Redshift to calculate age of stars In multiple articles I have seen the age of a star, within the milky way, referred to as its redshift (typically denoted by $z$). I know that $z$ can be calculated as $z=\frac{\lambda_{obsv} - \lambda_{emit}}{\lambda_{emit}}$ but what I can't find is how this is then used to calculate the age of a star within our galaxy? What characteristic does the redshift show us (spin, velocity, temperature, etc.) and how is this used mathematically to calculate age? Or is it just a method to determine where it falls on the HR diagram?

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.


HD 51684 - mass and distance Where to find, if it is possible, values of the mass and the distance of HD 51684, please? Many thanks.

HD 51684 is a magnetic Ap star. It has an entry in the recent catalogue of fundamental parameters for magnetic stars by Glagolevskij (2019). This catalogue lists it as having $T_{\rm eff}=7800$ K, and a mass of $2.3 M_\odot$.
These parameters are determined from the photometric properties of the star - i.e. they are not fundamental measurements of the temperature or mass.
The mass quoted by Kervella et al. (2019), in a study of binarity using Gaia data (HD 51684 is a known binary) is $2.195M_\odot$. However this is spurious precision; this mass comes from matching the inferred luminosity and temperature to stellar evolutionary models, not from a measurement of the binary motion.
The distance has been most precisely and accurately determined by Gaia EDR3
The parallax is $4.51 \pm 0.24$ arcsec. Roughly speaking, you can invert that to give a distance of $222 \pm 12$ pc.


How precisely is planetary tilt defined (the tilt direction, not just the angle)? The earth's axis is titled 23.4 degrees with respect to the ecliptic. Since this tilt is responsible for the seasons, it's clear that the tilt is in the direction away from the sun on the shortest day of year in the winter (on the northern hemisphere). That uniquely defines the tilt direction for me. How do we define the tilt direction for the other planets, though? What is the proper way of determining in what direction the tilt of a planet occurs?

A bit too simply (but not too much), it's the angle between the planet's rotational angular velocity vector and the planet's orbital angular momentum vector.

Where it gets tricky is dealing with little wobbles and such. Planets don't quite orbit in a plane because of gravitational interactions amongst the planets. This means the instantaneous (osculating) orbital angular momentum vector isn't quite constant, both in magnitude and direction. There are a number of different mean orbital elements that smooth out most of those tiny wobbles. One of those mean orbital elements sets (I'm not sure which one exactly) is used to determine the  mean orbital angular momentum vector.

Planets don't quite rotate nice and smoothly because planets aren't perfect spheres and aren't rigid bodies. This gives the Moon, the Sun, and other planets a handle by which they can exert tiny little torques in the planet in question. The response of the planet to these torques is somewhat arbitrarily divided into two categories based on frequency. Very slow responses are called precession; faster responses, nutation. The non spherical nature of a planet means the planet undergoes a small torque-free nutation as well as the torque-induced precession, nutation, and polar motion. In addition to these, there are a number of terms (all small) that don't yet have a very good model behind them. This oddball terms, along with the torque-free nutation, are collectively called polar motion. The motion is fairly smooth if one ignores nutation and polar motion, and the effects are all small.

Precession, while very slow, can be quite large in magnitude. The rotational angular velocity vector that is used in determining the axial tilt incorporates precession but smooths out nutation and polar motion.


What other “planetary formation mechanisms” exist beyond “gravitational collapse”? My curiosity was pricked by the an edit made to this post: What's the distinction between "planetary formation mechanisms" and "gravitational collapse"? sadly the edit was changing the original questions intend, so I had to roll it back.

But the question of the edit itself felt interesting to me aswell, so I'm reposting it here in another question:

What "planetary formation mechanisms" other than "gravitational collapse" exist?

"Gravitational collapse" is not the generally used name of any planet formation model. Gravitational disk instability may be what is meant.

In the context of the answer you refer to, the other, and probably dominant planet formation is known as the core accretion model.

Of course gravity plays a crucial role here too, but the distinction between this and direct collapse of an instability in a gas disc (what is perhaps being referred to as gravitational collapse) is that the major phase of gas accretion takes place after the initial formation of a rocky/icy core. This may account for the correlation between planet frequency and host star metallicity.


How are telescopes protected from blinding light and satellite glares? I imagine a situation must happen sometimes where a telescope watching a very dark object may catch a bright flash of light or a reflection off of a satellite. If that's even a concern at all (maybe all such telescopes are already designed to exclude that possibility), what countermeasures against blinding someone's eyes or frying the digital photo sensors are there?

Random bright objects can occasionally appear in astronomical images. Examples include asteroids and satellites. However, these objects are not moving at the sidereal rate across the sky, and thus unless the telescope is set to track them, they leave a trail across the image.

This is unlikely to cause damage and CCD detectors are fairly robust to saturation of exposures in any case. However, it may ruin the image.


When will all eight planets in our solar system align? Ignoring expansion of the universe, entropy, decaying orbits, and interference from any bodies colliding with or otherwise interfering with their orbits, will the eight planets known planets in our solar system ever align?

What is the "period" of the planets; how often would they align perfectly? And based on their current positions, how far off into the future is their next theoretical alignment?

This is a low accuracy - yet simple - answer

It allows you to calculate only radial alignment configuration of the planets.

If you would like an approximation, let's say, you approximate the position of the planets as hands in a clock, you could work the math out by something like this.

Assume $\theta_i$ is the initial angle for planet $i$ at time $t_0$ - measured from an arbitrary but fixed position, and $l_i$ is the length of the year - in days - for planet $i$.

Then it resumes to solving this system of equations:

$$
x \equiv \theta_i \left( \ mod\ l_i\right)
$$

From here you would then simply apply the Chinese Remainder Theorem.

Finding the minimum x, will give you the angle that the planet that at $t_0$ had angle $\theta_i = 0$ would have travelled until an alignment configuration was reached. Asuming you choose Earth as the mentioned planet, then divide that angle by a complete revolution ($360^{o}$) and you will get the number of years for that configuration to be reached - from the $t_0$ configuration.

The different $\theta_i$ in degrees for all the planets on 01 Jan 2014 - you can use this as your $t_0$:

\begin{align}
Mercury &\quad 285.55
\\Venus &\quad 94.13
\\Earth &\quad 100.46
\\Mars &\quad 155.60
\\Jupiter &\quad 104.92
\\Saturn &\quad 226.71
\\Uranus &\quad 11.93
\\Neptune &\quad 334.90
\end{align}

Source

The different $l_i$ in days for all the planets:

\begin{align}
Mercury &\quad 88
\\Venus &\quad 224.7
\\Earth &\quad 365.26
\\Mars &\quad 687
\\Jupiter &\quad 4332.6
\\Saturn &\quad 10759.2
\\Uranus &\quad 30685.4
\\Neptune &\quad 60189
\end{align}

Finally under an integer values aproximation and using this online solver for the system of equations the answer is $x = 4.0384877779832565 \times 10^{26}$ which divided by $360^{o}$ gives you roughly $$1.1218 \times 10^{24} \quad\text{years}$$

Edit 1

Just found this site you may like to play around with. It's an interactive flash application with the accurate position of the planets.

I also know that the all the information can be obtained from this NASA page and that is as accurate you can get, but it is just incomprehensible for me now. I will try to revise it later when I find time.

Also this book by Jean Meeus called Astronomical Algorithms covers all the fundamental euqations and formulas - it has nothing to do with programming algorithms though.

Edit 2

Seeing that you are a programmer, it could be worth for you checking out the NASA site I mentioned above, the data for all the planets can even be accessed via $\tt{telnet}$.
Or this Sourceforge site where they have implementations for many of the equations described in the book also mentioned above.


What happened to the third planet in the Kepler-47 system? In 2012, the discovery paper for the multiplanet circumbinary system Kepler-47 mentioned a potential third transiting planet in the system:

  A 0.2% deep transit-like event is present at time 2,455,977.363 (BJD) that is not caused by
  either of the two planets. A search for additional transits has revealed several more tentative
  transit events (12), but we caution that the star is faint (the Kepler magnitude is 15.178), there
  are large modulations due to star-spots, and the data contain correlated “red” noise, making
  small, non-periodic transit detection challenging. The marginal evidence at the present time is
  insufficient to place confidence on any additional candidate planet(s).

In 2013, Jerome Orosz gave a talk at the Second Kepler Science conference titled "The Confirmation of a Third Planet in the Kepler-47 Circumbinary System", claiming the detection of two further transits attributed to the third planet and giving the orbital period as 187.3 days and noting that the dynamical fit indicates that transits would stop for four years starting in July 2013.

In 2015, the discovery paper for Kepler-453 b was published, describing this planet as the tenth circumbinary planet discovered by the Kepler mission, which lists Kepler-47 d as the ninth of the previously discovered planets, giving a reference of "Orosz 2015, in preparation".

As far as I have been able to tell, there has been no further update since then - the Orosz 2015 paper never appeared. Is anyone aware of any further updates about this system and whether or not the third planet is still considered to be present?

Well it looks like it has finally showed up! The Institute for Astronomy in Hawaii has put up the following press release: Scientists Fill Out A Circumbinary Planetary System. Kepler-47d turns out to be the largest of the planets, with a radius 7 times that of the Earth.

The relevant paper is Orosz et al. (2019) "Discovery of a Third Transiting Planet in the Kepler-47 Circumbinary System".


VSOP87D: What are the purpose of the initial columns in the VSOP87D data? I am trying to write an app to determine planetary positions, using Jean Meeus' Astronomical Algorithms as a reference. His guidance on using the VSOP87D data is straightforward, as are many other sources that I googled. But all of these talk only about the last three columns of that data. What is the purpose of the preceding columns? Some are self explanatory (e.g., the simple numbering of each row), but others seem to have no explanation either in the file itself or the 'read me' documents that accompany it on the FTP site. I want to ensure I fully understand the data.

