special relativity - What is the equation of state for a relativistic fluid/gas?

Say we have a relativistic fluid/gas, as we have in some astrophyical systems.

Now let us write:

• $e$ - energy density in the fluid's rest frame.




• $P$ - pressure in the fluid's rest frame.




• $n$ - number density in the fluid's rest frame.




• $m$ - mass of the particles.

I know that for the non-relativistic case we have:

$$e=nmc^2+frac{1}{hat{gamma}-1}P$$

where $hat{gamma}$ is the adiabatic index. $hat{gamma}=1+frac{2}{f}$ for a gas with $f$ degrees of freedom.

For the ultra-relativstic case we have:

$$e=3P$$

My question is what is $e(P,n)$ for a relativstic case (which is the general case of the 2 limits shown above)? I would also like to know how to derive it.

---

Is the following way the correct way to do it ? :

The number density of particles is:
$$n=int_{0}^{infty} n_p(p) dp $$

The pressure is:
$$P=int_{0}^{infty} frac{1}{3} p v(p) n_p(p) dp $$

The energy density is:
$$e=int_{0}^{infty} epsilon(p) n_p(p) dp $$

where:

$$n_p(p)= (2s+1)frac{1}{ e^{({epsilon(p)-mu})/{k_B T}}+(-1)^{2s+1} } frac{4pi p^2}{h^3}$$

Here $s$ is the spin of the particles, for electrons $s=frac{1}{2}$.

$$ epsilon(p)=(m^2c^4+p^2c^2)^{frac{1}{2}} $$

$$ v(p)= frac{depsilon}{dp}=frac{p}{m}left(1+left(frac{p}{mc}right)^2right)^{-frac{1}{2}} $$

From calculating the three integrals above we can finally obtain $e(P,n)$.

• Can anyone confirm this is the proper way to do it, or am I missing something here?




• It seems as if those integrals cannot be solved analytically - is this
  true?




• Perhaps in this case there is no explicit formula for $e(P,n)$?

If we found evidence of life on Mars, how would we know that it originated on Mars rather than Earth?

Why would we assume that the early martian life originated on Mars,
rather than Earth?

There's still a whole lot we don't know. As Wayfaring stranger points out in the comments, Origin is a whole different question. It's possible that life originated outside our solar system and came to either Mars and/or Earth from outside the solar-system. I don't think anyone who studies this idea is "Assuming" life originated on Mars, only that the idea has a chance of being true.

Is it not likely that somewhere between 2.5-3 billion years ago, a
major volcanic eruption or meteor impact could have hit an earth
teeming with simple-life and sent it hurdling through space to Mars?

As I understand it, volcanic eruptions are unlikely to send anything into space unless it's a smaller sized moon. The escape velocity (young earth, maybe 10 km/s, young mars, maybe 3-4), volcanic eruptions, as far as I know, don't shoot things out at 10,000 - 20,000 MPH. But meteor impacts of sufficient size can do that.

Mars is a better meteor debris making target than the Earth cause it's smaller, so the gravity is lower and presumably it's mostly had a thinner atmosphere too. We've found martial meteors on Earth. We might not find any Earth meteors on mars cause it takes a much bigger impact to knock bits of rock off Earth and because the Atmosphere slows objects down, both coming in and going out.

I understand that my theory has the gravity of the sun working against
it, but I also see it not being entirely impossible.

The sun isn't as big a factor as you might think. Once something is knocked off a planet and it gets into orbit around the sun, gravitational assists can move it around further out or further inside the solar system. What presumably happens is that, with a big enough impact, many thousands if not millions of bits of debris get into solar-system orbit and from there, some of them land on other planets - probably much less than 1% of those hit Earth, but if it carries life that can survive the trip, all you need is one rock.

I mean, 100 years ago, one could assume that life originating on Mars
and then being sent this was was ridiculous and impossible but now
it's a rational, viable theory.

