Limb darkening in this image of today’s Sun

I was given a proper solar filter for my birthday about 1,5 month ago and I hadn’t taken a single photograph through it. Until today. Today, around noon, I took this (pretty awesome) picture of the Sun:

40 frame stack of the Sun at 12.00 GMT+1, as seen from Enschede

40 frame stack of the Sun at the 21st of nov. 2014, 12.00 GMT+1, as seen from Enschede

In this image (which I colored yellow-orange-ish because you probably feel more familiar with a yellow sun than a white one), the outer regions of the Sun appear clearly less bright than near the center. This is a well studied effect and is called ‘limb darkening’. In the Eddington approximation, the intensity depends on the angle between the outgoing light and the axis perpendicular to the surface. The equation that describes this intensity is not more difficult than one cosine and 2 fractions:


In the figure below, I have added 3 blue bars on the Sun’s disk, with their corresponding angles. The intensities given by measuring the mean pixel values that my DSLR gives me, are probably not the most accurate, but they are the best I can do (the intensity gradient is probably altered by the used solar filter and the DSLR settings). I have normalized the mean values with the maximum pixel value of 255 so that my measured values become 0.72 in the center, 0.60 at 71% of the radius and 0.28 at the edge of the Sun.

Measured intensities at different angles

Measured intensities at different angles

In the table below I have listed the measured values and those given by the equation I showed you. The fact that the value for theta=0 is the same was to be expected since I used this as the center intensity. The fact that the other 2 values correspond very good however, comes as quite a surprise. I didn’t really expect it to be this accurate, but the changes in contrast caused by the equipment and software seem to balance out pretty good.

table of measured and calculated values

Anyway, I just wanted to show you that the predicted limb darkening is apparent in the photo I made 🙂


Chandra’s and Hubble’s photos of Abell 1689

The Chandra X-ray Observatory and Hubble Space Telescope are two telescopes, orbiting earth, capable of observing in the x-ray and visible regime respectively. Over the years they have produced streams of absolutely stunning images of our beautiful universe. One of the images I like most is a combination of data from both telescopes:

This image of Abell 1689 is a composite of data from the Chandra X-ray Observatory (purple) and the Hubble Space Telescope (yellow)

This image of Abell 1689 is a composite of data from the Chandra X-ray Observatory (purple) and the Hubble Space Telescope (yellow)

It shows the enormous galaxy cluster Abell 1689 and apart from being visualy appealing, the image is also full of cool physical effects that I would like to point out. Let’s start with with the purple x-ray glow coming from the center of the massive galaxy cluster. It originates from extremely hot gas in the center of the galaxy cluster. Reportedly, the gravitational forces at play in that region cause the gas to heat to over a 100 million degrees Celsius. Also, the same purple region is predicted to contain large amounts of dark matter (matter we can’t directly measure, but has to be there in order for the gravitational fields to be as they are).

How intens the gravitational fields are in the center region of the cluster is also apparent from another, in multiple ways cooler, physical effect; gravitational lensing. The theory of gravitational lensing relies on Einsteins theory of general relativity. This may sound scary, but as long as we stay away from the math, there is nothing to worry about ;). To illustrate how this effect works I will borrow a figure from elsewhere on the webweb.

General relativity at work. Source:

Einstein’s theory of general relativity tells us that spacetime (simply picture this as space) is curved in the vicinity of very heavy objects. The huge galaxy cluster Abell 1689 significantly curves spacetime and this curved spacetime deflects light from its straight path as is illustrated in the image above.

Graphical representation of gravitational lensing by a galaxy cluster. Source:

The complex shape of the gravitational field in Abell 1689 bends light from galaxies behind it towards earth so that a single object appears to be at multiple different places at once. Taking into account that this lensing of course distorts the image intensely, what we expect to see are some vague blurry objects with odd shapes that don’t seem to belong there. This is exactly what is visible in the image that this article is about. In the image below (Hubble data only) I have highlighted the lensed images. Look them up in the original image.

Arcs that are lensed images of galaxies behind the galaxy cluster

Arcs that are lensed images of galaxies behind the galaxy cluster

One more effect I would like to point out is the diffraction due to the telescopic design. The brightest stars in the image are not simply bright dots as one would expect from a spherical star, but look more like crosses. These 4 ‘spikes’ that surround the center star are know as diffraction spikes. They are caused by the structure that supports the secondary mirror in the telescope. This structure is comprised of several (4 in the case of the Hubble Space Telescope) bars that keep the secondary mirror in its place as is shown in the graphic below.

Hubble Space Telescope’s optical design scheme

The diffraction is due to the interaction between light passing on either side of the support bars. But how is this possible if light moves in a straight line? Well, as light is not purely particle-like of nature, but also behaves somewhat as a wave, part of the incoming waves may ‘bend around the bar’ a bit. The diffraction pattern shows what is known as the ‘Fourier transform’ of the light. Which means that it shows the spectrum of frequencies present in the incoming light. This is also clearly visible in the image of Abell 1689. Below you see an excerpt of the bigger picture, clearly showing the different colors in the spikes.

Diffraction spikes due to secondary mirror support bars

Diffraction spikes due to secondary mirror support bars

Not only the Hubble telescope shows this diffraction pattern, but amateur telescopes with a similar design do to. In fact, my telescope has 3 such bars which shows 6 (albeit less pronounced) diffraction spikes around bright objects. A while ago I imaged Deneb, a blue-white supergiant star weighing about 20 solar masses, and the resulting image showed some cool diffraction spikes.

Single exposure of blue-white supergiant Deneb. Clearly visible are the 6 diffraction spikes due to the 3 bars that obscure the view.

Single exposure of blue-white supergiant Deneb. Clearly visible are the 6 diffraction spikes due to the 3 bars that obscure the view.

I hope that after reading this, you can appreciate the image at the top of this post as much as I do 🙂