Hubble law, cosmological redshift and distance I have got confused with some cosmology formulas which calculate distance.

The most famous formula is indeed the Hubble law which says (nowadays) velocity is proportional to Hubble's parameter at current time.
On the other hand we can find the velocity from redshift.

By Hubble law it is possible to evaluate distance but it's not clear for me that what does this distance mean...! As the photons, which are now observed, are belong to another redshift (larger) and by traveling toward Earth the have become "redder" because of expansion of universe, what does the observed redshift stand for?
The actual redshift of the galaxy at observation time (although we can't recieve the photons now), the redshift at emission time (that means we do not calculate current distance of the galaxy) or something else?

Thanks a lot.

I think that the wikipedia page on Hubble's law is reasonably clear. The distance in Hubble's law is the proper distance. This is the separation between two objects measured by observers at the same cosmic time. That is, if you imagine (instantaneously) stretching a load of metre rulers end-to-end from us to a distance galaxy, it is how many metre rules you would need. The velocity in Hubble's law is the rate of change of proper distance with cosmic time.

On large enough scales we find that that the ratio of velocity to proper distance is the Hubble parameter, which itself changes with cosmic time.

$$ v(t) = H(t) D(t)$$

In practice we cannot measure the velocity or proper distance in Hubble's law - we can generally only estimate these quantities for a galaxy as we observe it some time in the past, when it emitted the light we detect. This is why Hubble's law in terms of measured redshifts and estimated distances is only applicable over relatively small (cosmologically speaking) distances and for recession velocities much less than the speed of light.

For more distant galaxies the distance cannot be calculated from a redshift without a cosmological model for how the Hubble parameter changes with cosmic time. This in turn depends on the adopted cosmological parameters. See Evolution of the Hubble parameter

Proper distance is also discussed here.


Calculating the apparent magnitude of a satellite I'm writing a program that involves calculating the apparent magnitude of satellites from a ground location. I currently have the intrinsic magnitude of the satellites and the solar phase angle in degrees. I can't seem to find a formula that works.

I tried

magnitude = intrinsicMagnitude - 15 + 5 * Math.Log(distanceToSatellite) - 2.5 * Math.Log(Math.Sin(B) + (Math.PI - B) * Math.Cos(B));

(B is the phase angle)

...but it does not work (it's returning numbers like +30). I know it's wrong because I'm comparing it to heavens-above.com satellite passes.

intrinsicMagnitude = Visual magnitude at 1000km away (Use -1.3)

distanceToSatellite = Observer distance to the satellite in km (Use 483)

B = This is what I am trying to figure out.

In the paper it says what this is but it says some other things that I do not understand. The phase angle you use to get this should be 113.

The target output of this equation should be around -3.

This is for satellites with unknown size and orientation but known standard magnitude (Standard magnitude can be found on the satellite info page of heavens above, the number is called intrinsic magnitude)
The proper formula is

            double distanceToSatellite = 485; //This is in KM
            double phaseAngleDegrees = 113.1; //Angle from sun->satellite->observer
            double pa = phaseAngleDegrees * 0.0174533; //Convert the phase angle to radians
            double intrinsicMagnitude = -1.8; //-1.8 is std. mag for iss

            double term_1 = intrinsicMagnitude;
            double term_2 = 5.0 * Math.Log10(distanceToSatellite / 1000.0);

            double arg = Math.Sin(pa) + (Math.PI - pa) * Math.Cos(pa);
            double term_3 = -2.5 * Math.Log10(arg);

            double apparentMagnitude = term_1 + term_2 + term_3;

This will give the apparent magnitude of the satellite.
Note: I gave the formula in C#


Is it possible that the Sun and all the nearby stars formed from the same nebula? I was looking at a map of our local stellar neighborhood, and it occured to me, the stars are really close, if one compares them to the size of some nebulae. So can it be, that the Sun, Alpha Centauri system, and all the other stars in the neighborhood actually formed from the same nebula? Have we discovered any information about this?

