Some bits and pieces ...
Here is a montage of GALEX images (in the UV, obviously), alongside ones taken in "visible light" (and presented at the same scale). This makes the same point, and more: while spirals may, on average, have exponential profiles, in the V-band, quite a few clearly do not, in UV-bands. 
There's another aspect that may be important to keep in mind; here's Disney & Lang (bold added):
SDSS images of Local Group galaxies show that the uncrowded parts are easily resolved into stars; peering a bit further out and you come across objects like SDSS J143759.94+400622.3 (check out this SDSS image of it) and NGC 4395. The bright blue blobs are star-forming regions, perhaps like the Tarantula Nebula (in the LMC) or NGC 604 (in M33). While the underlying intensity profile in galaxies with lots of these sorts of regions may be exponential, what stands out - especially in bands which include the strong emission lines from such regions - is the blobs. The Visibility (to use the D&L term) of these kinds of galaxies is surely determined by the number, spacing, and surface brightness of the blobs, not the underlying galaxy! Classes of low SB galaxies unresolved into stars, which cannot already be seen in Schmidt surveys, are beyond hope of discovery by optical means alone.
 of course, a quantitative analysis would be needed, to show that the radial profile differs markedly from an exponential one, for these galaxies at least.
 a point already made by ngc3314, early on page 1 of this thread:
And what does this have to do with low SB galaxies? Not much - - except to stress, again, that D&L's analysis has some limits (possibly quite severe ones)[...] higher resolution lets us see the bright cores and star-forming regions, while redshifting ultraviolet into the visible band would exacerbate light loss due to dust while presenting us brighter and more compact star-forming regions when they are not heavily dust-reddened. One clear result of the Tolman dimming is that our optcally-chosen galaxy samples at high redshifts cannot fail to be biased in favor of galaxies or pieces of galaxies which are blue (UV-bright) of unusually high surface brightness.
Second para of Disney & Lang's Section V. ("HOW GALAXIES SINK FROM SIGHT"):
Except that, of course, at z = 0.5, what is the U-band  at z ~0 is observed in the V-band (more or less); and the V-band (at z~0) is now some non-standard band, neither R- nor I-band.The factor rapidly becomes very significant by comparison with the narrow FWHM (2.5 mag) of the Visibility Window. Even at z = 0.5 many of the most Visible galaxies that were in region A (Fig 1) at low redshift would be translated into region C and be far too dim to see.
Using the terms D&L introduce in Section III ("AN OUTLINE OF VISIBILITY THEORY"), , , and are all band-specific! This renders the rest of their argument moot ... unless they can show that these key parameters have the same values, irrespective of band. Of course they assume , which we already know to be, in general, an invalid assumption. 
Time to introduce J.L. SÚrsic. From the Keel document I linked to earlier, the SÚrsic profile:
where re is the effective radius.
Putting this into the form in D&L:
Which is the same as D&L's (2), where (I hope!):
To illustrate the band-specific nature of the intensity profile, consider the best-fit parameters in TvD11, to the stacked bright red ellipticals; from Table 1 ("Stack SÚrsic parameters"); the columns are, respectively Filter, SÚrsic index, and Effective radius (kpc):
u 3.94▒1.62 17.0▒9.80
g 4.03▒0.09 12.6▒0.19
r 5.50▒0.05 13.1▒0.10
i 4.86▒0.05 10.9▒0.06
z 4.91▒0.08 11.5▒0.12
Recall that D&L assume a SÚrsic index of 1, and a constant (band-independent) α; but note too that Section V explicitly does not cover this type of galaxy (that's left to Section VIII "HOW ELLIPTICAL GALAXIES SINK").
Incidentally, by ignoring the band-specific nature of the intensity profiles (including the fact that 'pure exponentials' may be the majority of galaxies in only a narrow wavelength range!), D&L's Section VI ("WHY HIGH REDSHIFT GALAXIES LOOK SMALL") and Section VII ("WHERE HAVE THE DESCENDANTS GONE?") become essentially moot too.
