24-hours on the Pleiades

This isn’t my deepest image of the Pleiades, but this dataset was taken over a couple of consecutive nights using the Sky90 array and the M26C OSC CCDs. This is 24-hours of total exposure time using 40-minute subs!

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Picture of the Week

This week we feature the Pickering’s Triangle region of the Veil nebula in the constellation Cygnus. This image comprises many hours of Sky90/M26C exposure.

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Picture of the Week

This week features the M44 (Beehive Cluster) “Stargate” in central Cancer. This is a composite image created using two outings (in different years) with the 200mm lenses and the M26C OSC CCDs in one year, and the 2600MC Pro CMOS cameras last year.

Over to the far left, in the middle, and towards the bottom, are two Carbon stars.

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Picture of the Week

This week we feature the bluest, brightest star in the sky – SPICA!

Always low down in the murk in the south, this is a difficult one to do justice as it always has a broad fuzzy appearance due to being low on the horizon. Even so, this is one of my favourite “single star” shots due to the uniqueness of the star.

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The Golden Solid Angle – Full Derivation

Title: The Golden Solid Angle: A Three-Dimensional Extension of the Golden Ratio

Abstract: The Golden Ratio (φ) and its planar analogue, the Golden Angle (α), are well-known mathematical constants that appear widely in Nature, design and geometry. In this short note, we introduce a three-dimensional extension: the Golden Solid Angle (Ω). Defined as the solid angle that divides the surface area of a sphere in the Golden Ratio, this construct is elegant, elementary, and – curiously – absent from both classical mathematics and natural phenomena. We derive its value, compare it to its lower-dimensional counterparts, and speculate on why it has gone unnoticed for so long.

1. Introduction

The Golden Ratio φ = (1 + 5) / 2 1.618 has fascinated mathematicians, artists, and scientists for centuries. In two dimensions, the Golden Angle α = (2 π) / φ2 2.4 radians = 137.5° is widely observed in phyllotaxis and other biological arrangements. This note introduces a natural extension of the Golden Ratio to three dimensions to form the Golden Solid Angle (Ω).

We define the Golden Solid Angle Ω, as the solid angle that divides the surface area of a sphere of radius r, 4 π r2 in the Golden Ratio Ω / (4 πΩ) = φ. Solving for Ω, we find:

Ω = 4 π / φ2 4.8 steradians. This solid angle has a direct geometric interpretation as the three-dimensional analogue of the Golden Angle: it partitions a spherical space in the same irrational ratio.

2. Comparison With 1D and 2D Golden Divisions

In one dimension:

If line of total length 1 + x is divided into two segments 1 and x such that

(1 + x) / x = x, then solving the resulting quadratic in x we find that x = φ.

In two dimensions:

If the circumference of a circle of radius r is divided in the Golden Ratio then:

1) 2 π r = 1 + φ = φ2

2) r α = 1, α = 1 / r

3) α = 2 π / φ2 2.4 radians = 137.5°

In three dimensions:

If the surface area of a sphere of radius r is divided into the Golden Ratio then:

4) 4 π r2 = 1 + φ = φ2

5) r2 Ω = 1, Ω = 1 / r2

6) Ω = 4 π / φ2 4.8 steradians

3. Why Has the Golden Solid Angle Gone Unnoticed?

Several factors may explain its absence in both mathematics and Nature, including:

1. Dimensional Asymmetry in Natural Growth: Most biological growth appears to follow either axial or spiral symmetry, not spherical symmetry. It seems that Nature rarely grows structures from a central point outward into a sphere.

2. Lack of Functional Utility: The Planar Golden Angle solves a clear optimisation problem such as stacking leaves to maximise the interception of sunlight. No corresponding functional benefit appears to arise from Golden Solid Angle partitioning of space.

3. Lack of Intuition: Solid angles are harder to visualise and are less intuitive than lengths or planar angles making them less likely to be explored in this way.

4. No Prior Need: Few problems, it seems, have called for partitioning a sphere in the Golden Ratio, and thus no demand has driven the creation of this constant.

4. Potential Applications and Future Work

While the Golden Solid Angle does not yet seem to have made an appearance in Nature, its mathematical elegance invites exploration in:

Optimisation of spherical sampling in computational geometry.

Radiation or signal dispersion monitoring.

Design and aesthetics of spherical objects.

Mathematical visualisation and education.

Further work could investigate whether near – Golden Solid Angles appear in viral capsid structures, geodesic domes, or particle emission patterns.

5. Conclusion

The Golden Solid Angle represents a natural and previously unexplored extension of one of mathematics’ most iconic ratios. Though it may not yet be found in Nature, its logical elegance and dimensional progression make it a worthwhile subject of mathematical inquiry.

Bibliography:

The Golden Ratio, Mario Livio, published by review, 2002, ISBN 0-7472-4987-3

Mathematics in Nature: Modelling Patterns in the Natural World, John A. Adam, published by Princeton, 2003, ISBN 0-691-12796-4

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Picture of the Week

This week we feature the Propus region of Gemini which of course contains the well-known Jellyfish nebula, and if your field of view is big enough, the Monkeyhead nebula and open cluster M35 (and neighbouring open cluster NGC2158) as well. This is a composite image combining 200mm lens data (the whole field of view) together with Hyperstar/Sky90 data for each nebula, and the open cluster, taken separately.

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On Being a Photon

Imagine you are a photon.


You suddenly come into existence, and without the passing of any time, and without travelling any distance, you suddenly disappear again. Did you even exist? Well you must have done because an astrophotographer on planet Earth captured you on his camera after you had travelled 10 billion light years from the galaxy he was imaging. So you existed for 10 billion years and travelled for 10 billion light years when you felt that you had not existed at all.

How do you reconcile this?

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Composite Arcturus

I just combined last night’s 24 x 5-minute subs 200mm lens single framer with an older 40 x 5-minute subs 2-framer also using the 200mm lenses. This is the resulting Arcturus composite.

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Arcturus

Managed to get some imaging done last night. Arcturus with the 200mm lenses and 2600MC Pro CMOS cameras. 24 x 5-minute subs. Note the “Napoleon’s Hat” asterism at the bottom of the lower spike slightly to the right.

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Picture of the Week

This week we feature the whole of Canis Minor, which is fortunately only 2 stars, but they are quite well separated. The 200mm lenses are of short enough focal length to capture both in the frame. There are only a small number of very small (Northern) constellations that can be imaged with 3-frames or less with the 200mm lenses, and Canis Minor is one of them.

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