Here is an example of the third row of VSOP87D for the planet Jupiter:

4510, 3, 0, 0, 0, 0, 2, -5, 0, 0, 0, 0, 0, 0, -0.00566632795, -0.00089195704, 0.00573610145, 1.44406205976, 7.11354700080

where I have replaced the various spaces with commas to save space.

I will note that the term .00573610145, for example, differs by by several orders of magnitude from that listed in Meeus' table, as do some other terms. But this does not seem to correspond to any of the numbers in the other columns, unless I'm missing the obvious.

The data is taken from the FTP website: ftp://cdsarc.u-strasbg.fr/pub/cats/VI%2F81/VSOP87D.jup

Thanks!

First, the data in most of these columns is not required for the computation of planetary positions using the theory if the 'second' form equation is used. The terms are, using the order of the example of Jupiter VSOP 87D's third line ...

4510 ...

"4" means the 'version code', i.e., this is the "D" version of the theory which allows computation of the heliocentric rectangular coordinates of the planet for the ecliptic of date. A "3" would be for the spherical heliocentric coordinates for the ecliptic of date.

"5" is the 'body' code, Jupiter being the FIFTH planet. Mercury would be planet '1', for example.

"1" is the 'index of coordinate', meaning in this example that this line is part of the 'longitude' coordinate calculation. Later lines show a '2' for the B (or latitude) coordinate or '3' for the R (radial or distance) coordinate)

"0" is the 'degree' or power of the time variable. Here, terms are multiplied by T to the ZERO power (or 1). Later terms are multiplied by successively higher powers of T.

3
... is simply the sequential number of this line in this series. Jupiter has 5 series (each multiplied by a higher power of T), so this number 'resets' for each series.

The next 12 terms, mostly zeros in this example, are the 'coefficients of mean longitude'. As near as I can tell, these coefficients are used to correct mean longitude quantities for secular perturbations. They are used only in the first form of the equation as shown below with the next two terms:

The next two terms are amplitudes S and K, used in the first form of the positional equation:

T**alpha * (S sin phi + K cos phi)

where phi = sum[i, 1 thru 12] [a(i) * lambda(i)] and lambda(i) values are given for each solar system body in the explanatory text.

The final 3 terms are Amplitude A, Amplitude B and Phase C for use in the second form of the equations to calculate position:

T**alpha * A * cos(B + CT)

... with T being time in Julian centuries, and alpha the 'power' of T.

My comment in the original question about the terms being 'off' by several orders of magnitude is moot ... for convenience either in calculating or printing, Meeus multiplied the terms by 10**8, then corrects for it later. This probably facilitated better computation on machines of the time it was written and also made for neater printing in the book.

The document provided by barrycarter, along with some other study of planetary motion allowed me to finally understand this, thanks!


How to calculate conjunctions of 2 planets So, the recent conjunction of Jupiter and Venus seems to have spawned lots of excitement over this "rare" event. But what I can't figure out, is exactly how rare it is. And I've seen such conflicting claims and calculations that I figured I'd better calculate it myself. The only problem is I'm no astronomer. So, does anyone know of a program that can calculate all the conjunctions between a date range that are within a degree of each other? Or does smart person know how to calculate it manually? :)
    I have a program called Stellerium which I've been using to view the various conjunctions that people have mentioned. I don't know if I can use it in the way I need it though. Any help? :)
Thanks

EDIT: http://wgc.jpl.nasa.gov:8080/webgeocalc/#AngularSeparationFinder lets you find planetary conjunctions online using NASA's data. It's still iterative, but fairly fast (since NASA uses fairly powerful servers, even for their websites).

Summary: I'm still researching, but there appears to be no well-known,
reliable non-iterative method to find conjunctions. Using the
iterative method and the C SPICE libraries, I created a table of
conjunctions for the visible planets (Mercury, Venus, Mars, Jupiter,
Saturn, Uranus) here:

http://search.astro.barrycarter.info/

Full "answer":

I am still researching a general answer to this question ("How to
calculate conjunctions of 2 planets"), but here's what I have so far.

The iterative method:

Compute the positions of the planets at regular intervals (eg,
daily). The "daily" works for planets (but not some asteroids and
definitely not the Moon) because the planets move through the sky
relatively slowly.
Find local minima in the daily lists.
For efficiency, carefully discard local minima that are too
large. For example, Mercury and Venus may approach each other, reach
a minimal distance of 20 degrees, and then drift apart. The 20
degrees is a local minima, but not a conjunction.
However, be careful when discarding minima. If you are searching
for 5-degree conjunctions, two planets may be 5.1 degrees apart one
day, and 5.2 degrees apart the next day, but less than 5 degrees
apart sometime in the interim.
For 5-degree conjunctions, you only need daily minima less than 8
degrees, and even that is overkill. The fastest a planet can move in
the sky is 1.32 degrees per day (Mercury), and the second fastest is
1.19 degrees per day (Venus). In theory, these movements could be in
opposite directions, so the fastest two planets can separate is 2.51
degrees per day. So, if two planets are more than 8 degrees apart
two days in a row, there is no way they could be closer than 5
degrees between the days.
In reality, planets maximum retrograde angular speed is slower
than the prograde maximum speed, so the 2.51 degree limit above is
never actually reached.
After finding local minima, use minimization techniques (eg, the
ternary method) to find the instant of closest approach.
I ended up using the C SPICE libraries, and found 32,962
six-degrees-or-less conjunctions between -13201 and 17190, averaging
about 1 conjunction per year. Of these, 2,185 occur between the "star
of Bethlehem" and the 2015 conjunctions:

http://12d4dc067e0d836f1541e50125c24a03.astro.db.mysql.94y.info/

This iterative process works, but can be tedious. Since planetary
positions are semi-well-behaved, you'd think there would be a faster,
non-iterative method. Well...

As https://math.stackexchange.com/questions/95250/ notes, people
have been interested in finding planetary conjunctions for at least
2000 years, probably longer. The first such known tool is the
https://en.wikipedia.org/wiki/Antikythera_mechanism. The "Operation"
section of the wikipedia page suggests this device was hand-cranked,
and thus iterative.
SPICE
(https://en.wikipedia.org/wiki/Spacecraft_Planet_Instrument_C-matrix_Events)
is one of the oldest digital computer attempts to compute planetary
positions, and it includes a routine to compute conjunctions:

http://emfisis.physics.uiowa.edu/Software/C/cspice/doc/html/cspice/gfsep_c.html

However, this also uses an iterative method, as the long description
of the "step" parameter indicates:

"step must be short enough for a search using step to locate the time
intervals where the specified angular separation function is monotone
increasing or decreasing. However, step must not be too short, or
the search will take an unreasonable amount of time"

By experimentation, I found that a step size of 6 days does not find
all Mercury/Venus minimal separations, although a step size of 1 day
does find these. In other words, reducing the step size from 6 days to
1 day found additional conjunctions, but reducing the step size to
well below 1 day did not produce any additional conjunctions.

Akkana Peck (akkana@shallowsky.com), creator of a well known script to find
conjunctions (https://github.com/akkana/scripts/blob/master/conjunctions.py)
tells me:

M[y script] iterates through dates.

[...]

In Astronomical Algorithms, Meeus has a chapter (17) on Planetary
Conjunctions, but the chapter is less than three pages and lacks
detail except that he's pretty clearly iterating and looking for
the smallest separation, same as my Python program does.

He does a little better in Astronomical Formulae for Calculators:
sounds like he's advocating iterating over big steps to find the
place where the separation stops decreasing and starts increasing,
then using "inverse interpolation" to find the exact time.

"Mathematical Astronomy Morsels" (also Meeus) talks starting on p.
246 about triple conjunctions (defined as two planets having several
conjunctions in a short time, due to retrograde motion, not three
planets all in conjunction at once) and gives some examples for very
specific pairs, like Jupiter and Saturn, with ways you could look for
those conjunctions.

It doesn't look like any of these books would be very helpful in
finding a non-iterative solution.

I haven't had a chance to read the books above, but did find:

http://adsabs.harvard.edu/abs/1981JRASC..75...94M

where Meeus confirms the standard iterative method, but also provides
a different, less accurate method. Unfortunately, Meeus only uses this
method to compute solar conjunctions, elongations and oppositions, not
interplanet conjunctions.

Jon Giorgini of NASA (jdg@tycho.jpl.nasa.gov) tells me:

As far as NASA computed planetary/stellar conjunctions/occultations, I
don't know of anyone within NASA that does that routinely. There is an
external network of volunteers that does that under the umbrella of
the International Occultation Timing Association (IOTA), and they have
developed pretty refined internal software for that purpose.

I contacted IOTA (http://occultations.org/) and they confirmed
they use an iterative method as well:

[...] the software package Occult does generate planetary conjunction
predictions - based on a two-body solution. The approach used is a
crude brute-force method of generating the planetary ephemerides on a
daily basis.  It is not particularly efficient - but it is sufficient
for its intended purpose. [...]

As a note, IOTA focuses on asteroid occultations, so computing
positions daily doesn't always work. Especially for near-Earth
asteroids, IOTA must iterate considerably more frequently.
I also tried contacting Fred Espenak, the creator of
http://eclipse.gsfc.nasa.gov/SKYCAL/SKYCAL.html, but was unable to
do so. Jon Giorgini tells me that Fred has retired.
I'm still looking, but my current conclusion is that there is no
good well-known non-iterative way to find planetary conjunctions. As
the image in my
https://mathematica.stackexchange.com/questions/92774 shows,
planetary separations aren't really as well-behaved as I had hoped.
I just got a reply from Arnold Barmettler, who runs calsky.com:

I'm using a very time consuming iterative approach to pre-calculate
Bessel Elements for conjunctions. This allows fast online calculation
for any place on earth.
Initial calculations are only done every few years, so CPU time does not
matter. This would change if I'd enter the business to calculate
asteroidal occultations.

re-iterating (pun intended) the same theme.