There are three main reasons why we can tell that local stars did not, for the most part, form from the same molecular cloud that the Sun formed from.
The first is that unless stars are born in a very tightly bound system such as a globular cluster (which the Sun is definitely not in), they will drift apart from their birth companions over time in slightly different orbits in the Milky Way galaxy. The Sun has gone around the center of the galaxy about twenty times since it was born; this is more than enough time for its initial cluster to become completely scrambled in azimuth (the angle in the plane of the galaxy). To see how this would work, imagine two stars with very similar orbits, one with a period of 200 million years and the other with a period of 210 million years. If they start off right next to each other, then after 2 billion years, the first star has made 10 complete orbits, while the second has made about 9.5 -- meaning it will now be on the other side of the galaxy from the first star.
You can also take the current velocities of nearby stars, and see that they can be rather different, which tells you they are not on very similar orbits. Alpha Centauri is moving about 20 km/s in our direction, while Barnard's star has a velocity in our direction of over 100 km/s (in both cases, they also have movement in perpendicular directions; they're not moving directly toward us). So these are stars which happen to be near each other now, but are on different orbits and therefore were almost certainly not near each other at any given time in the past.
The second reason is the one that David Hammen pointed out: stars born from the same clouds should have very similar chemical compositions ("metallicities" in astronomical parlance).
The third reason is that stars born from the same cloud should have the same age (to within a few million years, anyway), since molecular clouds tend to be disrupted by the star-formation process and don't keep forming stars for hundreds of millions or billions of years.
Looking at some of the very nearest stars: the Sun has what we define as "solar metallicity" and is about 4.6 billion years old. The stars of Alpha Centauri are slightly more metal-rich than the Sun and probably several hundred million years older. Barnard's Star is more metal-poor than the Sun (maybe only 1/3 the iron fraction), and about 5 billion years older. Sirius is significantly more metal-rich than the Sun (its iron fraction is about three times that of the Sun) and is only about 250 million years old.
So you can see that the Sun and the nearby stars are a heterogeneous group, and clearly not all born at the same time from the same patch of interstellar gas.


What is the dimensions of the smallest object detectable by an optical fiber from a specific distance? Optical fibers are well-known to observe astronomical objects, say, galaxies, to generate massive spectroscopic surveys. The galaxies are often very far from the optical fibers of a telescope. However, I'm thinking about the possibility of using optical fibers to observe much smaller objects (compared to galaxies) like satellites in much shorter ranges like from earth to GEO orbit.

In particular, given an object at the distance $r$ from an optical fiber mounted at the focal plane of a telescope, how big (in terms of dimensions) that object should be to be detectable by the signal coming from the optical fiber?  (I couldn't find any related specification in the web. Since I need to use the answer as a part of my research, any cited reference is extremely appreciated.)

Assuming you are using single mode fiber, the entrance (mode diameter) is small, usually a few microns (extremes can range from 1 to say 30 microns, but 2-8 microns is common for visible SMF).

The only constraint: you have is to get enough light into that small spot. You can do that with a single lens or convex mirror, a microscope, or a telescope, or even by proximity (hold the fiber so very close to a tiny object that enough light reaches the core, works for certain nanoparticle or biological experiments).

Size of target object: So far I can not think of any constraint on the size of the object. Really the question is how bright the object is and how sensitive the instrument is at the other end. If you have a high-dispersion spectrometer, you probably need a lot more light than if you are doing broadband photometry by counting photons with a photomultiplier.

Object is moving: It doesn't matter if the object is moving or fixed, as long as your optical system can move to track it so that the focused image falls on the core of the fiber. That's simply a mechanical and tracking issue not related to the fiber.

The big advantage of using a fiber is now obvious, you can move your telescope, but keep your measurement instrument somewhere else and fixed, and the flexible fiber will guide the light to it.

Further background

The trick is to put a very large telescope between the object and the fiber, and to use a robot to place hundreds or thousands of fibers at the focal plane, each at the exact location of a galaxy or a star you would like to collect. This way you can line up the other ends of the fibers along a single slit of a spectrometer and study the spectra of thousands or even millions of objects over time without having to point the telescope to each object one-at-a-time.

Fibers have some big advantages:

Multiplicity and configurability (as mentioned above)
Flexibility (literally) it allows the telescope and the instrument to be mounted separately and not have to be mechanically aligned and stabilized.
Mechanical stabilization of source for instruments: even if the telescope drifts and the object moves slightly with respect to the entrance of the fiber, the physical location of the fiber exit can remain fixed. So tracking errors don't affect the distribution of intensity within the entrance slit (virtual or real) of the fiber and therefore don't cause instrumental spectral shifts.

above: From the astrophysicist Ángel R. López-Sánchez's enjoyable A 2dF night at the Anglo-Australian Telescope

above:: From AAO's The stellar cluster behind Sirius

above: GIF from the video (with exciting music) A 2dF night at the Anglo-Australian Telescope