What - other than re-doing the main arguments in D&L - is left to consider then?
 Johnson-Morgan photometric system; ditto V-band, R-band, I-band
 Get it now Jerry?
The Galaxy Zoo forum is a terrific place to hang out, and feast your eyes on some interesting images (and discussions).
I expect that it could provide plenty of examples (or counter-examples) relevant to this thread. Not in any quantitative, survey-level sense, but as pointers to what it might be interesting to look for in the outputs from SDSS and COSMOS.
The results of the first Galaxy Zoo project - i.e. as described in the Lintott et al. paper I provided a link to earlier - are incorporated in DR8, which is very useful.
Oh, and the current Galaxy Zoo project a very cool one to participate in, involving classifying images from the Hubble, from some of the big surveys done with the HST (e.g. COSMOS).
That's how Section II ("THE NARROW WINDOW"), the start of the make part of the D&L paper, begins.As anyone who has looked for M31 can testify, the problem of detecting galaxies in the optical is not so much lack of light as lack of contrast against the foreground sky[M31 has a V mag of 3.4 which spread over its size of roughly 3 by 1 degrees amounts to a SB = 21.2 V mag per sq arc sec , where the sky is about 21.5 at a fair site]
Now I get 22.37 V mag per sq arc sec as the SB for M31, but that's close enough, given the "roughly".FIG. 3: The calculated Visibility Window for Exponential galaxies. The vertical scale shows the relative volumes within which galaxies with different surface-brightness-contrasts to the background (plotted horizontally) can be detected. Following the usual convention this contrast Δμ, in mag, is plotted from right to left with high surface brightness, i.e. high contrast galaxies to the left, low surface brightness galaxies to the right. The maximum heights of the two curves Vm (dashed or green) and Vθ (smooth or red) assume a sample for which Γc = π , typical of all Exponential galaxies, save those hundreds of pixels across. This is a typical Wigwam Diagram for the Visibility of galaxies of all kinds ( see later). Since the vertical scale is arbitrary the Wigwam Diagram is valid irrespective of Absolute Luminosity, just as it is valid irrespective of the absolute survey depth ( deepest isophotal level μc ) because the horizontal axis is given only in contrast Δμ ≡ (μc-μ0) where the latter is the central SB, measured in magnitudes. To be detected galaxies must lie inside the Wigwam, the shaded area marked A, which we call The Visibility Window. Note how narrow it is, with a FWHM of 2.5 magnitudes with a peak P at a contrast of 3.5 magnitudes .Because the Window is so narrow, redshift dimming will quickly move galaxies rightward and out of sight into regions C and even D. For future reference note that even the Vθ(smooth or red) curve, by itself, has a FWHM of only 3 magnitudes.
Of course, M31 is "hundreds of pixels across" , so the text for Fig 3 doesn't apply; however, if it did, what would Δμ be? To answer that, we need to know μ0 (we already know that μc is ~21-22).
Would any reader like to have a go at working this out?
FWIW, a blind guess by me would be μ0 ~ 5, giving a Δμ > 15, making M31 an extreme outlier!
Now suppose we took an image of M31, with a modern CCD-based SLR camera, attached to an equatorial mount (so the image won't smear during the shot), at a zoom that ensures M31 will be no more than ~100 pixels across. What would Δμ be?
For example, this by Moonhawk. The downside, of course, is that the images on the web are JPEGs, and they do not preserve flux. Maybe, however, the authors still have their original - digital - files, and might be willing to share them?
Astrophysically Motivated Bulge-Disk Decompositions of SDSS Galaxies, by C. N. Lackner and J. E. Gunn, makes an extremely good case for saying that characterising galaxies as either exponential or de Vaucouleurs (in terms of their surface brightness distributions) is too simplistic by far. And it's a great paper to read anyway.