Miscellaneous:

I used planetary system barycenters (the center of mass of a
planet and its moons) for an entirely different reason. If you ask
HORIZONS (http://ssd.jpl.nasa.gov/?horizons) for the position of
Mars and set the date, you'll see this notice:

Available time span for currently selected target body:
1900-Jan-04 to 2500-Jan-04 CT.

However, if you use Mars' barycenter, this becomes:

Available time span for currently selected target body:
BC 9998-Mar-20 to AD 9999-Dec-31 CT.

In other words, NASA computes the position of Mars' planetary system
barycenter for a much longer interval than they compute Mars' actual
position. Since I wanted to compute conjunctions for a long period of
time, I went with the barycenters (DE431 computes barycenters even
beyond 9998 BC and 9999 AD).

I've complained that this is silly, especially for Mars, since the
distance between Mars' center and Mars' planetary system barycenter is
only about 20cm (yes, centimeters, since Phobos and Deimos have very
little mass) per
http://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/aareadme_de430-de431.txt. However,
NASA apparently plans to keep it this way.

I ignore light travel time, which introduces a small error. Most
planetarium programs, like Stellarium, ignore light travel time by
default, although Stellarium has an option to turn it on.
I also ignore refraction, aberration, and similar minor effects.

More details on how I generated these tables (in highly fragmented form):

https://github.com/barrycarter/bcapps/tree/master/ASTRO

(README.conjuncts in the above is a good starting point)

Some of the "cooler" conjunctions I found are at:
http://search.astro.barrycarter.info/table.html. Here's a screenshot:


What area of Proxima Centauri would need to be occluded for the effect to be observable on Earth? One of ideas of how Breakthrough Starshot could communicate data from Proxima Centauri is to have a massive swarm of the solar sail powered craft to pivot (with their sails) synchronously, taking either 'sideways' or 'head on' orientation to the star - in effect either obstructing it or revealing for an observer 'behind' - on Earth. If the swarm is big enough, it would cause an observable dimming of the star - and timing the pivots right it could encode data in the star's visible luminosity.

The primary question of viability of this idea lies in 'how big a swarm?'. What percentage, or what absolute area or a star like Proxima would need to be occluded for the effect to be observable? We can notice transition of exoplanets quite reliably so that would be the upper bound, but what's the smallest object area that would make a detectable difference?

The photometric precision achieved by satellites like TESS is around 100 parts per million. The original Kepler mission achieved a bit better on bright stars - maybe 20 ppm, but the question of precision may be moot since stars themselves are quite variable;  McQuillan et al. (2012) https://arxiv.org/abs/1111.5580 suggest that the floor level of intrinsic variability in M-dwarfs may be as high as 1000 ppm.

Anyway, let's assume that a rapidly encoded signal at 100 ppm could easily be distinguished from stellar noise and maybe even be generous and say 10 ppm might be possible. That would mean that this fraction of the stellar surface would need to be obscured.

The radius of the star is 0.15 that of the Sun. So the physical area to be obscured is $\sim 10^{-5} \pi R^2 = 3.4\times 10^{11}$ m$^2$.

The web page referred to in the question suggests that Starshot consists of a thousand space craft with light sail areas of 16 m$^2$. It thus seems to fall short of providing detectable modulation by 7 orders of magnitude (unless I have misunderstood the proposal, which isn't mentioned on that web page).

There is also going to be the question of the low bit rate and what limited information could be transmitted using such a crude technique. Recall that the spacecraft will be travelling at an appreciable fraction of the speed of light, so won't have long to send their message.

A further point is that I don't think it can be assumed that the trajectory of the spacecraft will be such that they are travelling in a direct straight line
between the Earth and Proxima Centauri, which are of course in relative motion.
Thus the spacecraft would approach Proxima Cen "at an angle" as viewed from the position of the Earth some 30 years after the spacecraft were launched, unless the trajectories can be tuned quite carefully en-route.
Even if they were, the parallax motion of the Earth around the Sun would mean that the spacecraft would not necessarily be aligned be when they reached Proxima Cen, although I suppose the launch time and speed could be arranged so that was the case. However, from what I have read, the idea is just to pepper the inner au of the Proxima Cen system with these 1000 spacecraft and so only a few of them would actually be lined up so that they could obscure the star as seen from Earth.


Are all planets/galaxies moving away from *us*? From what I can see in a physics textbook page (on redshift), it seems to imply that all planets/galaxies emit red, rather than blue waves (for red-shift).

Wouldn't this mean that everything is moving away from us and that we're the centre of the Big Bang? (See quote from the book below)

On a similar note, does everything move at the same speed away from the centre of the Universe? - If not, then wouldn't that mean that we'd see a few planets/galaxies that emit blue, as they are traveling with a greater velocity from the centre than we are (with it being closer to the centre) and that they are 'catching up' with us?

It's not a big deal and I don't need to know it, but I would like to know, if possible.

~StratusGold

Quoting the textbook: "When we look at light from distant galaxies, we find that the wavelength has increased. The wavelengths are longer than they should be - they're shifted towards the red end of the spectrum. This is called red-shift. (...omitted...) Measurements of the red-shift indicate that these galaxies are moving away from us (receding) very quickly - and it's the same result whichever direction you look in" - CGP GCSE AQA 9-1 Physics Revision Guide, Topic 8 'Space Physics', Pg 102

No, not all of them. Only the ones very far away.

When you walk your dog, the dog is tied to you via the leash. It does not recede into the distance.

Planets in the Solar System are all tied up to the Sun via gravity. They don't recede into the distance.

All stars in our galaxy are bound together in a big clump (the galaxy) and so they stay together.

Finally, all galaxies in our neighborhood are also bound by gravity, so they stay together.

Beyond that range, things are not bound by gravity, so they keep running away from each other.

Gravity is like a leash. If things are close enough, they stay together.

Keep in mind that the universe does not have a center. It's infinite (as far as we can tell), and infinite things don't have centers. Rather, if you take any two distant galaxies, they all recede from each other. But the expansion does not go back in time to a common center.

Here's a comparison. Imagine you're a bi-dimensional creature; instead of having 3 dimensions, you only have 2. Like a stick figure drawn on a paper. You only exist in the 2 dimensions of the paper, and can't even imagine the 3rd dimension.

Now imagine a balloon. You draw a bunch of stick figures on it. Now blow air into the balloon. All stick figures recede from each other, but for them there's no center for that expansion - because all they know is the surface of the balloon. They don't know anything that's not on the surface. So they all see each other running away, but there doesn't seem to be a center for that expansion.

It's just an analogy, because the universe is not a balloon, but it will get you started.

Finally, red-shifted galaxies don't emit only red light. They emit the usual radiation like everything else. But because us and them are running away from each other, radiation from one side reaches the other side a little bit stretched out. For the visible part of the spectrum, it's like taking the spectrum and shifting it towards the red part.

That's all there is to it. It's still the usual radiation, but all wavelengths are a little bit blown up.


How to invert fits headers order (primary/secondary)? I have an image from Hubble telescope with two headers and this creates a problem because my program expects the NAXIS, NAXIS1, NAXIS2 in the first header, but in this image case, this info is in the second header.
So what I need is to invert the order of headers. i.e, SCI header should come first.
When I run:
from astropy.io import fits
image = 'image.fits'
fits.info(image)

I get
No.    Name      Ver    Type      Cards   Dimensions   Format
  0  PRIMARY       1 PrimaryHDU     722   ()
  1  SCI           1 ImageHDU       110   (1900, 1900)   float32

I already tried editing the first header, to add the needed info, but for some reason, it won't accept the changes.

EDIT:  I moved the last paragraph to the beginning because it's really a Python question, but I left my "manual" method answer in case it helps, too.
Forgot to add this link to Python code on Github.  It helped me get started with displaying images using Python and dealing with the headers without altering them.  On line 19, I changed it to "image_data = image_file[1].data" because the image was in the HDU [1] instead of HDU [0].
I've had this exact problem when dealing with TESS images - I can't open them in the "usual" software I use for all of my other fits images.  Inverting the headers won't solve it, either, because one of the keywords in the first header is needed.  If you are only trying to make the change for one image, it's easy to do manually.  You can edit the headers in Notepad or any other text editor.  Here's the general workaround I use if it's only one file (or a couple of files) - if you need to open a lot then this is too tedious.

Make a copy of the file and change the extension of the copied version to ".txt"
Open in a text editor like Notepad (I like Notepad++ better, but any text editor will work)
Turn on "word wrap" so you can see the header info, there aren't carriage returns or line breaks.  Each keyword line is exactly 80 characters long regardless of what info it holds.
Adjust the width of your text editor to get close to 80 characters wide and you'll see the headers "line up" pretty well but not perfectly.  There is almost always a situation where the entire 80 characters are used so there isn't a space between the last character of a keyword line and the first character of the next one which will throw off the word wrapping.  But that's ok, just frustrating.
Look through the header info until you see the beginning of the first extension (probably named "SCI").  It's fine to select and delete the keyword lines you don't need, the ones breaking it up into two HDUs, as long as you don't need the info.  Some of them will be entries such as "MISSION= Hubble" or something like that along with info about the pipeline that created it, etc.
When I've located NAXIS, I put the cursor in front of it and select everything up to the end of the first keyword line in the first HDU which should be "SIMPLE= 1" (or true). The first keyword line with SIMPLE= 1 is 80 characters long and MUST remain 80 characters long or it will break it, so I delte everything between NAXIS and the end of the SIMPLE line.  Look to see what the next line is after SIMPLE and delete to the first letter of that keyword. The SIMPLE keyword line is required and should be the first one. Make sure you don't add a carriage return or it'll break the file.
Save your changes and edit the extension to ".fit" or ".fits" and it should open fine.

As long as keep each chunk of keyword info 80 characters in length, you can manually force it to have only one header so you can view the image.  This is, of course, not practical for a large number of images and you could lose data that IS important to you such as the date, etc.
To view fits files without having to edit the header, check out SAOimageDS9.  It has never failed me in opening an image fits file with multiple HDUs.  If you are trying to work with large sets of images there are other workarounds depending on what you're trying to do.  Good luck, I hope this works for you!


