O.K. so this is getting quite a ways off deep-sky imaging, but I just clicked on a site that I thought was going to tell me about ultra-black materials and I was instead treated to a monologue of how the United States has already reverse-engineered alien technology (alien being defined as some entity from another world in this instance) for its own use.  This instantly reminded me of a hilarious incident on the TV some 15 or 20 years ago.

I was watching a programme on the same subject, basically how alien technology was being utilised by the United States government – and part of this programme was a live link to the States and a discussion with a well known Professor who was supporting the “alien technology” thesis.  He was there telling us how the strange craft coming out of Area 51 were the direct result of reverse-engineering crashed extra-terrestrial vehicles when……… suddenly…….. we lost the TV link.  Now call me a sceptic, but I have a real problem understanding how a country that has already reverse-engineered flying saucers for its own military use can possibly have trouble maintaining a video link for a few minutes.  Or maybe the break in communications was from our side

I have asked some friends to put up the Golden Solid angle on their sites to try and find where this might occur in Nature.  Some people in trying to help with a reply have gone astray with both the mathematics involved (which aren’t that complex) and the concept.  So here I will try to explain a little more about the Golden Solid angle (and solid angles in general as this doesn’t seem to be a generally understood concept).

An ordinary (planar) angle is defined by considering a circle of radius r (make r=1 for simplicity).  Now consider a length of arc on the circumference of the circle of length L, this will subtend a planar angle at the centre of the circle defined as L/r = L radians.  Now the total circumference of a circle is 2 x Pi x r so that if again we have r=1, then the total angle about the central point of a circle is 2Pi radians.  2Pi radians is therefore equivalent to 360 degrees, Pi radians is equivalent to 180 degrees, and Pi/2 radians is a right angle.  So far so good I hope.

Now let’s move onto the slightly more involved concept of a SOLID angle.  This is no more difficult in reality to the planar angle, it’s just that we don’t use it much (if at all) in every day life.

The unit of solid angle is the STERADIAN and it is defined as follows.  Consider a sphere of radius r, and consider some area on the surface of the sphere of area A.  Then the solid angle subtended by the area A at the centre of the sphere is A/r x r steradians.  The total surface area of a sphere of radius r is 4 x Pi x r x r so by using our definition of solid angle we see that the total solid angle about a point is 4 x Pi x r x r / r x r or simply 4Pi steradians (this is precisely why 4Pi turns up in the permeability of free space – but that’s another story).

Solid angle in steradians (or in square degrees) is of importance to astronomers too as it gives an indication of the size of an object in the sky – but as solid angle isn’t generally understood this also means that the apparent size of objects in the sky is also not well-understood.  When astronomers say that the Sun and Moon subtend about half a degree – they are talking PLANAR degrees and that the Sun and Moon are about half a degree in (planar) diameter.  That’s fair enough, but to put things into perspective we should know what looking out into one hemisphere means in terms of steradians (or square degrees) as it is only by looking at the “sphere of space” above us in this way that we can get some measure of how BIG our total field of view is.  A hemisphere is 2Pi steradians and if we convert this to square degrees we can get some idea of how big the celestial sphere is for an observer with a telescope with a typical field of view of 1 square degree.

We can go back to our PLANAR definition of angle to work this one out.

Pi radians = 180 degrees, so

Pi x Pi steradians = 180 x 180 square degrees, so

4Pi steradians = whole celestial sphere = 4 x 32,400/Pi square degrees  = 41,252.96 square degrees, so

2Pi steradians = celestial hemisphere = 20,626.48 square degrees.

So our observer with a 1 square degree field of view would have a roughly 1 in 20,000 chance of randomly hitting a selected object – it gives an indication of how BIG it is up there!

As a corollary:  1 steradian = 3,282.81 square degrees or equivalently 1 square degree = 3E-4 steradians.

Returning back to the Golden Solid angle!

We now consider a sphere whose surface area has been divided into two, one of area unity and one of area phi (the golden ratio or 1.618…) and the unity surface area will subtend some solid angle, let’s call it gamma, at the centre of the sphere.  In exactly the same way we define the Golden Ratio on the line, or the Golden angle for the circle, we can come up with an equation for the Golden Solid angle for the sphere:

(4Pi – gamma)/gamma = 4Pi/(4Pi – gamma) which is a quadratic in gamma which can be solved in the usual way to give:

gamma = 1.52786Pi steradians or 15757.2 square degrees.  If you (for whatever reason) wanted to take a slice through the sphere to see what PLANAR angle this solid angle corresponded to  you would get an angle of  152.7 degrees – though I’m not sure what use this information is except that it is NOT the same as the Golden planar angle of 137.5 degrees.

For completeness:  The solid angle corresponding to the Golden angle of 137.5 degrees is 1.275Pi steradians.

So I return to my original question – anyone seen the Golden Solid angle anywhere in Nature???

There’s plenty of clouds at the moment but I’m hoping for just a few gaps in the cloud tonight.  No Moon – and it should be the peak in the Geminid meteor shower (13th – 14th December 2009).  Went out for a bit of practice last night with the 40D piggy-backed on the C11 – just as well really as every picture was out of focus!  I know how to sort that particular problem out tonight – if it decides to clear.  Rather than squinting at the little dim LCD on the back of the 40D I shall hook up the laptop in “Remote Shooting” mode and use the Remote Liveview plus the magnifier to carefully focus (also through the laptop so no fumbly paws trying to do the job) the 40D.  When satisfied with the focus flick the switch from autofocus to manual (so things don’t change) and trigger the remote timer to take the frames.  It’s as much of a pallava as doing “real” deep-sky imaging with the main scope.  Please let’s have at least a couple of clear hours and plenty of meteors tonight – pretty please

Well as the weather is not allowing me to take outer-space images, it’s back to some inner space work again.  This time an opalescent beetle leg gets photomicrographed using a 23-frame focus stack put together using the Helicon Focus software.  SEM-like depth of field, but in full colour!

