Bias effects in galaxy detection

[Nereid]

Jerry, George, and BigDon (and any other interested reader lurker): do you think you could sketch the steps you think should be taken to work out an answer?

You a proposing a freshman astronomy approach to a problem that is much more complex than calculating the sky drop-out in different bandwidths.

You say this as if it's not worth doing, even as the first part to a more thorough examination of the question.

You also seem to have not really understood what I was outlining; may I ask how you came to conclude I was proposing to (merely) calculate "the sky drop-out in different bandwidths"

Look at the galaxies that we can see in the ultra deep Hubble fields. they are warped, torn and distorted.

Some are, some aren't.

How is this relevant to what I suggested, may I ask?

The lensing elements are both gravimetric and (likely) distorted by many pockets of intervening dust and gas.

Are they? How did you arrive at that conclusion, may I ask?

The Lyman forest tells us that the light is reaching us in tatters.

Colourful analogy; how is it helpful, may I ask?

Freshman calculations aside, the best we could hope for, is that the most distant events are directly mirrored by fairly local events and we can then make statistical assumptions based upon this premise.

It is? If you haven't done the work - of the kind I outlined - how would you know?

But current theory also dictates an evolution element; which is 'confirmed' by the fact that all of our freshman calculations indicate galaxies were brighter in the past than they are today.

I think you may not have really understood what you've been reading.

In any case, you seem to be saying that it's simply not even worth doing the kind of research I sketched; is that what you're saying?

We can accept this at face value;

Um, how, may I ask, did you so badly miss the central point of my posts?

but as our depth of knowledge about the spectral features of these most distant observations become clearer; if their root structure, metallicity and other evidences of evolution are found wanting; we should wonder how this new evidence fits with our best prior explanations.

It's like you're reading an 18th century novel, and I'm reading a 20th century thesis dissertation!

What has this got to do with testing Disney&Lang's ideas (per their paper)?

Do you think it a complete waste of time to test them?

Years ago, someone on this board discribed a scientist as someone who immediately explains the reason behind something that they told you was impossible in last year's lecture. This year, we are giving a Nobel prize this to three scientists who 'found' what Einstien called his biggest blunder. Something is missing in this equation, throwing in another ad hoc parameter is not the best solution.

And you've been saying things like this for years too.

When, may I ask, do you think you'll get down from your soapbox, roll up your sleeves, and start doing some real work?

What do I propose? I have already stated several times: Start with a shotgun of initial of possible theories and run the light through all the different wringers: Which one provides the best match with all of the modern evidence? Don't forget to remove your deeply ingrained preconceptions - which have been proven impossible - from your analysis.

OK, got it.

You clearly think the research I've suggested - in my various posts in this thread - is a complete waste of your time. Thanks for letting us know. Please don't waste any more of your time even reading the posts in this thread. Instead, why not concentrate on getting your proposal (now many years' old) into a form that you could - at some future time - start to think about how you could, perhaps, in another two decades or so, begin to test it?

[Nereid]

By chance, I was browsing astro-ph, and came across The Cosmic Origins Spectrograph.

The abstract includes this: "For faint targets, with flux F_lambda ~ 1.0E10-14 ergs/s/cm2/Angstrom, COS can achieve comparable signal to noise (when compared to STIS echelle modes) in 1-2% of the observing time." And I thought, Huh?!? the galaxy whose SDSS spectrum I linked to surely isn't 'faint', yet from ~6000 to 9000 Ĺ, F_lambda is ~1.3E10-16 ergs/s/cm2/Angstrom!

Reading the paper, I think I can see why SDSS galaxy spectra can be easily two orders of magnitude brighter than COS* ones (in F_lambda):


BTW, the galaxy is SDSS J104155.65+074513.9, and CAS gives its photometric pipeline outputs as:
fiberMag_r 18.62 mag
petroMag_r 17.22 mag
devMag_r 17.07 mag
expMag_r 17.41 mag
psfMag_r 18.60 mag
modelMag_r 17.07 mag
petroRad_r 5.810 arcsec

* interestingly, the COS 'aperture' is 2.5", close to that of the SDSS spectrograph (which is 3")

From here, the SDSS r-band filter has a FWHM of ~147 nm, and a central wavelength of ~629 nm.

SDSS J104155.65+074513.9's spectrum is more or less flat in the r-band; suppose F_lambda is 1.3E10^-16 ergs/s/cm²/Angstrom throughout*; that gives it an r-band flux (or flux density!) of ~1.9E10^-13 ergs/s/cm².

