From original burst, fraction of stellar mass still surviving on Main sequence

When you are calculating fractions, rather than absolute numbers, the value of $\Phi_0$ does not matter, since it will be a multiplying factor in both the numerator and denominator.

You have (almost, see below)  successfully got an expression for the denominator of your fraction.

The numerator is found by calculating an equivalent integral from your lower limit to an upper limit that is instead defined by the most long-lived main sequence stars that are still "alive" - i.e. those with a lifetime equal to the age of our Galaxy.

Finally, you were asked to find the fraction of stellar mass surviving, not the fraction of stars. The stellar mass existing between two mass intervals is
$$M_* = \int_{m_1}^{m_2} m\Phi(m)\ dm$$


Where can I find the previous years' International Astronomy Olympiad question papers?

A blog called Science Olympiad Blog claims to have the problem sheets for 2011-2014.


Does Jupiter help to protect the Earth against comets/asteroids impacts?

Wordy answer, mostly light on Math:

The key word here (and the article uses this word) is "long period comets".

First there's Jupiter impacts, but that's a relatively low percentage, cause even large Jupiter is quite small compared to it's orbit.  Even if you extend it out to it's Roche limit where a comet could break apart, it's still a very small target in the grand scheme of things.

But any comet that passes near Jupiter (but doesn't hit it, get captured by it or broken apart in it's Roche limit) gets it's orbit changed somewhat by gravity assist.  Jupiter's high mass and relatively high orbital velocity make it the gravity assist king of the solar-system.

The closer the pass to Jupiter, the bigger the change.  Gravity assists, change both direction and velocity.  They can't change only one.  It's safe to assume that half of those gravity assists add velocity to the comet and half reduce velocity.   Very loosely speaking, the direction of the comet affects it's Perihelion and it's velocity and any change in it's velocity affect's it's semi-major axis (explained a bit below).

The comet's orbital velocity is obviously dependent on how close the comet is to the sun.   (Kepler's law about equal areas over equal times), but any orbiting object also has an average velocity, which is closely tied to the semi-major axis and orbital period, and it's this average velocity and the velocity at Aphelion (it's slowest velocity) that matters for this discussion.

The orbital period of any object orbiting the sun is directly proportional to the 3/2 power of the semi-major axis.  The average velocity is very closely correlated to the square root of the distance between the semi-major axis and the Sun.

When you get comets with orbital periods of hundreds or thousands of years, their average orbital velocity, is relatively slow for a celestial object.  Pluto's average orbital velocity is about 4.67 km/s and it's minimum velocity at Aphelion is 3.71 km/s.  Source.

For longer orbital periods the orbital velocity drops, roughly by the square root of the relative distance, so a comet with an orbital period of 1,000 years (about 4 times Pluto's), would have an orbital velocity about 1/2 as much and a velocity at Aphelion of maybe 1.8 km/s or a bit less.

Escape Velocity for the comet requires just a 41.4% increase on Aphelion orbital velocity, so, with our 1.8 km/s orbital velocity long period comet at Aphelion, if Jupiter gives that comet a push of just about .75 km/s, that would be enough to push that comet out of orbit, where it escapes the solar system.

So, while it might seem logical that Jupiter could send a comet equally towards earth as away from Earth, the relatively small push that a long period comet needs to escape the solar system, makes that a far more likely scenario than a nearly perfect 100 foot put into the Earth's orbit.   The odds of a long period comet being thrown out of the solar system by Jupiter is relatively high, especially with multiple passes, where, hitting Earth is like hitting a 100 foot put blindfolded.   Earth is a small target.  Out of the solar-system is a large target.

For shorter period comets where the added velocity needed to escape the solar system is greater, this grows proportionately less true, but for long period comets which only need a small push to get thrown out of the solar-system, statistically, Jupiter tosses a lot more of those out than it sends towards earth.

The trick is to be very specific about what we're talking about.   If Jupiter was, for example, to migrate through the asteroid belt, or migrate outwards through the Kuiper belt, then it's size would send a lot of stuff in towards the inner solar-system.   It's not in and of itself, always going to protect Earth and reduce Earth-impacts.  It depends on where it is and what the orbits of the comets are, but specifically for long period comets, Jupiter is a-lot more likely to send them out of the solar-system than towards Earth.  Not all, just statistically more likely.

It's worth noting that there's a lot we don't know about the oort cloud and how many comets/icy objects are in it.   We don't know what percentage of long period comets are jupiter-crossers vs objects with more distant perihelions.   When a star or large rogue planet passes through Earth's Oort cloud, it can send some of the icy objects out there in towards the Earth.   (Planet 9 - if it exists, probably not so much, as that theoretical planet has probably already mostly cleared out it's orbit), so it's not a key player in sending stuff to the inner solar-systems, but a massive passing object that pass through the oort cloud can do that, at least, that's been theorized, though such events are quite uncommon because space is mostly empty.   When those rare events happens, then there's probably an increase in long period comets and when that happens, Jupiter, for the reasons described above, is pretty good at getting rid of them over time.

Scholz's star is thought to have passed within 0.8 light years of our sun some 70,000 years ago and such a pass might be close enough to send a lot of Oort cloud objects in towards the inner solar system (though 0.8 light years it's still a bit far, as the oort cloud is probably quite empty that far out.  For a really good number of new long-period inner solar-system comets you'd probably want a 0.1 lightyear pass, or closer . . . . but massive objects passing that close are extremely rare, and . . .  I digress).

But Scholz's star might have turned a number of Oort cloud objects into long-period comets that pass through the inner solar-system.  (We won't know for a long time, as it'll be perhaps a million years for those new comets to reach us), But assuming Scholz did that, any oort cloud objects entering the inner solar-system from that event would be extremely long period objects and Jupiter would be pretty good at (over time, it wouldn't happen quickly), but over multiple long period orbits which probably takes millions of years, Jupiter would be pretty good at removing most of those from the inner solar-system, but just by chance, it would probably send a few in Earth's direction too.   If there was no Jupiter, it would take a lot longer for those long period comets to get cleared out, so in that sense, it really does protect the Earth from Long period comets.   There would be a lot more of them if not for Jupiter (and to a lesser extent, Saturn) and more comets would mean more Earth impacts.


How to compare a reflector telescope to a refractor one?

Small and effective refractors are easier to make than small and effective reflectors, but reflectors are cheaper in general. Unless you have some special reason to want to avoid a larger aperture (eg because you are engaged in solar observation) or a special reason why you want the optical qualities of a refractor, you almost certainly want a bigger aperture and so for a fixed price you are likely to be better off with the reflector.

Not having a EQ mount may be a bit of an issue though.


Why doesn't the moon always follow the same path? Or why aren't the moon's apoapsis and periapsis fixed?

The Moon's orbit about the Earth is only approximately elliptical. The Moon's orbit precesses both axially and nodally, and the eccentricity of the Moon's orbit varies.

That the Moon's orbit precesses axially means that perigee sometimes occurs when the Moon is close to new, sometimes when it is close to half-full, and sometimes when it is close to full. Supermoons occur when perigee occurs close to when the Moon is full. The variation in the eccentricity means that apogee distance and perigee distance vary considerably.  Super-supermoons (my term) occur when perigee occurs close to when the Moon is full and when perigee distance is much closer than nominal. While supermoons occur about three of four times per year, super supermoons occur only once or twice a year.

With regard to eclipses, suppose that at the time of a full Moon that the Moon's orbit about the Earth is close to being aligned but not exactly aligned with the Earth's orbit about the Sun. This means that a month later, the Moon's orbit may still be close to being aligned with the Earth's orbit about the Sun. The large size of the Earth's shadow makes it possible for lunar eclipses to occur in two consecutive months.

Eclipse seasons, the times during which lunar eclipses and solar eclipses are possible, are about 34 days long, a bit longer than one month. This means that if a partial lunar eclipse occurs just as an eclipse season starts, a nice solar eclipse will occur half a month later, and a partial lunar eclipse will occur another half a month later. Similarly, if a partial solar eclipse occurs just as an eclipse season starts, a nice lunar eclipse will occur half a month later, and a partial solar eclipse will occur another half a month later.


Visibility of earth from moon during day-time of moon

Yes, the Earth would be visible even during day on the Moon. Without a (significant) atmosphere diffusing light, the day sky would be much like the night sky on Earth. If you were to look straight up such that you couldn't see the ground, you may not even notice that it was day at all (save for the Sun shining down at you).  This could actually be rather hazardous as the Sun would be more intense, but your eyes may not be adjusted to high light like they are on bright Earth days.


Would the electron cyclotron-maser emission mechanism affect Proxima b's ability to retain an atmosphere?

This mechanism, being actively emitting in the radio-wavelengths, is certainly negligible for the overall atmospheric energetics at Proxima b. One can conclude this by taking the band luminosities from the cited paper ($\rm 2.51\times10^{20} erg/s$, p.3, first paragraph) and compare them to the solar constant at the planets orbit, which should amount to $\rm9\times10^{23}erg/s$.
The latter is however only the thermal energy contribution given by the stellar effective temperature, there might be other significant chunks of energy contributed in the UV and X-bands.
The luminosities thus considered would be important for driving both bulk escape and jeans escape. Being negligible in the radio-range, shows that this process is unimportant for the retention of a bulk atmosphere via thermal mechanisms.
In terms of non-thermal mechanisms it's possible to imagine that there would be some resonant absorption of the ECM emission at the planetary radius, driving scattering of heavier species. Considering however the scalings of the cyclotron frequency $f=qB/{2\pi m}$, and the fact that the magnetic field is weaker at the planet compared to the emission region at the star, the scattered particles would have to be lighter than electrons. Those particles do not exist in atmospheres in significant numbers, only in particle accelerators.
Hence we can conclude that non-thermal processes driven by ECM heating will also not play a role for atmospheric escape.


Standard conditions for a heliacal rising

What I do is compute the limiting magnitude for the position of the object I am interested in, and subtract it from the magnitude of the object (I call this the magnitude contrast).  If negative, it should be visible with the naked eye.  I use the function limmag_jd() in the Fortran library libTheSky for this.

In order to find the best moment in a given night, I compute the contrast for say every 15 minutes, and find the minimum.  You can do this for a number of nights, until the best contrast found is negative, which would be the heliacal rising.

I'm not sure whether there is an easy way using a simple equation - the limiting magnitude will depend on things like the positions of Sun, Moon and the object in question, the phase of the Moon, the magnitude of the object, etc.


Why Venus evolved so differently from Earth?

