Michael Jackson – “Billie Jean” (1983)
• 57 – Procol Harum – “Whiter Shade of Pale” (1967)
• 56 – The Sex Pistols – “Anarchy in the U.K.” (1977)
• 55 – Little Richard – “Long Tall Sally” (1956)
• 54 – The Kingsmen – “Louie Louie” (1963)
• 53 – Percy Sledge – “When a Man Loves a Woman” (1966)
• 52 – Prince – “When Doves Cry” (1984)
• 51 – Grandmaster Flash and the Furious Five – “The Message” (1982)
• 50 – Smokey Robinson and the Miracles – “The Tracks of My Tears” (1965)
• 49 – The Eagles – “Hotel California” (1976)
• 48 – Simon and Garfunkel – “Bridge Over Troubled Water” (1970)
• 47 – The Jimi Hendrix Experience – “All Along the Watchtower” (1968)
• 46 – David Bowie – “Heroes” (1977)
• 45 – Elvis Presley – “Heartbreak Hotel” (1956)
• 44 – Ray Charles – “Georgia on My Mind” (1960)
• 43 – Little Richard – “Tutti-Frutti” (1955)
• 42 – The Kinks – “Waterloo Sunset” (1968)
• 41 – The Band – “The Weight” (1968)
• 40 – Martha Reeves and the Vandellas – “Dancing in the Street” (1964)
• 39 – Buddy Holly and the Crickets – “That’ll Be the Day” (1957)
• 38 – The Rolling Stones – “Gimme Shelter” (1969)
• 37 – Bob Marley – “No Woman, No Cry” (1975)
• 36 – U2 – “One” (1991)
• 35 – The Doors – “Light My Fire” (1967)
• 34 – The Righteous Brothers – “You’ve Lost That Lovin’ Feelin'” (1964)
• 33 – Ike and Tina Turner – “River Deep – Mountain High” (1966)
• 32 – The Rolling Stones – “Sympathy for the Devil” (1968)
• 31 – Led Zeppelin – “Stairway to Heaven” (1971)

There are some great songs in this list from 100 to 31, and I will blog about some of them over the coming months.

But, to start our countdown from 30 to 1, at number 30 in their list is ‘I Walk the Line’ by Johnny Cash. I didn’t really know much about Johnny Cash or his music until I saw the great 2005 movie about him, which is called ‘Walk the Line’ and stars Joaquin Phoenix and Reese Witherspoon. If you like this song, I recommend watching the movie.

At number 30 in Rolling Stone Magazine’s ‘500 Greatest Songs of all Time’ is “I Walk the Line” by Johnny Cash.

This song was written by Cash, was released in May 1956 and it reached number 17 in the US Billboard charts, and number 1 in their country music charts.

I keep a close watch on this heart of mine
I keep my eyes wide open all the time
I keep the ends out for the tie that binds
Because you’re mine, I walk the line

I find it very, very easy to be true
I find myself alone when each day is through
Yes, I’ll admit that I’m a fool for you
Because you’re mine, I walk the line

As sure as night is dark and day is light
I keep you on my mind both day and night
And happiness I’ve known proves that it’s right
Because you’re mine, I walk the line

You’ve got a way to keep me on your side
You give me cause for love that I can’t hide
For you I know I’d even try to turn the tide
Because you’re mine, I walk the line

I keep a close watch on this heart of mine
I keep my eyes wide open all the time
I keep the ends out for the tie that binds
Because you’re mine, I walk the line

Here is a video of this great song. Enjoy!

Which is your favourite Johnny Cash song?

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## Space debris and geostationary satellites

Last week, as part of the “tech hour” on BBC Radio 5’s “Afternoon Edition” on Wednesdays, I heard one of their reporters do an item on visiting a facility in Australia which is trying to track space debris and locate their positions precisely. After the item a listener asked about geostationary orbits, and it was clear from the reporter’s answer that he did not really understand them. Then again, if he is not a trained physicist or engineer, why should he?