From the SDSS DR8 page I linked to earlier:

A "maggy" is the flux f of the source relative to the standard source f₀ (which defines the zeropoint of the magnitude scale). Therefore, a "nanomaggy" is 10^-9 times a maggy. To relate these quantities to standard magnitudes, an object with flux f given in nMgy has a Pogson magnitude:

m = [22.5 mag] − 2.5 log ₁₀ f .

[...]

The standard source for each SDSS band is close to but not exactly the AB source (3631 Jy), meaning that a nanomaggy is approximately 3.631× 10^-6 Jy.

Almost there; what's an r-band flux (or flux density!) of ~1.9E10^-13 ergs/s/cm² expressed as an r-band (Pogson) magnitude (approximately)?

* it isn't, of course, but let's assume that cows are spherical

[StupendousMan]

The numbers posed earlier in this thread are off by quite a bit. It's not so hard to do the calculations -- I did it with students in class earlier this week.

A star of magnitude V = 0 has a flux density of roughly 3.6 x 10^(-9) ergs per sq. cm. per second per Angstrom. You can find this value in Allen's "Astrophysical Quantities", in other good textbooks, in many many websites, such as

http://www.sr.bham.ac.uk/~somak/constants.html

So, what does that mean? It gives the number of ergs of energy which will be collected each second if one points a telescope with 1 sq. cm. of collecting area at a star of magnitude zero, with a filter 1 Angstrom wide. Let's convert to the number of photons per second for that same telescope. The V-band is roughly -- very roughly -- 1000 Angstroms wide. The central wavelength of light in the V filter is roughly 5500 Angstroms. Now, physics tells us that the energy associated with a photon of wavelength lambda is h*c/lambda, where h = Planck's constant, c = speed of light, and lambda = wavelength in meters.

h * c / (lambda) = 6.626x10^(-34) kg m^2/s * 3 x 10^8 m/s / (5500 x 10^(-10) m) = 3.6 x 10^(-19) Joules

= 3.6 x 10^(-12) ergs

Okay. So, if we start with the flux density from a star of magnitude V = 0, we can compute the number of photons which will be collected by a telescope of collecting area 1 sq.cm. each second:

3.6 x 10^(-9) erg / (sq.cm.*s*Angstrom) * 1000 Angstrom / ( 3.6 x 10^(-12) erg/photon ) = 1 million photons

So, the rule to remember is that a star of mag V = 0 will yield about 1 million photons per second per square cm of collecting area. Well, above the atmosphere, and assuming perfect optics and detector, yadda yadda yadda.

Look at the galaxy mentioned in the post above: the "faint" galaxy is said to have a spectrum with a flux density of about 10^(-16) ergs per sq.cm. per second per Angstrom. That's roughly 10^7 times smaller than the flux of a V = 0 star. That means that the magnitude of that galaxy must be roughly 2.5 * log10 (10^7) = 17.5. Yup, that's a pretty faint galaxy.

This stuff is covered in advanced undergraduate observational astronomy courses -- not that there are many of those -- and graduate courses. Actually, a lot of it is learned on the street from colleagues, since there really aren't that many people who need to know in their regular jobs. You can find "exposure calculators" at the websites of many observatories which will do that calculations for you, taking a magnitude and figuring out how many photons per second will reach the detector.

[Jerry]

That is effectively an infinite universe of starting theoretical possibilities. Your low opinion of the intellect and intellectual honesty of my entire profession is noted, but surely you could propose a realistically attainable alternative?

This thread contains a discussion that is WAY TO GOOD to jumble it up with any more cross talk. Sorry for the jabber I will be brief:

The short answer is no. I have worked as exhaustively as I can to find alternatives that fits all of the known facts, and so have many others. But it is important to keep prodding, because many parameters have already been added, and they are not all right. Granting a Nobel prize for throwing in a new constant should not put the stamp of scientific approval on the latest tweak of the nob. Antagonistic science is good science insofar as it incourages an investment in new ideas, new designs, new approaches and most especially, new funding.

I think the lunar gravitational mapping probes (Grail) are an excellent example of how asking the right questions has lead to a new and clever test.

We need more of the same.

[George]

Can you expand a little please?

Here is what I did, though I see I goofed with the math, but the procedure is possibly correct if you only want a rough value. If I had the data set, I could do a more respectable job, if for some reason such a result would be useful.

Since E=hc/lambda , then the “blue” photons will have twice the energy of the far “red” ones, which are twice the wavelength. This will squash the blue end of the spectrum if we convert the curve to a photon flux curve. The peak shifts to the red. [The Sun’s shift is from 495nm (using a Planck peak @ 5850K -- the real peak is closer to 450nm, surprisingly) for a wattage curve to 695nm to a photon flux curve peak.]