A point on Tolman: isn't one of the four (1+z) factors just 'band shifting'? I.e. the 'surface brightness' is the total (apparent) energy per unit area (on the sky), and one of the (1+z) factors comes from the fact that the whole SED (appears to) moves redward? If this is so, then Disney and Lang's argument is even more flawed than I'd thought (they should be using (1+z)^3, as all observations of (optical) galaxies are made within specific (visual) wavebands).
Or do I misunderstand the Tolman surface brightness concept?
[Edit: the explanation below mistakenly derives a (1+z)^4 law for extended sources. The derivation is actually a correct derivation for POINT sources. To see the proper derivation for extended sources, look in post 103 below]
The four (1+z) factors in the Tolman effect explain why the surface brightness of a distant galaxy -- as measured in flux units, energy per unit area per unit time per unit solid angle -- decreases so quickly.
1 and 2. the inverse square law, decreasing flux with distance in the usual manner
3. time dilation: clocks on distant objects appear to run slowly, and so distant objects appear to emit fewer photons each second
4. decrease in energy of each photon due to the redshift of each photon
Note that this effect deals with measurements of energy flux: how many ergs per square cm per second are received.
In this context, people ignore the change in received flux due to the shifting of the object's spectrum through the detector's passband. That effect is very real, of course, but bringing it up opens a whole new kettle of fish. The discussion about (1+z)^4 centers on the properties of space and time, not properties of the source. In other words, people who talk about (1+z)^4 are interested in cosmology, not in the evolution of stars and galaxies.
There ARE plenty of people who do care deeply about the properties of galaxies and their evolution, of course. They will include in great gory detail the effects of the shifting of the source spectrum through the detector passband in these discussions.
One of the original references is Hubble and Tolman, "Two Methods of Investigating the Nature of the Nebular Redshift", ApJ 82, 302 (1935).
Last edited by StupendousMan; 2012-Jan-17 at 01:26 PM.
Now I'm rather puzzled and confused.
Disney has been doing astronomy for decades, full-time, and many of the published papers of which he an author concern observations of galaxies, in various wavebands. He's also written on cosmology (I have no idea about the co-author, R.H. Lang).
So for him to have written a paper (well, it's only a draft at this stage) whose main thrust is based on a fatal misunderstanding of the Tolman surface brightness relationship is, well, pretty astonishing, isn't it?
But more puzzling is that it seems he (Disney) has been working on this 'problem' for quite some time (e.g. "Some of these ideas were explored in 'The Visibility of High Redshift Galaxies' ( Phillipps, Davies & Disney 1990) which built on earlier papers in 1983 (Disney & Phillipps) and 1976 (Disney). However the highest redshift being considered there and then was 0.3!"); how could he have messed up so badly? I mean, if instead of being a (tenured?) professor, he was one of your (graduate) students, I doubt you'd've given him a pass if this were a term assignment, would you?
Forming opinions as we speak
Whoops. Antonsieb has it right: the apparent brightness of a point source will decrease as (1+z)^4. The short explanation I gave was, in fact, not appropriate for extended sources, but for point sources. It was my mistake -- I study supernovae, not galaxies, so I'm always thinking about those four factors.
Strangely enough, the apparent brightness of an extended source will ALSO decrease as (1+z)^4 in an expanding universe! The reasons are a little different. For an extended source, the simple inverse-law decrease in apparent brightness is countered by the increase in the source luminosity falling into the measurement aperture; that is, if we measure the amount of light in a circle of radius 3 arcseconds, then if a galaxy moves twice as far away, each little bit provides only 1/4 as much light, but at this larger distance, 4 times as many stars fall into our circle of 3 arcseconds. So, in a static universe, the surface brightness of an extended source would be constant.
In an expanding universe, the items 3 and 4 I listed in my original message, the time dilation and redshifting of each photon, would still apply. That would cause the surface brightness of a galaxy to decrease by a factor of (1+z)^2. But wait -- there's more! In an expanding universe, the apparent angular size of a galaxy does NOT decrease exactly as it does in a static universe. Instead of constantly shrinking and shrinking with distance, a galaxy will, at high redshift, start to increase in apparent angular size. It's hard to visualize (at least for me), but the basic idea is that when the galaxy emitted its light, it was much closer to us, which makes it apparent size (back then) larger than we'd otherwise expect.
