What is the most populated/numerous stellar system in which the orbits of all objects are known?

What is the most populated/numerous stellar system in which the orbits of all objects are known?

The answer is none. Other than our own solar system, astronomers don't know if they know all of the large bodies (aka planets) orbiting any given star system. Presumably other star systems have asteroids, comets, and other stuff. The orbits of those small bodies in star systems is unknown, and will remain unknowable for a long time.

We don't even know that for our own solar system. If we did, we wouldn't have to worry about killer asteroids and comets.

zoology - How do I save a coffee damaged wasp?

I found a wasp in one of my coffee cups about 1 hour ago. There was barely any coffee left but it had managed to get itself pretty much drenched in coffee. I took it out. Put it on a piece of paper and put a pile of sugar next to it. It started eating but it was in a pretty bad state. I came back half an hour later to check up on it and it had fallen down on the floor. It's state seemed to have worsened. Went from walking around to lying still and moving it's limbs.

What kind of damage did it suffer from the coffee and how would I go about maximizing it's chances for survival?

the moon - Smallest lunar probe that can be made using today's technology

A starting point is the smallest sample return probe ever built, the Luna 24.

Massing 5300 kg in lunar orbit it was pretty much bare bones. To see where we can improve this, we must first split the mission it its separate parts. This follows the standard procedure of doing the mission backwards. First, I would not choose ISS as the target, as braking into orbit and then rendezvous and dock with it is a complicated task. To simply hit the Earth is much easier.

Of the 514kg for the return stage of the Luna 24, around 300kg was propellant. Perhaps a slightly more efficient engine exist today, but the technology used now for landing and ascent would still be hypergolic propellants. That is the most important limit for scaling.

The re-entry capsule was only 34kg, but you might still be able to shave off a few kilograms. The main savings are in the remaining 180kg though, including electrical systems, control systems, the engine and the propellant tanks. With the miniaturization of electronic equipment since the seventies, and some new lighter materials, you may be able to squeeze everything required into a dry mass of 100kg. That is about 220kg at the lunar surface, roughly halved. A similar miniaturization of the descent stage and drilling equipment yields a spacecraft of about 2 metric tonnes in Lunar orbit.

To transport the spacecraft to the Moon, the minimal solution is a spiralling ion-craft. That is within current technology, take for instance the engine powering Dawn. At the cost of a long transfer time, the total mass in Earth orbit is only going to be around 3 metric tonnes minimum. However, considering the relatively high drag at the altitude of the ISS, the craft must start from a higher orbit.

Additionally, it is more effective to combine multiple related goals, like a lunar rover, and on-site experiments, than to launch multiple minimal missions with high risk of failure.

exoplanet - How would we detect an Earth doppelganger planet?

Using current technology (and by that I mean experiments and telescopes that are available now) we have not detected an "Earth-like" planet and we would probably be unable to detect life on Earth even if observed from a distance of a few light years. Therefore there is currently no prospect of detecting life on an "Earth doppelganger". I elaborate below:

1. No planets like the Earth have yet been detected around another star. That is to say, none that have a similar mass, radius and orbit at 1 au (or close to it) from a solar-type star. With current technology, it is just out of reach. Therefore any directed search for life on an Earth-like planet wouldn't actually know where to start. If you can't detect the planet at all then there is absolutely no chance of looking at its atmospheric composition to look for biomarkers (e.g. oxygen along with a reducing gas like methane, or chlorofluorocarbons from an industrial civilisation - Lin et al. 2014). The only exoplanets for which atmospheric compositions have been (crudely and tentatively) measured are "hot Jupiters". - giant exoplanets orbiting very close to their parent stars.




2. A "blind" search could look for radio signatures and of course this is what SETI has been doing. If we are talking about detecting "Earth", then we must assume that we are not talking about deliberate beamed attempts at communication, and so must rely on detecting random radio "chatter" and accidental signals generated by our civilisation.
   The SETI Phoenix project was the most advanced search for radio signals from other intelligent life. Quoting from Cullers et al. (2000): "Typical signals, as opposed to our strongest signals fall below the detection threshold of most surveys, even if the signal were to originate from the nearest star". Quoting from Tarter (2001): "At current levels of sensitivity, targeted microwave searches could detect the equivalent power of strong TV transmitters at a distance of 1 light year (within which there are no other stars)...". The equivocation in these statements is due to the fact that we do emit stronger beamed signals in certain well-defined directions, for example to conduct metrology in the solar system using radar. Such signals have been calculated to be observable over a thousand light years or more. But these signals are brief, beamed into an extremely narrow angle and unlikely to be repeated. You would have to be very lucky to be observing in the right direction at the right time if you were performing targeted searches.

