# Surface Brightness

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Hi

I am trying to tie up theory and reality, and I've realised my maths doesn't tie up with my experience.

Surface Brightness: This is a term used to describe just how bright (or dim) a Galaxy will appear. I found the usual unit is magnitudes per square arc second.

So, trying to locate a suitable target for my next shot, I worked out the surface brightness for all the Messier galaxies.

What I found interesting is that none of them have a surface brighness of 24mag/sqarcsec or dimmer, the really bright ones (M82, M51) seem to be quite active but are only ~21st Mag / squ arc sec. M31 and M109 (target of my latest effort) have surface brightness only 1 magnitude different, yet M31 is ~2Million light years distant yet M109 is ~55Million light years distant.

Now can M109 really be that much brighter?

Well a thought came to me: Inverse square law.

If we take M109 and doubled it's distance, it's apparent total brightness would reduce to 1/4, but so would it's apparent surface area, so we get the same surface brightness (per square arc second) regardless of distance... take it to 550million light years and it will still show up on the CCD, but hang on.. I don't see tens/hundreds of little galaxies in the background of my M109 shot

So.. why then do we not see all the galaxies out to the point where redshift starts influincing things in most amateur images..

Shoudn't most top grade images look like the Hubble Deep Field?

Is there something in the way dimming the far away galaxies?.. or are there simply not many galaxies in any one direction?

Derek

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so we get the same surface brightness (per square arc second) regardless of distance... take it to 550million light years and it will still show up on the CCD, but hang on.. I don't see tens/hundreds of little galaxies in the background of my M109 shot

You need an image scale large enough to resolve galaxies before the surface brightness figure matters. Very distant galaxies will appear as faint stars. Seeing smears reduce the resolution below what you would expect by applying Dawes Limit, except in the smallest scopes ...

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Thanks for the replies

I was aware of the Olbers Paradox, but I don't think that's really it.. related yes but not the driving force.

I was only concidering distances up to ~1billion light years, i.e. before redshift/expansion had made much of an impact.

Anyway I've made some more calculations and it looks like I should expect only 1 object on a 1 square degree image if we place the constraint that the object must be no more than 500million light years away... many assumptions made to get this, but at least the order of magnitude should be reasonable.

If we go to 5billion light years away, we get maybe 1000 objects, but by that stage all the photons are redshifted, and galaxies get very small.

Next penny to drop...

if a galaxy is emitting equal energy in blue, green and red light, by the time it's getting to us it will of course be redshifted, so that 'equal energy' will now be green, red and infrared. However photon colour is another way of describing photon energy... My camera that would normally expect a 22nd magnitude surface brightness to mean Xthousand red photons won't get those, in the red band it will get the green photons redshited to red... and green photons are fewer for the same energy, so it's brightness falls faster than one would expect just treating it in classical physics...

Size:

take M109 and place it at 550milion light years, it's size is reduced from 456 arc seconds long to 46 arc seconds long (still pleanty big enough to see).. but take it to 5billion light years and you have it at 4.6 arc seconds long.. a tad more challenging.

On the size scale it would seem that ~10 pixels ought to be the limit, below that you just get a spot. so 10 pixels at 1 arc sec per pixel takes us to 10 arc seconds, or about 2 billion light years.

How Many:

So how numerous is a 2 billion light year distant galaxy?

well the distance would suggest encompassing 64 times as many galaxies as at 500 million light years, so that would put the number per square degree around the 60 mark...

Or in the field of view of my quite small camera I've been using... about 2 objects. Ok.. so the real answer is very simple:

Not many galaxies per unit volume of space.

To see more I need:

More light / Better resolution / Wider field of view.

I think a large megapixel CCD is going to appear:D

Derek

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Well done Derek, that's a nice calculation. As you say, surface brightness is invariant with respect to distance, but angular size of objects gets smaller, and the amount of field that is actually filled by galaxy light is very small.

Another point to consider is that the galaxy census is limited by detectability. A few decades ago it was thought that the "typical" surface brightness of observed galaxies (somewhere around 22 mag/sq-arcsecond) was a general feature - but with better technology it became apparent that there are lots of low surface-brightness galaxies out there. The sky itself is about 24 mag sq-arcsecond so detecting galaxies fainter than that is a challenge. But adding these into the calculation would not much change your conclusion. Similarly, I'm guessing that very distant galaxies often have high surface brightnesses because of their active nuclei - but their angular size is so small that we have no chance of seeing them in amateur telescopes (apart from a few quasars) and since they're virtually point sources they aren't going to add up to much of a percentage of the field of view.

Of course, as you realise, if the universe were infinitely large, infinitely old, and randomly filled with infinitely many galaxies, then it would all add up, and we'd have Olbers paradox.

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It is the surface brightness per unit physical area which remains constant. So if you look at a 100x100 light year patch in a galaxy, its surface brightness in mag/sq.arcsec. would remain constant as the galaxy moved to greater distances. However, the area it subtends on the sky gets smaller and smaller.

If galaxies were infinitely wide objects of constant brightness then this wouldn't matter. But they are not - their light falls off towards the edges. As a result the surface brightness inside a pixel on your CCD pointing at the galaxy will NOT remain constant. It will go down drastically as the galaxy gets further away. This is why more distant galaxies become difficult to image.

NigelM

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It is the surface brightness per unit physical area which remains constant. So if you look at a 100x100 light year patch in a galaxy, its surface brightness in mag/sq.arcsec. would remain constant as the galaxy moved to greater distances. However, the area it subtends on the sky gets smaller and smaller.

If galaxies were infinitely wide objects of constant brightness then this wouldn't matter. But they are not - their light falls off towards the edges. As a result the surface brightness inside a pixel on your CCD pointing at the galaxy will NOT remain constant. It will go down drastically as the galaxy gets further away. This is why more distant galaxies become difficult to image.

NigelM

hmmm.. I follow your logic.. but I'm not sure I agree. However as I always try and leave room for self doubt.....

Derek

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