Venus orbits the Sun at 0.723332 AU. As solar radiation follows an inverse square law,  the intensity of the Sun at the top of Venus's atmosphere is almost twice that for the Earth ($\frac1{0.723332}^2\approx1.9113$). Even when the solar system was young and the Sun was considerably dimmer than it is now, Venus received 1/3 more solar radiation at the top of its atmosphere than the Earth does now. Unlike the Earth, Venus did not suffer a faint young Sun problem.

That faint young Sun was perhaps instead a blessing for Venus. Many have argued that shortly after cooling off, Venus started as wet planet, like the Earth. The intense and increasing solar radiation eventually resulted in Venus undergoing a wet greenhouse process. Instead of being a trace gas in the Earth's atmosphere, water vapor may well have been the dominant gas in Venus's atmosphere. Thanks to the potency of water vapor as a greenhouse gas, this resulting in a vicious positive feedback loop. The increased water vapor raised Venus's surface temperature, resulting in even more evaporation/boiling. Meanwhile, at the top of the atmosphere, solar radiation dissociated water vapor. The hydrogen slowly escaped, eventually leaving Venus more or less free of water in any form.

Water may be necessary as a lubricant for plate tectonics. Whether or not this is the case, Venus does not have plate tectonics. It instead has a stagnant lid geology, punctured occasionally by extreme vulcanism (Siberian traps level vulcanism, and then some). This eventually released some of Venus's deep carbon reserve into its atmosphere. The end result is a planet void of water with a very thick carbon dioxide atmosphere.


The redshift that is referred to is not a Doppler redshift, but a cosmological redshift. The difference is that the former is caused by the source moving through space, while the latter is caused by the "stretching" of the wavelength of the light as it travels through space.

The cosmological redshift happens gradually along its journey, and hence it is a measure of the distance to the source. But since traveling through space takes time, it is also a measure of the lookback time to the source — that is, the time that has passed since its emission. This notion of the term can thus be used as a timeline for phenomena in the Universe, and perhaps somewhat confusingly, it is sometimes used this way even when referring to local phenomena.

For instance, Earth was formed 4.54 billion years ("Gyr") ago. If some unrelated galaxy emitted light at the same time, and if that light reaches us today, then that galaxy must be at a particular distance$^\dagger$. During its journey it has been redshifted to $z \simeq 0.42$. Thus we may say that Earth was formed at redshift $0.42$.

Similarly, stars (in the Milky Way or elsewhere) that are said to have formed at redshift $z=10$ and $z=15$, can also be said to have formed 13.3 Gyr and 13.5 Gyr ago, respectively.

In the figure below, I plotted the relation between cosmological redshift and lookback time (assuming a Planck 2015 cosmology). Now hug each other, Rob and NotSoSN.

$^\dagger$5.4 billion lightyears, actually. Not just 4.54 billion lightyears, because space has expanded in the meantime.


What is the share of stars in total radiation input of Earth?

It is negligible.

The amount of energy per square metre per second is called the "flux". The flux of the brightest star in the sky, Sirius, is one ten billionth of the sun. That mean the energy the earth receives from the sun in one second is the same as the energy that it gets from Sirius in about 300 years. And Sirius (at magnitude -1.47) is by far the brightest star. All the others add next to nothing.


What percentage of the celestial sphere can the Moon cover?

If the Moon's orbit were fixed, the area covered by the Moon would just be $360^{\circ}*(0.5^{\circ}+2*1^{\circ})$. The 5.3 degree inclination relative to the ecliptic (or some other inclination relative to the equator) would not be a factor.

What is important to this question about the coverage is that the orbit changes over time. In particular, the position of the ascending node (where the Moon's orbit crosses the ecliptic) moves along the ecliptic with a period of 18.6 years (Wikipedia article on Lunar precession).

In a plot of the sky using ecliptic coordinates (so that the ecliptic is a straight line), the 5.3 degree inclination of the Moon's orbit to the ecliptic is shown as "year X" in my figure. The precession changes the orbit to positions such as year Y and year Z in my figure, and during the entire 18.6 year cycle, the entire band within 5.3 degrees of the ecliptic is covered by the Moon. In other words, the ascending node moves through all 360 degrees of ecliptic longitude. This results in the term 2*5.3 in your equation. (Note: I do not know if the precession is to the "right" or "left", so do not interpret my figure to mean that year Y is later than year X. Year Y could be earlier than X. I'm sure someone will correct me!)

If the same plot were made in equatorial coordinates (so that the equator is a straight line), the band covered by the Moon is of course the same on the sky. However, the precession along the ecliptic limits the ascending node of the Moon's orbit on the equator to a small range or right ascension; the ascending node does not travel around the entire equator. Representing that band using equatorial coordinates and determining the area inside the bands is much more difficult.


Where can I find a good solar irradiance chart for the infrared?

Thanks to @PearsonArtPhoto's comment data with 1 cm^-1 wavenumber resolution is available in the link labeled MODTRAN data here: https://www.nrel.gov/grid/solar-resource/spectra.html

Here's a quickie plot and script for the downloaded ASCII data.

There are six columns of spectral intensity data (Watts/m^2/nm) with labels 'MCebKur', 'MChKur', 'MNewKur', 'MthKur', 'MoldKur', 'MODWherli_WMO'. The plot shows them to be very similar in the thermal IR (not surprising) and differ substantially in the visible and UV where differences in the spectrometers will show up in the region rich with spectral emission and absorption lines. This is illustrated in the second plot, showing a randomly chosen zoom in on a few nanometers of visible light, and the Fraunhofer "A" line.

import numpy as np
import matplotlib.pyplot as plt

fname = 'AllMODEtr MODTRAN.txt' # MODTRAN data https://www.nrel.gov/grid/solar-resource/spectra.html

with open (fname, 'r') as infile:
    lines = infile.readlines()

lines = [line.split() for line in lines[19:-1]] # skip a few lines with missing data

print('line lengths: ', set([len(line) for line in lines]))

data = [[float(x) for x in line] for line in lines]

data = np.array(zip(*data))

print('data.shape: ', data.shape)

# data[0]: wavenumber (cm^-1)
# data[1]: wavelength (nm)
# data[2:8] MCebKur, MChKur, MNewKur, MthKur, MoldKur, MODWherli_WMO

labels = 'MCebKur', 'MChKur', 'MNewKur', 'MthKur', 'MoldKur', 'MODWherli_WMO'
wavelength = data[1]

if True:
    plt.figure()

    plt.subplot(2, 1, 1)

    for (thing, label) in zip(data[2:], labels):
        plt.plot(wavelength, thing)
    plt.xlabel('wavelength (nm)', fontsize=16)
    plt.ylabel('Watts/m^2/nm', fontsize=16)
    plt.yscale('log')
    plt.xscale('log')
    plt.xlim(190, 2E+05)

    plt.subplot(2, 1, 2)

    n700 = np.argmax(wavelength > 700)
    for (thing, label) in zip(data[2:], labels):
        plt.plot(wavelength[:n700], thing[:n700])
    plt.xlabel('wavelength (nm)', fontsize=16)
    plt.ylabel('Watts/m^2/nm', fontsize=16)
    plt.yscale('log')
    # plt.xscale('log')
    plt.xlim(190, wavelength[n700])

    plt.show()

if True:
    plt.figure()

    plt.subplot(2, 1, 1)

    n1 = np.argmax(wavelength > 516.5)
    n2 = np.argmax(wavelength > 519.5)
    for (thing, label) in zip(data[2:], labels):
        plt.plot(wavelength[n1:n2], thing[n1:n2], label=label)
    plt.xlabel('wavelength (nm)', fontsize=16)
    plt.ylabel('Watts/m^2/nm', fontsize=16)
    # plt.yscale('log')
    # plt.xscale('log')
    plt.xlim(wavelength[n1], wavelength[n2])
    # plt.legend()

    plt.subplot(2, 1, 2)

    n1 = np.argmax(wavelength > 757)
    n2 = np.argmax(wavelength > 760)
    for (thing, label) in zip(data[2:], labels):
        plt.plot(wavelength[n1:n2], thing[n1:n2], label=label)
    plt.xlabel('wavelength (nm)', fontsize=16)
    plt.ylabel('Watts/m^2/nm', fontsize=16)
    # plt.yscale('log')
    # plt.xscale('log')
    plt.xlim(wavelength[n1], wavelength[n2])
    plt.title('Fraunhoffer "A" line')
    # plt.legend()

    plt.show()


Does blackhole merging break their event horizon segregation?

We're also not physicians, not even Astrophysicians, but we're physicists.

One of the more famous physicists, Leonard Susskind, discusses in his excellent youtube lecture series on general relativity (I believe it was episode 7, which I've linked) that during a black hole merger the Schwarzschild surface (not a sphere anymore) deforms and 'comes out to get you'.

Otherwise one would run into the problem that early research into black holes had, when they were still called Collapsars. This problem was that a Collapsar would eternally collapse due to the gravitational time dilation and never actually form a Schwarzschild radius that the static solutions indicate should exist. Then also, black holes could never merge.

But this problem has been solved with the realization that the Horizon deforms during a merger. The Horizons themselves then merge, and A and B come into causal contact.


What is the format of the data from the JPL's HORIZONS system?

WebGeocalc (http://wgc.jpl.nasa.gov:8080/webgeocalc/) is actually a
fairly useful tool, and the next best thing to downloading the CSPICE
libraries, so let me provide a slightly more detailed answer to your
question.

When you first visit WebGeocalc, you will see something like this:

Because you are looking for perihelions, which involve distance,
scroll down and choose "distance finder":

Since you want to find the perihelion of Mercury, fill out the top
half of the form like this:

For this example, I'll find all perihelions from 1900 to 2201. A
perihelion is a minimum of distance, and since you're looking for all
perihelions, choose "is local minimum" on the lower half of the form,
so it looks as follows:

Scroll down once again (you can leave the remaining form values as is)
and click the "Calculate" button:

When you get your results, click "Save All Intervals" to save the
times of Mercury's perihelions. Note that each result is an interval,
but the interval is 0 seconds long, so it's actually a point in time:

Now click on "Calculation Menu" to get back to the main menu:

We'll now find Mercury's position at all of these perihelions so you
can see how the perihelion position changes.