The Golden Ratio (and the closely associated Fibonacci series) makes many appearances in the “living world” – here’s my question – not including Mathematics and man-made objects, does the Golden Ratio appear naturally in any inorganic systems?  There is a link between quasicrystals and the Golden Ratio, but I’m looking for a more direct link than these.  Once again, does anyone out there know of a clear example of the Golden Ratio making an appearance in a non-organic system?  If you do – please let me know ASAP

It will not have been mentioned before in this blog, but I like certain aspects of pure mathematics as much as I like deep-sky imaging.  I think most people will have heard of the Golden-Section, or the Golden-Ratio, and how it can be obtained by dividing a straight line up into two sections one of length unity, and the other of length tau or 1.61803398…  What is less well-known is that if you wrap the line round into a circle, so the circle perimeter is divided into lengths of unity and 1.618, then then angle subtended by the unity length of the perimeter at the centre of the circle is 137.507 degrees – or the Golden Angle.

That’s where the story seems to have been left, for a very long time, but I have to wonder, why?  We started with a line (one-dimension), then moved to a circle (two-dimensions), where’s the spherical case (3-dimensions)?  I did a long search a couple of years back and couldn’t find anything on this.  So I wrote a paper on “The Golden Solid Angle” for the Mathematical Gazette, which was in fact turned down as “although the result was new, just having a new result is not necessarily having something worthy of publication” – well that’s a new one for me!  So wishing to stake my claim as the discoverer of the Golden Solid Angle (sent to the Mathematical Gazette on Thursday 14th June 2007) here’s the thing explained for the first time below.

Divide the surface of a sphere into two regions, one of surface area unity, and the other of surface area 1.618…  The surface area of unity will subtend a solid angle gamma at the centre of the sphere.  By noting the total solid angle about a point is 4Pi Steradians, we can derive the following equation for gamma:

(4Pi – gamma)/gamma = 4Pi/(4Pi – gamma)

Giving a quadratic in gamma which can be solved in the usual way to give:

gamma = 1.52786Pi Steradians or 15757.2 square degrees.

Question is, does anyone out there know where the Golden Solid Angle, gamma, makes an appearance in the Natural world (or basically, anywhere)?  If you do then please let me know ASAP

Clear last night (although with a nearly full Moon) and managed to get a very seldom imaged iridescent reflection nebula appropriately named “The Butterfly nebula” in Orion.

O.K. it’s a picture of the Morpho Rhetenor butterfly wing (structural colour) but I can pretend I was actually out imaging for the first time in over a month.  It was certainly clear last night – but there was plenty of thin high cloud and the biggest light polluter in the sky made it pointless to set up

Great day today (no – not the weather!) – put the side panels on the mini-WASP array, and Brian May sent me a copy of his new book “A Village Lost & Found”.  This is a beautiful publication and you can see the attention to detail that has gone into every part of creating this book.  Wonderful job Brian – well done – would make the Thesis look like an almost trivial exercise by comparison

With the side panels now on, it’s a matter of getting the second observatory built and buying the cameras for the two Sky 90s and the guide scope.

I bolted the mini-WASP telescope frame to the Paramount versa-plate to get a feel what the finished system will look like.  The funny-looking device sticking out the front of the frame is a counterbalance arm for the weight of all the equipment that will sit on the back of the telescopes.  I have left the two side panels off in this photo.  I have also left out the 2 Sky 90s from the bottom pair of holes as I need to remove the Robofocus units from them before fitting to the frame – I might do this tomorrow and update the photos.  The two holes at the top are for 2 x FSQ106 telescopes to be bought at some future (unknown) date, and these will be used for narrowband imaging.

This is going to be some beast to set up prior to an imaging run and I’m glad I didn’t make provision for any more telescopes as I don’t think it would be viable to get them all sorted without losing a lot of good imaging time per session.

Just had two new publications for 2010 come through the post this morning – the Astronomy Now 2010 Yearbook & Patrick Moore’s 2010 Yearbook of Astronomy.  Why?  Because Noel & I managed to get an “Astronomy Now readers’ pictures of the year” with our February 2009 comet Lulin image and I wrote an article for Patrick’s 2010 Yearbook called “Hyperstar imaging at the New Forest Observatory” which includes Noel/Greg images both from the Hyperstar III and from the Sky 90/M25C.  I would also like to thank Keith Cooper for including Star Vistas in the “Books of the Year” section of the Astronomy Now 2010 Yearbook – thank you Keith

Well it was very nice to receive the above and get a little lift from this absolutely foul weather we’ve been suffering.  Monsoons and gales means I haven’t taken an image for a very long time now – the last one was a frame for the Heart nebula – and that was on Sunday 25th October 2009 – and I still need to try and get two more frames for that one  Suffering withdrawal symptoms, but that’s not too unusual for November, very often I’ve found November with all its promise of long evenings is actually a poor imaging month purely due to the weather.  Let’s hope these gales blow the clouds away and we get a couple of imaging evenings (at least) before we enter the final month of the year.