distances - How close would quasars have been to each other?

That's an image of two quasars, very close to each other, "separated on the sky by only about 70 thousand light years".

According to this paper the peak density should have been about $10^{-6}$ quasars per Mpc, resulting in a peak mean distance between two quasars of about 100 Mpc. (Density of $10^{-6}$ quasars per Mpc results in one quasar per cube of $10^6 Mpc^3$, hence on average one quasar each $sqrt[3]{10^6 mbox{Mpc}^3}=100 mbox{ Mpc}$)

With about 1 million known quasars in the observable universe I get a mean distance of about half that distance. (With a considered light-travel distance of about $4 mbox{ Gpc} = 4cdot 3.26 mbox{ Gly} = 13.04mbox{ Gly},~~$ a little less than the 13.81 Gly light travel radius of the observable universe, the volume of a sphere with this radius is about $frac{4}{3}pi(4cdot 10^3 mbox{Mpc})^3=2.68cdot 10^{11}mbox{Mpc}^3$. With 1 million quasars, it's about one quasar per $268000 mbox{ Mpc}^3$ or one quasar per $sqrt[3]{268000 mbox{ Mpc}^3}=64mbox{ Mpc}$ in one dimension. Due to non-constant quasar frequency over cosmic time, the minimum mean distance should have been below this overall average distance. Using the light-travel distance considers the higher density of the earlier universe better than the comoving distance of 13.8 billion years after big bang. 4 Gpc are used as light travel radius to restrict roughly to redshift z < 7, the range where quasars have been observed. A redshift calculator can be found e.g. here, take $H_0=70.4$, $Omega_m = 0.265$, $Omega_{Lambda}=0.728$, and $z=7$.)

The peak number density seems to be somewhere between a redshift of 2.0 and 2.5 (interpreting the paper). If it will turn out by future observations, that the peak number density has been earlier in the universe, we get much shorter peak mean distances.

So you have at least an order of magnitude: Somewhere between 150 and 300 million lightyears (between 50 and 100 Mpc, 1 parsec equals 3.26 lightyears) should have been the peak average distance between two quasars. Might be this estimate will shrink over time, as more (distant) quasars get known.

gravity - Do all the objects in the universe exert force on all other objects?

No. It's impossible for every object to interact with every other object, due to the assertion by general relativity, that the universe can, and is, expanding faster than the speed of light.

I then assume that the universe initially was expanding at, or close to the speed of light, and that it immediately after the big bang was expanding faster than the speed of light.

Some of the particles/forms of energy that would have reached us are also bound to have been "held back or deflected", even in the young stages of the big bang, and are now at a distance at which they can never reach us. They could have been held back by for example a black hole.

Potentially, if the expansion of the universe at one point was so slow that gravity from every particle had time to propagate to every other particle, then yes - every particle and energy in the universe affects every other particle.

general relativity - What do we mean by space is expanding?

Indeed space is not a physical entity (as far as we know). Saying space expands is another way to say that galaxies recede away from each other at a rate proportional to their distance.

The "expanding space" picture is unfortunately a source of endless confusion and misconceptions, as can be seen on the page of this very question: mpv mentioned that "The real expansion of space would manifest itself in the following way: you have 2 rockets in empty space, stationary with respect to each other. None of them starts its engines, but yet, they start moving away from each other."

This is plain wrong, the "expansion of space" is not some magical force that pulls objects away from each other. If you mean empty space as in no dark energy, no dark matter and no visible matter then the two rockets would move towards each other due to their mutual gravitational attraction (if you neglect their mass then they would not move at all). In the general case what will dictate the motion of the rockets is their gravitational attraction, the matter nearby (visible and dark) and dark energy (whatever that might be), the combination of which might pull them closer or further away from each other. The expansion of space is not an additional force. Galaxies don't keep moving away from each other because some undetected substance is created between them, but because they have an initial velocity which they acquired at the time of the big bang: without anything to stop them they keep moving away.

Joan.bdm said: "As I said in a large scale all objects recede from us, if it wasn't for the space expansion it would mean we are at the center of the universe."

The second part of the above sentence is wrong as well. As long as all galaxies recede from one point at a rate proportional to their distance, it follows that from ANY point all galaxies will be seen to recede at a rate proportional to their distance. This is easily seen with a little drawing (each * corresponds to a galaxy):

Before

After

If all galaxies are seen to recede at a rate proportional to their distance from the point of view of the galaxy at the center, then from any galaxy all other galaxies will be also seen to recede at a rate proportional to their distance.

I don't think we have any satisfying explanation as to why galaxies move away from each other at a rate proportional to their distance in the first place. The "expansion of space" is not an explanation, it is an analogy. A related and interesting read: Expanding Space: the Root of all Evil?

solar system - Do all stars have an oort cloud or is it a rare occurance?