In the light of the wrong answer given to the listener by the reporter, I’ve decided to blog about satellite orbits to clear up any confusion, and also to show that the mathematics of it is not that hard. This is the kind of problem that a student studying physics would be expected to do in their last year of high-school here in Wales. Mark Thompson, one of the astronomers who has appeared on BBC Stargazing Live, did clear up some of the confusion when he came on the programme later in the same hour, but he did not have time to go into all the details (nor would he want to do so on a radio show!).

## Low-Earth and geostationary orbits

For those of you not familiar with the terminology, a low-Earth orbit is typically just a few hundred kilometres above the Earth’s surface, which as I will show below leads to an orbit about the Earth which takes a couple of hours at most. These are the kinds of orbits which are used by most spy satellites, and also by the International Space Station (ISS) and, when it used to fly, by the Space Shuttle. The famous Hubble Space Telescope is in a low-Earth orbit. Any satellites in low-Earth orbits will experience drag from the Earth’s atmosphere.

Space “officially” begins at 100km above the Earth’s surface, but the atmosphere doesn’t suddenly stop at any particular altitude,it just gets thinner and thiner. In fact, the “scale height” of the atmosphere is about 1 mile, or about 1.5km. For each 1.5km altitude, the amount of atmosphere roughly halves (based on pressure), so at an altitude of 3km the air is $0.5 \times 0.5 = 0.25 (25\%)$ as thick, and if you go up to 10.5km which is roughly the altitude at which commercial aeroplanes fly, then the thickness of the atmosphere is about $0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 \times 0.5 = 0.5^{7} = 0.0078$ as thick, or $0.78 \%$ of the thickness at the ground. Even though the thickness of the atmosphere a few hundred kilometres up is very very thin, there is still enough of it to slow satellites down. So, satellites in low-Earth orbits cannot maintain their orbits without some level of thrust, otherwise their orbits would just decay and they would burn up in the atmosphere.

Low-Earth orbit satellites can be in any orbit as long as it is centred on the Earth’s centre. So, for example, many spy satellites orbit the Earth’s poles. If you think about it, if they are taking about 90 minutes to orbit (some take less, they will be even closer to the Earth), then each time they orbit they will see a different part of the Earth, which is often what you want for spy satellites. The ISS orbits the Earth in an orbit which is titled to the Earth’s equator, so it passes over different parts of the Earth on each orbit, but it is not a polar orbit. A satellite cannot, for example, orbit, say, along an orbit which is directly above the tropic of Cancer, and satellites do not orbit the Earth in the opposite direction to the Earth’s rotation, they all orbit the Earth in the same direction as we are rotating.

Geostationary orbits, on the other hand, are a particular orbit where the satellite takes 24 hours to orbit the Earth. So, as seen by a person on the rotating Earth, they appear fixed relative to that person’s horizons. Geostationary orbits are used for communication satellites so that, for example, your TV satellite dish can be fixed to the side of your house and point in the same direction, as the satellite from which it’s getting the TV signals stays fixed in your sky. They are also used by Earth-monitoring satellites such as weather satellites and satellites which measure sea and land temperatures. A geostationary satellite has to be at a particular altitude above the Earth’s surface, and it has to be in orbit about the Earth’s equator (so it sits above the Earth’s equator), and of course orbits in the same direction as the Earth’s rotation.

How do we calculate the altitude of these respective orbits, the low-Earth and the geostationary ones? It is not too difficult, as I will show below.

## The altitude of a satellite which takes 90 minutes to orbit the Earth

To make the mathematics simpler, I will do these calculations assuming a circular orbit. In reality, some satellites have elliptical orbits where they are closer to the Earth’s surface at some times than at others, but that makes the maths a little more complicated, and it’s not necessary to understand the basic ideas.

As I showed in this blog, when an object is moving in a circle it must have a force acting on it, because its velocity (in this case its direction) is constantly changing. This force is called the centripetal force, and as I showed in the blog this is given by

$F_{centripetal} = \frac{ mv^{2} }{ r }$

where $m$ is the mass of the object moving in the circle, $v$ is its speed, and $r$ is the radius of the circle.