While that's true, the "People used to think this was impossible" isn't a scientific approach for what might be true. We should determine what is possible and/or thought to be likely, based on physical evidence, not what wasn't understood 100 years ago. Your example is a good point on why it's important to keep an open mind about the unknown. You can still make theories based on evidence, and keep an open mind on the unknown. There's really no conflict between the two.

Considering such a scenario, how would scientist verify that these
life-forms didn't originate from Earth, rather than Mars? It seems to
me that this question is the first question that would need to be
asked and answered after the fossil-records were found and verified.

It's a good question.

The simple answer is that Mars cooled first and Mars (likely) had oceans first, so it's a better candidate to have developed life first though extremophiles can live in hot oceans, so . . . time will tell.

bioinformatics - Can I compare Shannon indices of metagenome gene data?

I'm not an expert on Shannon-Weaver Index, but according to wikipedia it is the same as exponentially transformed Renyi entropy. If it is the case, you can compare them since they are scale invariant summary statistics. If you want error bars, you can always try resampling methods such as bootstrapping. Hypothesis testing can also be done with bootstrapping, although the power of the statistical test may not be very strong and fail to reject the null.

temperature - Which moons have cold traps? (i.e. low ecliptic inclination in orbital and rotational axes)

Many of the moons of Jupiter and Saturn are tidally locked and probably move ice from their equatorial regions to their polar regions. Ganymede is the best example where you see bright polar ice caps and a dark equatorial zone. A migration process occurs on all of these moons in which light from the sun gets absorbed by water ice molecules and sends them on random jumps. After a long random walk, they eventually end up near the pole where they stay for a long time even if they are not in a perpetual shadow, and they stay there for nearly forever if they are.

But, if the moon has a very thick coating of ice everywhere, this process may not have dug down deep enough, so it is still covered with ice everywhere, in which case it is hard to see the effect happening.

bioinformatics - Generating custom human DNA sequences based on traits such as eye colour?

I'm wondering if it would be possible to create software (unless some already exists, but I couldn't find any) to generate human DNA (the base pairs on the double helix) containing genes representing specific permutations (eye colour, hair colour, etc.)?

Basically, something like the "character builder" from those "Saints Row"-style video games, except with actual human chromosomes, enabling you to essentially create a 'custom' human.

Of course, that is assuming that all human DNA has a common structure, and that the entire sequence can either be assembled from individual chromosomes, or by using a reference genome and modifying specific genes/chromosomes according to user input. Is this the case?

One setback here, of course, could be the number of bases in each chromosome, which ranges between 100 million and 250 million, and the (approximately) 23000 human genes - a lot of data to manipulate.

cell biology - How Do Large Ocean Viruses Form Their Own Organelles?

Several large viruses (Arslan 2011) form their own organelles within the amoebae they invade.

How do these organelles form?

---

Reference:

Arslan, D., Legendre, M., Seltzer, V., Abergel, C., Claverie, J-M. (2011) Distant Mimivirus relative with a larger genome highlights the fundamental features of Megaviridae. PNAS 108(42): 17486-17491 [DOI]

botany - Why do plants have pith and how is it useful to them?

The pith (medulla) forms part of the ground tissue system of a plant, and specifically it is the ground tissue which lies interior to a plants vascular tissues (xylem, phloem etc.)

The ground tissue system is responsible for much of a plants metabolic functioning, and contains various specialized cell types which aid in photosynthesis and storage of photosynthesis products. The pith is made from parenchyma cells.

If the universe keeps expanding, can we travel to the edge of the universe?

When we talk about the universe, we are really talking about one of two things:

• The observable universe, which is everything we can possibly see.


• The Universe, which is everything that has ever existed, currently exists, and will exist.

The observable universe has its own center, usually the Earth. It is a spherical region of everything that we can see, essentially anything whose light has reached us. We usually refer to this when we say things like "there are $10^{86}$ atoms in the universe."

In reality, everyone has their own observable universe, and it can change depending on where you are. An exoplanet far away has its own observable universe, and can receive light from different places. Essentially, you are the center of your own observable universe. As more light reaches us, our observable universe will grow (or expand, if you will). If this is what you are referring to, then your answer is here.