Hence my assertion that with current methods and telescopes there is not much chance of success. But of course technology advances and in the next 10-20 years there may be better opportunities.

The first step in a directed search would be to find planets like Earth. The first major opportunity will be with the TESS spacecraft, launching in 2017, capable of detecting earth-sized planets around the brightest 500,000 stars. However, it's 2-year mission would limit the ability to detect an Earth-analogue. The best bet for finding other Earths will come later (2024 perhaps) with the launch of Plato, a six-year mission that again, studies the brightest stars. However, there is then a big leap forward required to perform studies of the atmospheres of these planets. Direct imaging and spectroscopy would probably require space-borne nulling interferometers; indirect observations of phase-effects and transmission spectroscopy through an exoplanet atmosphere does not require great angular resolution, just massive precision and collecting area. Spectroscopy of something the size of Earth around a normal star will probably require a bigger successor to the James Webb Space Telescope (JWST - launch 2018), or even more collecting area than will be provided by the E-ELT in the next decade. For example Snellen (2013) argues it would take 80-400 transits-worth of exposure time (i.e. 80-400 years!) to detect the biomarker signal of an Earth-analogue with the E-ELT!

It has been suggested that new radio telescope projects and technology like the Square Kilometre Array may be capable of serendipitously detecting radio "chatter" out to distances of 50 pc ($sim 150$ light years) - see Loeb & Zaldarriaga (2007). This array, due to begin full operation some time after 2025 could also monitor a multitude of directions at once for beamed signals. A good overview of what might be possible in the near future is given by Tarter et al. (2009).

Spectral analysis of AGN (velocity dispersion of galaxy absorption, doppler shifts)

I was wondering what parameters I need to know/measure to calculate the velocity dispersion of a galaxy, specifically an AGN.

Also, I have spectra where there are blueshifted and redshifted components to certain emission lines in units of km/s.

For example, I have [O III] 5007Å but I have a blueshifted component centre at 5002Å and a redshifted component at 5011Å. I was wondering how I calculate the doppler shift again in units of km/s.

Many thanks.

electron - is there any theory or observational evidence that our universe is electrically neutral or not?

By observation, gravity dominates the universe on large scales. If there were a significant disparity in positive and negative charges then we would expect electromagnetism—which is approximately $10^{39}$ times more powerful than gravity—to dominate. So by this observation we can conclude that the universe is approximiately electrically neutral. Exactly why this should be so is not well-understood, and ultimately ties into the matter-antimatter problem, aka the baryon asymmetry.

Protons and electrons are produced through distinct processes. That we have many protons is not particularly remarkable, as this would come about by conservation of baryon numbers as other particles decayed. Why we have any electrons at all is a fair bit more mysterious.

In the early universe, we understand the energy was well above the electron-positron pair production threshold, and so the early universe had large amounts of both electrons and positrons. When the universe cooled below this threshold, the production would have stopped and we would expect the electrons and positrons to then collide and annihilate each other, leaving essentially none of either. But we obviously see large amounts of electrons: approximately as many electrons as protons by the aforementioned observational evidence. And we also do not see large amounts of antimatter—positrons in particular. If there were large amounts of positrons out there, we would expect to see tell-tale signatures in space as they annihilate with regular matter (or in any weak force interactions, such as would be evident in supernovae), and we have no such observations.

So somehow the early universe must have produced significantly more electrons than positrons. In other words, we have strong observational evidence that there must be an asymmetry in physics between matter and antimatter production. There are ways to work this into the theory, but the Standard Model by default does not support it, and experts have not really settled on any one particular modification.

Do star remnants actually burn?

Stars don't "burn", they undergo nuclear reactions that do not involve atoms and chemistry at all. The temperatures in the interiors of stars, and certainly in the interiors of stellar remnants like C/O white dwarfs, are far too hot (millions of degrees) for electrons to bind to nuclei and far too hot for molecules to survive. The energy required to break up carbon dioxide is 5.5 eV, which is easily provided unless a gas is cooled well below 10,000K.