For a satellite orbiting the Earth, the centripetal force is provided by gravity, and the force of gravity is given by

$F_{gravity} = \frac{ G M m}{ r^{2} }$

where $M$ is the mass of the Earth (in this case), $m$ is the mass of the satellite, $r$ is the radius of the orbit as measured from the centre of the Earth, and $G$ is a physical constant, known as Big G or the universal gravitational constant.

Because the centripetal force is being provided by gravity, we can set these two equal to each other and so we have

$\frac{ mv^{2} }{ r } = \frac{ G M m }{ r^{2} } \rightarrow v^{2} = \frac{ GM }{ r } \text{ (Equ. 1) }$

The speed of orbit, $v$ is related to how long the satellite takes to orbit via the definition of speed. Speed is defined as distance travelled per unit time, and as we are assuming circular orbits the distance travelled in an orbit is the circumference of a circle. So, we can write

$v = \frac{ 2 \pi r }{ T }$

where $T$ is the time it takes to make the orbit. Substituting this expression for $v$ in Equation (1) we can write

$\left( \frac{ 2 \pi r }{ T } \right)^{2} = \frac{ GM }{ r }$

Remember we are trying to calculate $r$ for a satellite which takes 90 minutes to orbit, so we re-arrange this equation to give

$r^{3} = \frac{ GM T^{2} }{ 4 \pi^{2} } \text{ (Equ. 2)}$

Notice that the only two variables in this equation are $r \text{ and } T$, everything else is a constant (for orbiting the Earth, obviously if the object were orbiting e.g. the Sun we’d replace the mass $M$ by the mass of the Sun). Equation (2) is, in fact, just Kepler’s Third law (which I blogged about here), which is that

$\boxed{ T^{2} \propto r^{3} }$

Now, to calculate $r$ for a satellite taking 90 minutes to orbit, we just need to plug in the values for the mass of the Earth ($5.972 \times 10^{24}$ kg), the value of $G=6.67 \times 10^{-11}$, and convert 90 minutes to seconds which is $T=5400 \text{ s}$. Doing this, we get

$r^{3} = 2.942 \times 10^{20} \rightarrow r = 6.65 \times 10^{6} \text{ m}$

But, remember, this is the radius from the Earth’s centre, so we need to subtract off the Earth’s radius to get the height above the Earth’s surface. The average radius of the Earth is $R = 6.37 \times 10^{6} \text{ m}$ so the height above the Earth’s surface of a satellite which takes 90 minutes to orbit is

$6.65 \times 10^{6} - 6.37 \times 10^{6} = 281 \times 10^{3} \text{ m } \boxed{ = 281 \text{ km} }$

which is roughly 300km.

## The altitude of a geostationary satellite

To calculate the altitude of a geostationary satellite we just need to replace the time $T$ in equation (2) with 24 hours (but put into seconds), instead of using 90 minutes. 24 hours is $24 \times 3600 = 8.64 \times 10^{4} \text{ s}$, and so we now have

$r^{3} = 7.53 \times 10^{22} \rightarrow r = 4.22 \times 10^{7} \text{ m}$

Again, subtracting off the Earth’s radius, we get the altitude (height above the Earth’s surface) to be

$4.22 \times 10^{7} - 6.37 \times 10^{6} = 3.59 \times 10^{7} \text{ m } \boxed{ = 35,900 \text{ km} }$

which is roughly 36,000 km.

## Summary

It surprises most people that geostationary satellites are so far above the Earth’s surface, some 36,000 km. There is no atmosphere at that altitude, so geostationary satellites really pose no threat in terms of man-made space debris. Their orbits will not decay, as there is no atmospheric drag. It is the low-Earth orbit satellites which are the source of our man-made space debris, and if their orbits are not maintained the orbits will decay and they will burn up in the Earth’s atmosphere. Of course, most satellites entering the thicker parts of the atmosphere burn up completely, but larger satellites have been known to have parts of them survive to hit the Earth’s surface. For example this is what happened to the Russian space station MIR when its orbit was decayed by switching off its thrusters.