And if you like that robot then consider the "magic fingers" at the focal plane of the Dark Energy Spectroscopic Instrument:

  DESI’s focal plane is designed to sit high atop the Mayall Telescope and to carry 5,000 robotic positioners, each holding a fiber-optic cable. Each of these fiber-toting robots is automatically positioned to fix on a preset sequence of individual galaxies and quasars so that the fibers can collect their light. The movements of these positioners must be carefully choreographed to avoid bumping into one another. The roundish focal plane, which measures nearly a meter in diameter, consists of 10 pie-shaped wedges that are fit snugly together. Each wedge holds 500 robotic positioners. The focal plane also contains sensors and light sources called field fiducials that help ensure the positioners are in proper alignment.

And also these robotic fibers in Australia's TAIPAN instrument at Siding Spring Observatory in North-Western NSW:


Is every star formed in an open star cluster? As far as I understand, an open cluster is formed from a single molecular cloud, with each star in the cluster having roughly similar age and properties.

Our Sun is not part of any star cluster, but it could have been formed in one. Is every star formed in an open star cluster, or can they develop independently?

The question is still an open matter of current research.

It seems to be true that the vast majority of star formation takes place in groups and aggregates of various sizes - from a few stars to millions of stars in "super" star clusters.
This is likely because collapsing clouds of gas are normally much more massive than a star and the collapse process reduces the Jeans mass and renders the cloud unstable to fragmentation into smaller cloud cores.

However, it seems that the vast majority ($>90$%) of star clusters/associations are either born in a gravitationally unbound state or become gravitationally unbound within a few million years. The gravitationally bound open clusters, who's exemplars include the Pleiades, are comparatively rare survivors (or at least partly survived) of this "infant mortality". So, in that sense we can say no, most stars are not born in open clusters, but the likelihood is that most were born in aggregates with near neighbours that went their separate ways shortly after birth.

Current thinking is that our Sun was born in a cluster of some ten thousand stars (Adams 2010). This is an argument based on the shaping of the early solar system by dynamical encounters and the early presence of radioactive nuclei that were likely injected by the explosion of a very nearby massive star (probably a cluster sibling).


Is the object a pulsar We have a star having 1.8 solar mass. Justify any four means (measurable/ observed properties) by which you can identify that the object is a Pulsar and not a white dwarf or a Black Hole.

So one obvious property would be the mass. Since mass > 1.4 solar mass, we can say it is not a white dwarf.

Also the mass is less than 3 solar mass, hence its not a black hole (I'm speaking loosely over here).

Also from periodic radiation patterns, we can say that the object might be a pulsar.

I need help with 2 other means to categorize this object.

As you said, the mass range of the object indicates that it could be a neutron star, rather than a black hole.
But also the density of the object is a clue. The density of a neutron star is much higher than that of a stable white dwarf.
Extremely regular electromagnetic pulses have to be observed. Known pulsars have a pulse period between 1.4 ms and 8.5 seconds.
The wavelength of the pulses is in the radio range.
Pulsars slow down gradually, at a known rate, so you can measure how the pulse period evolves over time, and check that is compatible with the slow-down rate of a pulsar.

Adapted from this page


Which redshift is used to determine the Hubbleconstant? I think they measure cosmological redshift to use in the Law of Hubble-Lemaître together with the distance to calculate $H_0$. Is this correct, or do they use Doppler shift (too)?
$H_0$ indicates how fast the universe is expanding, so I find it logical that we measure cosmological redshift. But the galaxies are receding (due to the Hubble Flow) away from us, so do they gain an additional Doppler shift?

The term "Hubble flow" refers to the homologous expansion of space and the resulting recession of all galaxies from each other (if they're not close enough to be gravitationally bound). This effect causes the "cosmological redshift", i.e. the redshift that light from distant galaxies attain as it travels through space.
In addition to this motion away from each other, galaxies have a so-called peculiar velocity, i.e. a motion through space. This motion adds an additional Doppler shift to the cosmological redshift, either to larger or smaller wavelengths, depending on the direction.
Whether or not you see these two types of redshift as fundamentally different, is not trivial I think. In most textbooks, they are described as two different things, the former having to do with the dynamics of the fabric of space, and the latter having to do with motions of emitters and observers. But in fact it may not be so different. For instance the Welsh cosmologist Geraint Lewis argues that, in some sense, the cosmological redshift can be interpreted as the sum of infinitely many infinitesimally small Doppler shift (Lewis 2016). On the other hand, the American physicist Sean Carroll argues that the notion of expanding space is nevertheless an extremely useful concept (Carroll 2008).