As I understand it, and without an appropriate book or teacher, the integration (ie area) of the curve represents the flux density because the power stated for the y-axis is per unit wavelength. [Be sure to extrapolate into the IR band to get the bulk of the complete spectrum.] So what we want out of this spectrum is the power (ergs per second) value that is a result of the integration. The Sun, for instance, has a peak sp. irr. value of over 2,100 watts m^-2 @ 1 AU, but we know the actual wattage (ie Solar Constant) is only ~ 1,361 watts m^-2 @ 1 AU. So, in the Sun’s case, the value is 64% of the peak.

Arbitrarily using this same percentage would give us about 9.5 E ^-17 ergs s^-1 cm^-2 for the example galaxy spectrum.

Now choose a wavelength that best represents the place the flux distribution is balanced. Say it is about 1500nm, which has a photon energy of 1.32E^-12 ergs. So dividing this energy into the power above (per cm^-2) yields a photon flux rate of 7.20E^-5 per cm ^-2. This means you would need a 1.3 meter aperture scope to get one photon per second.

Obviously, this is a very rough value given the lack of effort to get hard values, but it should be a reasonable approach to the problem.

It is much too late to study the other related posts by the more credible posters, but one concern I have regarding Vega is that it is a “blue” star (9600K) so there will be fewer photons coming from it compared to a red star, for instance, having the same apparent magnitude -- perhaps by a factor of 2 to 5, which isn't much of a variance I suppose.

I hope the others will review this approach since it is just something I did on a wing. [It was a wing that stayed on the ground for a long time, admittedly. ]

[George]

The rule I heard from Joe Wampler in grad school was V=0 corresponds to 1000 photons/(cm² second Angstrom) in the middle of the V band. I once did a more careful calculation and got something like 1090, but adding significant figures starts to need so many details (spectral shape, filter passband) that it ceases to have very general use.

That makes sense given the value StupendousMan gives us in his post, namely 3.6 x 10^(-9) ergs per sq. cm. per second per Angstrom for V=0, (within the V band, not the entire spectrum).

As he did, 550nm does give us 3.6 x 10^(-12) ergs per photon, so the simple division does give 1000 photons per second per cm. I do think, however, that the SED of the object will vary the flux since fewer blue photons are needed to carry the same amount of energy carried by red ones.

Okay. So, if we start with the flux density from a star of magnitude V = 0, we can compute the number of photons which will be collected by a telescope of collecting area 1 sq.cm. each second:

3.6 x 10^(-9) erg / (sq.cm.*s*Angstrom) * 1000 Angstrom / ( 3.6 x 10^(-12) erg/photon ) = 1 million photons.

[my bold] Multiplying by the bandwidth will not work for SEDs because integration is required to determine the actual flux density. At least I am fairly sure of this. In other words, the y-axis values are in some sort of derivative form when they use the per wavelength term. [There is likely a more appropriate way to say this and I'd like to know what it is, assuming I'm right, of course.]

[ngc3314]

[my bold] Multiplying by the bandwidth will not work for SEDs because integration is required to determine the actual flux density. At least I am fairly sure of this.

and to add one more complication (that really does seem to be talked on the street but seldom written): when the detector counts photons, you have to do the integration after converting the spectrum into photon units, because, for example, a CCD counts one photon (i.e. one electron at readout) no matter what the photon energy is, so that a blue photon has more leverage in energy space. This matters in a differential sense when the spectral shapes of two objects being compared are very different; the mean energy of V photons from a red object is distinctly smaller than for a blue object. This matters less for narrow bands and more for broad bands; for years, X-ray astronomy largely consisted of modeling the range of contributions of a few models allowed by an instrumental response that was an order of magnitude wide in energy. (No, in comparison to that, the magnitude scale in optical astronomy introduces no complications...)

[George]

and to add one more complication (that really does seem to be talked on the street but seldom written): when the detector counts photons, you have to do the integration after converting the spectrum into photon units, because, for example, a CCD counts one photon (i.e. one electron at readout) no matter what the photon energy is, so that a blue photon has more leverage in energy space. This matters in a differential sense when the spectral shapes of two objects being compared are very different; the mean energy of V photons from a red object is distinctly smaller than for a blue object.

So you are saying that the electrons aren't colorful enough? [If blue and red electrons emerged from sensors, life would be easier.]