The only parts of a star where chemistry can occur is in the outer atmospheres where temperatures can drop to thousands of degrees and where atoms and partially ionised atoms can exist. Here, yes, then it is possible for carbon atoms and oxygen atoms to interact, but that would mostly produce carbon monoxide. This molecule is produced and can survive between temperatures of about 1100K and 3500K in the atmospheres of cool M-dwarfs and brown dwarfs and the atmospheres of red giant stars.

These chemical reactions are utterly negligible in terms of their energetics compared with the fusion reactions that power a star.

astrophysics - Circular formation around the moon

I'm guessing what you see is the moonlight being scattered by the hexagonal ice crystals in cirrus or cirrostratus clouds, which lie at very high altitudes, 5-6 km and above. The light is scattered by roughly 22º, and because of the slight wavelength dependence, the halo actually has rainbow-like colors, although often they are so faint that you just perceive it as white.

Read more here.

EDIT: My first was drawing was made with my head under my arm, and had the angle wrong. Rob Jeffries made me aware of this, and also pointed out that larger angles are possible, up to 50º. Moreover, I drew the colors in the opposite order. Since blue light is deflected slightly more than red light, the blue light that reaches you is deflected from the part of the clouds that are more distant from the Moon.

botany - Why are some plants frost tender?

All cells are susceptible to cold, but freezing is particularly difficult. If the water in the cells freeze, it would usually burst the cells (water expands when it turns to ice) and tearing the cell membrane (totally killing it).

Some plants and animals have adapted to colder temperatures by putting antifreeze proteins in their cells. A lot like adding antifreeze to the radiator of a car, this lowers the freezing temperature of the water in the cell.

So the answer is that some plants have adapted to colder climates with the help of additional genes. There are other things that have to change too - the chemical reactions in the cell will go haywire when temperature changes too, so lots of genes must make adjustment for a fully cold adapted plant.

Is it plausible that Mars could have been one of Jupiter's natural satellites at one time?

I'm going to say no, for a few reasons, but if anyone wants to give a more detailed answer, feel free.

1) Planets and moons that form inside the frost-line have much less water and other ices than those that form outside. Mars likely once had water and oceans but if it had formed outside the frost-line it should have a lot more water and a lower density. The frost line moves away from the sun as the sun grows more luminous and planets move, so there is a degree of imprecision with this argument, but I believe this on it's own is a pretty good reason why it's not likely.

2) A moon that escapes Jupiter's orbit should enter a near Jupiter orbit initially, and at some point later, it could receive a gravity assist and be propelled either in towards the sun or away, but it's unlikely that Mars could have obtained it's relatively circular orbit had it originated as one of Jupiter's moons, unless Jupiter was much closer to the sun when Mars escaped.

3) it's difficult to see how the asteroid belt could have stayed around if Mars originated outside the asteroid belt and moved inside it. Mars passing through the asteroid belt would have caused significant disruptions to it.

astrophotography - How accurately can you tell the time using a photo of the stars?

To make a clock you need something that changes predictably with time, and stars are nearly constant, making them poor clocks.

The stars do move across the sky each night, but since the morning stars of winter become the evening stars of spring, and the sky looks different in different locations you need to know the date of the image, the location of the photo, and the exact direction of the photo. That could give sub 10 min accuracy. Unfortunately while date and location may often be recorded with a photo, exact direction is not, unless the horizon is visible in the image.

Without that information, and it is possible to 'get lucky'. For example, if there are two or more planets in the shot, their locations could be measured to give at least the date of the photo. Similarly if a rare event like a nova is visible that can offer dating information. If the shot happened to catch a bright meteor, and the date was known well enough, you could get almost 1 second accuracy. Variable stars might also give the chance of at least dating the image.

With no planets you can use the very slow proper motion of the stars. With a good enough telescopic shot of the night sky you can be quite accurate, but with a wide view of a camera you wont be able to get enough accuracy to give any useful date.

The exposure time of a typical camera is to short to capture any stars. If you just point and shoot at tye sky, you just get black. To capture images of stars you would need a timed exposure. A telescope would help too.

So if you photographer wanted to create an image which could be timed accurately, and they had time to prepare for it, or it was a lucky shot this might be possible, otherwise not.

biochemistry - Evolutionary origin and exogenous cues of ~28 day infradian rhythm?

A double-blind, prospective study during the fall of 1979 investigated
the association between the menstrual cycles of 305 Brooklyn College
undergraduates and their associates and the lunar cycles.

