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Image processing

28/5/2017

A long time since my last blog post! Busy at work, at home and studying. To calm my nerves I've been processing pictures of little bits of the heavens. Not my own images - the summer night sky is getting too bright - but data collected by various sky surveys and available online through the ESO Online Digitized Sky Survey.

A number of surveys are available and offer various degrees of sky coverage; from 45% of the sky (in blue filtered light) to 99% (in red and infrared). The web interface allows selection of a target or specific coordinates. The images can be displayed as a GIF or downloaded as GIF or FITS. The FITS file format contains a wider range of pixel intensities than GIF. I use FITS Liberator to select the best "window" to view the FITS data.

The images are scans of photographic plates taken by some very big telescopes. Many of them were obtained with the 1.2 metre UK Schmidt Telescope in Australia during the early 90s. Being scanned from the original plates means there are some interesting blemishes in some of the plates. Hairs, fingerprints and other defects can be found if you look carefully enough :-) They can, of course be photoshopped out these days!

Here is what typical GIF images look like:

M33 (Red)

M33 (Blue)

There are subtle differences to the images; the blue filtered image shows the structure in the spiral arms more easily because of the hot (bluish), young stars there. The red filtered image shows the background glow of the disk - containing many more low mass, cooler (redder) stars.

These are grayscale images. To make a natural colour image we'd need an image taken with a green filter! However, no images in green were taken.

Just for fun...I wanted to see some colour images so I synthesised my own green image. This was done by averaging the pixel intensities in the red and blue images. For example, if a pixel is very bright in blue (say a value of 3000) and dimmer in red (say, 1000) then the I'd interpolate the green value to be (3000+1000)/2 = 2000. Doing this for every pixel generates a synthetic green image.

Armed with the red-green-blue (RGB) images it's a simple matter to blend the images to get a colour image:

M 33 natural colour image. Composed from red, blue and synthetic green channels. Credit: STScI Digitized Sky Survey. Adrian Jannetta.

The full res version can be viewed here. And you really should look at it! The amount of detail in the images is breathtaking.

I cropped this from the image above. A couple of things to notice: the pink/orange object is called NGC 604 and is a glowing cloud of hydrogen heated by a great cluster of hot stars at its centre. It is the same kind of object as the Orion Nebula but 40 times the size!

Note the graininess of the image; the telescope has resolved the brightest of the giant stars in the galaxy from a distance of 3 million light-years.

So this has become my most recent astronomical diversion. You can see some of my other results on Flickr.

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Venus and Neptune

8/1/2017

An interesting planetary conjunction for telescope users on January 12th. Brilliant Venus passes within half a degree of the distant ice giant Neptune.

The southwestern aspect of the sky, just before 7pm, looks like this:

This Stellarium rendition of the sky shows that Mars is not too far away from Venus in the sky as well. And the planet Uranus (needing at least binoculars) is high in the sky in the constellation Pisces.

Zooming in a bit on Venus to simulate the telescopic view:

Venus is 51% illuminated (very nearly half-full) and shining at magnitude -4.4. The distance to Venus is 102.1 million km. By contrast, Neptune is 45 times further away (4.6 billion km) and shining at magnitude +7.9.

It's an interesting scene to visualise on software (like Stellarium, above) but it might much more difficult to get a real picture. Venus is around 80,000 times brighter than Neptune. Exposing to capture the phase of Venus won't be long enough to capture a glimpse of Neptune. However, a greatly overexposed Venus should allow Neptune to be picked up in the same view.

The best view - if you can do it - will be with your own eye at a telescope eyepiece.

Bright Planets in 2017

10/12/2016

Nearly Christmas, so time for the annual planetary elongation chart!

A high quality PDF version with much more information can be downloaded here.

The planets shown are Mercury (m), Venus (V), Mars (M), Jupiter (J) and Saturn (S).

This chart shows the positions of each planet relative to the Sun (middle) all through the year. The vertical axis represents the days and months of the year. The diagonal bands represent constellation boundaries. The wavy yellow band is a region close to the Sun in which it would be difficult to observe the planets. The wavy yellow line represents regions of the sky rendered invisible because of the proximity of the Sun. The shape of that wavy line explains, for example, why Mercury is easier to observe in the March evening sky than the July evening sky (even though it is further from the Sun in July). Also, the chart behaves like a game of PacMan; any planets reaching opposition 180 degrees west of the Sun wrap straight over to the evening sky on the far left. Think of this chart as being like an unwrapped cylinder!