I think I see your point. If all you have is an electron count for a given sensor reading, say within the V-band, then how would one know what percent were from "blue" photons and what percent were from "red" ones, as well as the other colors? Even if we knew the reactiviness of blue ones vs. red ones with the sensor, we still wouldn't know what the actual spectral flux was. If, however, we already knew the object's surface temperature we could apply a Planck distribution to the spectral sensitivity of the sensor and make the adjustments. Am I close? [It's not quite as bad as, "If we had some ham, we could have ham and cheese, if we had some cheese. ]

[Nereid]

Obviously, my memory isn't what it used to be.

At a level much below that we are concerned with in this thread (i.e. ~1%), there are - even for SDSS - unresolved issues. The AB system is what SDSS uses* (Oke & Gunn (1983)); not exactly the same as that in the link in StupendousMan's post?

George: I think there are some potentially very serious challenges to working from a blackbody spectrum, even an 'equivalent temperature' one. Even for the Sun, there's limb darkening, and the fact that, at the limb, a line of sight integrates over different photospheric depths than at the centre, so even if the SED at any depth in the photosphere were a blackbody (which it isn't), the integrated (whole solar disc) SED will not be all that close to a blackbody. When it comes to certain galaxies, this approach can be hopelessly wrong! Consider, for example, SDSS J085604.39+344234.8: the continuum is vaguely blue, but the narrow emission lines surely totally dominate the SED.

* "by which a magnitude 0 object should have the same counts as a source of Fν = 3631 Jy"

[StupendousMan]

If you want to get the answer right to within a factor of 2 or 3, you can use the quick sort of calculation I demonstrated a few posts back.

If you want to get the answer right to within a few percent, you need to

a) start with a spectrum of the object in question; you can use spectra from Pickles http://cdsarc.u-strasbg.fr/viz-bin/Cat?J/PASP/110/863
b) find a good model for the instrumental bandpass (filter + air + detector ...)
c) break the bandpass into small pieces -- say, 1 Angstrom wide
d) compute the number of photons per sq.cm. per second per Angstrom in each piece
e) add up all the photons from all the pieces

In other words, one must numerically integrate the spectrum across the bandpass. People don't use the blackbody approximation for this sort of calculation, because there's no point to going to all this trouble without using a more accurate spectrum.

[StupendousMan]

At a level much below that we are concerned with in this thread (i.e. ~1%), there are - even for SDSS - unresolved issues. The AB system is what SDSS uses* (Oke & Gunn (1983)); not exactly the same as that in the link in StupendousMan's post?


* "by which a magnitude 0 object should have the same counts as a source of Fν = 3631 Jy"

There are two main types of photometric calibration in the optical: "Vega-based" and "AB". "Vega-based" systems assign a magnitude of zero (or something very close to zero -- you can ask me to explain but it will take a while) to Vega in each passband. So, to a good approximation, Vega has B = 0, V = 0, R = 0, I = 0, etc. This is easy to do in practice, because the calibration is based on a real source that everyone can observe. On the other hand, it's a pain to compute the number of photons per second from a star, or the number of ergs per second, because there's no theory behind the numbers. The number of ergs per second per sq. cm. per second from a V = 0 star isn't the same as the number for an R = 0 star, or for an I = 0 star, etc.

"AB" systems, on the other hand, are more theoretically grounded. The idea is that a star of magnitude zero in any band will produce the same flux density. It is much easier to figure out how to convert from magnitude to flux density in this sort of system, because you only need to know one conversion factor: it will be the same for all passbands. On the other hand, since it is nearly impossible to MEASURE the actual flux density from a star with high precision, these systems have problems assigning magnitudes to stars; there's always the possibility of a significant systematic error. The SDSS magnitude system is one of the AB systems -- Nereid's post provides the zero-point conversion factor for it.

It would be nice if someone would fly a small telescope above the atmosphere and measure the flux density of a few bright stars against some fundamental flux calibration system -- everyone in the entire astronomical community would benefit from this. Unfortunately, funding agencies would rather give money to someone who can "do science" by making measurements of some galaxies or planets or whatever, than give money to calibration efforts. I guess they figure that calibration is unimportant.

[George]

I think there are some potentially very serious challenges to working from a blackbody spectrum, even an 'equivalent temperature' one.

The blackbody approach is never going to be very accurate. The closer a given SED is to a Planck distribution, the better the accuracy, but even then it is obvious I took the "shooting from the hip" approach by guessing a weighted midpoint for wavelength, etc.

Even for the Sun, there's limb darkening, and the fact that, at the limb, a line of sight integrates over different photospheric depths than at the centre, so even if the SED at any depth in the photosphere were a blackbody (which it isn't), the integrated (whole solar disc) SED will not be all that close to a blackbody.