Awesome question, especially since we know so little of the answer.

Nobody knows for sure how the Oort Cloud formed - I'll put that out there right now - but the current hypothesis is that it was originally part of the Sun's protoplanetary disk. All of the ice and rock coalesced into small bodies - proto-comets, if you will. While these bodies were much closer in to the Sun than they are today, they were tossed far out by gravitational interactions with the gas giants. Other interstellar comets could also have been captured by the Sun, adding to the population.

So why is the Oort Cloud spherical? After all, the protoplanetary disk was just a flat disk. Why were the orbits of the objects perturbed? Well, the Oort Cloud objects are only loosely bound to the Sun - relatively, that is. They can be influenced by passing stars or other objects. It appears that galactic-scale tidal forces, combined with the influence of passing stars, molded the Cloud into its current spherical shape.

So what does this all tell us? Well, we know other stars have protoplanetary disks, right? Some also have exoplanets - gas giants like Jupiter. They are also subject to tidal forces and the passing of nearby stars. So, theoretically, there's no reason why other stars shouldn't have Oort Clouds.

So can we find them? The answer is, most likely, no. Here's why. According to Wikipedia,

The outer Oort cloud may have trillions of objects larger than 1 km (0.62 mi), and billions with absolute magnitudes brighter than 11

An absolute magnitude of 11 is very dim. Now, the object's apparent magnitude is how it would look from a given distance; the absolute magnitude is how it looks from a distance of 32.6 light years. So these objects, to us, have an apparent magnitude brighter than 11.

The point of that poorly-explained interlude is that these objects are faint. Very faint. And objects in Oort Clouds around other stars would appear even fainter. Using the distance modulus, we can calculate the apparent magnitude of an object if the distance to that object and its absolute magnitude are known:

$$m-M=5(log_{10}d-1)$$

(from here)

where $m$ is apparent magnitude, $M$ is absolute magnitude, and $d$ is distance.

Given an Oort Cloud object $x$ light-years away, you can figure out how bright (or dim) it would appear. Try this with the distances of nearby stars, and you'll realize how dim objects in these stars' Oort Clouds would be.

As a final note: We don't know for sure if other Oort Clouds exist. From what I've been able to find, we don't have sufficiently powerful telescopes to observe these hypothetical Clouds, and so we don't (and may never) know if they exist.

I hope this helps.

This paper was instrumental in this answer. Start at page 38 for the relevant information. This page, too, has some good information.

Edit(s)

As I found from a link from an answer to this question on Physics, we've found Kuiper-Belt-like disks around other stars. This means it is certainly plausible for these stars to have Oort Clouds, too. And exocomets have been detected, which is another good sign.

radio astronomy - Is broadcasting the location of Earth to potential extraterrestial civilization regulated?

No, except in the sense that national broadcasting and so on is regulated. Our location can be fixed, as you suggest, from simply looking at where the signals come from.

Perhaps, though, you mean is it sent out "regularly" (i.e., at specified intervals). In truth we have been sending very powerful signals continuously into space since the dawn of the television age. Those signals may now start to wane in strength as we move to digital terrestrial television but that won't bother any aliens for some time yet.

What particles does a black hole emit when it evaporates itself?

Although black holes are widely believed to emit Hawking radiation, it should be stressed that it has not actually been observed (yet?). The Hawking radiation should consist of electromagnetic radiation/waves that have a near-perfect black body spectrum, which is at a temperature that is inversely proportional to the mass of the black hole - the smaller the black hole, the higher the temperature.

The radiation is caused because "vacuum" is not empty when considered in quantum mechanics. Particle/anti=particle pairs are created for brief moments of time, before they annihilate again. Close to the event horizon of a black hole, it is possible that one particles from a virtual particle pair travels within the event horizon of the black hole and cannot escape to annihilate with its anti-particle. In principle, these particle/anti-particle pairs could be any types of particle, but in practice they are more likely to be the lightest particles. The lightest charged particles are electron/positron pairs, since these need to "borrow" less energy from the vacuum to be created. I do not think it is essential for one of the pair to disappear, since "Unruh radiation", a close relation of Hawking radiation should also be seen whenever there is acceleration with respect to the vacuum.

To stay out of the black hole event horizon the particles must be accelerating. Accelerating charged particles locally emit electromagnetic radiation, which is then gravitationally redshifted when seen by an observer a long way from the black hole. It turns out that to be in a thermal equilibrium, the radiation must have a blackbody spectrum form.

The temperatures and amount of radiation emitted are very small for stellar-sized black holes. The temperature of a non-rotating Schwarzschild black hole is given by
$$ T = frac{6.2times 10^{-8}}{M} K,$$
where the mass $M$ is in solar maases.