To find Mercury's position at these times, first click on "State
Vector", which will take you to a form. Fill out the top half of the
form as follows:

Fill out the lower half of the form as follows, dragging the results
window (for the times of Mercury's perihelions) into the input times
window:

Click "Calculate" to receive your results:

See also http://naif.jpl.nasa.gov/pub/naif/toolkit_docs/Tutorials/pdf/individual_docs/36_webgeocalc.pdf

NOTE: if you're going to do extensive calculations, you'll probably
want to download the C SPICE library and kernels:
http://naif.jpl.nasa.gov/naif/tutorials.html

EDIT: I emailed Jon D. Giorgini (jdg@tycho.jpl.nasa.gov) re the open-sourceness of HORIZONS, and got this answer (which I'm not sure is helpful, but...):

Horizons output is copyrighted in the sense one can't take a
table output by Horizons and re-publish it as one's own.
JPL is a contractor organization and so the output is not prepared
by an officer or employee of the US government, and it is not
transferred to the US government, so copyright remains relevant.

The data values themselves, say the RA/DEC of Pluto at
some instant, are not legally copyrighted, though it would
be unethical to publish it without citation.

Legal copyright protection is for fixed representations
analogous to a book; the book has copyright protection, but
an individual word extracted from the book (i.e., into a
different representation) is not itself protected.  Up to
a point

Note that Horizons reads SPICE files for planets, natural
satellites, and spacecraft, then derives results it outputs
from scratch from that basis. But for comets and asteroids,
it creates all information on demand from scratch.

Further, we (JPL Solar System Dynamics) put the planet/satellite/
spacecraft information into the SPICE format to begin with. So
a SPICE file might be like the blank pages of a book; not
copyrighted until the original work is written into it, but not
itself copyrighted (beyond the source code, of course).

But if you need a more trustworthy legal opinion for a specific
situation, I can put you in touch with legal staff. If this is an
actual issue and not just a point of curiosity, that's the way to
since perhaps other rules would apply to your actual situation.


Theoretically, can a solar system evolve around lagrange points

Yes, provided the star is big enough, the "Jupiter" is big enough and the planet is small enough.

The details are discussed herein: "Is there a ceiling for stable L4 or L5 masses?"

To summarise: The sun needs to be 25 times more massive than the giant planet, and the small planet needs to have negligible mass.

It is worth noting that the minor body could not be said to be dominating its orbit, and so would be considered a dwarf planet by the IAU.


Meaningful theories of the shape of the universe

I disagree with your premise. First, the universe being infinite does not imply anything at all about its spatial curvature without additional assumptions. Second, even with the usual cosmological assumptions of large-scale homogeneity and isotropy, being infinite would not imply spatial flatness, as negative curvature would still be possible.

  On the contrary, "the universe is finite and unbounded, compact, closed in all directions" is indeed a meaningful and acceptable theory.

This this alternative still makes grand assumptions about the structure of the universe beyond what we can observe, I don't see why you attribute more 'meaning' or 'acceptability' to it. At the end of the day, all we really know is that the bit of the universe around us is very close to being flat. Any statement whatsoever about the larger structure of the universe is going to introduce some assumptions not backed by observations, and you haven't provided much guidance to what kind are 'meaningful and acceptable'.

  Is there any other meaningful alternative?

You seem to be using that word in an idiosyncratic sense I'm not familiar with. In any case, you're mistaken about what "the universe is infinite" implies.


Semi-major axis vs. apogee and perigee of orbit

That's Wikipedia for you. The linked page demonstrates that Wikipedia is not a reliable source. Here are some of the problems:

The values for aphelion, perihelion, semi-major axis, and eccentricity are inconsistent with one other.
The values for perihelion, semi-major axis, and eccentricity are unsourced.
The values for perihelion, semi-major axis, and eccentricity have far too much precision. Sedna was discovered 17 years ago, corresponding to about 0.15% of Sedna's orbital period. That's far too short of an arc length to justify five or six places of accuracy.
The value for semi-major axis is inconsistent with other sources.
The value for aphelion is poorly conducted original research. Footnote 5 (the reference for the aphelion value) states that the source is from osculating orbital elements about the solar system barycenter as retrieved from JPL Horizons.
There are two key things wrong with this. One is the use of osculating elements, which can be deceiving. The other is using orbital elements about the solar system barycenter rather than about the Sun. This is just wrong.


Type I delta-Cepheid star, calculation of apparent maginute and luminosity

Pay attention to your signs: Remember that the value of $M$ is negative so on the second equation when you move $M$ to the right hand side, is $ 5 \log()−5- M$ giving you an  apparent magnitude of $\sim -6$ which is acceptable as cepheids are the brightest stars in the galaxies


This is a tricky question because a lot of pros and cons come into play. Terraforming isn't a "yes or no" answer as much as it's a pros vs cons answer and as you'll see.  This answer might be too broad for this site, but that's for the powers that be to decide.
The short answer is mostly no.   There's several reasons why a large impact wouldn't be ideal and probably, adding water to Venus hurts more than it helps because water is a strong greenhouse gas.
The longer answer:
Any collision of that size is problematic.   It doesn't take that large an object to kill nearly all life on Earth or set the oceans boiling.   Just 20 or 30 miles across might be big enough for that.   A moon over a thousand miles across would vaporize the surface of Venus and create enormous blowback, losing a lot of the material into space that you wanted to add.   An impact that large might create a moon around Venus - and that in and of itself might not be so bad, unless the moon's orbit was unstable and then you could have a inner solar-system rogue moon that might destroy Earth one day.
An impact of the size of Triton on Venus could also create a lot of debris that might threaten Earth with multiple dangerous near Earth objects that would need to be tracked and perhaps deflected.   So, it's not a good idea to hit Venus with something extreemly massive.
Moving Moons is also hugely difficult.   Moons with the amount of water you're considering are enormously massive and energy intensive to move and worse, Moons are in their planet's gravity wells, so you have two gravity wells to work against.   You know the old joke, I used to walk 10 miles to school, uphill both ways.  Gravity wells actually are like that - uphill in both the towards the sun and away from the sun.   Sending something into Venus requires a lot of acceleration.
A quirk about gravity wells is that, far away is much easier.   It's much easier to push a distant Kuiper belt object with a highly eccentric orbit into Venus than it is to push a much closer asteroid belt object or gas giant planet's Moon into Venus.  Probably orders of magnitude easier.
It's still an enormous amount of energy to move anything that massive, but a distant, eccentric orbit Kuiper belt object would be the place to start.
Also, many small impacts is almost certainly better than 1 large impact in terms of retaining the materials.  Unless your goal is to blow away some of Venus' atmosphere, which might not be a bad idea, but lets save that for another question.
Part II.  Just add Water.
Venus needs water, but it probably needs other things first.   It's too hot and water on Venus would evaporate and become a greenhouse gas.   Adding that much water to Venus would only increase it's runaway greenhouse effect and that's the opposite of what you want.
An additional problem is that the icy bodies you mention also have CO2 and adding more CO2 to Venus would only increase it's problem of too much CO2.
Venus needs to be colder.  That's step one.   It's vast amount of atmospheric CO2 needs to be reduced.  That's also step one.   While water is necessary for photosynthesis, there's no reliable way to make photosynthesis happen on Venus unless it gets colder first, or the photosynthesis is done in air-ship but that's probably too little, too slow and too expensive.
Venus is also quite dark on it's surface due to constant cloud cover so artificial light would probably be required too.  If you could get photosynthesis going on Venus on a large scale, then adding lots of water would help, but that's a pretty big if.  Water in tandem with photosynthesis would be a nice first step.  Water by itself would do very little and probably make things worse.
Another problem, though it might be slow enough not to matter when you're talking oceans of water (and Triton has perhaps 5-10 times as much water as all the oceans on Earth), so this might not be an issue at all in your scenario, but Venus, like Mars, loses water into space.  It's gravity is too weak for it's temperature and lighter molecules like Water, Methane and Ammonia escape from Venus into space and that would be true even if you gave Venus a strong magnetic field.  It's a product of temperature and escape velocity, called Jeans escape.
There are other theoretical methods.
Deliver a hundred million billion tons of calcium to Venus' surface or mine it from Venus' crust might work and let the calcium absorb the CO2.  Delivering that much calcium would be next to impossible and mining on Venus would be enormously difficult due to it's very high temperature, so neither of those are easy.
Most rocky planets have plenty of Oxygen stored in the rocks that make up their crust and Venus already has plenty of Nitrogen. What Venus really lacks is hydrogen, so any hydrogen rich elements might be useful in the long run, but the first step is some kind of process to reduce the CO2 and get the planet to begin to cool down.  Maybe a sun-shield but one that size would be a very expensive project and very far outside of our ability right now.
Removing what's basically an ocean's mass of CO2 is very difficult.  CO2 is a very stable molecule and not easily removed from an atmosphere.


How to query a region on the sky when given only ecliptic coordinates?

This is because the ICRS has an origin at the barycenter and this is not the same as the center of the Sun (the heliocenter). The heliocenter is offset from the barycenter (although it's still within the radius of the Sun) and varies with time, primarily due to the perturbations of Jupiter and Saturn.

If you use BarycentricTrueEcliptic it will work fine:

from astropy.coordinates import BarycentricTrueEcliptic
from astropy import units as u

cvz = BarycentricTrueEcliptic(lon=0*u.deg, lat=-90*u.deg)
cvz.transform_to(ICRS)
Out[9]:
  <ICRS Coordinate: (ra, dec) in deg
  (89.9999853, -66.56071866)>

Which agrees with the Coordinate conversions using the LAMBDA website at NASA's HEASARC


Convert coordinates between RA/dec and WGS 84 (SRID=4326)

Assuming negative longitude in WGS84 is west of Greenwich than you would get LONGITUDE = RA*15 and LATITUDE = DEC. The celestial longitude RA=0 through the vernal equinox would then be projected on the meridian of Greenwich.

If you subtract 180 from the LONGITUDE then you would just rotate the celestial sphere by 180°. The meridian of Greenwich would then equate to RA=12h.

If you want the stars to move with the rotation of the Earth, then you would use LONGITUDE = RA*15 - THETA*15, where THETA is the sidereal time at Greenwich in decimal hours.

You should be aware that for the celestial sphere you look from the centre towards the inside of the surface of the sphere, while for geographical purposes you look from the outside (above) towards the spheroid (down). So if you project the stars on a globe like this, you will notice that the constellations will look inverted from what you're used to in star maps. If you find one of those old celestial globes, you will also see the inverted constellations. For instance on this image, you'll see Leo to the left of Virgo and Hercules, while on most star maps, Leo will be to the right.

If you want the constellations to look 'right' then you would need to use LONGITUDE = -RA*15 (mind the minus sign). But then you would also need to invert the rotation of the Earth as well.