If you are referring to the latter, then there is a totally different answer and "expansion" refers to something completely different. The Universe (notice the capital "U") is all of space and time and its contents. Anything that has existed, will exist, and currently exists is part of it.

The Universe is thought to be infinitely large, so it can't have a center. The center of something is the point equidistance from the edges, but if something spans infinitely long, it would just keep going. It wouldn't have an edge, and thus it wouldn't have a center. You couldn't find the point equidistant from the edges if it just spans infinitely.

The Universe is not like a ball. Rather, you can think of it like a flat grid, and its "expansion" just means that the distances between objects on the grid are getting larger. In essence, more space is being created between the objects. That's what we mean by expansion — that objects are moving away from each other, since more space is being created between them.

Here's an easy analogy: imagine you are walking your dog. Suddenly, the ground begins expanding between you. You and your dog will separated and continue receding away from each other.

That's essentially happening everywhere: space is expanding between everything, so we are drifting away from other galaxies. The Universe is infinite, and we can constantly drift apart from other objects because space is being created in between us. Here's a GIF I made that might help you get it:

You can see how the galaxies drift apart as the space between them increases. And this happens everywhere in the Universe. The Universe is infinite, but more and more space is being created between matter.

Fun fact: These objects can actually drift away from each other faster than the speed of light. That is, light from them eventually won't make it to us, since they'll be drifting away too quickly.

Now, this doesn't actually go against Einstein's theory that the speed of light is the fastest thing in the Universe. Einstein said that nothing can travel through space faster than light — but here, space itself is actually being created between the objects. Distances are increasing because space itself is dilating, and thus we can drift apart from other objects faster than light.

homework - Does Human Female Meiosis II occur after fertilization with sperm?

Meiosis, as you know, have two stages, Meiosis I and II. The oocyte is arrested during metaphase II of MEOISIS II. This arrest is facilitated by a complex called "Cytostatic Factor" (CSF).

After fertilization, the sperm induces a rise in intracellular calcium ion which activates and enzyme, Calmodulin Kinase II. This complex, through a series of phosphorylation and ubiquitination, degrades the CSF comples and in turn, activates APC (Anaphase Promoting Complex). APC will then degrades cyclins, securins and this will promotes the completion of Meiois II.

Sperm must ignite some process in female that puts female meiosis II going on before sperm can fuse with egg.

I think the statement is a bit incorrect. The sperm will ignite (I prefer induce) the above changes and fuse at the same time. Because, what starts all the process above is part of the sperm's cytoplasm that need to be assimilated into the oocyte's cytoplasm.

I hope this clears it all up. Or if you are interested, I can provide the long list of signalling pathways that leads to zygotic development.

structural biology - How does one calculate the resolution of a crystal structure?

Hi sorry i missed this one - not too hard for "biology"

If you look at a protein crystal (or any crystal really) in an x-ray beam, it scatters lots of spots (diffraction reflections). If you look at a picture of crystalline diffraction, at larger angles from the center of the x-ray beam, the reflections get weaker and weaker and basically just stop, if the wavelength is short enough (in all crystallography labs its plenty short - from 1.5 to 0.9 Angstroms).

The resolution is marked by the angle of scattering to the last measurable spot (aesthetics can vary here, but there is little variance from personal judgement here). Once you have the angle of scattering, you can calculate the resolution from the Bragg scattering equation:

lambda = 2d sin(theta)

which is rearranged to solve for 'd'

d = lambda / 2 * sin (theta)

where lambda is the wavelength of the incident radiation and theta is the scattering angle.
d = the apparent width of the 'slit' which caused this highest resolution reflection is called the 'resolution' of the X-ray scattering experiment.

There is one part that's a bit tricky as in the typical diagram, depicting Bragg reflection/scattering, you would take the incident x-ray beam as first beam and the scattered angle as the beam of the high resolution scattering, which is more easily measured as 180 - 2* theta.