.... Approximately 1/3 of the subjects had lunar period cycles, i.e.
a mean cycle length of 29.5 ± 1 day. Almost 2/3 of the subjects
started their October cycle in the light 1/2 of the lunar cycle,
significantly more than would be expected by random distribution. The
author concludes that there is a lunar influence on ovulation.

(Menstrual and Lunar Cycles, Friedmann E., American Journal of Obstetrics and Gynecology, 1981)

Another source supports this conclusion, finding that "a large proportion of menstruations occurred around the new moon."

Somewhat related, this study found that light exposure shortened menstruation cycles.

In summary, there seems to be a good amount of data suggesting that lunar cycles do in fact calibrate the length of human menstrual cycles to some degree.

cell biology - X chromosome "weight"?

According to Wikipedia, the X chromosome has approximately 153 million base pairs, while the Y chromosome has only 60 million base pairs. Thus, the difference is roughly 93 million base pairs.

My question is: Could this "extra" 93 million base pairs in virtually every single female cell of the human body slightly contribute to the "extra" relative weight of the female as a whole, in comparison to a male (of course) with a similar amount of cells? Or would this weight be so small that we can consider it negligible?

molecular biology - How do I clean and calibrate pipettes, and how often should I do it?

How often you should calibrate your pipettes depends on the tolerances of your application. For some applications, like quantitative PCR setup, one may care a great deal; for general lab work, one can probably be less particular. You can get a sense of how accurate your pipettes are by pipetting DI water onto the platform of a sensitive (and calibrated!) balance. Comparing the random and systemic error you get while doing that over several trials to the specifications published by the manufacturer will give you a good idea of whether pipette calibration will help and help give you a sense of how much you care. Note also that the error may vary with the volume; you should test error over the entire volume range you intend to use the pipette for.

How to clean and calibrate pipettes is specific to each design. Some pipettes, like Gilson's, are not intended to be calibrated by the user. Others are. The pipette user guide published by the manufacturer should tell you.

For some demanding applications, even well-calibrated and well-handled micropipettes may not be sufficiently accurate. In particular, I've made standard curves by serial 10-fold dilution for quantitative PCR both a) relying on my micropipettors to be correct and b) trusting pipettes only to get me into the right ballpark, and intentionally undershooting and correcting volumes with sequentially smaller pipettes until the reading on the balance is correct, if that's at all clear. In both cases I was using 900 ul of diluent and 100 ul of the previous dilution, which I feel are comfortably large volumes to handle. I found that the latter curve gave me noticeably better regressions.

biostatistics - Is bootstrapping an acceptable way to determine standard error for binding constants?

My understanding of bootstrapping is that you may estimate variance (and thus standard error of the population mean) iff your measurements are independent and have the same population distribution, in which case a number of sampling-with-replacement calculations can be done. I suspect that this method of estimation would be less desirable with small n values because of the effect that outliers or large standard deviations may have on the calculation. If you have a large dataset then it shouldn't be an issue.

planet - Why don't storms on gas giants move to the poles, like hurricanes on Earth do?

The answer is the Coriolis effect, on Earth this produces cells within which storms move, converging towards the cell boundaries as you can see below.

Jupiter however spins much faster that the Earth which produces a stronger Coriolis effect and thus more cells. This is the reason why there are so many different coloured bands on Jupiter (see image below). So storms like the big red spot don't have as far to move laterally within these cells compared to hurricanes on Earth (in comparison with the planets size).

general relativity - Quantum Mechanics after the detection of Gravitational Waves

The impact of this measurement on the status of quantum gravitation is exactly zero.

The proper statement of the incompatibility of general relativity and quantum mechanics is that the quantum field theory of general relativity is not renormalizable. Renormalizability essentially means that the theory is well-defined at all energy scales, which seems like a reasonable demand on a proposed fundamental theory.

So what we know is that taking classical general relativity and quantizing it, we do not get a fundamental theory of quantum gravitation. This does nothing to rule out other proposed quantum theories of gravitation, for example, LQG or string theory.

Furthermore, the way physics works is that new theories must reduce to old ones in the domains of applicability of the old theories. Whatever the correct quantum theory of gravitation, its low-energy limit should be quantized general relativity, and the classical limit of that is classical general relativity. It's just not true that you have to choose between general relativity or quantum mechanics.