The bottom line is that this increase in apparent angular size with redshift adds another factor of (1+z)^2 to the equation. The apparent surface brightness (measured by energy) of extended objects in an expanding universe decreases as (1+z)^4, AND the apparent brightness (measured by energy) of point sources also decreases as (1+z)^4. Kind of neat, now that I write them both.
These simple calculations assume that the spectrum of the source object doesn't change over the range of observed wavelengths, regardless of the redshift. In real life, the energy emitted by source objects can change quite sharply over the wavelengths sampled in, say, the observer's V-band as redshift changes from z=0 to z=7; so in real life, comparing observations to theory is much more difficult.
Wait a minute! Something cannot be right here, because in "point sources" we are integrating over the entire angular extent so that cosmological departures from inverse-square (heuristically due to changes in apparent size between emission and observation) don't matter. In common jargon, this relates to the difference between luminosity distance and angular-diameter distance. All this is easiest to work with for bolometric fluxes - there one has to deal with (1+z) factors for photon arrival rate and energy, but not for narrowing or shifting of the filter bands in the emitted frame.
Earlier in this thread one of the Lubin and Sandage (2001) papers, on an observational test of the Tolman surface brightness test, was referenced. There are altogether four in the series. The first begins with this (some formatting is lost):
The critical point - that underlies Disney&Lang's apparent fatal flaw - is that the key dimension of "luminosity" is energy (per unit time, in the observer's frame).Originally Posted by Lubin&Sandage
This luminosity is band-independent; it is a total energy, integrated over all wavelengths (frequencies) of light (electromagnetic radiation).
Disney&Lang, however, conflate the luminosity of the Tolman surface brightness test with the luminosity of galaxies in a very, very narrow slice of the electromagnetic spectrum (basically, little more than 'the visual', from the atmospheric UV cutoff to ~1 micron), e.g. "Fig 1 illustrates what happens for objects with an exponentially declining light distribution ( virtually all galaxies bar Giant Ellipticals; see later)."
NGC3314 mentioned "k-corrections", which is highly relevant; Disney&Lang use the phrase precisely once, on p26, and promptly proceed to ignore it ("Given uncertainties as to which is the correct model, and K-corrections, dust and Evolution, this approximation is more than satisfactory").
As I currently understand it, Disney&Lang say - among other things - that the surface brightness of a galaxy like the Milky Way will drop by (1+z)^4 when observed in a band which shifts with z (as such a galaxy is 'placed' some distance away from us); if we use V as our band to perform observations, then as our test galaxy gets further and further away, the band in which we observe it becomes shifted further and further into the red. Here's where their conflating of the two different definitions of luminosity causes their fatal flaw: redshifting is, itself, one of the (1+z) factors!
In addition, as I pointed out earlier, the sky is band-dependent; the limiting surface brightness - due to the sky - is not the same in all bands. So, the extent to which a galaxy's surface brightness falls below the sky, as its apparent surface brightness declines (by a factor of (1+z)^3, in a band which tracks z), depends critically on what the surface brightness of the sky is, in the two different bands.
I wrote: measured bolometric flux of point source goes like (1+z)^4. Measured bolometric flux of extended source goes like (1+z)^4 also, but for different reasons.
Indeed, the spatially integrated bolometric flux of point and extended sources must have the same behavior. But for extended sources, it's more usual (and germane to galaxy detection) to deal with surface brightness, at some reference radius or within some metric radius. For each, spatially integrated bolometric flux does the same thing. For some specific numbers, I took Ned Wright's cosmology calculator with "consensus" WMAP parameters plus flat geometry, and consider an object with bolometric magnitude Mbol=-24. The apparent bolometric magnitude mbol is just -24 + 5 log (D/10 pc). To show surface brightness, let this flux be uniformly distributed in a circular disk with radius 10 kpc (thus over an area pi*(10/angular scale)^2 arcsec^2). The table lists z, luminosity distance, angular scale in kpc/arcsecond, mbol, and bolometric surface brightness in mag/arcsec^2.