Places where the lines intersect are planetary conjunctions --- often beautiful (but not significant) events where the planets appear close together in the sky. There will be several notable conjunctions during the year. Details below.

Supermoon

13/11/2016

The elliptical orbit of the moon means that there is one moment per lunar month when it is closest to Earth. Astronomers call this point the perigee of the orbit. The apparant size of the moon depends on how far from Earth it is, so it appears bigger in the sky at perigee.

Difference between moon at perigee (nearest) and apogee (furthest) from Earth.

When perigee happens at full moon then the moon is obviously bigger and brighter than full moons which don't happen at perigee. In recent years, the phrase supermoon seems to have emerged to describe this situation. It's a little crazy that this phenomenon receives the attention that it does!

Perigee sounds like an easy idea - just the smallest distance between the Earth and moon. However, the Earth and moon are spheres so distance is usually measured from the centre of Earth to the centre of the moon. If you really want to see the biggest possible full moon from the surface of the Earth then a little more investigation is needed!

My copy of SkyMap Pro software tells me that the perigee moon on November 14th 2016 occurs at 11.21am. The perigee distance (moon centre to Earth centre) is 356,509km.

At the time of perigee the moon is below the horizon in Northumberland (and the wider UK). In other words - there's nearly a whole planet Earth between Northumberland and the moon. The moon is actually around 359,000 km from Northumberland at perigee!

The situation improves if we wait a bit.

Here's a plot of the distance between Northumberland and the moon (well, the centre of the moon) over the next few days.

The varying distance is due to the fact the Earth is spinning on its axis. The Earth itself carries Northumberland towards and away from the moon once a day.

In the hours that follow the moon slowly begins to move away from perigee BUT the rotation of the Earth will carry Northumberland closer to the moon. And at a faster rate than the moon is receding. On Monday night, with the moon climbing high into the sky we get our local supermoon: at 12.10 am on Nov 15th the moon will be 351,951 km from Northumberland.

If you look at the graph you'll see there was another minimum before perigee - about 24 hours earlier. The moon was only marginally further away.

Superhype!

Supermoons are overhyped. The oft quoted figure of 10% larger, 30% brighter sounds significant but is very difficult to see in practice because (a) it's comparing apogee full moon with perigee full moon: you never get to see that comparison side by side, (b) the moon illusion complicates the visual appearance just after moonrise (even apogee full moons look big to some people) and (c) the transparency of the atmosphere can markedly change the brightness of the moon and an observer's perception of it!

Even the claim of "no better supermoon until 2034" fails, because the perigee distance referred to in that claim is for moon centre to Earth centre. Actually, for Northumberland (and the rest of the UK) the supermoon on January 1st 2018 will be marginally closer than this one!

Solar structure diagram

6/11/2016

Another day, another diagram. This is from a presentation about the Sun which I'll shortly be giving at Berwick Educational Association.

The diagram is fairly simple. A series of overlaid wedges. A nice thing about PSTricks is that there's a package for almost everything. In this case the wedges can be colour shaded (using pst-slope) to give the impression of a decreasing temperature gradient.

The lines in the radiation zone should ideally be random-walk zig-zags to represent the path of a typical photon! I haven't worked out how to do that so for now they're wavy lines (using \ncsin). The convection lines are done using \pccurve and specifying the angles leaving and entering the nodes at the end points.

The other useful feature of LaTeX is the \multido command; it's necessary to just specify one command (for say, one convection loop) and let \multido put the shape at regular angle increments at a fixed radius.

Here's the code.

Polynomial division and tangent lines

5/11/2016

Six weeks into the first semester of the maths course I teach and my students get introduced to polynomial division, factor theorem and remainder theorem.

I read some time ago of a geometric application of polynomial division to finding the equations of tangents to points on a polynomial curve.

