Counting Photons (How many astronomers does it take to make a cup of tea?)


This article continues the theme set in "The Scale Of The Universe" and attempts to draw a real world analogy around an astronomical concept.

Most of us have also heard tell of photons. None of us have ever seen a single photon... it doesn't work like that. It is beyond the scope of this article to explain what a photon is in detail and how it gets produced. We ony need a few background comments. Look at the lightbulb in your room. Vagaries in local power supply permitting, it appears that the light issues forth in one continuous smooth stream of energy. Quantum theory tells us that its not quite as simple as it seems. Energy comes in little indivisible lumps called photons. Like atoms and matter, light comes in chunks. One photon is one quantum. Your lightbulb is constantly spraying you and the room with little packets of energy. Quite a lot of little packets of energy in fact.

What about light waves? Well, this is were it starts to get weird. Some ways of examining light (eg with a diffraction grating) will show the wave nature of light, whereas other ways (with a ccd camera for example) show the lumpy nature of light. Light does both at the same time.

Photons are, as mentioned above, extremely small lumps. So much so, that, like a river of water, it's hard to imagine it being made up of lots of little bits. Later, we will explore how many of these little blobs of light are involved in our astronomical activities.

How Big is My Telescope?

All of us, unless we've had a misfortunate illness or accident, come with some built in astronomical detectors... eyeballs. Most astronomers soon tire of these, and aspire to something a lot larger. The reason for this is our lumpy photons. The wider the aperture of your device, the more photons you collect. The more photons you get, the more chance you have of seeing something useful.

A typical amateur instrument will have an aperture of about 6 to 8 inches (150mm to 200mm). If we estimate a dark adapted eyeball has an aperture of 6mm, then our 200mm scope collects in the region of one thousand times more photons than the Mark One eyeball.

The most advanced professional instruments are somewhere in the region of 8 metres across. So, do the sums. An 8 metre scope collects around 1600 times more light than our amateur instrument. Start comparing it to an eyeball, and the numbers get very meaningless very quickly.

I bet you thought the ratio was going to be a lot bigger than that? I certainly did. I've often wondered how many amateur astronomers there are in the world with telescopes. It's certainly hundreds of thousands, if not millions. It is a question that cannot be answered, but I'm sure it's more than 1600.

This demonstrates that the light gathering power of the amateur community is vastly greater than that of the professional astronomers. Later on we will see what happens if we tried to link all these amateur scopes together...

How hot is a photon?

Photons carry energy. As you may recall from school, energy is measured in Joules, calories, ergs or a number of other obscurely named units. We will stick to the "modern" unit - the Joule. One Joule is not a lot of energy. An ice cube from a cold drink will melt quite readily in a mouth or hand (or other part of the anatomy). However, to melt the average icecube, you need about ten thousand joules all working together.

Joule himself, for those of you who like to know about such things, was an English physicist who discovered the first law of thermodynamics which describes the conservation of energy.

A Joule is just a measure of an arbitrary amount of energy. Photons are little lumps full of energy. The obvious question is: How many joules in a photon? Answer: Not many. The correct answer is more involved. Light comes in different wavelengths - this gives rise to the colours we see. The energy content of one lump of light depends on its wavelength. Trust me, please, lumps have a wavelength. If we tried to continue this thread taking into account all these different wavelengths things would get horribly mathematically involved. So we won't. Henceforth, all light is cyan (blue/green), and comes in at about 500 nanometres.

Whats a nanometre? Its very small, and for the sake of this argument, utterly unimportant, so forget I said it.

One photon of cyan light has the following energy content:

0.000000000000000000397 Joules.

To save trees, such numbers are written as 3.97x10E-19 Joules. This number is, as you will have spotted, utterly meaningless to the average brain. How many photons would you need to make up a Joule? Answer: Many. Let us scale up each photon into a little cube 1 millimetre on each side. Gather together enough of these 1mm photons to make up a Joule of energy. Put them end to end. Where would the line reach? The Earth to the Sun ? Oh yes. Back again? Yes. The line would be long enough to do the Earth-to-the-sun and back journey over 8000 times - ie 16000 astronomical units. Thats a lot of photons. I did say they were very small. The total number of "photons" is 2.5x10E18. Thats 2.5 million million million photons.

I tend to find the astronomical unit quite a large item to deal with - and trying to swallow 16000 of them at once is beyond me. Photons are so small, and carry such a tiny amount of energy, it's very hard to imagine them in a real world context. Instead of putting all our photons end to end, we try to construct a big cube out of them. This is a useful exercise, as using cubes tends to make the numbers a lot smaller. The sides of this cube would be a far more comprehensible 1.3km long. Thats a box 1.3km long by 1.3km wide by 1.3km high, filled with little 1mmx1mmx1mm cubes.