Mass accumulation rate and orbit-time of earth According to me following reason of mass accumulation of earth has been completely overlooked so far.
The sunrays that fall on the oceans travel deep into water and finally get converted into mass which eventually results in "mass-accretion of the earth". So earth is continuously getting heavier due to the mass being added to it due to this reason year by year.
Therefore, should this also not be one of the reasons why the orbit-time of the earth (time taken by it to take one round of the sun) is bound to change (increase or decrease) with time?

OK. Let's do the numbers. The Earth presents an circular "profile" of radius about 6 million meters to the Sun so it's cross-sectional area is about $10^{14} m^2$. The solar irradiance at the top of the atmosphere is about 1400 $W/m^2$ so the total power received from the sun is about $10^{17}W$, by coincidence, more-or-less equal to $c^2$ in SI units. So the mass equivalent of the sunlight received by Earth is about $1 kg/s$ or about $3 \times 10^7 kg/year$. This is about one part in $10^{17}$ of the mass of the Earth per year, so even if were all absorbed and none of it re-radiated (which would be thermodynamically impossible) the increase in mass is so small that the orbital change would be unmeasurable. In fact as @James K has already pointed out in a comment, more or less the same amount of energy (and therefore mass) is radiated away as heat radiation, so even this tiny change does not really occur.


How is dark matter different from ordinary (visible) matter? How is dark matter different from ordinary (visible) matter?

1) Does dark matter consists of invisible astronomical objects like white dwarfs, black holes?

2) Has dark matter not been directly observed?

Is there a better explanation for dark matter. I could not understand completely the difference between dark matter and ordinary matter.

There is certainly one and possibly two types of dark matter to solve two dark matter problems.

The first dark matter problem is that most gravitating matter is dark. Most of the inferred mass that appears to be responsible for the observed gravitational dynamics of large scale structures like galaxies and clusters of galaxies cannot be seen (i.e. is dark). That is, we can't detect it through visible light or any other electromagnetic waves.

Now it is possible that you could make up this mass out of very low mass stars,   cold white dwarfs, planets, black holes, lost golf balls or indeed any other form of cold, non-luminous matter. Such objects (bar the golf balls perhaps) certainly do exist, are known as baryonic dark matter (because they are made of "normal" protons and neutrons - baryons), but there is unlikely to be enough of this stuff to explain the observations.

The second problem is that most of the dark matter must be non-baryonic. This is deduced from the lack of sufficient plausible baryonic dark matter to explain the dynamics of galaxies and clusters, but also from estimates of the primordial abundances of helium, deuterium and lithium. Big bang nucleosynthesis depends on the density of baryonic matter compared to the universe as a whole and it looks like baryonic matter can only be responsible for a fifth of the dark matter.

Thus the bulk of the dark matter is supposed to be some non-baryonic stuff, probably in the form of particles that do not interact via the electromagnetic interaction and therefore do not emit or absorb light.


How to view Jupiter with 4.5" Newtonian Reflector I have a 4.5" Newtonian Reflector and I took a look at Jupiter the other night.

I was able to get a reasonably sharp view of it and see a number of it's moons, but I was unable to see any surface details, it was pretty much just white glare.

I'm not sure if it's possible to distinguish any surface details with the telescope I have, so any advice on best setup would be useful.

I have a few accessories:

x2 barlow lense
10mm eye piece
25mm eye piece
6mm eye piece
moon filter
color filters

Would anything else, other than a more powerful telescope, help me?

I was unsure if I might have an alignment issue that was causing this, but as I could make out the moons as sharp dots on the planetary plane, I assumed not.

Any advice would be welcomed.

Jupiter is a very bright object, so maybe the best procedure is to use your filters, they might help with the glare. However, I think you have either a case of low resolution or distortion. Generally, the larger the diameter of the telescope, the better is the resolution (more details here). Resolution defines how much detail you are able to resolve with the telescope.

Changing the eyepieces or using Barlow lenses can amplify the images, but it doesn't improve resolution (you'll just see a bigger blob - but still, it's cooler than just a small speck).  There are also other things to check:

Was it windy? Too much wind can, usually, mean a turbulent atmosphere, and it hampers observations. Another way to check for turbulence is to look at stars with the naked eye: if they are twinkling too much, the atmosphere is very turbulent
Was Jupiter near the horizon? If so, maybe you should try observing it when it's higher in the sky. When looking near the horizon, light has to pass through a large column of air (a parameter known as airmass by astronomers), and it distorts images
Is the telescope well collimated? Poor collimation (mirror alignment) usually results in tail-like structures in sharp objects, almost like a comet. This is probably not the case if the scope is new
What is the quality of the optics? Some telescope manufacturers ship their products with very low-cost, low-quality optical components (eyepieces, filters etc.)

Also, if your objective is to observe mainly solar system objects, I suggest using a refractor telescope, they are wonderful for that. Additionally, this video by Phil Plait on Crash Course might help you and give invaluable information.


Do other stars have Oort clouds? Stars have in the past passed within 1 light year from the Sun, and will do so in the future too. If they have Oort clouds, would there not be interactions between their Oort cloud objects and the planets of the Solar system? For example, the outer edge of the Sun's Oort cloud is approximately 100,000 AU away. So, if another star comes at a distance of 1 light year from the Sun, its Oort cloud should similarly go through the inner solar system. Does this actually happen?

The other question has some very good detailed answers, worth reading, but briefly and more specific to your question:

  Would there not be interactions between their Oort cloud objects and
  the planets of the Solar system? For example, the outer edge of the
  Sun's Oort cloud is approximately 100,000 AU away. So, if another star
  comes at a distance of 1 light year from the Sun, its Oort cloud
  should similarly go through the inner solar system.

While very little is actually known, it's not crazy to think that smaller stars might have smaller oort clouds and larger stars, larger ones, though there's a great deal of uncertainty on what an average oort cloud is, how far an average one extends, and how densely packed they are.

First part first, how often does a star pass within one light year of our sun?

Not very often.   Ballpark, maybe every million to couple million years or so.  The last time was 70,000 years ago, a small star, 15% the mass of the sun.  See here.

In the fly-by section of that same wiki-article, it says a star passes that close (0.82 light years or less) about once every 9 million years.

Scholz's star is quite small, only about 15% the mass of our sun, and it's unclear if it had an Oort cloud that extended .82 light years at all, so while it passed through our Sun's Oort cloud, it's unclear if the inner solar-system passed through it's Oort cloud, assuming it has one.  It's also possible that the sun stole much of Scholz's star's oort cloud during the fly-by, but that's just speculation.

Did Scholz's star send Oort cloud objects heading towards Earth?  That's a good question and the answer is, probably, but we don't know, because if it did, it would take longer than 70,000 years for any re-directed Oort cloud comets to reach the inner solar-system.   By the same article, an estimated 2 million years for re-directed Oort cloud objects to reach the inner solar system, so, in about 1.93 million years (give or take), we'll start to see what Scholz's sent our way.    :-)

In 1.36 million years, Gliese 710 could pass within 1 light year of Earth.  The estimate is 1.1 light years plus/minus .577 light years.  Gliese 710 is larger than Scholz's, about 60% the mass of the sun but we know nothing about how large or how dense it's Oort cloud might be.   But, very generally, it seems likely that the Sun has passed through another stars Oort cloud at some point, perhaps every 1-10 million years, perhaps every hundred million years or so, but it seems reasonable to me that has happened and will continue to happen, on rare occasion.

2nd question, what would happen if the inner planets passed through another star's Oort cloud.

If we use our Oort cloud as a model, and, warning, these estimates are enormously ballpark, but, a trillion Oort cloud objects 1 KM or larger and the Oort cloud extends between 100,000 and 200,000 AU (lets say 150,000 as a middle estimate), that works out to one 1 KM or larger object per cubic region of space about 15 AU across.   That's 3/4ths the distance between the Sun and Neptune, so if you figure just one, 1 KM or larger comet or object in a 3D space over such a vast distance, most would pass by the inner planets unaffected.   The occasional one would get it's orbit measurably altered with a fly-by of one of the gas giant planets and rarely, there would be an impact.  Now presumably there's more smaller stuff in Oort clouds too, so there should be smaller impacts more frequently, though there's almost nothing that's actually known about how dense Oort clouds are, so anything beyond "infrequent", is impossible to say.

Passing through another Stars Oort cloud should take tens of thousands, perhaps 100,000 years years or so, so there probably would be the occasional impact on a planet here and there, but on a human time scale, significant impacts should be very rare.


When is Gamma Draconis closest to the zenith in London on April 4th I am reading Roobert Hooke's paper An attempt to prove the motion of the earth from observations (1674, faksimile). Hooke writes

  The principal dayes of doing which will be about the 4 of April, when our Zenith passeth by the said Star at midnight

The said star is Gamma Draconis, and the observations are made in London. As far as I see it, the star is closest to the Zenit at 6 in the morning in London at April 4th.

So, why does Hooke say at midnight? Is this an expression of the 17th century?

I read this as "Benenaim" (another name for Alkaid) too (not Etamin, which is another name for Gamma Draconis), and that star does indeed transit the meridian near midnight on April 4th (12:44am to be more precise). Note that the hour angle is 0 in the Stellarium screenshot below, and the star is only 35 minutes from the zenith position:


Are accretion disks the only way stellar planetary systems form? The accepted theory for the formation of our Solar System is from an accretion disk.  A molecular cloud is thought to have collapsed under its own gravity and formed a disk.  Most of the matter in the cloud formed the Sun, and the remaining matter was either ejected or formed the planets and other bodies (e.g. moons, asteroids, comets).
Question:  Now that we've discovered many extra-solar planets, are there theories differing from the accretion disk theory for the formation of planetary stellar systems?

The current paradigm for understanding the formation of the solar system (including the planets) is nebular theory, which is thought to be applicable to the rest of the Universe.