What if there was an asteroid belt with the mass of a planet?

The mass of a planet? There probably was, billions of years ago, and perhaps still is, but the debris of this planet-in-the-making has been shattered and dispersed by impacts. We know from fragments which have struck the Earth that the planet which was forming there had grown large enough to have a molten interior where the heavy metals, mostly iron and nickel, had sunk to the centre and differentiated from the rocky mantle. A massive impact must have shattered this forming planet about 4 or 5 billion years ago, and the debris has become so dispersed that it is difficult to assess how much there originally was. Some was kicked out of the asteroid belt completely, the rest is still there but much of it so finely shattered and dispersed that we can't see it (the asteroid belt covers an enormous area, much larger than the orbit of the Earth-moon system).

The resulting absence of a body with substantial gravitational field to attract comets & other objects has enabled them to pass on by, perhaps to impact the Earth and inner planets. Bodes Law says here should be a planet there, but whether it would have been a Mars-sized planet or not, nobody knows. The differentiation of nickel-iron from the rocky material becomes hard to explain without a fairly substantial planetesimal there to separate them, and in our solar system at least, Bodes Law seems fairly reliable.


How far should the source be, for the gravitation waves to be visible on Earth?

The waves pass by at the speed of light. So you you would'nt see ripples, they would pass too fast, and remember the waves would be passing through you too. The wavelength was (relativly) long about 3000km. The wave doesn't pass you, you are inside the wave.
The amplitude of the waves detected by LIGO was small, one part in $10^{21}$, Now while the intensity of the waves follows an inverse square law, because intensity is proportional to amplitude squared, the amplitude is inversely proportional to distance from source.
We are familiar with strains that are one part in 1000, that is a variation of one millimetre for every metre. To get that we would have to be a lot closer to the source, $10^{18}$ times closer. Since the black holes were about 1 billion light years distant, to get a strain of 1 part in 1000, the black hole merger would have to be about 10000km away
Now if you are 10000km from a merging pair of 30 solar mass black holes, the strain of gravitational radiation is the least of your problems. If it is any consolation, your problems won't lost long.


Centre of the singularity that is the big bang

Since the Universe evolved from something like a singularity, where all of space was contracted to a point, yes, we are at the center of the Big Bang. So is every other point in the universe.

The CMB represents the light from a universe-filling plasma that lasted for roughly 378,000 years. It is a feature of the observable universe, rather than something that is actually there in the present universe; in particular, we can never reach it, as it's the same distance away from every point in the universe. As such, there's not much sense in talking about what's behind it, much like talking about what's behind a shadow.


How can absolute value of azimuth exceed distance from pole?

It turns out I was right when I thought I was wrong.

Consider the azimuth circle for 85 degrees elevation. It's very small. For example, 85 degrees elevation and 0 azimuth is only 10 degrees away from 85 degrees elevation and 180 azimuth.

Therefore, the azimuth circle for a given elevation is smaller than a great circle. It's only cosine(elevation) the size of a great circle.

At 35 degrees, the azimuth circle is cosine(35 degrees) = 0.819152 the size of a great circle, so 4.17 degrees in azimuth = 4.17 degrees * 0.819152 = right around 3.5 degrees on a great circle.


What can be learned from, or noted in this LIGO Orrery video?

The SXS consortium has a page linking  a number of their videos. From the text around the videos it seems they use colour to encode various different scalar quantities in different videos. For instance

  The colors encode the lapse function — the slowing of the rate of flow of time.

And

  Colors indicate the vorticity of the apparent horizon

and

  with the color scale giving the density of matter within the star [this is a neutron star merger, not black holes]

Unfortunately, the LIGO gallery has not preserved this information (nor  the links to published papers where more details seem to be given) so it is hard to say in this specific case. Also simply understanding what many of these quantities are needs a fair level of differential geometry.

You asked:

  Why can I see some evidence of rotation in the resulting merged object for a few rotations, then it disappears?

Since we don't know what quantity is being depicted, it's a bit hard to be very precise, but there is an established concept of "ringdown".

You may have heard the phrase "black holes have no hair" meaning that from outside everything about a (non-quantum) black hole is determined by its mass, spin and charge, with no details (the hair) remaining. This is not exactly true. What is true is that any "hair" a black hole has is quickly radiated away as gravitational waves so that it rapidly approaches the ideal state given by its mass, spin and charge. This phenomenon is called "ringdown" and is what you are in seeing in the first moments after the collision. It can be detected in the gravitational waves.


Assuming you have downloaded their software from the link you provided and have astropy installed, you should be able to do:

from wssa_utils import wssa_getval
from astropy import units as u
from astropy.coordinates import SkyCoord

# Declare your wanted point in Galactic co-ordinates
c = SkyCoord(l=0, b=30, unit=(u.deg,u.deg), frame='galactic')
ra = c.icrs.ra
dec = c.icrs.dec
vals = wssa_getval(ra, dec)

The results look like they will be in MegaJanskys per steradian, which would be u.MJy/u.sr in astropy units.
For UV data, I'm not sure what is best, maybe GALEX ?


Binaries consisting of a black hole and a non-black hole?

Black hole and main sequence star/giant star

We can observe binary systems containing a black hole by looking for emissions from accretion disks which may form when matter is transferred from the companion star. X-ray binaries and microquasars are particularly notable types.

The compact object doesn't have to be a black hole - neutron stars also sometimes show up - but in some cases, measurements indicate that the remnant is a black hole. In other cases, either the mass of the compact object is poorly constrained or could still be a massive neutron star, so for many systems, the nature of this object remains unknown.

Examples:

Cygnus X-1: The companion star here is HDE 226868, a massive O-type star.
V404 Cygni: The companion star is a K-type star (much less massive than HDE 226868); the system forms a microquasar, with occasional high-energy x-ray emission.

Black hole and white dwarf

It seems a logical assumption that black hole/white dwarf systems should be possible, and indeed they are. Mass loss through accretion by a companion object can change a star's evolutionary future (see the Algol paradox for a particularly weird case), but for many systems with a black hole, the "normal" star should still progress through the main sequence and post-main-sequence evolution, possibly becoming a white dwarf.

Matter transfer can still happen at this stage, of course, so emission may continue, although the nature of the radiation could be changed. It's true that there aren't many known systems of this type, but there almost certainly more.

Examples:

X9: The companion star is a white dwarf orbiting extremely close to the black hole. Bahramian et al. are confident that the companion is a white dwarf, and the main object is likely a black hole.

Black hole and neutron star

These type of systems are possibly the most interesting from a gravitational point of view. They should produce gravitational waves, which could be detectable by LIGO.1 LIGO has yet to observe such a system, which seems a bit puzzling, although something like the blue supergiant in Cygnus X-3 could evolve into a neutron star in the future. This is a case where more data and observations are needed.

1 You've probably heard the rumors that LIGO detected a possible neutron star-neutron star merger, which would imply that its sensitivity to neutron star-binary systems could indeed be high enough for detections, as expected. It turned out that it was simply a binary black hole system[1] - not quite as exotic as some people were hoping for.


If gaseous planets have solid core, can they theoretically be considered as rocky planets?

As a general rule, a gas giant planet is large enough that it retains a lot of hydrogen and helium, such as Jupiter and Saturn.  Their chemical makeup is similar to the universe, nearly 90% hydrogen.

Ice giants, like Uranus and Neptune are made up of material more similar to comets, lots of ices like H20, CO2, CH4, NH3 are the primary 4 ices that form into bodies beyond the frost line, but neither Uranus or Neptune have much free hydrogen or helium, either because they're too small or because of where they formed, to close to the sun where it was too hot for them to retain or keep the lightest 2 gases.

As a general rule, in addition to your definition, one possible definition, of a gas giant is an abundance of hydrogen.

When we get a bit of better look at planets in other solar systems, some of the terminology might need to change.    Super-Earths with a lot of water would be similar to hot Neptunes.   Any very massive planet that is able to collect and retain massive amounts of hydrogen will look like a gas giant, whether it has a super-earth origin or not.    Perfect definitions aren't possible.


Is there a formula to calculate any star mass from Luminosity, Radius, and/or Temperature (K)?

I think the issue you are going to have here is that the position of a star in the HR diagram - which amounts to saying what the (2 dimensional) spectral type is - does not just depend on its mass.
The Vogt-Russell theorem says that the position of a star on the HR diagram depends on its mass, its composition and crucially, how its chemical elements are distributed in the interior. Thus we know for example that we could find a 2 solar mass star on the main sequence with a spectral type of say A2V but it is also possible for a K0III giant to be a star of 2 solar masses.
So there really isn't any shortcut or quick formula that is going to give you a route to finding the mass based on a spectral type (or equivalently, based on a luminosity and effective temperature).
If you know the star is on the main sequence then you can use an approximate luminosity-mass relation (although you should be aware that stars change their luminosity during their main sequence lifetime and so the luminosity of a main sequence star depends on its mass, age and its initial chemical composition).
If your star is a giant or at some intermediate evolutionary stage (subgiant, or some short-lived supergiant phase) then it is really difficult to estimate the mass of such an object without also knowing its age. For example if you know a giant star is a billion years old, then there is only a small range of masses which will be in the relatively short-lived giant phase at 1 billion years old.


Does gravitational lensing cause a black-hole to be the main 'source' of light in a given area?

So the question is, Can black holes be really be extraordinarily bright due to lensing of background objects?

Let's first also specify that these black holes must certainly be super massive. If not, we're talking about stellar mass black holes (or intermediate mass black holes), and the lensing signal would be much weaker. More mass = more lensing generally speaking (there are some exceptions, e.g. - microlensing, where the lensing is done by stars in our own Milky Way galaxy and the signal is enhanced because the source and lens are aligned nearly perfectly).

This is an important statement to make because virtually every galaxy has been identified with a super massive black hole at its center. The reverse is also true - every super massive black hole is associated with a galaxy. It's also important to note that, realistically speaking, galaxies have total masses of about $\sim10^{11} - 10^{12} M_{\odot}$, whereas super massive black holes typically have masses of millions to hundreds of millions (maybe even billions) of solar masses. This is a smallish fraction of the total mass. This means that most of the lensing will be done by the galactic halo and not the super massive black hole.

If we modeled the center of the lens galaxy as a point-lens, and the galactic halo as a singular isothermal sphere, the relevant question I would ask is what are their Einstein radii (which is a measure of how effective or efficient they are at lensing) individually, and what are they in combination? Essentially, how much does the existence of a central super massive black hole matter to the system as a whole. Strong lensing features (arcs, rings, or multiple images) generally occur at around the Einstein radius of the object.