So this measurement of a prediction of classical general relativity does absolutely nothing to show that no quantum mechanical model of gravitation exists. It couldn't, because we already have a quantum mechanical model of gravitation: quantized general relativity. It's not as "nice" as we would like, but that really only rules it out as the fundamental theory.

molecular biology - If inhibiting S6 kinase decreases protein translation, then could inhibiting S6 kinase could possibly slow down long-term potentiation in neurons?

I can't rule it out, but it sounds a lot like trying to tune a piano with sledgehammer.

Neuronal LTP depends on protein translation, but so does absolutely everything else in the cell. Inhibiting protein synthesis at the ribosome will block the formation of all proteins, not just the ones responsible for LTP. Unless there's a link I don't know about between LTP and total levels of protein translation, you're really going to want to look into inhibiting the production of proteins specifically responsible for LTP and not protein synthesis in general.

astrophysics - Are there any electrons inside the sun / a star?

Yes, there are electrons inside stars, they have not gone anywhere.

I must say your new theory of electromagnetic force does not make any sense to me, but there are actually a few cases where electrons "disappears" from a star:

First off, the nuclear fusion of 4 hydrogen atoms into one helium atom actually consumes electrons. Let us count the particles before and after:

Before:

• 4 electrons -4


• 4 protons +4

After:

• 2 electrons -2


• 2 protons +2


• 2 neutrons 0

Please note that the overall charge has not changed, and can not possibly do that by any means. The charge is always preserved.

The other way electrons can disappear from a star is through the formation of a neutron star. There, the particles has are pressed together so strongly that the protons and electrons have merged into neutrons. Note how the charge is preserved here too.

digestive system - What is an simple way to burn glucose for visible effect?

I want to make a partially working model of the digestive system that could digest complex carbohydrates. My ultimate goal is to be able to cut up some bread, put it into the model, operate it, and eventually see some effect equivelent to glucose being used in a muscle. I obviously have a very limited buget. How could I simulate glucose being burned in some way that has a visible effect?

Edit: I'm specifically looking for some way glucose can be used to achieve some visible effect that makes sense as a metaphor for combustion, without dangerous/expensive chemicals.

receptor - Why doesn't a substance like loperamide promote analgesia?

This is more of a chemical question I think, but the gist of it is that, unlike its predecessors difenoxin (Motofen) and diphenoxylate (Lomotil), it was specifically designed by Dr. Paul Janssen and other researchers at Janssen Pharmaceutica to have limited bioavailability from the gut (it is easily first-pass metabolized through cytochromes), as well as being too lipophilic to be dissolved in water and thus administered intravenously (and thus, junkies hoping to get a "hit" from loperamide through injections will be sorely disappointed), while still retaining its ability as a μ-opioid receptor agonist to the receptors in the myenteric plexus (resulting in the slowing of peristalsis). (Difenoxin/diphenoxylate in fact is administered with small doses of atropine to discourage abuse.)

In addition, efflux mediated by P-glycoprotein in the blood-brain barrier helps retard the entry of loperamide into the CNS. The concomitant administration of a P-gp inhibitor like quinidine (a stereoisomer of quinine used as an antiarrythmic) with loperamide, however, will cause respiratory depression and other CNS effects that are characteristic of μ-opioid receptor agonists.

Can we simulate Earth's gravity in space?

Simulating gravity in space basically means simulating weight, which requires acceleration. So basically, the question is, how do we create acceleration in space.

The easiest method for simulating gravity in space is by spinning the space station. In this case, the reaction force to centripetal force substitutes the force of gravity, which pushes the inner contents to be pushed against the outer edge, giving a sensation of weight. This method is theoretically sound and can be used to simulate gravity.

The problem with doing this in ISS boils down to two things- size and shape.

The best shape for a space station having a simulated gravity is something like a doughnut spinning about its axis, so that the occupants are pushed against the outer wall. The shape of ISS is nowhere near this.

If one excludes the solar panels and the truss structure, the ISS is around 45m along the axis of the habitable modules, if we spin along the axis of the truss structure, this gives a radial distance of ~22.5m.