Question: Now that we've discovered many extra-solar planets, are there theories differing from the accretion disk theory for the formation of planetary stellar systems?

Not in terms of modern scientific theories - at least not that I am aware of, but this is outside of my expertise. I am willing to be wrong here. My point here is that if you consider the long history of proposed hypotheses of the formation of the solar system (which includes the planets) as "alternatives," then you could consider some of them in some special cases, perhaps, but this is pure speculation. Most of these historical hypotheses have been ruled out. There are issues for the contemporary nebular theory, for examples, explaining the origin of planetesimals, and explaining the discrepancy between the observed and theoretical angular momentum of the solar system.
Regarding exoplanets, I am also not aware of any system that currently definitively challenges the nebular/accretion formation paradigm (again I could be wrong here). However, mechanisms such as disk migration are used to explain the properties of many exoplanets.


How did pre-Earth's mass and orbit change at the Moon formation event? The popular, but still disputable, Moon formation theory is that a pre-Earth was hit by a Mars sized planet. The mass ratio of Earth:Mars:Moon is roughly 100:10:1.

Does this mean that pre-Earth was roughly 9 Moon masses (~10%) lighter
than today's Earth? Or could much mass have been ejected from the
Earth-Moon system?
Did the impact substantially and immediately change Earth's orbit,
such as its average distance from the Sun?
Is there a better planet name for "pre-Earth", like there is for
Theia? It was after all another planet than Earth is.

Does this mean that pre-Earth was roughly 9 Moon masses (~10%) lighter
  than today's Earth? Or could much mass have been ejected from the
  Earth-Moon system?

I think that's likely about right.  In a nutshell, there's two basic questions here.   How much mass was lost as a result of the giant impact and how much mass was gained after the giant impact during the late heavy bombardment and following collisions.

We know that smaller bombardments on Mars has resulted in debris landing on Earth, so it seems highly likely that some debris was lost when Theia and Earth collided, but estimating how much is tricky and you'd really need to be well versed in the models to have a pretty good guess.

One theory is that Theia formed in an Earth Trojan point so it shared the same orbit around the sun that the Earth did.   Source   This would cause a slower impact than a more oblique approach where the velocity of impact wouldn't be much greater than the Escape Velocity of Earth.   If the impact velocity is only a bit above the necessary escape velocity and if a fair bit of energy is transferred into orbital momentum and heat, it's unlikely that a high percentage of debris would reach escape.

Now if it was a faster impact from a more oblique orbital approach or a head on collision that would create greater compession and rebound, those two scenarios would likely lead to more material escaping.   I can't give you precise numbers, this is more of general thinking about what might happen.

Again, according to Wikipedia about 20% of the mass of Theia entered the earth's orbit after the impact.    Orbital Velocity has to be at least Escape Velocity over the square root of 2 and less than escape velocity.   Source

So if we use 11 KM/s as an escape velocity estimate, anything that flew off the impact at less than 7.5 KM/s would have returned to Earth pretty quickly.   A bit above 7.5 and maybe about 9 or so KM/s would be low Earth orbit and about 10 KM/s high earth, with 11 KM/s, likely escape and enter in a near earth orbit around the sun.     You'd need a bit faster than that to have a truly outside of earth orbit.

What percentage of Theia flew off at those velocities is hard to say, but I think it's likely that much of the planet slowed down upon impact and not too much escaped the Earth's Hill sphere, but I'd be guessing if I tried to put percentages to that.

  Did the impact substantially and immediately change Earth's orbit,
  such as its average distance from the Sun?

Because it was a (I assume) a Trojan object, the 2 young planets would have shared the same orbit and so there probably wouldn't have been a big change.  Orbital momentum should be largely be conserved though some orbital velocity might be converted to rotational momentum, I don't think there would be a huge change.   Similarly, if any debris from the impact flew off in one particular direction, the remaining material would react with similar momentum in the opposite direction, but having much greater mass, the effect would be small.

Certainly some orbital change probably happened, but compared to the entire Earth, it was probably small.

Now, if the impact was different than theorized, that might change the answers quite a bit.    (My 2 cents anyway)


Whose name is Minkowski 92 bearing? Minkowski 92 is a protoplanetary nebula. But after which Minkowski is it named? Same question for Minkowski's object (part of NGC 541).

Rudolf Minkowski.

Its not likely to be the mathematician, Hermann. Nor the physiologist Oskar.  There's only one Minkowski who was an observational astronomer.

The paper describing M1-92 was https://iopscience.iop.org/article/10.1086/125855

The 2006 paper on Minkowski's object (Croft et al) cites Minkowski, R. 1958, PASP, 70, 143


Are those noise or stars? So there is this huge picture by Hubble.

Better yet, with the zoom tool

And if you zoom to the maximum level, you can see a seemingly uniformly distributed dots of orange and blue. Are those actually stars and maybe galaxies behind them and not just camera noise?

Furthermore I'm trying to get a grasp on how large portion of the sky this photo represents.

  At 3.4, the apparent magnitude of the Andromeda Galaxy is one of the brightest of any Messier objects, making it visible to the naked eye on moonless nights even when viewed from areas with moderate light pollution. Although it appears more than six times as wide as the full Moon when photographed through a larger telescope, only the brighter central region is visible to the naked eye or when viewed using binoculars or a small telescope.

Let's say this is about the half of the diameter of the galaxy so this photo is about ~3 Moons wide, is that correct?

This is most likely not noise.

Photographic noise is generally going to be limited to roughly single pixels. This is because much of it is generated at the individual pixel level (often a single color, so not even an entire pixel) on the sensor. There is also some relationship to this and overheating - I'm not familiar with the cooling systems of the Hubble, but presumably it's sophisticated enough to prevent serious noise from overheating.

When examining the photo, you can see that these are not individual pixels but rounded seemingly point-source light. It's hard to say whether this is an accurate representation of their actual angular diameter, as they may have acquired some of that rounding by diffraction. Regardless, this suggests that these are not artifacts from the sensor but are actual light sources. For a long exposure photograph, this also means that they are a persistent light source. The colors that they are composed of also fits the colors we would expect to see from stars.

Unless there has been some very heavy editing done with the intention of representing noise as stars, I think it's fairly safe to say that these are all stars or other star-like light sources.


What fraction of galaxies host active galactic nuclei? I've been saying "some". I can't find a reference for a particular percentage.

This seems to address the question to some extent; https://arxiv.org/abs/astro-ph/0108389 would "roughly 1%" be in the right ballpark for an answer then?

pesa points out this is redshift-dependent. I am looking for a ballpark figure of the fraction of AGN at $z < 1$.

As Pela said in the comment, it is highly redshift dependent so there is no straight answer. "Some" is the safest option.


How can I calculate moon phase and height? How could I know what phase the moon is in when it's high in the sky during sunrise?

How could I tell in general what its height in the sky will be at any given time?

The first question is quite simple if you imagine how the Earth is looked from above. To help you with that I've added a little drawing below.

Probably you know that the Sun rises on the East and sets on the West.  Imagine now how the sun illuminates the Earth: Europe is in the afternoon when the East cost of US is waking up. Imaging this you can see that the Earth is turning anti-clockwise if looked from the north.

Now, imagine you are somewhere in US (so we can use the drawing from below), the sun is rising (on the East) and the moon is above you (something close to the zenith).  This means the moon the configuration of the Sun-Earth-Moon system is something like in the drawing.  As you can see in the drawing I've kept in white what's illuminated from the Sun.  In Spanish we use to say that the moon is a bit of a liar, this comes because it shows a C shape when is waxing (decreciente) and a D shape when is crescent (creciente), so just the opposite letter of its action (don't know if there's a similar expression in other languages, and notice that the moon is not a liar for the Spanish speakers in the Souther hemisphere).  So, coming back to the drawing, you can imagine that from the red-star position you will see the moon with a C shape, therefore is waxing... which means in 7 days or so would be new moon so is going to move between the Sun and Earth.. and we can see how the moon rotates around the Earth in the same direction than the earth rotates around itself.

I don't really understand your second question, so I cannot help much with that.  But, if you mean to get a roughly calculation as on where the moon is at a certain time knowing its phase you can calculate it by knowing that the Earth spins 15°/hour and that the moon positions for each phase.  Therefore, new moon the moon will be closer to the zenith at noon (local solar time), 6am for waxing moon, 12 am for full moon and 6pm for crescent moon. However, this is just a roughly approximation as the moon movement happens on a plane which is 5.14° inclined from the ecliptic plane which corresponds to positions between 18.29° and 28.58° latitude on Earth.  So the same phase moon can be at different heights at different times of the year at the same hour.


What direct or indirect observations of dust can one make by eye or with binoculars? The solar system and space in general has plenty of dust, and dust has major impacts on observing both visually and in radio astronomy.

If I wanted to "see dust" in space, or see its effects, using my eyes or at most a pair of regular binoculars, where could I look and what would I look at in order to say to myself "yep, that's dust!" or a direct result of dust's properties in space?

NOTE: It doesn't necessarily have to be interstellar dust!

If you can see the Milky Way, then in Summer through early Autumn you should be able to see the "dark rift" which runs through the middle of the milky way from Saggitarius to Deneb in Cygnus. This is a dust band in the Milky Way.

If you live far enough South, you can see the Coalsack, a quite dense dust cloud near the Southern Cross, and also clearly visible without optics.


I am looking for the SKYMAP Api (To create nightsky/star maps) The ones used by underluckystars.com and thenightsky.com

I have found several but I am having difficulty finding a working updated version, has anyone come across this API, or know where I can find it?

Those vendors don't need an API, and neither do you.
What their "how it works" pages do talk about are star catalogs and map projections.

The Bright Star Catalog has about 9000 stars down to magnitude 6, as faint as most people can see in a dark sky without optical aid.
The Hipparcos catalog has about 100k stars down to magnitude 9 or 10.
The key data for a star map are equatorial coordinates (right ascension, declination) and V-band apparent magnitude.
For an all-sky map, which typically uses a stereographic projection to plot everything within 90° of the zenith (RA = sidereal time, Dec = geographic latitude), transformation to horizontal coordinates may be a useful intermediate step.