The power emitted by a black hole as Hawking radiation is
$$ P = frac{9times 10^{-29}}{M^2} W$$

Very bright star in the east at northern hemisphere. What is it?

As other people have pointed out, it is hard to work out which star it is, without knowing your general location. However, after checking on Stellarium, there seem to be a couple of likely suspects:

• Sirius - the brightest star in the sky. I've seen it myself - and on a good, dark night, it can really stand out.


• Jupiter - the king of the planets is also rising at about the same time. It is brighter than any star in the sky, by a wide margin (though fainter than Venus), and it can really stand out.

Other than that, there aren't really that many objects rising in the East at the time you specify that could really stand out.

There are a couple of useful ways to tell the two apart:

• Sirius is a bright white object - perhaps with a subtle bluish tinge to it, whereas Jupiter has a slight yellow tint to it.


• Jupiter is currently rising in the North-East, and can get very high in the sky at the moment from the northern hemisphere, whereas Sirius rises in the South-East, and doesn't get that high (though that does depend on location).


• Sirius tends to twinkle, and 'flicker', as its light is disturbed by air currents, whereas Jupiter remains very steady - perhaps not twinkling at all.

As mentioned earlier, the best method is usually to use software like Stellarium, which will tell you exactly where everything is, and hopefully give you a definitive answer to which object it is.

data analysis - Correlation of planet sizes with star sizes?

I think there are a number of studies that look at these statistics; I'll try to dig some out. In the meantime I can give you some further things to ponder.

First, let's assume a null hypothesis that the statistics and mass distribution of planets were independent of stellar mass - what would we observe?

Well one selection bias you don't mention is that the likelihood of transit detection depends on the ratio of planet to star size. This means you are less likely to see small planets around F-stars than M-stars. Or to put it another way, you expect the fraction of large planets to increase with stellar mass - as you have found.

It is quite likely that the material available to form planets is correlated with the stellar mass - in other words there is a correlation between stellar mass and protoplanetary disc mass. In which case it could be quite difficult for smaller stars to have the requisite protoplanetary disc density to form giant planets before the disc disperses.

surface - Would bagging an asteroid destroy valuable science about it?

The simple answer, unfortunately, is yes.

In fact, it is a resounding "yes". Option "A" uses a large "bag" to enclose the asteroid, and then to tow it to lunar orbit (or another feasible location). There's a problem, though, which is that simply surrounding an asteroid with a cylinder won't capture it. You have to "tighten" the "bag". This is implied in a graphic here and is explained in an article in Popular Mechanics ("How to Mine an Asteroid", Aug. 2012), and shown in a video on NASA's website. Basically, a series of "arms" constrict the "bag" to form-fit the bag to the asteroid's surface. The procedure could create many indentations on the asteroid's surface, and, as you said, destroy some of the most valuable information. The footprints of the astronauts who walked on the moon will stay there for many years; any indentations on an asteroid will be impossible to fix.

However, this entire idea is far from finished. NASA will not even decide between options "A" and "B" until "late 2014". So we won't know the precise details for a while.

So why go ahead with the whole thing if we're going to lose some of the best information we could get? Well, the Initiative isn't motivated purely by surface features. Part of it is spurred on by plans to pioneer new technologies needed to get humans to Mars. Other tangential benefits include diverting an asteroid that could be headed for Earth (although the asteroid candidates for this mission are much smaller than any of their more dangerous cousins; also, the Asteroid Initiative is under "Asteroid and Comet Watch" on NASA's website - right near plans to blow up a dangerous asteroid!), or even mining an asteroid for metals or resources for astronauts (such as water).

Also, this mission isn't the only way to study an asteroid (although it might be the best). We could always send a lander to an asteroid - after all, we can land on a comet (erm, well, we're trying - I'm rooting for Rosetta and Philae!), which would make it simple to collect samples before heading home . . . although the distance to travel would be a lot longer than just to lunar orbit.

All of this comes together to mean that NASA won't worry itself over losing data on the surface of the asteroid. Would that information be nice? Oh, yes. But given the other good things coming out of the mission, the pros here outweigh the cons. It's simply one of the trade-offs that might have to be made for this mission.

---

Sources:

Planetary.org

Space.com

NASA

"How to Mine an Asteroid". Popular Mechanics. August, 2012.

orbit - Is there a standard mapping of symbols to terms for celestial and orbital mechanics

Is there a standard mapping of symbols to terms for celestial and orbital mechanics (and physics in general)?

For example here are some ambiguities

Angular Momentum: L, H, J, $Gamma$,

Orbital period: T or P,

M: mean anomaly or mass of the Sun,

E: energy or eccentric anomaly,

p: linear momentum or length of semi-latus rectum,

a: acceleration or length of semi-major axis,

e: eccentricity or base of natural logarithms,

gravity - How massive does a planet need to be to create gravitational lensing?