Exoplanet Temperature Calculations - Astronomy

I was given an exoplanet similar in size and distance to host star to our own earth. It's orbiting a star with luminosity six times our sun, the greenhouse coefficient 0.3, bond albedo 0.3.

I need to calculate the approximate surface temperature (i.e. close enough to know whether it is in habitable range). Thanks in advance (this question was on a past exam and we never got our test back)

zoology - Why do animal teeth get darker if exposed?

I was wondering why the teeth of cats (And dogs, as far as I know, plus possibly some other animals or even humans) get darker if they are exposed to air / light?

Example:

(See the lower end of the tooth, which sticks out if the mouth is closed. No, the cat was not resisting, just slightly annoyed.)

So, my question is:

• Why does this happen?


• Would it happen to human teeth, if they would stick out of the mouth like the ones of cats do?

My guess is that it has something to do with the teeth reacting to air and corroding, but all my knowledge about biology comes from two years of biology in school, so I'm asking here.

organic chemistry - How does Polytetrafluoroethylene (Teflon) differ from Polyvinylidene fluoride (PVDF) as a protein binding membrane material?

PTFE and PVDF (durapore) are both used in protein binding filter membranes (Millipore specifically). Chemically speaking the two polymers differ quite significantly due to the additional fluorides and molecular weight. Their protein-binding properties also differ. I'm curious about how the chemical differences affect their properties as a membrane material.

From what I understand, the durability between the two is quite different. The PTFE membranes are used for the filtration of organic solvents where as the PVDF membranes are generally for low protein binding. Furthermore the PTFE membranes typically require a polyethylene support.

I don't believe that the degree of polymerization or the architecture; if we were dealing with a copolymer I expect differently. Based on my molecular intuition, I would imagine that this is completely based on the chemistry of repeat unit.

My question concerns a few things. What makes the PTFE a better solvent resistant polymer? The other is how does the reduced number of fluroides result in reduced protein binding?

Teflon/PTFE

PVDF

How to calculate when the moon is highest in the sky to the earthbound observer?

In winter, the full moon is opposite the sun, and as the sun is low, the full moon is high.

In summer the full moon is low (for the same reasons). The crescent moon is high in summer (and low in winter) but as the crescent moon is near the sun, it is normally not visible during the day.

During spring and autumn, the sun, and the moon follow roughly equal paths, with no phase of the moon being higher in the sky.

Third quarter, being at right angles to the sun will be at an intermediate altitude, in both summer and winter.

For exact calculations either use technology, or a set of astronomical tables and a sharp pencil!

statistics - Is it necessary to conduct a power analysis before beginning an experiment?

Due to my own woeful ignorance on the subject, I have been reading up on statistical methods recently. From what (little) I understand, the real answer to this question is:

Yes, but only if you are doing Neyman-Pearson hypothesis testing

and

Absolutely not, if you are using Fisher p-values

That is, the question isn't formulated correctly, because power analysis is only valid under one statistical framework (Neyman-Pearson). And you are probably not using that framework.

In my experience, most experimental biologists use Fisher's p-value, which gives the probability of the data (or more extreme data) assuming that the null hypothesis is true. Under Fisher's framework, among other drawbacks, there is no quantitative measure of the test's power. However, it has the benefit that it allows scientists to do something close to what we would like to do--that is to draw conclusions from evidence obtained in individual experiments.

The Neyman-Pearson framework does included the idea of a test's power, because you must formulate an alternative hypothesis as well as desired alpha and beta error rates before starting your experiment. However, it mostly denies us the ability to make inferences from individual experiments, and for that reason appears less suited to experimental science. To quote from Goodman (see below), under Neyman-Pearson, "we must abandon our ability to measure evidence, or judge truth, in an individual experiment."

There is no clear right frequentist framework, although what is clear is that you cannot mix Fisher and Neyman-Pearson. Finally, although it doesn't really address your question directly, it seems wrong not to mention Bayesian methods as an alternative to these two frequentist frameworks, which comes with its own baggage.

