Stellar Magnitudes

The magnitude scale measures how bright an object appears to be to a human observer. Because the eye perceives brightness in a non-linear way (it is very sensitive to low light levels but becomes desensitized at high levels), a magnitude value represents the ratio between the brightnesses of two objects, not the difference. In fact, the scale is defined such that a magnitude difference of 5 represents a brightness difference of 100-fold. Sorting out the numbers, we find that a magnitude difference of 1 represents a brightness difference of 2.512 times: for every additional magnitude the brightness difference is multiplied by 2.512. Thus the difference of 5 magnitudes mentioned earlier is 2.512 x 2.512 x 2.512 x 2.512 x 2.512 times brighter, which is the factor of 100 required. The mathematicians among you will recognise that the defining equation for magnitudes is therefore that 100 = 2.512⁵. If we generalise the equation to R = 2.512^M, where R is the brightness ratio and M is the magnitude difference, we can take logarithms of both sides and re-arrange things a bit to find that M = log₁₀(R)/log₁₀(2.512). Because log₁₀(2.512) equals 0.400, this can be simplified to M = 2.5 x log₁₀(R), which is the usual formula for determining magnitude differences from brightness ratios.

Slightly perversely, in the magnitude scale the higher the number the less bright the object is. The whole scale is calibrated by reference to the bright star Vega (Alpha Lyrae) which is defined to have a magnitude of exactly 0.00. Very bright objects can thus actually have negative magnitudes - for example Sirius, the brightest star in the sky, is magnitude -1.44 which makes it 3.77 times as bright as Vega. The reason for this reverse-scale system is historical. The astronomer Hipparchus was the first to group stars according to their brightness, and he did so by referring to the very brightest as being of "first magnitude" and so-on down to sixth magnitude. This naming scheme was then simply carried over when Hipparchus' classification was translated into the modern, mathematical, form.

There are clearly far more dim stars than bright ones, but it is surprising just how many more. In fact, for each magnitude that we go down in brightness there are about 3 times as many stars. Each of them is 2.5 times less bright than those in the previous magnitude "slice", of course, but as 3 is larger than 2.5 the total amount of light they send us is greater. This leads to the surprising result that the faint stars - so faint they can only be seen with a telescope - contribute more light to the night sky than all the bright stars put together! We don't notice this because the light from all the stars we cannot resolve with the naked eye is spread out over a very large area whereas that from the bright stars is highly concentrated in distinct points. The effect is noticeable in the Milky Way though, where the faint stars in the far reaches of our galaxy together contribute to this diffuse misty band in the sky.

Apparent and Absolute brightness

Note that in the first sentence above I said that the magnitude scale measures how bright an object "appears to be". Objects in the sky are at vastly different distances from us and so an intrinsically faint nearby object could appear to be brighter than a truly brilliant object an enormous distance away: think of the difficulty of distinguishing between a nearby torch and a distant car headlight at night. Stars thus have two magnitude values: apparent and absolute. The apparent magnitude is the value normally used on star charts and in astronomy programs as it is the more useful from a practical, observational, standpoint. The absolute magnitude is the brightness an object would have if placed at a fixed, standard, distance from us - it can thus be used as a true comparison of "real" brightness.

Limiting magnitude

Against a very dark sky the unaided eye can see stars down to about magnitude 6, but the practical limit is rarely greater than magnitude 4.5 due to light pollution and dust etc. With a reasonable pair of binoculars you can go down to about 9.5. The digital camera I use will reveal stars of well below magnitude 9 (as long as the "stacking" technique is used) and my telescope plus webcam will reach about magnitude 11: being electronic, both these instruments are ultimately limited by the effects of noise more than the optics, however. The great advantage of the telescope over the camera, though, is the exposure times. The camera needs 4sec at 800ASA equivalent to get down to its limit but due to the much greater light-gathering power of the telescope the webcam will do so in fractions of a second: this avoids the "trails" seen in the long exposures necessary with the camera. The other difference is the resolving power i.e. the amount of detail visible. Again, the telescope is far superior due to its larger-scale optics - comparison of the pictures on the Jupiter page will show this very well. Set against this is the great difficulty of getting the object you want into the webcam's field of view, which with the telescope is fixed and very small. The camera is much easier to line up as its magnification can be set very low to find the object then slowly stepped up to increase the object size while keeping it in the frame. Its ultimate field of view is, however, much larger so things will never get as big as with the telescope. In many ways the two compliment each other - the telescope is best for planetary work for example but the camera scores where ultimate resolution is less important than convenience.

Much of the above discussion about exposures etc. could of course be avoided if the camera were mounted on an equatorial mount which was then motor-driven to track the stars, but that would be a whole different ball-game - much too easy! Still, there's always Christmas I suppose .....

Practical example

This picture is an example of what the digital camera will do. It's a stack of four images taken at 200mm plus x4 digital zoom at 400ASA (to reduce the noise problem) but with an exposure time of 8secs to compensate - this has the (in this case) useful side-effect of allowing one to distinguish stars from noise due to the trails they make. While it was only really intended to be a test shot I thought I might as well take something interesting, and the "something" is the odd blob to top left. It's the most distant object visible with the naked eye - the Andromeda Galaxy [also known as M31, being the 31st entry in the catalogue of such things compiled by the French astronomer Messier]. Despite being the nearest spiral galaxy to our own, it would still take 2.5million years to get there even if one could travel at the speed of light! The "blob" is actually just the bright core of the Galaxy: its full extent is about twice the width of this picture!

Although quite bright in this image, none of the stars shown here are even approaching naked-eye visibility. The brightest star (upper right) is just magnitude 7.1, the one nearest to the bottom is 7.6, the pair towards the middle-bottom are 7.4 and 8.8 and the one at bottom-right is 8.5. The triangle of stars to the left of the bright one and those to the extreme left and right of the frame weren't listed in my "standard" astronomy programs (which only go to mag. 9) so I downloaded an update to one of them to take it to mag. 11.5. This showed all of the unlisted stars, from mag. 9.2 to 9.4. Well, not quite - the right-most one of the triangle wasn't there! This was rather a puzzle, so firstly I confirmed it could not be an asteroid or variable star (by checking with the same program). I then looked to see if it was there in other images of the area I had taken some nights afterwards, and indeed it was. This ruled out such things as aircraft, satellites and weather balloons, plus the possibility that the trail might have been some odd artefact of the digital processing. I was just beginning to think something strange was going on when I had a flash of inspiration and turned on the "show nebulas" option, whereupon a large blob appeared exactly in the position of my "phantom star"! It wasn't a star at all, but another galaxy: M32, in fact! This is one of two companion galaxies to M31, the other being M110. I did have a search for M110 (well off the top of the frame in this shot) but didn't find anything definite, probably because it was too diffuse to stand out from the noise background. So, problem solved - I'd captured two galaxies for the price of one!!