You could google "Maccone focal" for the most ardent proponent of this project (and I love it too, the most furthest meaningful mission possible within a life time, and what a view!) But others may have variations of his idea. One should be a bit wary about enthusiasts.

Here's a link to some article where Maccone is cited to explain that planets need 6-17 greater distance to become gravitaional lenses than the Sun needs. Neptune surprisingly happens to be next best after Jupiter: http://www.centauri-dreams.org/?p=15290

Paper (might not be available for free):
Maccone, “A New Belt Beyond Kuiper’s: A Belt of Focal Spheres Between 550 and 17,000 AU for SETI and Science.”

Here are slides from one of his presentations where he explains his calculations more generally, but I think only for the Sun:
http://www.spaceroutes.com/astrocon/AstroconVTalks/Maccone-AstroconV.pdf

And then there's micro gravity lenses! A very different thing in terms of our observation. I now think my answer above misunderstood what you were looking for.

What implications does younger volcanism have on the Moon's thermal evolution?

In the paper Evidence for basaltic volcanism on the Moon within the past 100 million years (Braden et al. 2014), suggest that features such as Ina (image below) represent a far more recent age of volcanism (relatively speaking, around 100 million years).

Image source: NASA Science News, where they have a different non-volcanic theory. However, this question is not about whether the features are volcanic or not.

From the Braden et al. article abstract:

The morphology of the features is also consistent with small basaltic eruptions that occurred significantly after the established cessation of lunar mare basaltic volcanism. Such late-stage eruptions suggest a long decline of lunar volcanism and constrain models of the Moon’s thermal evolution.

Assuming the latest models indicate volcanism as recent as within the 100 million years ago, what are the implications for lunar thermal history of younger volcanism?

dark matter - Density of hydrogen between galaxies

Hydrogen clouds don't even make up a small component of dark matter, because hydrogen is not dark. The image below depicts the emission spectrum of hydrogen in the visible regime (a.k.a., the Balmer series).

The emission spectrum of dark matter on the other hand would be completely black.

Dark matter is gets its name from the fact that it doesn't absorb or emit light, i.e., electromagnetic radiation. From an elementary particle standpoint, this means dark matter particles can't interact with photons via the electromagnetic force at all, so dark matter particles must be electrically neutral, which electrons and protons certainly are not. No electrons are protons means no hydrogen atoms.

gravity - If the hammer and feather move at the same speed why do comet and the tail particles move at different speeds?

There are a few more elements which you might consider. Like, when dropping a hammer along with a feather in earth, the feather descends slower than the hammer. Reason: although the force of gravity is same, their mass, and the resistance from external forces faced by them is an entirely different story. The air between the object and the ground must also be taken into consideration. Whereas, moon lacks atmosphere and hence they fall at relatively same speed (with only low gravity acting upon them).

As in http://en.wikipedia.org/wiki/Comet_tail

As a comet approaches the inner Solar System, solar radiation causes
the volatile materials within the comet to vaporize and stream out of
the nucleus, carrying dust away with them. The streams of dust and gas
thus released form a huge, extremely tenuous atmosphere around the
comet called the coma, and the force exerted on the coma by the Sun's
radiation pressure and solar wind cause an enormous tail to form,
which points away from the Sun.

Similar to the air acting on the detached feather (as said above), the solar radiation pressure and solar winds act on the tail debris of the comet. For instance, you drop an article from a moving vehicle, it falls to the ground rather than flying in the air at the same speed. Deceleration and opposite forces like solar winds cause it to slow down.

UPDATE: The reason why mass is taken into consideration is due to the fact that mass is directly proportional to gravity. So, increasing the mass means increased gravitational pull. That is why heavier objects like hammer fall faster than lighter objects like feather. Explaining this would be a matter of physics and not astronomy which is irrelevant to this website and should be discussed in the physics website.

geology - Why is the Earth's center still hot after millions of years?

There is a nice article from Scientific American, but the main point is:

There are three main sources of heat in the deep earth: (1) heat from when the planet formed and accreted, which has not yet been lost; (2) frictional heating, caused by denser core material sinking to the center of the planet; and (3) heat from the decay of radioactive elements.

Other planets may have more or less radioactive material and this will make a difference over time as to whether they continue to have a molten core, or become volcanically and tectonically inactive.

orbital migration - How has the Earth's orbit changed over hundreds of millions or billion of years?