z DL,Mpc ang scale, kpc/"" mbol Bolometric SB
0.01 42.6 0.20 9.14 18.88
0.1 455 1.82 14.29 19.23
0.25 1249 3.87 16.48 19.78
0.5 2822 6.08 18.25 20.57
0.75 4638 7.34 19.33 21.24
1.0 6634 8.04 20.10 21.82
1.5 11008 8.54 21.20 22.78
2.0 15733 8.47 21.98 23.56
(Rats. Where's a proportional font when you need it?)
At low z there is the familiar Euclidean limit - inverse-square for flux, constant surface brightness. But things get rapidly worse at cosmologically important distances, leading to almost a 5-magnitude drop in SB by z=2.0 (the Tolman signal would be 3^4=81 times or 4.77 magnitudes, a good match to the calculated 23.56-18.88=4.68 from the table, noting that the top point is not at z=0).
Consider an 'exponential' galaxy. Viewed locally (within the Local Group say, but from the outside), in the V band, its effective radius (Re) is 1 kpc. Assume the 1 kpc, at the distance we've observing it, is the same size as the seeing. Assume that, like SDSS, we can measure/detect the galaxy out to 3Re (in 2D), and truncate it to zero at 4Re. Let A be the brightness of the central, 1Re, point.
In a flat, Euclidean universe, when the galaxy is far enough away that 3Re corresponds to the seeing, what is the brightness of this (almost) point source (compared with A)?
In our universe, if the galaxy has a redshift of z, then one of the (1+z) effects is that we should observe the galaxy in a V band redshifted by z. If this were the only effect, then the galaxy would appear,when viewed in the redshifted V band, to be just like the same galaxy in a flat universe (observed in the V band).
The aberration, which comprises two of the (1+z) factors: compared with the same galaxy in a flat universe, the galaxy looks bigger; specifically, the seeing corresponds to 3/(1+z) Re, and the galaxy is clearly not a point source (assuming the redshifted V band has the same sky limit; i.e. we can measure/detect to 3Re, and truncate to zero at 4Re). But wait, can we still measure/detect out to 3Re? The 'fuzz' around the point souce that is now our galaxy - from 3/(1+z) Re to 3Re - has a lower surface brightness than that part of the galaxy when it was nearby (correcting for redshifting the V band).
Now add the last (1+z) factor; this simply makes every pixel dimmer.
If the logic is OK, it's time for some numbers ...
Some sanity checks:
z DL,Mpc ang scale, kpc/"" mbol Bolometric SB
0.01 42.6 0.20 9.14 18.88
0.1 455 1.82 14.29 19.23
0.25 1249 3.87 16.48 19.78
0.5 2822 6.08 18.25 20.57
0.75 4638 7.34 19.33 21.24
1.0 6634 8.04 20.10 21.82
1.5 11008 8.54 21.20 22.78
2.0 15733 8.47 21.98 23.56
In a flat, Euclidian ("flat") universe, the last column (bolometric SB) would be:
Those are calculated values (!), using as inputs the values in the second (DL,Mpc) and fourth (mbol) columns. They also validate ngc3314's conclusion ("a good match to the calculated 23.56-18.88=4.68 from the table, noting that the top point is not at z=0").
Now suppose our uniform 10 kpc disk galaxy emitted all its electromagnetic radiation as/at [OIII]5007. In a perfect astronomical observer's world, our galaxy would have a total, integrated magnitude of infinity (zero flux) in all but one of the five SDSS bands (and infinity in all, in the last two cases); the bands in which it would be visible would be, respectively, g, g, r, i, z, and z. In all bands except the z it would be detected, easily (though in the i band it'd be faint; it'd be too faint to be detected in the z band, even at z=0.75).