My best guess:

Quite honestly I don't see the central super massive black hole mattering all that much when it comes to lensing. Many of these mass profile models for the lens galaxy halo are singular, or rise very rapidly to a central core. Furthermore, I've never really heard of a situation where a lone super massive black hole (not associated with a galaxy - call it a 'rogue' smbh if you will) has been found floating around in space to do this sort of lensing. They generally hide at the centers of galaxies, or show themselves only if they're actively accreting material. Correct me if one has been found (maybe it would come from a merging of two smbh's where one is kicked out of the system).


Are black holes considered part of baryonic content of universe?

Black holes formed after the epoch of primordial nucleoynthesis would be part of the baryonic matter, since they would be mostly formed of baryons. It is possible or even probable that every black hole must contain some non-baryonic matter too, since there is nothing to prevent it being accreted, in small amounts, directly into black holes. However, the cross-section for that is very small because dark matter falling towards a black hole has no means to lose angular momentum. Unless there is almost a direct hit within the photon sphere then it won't be accreted. Normal matter can form an accretion disc and lose energy and angular momentum through viscosity, radiation and other processes.
Since the fraction of dark matter that is in black holes formed since the big bang is negligible, then the approximation that their mass consists only of baryons is hardly a big deal when it comes to doing the mass budget for the universe.
Primordial black holes on the other hand, that were already present at the epoch of cosmological nucleoynthesis, would not be counted as baryonic matter and it would be possible that they could contribute significantly to the required non-baryonic component, if they exist.
Could there be dark matter black holes? is essential reading.


Is the equinox Sun always at the same zodiac sign in the ecliptic belt?

As you said, the Sun takes a year to cycle the Zodiac, and it takes a year between two spring or autumn equinoxes. Both statements are true, but "year" in them means two different things. It takes a sidereal year to cycle the Zodiac, but it takes a tropic year to cycle between equinoxes. Since the tropic year is about 20 minutes shorter, the equinoctial points move across the Zodiac cycling it in about 25,000 years.

The calendar year we use in the Gregorian calendar is designed to approximate the tropic year to have it synchronised with seasons.

The reason for all it is that the Earth's axis is not fixed but its orientation slowly rotates.


Can lunar atmospheric density, as it exists today, increase naturally?

There's not enough gassible (is gassible a word?   I'm guessing not), OK, there's not enough of the right elements to form enough gas on or in the moon to create any kind of atmosphere that would measurably slow down falling objects (say, enough to land a probe with a parachute, like they do on Mars).   Atmospheres are heavy.  They may not seem heavy but they are.   Mars' atmosphere has a mass of about 25 trillion metric tonnes and that's a very thin atmosphere compared to Earth.

The Moon, with roughly 1/4th Mars' surface area, would need about 1/4 of that mass in it's atmosphere to have an atmosphere with similar air resistance to Mars, somewhere in the 6-7 trillion metric tonne range.

There's some ice on the moon in craters that never get direct sunlight, but not nearly enough ice to make that kind of an atmosphere.   The Moon also has very infrequent volcanic eruptions, but while some gas is likely released, again it wouldn't be nearly enough.

One possible scenario would be a large asteroid or comet impact that could both release some trapped gas from below the surface and release gas from the impactor.   The problem with such a scenario is that the heat from the impact would likely send much of the gas away from the planet at above escape velocity shortly after impact.    The escape velocity of the Moon is 2.38 km/s and at the top of this theoretical atmosphere it would be a little bit less than that.  A large impact from a comet is the best way I can think of, but that wouldn't be very efficient.

Even if you found a way to create that kind of atmosphere on the Moon, it wouldn't last very long.  The Moon is simply too small, with too little gravity and it's too close to the sun to maintain an atmosphere, unless that atmosphere was Xenon or if you want to be a bit more daring, sulfur-dioxide (acidic when it comes in contact with water and stinky) or sulfur-hexaflouride, which isn't toxic in and of itself but becomes very toxic when exposed to UV light from the sun, so, lets not use either of those, but Xenon is safe, so we can use that.

see atmospheric escape velocity.

So the answer to your question is yes, the Moon could be given a thick enough atmosphere to slow falling objects down, but it would probably need to be done artificially, as there's no natural process I can see that would do that, and ideally, you'd use a very heavy and non toxic gas like Xenon.    All the Xenon in the Earth's atmosphere wouldn't be enough.   A rough calculation, Earth's atmospheric Xenon comes to less than a billion metric tons, which is still several thousand times too little mass to create a mars like atmosphere on the Moon.   Now if you harvested all the Xenon from Jupiter and/or Saturn and moved that to the moon, then you might be able to create a somewhat long lasting atmosphere on the moon that would provide some measurable air resistance.   It wouldn't be breathable, but it wouldn't be toxic, which is a plus and with enough Xenon, you might be able to get around on the moon with just an oxygen tank and no pressure suit.  Buildings with Oxygen/Nitrogen would need to be air-tight but not vacuum resistant, which is far easier to build.   Enough Xenon would make living on the moon quite a bit easier.   Still wouldn't be easy, but definitely easier.


Does the Sun and Moon basically appear to "move on the sky" identically regardless of location and date?

The Moon's one month orbit makes the Moon rise about 50 minutes later each day for people who aren't at extreme latitudes. (I'll examine extreme latitudes later). The Moon's orbit is inclined by 5° with respect to the ecliptic. This means that the new Moon more or less follows the path of the Sun that very day, while the full Moon more or less follows the path the Sun took six months earlier. This is why the winter full Moon rises so high in the sky in temperate latitudes.
What about more extreme latitudes? The Amundsen-Scott South Pole Station is the epitome of extreme latitudes. At the South Pole, the Sun sets six months after it rises while the Moon sets two weeks after it rises. In winter, the moon rises about a week before full Moon and sets about a week after full Moon. In summer, the moon rises about a week before new Moon and sets about a week after new Moon.


Average number and type of planets orbiting given star types

I wrestled with this one a bit, but because there's no real scientific answer to it, your best bet might to look at the 6 or 8 best known other solar-systems and build off that.   There's no right answer, so a base off evidence approach is probably the way to go.

Or, if you want to base it off nonsense, but sound impressive, you could use the Titius-Bode law.   But I think that's a bad idea, unless large gas giant planetary migration is kept to a minimum, because a migrating Jupiter would make mincemeat of that "law".

The easiest exoplanets to find are planets with short orbits and the easiest stars for us to find planets around are smaller stars.

With those limitations to your view, given that large planets are easier to see, it's hard to say how many smaller ones are out there or how many distant planets from their star.   There's basically zero information to build on regarding those 2 points.

A few generalities though, just for fun.

Rocky planets like the 4 inner planets in our solar system form inside the the Frost Line.  These planets are very low water by percentage.    Now, you could look at the Earth and say "there's tons of water on Earth", but there's really not, at least, not in terms of water to silicate ratio.   Much of Earth's water is on the surface, so we see lots of water, but Earth is basically Iron, Nickle and Silicates with less than 1/3 of 1% of water by volume (even less by mass).   By volume, (using the 860 mile diameter sphere pictured below and from this article).  Earth is about 1 part in 800 water by volume.

There may be as much or more water trapped in Earth's mantle than in the oceans, so the ratio might be closer to 1 part in 400 or there abouts, but the point remains the same.   If Earth is 0.25% water by volume, it's mostly made of stuff that isn't water.

Ceres, by comparison, may be as much as 25% water.  See here and here.  That's 100 times as much water to dry material ratio.   Outside the Frost-line you're going to get very different types of planets.

Over time, you have to figure atmospheric changes, which planets will lose their atmosphere and which planets will heat up like a greenhouse.

Now lets take 3 types of stars, one with 1 solar mass like our sun, one with 0.4 solar masses (red dwarf) and one with 2.5 solar masses (very bright star).

Our sun has a lifespan of roughly 10 billion years and a frost-line, currently at about 2.7 astronomical units (roughly Ceres distance from the sun), but when the sun was young and the planets were forming, our sun was some 30% less luminous, so the frost-line would have been (.7)^.5 * 2.7, or about 2.2 AU.   Now different frozen gases have different frost-lines, but this is just an approximation.   The fact that Ceres is about 25% water even though it's losing water to empty space implies that it formed outside the frost-line where frozen water was abundant.   Ceres is just one example.  All the large moons of the outer planets are also ice and rocky composites.

So, take our 0.4 solar mass red dwarf - a very common type of star.  It's luminosity (0.4)^4th power, is about 1/39th as bright, so it's frost-line would be a bit over six times closer in, less than 0.4 AU.   With less gravitation it might not have a much smaller disk of spiraling matter at the time of formation, so a much larger percentage of the material that forms our red dwarf's solar-system would probably form outside the frost-line, which would imply, fewer 99% dry rocky worlds and more wet (25% water) worlds.   We have many worlds like that in our solar system, but they're all moons or dwarf planets.   Ceres, Ganymede, Europa, Titan, Triton, many others are icy-rocky composites that formed beyond our solar-systems frost-line.

Many of the Kepler observed exoplanets have densities that suggest "water-worlds", many of them larger than Earth.   This fits with the red-dwarf, closer in frost-line argument.   We have no way of knowing how many rocky worlds smaller stars have, but I think it's a safe bet that they have a lower ratio of 99% or greater dry material worlds like our 4 inner planets simply because of the smaller zone inside the frost-line.

The reverse is true for the 2.5 solar mass star.   It also has a short lifespan of 10 billion / (2.5)^3, or about 640 million years, so there's less time for the planets to form, get past any early bombardments and cool down.    And the frost-line for our 2.5 solar mass star would be about 14 AU when the star is in it's early life, some 6 times further than the frost-line in our solar-system, so that's likely a much greater "dry" region for planet formation.   You probably get a higher percentage of rocky worlds around larger stars, so a star 1.5 or twice as bright as our sun could (in theory) have 6 or 8 rocky worlds of varying sizes ranging from hot to cold.  At least, I think it could, though I'm pretty sure this has never been observed.

Gas giants are fairly simple in nature.   All a gas giant needs to form is sufficient gravity to retain it's hydrogen and helium and a source of hydrogen and helium to collect.   A gas giant is more likely to form outside the frost line, but there's no reason why a gas giant couldn't form inside the frost-line.  You'd need about 8 earth masses of a rocky world at Earth temperatures to have a sufficiently high escape velocity that it could retain hydrogen and Helium and in theory, grow into a gas giant, but further away from the star where there's less solar heating, the amount of mass needed goes down with cooler surface temperatures, and with the additinoal mass of ices outside the frost line, it's probably far easier for gas giants to form outside the frostline than inside.