Fusing and buring of Helium core/shell In the book "Horizons_ exploring the universe-Cengage learning (2018)", p196, it states that

The nuclear reactions in a main-sequence star’s core fuse hydrogen to produce helium. Because main-sequence stellar cores are
cooler than 100,000,000 K, the helium can’t overcome the Coulomb barrier to fuse in nuclear reactions, so it accumulates at the
star’s center like ashes in a fireplace.

But if the temperature is not high enough to fuse helium, then how does the ending stage of a main sequence star is by burning its helium shell? Is fusing different from burning? If so, why doesn't a helium core star burning its core, even if it cannot reach the temperature about the Coulomb barrier?

The helium core becomes significantly denser and hotter after a star has left the main sequence.
Providing the overall mass of the star is greater than about 50% that of the Sun, the helium core will become hot enough to begin fusion whilst the star is a red giant.
From there, the star will pass through phases of burning helium in its core, then hydrogen and helium in shells around an inert core of carbon and oxygen.


Is there a way to estimate or calculate the tidal range induced on a water-bearing planet? Consider a system in which a central star is orbited by a planet with liquid water oceans, which is itself orbited by a moon.
Given the masses and distances between these three objects, is there some formula that outputs the minimum and maximum tide heights the planet's oceans cycle through for every orbit of the moon?
For simplicity, the effects of local topography on the tides are being ignored.
Also, if there is such a formula, could it be applied to solar systems in which there is more than one central star and/or more than one moon orbiting the planet?

Wikipeida (quoting Icarus ) gives
$$\frac{15}{8}\frac{mA^4}{Mr^3}$$
Where $m$ and $M$ are the masses of the moon and planet, respectively; $r$ is the orbit radius of the moon and $A$ is the radius of the planet. For Earth this is a little less than a metre.
Systems with multiple moons (or moon and sun, like on Earth) will have multiple bulges which can add up.
This tells you very little about the sizes of actual tides, which are strongly magnified by topography. It predicts tides of 0.7 m, whereas real tides have ranges of between 0 and 16 m.


SDSS Image FITS files have negative values. What do these negative values mean? I am new to using SDSS image files and have noticed that some of the values in the FITS file are negative. Can somebody explain to me what these values actually mean?

Without more information about the specific nature of the fits file, my immediate guess is that you're using a file that has already been corrected for things like bias and flat fielding.

It is true that for a simple exposure (what we might call the "science" image), the fits image should have only positive values as each pixel is a (proxy) count of how many photons hit it and how can you have a negative number of photons hit your detector? But often astronomers need to clean up their science image from various sources of noise. One such example is cleaning it with a bias frame. A bias frame is an exposure with the shutter closed and the exposure time set to zero. Essentially you just pull the signal from the CCD without taking a picture. In a perfect world, the CCD will be blank and empty and there will be nothing to read out if you never exposed, but in the real world things are messy and you have lingering electrons all over the place. Reading a bias frame allows you to get some estimate of this noise that can inherently exist in a CCD. An astronomer will then take this bias frame and subtract it from the science image to remove this noise, but of course this can result in your science image having negative pixel values. This usually isn't a problem though.

This resource looks pretty comprehensive about the basics of cleaning science frames.


Was the cosmic microwave background (CMB) radiation originally gamma rays? I believe that sometime after Big Bang the entire universe was filled with high energy particles. Now we fast forward to the present; most of these particles still linger and managed to avoid being absorbed, and also they lost quite a significant amount of energy and became today's "noise" which can be picked up by our TV antenna. Is this true, or am I mistaken?

The cosmic microwave background is a result of an almost perfect blackbody emitter. That means the spectrum covers a broad wavelength range with a peak that is given by Wien's law:
$$\lambda_{peak} = \frac{2.9\times 10^{-3}}{T},$$
where the wavelength is in metres and temperature in Kelvin.

The microwave background is formed at the epoch of (re)combination, when protons and electrons combine to form hydrogen atoms and the Uinverse effectively becomes transparent to the radiation field within it. This occurs at approximately 3000 K. So at that time, the cosmic microwave background had a peak wavelength of 970 nm, which in the near infrared part of the spectrum.