In a flat universe it would be an SDSS point souce in all cases except the first three, and in the third it'd be marginal (assuming a seeing of 1.4" in all bands); in the real universe, it would be a marginal point source in the fourth (z=0.5) case, and an SDSS point source at all greater z's.
Assuming a PSF of 0.05" for the HST, it would be easily resolved in all cases, in a flat universe and the real one. Whether it would be too faint to be detected I haven't checked, but I expect it to be easily detectable.
Now real galaxies emit light at many wavelengths, and that light is not emitted uniformly across a circular disk (the 'uniformly' is unreal, the 'circular disk' is not). The effect of the first is, obviously, to make our galaxy fainter than its apparent bolometric magnitude, and for it to have a non-infinite magnitude in all bands; the effect of the second is what the Disney and Lang paper is about.
A word about why I like ngc3314's approach.
His 'galaxy' is so horribly simplified it is almost a joke, wrt real disk (exponential) galaxies.
However, it's very easy - and intuitively straight-forward - to quickly turn it into a more reasonable toy (or cartoon) galaxy.
For example, add a bulge by adding a similar circular disk, with a radius of, say, 1 kpc.
Represent the colours of a real galaxy by splitting the -24 bolometric mags across five lines, centred in each of the five SDSS bands.
Represent an exponential disk by three co-centred disks of radius Re, 2 Re, and 3 Re (or some similar combo).
What say you, dear reader?
I made a mistake in my calculations in my last post. The SDSS seeing is 1.4" FWHM, and the -24 absolute (bolometric) mag disk has a radius of 10 kpc. Thus it remains distinguishable from a point source even at z=2 in the real universe; in a Euclidean universe (what I called "flat" in my last post), it becomes a point source at z=0.5.
One other thing to add: the venerable RC3 includes a field called "Log_D25", which has to do with the size of each galaxy in the catalogue as measured at the B-band 25th magnitude per arcsec squared isophote. In the toy universe I'll be exploring, astronomers can robustly detect and measure galaxy isophotes to 25 mag arcsec^-2, in all bands (and with the HST they can do a lot better). In the example in my last post, then, the galaxy would be observable, as an extended source, at all z's1.
Now consider the same, uniformly luminous, circular disk, but with an absolute bolometric magnitude of -17.7.
Here's the same table as before, with just the columns of interest:
z....mbol Bolometric SB
0.01 15.46 25.19
0.10 20.60 25.54
0.25 22.79 26.1
0.50 24.56 26.89
0.75 25.64 27.56
1.00 26.42 28.14
1.50 27.52 29.1
2.00 28.29 29.95
Now suppose this galaxy emits all its electromagnetic radiation at 265.0 nm. In the SDSS system, it is not visible at all until it has a z of 0.25 (among the eight cases above), and then only in the u-band. Its SB is too low for it to be detected, and even if it were a point source, it'd be too faint for the SDSS to have assigned it the honour of being a photometric object. At z=0.5 it would be detectable in both the u and g bands (if only it were brighter)! Why? Because all the SDSS bands overlap somewhat, and at 397.5 nm neither of the (u and g) filters transmits much. However, I'm assuming a perfect astronomical observer, so I'd have to decide to make it one band or the other ... If the galaxy were brighter it'd be visible in the g-band (z=0.75 and 1.00), r-band (1.5), and i-band (2).
1 Sorting out the apparent discrepancy of having a galaxy, as a point source, being too faint to be detected (e.g. SDSS g-band limit is 22.2 mag) , yet as an extended source being easily detectable will be left to another day.
Next, a somewhat brighter galaxy.
One more 'full treatment' galaxy: circular, uniformly luminous disk of radius 10 kpc; Mbol = -22.3
z....mbol Bolometric SB
0.01 10.86 20.59
0.10 16.00 20.94
0.25 18.19 21.50
0.50 19.96 22.29
0.75 21.04 22.96
1.00 21.82 23.54
1.50 22.92 24.50
2.00 23.69 25.30
Oh what a difference a mere 4.6 mags makes! This galaxy is easily detectable out to z=1, at 1.5 it would be too faint to detect if it were a point source in any SDSS band1 (but still marginal as an extended source), and invisible at z=2.