First, I know that modeling orbital mechanics of 8 planets is hard, but there are some theories out there, for example, Jupiter is thought to have moved in towards the sun then started moving away. Article

and Uranus and Neptune may have switched spots Article

Is there any pretty good evidence on how the Earth's orbit has changed over time. I remember reading some geological evidence that a year used to be longer, implying that the Earth used to be farther form the sun, but I've since been unable to find that article, and for purposes of this question, lets count a day as 24 hours even though a day used to be quite a bit shorter hundreds of millions or billions of years ago. - footnote, I've still not been able to find that article but it occurs to me, it could have been counting shorter days, not longer years - so, take that part with a grain of salt.

Are there any good studies out there on how many 24 hour days were in a year, 100, 300, 500, 800 million years ago? or 1 or 2 billion years ago? Either geological or orbital modeling? Preferably something a layman can read, not something written by and for PHDs?

or any good summaries, also encouraged. Thanks.

I also found this article, but it seems more theoretical than evidence based. http://www.futurity.org/did-orbit-mishap-save-earth-from-freezing/

life - Landing on an "earth-like" planet (practical application)

This may not belong here in astronomy and my apologies if that's the case.
The hypothetical:
We find, and are able to travel to (in a reasonably short amount of time), a planet that is for all intents and purposes a second earth: it has close to the same atmosphere, gravitational force, a stable orbit, land and liquid water, preferable temperatures, vegetation and other life , etc...

From a 'War of the Worlds' approach to this, wouldn't it stand to reason that we still would not be able to land on the planet and exit our spacecraft without suits to protect us because of the fundamental aspect of our immune systems? That is, our bodies have adapted and continue to adapt to Earth's lifeforms but have absolutely no tolerance to another planet of almost identical properties and therefore would perish shortly after exposure, yes? So I wonder, does anyone know if there is a plan for the hypothetical scenario to gradually expose individuals to the new environment and if so, what is the plan? For example, we could try to collect at least some of the potentially hazardous bacteria and viruses for vaccination? (maybe, not sure how that works even on Earth)

gravity - Satellite's orbit - Astronomy

Orbits around Earth need to be within its Hill sphere. Earth's Hill sphere is the region where Earth's gravity field dominates over the gravity fields of Moon and Sun. For Earth the radius of the Hill sphere is about 1.5 million kilometers. But for long-term stability the satellite needs to be well within the Hill sphere, less than half the maximum radius.

The escape velocity from Earth's surface is about 11.2 km per second. That's the velocity an objects needs at the surface of Earth, relative to a non-rotating coordinate system, to escape the gravity field of Earth, no atmospheric drag assumed.

The escape velocity is dependent of the distance from Earth's center of gravity, the further away from Earth the slower the escape velocity. Its roughly proporional to the inverse of the square root of the distance from Earth's center, as long as points above Earth's surface are considered.

galaxy - Explaining Dark Matter and Dark Energy to layman

Dark Matter
Your understanding of dark matter isn't bad, but here's a few clarifying details.
Orbits: The speed of an object's orbit is related to 2 things: the radius of its orbit and the mass inside of it. In the solar system, over 99% of the mass is concentrated at the centre, so radius is the dominant effect on orbital speed. As we look at planets further away from the sun, their orbital speed decreases. In the galaxy, it's a similar story, but with increasing orbital radius, you're also getting more and more mass inside of your orbit since the galaxy is full of other stars. Even so, we would expect orbital speeds to drop as you look at objects toward the edge of the galaxy, since the stars get more diffuse as you move out and not enough mass is inside the orbit to compensate for the increasing radius. Instead, we see that after a certain point, the orbital speeds stay the same, suggesting that there is more mass there than we can account for with stars and other visible objects.

Here is an example of a so-called rotation curve of the spiral galaxy NGC 3198:

Without dark matter, we would expect the speed of objects as a function of radius (from the center of mass) to follow the curve labeled 'disk'. However, what we see appears to be the sum of the two contributions (the disk and the 'dark matter halo' which surrounds this disk), which is overplotted with actual data.

We also have evidence of this extra mass from gravitational lenses. As light passes by a large object like a galaxy cluster, its path can get bent by that object's gravity. We can see this effect when a galaxy cluster is in front of an even more distant galaxy, the light from the background object gets magnified and distorted. We can calculate the path the light must have taken to appear as we see it, and the mass needed to bend it that way can't be accounted for with just the stars and gas we see in the lensing cluster. This again suggests that there is extra mass that we can't see.

Dark Energy
You're not too far off with this one, with one major distinction: Dark energy is the 'force' that's causing the universe to accelerate its expansion. We expect the universe to be expanding from the Big Bang cosmology model, but we'd expect it to be slowing down as the mutual gravity of everything in the universe acted upon each other. Dark energy appears to be a "pressure" on every part of space to expand, but it's very weak and has been dominated by gravity and other factors throughout most of the universe's history. It's only in the past few billion years it's started to take over.