I assume the SED (Spectral Irradiance for the Sun -- an extended object) for the Sun is an integrated SED, which is why they measure the distribution as an irradiance. [I don't think I ever asked if this is the correct view. Am I right?]

When it comes to certain galaxies, this approach can be hopelessly wrong! Consider, for example, SDSS J085604.39+344234.8: the continuum is vaguely blue, but the narrow emission lines surely totally dominate the SED.

That is a nice example of a SED that if converted to photon flux density would likely be flat as a pancake since the blue end gets cut in half compared to the far red end since the wavelength for blue is only half that of blue. So, perhaps, in this case the incident photon flux will be easier to calculate. But what comes out in electron counts is where the devil seems to be having fun.

I mention the photon flux more as novelty and look to ngc3314 and StupendousMan for the real world stuff.

[Nereid]

What? A whole week has passed?!?

Well, I've been having fun, trying to get my head properly around "flux", "flux density", F_ν, F_λ, f_ν, and f_λ, "intensity", "magnitude", "surface brightness", etc, etc, etc. All this as a prelude to being able to start - yep, that's right, merely start - to look at how some actual galaxies behave, when observed by different survey-grade facilities, at different wavelengths. My starting point is (or will be, one day) the Tal and van Dokkum 2011 paper on stacked LRGs I have mentioned several times.

Here is a nice summary, with lots of the key formulae and concepts bundled into a few short (PDF) pages: ASTROPHYSICAL INFORMATION DERIVED FROM THE EM SPECTRUM.

I've also learned a lot about how to extract data from the online SDSS databases, using CasJobs, Yay!

[StupendousMan]

Way to go, Nereid! Keep up the good work.

It takes a typical grad student 3-6 years to absorb enough of this material to do original research. Sounds like you're ahead of schedule :-)

[Nereid]

Thanks Stupendous Man.

I'd like to start with some simplifying assumptions.

The TvD11 stacked LRGs have r, i, and z (μ_λ vs radius) profiles that are pretty similar; let's assume they are the same [1]. Let's also assume that these galaxies have a smooth continuum all over (i.e. the spectrum taken through a pinhole - circular aperture of 0.01" radius, say - is the same, no matter where on the galaxy it's taken, modulo the integrated flux), no pesky absorption or emission lines to have to worry about. Further assume that the SDSS ugriz magnitudes are AB magnitudes (there is a small difference, but at the ~1-3% level), i.e. their zero points have the same f_ν. That means - I hope! - that we can move the stacked LRG (treating it as a single galaxy, not an ensemble) closer to us, and further from us, and we can construct an artificial filter/passband/waveband within which colour and surface brightness changes should be easy to handle ... as long as the artificial filter doesn't go longer than the red end of the i-band, or shorter than the blue end of the r-band.

But first, a question: TvD11's Figure 6 y-axis is labelled "μ_λ [mag/arcsec²]"; however, SDSS ugriz magnitudes are AB magnitudes! Why μ_λ? Why not μ_ν?

Anyway, in a Euclidean universe, surface brightness is constant, it does not change as the source gets closer or further. If our test LRG is unchanged, we can determine the apparent effective radius, in arcsecs, at any distance, from its (μ_λ vs radius - in arcsecs) profile (this is true no matter what the geometry of the universe! it follows from the definition of 'effective radius'). In such a universe, then, our trusty LRG becomes marginally detectable, as a galaxy, when its apparent diameter, in arcsecs, is the same as the seeing. And what is its apparent diameter? Twice the distance, in arcsecs, from the nucleus to the point where the surface brightness equals the sky. [2]

So, first, in a Euclidean universe, how big does our LRG appear, at z = 0.01? At what z does it become indistinguishable from a star?

Anyone - other than Stupendous Man, NGC3314, parejkoj, etc - like to try to work that out?

[1] Let's also assume the effective radius is the same, in all three passbands; TvD11 found small, but significant, differences

[2] These two aspects can be tweaked somewhat: a cleverly designed survey may be able to tell the difference between a point source and an extended source somewhat more finely than the average seeing; the outer limit of an object may be traced to somewhat below the sky. These are, however, refinements that we can add later.

[Jerry]

Way to go, Nereid! Keep up the good work.

It takes a typical grad student 3-6 years to absorb enough of this material to do original research. Sounds like you're ahead of schedule :-)

...unless you cheat and read the last chapter first.

[StupendousMan]

Thanks Stupendous Man.


But first, a question: TvD11's Figure 6 y-axis is labelled "μ_λ [mag/arcsec²]"; however, SDSS ugriz magnitudes are AB magnitudes! Why μ_λ? Why not μ_ν?