If you like working in Python, either Skyfield or Astropy/Astroquery can help you retrieve and process the data but will not plot the map.


What is the standard reference point for measuring speed? Speed, as we know, doesn't exist without first having a reference point. We then say that the reference point isn't moving at all, and speed is then measured in relation to the reference point.

What is the standard reference point when determining the speed of objects in astronomy? Earth?

As Einstein realized and like you correctly state, you indeed can't measure the speed of an object by itself since it has to be measured relative to something else.

As a logic result, if you ask the question "How fast is X moving?", you will have to specify that you want the speed with respect to another object because motion cannot be measured without a reference point.

Some examples:

If you ask how fast Earth is moving with respect to its own axis, the center of the Earth will be your reference point.
If you want to know how fast the Earth orbits the Sun, the Sun will be your reference point.
If you want to know how fast the Moon's distance from Earth increases, Earth will be your reference point.
If you want to know the speed of our solar system in the milky way galaxy, the center of the Milky Way will be your reference point.

"standard" reference point

If no reference point is given, it could be assumed that the "standard reference point" is the location of the observer. In the realms of astronomy, that would be the location of the astronomer.

It should be noted that the astronomer could well be on an space mission outside Earth's atmosphere, or he/she could be using a telescope in Earth's orbit. Therefore, "Earth" can not generally be assumed to be a standard reference point for astronomy because, depending on the precision of the measurements and the location of (for example) the telescope, assuming "Earth" may result in inexact measurements.

EDIT

As @TidalWave correctly commented, there's also the International Celestial Reference Frame (ICRS) which can help you find reference points, calculate distances, etc. according to a celestial reference system, which has been adopted by the International Astronomical Union (IAU) for high-precision positional astronomy. The origin of ICRS resides at the barycenter of the solar system.

Wrapping it up:

If no reference point is defined and if we can assume that the astronomer works according to the rules and definitions of the International Astronomical Union (which would be the regular, if not ideal case), the International Celestial Reference Frame provides you with a "standard reference point" at it's center (which is the barycenter of the solar system).

In rare cases where IAU compliance can not be assumed (something that "might" happen in amateur realms), it has to be assumed that the reference point is the point of observation.


Is a double quasi-binary system valid? I was messing around with 4-body problems, and discovered a system where a close binary and a regular binary pair orbit each other. The regular binary gets disrupted every few orbits and one star orbits the close binary, while the other orbits farther out. Then, as the outer star reaches periastron, the inner star gets captured by the outer star and the cycle begins again, shown in this video (the stars here are color-coded).
Is this possible and is stable in the long-term (without any external perturbations), and have we discovered anything similar to this?

As far as I know, all known old multiple star systems with stable orbits have highly hierarchical structures.
Binary star pairs in those systems orbit much closer to each other than they do to other binary pairs or single stars within the system.  In such a system it would be very unlikely for a star which is part of one binary to be temporarily captured by another binary and then return to its original binary.
Exceptions are multiple systems of very young stars which have formed recently.  The orbits in those systems are thus very new and have not yet been regularized.  It is believed that the stars in those systems will eventually move into hierarchical orbits or be ejected from those systems.
So I think that your system seems very improbable.  However, you won't know for certain unless you can run highly accurate simulations covering billions of years of evolution of the orbits in your system.


How Would a Neutron Star Actually Appear? Having seen many pictures produced by artists of neutron stars and planets that orbit some of them, I was wondering how a pulsar would appear to a human being, in visible light (assuming the intense radiation etc. doesn't kill us in the process).

As I understand, the pulsar's beam is projected from the star's magnetic poles rather than rotational poles, which are not necessarily in line with each other. Given that pulsars rotate extremely quickly and the beam could be visible across vast distances - such as if it were shining through the pulsar's nebula - would it appear as a straight line, curved line or perhaps a cone? This is assuming the beam can be seen in visible light.

Given the incredible density of neutron stars and their small physical sizes, would the night sky be visibly distorted to the point where (for example) just after sunset on a hypothetical planet, one could possibly observe other planets near or behind the star that would otherwise be blocked by it?

Given their small surface areas, would a neutron star still appear as luminous as say, the Sun, at a similar distance? How close would you have to get to a neutron star for its apparent magnitude to match the Sun's from Earth?

Your question is too general, you need to get to specific examples.

First, very few neutron stars are pulsars. Pulsars are either a brief phase during a pulsar's spin-down at the start of a neutron star's life, or they are the product of the spin-up of a neutron star in a binary system. Most neutron stars fall in neither of these categories.

A standard neutron star will look like any other star at a similar temperature. Most of them will be very hot indeed - 100,000 K or more, though the cooling histories of neutron stars are still uncertain and depend on some exotic physics. Such an object is "white hot" - it emits black body radiation at all frequencies visible to the eye (as well as lots more at UV wavelengths).

How close would you have to get for it's apparent luminosity/magnitude to match the Sun? Well that depends on the size and temperature of the neutron star. Most are thought to have a diameter of 20 km. The way you would do the calculation is equate the blackbody radiative flux per unit area at a given distance to the solar radiation constant of about 1300 W per square metre. However, there are two wrinkles for a neutron star: First, the radiation is gravitationally redshifted, so the temperature we measure is lower than the temperature at the surface. Second, General Relativity tells us that we can see more than just a hemisphere of the neutron star - i.e. we can see around the back - and this increases the flux we observe. These are roughly factor of two effects, so just to get an order of magnitude estimate, ignore GR and assume a 10 km radius NS with $T=10^{5}$ K.

Using Stefan's law for a blackbody, then at a distance $d$, we have that
$$\frac{4 \pi r^2}{4\pi d^2} \sigma T^4 = 1300\ W\ m^{-2},$$
where $\sigma$ is the Stefan-Boltzmann constant.

For $r=10$ km, then $d=7 \times 10^{8}$ m, which is coincidentally about a solar radius. Of course this distance depends on the square of the temperature, so a younger NS with $T=10^6$ K, then $d \sim 1$ au.

These are the distances where the total flux at all wavelengths would be similar to that from the Sun. To do the calculation just for the visible range we need to account for the bolometric correction, which converts a visual magnitude to a bolometric magnitude. The bolometric correction for the Sun is $\sim 0$, whereas the bolometric correction for a very hot star could be -5 mag. This means that only 1% as much flux from the hot neutron star emerges in the visible band compared with sunlight. This means that the distances calculated above, if we require the visual brightness of the neutron star be similar to the Sun, must be reduced by a factor of 10.

To turn to pulsars. Note that the pulsed radiation does have an optical component and pulsed optical radiation has been seen from a number of pulsars. Optical synchrotron emission would just appear to be a periodic, intense brightening of the pulsar, as the beam sweeps across the line of sight. If you were not in the line of sight, then you would not see the pulsed optical emission. If you could observe the beam passing through nebulosity or some other medium around the pulsar then yes there may well be some effects you could see in terms of ionisation or scattered light coming from along the beam path.

Lastly, the gravitational lensing effect. Yes, this should be strong close to a neutron star. The deflection angle (in radians) is given by
$$ \alpha  = \frac{4GM}{c^2 b},$$
where $b$ is how close the light passes to the neutron star and $M$ is the neutron star mass. Expressing $b$ in terms of the 10km radius of the neutron star:
$$ \alpha \simeq 0.83 \left(\frac{M}{1.4M_{\odot}}\right) \left(\frac{b}{10 km}\right)^{-1},$$
where strictly speaking this formula is only valid for $\alpha \ll 1$.

So consider a planet directly behind the neutron star at a distance of 1 au. The light from this would only need to be bent through an angle of $\sim 2 \times 10\ km/1\ au \sim 10^{-7}$ radians in order to be seen from a planet diametrically opposite at a distance of 1 au. So this is easily possible. However, the image would likely be highly distorted, especially if the neutron star was spinning. It would not look dissimilar to this simulated black hole image, but with a bright neutron star in the middle rather than a black disc.


What were the 2 satellites I saw this morning? This morning at 6:03am Israel time (3:03 UTC) I saw, simultaneously, what appeared to be 2 satellites streaking across the sky directly overhead. If I remember correctly, they were both coming from NW (but it was early).

Does anybody know what they were, or how I can find out?

I live in Mitzpe Netofa, lon/lat about 35.39/32.80.

The best way to figure it out would be to use the site "In the Sky" (https://in-the-sky.org/). You can enter time, date and location to identify satellites. This would at least be able to tell you if what you saw was a satellite.


Wher can I find a list of blueshifted regions in the Milky Way? It would be nice to get a relation between the blueshifted regions and the distance to earth. I'm searching especially for objects in other galactic arms.

I don't think there is a catalogue containing information about blue/red shift per region of the sky. You probably need a catalogue containing blue/red shifts per object and then calculate the average blue/red shift per region.
Most catalogues with blue/red shift are however targeted towards galaxies. They may also contain stars but you will have to filter the stars from the background galaxies. If I remember correctly, the Sloan Digital Sky Survey (SDSS) contains a flag specifying whether an object is a galaxy or a star. So you might use the SDSS data to get a large set of stars with blue/red shift. This flag has been set automatically using pattern recognition. I do not know how good the classification was.

You can access the data at the SDSS SkyServer. There are a lot of objects in SDSS so you will need time to acquire the data and knowledge about SQL to query the database.

Even then, you still don't have information about the spiral arm in which the object is located although you might use a method such as was used by Belokurov et al. (2006) to detect the field of streams to separate the different spiral arms.


North Pole Right Ascension/Declination to axial tilt conversion Wikipedia , on several asteroid articles, claims that right ascension and declination of the North Pole can give you the axial tilt of the object. What formula is used to convert these 2 values into the axial tilt? (Ceres, for example, has an R.A. Of 291 degrees and a declination of 59 degrees; its axial tilt is approximately 3 degrees. How is this figured?