So, how many photons do we get in our telescope?

Let us start this off with something close to home - our lightbulb again. My lightbulb is rated at 100 Watts. Whats a watt? A Watt is a measurement of the rate of flow of energy. 1 Watt is 1 Joule per second. Ignoring heat conduction and efficiency, my light bulb generates 100 joules of light per second.

Not all these photons are hitting me. They are flying out of my lightbulb in all directions. Doubtless some of them are bouncing off to annoy a nearby astronomer. Or not, looking at the clouds. Now we bring the telescope back into the equation. I put my telescope 100 metres from my lightbulb and count how many photons fall down its 200mm aperture. Working out the sums, we find that of all the photons coming out of the lightbulb, only one in 4 million reach my telescope. Mind you, thats still 6.3x10E13 Photons per second.

For the mathematically curious, it's worked out like this...
Energy in a photon = 3.97x10E-19 Joules
Photons in a Joule = 2.52x10E18
100 Watts = 2.52x10E20 photons per second.

Now, those photons are assumed to be flying out of the lightbulb equally in all directions. We are 100m from the lightbulb. So lets draw an imaginary sphere 200m across with the lightbulb in the middle.
This sphere has a surface area of 100*100*PI*4 = 125640 square metres.

The surface area of our telescope is 0.03141 square metres.
So, of all the photons flying out of the lightbulb, we will only catch one in 125640/0.03141 = 4000000.
2.52x10E20 photons per second divided by 4 million = 6.3x10E13 photons per second.

Let's try our Sun. The Sun generates about 4x10E26 Watts. We all know not to point our telescopes at the sun without a "photon filter" on the front... so lets move our Sun to a safe distance - 200 light years - the distance to the star Mirach in Andromeda. Jiggle the sums, and the number of photons that reach my telescope is about 700000 photons per second. Happily we now have a more comprehensible number. How large is our box of 1mm "photon" cubes now? It works out to about 10cm a side.

Now, how about a nice cup of tea?

Any tea lover will know that you need boiling hot water to make tea. A nice mug of tea needs about 300 millilitres of boiling water. To do this you need 125000 Joules of energy. Looking up the page we see our lightbulb kicks out enough photons to boil our cup of tea in just over 20 minutes. Sadly, due to efficiency and so forth, it doesn't always work out like this - hence the need for kettles. Regardless of this, we will use this number anyway.

So, now to answer the question. How many astronomers does it take to make a cup of tea? Well, each of our 8 inch telescopes are collecting 700000 photons a second from the distant star.

700000 photons per second = 2.779*10E-13 joules per second.
Energy gathered by 1 million such telescopes = 2.779*10E-7 joules per second.
Time take to make cup of tea = 140,000 years!

So next time you are waiting for the kettle, just stop and think!

Imaging with CCD Cameras

CCD camera are quite simple devices. They have a grid of light sensitive elements which can collect photons and turn them into an electrical signal for our computers to enjoy. These photons come a long way to reach our cameras. Imagine a galaxy like our Milky Way situated 50 million light years away. The total power output of such galaxies is in the region of 5x10E36 Watts. How many of those photons finally reach our telescope? Applying the maths, it is about 150000 per second.

A galaxy is not a point source. A CCD picture of this nice circular galaxy might be 200 pixels across. This is an area of about 30,000 pixels. So each pixel is getting about 5 photons per second from the distant galaxy.

CCD Cameras have limits. A pixel can only collect so many photons before it's "full". At which point the pixel is saturated, and stops giving a useful result. Modern CCD cameras can collect about 80,000 photons per pixel before they saturate. So we can expose our galaxy for about 4 hours before the picture becomes a solid white blob.

The light from this galaxy is of a similar order of magnitude to that from the star used above to make the cup of tea. In terms of human enterprise, making a cup of tea is not considered a undertaking of global proportions - we have far more energy hungry processes at work. Try and calculate how much water you could boil using the energy consumed driving to work. Mankind has been observing distant galaxies with telescopes for over 100 years - but gather together all the photons from all these observations, both visual and photographic and you cannot even make a single cup of tea.

Please note

The maths used in these examples is quite rough. The different energies of varying wavelengths of light will affect things by at least a factor of 2. Also, CCD cameras are not 100% efficient - they do not convert every photon into a signal. Another factor is dust - this tends to get in the way of photons and block some of them out. However, the accuracy is sufficient to illustrate the point