It's simply a practical issue. Most passbands in the optical are described in terms of transmission vs. wavelength. So, if one wants to convolve some spectrum with those passbands to compute synthetic magnitudes, it's much easier if the spectrum is expressed in terms of wavelength.

[ngc3314]

It's simply a practical issue. Most passbands in the optical are described in terms of transmission vs. wavelength. So, if one wants to convolve some spectrum with those passbands to compute synthetic magnitudes, it's much easier if the spectrum is expressed in terms of wavelength.

And then there are those astronomers (some of them Quite Eminent) who made a habit of plotting spectra as F-nu versus wavelength. Where are the torch-waving villagers when you need them? Or is it too much of an idiosyncracy to want the integral under the curve to have an obvious meaning? (OTOH, I do see use for those nu*F-nu plots as long as it's versus frequency).

[George]

Here is a nice summary, with lots of the key formulae and concepts bundled into a few short (PDF) pages: ASTROPHYSICAL INFORMATION DERIVED FROM THE EM SPECTRUM.

That's helpful.

You're right, ngc3314, about the 1000 photon flux a zero mag. The link shows that, in georgeeze, if some punk -- in your inertial frame -- was shining a green laser at you, and its mag. was 0, then you would be receiving 1,005.1 green (550nm) photons per sec. per cm². At least that's my interpretation of Zero Point monochromatic app. mag.

[George]

And then there are those astronomers (some of them Quite Eminent) who made a habit of plotting spectra as F-nu versus wavelength. Where are the torch-waving villagers when you need them?

Give me a sign; I'll march! I've seen those labels and they are salt on the wound because they don't make sense unless you do integrate -- so they need to use appropriate labeling. In some cases they don't even show a derivative form for the y-axis. Heliochromology almost died during its infancy because of this. Hmmmm, that might actually explain why they do that, I suppose.

[Nereid]

But first, a question: TvD11's Figure 6 y-axis is labelled "μ_λ [mag/arcsec²]"; however, SDSS ugriz magnitudes are AB magnitudes! Why μ_λ? Why not μ_ν?

It's simply a practical issue. Most passbands in the optical are described in terms of transmission vs. wavelength. So, if one wants to convolve some spectrum with those passbands to compute synthetic magnitudes, it's much easier if the spectrum is expressed in terms of wavelength.

And then there are those astronomers (some of them Quite Eminent) who made a habit of plotting spectra as F-nu versus wavelength. Where are the torch-waving villagers when you need them? Or is it too much of an idiosyncracy to want the integral under the curve to have an obvious meaning? (OTOH, I do see use for those nu*F-nu plots as long as it's versus frequency).

Thanks StupendousMan, thanks ngc3314.

I'm still puzzled. Here's why (bear with me please; this is an interactive session, in which I try to write in [ t e x ]):

First, the standard definition:



where is the integrated magnitude of the source and is the angular area of the source in units of arcsec².

Presumably the following is also correct (is it?):



where is the integrated magnitude of the source and is the angular area of the source in units of arcsec².

How do these relate to spectral flux density, or ?

Well, observations are made in bands (wavebands, passbands):



where is the system response function. Now if is perfect, then:



Presumably the following is also correct, for a perfect system response function (is it?):



Almost there.

For a perfect band, the limits of integration are and ; they are also, for , and .

And, if it's the same band: , .

Now a band is a band is a band; assuming perfection (including the definition of a particular band!), if = 12.3 mags (say) for a particular star, does it make sense to write mags (or mags)?

[Nereid]

Thinking about this another way ...

The perfect filter allows all the wavelengths (frequencies) of EM radiation, within a specific range, through, perfectly, and none whatsoever outside that range. The perfect detector produces an output that is linear with respect to the total energy that comes through the filter (directly proportional, constant = 1). Situated above the Earth's atmosphere, oriented normal to the incoming wavefronts, with an area of 1 cm², the system (filter plus detector) outputs values of energy per unit time.

In what way does it make sense to refer to these energies (per unit time, per ...) - as 'wavelength' mags ()? as 'frequency' mags ()?

[StupendousMan]

First, I don't understand the first two equations you wrote in message 81. The letter "m" is used for magnitude -- okay. The letter "mu", in astronomy, sometimes means surface brightness, in magnitudes per square arcsecond. The letter "Omega" can mean the surface area of an object .. well, maybe. But the three quantities aren't related by the equation that you gave, as far as I can tell, and I don't see why one would bring surface brightness into this discussion, anyway.