For example: given the curve \[ f(x) = x^3 + x^2 -4x - 6 \]

We can find the equation of the tangent at the point \( x = 2 \) by dividing \( f(x) \) by \( (x-2)^2 \).

Therefore, the equation of the tangent is \( y = 12x - 26 \).

The claim is that for any polynomial curve \( y = f(x) \) then the tangent line at the point where \( x = a \) can be found by finding the remainder after carrying out the division \( f(x) / (x-a)^2 \).

Here is my attempt at proving the result. First a picture of a function with a tangent to the point \( x = a \).

First, note that polynomial division results in a quotient and remainder. In this case:
\[ \frac{f(x)}{(x-a)^2}=q(x)+\frac{r(x)}{(x-a)^2} \]
which can be easily rearranged to give
\[ f(x) = q(x) (x-a)^2+ r(x) \]
A couple of things to note. First \( r(x) \) must be linear, since it is a degree less than \( (x-a)^2 \).
Second, at the point \( x = a \) we find that \[ f(a) = r(a) \].
In other words, \( f \) and \( r \) have the same value at \( x = a \).

The next bit of proof uses some differentiation. Applying the product rule we get \[ f^{\prime} (x) = 2 q(x) (x-a) + q^{\prime} (x) (x-a)^2 + r^{\prime} (x) \]

At the point \(x = a\) we note that \[ f^{\prime} (a) = r^{\prime} (a) \]
In other words \( f^{\prime}(x)\) and \( r^{\prime}(x) \) have the same slope/gradient at \(x = a\). This is necessary if \( r(x) \) is a tangent line.

Since \( r(x) \) is linear, it has the form \[ r(x) = mx + c \] and we know straight lines are completely specified by knowing their gradient --- \( f^{\prime}(a) \) --- and a point --- \( (a,f(a)) \) --- on it.
Therefore \( y = r(x) \) must be a tangent line to the curve \( y=f(x) \) at the point \( x=a \).

Hubble's Variable Nebula

5/11/2016

A clear night earlier in the week meant I could collect some photons for a couple of hours with the Meade LX10. Just before packing up I went for some quick shots of NGC2261 in the constellation Monoceros.

After processing this is what I had.

This little triangular patch of light has an illustrious back story: it was the first object to be photographed by the famous 200 inch Hale telescope at Palomar. The star at the right vertex of the nebula is a variable star named R Monocerotis. In 1916 the eminent astronomer Edwin Hubble noticed that the nebula itself was subject to significant changes in brightness and appearance over periods of days and weeks.

Amateur astronomers are able to target this nebula fairly easily. There are several good examples of how the nebula varies over time.

The variations are attributed to small, opaque clouds passing between the star and the more distant material of the nebula. The resulting moving shadows dramatically change the nebula's look over fairly short periods.

After this initial test of finding and imaging the nebula I think making my own animation of changes to Hubble's Variable Nebula might well be a good project for the winter!

Waning Crescent

26/10/2016

I woke at 4.30am and noticed the clouds from the previous night were clearing. By 5.30am I was outside with the telescope and CCD camera and grabbed a series of movies which were assembled into this:

Click here to see the high resolution version on Flickr. You should click to zoom in when you get there!

This is how the moon looks 25 days after new moon. The phase (illuminated fraction) of the moon is 18%. We're in the period between full moon and new moon where the illuminated portion is getting smaller (called waning).

I've not taken a picture of the waning crescent at such high resolution before. It requires a clear sky and a very early morning....and I'm not a morning person. However, to see the Bay of Rainbows (the semi-circular shaped feature towards the top) as the shadows lengthen at local sunset was a joy!

LaTeX Graphs

8/10/2016

This is a little experiment to test out LaTeX code posting on the blog.

This generates a labelled sine curve:

Lunar Apennines

8/10/2016

I've reprocessed an old picture of the Apennine mountains on the moon using Adobe Lightroom. Here is the result:

This image was originally captured on March 10th 2013 using the QHY5 Mono camera at prime focus of the trusty Meade LX10 (8 inch) telescope. Probably a stack of the best 10% of a thousand frames.

This is undoubtedly one of my favourite features on the Moon. The total length of the range is about 600 km (370 mi), with some of the peaks rising as high as 5 km (3.1 mi).

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