Further reading from people that understand this much better than me:

Michael Lew's answer to "Setting the threshold p-value as part of hypothesis generation" at Cross Validated

Michael Lew's answer to "What are common statistical sins" at Cross Validated

Hubbard, Raymond, and M. J Bayarri. “Confusion Over Measures of Evidence ( p’S) Versus Errors (α’S) in Classical Statistical Testing.” The American Statistician 57, no. 3 (August 2003): 171–178. (Working Paper PDF)

Arguments for Bayesian statistics:

Goodman, Steven N. “Toward Evidence-Based Medical Statistics. 1: The P Value Fallacy.” Annals of Internal Medicine 130, no. 12 (June 15, 1999): 995–1004.

Jaynes, E. T. Probability Theory: The Logic of Science (Online version of some parts)

Transcription and translation of prokaryotic operons

As you pointed out, the repressor gene lacI is transcribed as a one mRNA, and three structural genes: lacZ, lacY and lacA are transcribed into a single polycistronic mRNA. The two mRNAs are translated independently of one another.

The polycisronic mRNA is not broken into pieces. Rather, it is translated by ribosomes (at least three, explanation below), giving rise to three individual proteins.

How from one mRNA three protein products are translated? The polycistronic DNA consists of a promotor (Plac), which ensures that transcription will be initiated, and three open reading frames (ORFs) for for lacZ, lacY, and lacA. Upstream of each of the ORFs, ribosome binding sites (RBSs) are located. Translation is initiated by the binding of the ribosome with both the RBS and the start codon. Thus, multiple ribosomes can bind upstream of the three ORFs and thus resulting into three proteins.

There is not very thorough literature on the translation of polycistronic RNA in prokaryotes, mainly because the condition which is necessary for translation is the RBS and there is nothing specific for polycistronic translation. A brief overview on the subject can be found in several places, as I recommend the following:

1. Ralston, A. (2008) Operons and prokaryotic gene regulation. Nature Education


2. From Genes to Genomes: Concepts and Applications of DNA Technology


3. Molecular cell biology


4. Analysis of Genes and Genomes

Dark matter inertial mass - Astronomy

If the inertial mass is not equal to the gravitational mass, it would be equivalent to the gravitational constant, G, being different, or perhaps non-constant, for dark matter. If that were so, the manner in which dark matter would orbit the milky way would be different, leading to a different distribution of dark matter

We know from the rotation curve of the milky way roughly how dark matter is distributed in the milky way. Our models are consistent with the inertial mass of dark matter being equal to the gravitational mass.

This is, of course, far from "proof". As we don't even know what dark matter is, it is impossible to be certain of any of its properties. And the equivalence of inertial and graviational mass is not "proved" even for normal matter (but no experiment has ever detected a difference) But it would be exceedingly surprising if gravity affected dark matter differently, and there would need to be strong evidence for that. As it stands, there is none.

If I graft two trees together while young, will they grow as one plant?

If two trees grow close enough together so that their trunks touch each other anywhere along the length of the tree, then they will eventually fuse. This generally only happens at the trunk because, unlike small branches, the trunk really can't be pushed out of the way as easily. It doesn't necessarily need to be two trees of the same species either.

There used to be a fused sycamore-maple on my school campus (it was damaged in Sandy and was cut down). They weren't completely fused together, but you could see a joint at the base and about 20 feet below the canopy where the trunks essentially became the same. There was no distinction between the two separate trunks.

But to finally answer your question; when the trees fuse they pretty much become conjoined twins. I'm not sure if they transfer genetics to each other, but they do share resources.

intergalactic space - Would being ejected from the Milky Way Galaxy have any major impact on life on Earth?

I can think of two ways this benefits life.

• Less chance of a passing star disrupting our Oort Cloud and sending
  deadly comets crashing into the Earth.


• Less chance of nearby supernova destroying our ozone layer.