It works something like this. Imagine you have a 1-D region of spacetime 1 unit long (just for convenience's sake). Here it is: |--------|
Now this region has this dark energy 'expansion pressure' acting on it. Let's say it's 0.07 units per year per unit. This means that every year, this region of spacetime gets bigger by 7%. In 10 years, the length will be double:
|--------|--------|
Now the thing is, each of these regions has the same expansion pressure as the original one! So both will double in 10 years, and then those will double, and so on. So what happens is you get a small expansion locally, but the further away something gets the faster it's accelerating away from you. The real effect of dark energy is orders of magnitude smaller, something like 67.15 ± 1.2 (km/s)/Mpc (Wikipedia), but it still means that any galaxies more than 4.5 Gigaparsecs away from us are currently being carried away from us faster than the speed of light. (We can still see them right now because the light we see was emitted long before the expansion rate got that high.) The expansion adds up over the huge distances we see in the universe.

Dark energy doesn't affect things like planets, solar systems, and galaxies because the effect is so weak on small scales that gravity more than counteracts it. It's effect can mostly be seen over the vast empty stretches of space between galaxy clusters.

Hope that helps!

universe - The reason behind Big Bang

WHY implies a reason, which is venturing in to the religious realm.

The circumstances under which an event such as the Big Bang occurs are guess work at best. Since the Big Bang is a singularity from which time itself started there is no real before the big bang. That's the first problem. The second is that because of the spatial singular nature of the big bang no information from 'before' it could ever pass through to our current universe and provide us with any hints.

There are a number of oddball theories that explore the idea of physics before the big bang. Some investigate what occurs on scales of time shorter than that of the Planck time, where quantum mechanical processes may give us hints in to the cause or mechanics of a big bang event. Proving or, indeed, disproving these theories is currently impossible experimentally.

Does Doppler shift affect apparent pulsar frequency?

We know that redshift and blueshift is the result of the frequency of light waves reflected or emitted by an object lengthening or shortening due to the relative velocity of the observer moving away or toward the object. This is the Doppler effect.

However, it was stated in response to a question I asked on a sister site that the Doppler effect would also result in the apparent frequency of a pulsar's pulses being longer or shorter due to the relative velocity of the star. Given that the speed of light is constant regardless of the frame of reference (a concept I have a really hard time understanding by the way), is this correct?

My thinking is that it is incorrect. Using the analogy of a launcher lobbing medicine balls at me at a fixed rate, I can "see" the rate of launch by the frequency of when I get hit. However, if the launcher is moving away from me, the velocity of the balls lobbed at me will be slower (relative to my frame of reference), and thus will hit me with less frequency. However, if the launcher were to adjust its firing speed (i.e. speed of the ball relative to itself; frequency remains constant in this analogy) so that the balls always hit me at the same speed (thereby simulating the concept of the always-fixed speed of light), the time between them would not change.

So, given the always-constant speed of light, is it correct that a pulsar's pulse rate would appear to be different at different relative velocities?

galaxy - What happens if two galaxies collide very quickly?

Have 2 people stand in opposite corners of a large empty room each with a jar of marbles. Have them roll these at each other all at once as fast as they can, and you will get a pretty good answer to your question.

97% (or thereabouts) of a given galaxy is empty space. Most of one is going to pass harmlessly through the other, though you are likely to get a few collisions. At or near the speed of light, gravity will have very little effect as well, so a few orbits will be perturbed, but any given star will hardly notice.

I suspect (though I haven't the expertise to say definitively) that the radiation increase during the ~100,000 years or more that this collision would take (assuming the galaxies were of approximately the size of the Milky Way and that they hit exactly edge on) would be significant to any creatures living in your galaxies.

If by the off chance (I'm talking about lightning-striking-a-shark-currently-eating-a-recent-lottery-winner kind of chance) the supermassive black holes at the center of these two galaxies collided, the speed would be no obstacle, and they would combine. The angular momentum would certainly rip the new black hole from one of, or likely both, of the colliding galaxies. This would be bad for them. I propose sending this as a "What If?" question to XKCD. He seems to have the time to do the math on questions of this nature and follow as far as it leads.

How is the maximum density in a circumstellar disk determined?

I'm working with some equations to model the evolution of a circumstellar disk. One of the equations is
$$rho(r)=Ce^{-frac{(r-r_{peak})^2}{2 sigma ^2}}$$
where $rho$ is density, $r$ is the distance from the center, $C$ is a constant, $sigma$ is one standard deviation, and $r_{peak}$ is the radius at which the density is at a maximum.