The next thing to consider is Planetary Migration.

Take our Solar-system for example.  There's a theory that Neptune and Uranus may have formed formed inside Jupiter and Saturn, but Jupiter and Saturn pulled them outwards as they drew inwards, perhaps because in the early solar-system there was a period when Jupiter and Saturn were in resonance.

It's also possible that there were 3 large gas giant planets in our solar-system initially.   (not counting Neptune and Uranus as "large" gas giants).   The 3 gas giant model helps explain the formation and water content of the 4 inner planets.

And it's also thought that Jupiter migrated inwards towards the sun, then back outwards.   If Jupiter had kept migrating inwards, which may be the case for several hot Jupiter planets that have been observed, then there's no telling what would have happened to the inner planets, perhaps scattered every which way, some out of the solar-system, pushed too close to the sun or swallowed by the migrating hot Jupiter.  A migrating hot Jupiter would wreck havoc with smaller inner planets as they crossed orbits.

As it is, our own Jupiter may have tossed a large gas giant planet completely out of the solar-system and perhaps made Neptune and Uranus switch places.    (all this is just numerical model based hypothesis, but several articles have been written about these ideas).

Here's a very cool video, if you have some time that runs models on planet formation on "sol" like systems, estimating planet size, location and including how much water the inner planets are likely to get from comets.   Earth, it's thought, got most of it's water from comets and asteroids, largely thanks to Jupiter's migrations.

If you skip forward to about 20 minutes 40 seconds in, you'll see he's computer modeled several different scenarios on planet formation with different gas giant combinations.   The chart shows eccentricity on the y-axis and distance from the star on the x-axis.  As eccentricity goes to one (top of the y-axis), the object is either sent outside the solar-system or close enough to the sun to get vaporized, so basically the dots you see flying up and out of the chart are leaving the solar-system one way or another.

There are, ofcourse other factors, mostly unknowns, like how much material is likley to be available for planet formation when a cloud of matter forms into a solar-system.   Our sun, for example, is something like 99.8% of the mass of the solar system with just 0.2% making up all the other stuff, but I don't know and I don't think anybody knows if that 0.2% is standard, higher than average or lower, and how larger stars might have different ratios than smaller onces, or how much of the material that gets blown off a star in it's young life gets captured by it's orbiting, forming planets.

It's entirely possible that more planetary material can lead to larger but fewer planets as larger planets tend to sweep out a bigger area where less planetary material over the same area might actually give you more smaller planets, but how many planets are likely to form in an average star system, based on size of the star, I don't think anyone knows.   But predictions can be modeled on what happens after the planets form.

What Kepler can tell us:

Most red dwarf stars have planets and at least 25% have planets in their habitable zone.   Source.   (Red Dwarfs are the easiest planets to look at and look for stars).

From the article:

  The new finds imply that virtually all red dwarfs throughout the Milky
  Way have planets, and at least 25 percent of these stars in the sun's
  own neighborhood host habitable-zone "super-Earths," researchers said.

A simple (2-5 inner, 3-5 outer) won't cut it if you're looking for accuracy, but you might be able to run some models, based on how much you (randomly) decide that the gas giants move around and how much material you randomly give the planets.   Heavier planets would need more space around them.

As for mining.  Venus would be a nightmare - too acidic.   Water worlds I would think would be terrible.   No solid platform to build on.   Ice worlds also not great.  Planets with past volcanic activity would be ideal and smaller planets are probably best, as most of the energy with mining probably goes into lifting the material into space, not actually the mining part.   Smaller planets could more easily build a space elevator.   Mars (volcanoes), maybe Mercury would probably be the best mining planets in our solar-system, unless you're mining for 3-HE, in which case, I'd look at the gas giants.   The moon is (I think), a terrible place to mine 3-HE, it's got so little of it, but the Moon's advantage is that it's close.

All that said, I'd just look up 6 or 8 fairly well studied solar systems and build based on those, add some common sense, like a hot Jupiter would remove half the smaller planets, and go with that, because there is no real knowledge on what the most likely scenarios and probabilities are.

Too long? / too many guesses?.  I think about exoplanets a lot.  It's one of my favorite subjects.  I'm very much looking forward to the James Webb Telescope's launch.


Would the Event Horizon Telescope be able to produce a superior image of Betelgeuse?

Presumably, yes, EHT observations could improve on existing radio observations of Betelgeuse (e.g. recent ALMA images and comparatively ancient VLA images). Any observations would likely be targeted at known photospheric mm or super-mm emission from the star, mapping radius and temperature changes. The ALMA data showed the existence of a spot $\sim1000$ K hotter than neighboring regions, which would ideally teach us something about convection inside the star.

With a (current) resolution of a couple tens of microarcseconds at $\lambda\sim1.3$ mm, the EHT would indeed improve on existing interferometer results by orders of magnitude (which I believe, in the case of ALMA, are the highest-resolution images of Betelgeuse, period, at any wavelength).


Is there a consensus as to where terrrestrial planet atmospheres in our solar system came from?

Three decent answers in comments converted to a community wiki.

A quite recent study about the origin and development of terrestrial planet atmospheres is the 2015 paper "The Atmospheres of the Terrestrial Planets: Clues to the Origins and Early Evolution of Venus, Earth, and Mars" (Baines et al.) includes a fairly comprehensive analysis of chemical evidence from Earth and from Martian meteorites and observations of the atmosphere of Venus.

The general consensus is that much of the atmospheric gases were brought through comet and asteroid impacts.

The pre-planetary disk would have been centrifugally sorted. Many iron, stone, carbonacious meteorites, watery, ice-iron-rock agglomerates would have brought a very diverse amalgam of different stones, ices, and room temperature gas elements, throughout the formation. CO2 and N2 objects are a bit less stable in direct solar radiation at 1 AU. CO2 and N2 would have pounded into the world in considerable quantities throughout the accretion process, with a wide range of particulate sizes from microns to kilometers in diameter without a marked difference in element type from start to end.


Did the night sky ever change in recorded history?

There are a few ways to think about this question:

Do the stars change positions in the sky ... such that the layout of say... major stars in constellations appear to move over time?

Are there events that cause sudden changes (changes you might notice overnight ... or within a few weeks or months)?

How long does it take to notice a change?

Precession
The Earth's axis mostly stays oriented in the same direction as we orbit the Sun.  This is why we can get away with saying that Polaris is the "North Star" ... because it appears to be almost directly above Earth's North Pole regardless of what time of the day or night ... or what day of the year you choose to observe (it is roughly within 2/3rds of a degree from the pole).
But it turns out the Earth's axis does "wobble" like a spinning top.  This wobble is very slow and takes thousands of years to complete a cycle (nearly 26,000 years).  At the end of this century (roughly the year 2100) Polaris will be a little closer to the pole ... only 1/2° away from the true pole (today it is about 2/3rds° away from the pole).  That is as close as it will get ... and then our pole's precession cycle will start taking it farther away from the pole.
This "precession" shift causes all stars in the sky to shift their position very slightly from year to year and it's mostly not enough to notice in a single human lifetime ... but it is easily noticed when measured across thousands of years.
There are other more subtle cycles, but Precession is noticeable across large amounts of time (thousands of years).  The position of the Sun in the sky at the equinoxes changes every year and this causes the Sun to appear to be moving across the zodiac over thousands of years.
The sky has a coordinate system used to catalog the locations of stars and other objects.  That coordinate system uses a "declination" to measure where a star is located relative to Earth's North/South ... it works much in the same way that latitude measures geographic locations on Earth.  There is also a "right ascension" value that works a bit like longitude on Earth.  Except Earth is spinning (and longitudes remain fixed to the planet) ... so we needed a way to mark a fixed position in space to have a sort of celestial longitude.  That measurement is called Right Ascension and it needs a "0" position.  So the zero position is established by the location of the Sun in the sky at the Vernal Equinox (the start of Northern Hemisphere Spring).   It even got a name... the "First point of Aries" ... named because, when it was established a few thousands years ago, it was in Aries.  Today ... it's migrated all the way over to Pisces (and is getting closer to the boundary of Aquarius).
These are changes caused by Precession and noticeable ... but mostly only when measuring over hundreds of years or thousands of years.  It is actually noticeable just from year to year ... but only with precise measurements.
Precision causes the entire sky to shift in the same direction (and by a very subtle amount).  This means that the locations of the constellations will shift ... but it wont explain changes to their shapes.   For that, we have to look at other factors.  In reality it isn't the "sky" that is shifting ... but rather the orientation of Earth's axis that changed.
Proper Motion
Stars are moving ... and they don't all move in exactly the same direction.  But the stars are so distant that the movement is mostly only noticeable with precise instruments.  Regardless, they do move.
Here for example, is the Big Dipper asterism, but I've turned on Proper Motion vectors ... the vectors (shown in a light brown color) indicates the direction of that star's motion (relative to us) and longer vectors indicate great proper motion.

Notice that many of the stars of the Big Dipper are all moving in the same general direction at about the same speed ... including Alcor/Mizar, Alioth, Megrez, Phecda, and Merak.  In this diagram, those stars are all moving up and to the left (relative to this frame).
But more importantly... notice that Alkaid and Dubhe are going in roughly the opposite direction.  These stars are moving down and to the right.  This means that over thousands of years, this asterism will become distorted and will eventually not resemble the "Dipper" shape anymore.
Incidentally, the stars in the dipper that are going in the same direction are all part of an Open Cluster called the Ursa Major Moving Group.  It isn't as obvious that it's an open cluster because it is relatively close (not all stars in Ursa Major are members of the cluster.)
Stars that physically closer to our Sun will may appear to have much faster proper motion.   It isn't so much that the stars are necessarily physically moving faster ... but since they are close, the changes are easier to spot.
Here, for example, is the proper motion of Rigel Kentaurus (aka Alpha Centauri).