On the other hand, if our Solar System passes near a relativistic jet emanating from the central black hole of a galaxy, Earth will get hit with high speed particles that will disrupt the magnetosphere and ozone layer. Definitely not good for any life on Earth at that time.

Getting ejected from the galaxy also means there is less chance of future humans exploring the galaxy by hopping to nearby stars. We'd only have our solar system to explore and colonize. There would not be another star system within a hundred thousand light years.

pulsar - radio white dwarfs

We have discovered about 2500 radio pulsars. Part of them show pulsations in other bands like X-ray and gamma ray. The radiation mechanism remains unclear now.

I wonder whether there are radio pulsations caused by the spins of white dwarfs. It is the best to give reference papers.

Does the gravity of the planets affect the orbit of other planets in our solar system?

It does - although the term 'disrupt' may be a bit too strong to describe the effect; personally, I think 'influence' would fit better.

An interesting consequence of such iterations is something called orbital resonance; after long periods of time - and remember that the current estimate for our planet's existence is 4.54 billion years - the ebb and flow of tiny gravitational pulls cause nearby celestial bodies to develop an interlocked behavior. It's a double-edged sword, though; it may de-estabilize a system, or lock it into stability.

Quoting the Wikipedia entry,

Orbital resonances greatly enhance the mutual gravitational influence
of the bodies, i.e., their ability to alter or constrain each other's
orbits.

Another gravity-related effect (although, as pointed out by Dieudonné, present only on our solar system between bodies that have very close orbits like the Earth-Moon and Sun-Mercury systems) is known as Tidal locking, or captured rotation.

More about orbital resonance on this ASP Conference Series paper: Renu Malhotra, Orbital Resonances and Chaos in the Solar System.

temperature - Why is the Boomerang Nebula colder than the CMB?

The Boomerang Nebula (or Bow Tie Nebula) is a cloud of gas being expelled from a dying low-mass star, at $164~mathrm{km}~mathrm{s}^{-1}$. In general, when a gas expands, it cools (see extended explanation below). If the gas were optically thin to the CMB — that is, if it were sufficiently dilute that CMB photons could easily penetrate — it would quickly get reheated to the temperature of the CMB, i.e. 2.7 K. However, the Boomerang Nebula is optically thick (dense), so the CMB hasn't yet had the time to heat it. The temperature in its outer parts, however, is higher, and as the nebula expands further, it will eventually (on timescales of, say, 1000s of years) be heated not only by the CMB, but also by the central white dwarf, i.e. the remnant of the star that produced the nebula.

Why does an expanding gas cool?

The usual approach to explain this is to consider a gas in a piston. When the volume is increased, the gas molecules do work on the piston, and hence lose energy, so temperature decreases. However, in the case of the Boomerang Nebula, there are no walls on which the gas can do work.

Cosmologically speaking, the nebula expands rather quickly (it has "only" been expanding for ~1500 yr). Assuming that it doesn't have the time to exchange energy with its surrounding, the expanding is thus adiabatic. For an ideal gas undergoing a reversible adiabatic expansion (or contraction), we know that
$$P V^gamma = mathrm{constant},$$
where $P$ and $V$ are the pressure and volume of the gas, respectively, and $gamma$ is the adiabatic index. For a monoatomic gas, $gamma = 5/3$, but here there are probably also molecules, so it's probably somewhat higher. At any rate, it's higher than $1$, which is the important thing for us, as we shall see below.

Now the temperature $T$ of the gas can be obtained from the ideal gas law:
$$P V = N k_mathrm{B} T.$$
Here, $N$ is the total number of particles, and $k_mathrm{B} = 1.38times10^{-16}~mathrm{erg}~mathrm{K}^{-1}$ is a constant (Boltzmann's, to be specific). Even for non-ideal gasses, this relation is usually a pretty good approximation. Combining these two equations, we see that
$$T V^{gamma-1} = mathrm{constant},$$
and since $gamma gt 1$, it is evident that if $V$ increases, $T$ must decrease.

human anatomy - Can any other animal choke on food?