Now suppose this galaxy emits all its electromagnetic radiation at 666.5 nm. In the SDSS system, it is not visible beyond z=0.25 (among the eight cases above), and it moves through the bands quickly, going from r (at z=0.01) to i (at 0.10) to z (0.25).
You may be wondering why I've chosen these two toy galaxies; why, for example, pick Mbol = -17.7 and -22.3? Well, there is a method to my madness ... but first I've a few other galaxies to present for your pleasure.
Finally, a request: would some kind reader(s) please check my calculations (etc)?
1 Sorta; while the SB remains well above the threshhold, as a point source it'd be undetected by SDSS in both i and z bands at z=1 and z=1.5, and also at z=0.75 in the z-band. Of course, with an integration time per band of not even a minute (except for Stripe 82), and a wimpy 2.5m telescope, SDSS is far from being the deepest ground-based survey!
Thanks Jerry! The cows are actually spherical, but because we observe them from a great distance they appear circular.
Three more galaxies, but this time with data for only z=0.01, 0.25, 0.5, 1, and 2.
The first has an Mbol of -19.7:
z....mbol Bolometric SB
0.01 13.46 23.19
0.25 20.79 24.10
0.50 22.56 24.89
1.00 24.42 26.14
2.00 26.29 29.90
Hmm, is it a coincidence that these values are just the same as those for the Mbol of -17.7 galaxy, only 2 mags brighter? In any case, it would be detectable out to z~0.5.
Suppose this galaxy emits all its light at a wavelength of 350.0 nm; locally it would be visible in the SDSS u-band, at z=0.25 and 0.5 it'd be a g-band object, by z=1 an i-band one, at at z=2, invisible.
What if Mbol = -21.9?
z....mbol Bolometric SB
0.01 11.26 20.99
0.25 18.59 21.90
0.50 20.36 22.69
1.00 22.22 23.94
2.00 24.09 25.70
Much like the second galaxy; easily detectable out to z=1, invisible at z=2.
And if this galaxy shines only at 558.3nm? Then it starts as an r-band object, becomes an i-band one at z=0.25, a z-band one at 0.5, and is invisible thereafter.
Finally, for now, Mbol = -21.3. As this is just 1 mag fainter than the second galaxy (Mbol = -22.3), I'll leave you, dear reader, to work out how far out it has to be before it 'has completely sunk' (to use D&L's term). Let's see what happens if this galaxy emits all its light at 460.0 nm: locally it's a g-band object, at z=0.25 an r-band one, at 0.5 an i, at z=1 a z-band object, and invisible at z=2.
Next: what if all five galaxies were, in fact, just the same galaxy?
But could you see it, with your own eyes, on a clear, moonless night? After all, its surface brightness is only 18.81 mag/arcsec^2, despite the fact that, at 10 pc from us, its total light would be a mere 2 magnitudes (or so) fainter than the Sun.
Obviously, only if it emitted light in that part of the electromagnetic spectrum your eyes are sensitive to (that would rule out the first of my galaxies, even if it weren't so faint; none of its light - which is in the UV - would even get to the ground). Suppose all the light was emitted close to the part of spectrum which our eyes are most sensitive to. For dim light, our eyes' PSF is approx 4' (radius), say. In this case, the centre of this magnificent disk galaxy, a mere 10 pc away, would be visible with the unaided eye ... just. I get an integrated magnitude of ~5.7! (please check my calculations).
A quick way to check the scaling: the full Moon, at apparent visual magnitude -12.7 and angular area around pi/16~0.20 square degrees, has surface brightness of V=-12.7+2.5 log (0.20) = -14.44. Or in the usual galaxy units of equivalent magnitude, we have to add 5 log (3600) = 17.78 to get V=3.3 per square arcsecond, which makes sense from comparing visibility of planets and bright stars near occultation. So our toy galaxy is 15.5 magnitudes fainter than that, which would, for example, correspond to a source with the Moon's angular size but visual magnitude 2.8. That's not too different from the Small Magellanic Cloud (NED lists integrated V=1.87 over an area 185x320 arcminutes ~ 12.9 square degrees, which would mean V=22.4 per square arcsecond (of course, the inner parts which are visually obvious are rather brighter than this). For such extended sources, one can do much better than the otherwise equivalent point-source limit (visually and instrumentally) because detection averages over a much larger region than the PSF.