The epoch of (re)combination occurred at a redshift of approximately $z=1100$. What this means is that since then, the wavelength of the photons has been stretched by a factor of $1+z = 1101$, so that $\lambda_{peak} \simeq 1$ mm, and this corresponds to a temperature of $T=2.7$ K. I.e It is near infrared photons emitted at the epoch of recombination that have their wavelengths stretched by the expansion of the universe, and are seen today as microwaves.

However, prior to the epoch of recombination, the Universe would still have been full of radiation, and as we go back in time towards the big bang, that radiation field becomes hotter and hotter, and the peak wavelengths get shorter and shorter.

If we look at models for the thermal history of the Universe, then at about 1 second after the big bang, the Universe would have been at about $10^{10}$ K and the radiation field would have been in equilibrium with the matter at this temperature. The formula above then gives $\lambda_{peak}=2.9\times10^{-13}$ m, which is definitely in the gamma-ray regime. But let me emphasize that these photons are emitted then absorbed many times and we do not see them now.

The "particles" that you are talking about are photons of light.


Cancelling out earth rotation speed, Altazimuth mount I have a Dobsonian telescope.
It is using Altazimuth mount.
Basic idea of using it is to target the object by moving telescope vertical axis perpendicular to the ground, and an elevation axis that is parallel to the ground.
I have installed two step motors to automate the movement along both vertical and elevation axis.
I would like to find out how can I cancel out earth rotation speed by simultaneously moving both vertical axis and elevation axis motors.
Idea behind it is to point telescope at the object and press the button.
Then the step motors driver software would follow the object as the earth rotate.
I will quote few lines from The Basic Astronomical Telescope Mounting Designs to help me explain what I am trying to achieve:
[Using altazimuth telescope...:]

If you happen to be observing from the North or South Pole, the vertical axis would be aligned with the Earth's spin axis. The nice thing about that would be that when you found an object to observe, rotation in only the vertical axis would be needed to keep the object in the field of view. Rotating at the Earth's spin rate in the opposite direction as the Earth's rotation would keep and object motionless in the eyepiece.
However, for any other latitude on the planet, the vertical axis is not aligned with the Earth's spin axis. This means that to keep an object in the field of view requires motion in both axes. The motion rates will change over time as the elevation angle changes. Tracking objects near the horizon requires mostly changes in elevation, and tracking objects more straight up requires mostly changes in azimuth.

I need to find mathematical algorithm that will help me solve the problem described in the second paragraph.
Hope this is clear.

Given ra, dec, lat, lon in radians, and d in number of fractional days
since '1970-01-01 00:00:00 UTC', the azimuth of a star is:

$-\cot ^{-1}(\tan (\text{dec}) \cos (\text{lat}) \sec (\text{d1}+\text{lon}-\text{ra})+\sin (\text{lat}) \tan (\text{d1}+\text{lon}-\text{ra}))$

and the elevation is:

$\tan ^{-1}\left(\frac{\sin (\text{dec}) \sin (\text{lat})-\cos (\text{dec}) \cos (\text{lat}) \sin (\text{d1}+\text{lon}-\text{ra})}{\sqrt{(\cos (\text{dec}) \sin (\text{lat}) \sin (\text{d1}+\text{lon}-\text{ra})+\sin (\text{dec}) \cos (\text{lat}))^2+\cos ^2(\text{dec}) \cos ^2(\text{d1}+\text{lon}-\text{ra})}}\right)$

where d1 is: $\frac{\pi  (401095163740318 d+11366224765515)}{200000000000000}$

(you will need to resolve the ambiguity in the arc(co)tangent, but
this isn't difficult).

These formulas aren't as daunting as they seem, since, for you, lat,
lon, ra, and dec will be fixed, and the only thing that changes is d.

Hope this helps, but I'm worried that it just demonstrates how complicated these formulas are.


Calculate the orbital phase when plotting the Radial Velocity data of a transitting exoplanet I'm not quite sure how I find the orbital phase  ($\phi$) of a transitting exoplanet. For example, I have this RV data in this article:

First of all, I don't know how that phase or "cycles" is derived. But that's not the phase $\phi$ I'm looking for but this (in the same article):

As you see, that phase goes from 0 to 1, and I know it has to do with the time of passage from the periastron and the data time (the HJD column), but I'm not able to derive this number in the interval [0,1]. So, if we assume the orbital parameters are known, how can we derive that number?

The phase as plotted on the graph is constrained in the range [0,1) to plot everything in one orbital cycle (this makes it easier to see the shape of the RV curve). The phase can be obtained by taking the fractional part of the total number of cycles. So the first row in the table with the cycles number 28.677 corresponds to phase 0.677.