Moving on, I don't think that it makes sense to talk about "wavelength" magnitudes or "frequency" magnitudes. We collect energy or photons from a distant star, and measure the amount of energy per unit area per unit time. We then assign a number called the magnitude to each star, based on how much energy we receive. Once you choose some particular passband, it doesn't matter whether you describe it in terms of frequency or in terms of wavelength: the amount of energy which is collected from a given star through that passband is the same.

I get the feeling that I'm missing the thrust of your questions. If you ask in a different way, maybe I'd be able to understand the questions better. Sorry :-/

[Nereid]

First, I don't understand the first two equations you wrote in message 81. The letter "m" is used for magnitude -- okay. The letter "mu", in astronomy, sometimes means surface brightness, in magnitudes per square arcsecond. The letter "Omega" can mean the surface area of an object .. well, maybe. But the three quantities aren't related by the equation that you gave, as far as I can tell, and I don't see why one would bring surface brightness into this discussion, anyway.

Thanks!

My post started out as one thing, but quickly became all about trying to get [ t e x ] tags to work properly.

What I am working towards is my own answer to an earlier question, here re-phrased using tex tags:

TvD11's Figure 6 y-axis is labelled " "; however, SDSS ugriz magnitudes are AB magnitudes! Why ? Why not ?

The first equation, in my earlier post, comes from the source I linked to earlier, the University of Virginia's "ASTR 511/O’Connell Lec 2"; in hindsight I should have *quoted*, and provided a source (the only parts I omitted are the heading - "Surface Brightnesses (of extended objects):" - this, at the end of the first bullet: "1 arcsec² = 2.35 X 10^-11 steradians", and the second bullet [1])

Whatever; the O'Connell document seems to have left out (i.e. implied) this: "at λ" (i.e. the definition of the surface brightness of an extended object, at a wavelength of λ, is ...).

Moving on, I don't think that it makes sense to talk about "wavelength" magnitudes or "frequency" magnitudes. We collect energy or photons from a distant star, and measure the amount of energy per unit area per unit time. We then assign a number called the magnitude to each star, based on how much energy we receive. Once you choose some particular passband, it doesn't matter whether you describe it in terms of frequency or in terms of wavelength: the amount of energy which is collected from a given star through that passband is the same.

Yes, that's the feeling I had earlier ... but the axis-label in Tal & van Dokkum's paper's is as I've quoted it, and the O'Connell document uses the same term.

Most of the rest of my post is about showing that, just as you write, it doesn't make sense to talk about λ magnitudes or ν magnitudes, if you're working with bands [2]. So, the λ subscript in TvD11's Figure 6 is irrelevant.

I get the feeling that I'm missing the thrust of your questions. If you ask in a different way, maybe I'd be able to understand the questions better. Sorry :-/

I hope it's clearer now ...

[1] The second bullet reads:

μ is the magnitude corresponding to the mean flux in one arcsec² of the source. Units of μ are quoted, misleadingly, as “magnitudes per square arcsecond.”

[2] Clearly specifying, or defining, the magnitude system is important of course!

[ngc3314]

As I think about it, magnitudes would be usually defined only for a particular band (UBVRI, ugriz, whatever), and then the wavelength-frequency distinction would be mot - the band is what it is. The only way I can think of for m-lambda to make sense would be a sort of AB magnitude analog where data at different wavelengths are being combined-compared (such as because of redshift changes). There is a common UV magnitude system which is defined like m(UV)=constant - 2.5*log(F-lambda) across wavelengths (although one must still specify where it's measured for useful comparisons).

[StupendousMan]

TvD11's Figure 6 y-axis is labelled " "; however, SDSS ugriz magnitudes are AB magnitudes! Why ? Why not ?

I've just scanned the TvD11 paper, and can't find any explanation for the y-axis label in their Figure 6. I can only speculate that they mean to say, "the quantity plotted on this axis represents the surface brightness in u-band magnitudes per square arcsecond for the u-band points on the graph, and surface brightness in g-band magnitudes per square arcsecond for the g-band points, etc."

If I were reviewing the paper, I'd ask them to change that label or at least try to explain what they mean.

[Nereid]

Thanks ngc3314, StupendousMan.

The little excursion into understanding flux, intensity, magnitudes etc may have seemed a but of a diversion, a luxury, even irrelevant.

However, for me it was not. It allowed me to discover an error, an approximation too far, (or something similar) in the D&L paper.

At the bottom of page 10, top of page 11 (may not work perfectly, tex tags inside a quote, etc):

To keep things simple we consider only exponential galaxies and ignore Tolman dimming and cosmology for now (see later). If we adopt de Vaucouleurs (1959) 2-parameter intensity I(r) profiles for galaxies, i.e.