Using an apparent visual magnitude (i.e. V-band) of -12.7 for the Moon, and a half degree diameter, I get V=3.3 per arcsecond squared (to one decimal place). A patch of sky the size of the Moon, at the centre of our toy galaxy, 10 pc away, assuming it emits all its light in the V-band, would have an apparent V-band magnitude of 2.8.
For the purposes of this thread, just how much better, and why, are important.For such extended sources, one can do much better than the otherwise equivalent point-source limit (visually and instrumentally) because detection averages over a much larger region than the PSF.
Earlier, I wrote:
A circular area, radius 2.1", of a patch of sky with a uniform surface brightness of 25 mag/arcsec^2, has an integrated magnitude of 22.2.Originally Posted by Nereid
First, though, let's explore (observe) my toy
galaxiesgalaxy a bit more.
Put all five of the galaxies together. They would have an Mbol of -23.1, and the bolometric SB locally would be 19.7 mag/"^2.
So, how easy is it to detect a faint object against the sky?
If the object's surface brightness is larger than that of the sky, then it's easy: it sticks out in an obvious way. Done. Example: a perfectly still lake has a surface which is perfectly flat. A turtle sticks its head up so that it is 1 inch above the surface. "Look -- there it is!"
If the object's surface brightness is less than that of the sky, it's still possible. The key is whether the surface brightness of the object is larger than fluctuations in the sky background. If the answer is "yes", then it's still possible without great effort. Just model the background, subtract it, and look for items which exceed the level of the residuals. Example: a lake has waves which are amazingly perfect sinusoids -- there are no ripples or foam, just a smooth surface which rises and falls in a perfectly regular way with an amplitude of 1 foot. A turtle sticks its head up so that it is 1 inch above the surface. That does not stand out at first glance, because the waves are much larger. But, if one takes a picture, makes a model of the waves, and subtracts the model -- the leftovers will be a perfectly flat surface. The turtle's head is the only positive residual from the model. "There it is!"
If the object's surface brightness is smaller than fluctuations in the background, then detecting it is hard. Example: a lake has realistic waves, which are basically sinusoidal, but with small ripples and foam irregularities of size around 2-3 inches. A turtle sticks its head 1 inch above the water. You take a picture, make the best model you can, subtract it from the picture ... and the leftover water surface still has ripples and holes which are larger than 1 inch in size. "Where is that turtle?"
Am limited today, and tomorrow, so just a couple of brief words.
Thanks SM, and ngc3314,;this aspect (instrumental detectability of low SB objects) is something which I will certainly be interested in exploring more deeply, but later.
Any word from the others who've posted in this thread previously? Recently Jerry, ngc3314, StupendousMan, parejkoj, and antoniseb have posted; what about you others?
Specifically, George: what do you think about the colours of the toy galaxy?
Suppose T1 - toy galaxy #1, introduced by ngc3314, with an Mbol=-24, uniformly luminous circular disk, radius 10 kpc - were the surface of a sphere, at a distance of 10 pc from us, emitting all its electromagnetic radiation in the V-band. Would the dark, cloudless/clear, moonless night sky be sufficed with a uniform glow, of apparent magnitude ~2.8?
I'm not sure how much cloud/fog/smog would be needed, or how early in the morning/late in the afternoon (etc), but the Sun's apparent visual magnitude is surely often very close to -24; spreading this much light over the whole sky (both day and night) would - if I understand it correctly - merely make the sky glow like the diffuse light from a nearby large city (in my personal experience, it doesn't take much to drown out the SMC, and M31 is even easier to sink).