If the function was of the form
$$rho(r)=Ce^{f(r)}$$
where $f(r)$ is a function of $r$, I could find the maximum easily by finding
$$rho'(r)=Cf'(r)e^{f(r)}=0$$
and solving for $r$. However, this appears to be impossible in the current case because $rho(r)_{peak}$ is already in the equation, at $r_{peak}$.

How is $r_{peak}$ determined in a given scenario? Is it determined experimentally?

matter - If oxygen were pumped into space where would it go?

Gases tries to occupy all the volume of the container in which they are encased in equilibrated pressure. This happens because the gas molecules are always colliding and kicking each other in a 3D high velocity brownian motion. The result is that the gas molecules occupies all the container's volume homogeneously.

So, if the sealed container is opened up in space, the oxygen would quickly leak out, because the molecules near the vacuum edge would be instantaneosuly kicked out by the gas innermost molecules. This would happen rather quickly and in a chained manner, but is dependent on the size of the open hole. Macroscopically, the result would be that all the gas would be running out in a form of a wind/jet out of the open hole into outer space. This would only stop when all the molecules becomes too far apart from each other to have any further intermolecular interaction.

This means that the oxygen (or any other gas) would be dissipated into outer space. The gas molecules would initially be spread out by the intermolecular collisions and then would go outwardly due to the momentum acquired from each individual gas molecule.

When freed in outer space, each molecule would eventually reach something else to interact, so they could be attracted and retained by some gravity well (most likely from a planet or moon), or interact with some magnetosphere (if near a planet) or interact with some other molecule wandering around, or with a cosmic ray or be catched by the solar wind particles (or any of the equivalent if this happens near some other star).

If this happens in intergalactic space where there is no body with appreciable gravity or magnetic field nearby, no stellar wind and the cosmic rays are very sparse and rare and the very few that exists are randomly directed, then the molecules would just spread out in virtually linear trajectories and travel the intergalactic space lonely, quietly and virtually undisturbed for some billions or trillions of years. In their journey, each molecule would only very occasionally interact with some other molecule (likely to be hydrogen) or some other lost particle wandering around.

star - Triangular Asterism Trigonometry

Using the SIMBAD database for Alnitak, Saiph, and Sirius -- i.e.:

Alnitak: FK5 coord. (J2000): RA = 05 40 45.527 DEC = -01 56 33.26;
Saiph; FK5 coord. (J2000): RA = 05 47 45.389 DEC = -09 40 10.58;
Sirius; FK5 coord. (J2000): RA = 06 45 08.917 DEC = -16 42 58.02;

What is the trigonometric relationship in degrees -- length of sides and inclusive angles to four decimal places -- between these three stars when they are used to form a triangular aterism?

orbit - Is the Apophis asteroid a concern?

Apophis is of no real concern to us as far as we can tell. That said, it doesn't mean another asteroid doesn't have our name on it...

According to the most reliable data we have regarding this particular asteroid it only has a 1 chance in about 250,000 of hitting the earth. University of Hawaii states:

“Our new orbit solution shows that Apophis will miss Earth’s surface in 2036 by a scant 20,270 miles, give or take 125 miles,” Tholen said. “That's slightly closer to Earth than most of our communications and weather satellites.” He credits the large telescopes and superb atmospheric conditions on Mauna Kea for being able to make these determinations.

Considering how big space is, this is pretty much a hit in cosmic terms. Keep in mind that this information has been readily available since 2009 at the NASA website as well.

lagrange point - L4 and L5 stability

It does indeed seem counterintuitive that $L_4$ and $L_5$ would be at the same time both high points of potential as well as stable points in the system. In fact, a quick look at an example contour plot with all five Lagrangian points demonstrated would also suggest that $L_4$ and $L_5$ would be unstable:

In the picture you can see that $L_1 - L_3$ are placed in saddle nodes, and intuitively one concludes that the smallest perturbation would set them "falling" in either direction, i.e. they need station keeping.

However, $L_4$ and $L_5$ would still seem to be unstable, so what gives? What makes $L_4$ and $L_5$ stable is that each of them is located equidistant from both of the masses. This leads to the gravitational forces from each of the bodies towards $L_4$ and $L_5$ to be in the same ratio as the two bodies' masses, hence the resultant acceleration points towards the barycentre of the system (which is also the centre of rotation for a three-body system).

This resultant acceleration, and by extension force, leads to the object in either $L_4$ or $L_5$ to in fact have just the correct amount of acceleration / correct force towards the system's barycentre so that it will not fall out of orbit (it "falls" towards the barycentre as the system rotates, like Earth similarly falls towards Sun throughout its orbit), as counterintuitive as it sounds at first. (Furthermore, this effect is not only limited to bodies of negligible mass, but is independent of mass, which is why you can have more massive bodies orbiting in $L_4$ and $L_5$ like Jupiter's trojans, as well as smaller, artificial satellites.)