In this frame (current time) the star is in the upper-left section of this chart.  It is in Centaurus (hence its name) but near Circinus.  But notice it has a VERY long proper motion vector which is headed toward the right side of this chart in the direction of Hydra.
Its proper motion vector is much longer than the other stars ... but Rigel Kentaurus is just a little over 4 light years away (one of our closest neighbors) -- so its relative motion is more noticeable.
While Rigel Kentaurus makes the top-20 list for stars with the highest proper motion, it's not the winner ... that spot goes to Barnard's Star (in Ophiucus).  But Barnard's Star isn't very bright... at magnitude 9.5 it isn't noticeable to ordinary human vision and requires a telescope to see it.  But it is moving just a little under 3x faster than Alpha Centauri A & B.
If you're interested, here's a top-20 list:  https://www.cosmos.esa.int/web/hipparcos/high-proper-motion
Transient Events
Certainly events such as bright comets would change the sky -- but those aren't stars.  Super-Novae events, on the other hand, are stars ... and these are easily noticeable if they happen in our home galaxy.  They don't happen very often.  The estimate is that, on average, they happen in a galaxy our size about once every hundred years.  But's not like a clock where they happen with periodic regularity ... huge amounts of time may pass without any of them ... and then several could occur timed much closer together.
Since Super-Novae events can be quite bright, they can even be visible in the daytime sky.  They don't last particularly long ... they quickly brighten up ... then begin to dim over weeks and ... after a few months they may no longer be noticeable (but they will leave behind a Super Nova Remnant -- deep sky nebulae such as the Crab Nebula and Veil Nebula are examples of Super Nova Remnants.)


What would the effects be on Earth if Jupiter was turned into a star?

Before I start, I'll admit that I've criticized the question based on its improbability; however, I've been persuaded otherwise. I'm going to try to do the calculations based on completely different formulas than I think have been used; I hope you'll stay with me as I work it out.

Let's imagine that Lucifer becomes a main-sequence star - in fact, let's call it a low-mass red dwarf. Main-sequence stars follow the mass-luminosity relation:

$$\frac{L}{L_\odot} = \left(\frac{M}{M_\odot}\right)^a$$

Where $L$ and $M$ are the star's luminosity and mass, and $L_\odot$ and $M_\odot$ and the luminosity and mass of the Sun. For stars with $M < 0.43M_\odot$, $a$ takes the value of 2.3. Now we can plug in Jupiter's mass ($1.8986 \times 10 ^{27}$ kg) into the formula, as well as the Sun's mass ($1.98855 \times 10 ^ {30}$ kg) and luminosity ($3.846 \times 10 ^ {26}$ watts), and we get

$$\frac{L}{3.846 \times 10 ^ {26}} = \left(\frac{1.8986 \times 10 ^ {27}}{1.98855 \times 10 ^ {30}}\right)^{2.3}$$

This becomes $$L = \left(\frac{1.8986 \times 10 ^ {27}}{1.98855 \times 10 ^ {30}}\right)^{2.3} \times 3.846 \times 10 ^ {26}$$

which then becomes

$$L = 4.35 \times 10 ^ {19}$$ watts.

Now we can work out the apparent brightness of Lucifer, as seen from Earth. For that, we need the formula

$$m = m_\odot - 2.5 \log \left(\frac {L}{L_\odot}\left(\frac {d_\odot}{d}\right) ^ 2\right)$$

where $m$ is the apparent magnitude of the star, $m_\odot$ is the apparent magnitude of the Sun, $d_\odot$ is the distance to the Sun, and $d$ is the distance to the star. Now, $m = -26.73$ and $d(s)$ is 1 (in astronomical units). $d$ varies. Jupiter is about 5.2 AU from the Sun, so at its closest distance to Earth, it would be ~4.2 AU away. We plug these numbers into the formula, and find

$$m = -6.25$$

which is a lot less brighter than the Sun. Now, when Jupiter is farthest away from the Sun, it is ~6.2 AU away. We plug that into the formula, and find

$$m = -5.40$$

which is dimmer still - although, of course, Jupiter would be completely blocked by the Sun. Still, for finding the apparent magnitude of Jupiter at some distance from Earth, we can change the above formula to

$$m = -26.73 - 2.5 \log \left(\frac {4.35 \times 10 ^ {19}}{3.846 \times 10 6 {26}}\left(\frac {1}{d}\right) ^ 2\right)$$

By comparison, the Moon can have an average apparent magnitude of -12.74 at full moon - much brighter than Lucifer. The apparent magnitude of both bodies can, of course, change - Jupiter by transits of its moon, for example - but these are the optimal values.

While the above calculations really don't answer most parts of your question, I hope it helps a bit. And please, correct me if I made a mistake somewhere. LaTeX is by no means my native language, and I could have gotten something wrong.

I hope this helps.

Edit

The combined brightness of Lucifer and the Sun would depend on the angle of the Sun's rays and Lucifer's rays. Remember how we have different seasons because of the tilt of the Earth's axis? Well, the added heat would have to do with the tilt of Earth's and Lucifer's axes relative to one another. I can't give you a numerical result, but I can add that I hope it wouldn't be too much hotter than it is now, as I'm writing this!

Second Edit

Like I said in a comment somewhere on this page, the mass-luminosity relation really only works for main-sequence stars. If Lucifer was not on the main sequence. . . Well, then none of my calculations would be right.


Is there a legend for these 88 HST images?

To answer my own question: no, not that I can find.

Here is an attempt to identify them by image search.  Let me know if there is an image/legend match that seems wrong.  I can upload a higher resolution (466x426 px) version of the image.

The images were cropped from the original image when it was oriented so the the 'HUBBLE SPACE TELESCOPE' message read from the bottom to top of image, and I will try to identify them in the same order.  As such, we do 1 column at a time from top to bottom, starting from the right most.  The column is marked in red above/below.

But first, here is the captioned video created from combining the captions with a video slideshow of the images!

Column 1

Gravitational lensing of distant galaxies.
Red Rectangle Nebula.
The Sombrero Galaxy: Messier 104.
Doradus Nebula.
Stingray Nebula.
Elliptical Galaxy NGC 4881 in Coma.
Galaxy NGC 4214.
Glowing Eye of NGC 6751.
Collection of galaxies of different types, from different ages in the universe.
Uranus: Springtime in the Southern Hemisphere.
The Lunar Aristarchus crater imaged using ultraviolet and visible wavelengths.

Column 2

Supernova remnant N132D, in the Large Magellanic Cloud.
Ring Galaxy AM 0644-741.
Bug Nebula NGC 6302.
Planetary Nebula M2-9.
The Cone Nebula in NGC 2264.
Young planetary nebula MyCn18.
A stellar jet in the Trifid nebula.
Galaxy Pair NGC 3314. A face-on galaxy lying precisely in front of another spiral galaxy.
Comet LINEAR dispersing after breaking up.
A 5000 light year long jet from the giant elliptical galaxy M87.
The Massive Stars of 30 Doradus as seen in the star cluster R136.

Column 3

NGC 346 is an open cluster with associated nebula located in the Small Magellanic Cloud.
The Cat's Eye Nebula or NGC 6543, is a relatively bright planetary nebula in the northern constellation of Draco.
This infrared view of Uranus reveals that the planet is surrounded by its four major rings and by 10 of its 17 known satellites.
Galaxy NGC 4013. An amazing \"edge-on\" view of a spiral galaxy 55 million light years from Earth.
The inner ring of NGC 1512.
Detail of the Omega Nebula.
This close-up of the red planet Mars was taken when it was just 55,760,220 km away.
Stellar Spire in the Eagle Nebula
Interacting galaxies NGC 1409 & 1410.
NGC 1850 - Star clusters in the Large Magellanic Cloud.
Io, Ganymede, and Callisto casting eclipse shadows on Jupiter.

Column 4

Crab Nebula Mosaic
N44F's central star is wrapped within a roughly spherical shell of cool orange gas, with fingerlike columns of cool gas and dust protruding from the interior wall.
The Whirlpool Galaxy (M51) and companion galaxy.
Aurora at Jupiter's north pole as seen in ultraviolet light.
Bow shock near young star LL Ori.
The Black Eye Galaxy (also called Evil Eye Galaxy; designated Messier 64, M64, or NGC 4826)
A color-composite image of the Pleiades, or Seven Sisters (Messier 45 or M45), an open star cluster.
Orion nebula mozaic.
The Eskimo Nebula (NGC 2392).
Galaxy collision in NGC 6745.
Rich star fields and glowing hydrogen gas silhouette dense, opaque clouds of interstellar gas and dust in a bright star forming region in Centaurus

Column 5

Aristarchus Plateau on the Moon.
The Tadpole Galaxy features a trail of stars about 280,000 light-years long and massive, bright blue star clusters.
Infant Stars in the Small Magellanic Cloud.
The Ghost-Head Nebula: NGC 2080.
The Pencil Nebula (NGC 2736) is a small part of the Vela Supernova Remnant, located near the Vela Pulsar in the constellation Vela.
Flash from star V838 Monocerotis echoes through space.
Rampaging Supernova Remnant N63A.
The center of globular cluster Omega Centauri.
The Bubble Nebula (NGC 7635), also called the Sharpless 162, or Caldwell 11, in the constellation Cassiopeia.
The center of NGC 4414, an unbarred spiral galaxy about 62 million light-years away in the constellation Coma Berenices.
NGC 3132, also known as the Eight-Burst Nebula, the Southern Ring Nebula, is a bright and extensively studied planetary nebula in the constellation Vela.

Column 6

Messier 22 (also known as M22 or NGC 6656) is an elliptical globular cluster in the constellation Sagittarius, near the Galactic bulge region.
Star forming region N11B lies in the Large Magellanic Cloud.
The Homunculus Nebula is a bipolar reflection nebula surrounding the massive star system Eta Carinae.
Hoag's Object: a Ring Galaxy.  On the outside is a ring dominated by bright blue stars, while near the center lies a ball of much redder stars.
Detail of the Keyhole Nebula. The small nebula to the upper left has been nicknamed \"finger of God\" or \"God's birdie\", due to the gesture it appears to be making.
Reflection nebulosity near Merope (IC 349).
NGC 1569: Starburst in a small calaxy.
NGC 1300 is a barred spiral galaxy about 61 million light-years away in the constellation Eridanus.
The Lagoon Nebula
Planetary Nebula NGC 6543, the \"Cat's Eye Nebula\" is thought to have formed one one star of a double system went supernova.
NGC 1999 is a dust filled bright nebula with a vast hole of empty space represented by a black patch of sky.

Column 7

This glowing region reveals arcs and bubbles formed when stellar winds - streams of charged particles ejected by the Trapezium stars - collide with material.
Auroras at Saturn's north and south poles.
Sheets of debris from a stellar explosion (N 49, DEM L 190).
The Horsehead Nebula.
The Retina Nebula: dying star IC 4406 has a doughnut shaped nebula seen edge on.
The center of NGC 1275 (also known as Perseus A or Caldwell 24), which is a type 1.5 Seyfert galaxy
NGC 4622 has bright twisting arms containing newborn stars and lies 111 million light years away.
Double Bubble' of gas and dust in the Large Magellanic Cloud.
Saturn displays its familiar banded structure, with haze and clouds at various altitudes.
Comet-like filaments along the inner rim of the Helix Nebula's gas ring.
Galactic nebula NGC 3603 shows various stages of the life cycle of stars.