For simulating earth's gravity in ISS, we will require,

$omega^{2} r = g$

$omega = sqrt{frac{g}{r}}$

For g = 9.8 $ms^{-2}$,

$omega = sqrt{frac{9.8}{22.5}} = 0.66 rad/sec = 6.3 rpm$

The problem is that at this rotational speed, the head and foot of the astronomer will have different linear velocities. The head of a 1.5m tall astronaut will be moving at 13.86 $ms^{-1}$, while the foot will be moving at 14.85 $ms^{-1}$, never mind they are standing at the bottom of a barrel. This will lead to Coriolis effect, which will be uncomfortable.

So, while earth's gravity cannot be simulated in ISS due to various constraints, we can settle for reduced gravity (with reduced, ~2 rpm) in an ISS sized spacecraft.

human anatomy - Circulation through the liver in light of drug metabolism

I have a lingering question which stems from an answer that I gave to What hydrolyses aspirin within the digestive tract and blood stream?

When a drug or any other substance is absorbed into the bloodstream in the stomach or small intestine, it ultimately passes through the hepatic portal vein and into the liver sinusoids, where it is processed by hepatocytes and introduced into the general circulation via the vena cava. In terms of metabolism, this is what causes a "first-pass" effect for drugs that are ingested.

For drugs that are delivered either by intravenous, intramuscular, or sub-lingually (as in the other Biology.SE question), this first-pass effect is avoided, and the drug is introduced into the general circulation without being metabolized by the liver first.

Even though the first pass is avoided, the blood in the body still makes its way back through the liver eventually via the hepatic artery, which is a branch off of the celiac artery.

The issue I still have is, does the incoming blood from the hepatic artery merge with the blood from the hepatic portal vein? If not, does the blood from the hepatic artery still interact with the hepatocytes in some way? (it makes sense that it does, and I have also read that one of the main functions of the hepatic artery was to deliver blood supply for the liver's metabolic needs) If this is not the case, where in the body would these drugs that were introduced via IV, etc., be metabolized?

fundamental astronomy - What exactly is 'eastward movement'

From the point of view of an observer hovering above the Earth's surface, the rotation of the Earth is eastward. That is, points on the surface appear to move in an easterly direction as the Earth spins.

Extending this to the orbital motion, there are at least two possible interpretations. One is that the spin angular momentum and orbital angular momentum are in the same direction. This is roughly true (and certainly closer to true than the opposite). Another interpretation would be that the Earth moves around in its orbit such that it heads in an easterly direction when referred to a longitude on the Sun. This is also approximately true.

If you are asking what is a general definition of the north pole, then I suppose it could be the direction of the angular momentum vector. Then yes indeed, the right hand grip rule tells you the spin is eastward as it relates to north.

milky way - What will the mass of the new galaxy be?

The mass will be slightly less than the combine masses of the 2 galaxies since some of the stars will be hurled away.

Since the disks of the galaxies are at an angle to each other, the volume would be (roughly) the volume of the 2 galaxies as they collide. Eventually, the volume will decrease (I'm guessing to between 60 to 70%) as the stars adapt to their new environment and the central massive black holes combine.

Planets classification by density - Astronomy

There is no 1:1 mapping between density and composition/structure. You have to look at detailed planetary models. For example, some hot Jupiters are extremely dense ($geq 10$ g/cm$^3$) but they are undoubtedly gas giants. The origins of this diversity are the source of much speculation and theory, but are certainly within the realms of known physics.

An example of the difficulties can be gleaned from this plot from Lissauer et al. (2014) that shows the mass-radius plane for small planets together with the loci of models for various compositions. Lines of constant density are drawn on the plot too, but notice that a planet of a given composition can have a range of densities depending on its mass.

The situation appears even more strange with giant planets. Generally speaking, more massive giant planets are more dense, which is as expected from basic theory of degenerate gases, but objects of a given mass can have almost an order of magnitude spread in density. Below is a plot I made from exoplanets.org. Almost certainly, the change in the nature of the relationship at around 0.1 Jupiter masses marks the transition from gas giants to ice giants and other rocky-type planets, but notice that gas giants can be just as dense as smaller planets. i.e. A classification based on just density wouldn't work, but mass and density gives more of a clue.

radiation - Is there a direct relationship between an isotope's neutron count and radioactivity?

As @dmckee says the problem is complicated. It is complicated because it is not a solution of a potential describing one force, but a balance between electromagnetic forces and the strong force that is keeping the quarks within the nucleons. (In the nucleus the strong force is like a type of Van der waals potential, a higher order interaction, overflowing from the QCD dynamics of the nucleons). In addition there is the fermi exclusion principle because both protons and neutrons have spin 1/2.