Update: As to what the cycles number is, it is a measurement of time since a reference time (usually the transit time), counted in units of the orbital period of the planet. In the case of a transiting planet, the orbital period will be known from the transit interval, so it doesn't need to be derived from the RV curve.

In this case, the first row in Table 1 has time $t = \mathrm{HJD}\ 2455107.37937$, while in Table 2 the transit time is given as $E = \mathrm{HJD}\ 2454967.27571$ and the orbital period as $P = 4.885525\ \mathrm{days}$. So the number of cycles can be calculated as $\mathrm{cycles} = (t-E)/P$ giving 28.677 (to 3 decimal places).


What is exactly the "longitude of the perigee" Some moon phase calculation algorithms (apparently derived from Duffet-Smith's book, example here) seem to use a parameter called "longitude of perigee at epoch". What exactly is this?

Can I assume this is the same as the longitude of the periapsis? Can I just calculate it adding the longitude of the ascending node and the argument of periapsis together?

Unfortunately, all the data that I find on this parameter is either related to Duffet-Smith's book or algorithms based on it. So, I have no idea on how this can be calculated for a specific epoch.

Perigee is the Earth-specific name for periapsis. People use longitude (which is a composite angle rather than an angle) because this solves the problems of circular and equatorial orbits.

The reason you need to use an epoch time to specify the Moon's argument of perigee and longitude of ascending node is because the Moon's orbit about the Earth precesses. Neither the argument of perigee nor the longitude of ascending node is constant. Instead, they are functions of time (and hence, so is longitude of perigee).

In the case of an inclined, non-circular orbit, "longitude" is the sum of the right ascension of the ascending node (which is measured on the fundamental plane of the reference system) and some angle (or angle-like measure such as mean anomaly) on the orbital plane. Thus longitude of periapsis (or longitude of perigee in the case of an object orbiting the Earth) is the sum of the object's right ascension of ascending node and it's argument of periapsis.

The use of "longitude" as orbital elements are for two reasons: To be able take advantage of Hamiltonian mechanics (this was done by Delaunay, resulting in the Delaunay orbital elements), and to address problems related to orbits with very small inclinations and/or very small eccentricities (this was done by Poincaré, resulting in the Poincaré orbital elements). Anomaly and longitude are synonymous in the case of an orbit with zero inclination.

While it is invalid to add two angles on different planes and treat the sum as if it were an angle, it is not necessarily invalid to treat that sum as what it is, a composite angle or dogleg angle, which is exactly what the Poincaré orbital elements do. ("Dogleg angles" aren't angles. Analogous terms include "dwarf planets", which aren't planets, and "red herrings", which oftentimes are neither red nor herrings.)


Do planets form around lone stars, multiple stars, or both? Do we know this yet? Since about 1990 astronomers been able to detect planets around other stars, using a couple different techniques, which is amazing.

By this point, do we know whether planets form only around single systems, only around multiple systems, or both??

(Surprisingly you can't find this general issue explained in popular writing. I think it's "obvious background knowledge" to experts, hence just doesn't get outlined explicitly in popular writing.)

So, as of mid 2016:

(i) About how many exoplanets do we know of (say, with good certainty),

(ii) In fact, how many (if any) of these are around single-star systems?

(iii) How many (if any) are around multiple-star systems?

If indeed it is possible to have planets in multiple systems: is it just a planet going around one of the stars, or is "all the way around both" known?

There are currently 3424 planets identified (including those in the Solar System and unconfirmed exoplanets). Of these, 3310 are confirmed exoplanets.

Out of the 3424, 180 are in binary (or multiple) star systems. Of those, 28 are circumbinary. I am not certain if any circumternary planets have been identified, though I know some circumternary disks have.

It is more difficult to identify circum-n-ary planets than those that only revolve around one star, so these numbers might not precisely represent the actual proportions.

Source:

Open Exoplanet Catalogue


How to convert ra&dec between different epoches? For example, how to convert B1950.0 into J2000.0?

Do we need to know the proper motion of the star and the time it was observed?

The difference between B1950.0 AND J2000.0 is not really concerned with proper motion., they refer to the definition of the coordinate system. Thus a star with zero proper motion would still have different B1950 and J2000 coordinates.