The veterinarian in our group offers this: For humans, who choke much more frequently than other mammals, it is likely to be a cognitive problem. We talk and eat at the same time and so give ample opportunity to allow food passed the epiglottis and choking. Animals do choke - dogs can, cats can. Not all animals can vomit and this is particularly a problem for horses, whose stomachs will rupture instead of releasing contents through the mouth.

imaging - Photodetector Question: Does converting an RGB image to grayscale produce the same result as using a grayscale image detector?

OP here. For those who care, I think I figured out a solution; for my particular situation at least.

I'm pretty sure that as long as I use reference images for each wavelength I will be able to measure the reflectivity of my samples using RGB images converted to grayscale. Much like in spectrophotometry, I will be calculating the following ratio:

sample(wav)/ref(wav) = reflect(wav)

Where...

sample(wav) is a grayscale pixel value as a function of wavelength for images of the sample.

ref(wav) is a grayscale pixel value as a function of wavelength for images of the reference.

reflect(wav) is the reflectivity of the sample as a function of wavelength for an arbitrary pixel.

FYI my samples are Silicon wafers and the reference I will be using will be an Aluminum mirror. Both are specularly reflective.

Now I will attempt to prove that this makes sense mathematically...

Let's say at a particular wavelength there is some intensity reflected from the sample, I_sample(wav), and some intensity reflected from the reference, I_ref(wav). We want to measure something proportional to each of these intensities at each wavelength. Let's see if the RGB camera is going to screw us over or not...

The red, green, and blue filters each have there own transmission as a function of wavelength. Let's call these R(wav), G(wav), and B(wav). Thus the R,G,B values (0-255) that get recorded for the sample are proportional to R(wav)*I_sample(wav), G(wav)*I_sample(wav), B(wav)*I_sample(wav). And similarly for the reference we have R(wav)*I_ref(wav), G(wav)*I_ref(wav), B(wav)*I_ref(wav).

Converting these images to grayscale involves weighting the R, G, B values in the following manner: gray = aR+bG+cB. Which will give us the following grayscale pixel values:

sample(wav): a x R(wav) x I_sample(wav) + b x G(wav) x I_sample(wav) + c x B(wav) x I_sample(wav)

ref(wav): a x R(wav) x I_ref(wav) + b x G(wav) x I_ref(wav) + c x B(wav) x I_ref(wav)

A factor of aR(wav) + bG(wav) + c*B(wav) can be pulled out of each, and then if the ratio of sample(wav)/ref(wav) is calculated the common terms cancel leaving I_sample(wav)/I_ref(wav). This is reflectivity! Voila!

Hope this is correct... I have to go, but will continue to think about this, and will edit if I realize anything is wrong.

mass - What type of star does theory predict should be the most massive?

Since you phrase your question "What type does theory predict?", I guess the answer must be the so-called Population III stars, which are thought to be the first generation of stars, born from zero-metallicity gas. With no metals, it is difficult for the gas to cool. The mass of a star is given by the Jeans mass of the collapsing cloud, which is proportional to $T^{3/2}/rho^{1/2}$. This temperature dependence means that the cooler the cloud can be, the smaller clumps the cloud is able to fragment into. Hence, if it's difficult to cool, the cloud has a large Jeans mass, i.e. only large cloud collapse and form stars (this explanation is rather simplified and doesn't take into account stuff like shocks, turbulence, etc., but captures the basic physics).

Moreover, without metals to act as absorbers, the radiation may escape the star more easily, i.e. without interacting with the stellar atmosphere, so mass losses may be less significant and the star is expected to maintain its mass.

Pop III stars are expected to have masses of several hundred to 1000 $M_odot$. With such large masses, they burn the fuel fast ($sim$$10^6$ yr), but their spectra are extremely hard, and they should be able to ionize helium, so one way to get a hint of the existence of these guys would be by detection of the He $lambda$1640 Å line.