The time it takes a satellite to orbit the Earth is entirely dependent on how far from the centre of the Earth the satellite is. Low-Earth orbit satellites orbit the Earth in a matter of hours, many less than two hours. Geostationary satellites have to be about 36,000 km from the surface of the Earth in order to take 24 hours to orbit. All satellites orbit the Earth in the same direction as the Earth is rotating, except for satellites in a polar orbit which essentially allow the Earth to rotate below them as they go around the poles. Geostationary satellites have to orbit above the Earth’s equator, but other satellites can be in an orbit which is inclined to the equator, as long as the centre of their orbit is the centre of the Earth.

Finally, coming back to the issue of space debris; one of the reasons that space debris is so dangerous to other satellites and astronauts is that objects in Earth-orbit are moving very quickly. For example, for an object taking 90 minutes to orbit the Earth, such as the orbit the ISS is in, we can work out how quickly (in metres per second) it is moving quite easily.

If $T=90 \text{ minutes } = 5400 \text{ s}$ and the radius of the orbit (as we showed above) is $r = 6.65 \times 10^{6} \text{ m }$ then

$v = \frac{ 2 \pi \cdot 6.65 \times 10^{6} }{ 5400 } \; \; \boxed{= 7737.6 \text{ m/s } }$

so, nearly 7,740 m/s or 7.7 km/s or nearly 28 thousand km/h, which by anyone’s standards is FAST!

The damage a piece of space debris can make is dependent on its kinetic energy (KE), which is given by

$\text{Kinetic energy } = \frac{ 1 }{ 2 } mv^{2}$

An object with a mass of 1kg moving at 7,740 m/s will thus have a KE of

$KE = \frac{ 1 }{ 2 } 1 \cdot (7740)^{2} = 30.0 \times 10^{6} \text{ Joules}$

(or 30 Mega Joules). This is a huge amount of energy, because the object is moving so fast. By comparison, a car, which we will assume to have a mass of 1,000 kg, moving at 100 km/h will have a kinetic energy of $0.77 \times 10^{6} \text{ Joules}$, or nearly 0.8 Mega Joules, so about a factor of 40 (forty!) less than a 1kg object in low-Earth orbit. This is why such small objects can cause such damage!

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## Should Ched Evans be allowed to return to football?

It was announced last week that convicted rapist Ched (Chedwyn) Evans, who has served half of a 5-year sentence for raping a teenage girl, is to start training again with his old club Sheffield United. Towards the end of last week I posted this screen-capture from the BBC news story on my Facebook account, with the question “Should Ched Evans be allowed to return to football?”. Quite a few interesting comments ensued, so I thought I would post the same question here on my blog.

Ched Evans, a convicted rapist, is to start training again with his old club Sheffield United

To give a little of the background to those reading this who know nothing of this story, Ched Evans, who played professional football for Sheffield United and Wales, was convicted in April 2012 of raping a 19-year old woman in a hotel room in North Wales in May 2011. The details of the rape are rather sleazy, but if you wish you can read more about them here. He served half of his 5-year sentence (being released early is normal for good behaviour), and he was released a few weeks ago. Evans maintains his innocence, and although so far his appeals to the legal system have failed, he does have a “judicial review” in motion, which will re-examine the evidence on which he was convicted. In addition to maintaining his innocence, he has so far shown no remorse for what happened to the 19-year old woman.

It would seem that the Professional Footballers’ Association, the players’ union, have put pressure on Sheffield United to allow him back to play there. The club have not announced what they will do longer-term, but for the present they have allowed him to come back and train with them. As a consequence of their decision to do this, several trustees of the club have resigned, and some of their sponsors have threatened to withdraw their sponsorship. Olympic gold medallist Jessica Ennis-Hill, after whom a stand is named at the club’s ground, has said that if he is re-signed by the club she will ask for her name to be removed from being associated with the club.

What do you think? Should a convicted rapist be allowed to return to playing professional football? He has served his sentence, so shouldn’t he be allowed to just get on with his former life as he has “paid his debt to society”? Or, given the nature of his crime, should he be precluded from returning to such a high profile (and well-paid) career?

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