This is wrong, but may help someone find the right answer:

You can convert right ascension and declination to a 3 dimensional
unit vector in the J2000.0 ICRF reference frame using the standard
formula for converting spherical coordinates to rectangular
coordinates and using a radius of 1:

{Cos[dec] Cos[ra], Cos[dec] Sin[ra], Sin[dec]}

The ra and dec of the north ecliptic pole is ra,dec of
{270, 66.5607083333} per Wikipedia, so the unit vector representing
the north ecliptic pole is:

{0, -0.3977771648286046, 0.9174820582119942}

As it turns out, there is conflicting data for Ceres, perhaps because
of the semi-recent DAWN flyby.

Instead, I'll use Vesta, where wikipedia explicitly states (https://en.wikipedia.org/wiki/4_Vesta#Rotation):

north pole pointing in the direction of right ascension 20 h 32 min,
declination +48 [... which] gives an axial tilt of 29 [degrees]

Converting to degrees, Vesta's north pole's ra,dec is
{308,48}. Using the formula above to find the unit vector, we get:

{0.411957936296447, -0.527282113378167, 0.743144825477394}

If we take the dot product of two vectors and divide by the product of
their lengths, we get the cosine of the angle between them. In this
case, both vectors have length 1, and the dot product is 0.891563
whose arc-cosine is right around 27 degrees.

So, if Vesta's orbit were the same plane as Earth's orbit, the axial
tilt would be 27 degrees.

However, since Vesta's orbit is inclined 7.14043 degrees to the
ecliptic, this answer is incorrect.

I don't think you can find the correct answer without using Vesta's
inclination and the longitude of its ascending node.

Both of these values are known, but I can't figure out how to use them
to get the correct asnwer.


Why do post main sequence stars enter the red giants branch? I am an early graduate student in astronomy and have hard time understanding why do post-MS stars move up the RGB.

Here is what I understand about post main sequence evolution of stars. As their hydrogen core is exhausted, the core shrinks under its own gravity. This lets a region of previously too cold hydrogen enter hotter regions, thus starting a hydrogen shell burning process. While this process can keep the luminosity constant, it expands the envelope, hence lowering the temperature and moving the star to the right in the HR diagram.

Then suddenly, the star starts moving up on the RGB: slight or no temperature change, but sudden increase in luminosity. What makes the star move up the HR diagram all of the sudden? This is before the helium flash when the core starts burning helium.

Thank you!

If your question is why the star starts moving up the red giant branch, it's in essence because of the behaviour of the surface opacity and the development of a substantial convective envelope in order to meet the surface boundary condition. It's basically the Hayashi track in reverse.

We can say this because if you create a model of a star and artificially suppress convection in the equations, then the model just keeps getting colder and colder at roughly constant luminosity, instead of ascending the giant branch.  See e.g. Fig. 2 of Stancliffe et al. (2009). Note that the star still becomes large in terms of radius, just not bright.


22° Sun halo today in Zimbabwe. But there's a moon like thing inside the halo beneath the sun that cannot be seen with the naked eye, what is that? Guys what is that moon like thing inside the halo?

It's a lens flare, caused by scattered sunlight within the camera's lens system. It looks very much like one of the examples on the Wikipedia page:

By Hustvedt - Own work


How can I map the zenith for an observer at a given time to equatorial coordinates (as would be measured in J2000)? I'm writing a script to generate a star chart for a given observation point (latitude and longitude) at a given time.  I've found a database of stars with their right ascension and declination from J2000, and with that written a program that displays a star chart with the center being any right ascension and declination that I feed into it.  I've also written a program that will convert local time into local sidereal time. Now I just need to figure out how to determine the zenith for an observer in J2000 right ascension and declination given their longitude, latitude, and local sidereal time.

I've found a wikipedia page that gives formulas for converting between celestial coordinate systems, but the formula for converting equatorial and horizontal coordinates doesn't seem to depend on the sidereal time, which I imagine it should for my case.

I've tried finding a vector that represents the observer's zenith at the given time, then rotating it about the earth's rotation axis by the sidereal time, but that's giving incorrect results.

Zenith right ascension = sidereal time.

Zenith declination = geographic latitude.


What is the difference between a moon and a random chunk in the rings Jupiter has 79 (known) moons, Saturn 82, Uranus 27 and Neptune 14 (numbers come from Wikipedia). These planets also all have rings. The rings are made up of chunks of rock and ice. There also are satellites in between the rings. What distinguishes a moon from any other chunk in orbit around a planet?

If a body is in orbit around a planet, how do you know if it's a moon or if it's part of the ring? Are there orbital characteristics that differ between the two? Is there a lower bound on size or mass for something to be a moon? Is it the density of surrounding bodies? Is there no formal (e.g from the IAU) definition?

Your question:

  There also are satellites in between the rings. What distinguishes a
  moon from any other chunk in orbit around a planet?

Wikipedia should be taken with a grain of salt, but they define moons or moonlets within a ring system as creating a gap or partial gap, sometimes described as looking like a propeller.

  In 2006, four tiny moonlets were found in Cassini images of the A
  Ring.[44] Before this discovery only two larger moons had been known
  within gaps in the A Ring: Pan and Daphnis. These are large enough to
  clear continuous gaps in the ring.[44] In contrast, a moonlet is only
  massive enough to clear two small—about 10 km across—partial gaps in
  the immediate vicinity of the moonlet itself creating a structure
  shaped like an airplane propeller.[45] The moonlets themselves are
  tiny, ranging from about 40 to 500 meters in diameter, and are too
  small to be seen directly.[9]

A moon, I would think, would need to clear out an entire gap within a ring.   Clearing out a partial gap or "propeller" as noted above can be as small as 40 meters.  Often the moonlets themselves aren't visible but the propellers they create in the ring system is visible.

Some more details and images here and here


What is the explanation for rapid inflation just after the big bang? I often hear about cosmic inflation occurring rapidly after the Big Bang, but have never came across any explanation for it. What supposedly caused this inflationary period in the early universe? Also, my assumption is that the laws of physics, i.e. light speed and relativity, did not exist during inflation, but were after effects. Is that a correct assumption? Thanks!

With normal matter, the strength of gravity depends on the stress-energy tensor, which, in an isotropic homogeneous universe, has trace of $\rho + 3p$.  Note that the pressure from relativistic particles (photons and neutrinos) enhances the mass density in slowing down expansion.   In inflation theory, it is assumed that there was a scalar field (perhaps) in a small section of the universe that began in a false vacuum state. That is, the field's energy density remained roughly constant as the universe began to inflate, and therefore the total energy of this field grew as the volume of the universe grew.  The field's pressure was negative and greatly overwhelmed the inward force of the density of normal matter and pressure. It is this negative pressure that rapidly expanded this section of the universe by many factors.
Energy was conserved because work was done by the gravitational field as it pressed inward against the field's outward pressure; the gravitational potential became very negative, while the energy density in the field became very large.  Eventually, the field decayed away, heating the particles, which resulted in the traditional Big Bang conditions of a hot expanding universe.
Throughout this theory, standard physics is expected to hold with just the addition of this new scalar field.  It is introduced because it helps to explain why the entire visible universe today is so homogeneous even though there are subregions that were previously thought to have not been in causal contact until quite recently.  It also explains why the universe is very nearly flat.  It converted a small section of the universe to extremely large radii and thus small curvature.
You can read about this from one of its inventors here


Retrogradation movement of planet Mars relatively from Earth by Copernic I  have a simple question : on the following figure:

I don't understand on the right figure why there is a progressive shift on the right when we start from step 1) to step 9). I guess there is an angle between the rotation plane of Mars and the rotation plane of Earth. Otherwise, we could'nt see clearly the recessing movement if the 2 planes were identical, could we ? We would just draw in the sky a simple line which would be the projection of the curve (on right figure) on a single Oy axis.
Anyone could explain me if this difference in two rotation planes is the cause of the shape of this curve (which is also due to the relative position of Mars from Earth) ?
Any help is welcome.

You are right.  If Mars orbited in exactly the same plane as the Earth, instead of an S or a loop, we would see Mars moving prograde relative for the stars along the ecliptic, then slowing and stopping, moving retrograde for a few months, as Earth overtakes it, still on the ecliptic, then moving prograde again.
But Mars doesn't orbit in the same plane, so it has some motion perpendicular to the ecliptic. when these motions are combined, the usual effect is a "loop" or sometimes an "S".
(note that the axial tilt of the Earth is not relevant here)


Alignment of Semi-Major Axis of Orbits So I'd like to ask whether the semi-major axis of planetary orbits are aligned? Logically, they shouldn't be, but many websites act as if they are. If they're not aligned, what's the angle between the reference xy axis and the semi-major axis of the orbits of planets?

Thanks!

The major axes of the planets are not aligned.

The major axis passes through the perihelion, the solar system barycentre and the aphelion. So by looking up the Longitude of perihelion we can find the direction of the major axis:

planet  Longitude of perihelion
M       77
V       132
E       103
M       -24
J       15
S       93
U       171
N       45

There is no evidence of alignment.


What is a dead comet? How is a Dead Comet different from the normal comet?

How are they formed?

And why is the Halloween asteroid 2015 TB145 called a dead comet?.

A comet is usually characterized by its tail. A dead comet has lost all its ices and gases (responsible for producing this tail), leaving just a rocky core.

The Halloween comet is such a dead comet, in that it has no tail, but furthermore it resembles a skull, making it particularly relevant for Halloween.


Has the Oort cloud moved? Planets have migrated throughout the history of the solar system, and some planets could even have swapped positions. Assuming the Oort cloud does exist, has it always been at its current distance from the Sun? Or did it used to be closer? Or further away?

The Oort cloud is more of a theoretical construct. It's a catch all, in effect, for small icy fragments anywhere from 2000 AU to over 100,000 AU.
From NASA:

The Oort Cloud lies far beyond Pluto and the most distant edges of the Kuiper Belt. While the planets of our solar system orbit in a flat plane, the Oort Cloud is believed to be a giant spherical shell surrounding the Sun, planets and Kuiper Belt Objects. It's like a big, thick bubble around our solar system, made of icy, comet-like objects. The Oort Cloud's icy bodies can be as large as mountains — and sometimes larger.