Source: Lagrangian point - Wikipedia

exoplanet - If Alpha Centauri A's solar system exactly mirrored our own, what would we be able to detect?

This is a broad question and too broad for me to answer comprehensively. It should be broken down into doppler methods, transits and direct imaging; and that's before we get to questions of detecting Kuiper belts, radio emission etc.

I'll stick for the moment with what I know about detection of planets using the doppler wobble technique.

Doppler Technique

The reflex radial velocity semi-amplitude of a star for the case of a planet of mass $m_2$ orbiting a star of mass $m_1$, in an elliptical orbit with eccentricity $e$, and orbital period $P$ and with an orbital axis inclined at $i$ to the line of sight from Earth is:
$$ left( frac{2pi G}{P}right)^{1/3}frac{m_2 sin i}{m_1^{2/3}} (1-e^2)^{-1/2}. $$
A (very) detailed derivation is given by Clubb (2008).

So I built myself a little spreadsheet and assumed that all the planets were seen optimally at $i=90^{circ}$ (they could not all be seen optimally, but the smallest inclination would be about $i=83^{circ}$ for Mercury, so it doesn't make too much difference) I'll also assume the mass of Alpha Cen A is about $M simeq 1.1M_{odot}$.

The results are

Planet | RV semi-amplitude (m/s)

Mercury | $8.3times 10^{-3}$

Venus | $8.1times 10^{-2}$

Earth | $8.4times 10^{-2}$

Mars | $7.5times 10^{-3}$

Jupiter | $11.7$

Saturn | $2.6$

Uranus | $0.28$

Neptune | $0.26$

The limits of what are possible are well illustrated by a planet around Alpha Cen B, claimed to be in a 3 day orbit and with a mass similar to the Earth (Dumusque et al. 2012, and see exoplanets.org). The radial velocity semi-amplitude detected here was $0.51pm 0.04$ m/s, and some spectrographs, notably the HARPS instruments, are routinely delivering sub 1 m/s precision. Thus Jupiter and Saturn would be detectable, Uranus and Neptune are right on the edge of detectability (remember you can average over many RV observations), but the terrestrial planets would not be found (Earth detections would require precisions below 10 cm/s. Remember also that the weaker signals would have to be dug out from the larger signals due to the Jupiter- and Saturn-like planets.

However, there is a second limitation: to find a planet using the doppler method you need to observe for at least a significant fraction of the orbital period. Given that current m/s precisions have been available for only $sim 5$ years, it is unlikely that Saturn would yet have been detected.

A picture that illustrates the situation can be obtained from the exoplanets.org website, to which I have added lines that approximate where RV semi-amplitudes would be for 10 m/s and 1 m/s precision (assuming the Alpha Cen A mass and circular orbits). I've marked on the Earth, Jupiter and Saturn. Note that few objects have been discovered below the 1 m/s line. Also note the lack of planets between the 1 and 10m/s lines with periods longer than a couple of years - the recent increase in sensitivity has yet to feed through to lower mass, longer period exoplanet discoveries.

In conclusion: only Jupiter would have been so far found by the doppler technique.

Transit techniques

I'll also add a few comments about the transit technique. Transit detection will only work if the exoplanets orbit such that they cross in front of the star. So high inclinations are mandatory. Someone who is better at spherical trigonometry should use the published data for the solar system to work out how many (and which) planets transit in some highly optimal orientation. Given that the planets have orbital inclinations with a scatter of a few degrees, then some straightforward trigonometry and a comparison with the solar radius, tells you that these orbits will generally not all transit for any particular viewing angle. Indeed a number of the Kepler-discovered multiple transit systems are much "flatter" than the solar system.

The Kepler satellite is/was capable of detecting very small transiting planets thanks to its very high photometric precision (the dip in flux is proportional to the square root of the exoplanet radius). The picture below, presented by the NASA Kepler team (slightly out of date now), shows that planetary candidates have been discovered that are down to the size of Mars. However these tend to be in short period orbits because a transit signal needs to be seen a number of times, and Kepler studies this patch of sky for about 2.5 years (when this plot was produced).

So from this point of view, possibly Venus would have been seen, but none of the other planets could be confirmed.

However, there is a wrinkle. Alpha Cen A is way too bright for these kinds of studies and way brighter than the Kepler stars. You would have to build a special instrument or telescope to look for transits around very bright stars. Some of this work has been done by ground-based surveys (mainly finding hot Jupiters), but a new satellite called TESS (Transiting Exoplanet Survey Satellite, launch perhaps in 2017) will do a more comprehensive job. This is a two year mission, so might be capable of detecting the Earth and Venus (and possibly Mercury), but Mars would not produce multiple transits on this timescale.