Column 8

Thousands of stars are forming in the cloud of gas and dust known as the Orion nebula.
A wheel galaxy around 500 million light-years from the Earth.
Stellar spire of the Eagle Nebula
The Circinus Spiral Galaxy
The Egg Nebula
Gaseous streamers flutter from a group of stars.
The Seyfert's Sextet.
Bubbly ocean of glowing hydrogen, oxygen, and sulphur gas in the extremely massive and luminous molecular nebula Messier 17.
North Eastern region of NGC 6369: The Little Ghost Nebula.
Double galaxies of NGC 4676 in collision.
Star clusters born among the interacting galaxies of Stephan's Quintet.


Planets and moons reflect sunlight, as mentioned in this comment by AtmosphericPrisonEscape.

Now I'll add a bit more info to make this post more informative :)

That also means that just like the moon, the planets have phases, but they're just too subtle to see with the naked eye, and for any planet further from the Sun than the Earth is, we don't see all the phases.  They're full when the planet is furthest from us (on the other side of the Sun), so it'd be hard to tell the phase just from how bright the planet appears.

But through a telescope, you can easily see the phases of Venus, and you can tell it's smaller when it's full (because it's farther away):

Source: Wikimedia Commons


In a binary star system, what relation determines the eccentricity of the three orbits (for $m_1$, $m_2$, and the reduced mass)?

Isn't it just conservation of momentum? Without any external forces, the centre of mass of the binary system must stay in the same place.

For example, imagine that you had a binary system, where one star had a circular orbit and the other, with equal mass, was eccentric. Clearly, as the orbits proceeded, the centre of mass, which is half way between the stars, would have a position which oscillates with time. But with no external forces applied to the binary system, this is forbidden by the conservation of linear momentum.


what is the resolution of the magnitude that is distinguishable to the naked eye?

While the eye is terrible at determining apparent magnitude or brightness of stars due to our adaptive iris, it's perfectly capable of discerning relative brightness between two objects in the same visual field.

Also, the ancient Greeks could discern six magnitudes of star brightness by realizing that brighter stars look bigger in the night sky.  The word 'magnitude' means 'bigness' or 'size'.

Magnitude - Wikipedia

This wasn't the most accurate system, but it was good enough to be able to discern about six different magnitudes. The system dates back about 2,000 years, to either Hipparchus or Ptolemy.   So there's no question that ancient greeks had at least a rough categorization of stars by brightness.

Now, could they tell when a planet was closest?  Easily.  The difference in magnitude between closest and farthest distance from the Earth for most of the planets Ptolemy could see was very large.  For example, Mercury at its brightest as seen from Earth has an average apparent magnitude of -1.89, and at its faintest, +5.93.  A change in magnitude of that amount would be easily discernible by the human eye. Venus differs by about two magnitudes.  Mars almost five magnitudes.  Jupiter about 1.3.  Saturn is the only one with a difference of less than one magnitude, making it uncertain if the ancient greeks could see that difference.

Calculating Apparent Magnitudes of Planets

If the ancients couldn't determine the absolute magnitudes of the planets, they could certainly compare them to the magnitudes of known stars and determine if they were brighter or dimmer.  Given the huge magnitude swings of the inner planets, determining when they were closest would have been relatively easy, within some margin of error.


At the center of our galaxy is a powerful radio source named Sagittarius A*, which is believed to be a super massive black hole (SMBH). This blackhole would contain far more mass than your run-of-the-mill supernova remnant. Our galaxy is believed to contain a SMBH containing the mass of likely a bit above 4 million times the mass (Gillessen) (2) (Ghez) of our Sun. For reference, I don't think we've ever discovered a star more massive than 600 times that of our Sun.

It's also important to understand that while many people look at blackholes as mystical or all-consuming, they actually have to follow the same rules as everyone else in the stellar neighborhood. The stars that make up our galaxy don't fall into the black hole for the same reason our planet doesn't fall into the Sun. Our star orbits black hole, our star system's velocity in equilibrium with the attractive force of the galaxy's center of gravity. This should hopefully resolve point 3.

For point 1, we should make clear that the 'black' part of the black hole is only true once you cross the event horizon. This is the case because at this point the escape velocity to escape the gravity of the black hole because greater than the speed of light. The light that isn't within the event horizon and is moving away from it is free to escape. So we can see light around it.  But why is there so much light? Well, as it happens there are a lot of rather young and large stars in this area. It is not completely understand why this is the case. Lots of stars, lots of light! There are other factors that can contribute to this too, such as there simply being lots of stars between us and the center, not just in the center itself. The accretion disk of a black hole can also be exceptionally bright. Hopefully that clears up part 1.

Now for part 2. As far as I know, we don't really have any way of determining exactly where our SMBH came from originally. Black holes aren't necessarily formed just from a supernova event, there are a handful of other ways they can be created in nature. What is apparent, however, is that SMBHs contain far too much mass to be from a single star. It has probably consumed plenty of other black holes to grow to what it is now.

One interesting and notable difference between the comparison of a star system and a galaxy is the distribution of mass. While our Sun is believed to contain 99.8% of the mass of our solar system, the SMBH at the center of the Milky Way is not nearly as massive as the total mass of the Milky Way. The ratio can vary a lot, and there are some galaxies which are believed to host no SMBH at all.

Gillessen, Stefan et al. (23 February 2009). "Monitoring stellar orbits around the Massive Black Hole in the Galactic Center". The Astrophysical Journal 692 (2): 1075–1109.

Ghez, A. M. et al. (December 2008). "Measuring Distance and Properties of the Milky Way's Central Supermassive Black Hole with Stellar Orbits". Astrophysical Journal 689 (2): 1044–1062.


How many galaxies disappear beyond the Hubble Bubble horizon every year now?

This is a tricky question to answer, because it has some false and imprecise assumptions baked into it. The short version is: I don't know at what rate stuff is currently crossing the cosmic Event Horizon. But future telescopes will not see these galaxies redshift themselves into oblivion, because this will happen in the infinitely far future, literally. Cosmological redshift is equivalent to relativistic time dilation; events that are redshifted from our point of view also play out slower, and infinite redshift means they come to a complete halt.

It is also not correct that the radius of the observable Universe is expanding at the speed of light. There are two different radii that could be candidates for the title "limit of the observable Universe"; they are very different but both expand faster than light.

Your intuition is right, however, that the distance at which galaxies recede from us at exactly light speed - the one we call the Hubble Sphere - has been expanding and since approaching us again. It reached its maximum when the cosmic expansion went from decelerating to accelerating, when the Universe was around half its current age. The Hubble Sphere, however, is not the limit of anything.

One of the two horizons, the one that seems to be the one you have in mind in the beeginning of your question, is the Event Horizon. This is the current distance, beyond which light emitted at this cosmological time will never have the chance to reach us. This horizon receding from us in proper distance, and since it is located outside the Hubble Sphere, these regions of Space are actually receding from us faster than light.
In co-moving coordinates, the event horizon is approaching us. Co-moving coordinates is a coordinate system that stretches along with the expansion of the Universe, such that a galaxy that is not influenced by anything else than cosmic expansion will have constant coordinates in time. Because the cosmological expansion is accelerating, it gets harder and harder for light at a fixed co-moving distance to reach us until it gets impossible and the Galaxy redshifts to infinity. This  is equivalent to saying that the Event Horizon has moved inwards past this galaxy in co-moving coordinates. In proper coordinates, it amounts to saying that the galaxy overtakes the Event Horizon.
Ours, by the way, is currently at a redshift of around 2, around 18 billion light years away. The light we receive from this distance is only slightly redshifted now, but the light that is emitted now will be redshifted to  infinity when it reaches us at infinite time.

There is also another horizon, however, which does expand forever, also in co-moving coordinates. That is the current co-moving distance at which light emitted at the time of Big Bang reaches us just now. In co-moving coordinates, this horizon expands slower and slower, which is why the CMB gets more and more redshifted. Objects currently on the Particle Horizon are a long time out of our Event Horizon - we can only ever see the very beginning of their history, before they redshift into oblivion. And the objects shifting into our particle horizon will show us still less of their history before having left the Event Horizon for good (yes, language gets hairy when talking relativistic time).

So it is hard say what the limit of the observable Universe is, it is simply not well defined. The Event Horizon has always been growing in proper distance but shrinking in co-moving distance. The Particle Horizon has always been growing both in co+moving and proper distance. The Event Horizon has always been expanding faster than light, and always will. The particle horizon started out slower,  but very soon overtook lightspeed in expansion velocity.

Interestingly, all objects with a redshift higher than $z \sim 1.5$ are not just receding faster than light, but they always have been! Yet we observe them with no trouble.


Is metallicity low at the central region or nucleus of the Milky Way?

The stars in the Galactic bulge are predominantly metal-rich (by that I mean have a metallicity similar to the Sun or even a little higher).

Even though these stars are predominantly old, the bulge is thought to have formed extremely quickly and the interstellar medium from which the stars were formed would have been enriched with metals very quickly.

Here is a plot from Zoccali et al. (2009) (which I recommend reading). It shows the metallicity distribution of many stars in the Galactic bulge, measured using high-resolution spectroscopy. It shows that the highest metallicity stars are towards the middle (it is very hard to get samples right in the middle because of extinction) and the averge metallicity falls as you mover further from the centre (the samples with more negative Galactic latitude $b$.). The percentages in the plots are the estimated contamination from the disk population in each sample.

The dependence of metallicity on height and radial coordinate in the Galactic disk is still keenly debated. There is general consensus that the metallicity falls with radial distance from the Galactic centre and with height above the Galactic plane. The gradient is of order a few hundredths of a dex per kpc between a few kpc and 10-12 kpc from the Galactic centre. It may flatten beyond this. You could have a look at the discussion in Cheng et al. (2012), but there are many other attempts at parameterising the gradient and it is a technically difficult thing to do.


Interestingly enough, Neil deGrasse Tyson, an american astrophysicist, wrote a book titled Death by black hole.

In this video he goes on to explain the details of what you asked, the subjective experience of the astronaut being sucked into the black hole. It is definitely painful!