All these have been approximated with the shell model of the nucleus, and you could maybe spend some time reading the link.

The shell model is partly analogous to the atomic shell model which describes the arrangement of electrons in an atom, in that a filled shell results in greater stability. When adding nucleons (protons or neutrons) to a nucleus, there are certain points where the binding energy of the next nucleon is significantly less than the last one. This observation, that there are certain magic numbers of nucleons: 2, 8, 20, 28, 50, 82, 126 which are more tightly bound than the next higher number, is the origin of the shell model.

Note that the shells exist for both protons and neutrons individually, so that we can speak of "magic nuclei" where one nucleon type is at a magic number, and "doubly magic nuclei", where both are. Due to some variations in orbital filling, the upper magic numbers are 126 and, speculatively, 184 for neutrons but only 114 for protons, playing a role in the search of the so-called island of stability. There have been found some semimagic numbers, notably Z=40.2 16 may also be a magic number.3

So there are stable nuclei and the various models do a good job of predicting them.
There exists a band of instability for the various isotopes and the island of stability for high Z.

So the answer is no, there is no general rule, except solutions of the shell model, though adding or subtracting neutrons to a stable isotope one expects a high probability that it will become unstable,as an examination of the table of nuclides shows..

cosmological inflation - Intuitive explanation for why the universe is flat

The CMB lets us measure how close to flat the universe is right now.

On the other hand, inflation tries to explain how we got from whatever the early universe was to right now.

The motivation for the latter being that even extremely small deviations from perfect flatness in the early universe should have resulted in very obvious deviations from flatness today. That we don't see obvious deviations then suggests the universe was amazingly close to perfectly flat early on, and such perfect tuning seems bizarre. Inflation basically lets a (in some sense "very") non-flat early universe get flattened to this precision, thereby resolving the "problem" of having an almost perfectly flat early universe. It takes the situation from an extremely narrow range of initial conditions for the flatness of the universe and gives us a broad range of equally valid possibilities, instead.

Inflation is also commonly used to explain the remarkable homogeneity of the CMB, which suggests that parts of the universe that cannot ever have received light speed signals from each nevertheless achieved thermal equilibrium (an at-most light speed process). Inflation throws in a super-luminal expansion to allow these regions to be within light speed communications of each other in the very early universe, with just enough time to achieve thermal equilibrium, and then inflates them beyond light speed communication.

stellar structure - How well can we in principle determine $T_{textrm{eff}}$ of a star?

This is a question about the basics of astronomy, which I have never happened to see a good discussion for. It is about how well would we be able to measure effective temperature of a star, if we had any arbitrarily perfect measurement devices.

Here is some context. Canonical definition of $T_{textrm{eff}}$ of a star is based on its bolometric luminosity $L$ (total electromagnetic energy radiated by the star per unit time) and its photospheric radius R (radius, at which the optical depth at a given wavelength is equal to unity). This way, the definition specifies $T_{textrm{eff}}$ through $L=4pi sigma R^2 T_{textrm{eff}}^4$, where $sigma$ is Stefan-Boltzmann constant.

The definition clearly alludes to black-body law. Many stars, including our own Sun, have a spectrum that does not follow it. For this reason, one often talks about another effective temperature, which is the temperature of stellar material at photospheric radius, and which can be determined by examining stellar spectrum. There are a few more complications to that, but let's put them aside.

Determining $T_{textrm{eff}}$ is extremely important in characterising stars, therefore there exists a variety of methods of measuring it, and naturally researchers strive for obtaining the best possible precision.

Hence, the question: How well can one in principle measure $T_{textrm{eff}}$, if one could have arbitrarily perfect instruments?

Edit: I would like to see a quantitative estimate in your answer. Is the best possible precision for $T_{textrm{eff}}$ of order $10textrm{K}$, or is it $1textrm{K}$, or some $10^{-4}textrm{K}$, or can we measure it arbitrarily well?

Here are just a few sources of uncertainty/arbitrariness: convection in stars, dependence of photospheric radius on wavelength, limb darkening, stellar variability, to name a few.

I would encourage the answers to be in the format "Source of uncertainty" - "Simple derivation" - "Estimate of the effect". If there are more than a few estimates, I will add a summary of them in the question or in a separate reply. Please, also feel free to edit the question if you might like to.