Usually (but not always), if you quote a position as equinox 1950 you are implicitly assuming that this is epoch 1950 (unless otherwise stated). And similarly for J2000, the implicit assumption is that the coordinates refer to epoch 2000 unless you specify otherwise.

Thus there are two things you needs to do. One is to apply a correction for (roughly$^{*}$) 50 years of proper motion (if indeed you only have the epoch 1950 position; or maybe 66 years if you want the coordinates of the star now). Second is to apply (roughly) 50 years of precession to the coordinates to put them in the J2000 system.

This description seems reasonable.

$^{*}$ There is some detail about B= Besselian years and J = Julian years which you can read about here.


Oldest Reference to Astronomical Seeing I'm writing a paper on astronomical seeing. Sir Isaac Newton identified both the phenomenon and origin of astronomical seeing in his Opticks. He writes:

If the Theory of making Telescopes could at
length be fully brought into Practice, yet there would
be certain Bounds beyond which Telescopes could
not perform. For the Air through which we look
upon the Stars, is in a perpetual Tremor; as may be
seen by the tremulous Motion of Shadows cast from
high Towers, and by the twinkling of the fix'd Stars.

is anyone aware of any older reference from Gilbert, Galileo, Harvey, Ptolemy, Copernicus or Kepler?

Robert Hooke's Micrographia from 1664 has a detailed discussion of "seeing"

The table [of contents], which is at the end of the book, explains that pages 230-232 discuss:

  that the Air near the Earth is composed of parts of differing density...this property produces the effects of waving and dancing of Bodies; and of the twinkling of the Stars


Why is the detection of gravitational waves such a "Big" deal? I'm a non-scientist. Today's announcement of LIGO's detection of GW waves has been heralded as "huge" –on the same level as the Higgs Boson detection. I realize this is the last remaining part of Einstein's theory of general relativity to be directly observed. Is that why it's such a big deal? Does this mean the theory is now on pace to become the  law of general relativity?

Side note: I had the privilege to visit and take a general tour LIGO in WA in 2000 or so and found it quite fascinating from a layman's pov. I've been waiting for this day ever since.

The way I see it, there are three reasons it's such a big deal.

The first, as you say, is the (further) confirmation of Einstein's theory of gravity. Newtonian gravity doesn't have gravitational waves. Their existence was already quite established by Hulse and Taylor's discovery and analysis of the binary pulsar PSR B1913+16.  The system's orbit is decaying almost exactly as predicted by General Relativity.  (Note that in the classic figure, the curve is not a fit: that is the prediction!) The direct detection of gravitational waves, however, better confirms that the waves behave as we expect they would, in the sense that the observations match the so-called "chirp" that we expected.

The second reason is because this is now a new way of measuring things in the Universe. Note that the announcement told us (roughly) the masses of the two black holes.  It's actually very difficult to weigh black holes!  For example, this 2010 paper discusses constraints on 23 stellar mass black holes, all from the fact that they're in binary systems, and the motion of the partner can be observed.  Even then those are mostly mass functions that still depend on the inclination of the orbits, which is often unknown.  So the gravitational waves give us a nice measurement, not just confirmation that the phenomenon exists.

With the method proven, there are some fascinating potential applications.  One of my favourites is that LIGO (or similar) could potentially detect the formation of a black hole in a core-collapse supernova.  Just like the detection of neutrinos from SN1987A confirmed some of our basic ideas about these events, detecting the gravitational waves from the birth of a black hole would inform our understanding further still.  And it might even tell us, for example, how massive the black hole is at birth.  There's no end to the novel ways that our theories could be tested!

The final reason is simply the astonishing technical achievement.  They measured motions of at most a few thousandths of a femtometer over the 4km arms.  That's mind-bogglingly precise.  Hopefully this will give more impetus to other gravitational wave detectors.  Here's hoping that eLISA will fly!

The parallel with the Higgs boson is quite apt.  In both cases, we built a big machine to find something that we were pretty sure should be there. (Though the LHC was a tad more expensive...)  Now it's a case of using the new tool to observe the Universe, and seeing if the waves behave as we think they should. And who knows, maybe some completely unexpected signal will appear to confound us completely.


Do most planetary systems have fewer planets than the solar system? It is my impression that most planetary systems have, at most, three to four planets. Is that real? Or is it just that we lack the ability to detect most planets?

Perhaps the way to answer this is ask - could we detect the planets in our solar system if we were looking at the Sun, using current technology, from distances of many light years?