But this is not correct; the "2-parameter intensity I(r) profiles for galaxies" are band-specific!

So, more accurately, for the V-band there's:


for the B-band there's:

(you get the idea).

Does this matter? Well, yes, it does! Why? Stay tuned!

[ngc3314]

Expanding a bit on your last point, most kinds of galaxies have color gradients, either as a result of differing star-formation histories or changes in metal abundance. At their most extreme, they can make a galaxy look completely different between ultraviolet, optical, and near-infrared bands, something known in the trade as the morphological k-correction. An example is shown in this series depicting M81 across the spectrum. In the UV, the central bulge nearly vanishes, while in the near-IR we barely see the star-forming regions in the arms. At the least, this means that different light-distribution laws would be needed to model the detectability of such galaxies at low and high redshifts, unless one had data at closely matching emitted wavelengths (which accounts for the attention given to doing the Hubble Deep Fields in the near-IR, and the near-IR images of the giant CANDELS Hubble project).

[Nereid]

Expanding a bit on your last point, most kinds of galaxies have color gradients, either as a result of differing star-formation histories or changes in metal abundance. At their most extreme, they can make a galaxy look completely different between ultraviolet, optical, and near-infrared bands, something known in the trade as the morphological k-correction. An example is shown in this series depicting M81 across the spectrum. In the UV, the central bulge nearly vanishes, while in the near-IR we barely see the star-forming regions in the arms. At the least, this means that different light-distribution laws would be needed to model the detectability of such galaxies at low and high redshifts, unless one had data at closely matching emitted wavelengths (which accounts for the attention given to doing the Hubble Deep Fields in the near-IR, and the near-IR images of the giant CANDELS Hubble project).

A somewhat more detailed examination of these points may be found here (the author is a certain Bill Keel!), and here.

In addition to this, consider mergers. The first Galaxy Zoo paper (DR1, link to follow) found that 'mergers' constitute a minor, but not totally insignificant, fraction of SDSS 'galaxies' with z < 0.25.

The key points, so far as the D&L paper is concerned, would seem to be:
-> it's not just "Giant Ellipticals" which have a radial intensity profile closely approximated by β=4, but (local) ellipticals in general, and the bulges of (local) spiral galaxies (for bands such as B, V, g, r)
-> the observed profile for (local) spiral disks has a projection component, and edge-on disks have a rather different profile (again, for visual/optical bands)
-> "indeed many galactic disks are seen to deviate from (2) [the exponential profile] at smaller radii" (from the second source)
-> perhaps the most important - empirical - radial profiles to start with are those of local galaxies (as determined by their redshifts) in various UV bands, including those which have the rest Lyman-α (and higher) within them.

(more later)

[Nereid]

Section IV ("IMPRISONED BY LIGHT") looks, at skim-level, pretty impressive, and its conclusion* inescapable.

However, this whole section uses the material developed earlier, without any explicit recognition of its assumptions! This may - I haven't done enough work on it yet to be sure - prove fatal to its stated conclusions. In particular, the band-specific assumption seems to have been ignored, and the explicit "we consider only exponential galaxies and ignore Tolman dimming and cosmology for now" assumptions not revisited/restated/added-in/etc!

This is particularly ironic, as the very next section (V "HOW GALAXIES SINK FROM SIGHT") starts with these words (bold added):

The Visibility Window depicted in fig 3 is immutable, mathematical and pinned in local coordinates because it shows the contrast to ones local sky, be it on the ground or in space. What we need to calculate next are the properties, in particular the sizes and intrinsic SBs, of the kinds of galaxies, seen at different redshifts, which will make it through that narrow window, particularly near its peak, taking into account the Tolman effects described above, which both dim a galaxy and increase its apparent size.

Can you spot the - potentially fatal - inconsistency?

Disney & Lang calculate "The Visibility Window depicted in fig 3" assuming "only exponential galaxies and ignore Tolman dimming and cosmology" (and that the intensity profiles - especially the exponential one - are universal, and not band-specific). Then, assuming this result "is immutable" [sic!], they then proceed to take "into account the Tolman effects". To be quite clear, I haven't (yet) gone through their subsequent work; however, at first blush this looks like a fatal logical flaw. A more rigorous - and convincing - approach would be to re-work the earlier sections, starting with the more general (band-specific) intensity profile and Tolman dimming and cosmology.

(to be continued)

* "For all practical purposes then we are implacably imprisoned in our cell of light. 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. It follows that large hidden populations of low surface brightness galaxies, both near and far, cannot be ruled out by optical observations alone. This is a much stronger statement than could have been made before and it relies on the arguments which led to eqn.(26)"