Star Vistas

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!

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 will get a planar (solid) 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 – which IS interesting.

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

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

Amazon U.K. are offering Star Vistas at a nice reduction right now – only £21.69, a 40% saving on the RRP!  Get the ideal bloke’s Christmas present now while it’s being offered at such a good price  In fact buy 5 and give some as pressies to your Astronomically-inclined friends

The Double Cluster in Perseus (image a couple of entries below) was the Meridian News weather picture for the evening of 10th October 2009.  I shall prepare a short video of this a little later and post up on the New Forest Observatory website.

With the recent run of clear dry weather we have had recently I have managed to complete a few more deep-sky images listed on the white board.  We now have well over 50% of the deep-sky images required for Star Vistas II completed.  I expect 100% completion by summer 2010 meaning a possible publication date for Star Vistas II of summer 2011

The third clear Moonless night in a row had me out imaging NGC457 in Cassiopeia also called the Owl or ET cluster.  This beautiful little open cluster will form just one of the approximately 64-frames Noel and I are putting together that will cover the whole of Cassiopeia at this resolution.  Needless to say NGC457 is another one for Star Vistas 2

We had three days of clear dark skies on the trot – is this a record?  On one of the evenings I managed to catch the Pacman nebula, on another one I returned to one of my old favourites – the famous Double Cluster in the constellation Perseus.  Noel is the world’s expert when it comes to pulling out the best from star fields and star clusters – so Noel also always likes to see Double Cluster data   Here is another image for Star Vistas II from Noel & Greg – the Double Cluster in Perseus.

Noel has just processed a recently taken image – this time it is NGC281, the Pacman nebula in Cassiopeia.  Twelve 15-minute subs went into this one, and we have yet another image ready for Star Vistas II

The new wide bandwidth hydrogen-alpha filter was used for the first time to take an image of the Sadr region in Cygnus.  I’m quite pleased with the result, but I don’t think it is overwhelmingly useful.  May as well use the IDAS light pollution filter and take a full colour image in one go.