The inner edge of the Oort Cloud, however, is thought to be located between 2,000 and 5,000 AU from the Sun, with the outer edge being located somewhere between 10,000 and 100,000 AU from the Sun.

While the inner edge of such a cloud may have been closer or further in the distant past, as it has been disturbed by the planets, the idea of that region of space having icy planetesimals that may become comets doesn't really have any physical constraints other than those that Opik-Oort decided upon, unlike the Kuiper Belt, which does have measured boundaries.
Image of Oort Cloud from study.com:


Who discovered Wolf 359? I always assumed that Wolf 359's discoverer had been Max Wolf, but I just found out that he simply measured its high proper movement and included it in his star catalog.

Since Wolf 359 is a very faint star that can only be observed with a large telescope it is clear that it was unknown in ancient times and that some astronomer must have been the first to observe its existence.

Are there any data on who was the first observer of Wolf 359? If not, in which star catalogue or other map was it first included?

For me, it looks like Max Wolf discovered the movements of the Wolf 359, see the (German) original publication from 1918 "Zwei Sterne mit großer Eigenbewegung in Leo," but he does not mention which previous observations existed:

Fun fact: The object was observed from Königstuhl, a peak next to the city of Heidelberg, Germany. That was back in the days when light pollution was not an issue yet.


Examples of astrophysical objects NOT in virial equilibrium? Astronomers always talk about astrophysical objects being in virial equilibrium (e.g., a stellar system, or a disk of gas within a galaxy, etc.). But I never hear about thermodynamical equilibrium.

What are some examples of objects that are not in virial equilibrium?

One example question: suppose that what you thought was a disk of rotating gas within a galaxy (as revealed by spectroscopy) is actually a bipolar outflow of gas. Would the bipolar outflow of gas be in virial equilibrium?

Is a radio jet (length of hundreds of kpc) from a supermassive black hole in virial equilibrium?

Not clear what the initial part of your question means. Objects can be in virial equilibrium without being in thermal equilibrium.

A clear exception to the virial theorem would be any system that is gravitationally unbound. So you couldn't apply it to a supernova explosion or a dissolving cluster of stars.

The examples you quote are not self-gravitating, bound systems, so no, the virial theorem would not apply to them.


How many stars are there max and average per galaxy? What is the range of the number of stars possible in a galaxy? What is the rough average?

Googling leads to vague answers, things like "billions upon billions". But what is a more pinpointed set of numbers?

The term galaxy can not be given an approximate number of stars since our galaxy has a 27,000 light-years radius while a dwarf galaxy is typically 130 light-years. On the other hand, for and old elliptical galaxy, it can happen that some of the stars are already dead and the creation rate is almost zero, but since those are probably made out of 2 spiral galaxies colliding, the number would be around the sum of the two initial spirals. For some young spiral galaxies, 2-3  stars per year are being created and very few die.

So there are many factors that play a role. Ironically, for our galaxy the number of stars is pretty uncertain since we are inside of it, and we can't infer it from the apparent mass knowing how it rotates, because the total mass and the dark matter both play a role on it. Plus the star mass range can be as wide as 0.1 to 150 (!) solar masses. So, an order of magnitude for our galaxy? $10^{11}$  assuming the population of stars is consistent with the H-R diagram and that half of the population are binary stars.


What is the dielectric constant of a star? What is the dielectric constant of a star, especially its corona? Is it of order 1 or is it quite large?

How much do "impurities" (elements other than H and He) affect the dielectric constant?

Disclaimer
This not (yet) a full answer, since I only  focus on approaching part 1 of the question, namely the calculation of the dielectric constant of the corona.
Dielectric constant of the corona
The Sun's corona

... extends millions of kilometres into outer space and is most easily seen during a total solar eclipse [...]  measurements indicate strong ionization in the corona and a plasma temperature in excess of $1\,000\,000 {\rm K}$ much hotter than the surface of the Sun.

So let's dig a bit into plasma physics.
According to I. M. Podgornyi's article Determination of the Dielectric Constant of a Plasma, in the absence of a magnetic field,  the dielectric constant $\varepsilon$ of a plasma can be written
$$ \varepsilon(\omega) = 1 - {{4\pi {e^2}n} \over {{m_e}{\omega ^2}}} \cdot {1 \over {1 - i{{{v_{col}}} \over \omega }}}$$
where $\omega$ is the angular frequency of the electromagnetic wave, $e =  1.60217662 \cdot 10^{-19} {\rm C}$ is the charge of an electron, $m_e = 511 {\rm keV} = 9.10938356 \cdot 10^{-31} {\rm kg}$ is the mass of an electron, $i$ the imaginary unit.
The frequency of collisions $ν_{col}$ with charged particles is given by
$${v_{col}} = {{{{2.10}^{ - 5}}{n_i}Z} \over {T_e^{3/2}\left( {\rm eV} \right)}}.$$
Here, we have $T_e \approx 80\ldots90 {\rm eV} = 10^6 {\rm K}$ the temperature of a plasma given in electron volt, and (average?) $Z$ the charge number.
The next step towards an answer would be to figure out all parameters in this equation, in particular the average composition of the plasma (in terms $n$ and $Z$).
I did not yet manage that (yet), see also my question Physical properties of the stellar corona?
References
The following publications might be helpful for determining $n_i$ and $Z$ for some or all consitutents of the coronal plasma:

Mercier, Chambe: Electron density and temperature in the solar corona from multifrequency radio imaging could help us with $n_e$.

Bo Thidé: Radio Studies of Solar-Terrestrial Relationships shows a graph with the radially dependent plasma frequency of the solar corona at a given temperature.

V. G. Ledenev, V. V. Tirsky, V. M. Tomozov: High-frequency waves in the solar coronal plasma looks like saying something about $\omega$.

Related

What is the temperature of the solar atmosphere (the corona) and how is it measured?


Public Data Set on Atmospheric Absorbtion / Extinction Is there a publicly available data set for mean atmospheric absorption / extinction? I would like to be able to process and plot a graph similar to this image from NASA on wiki commons.

The HITRAN database appears to offer individual lines and might possibly be used to compute mean absorption at standard atmosphere. However, the process is unclear and the data search appears to be broken.

Spectral Calculator by GATS also appears to have done the calculations for various configurations pulling from the HITRAN database and using standard atmosphere, but access to transmittance and absorption plots is limited to subscribers paying $50 per month.

A lot of individuals seem to be generating or copying these plots, but very few citations are given. Furthermore, indications are that there may be a fair amount of variability depending on conditions so a plot based on standard or mean conditions looking upward or downward through the atmosphere would be helpful.

One source with absorption data worth mentioning is hosted by the Gemini Observatory at https://www.gemini.edu/observing/telescopes-and-sites/sites#Transmission
There are plain-text files with e.g. for the range 0.9-5.6 microns as measured on Mauna Kea.


How does a gravity slingshot actually work? From what I know of elliptical orbits, an object speeds up near the periapsis and slows down at the apoapsis, much like we learned in high school physics how a sphere would roll down and back up a valley in a frictionless vacuum: The height is inversely proportional to the speed.

The "gravity slingshot" maneuver we've seen in sci-fi and even used by our own spacecraft relies on the physics of hyperbolic orbits, where an object both enters and exits the orbit before making a single lap around the planet/moon/etc. Since gravity pushes the craft toward that body both while they're heading towards and away from it, shouldn't the craft's velocity be the same at (for example) 1 megameter before the periapsis as 1 megameter after? If so, then the gravity slingshot maneuver should only have the end purpose of redirecting the craft's trajectory, not of increasing its speed, as the name implies.

My understanding in a simple diagram:

Here's an intuitive understanding without math or physics explanations (others will provide that stuff here):

You are right that approaching and leaving the vicinity of a planet in itself adds up to zero effect. Gravity assist is the effect of being "dragged along" with the movement of the planet. If a spaceship approaches the planet from behind in its orbit, it will be dragged along and accelerated. If a spaceship approaches from infront of the planet in its orbit, the spaceship will slow down as the moving gravity field of the meeting planet pulls it backwards.


Does the Sun have permanent geographical features? Equivalent formulations of this question:

Would it make any sense to draw a map of the Sun?
Is the Sun heterogeneous with respect to latitude and longitude? (I know that it is heterogeneous with respect to altitude/depth)

I know that the Sun does not rotate uniformly, so any map would line up differently as time passes.  However, this does not rule out a slice of the Sun at a given latitude having permanent geographical features, since this slice does rotate uniformly.

Of course, even geographical features on Earth are not really permanent, they change on long time scales.  I don't expect the Sun to have any eternal features, I'm really just thinking about features that change on long time scales, or at least longer than a few months (e.g. not sunspots).

As you suspect the Sun does not have any permanent surface features. Up till 1951 the longest actually observed sunspot group lasted 134 days see http://adsabs.harvard.edu/full/1951ASPL....6..146P. That duration gives you an idea how long sunspot groups last. There are associated with coronal holes which would last about as long. However, the structure (shape size) of the group and any coronal hole would be constantly changing.

Temporary maps are made and there is a heliographic co-ordinate system, with a latitude and longitude.

The sunspots show the rotation of the Sun, and so a north and south solar pole can be found. North is in the same side of the ecliptic as the Earth's north pole).

Solar longitude is much more arbitary. The zero helographic meridian is defined as the plane (solar pole to solar pole) passing through where the solar equator crossed the plane of the ecliptic (earths orbital plane) at Greenwich mean noon on January 1, 1854. see http://wso.stanford.edu/words/Coordinates.html
Since then astronomers have just used a constant rotation rate based upon the suns equatorial rotation rate. These are called Carrington Solar Coordinates

A similar system is used for the longitude of the gas giant planets like Jupiter and Saturn.