The short answer is that we could detect Jupiter using the Doppler radial velocity technique, if we observed for more than 10 years (at least one orbit is required). If we were lucky, and the orientation is right, we might then also be able to detect a transit of Venus or the Earth, using a satellite observatory like Kepler. Kepler could detect Earthlike planets by the transit technique, but the solar system is not "flat" enough that you would observe multiple transiting planets.

So the answer is that we would currently have seen Jupiter and maybe one other planet. Therefore we cannot at the moment conclude that 8 planets is an unusually high number; it may be quite typical. Although we do know that solar systems can be much more densely populated with planets (in their "terrestrial planet zones") than our own.


How much mass do the volcanoes of Io erupt past escape velocity? While trying to do some calculations to answer this quesion, I got stuck missing a crucial piece of data: I have no clue how much mass the volcanoes of Io manage to throw out of the gravity well of the Jupiter moon.

It is clear that most of the erupted material falls back to the surface, and the various velocity estimates I have found are quite a bit lower than the escape velocity (~2.56 km/s). For instance "up to 1 km/s".

One number that is possible to find is how much mass the magnetosphere of Jupiter strips of Io. Most are listing that as 1000kg/s (example). That is about as much as the wheat production of Canada.

Are the volcanoes directly throwing away a lot more than that? A lot less? Not at all?

The velocity required to escape the gravitational attraction of a massive body is given by the following equation:

$$ v_{\mathrm{escape}} = \sqrt{\frac{2GM}{R}}$$

where $G$ is the gravitational constant ($G = 6.67 \times 10^{-11} \; \mathrm{Nm^{2} {kg}^{-2}}$), $M$ is the mass of the body from which you are escaping, and $R$ is its radius.

Inputting the values for Io's mass and mean radius, $M = 0.015 \, M_{\oplus}$ and $R = 0.286 \, R_{\oplus}$,1 we get an escape velocity of

$$ v_{\mathrm{escape}} = 2560 \; \mathrm{m/s}$$

However, the explosive ejecta are ejected from the top of volcanoes, so we should strictly add this to our radius. The tallest volcano on Io is approximately $2.5 \;\mathrm{km}$ above the surface; including this we get a marginally lower velocity

$$ v_{\mathrm{escape}} = 2559 \; \mathrm{m/s} $$

which is higher than the $\sim 1000 \; \mathrm{m/s}$ maximum velocity of ejecta calculated in McEwen & Soderblom (1983). Therefore, no mass is ejected from the surface of Io through volcanic eruptions.

For comparison, the earth's escape velocity is much higher, $11.2 \; \mathrm{km/s}$.  As discussed in the linked paper, the most extreme ejecta can reach heights of $500 \; \mathrm{km}$ before falling back to the surface.

1where $M_{\oplus} = 5.972 \times 10^{24} \; \mathrm{kg}$ and $R_{\oplus} = 6371 \;\mathrm{km}$


The contents of potential "message" in the Wow! signal lost? The Wow! signal is still an enigma.

I understand the incredibly high signal strength it entails. This during some time-period because the sweep of the alignment (or rather the receiving station). That could explain the rise and falloff. This because the horns tracking of that part of sky, once, because of earths movement.

The signal never returned. Everybody understands the narrow band of the signal, and the frequency of the hydrogen line. This incredible signal strength was logged, but only that, the spectacular strength.

But the contents of said signal was not logged, it was not even capable of recording that. Is this right? Is there a possibility that the burst could have carried information, however unlikely? It could have carried a one-time "message"? Is this correct?

I'm not an expert in this, but it's a fun little blip in the history of SETI.  Pretty much the only blip I think.

  I understand the incredibly high signal strength it entails.

I wouldn't call it "incredibly high".  It peaked at 30 times normal. source, and that's inside the "waterhole" a frequency range where the background radiation is the lowest in the cosmos.  30 times a normally very low signal is still pretty small, I think.  It was unusual, no question, but looking at the sheet of paper most often reprinted in articles about this, each space is 12 seconds, there are a handful of visible spikes.

Just on that one printout there's a 3,3,2,space,7 and a 3,6,1,3 and a 4,3,4,4
The Wow, 6,E,Q,U,J,5 or 6,15,26,30,20,5 compared to those numbers is maybe 6 or 7 times larger than some fairly common spikes.   Statistically speaking, that's high, but not enormous.  It's still an unexplained oddity, but it wasn't an "incredibly loud" signal, just unusually loud.

In 1977 they couldn't turn the dishes, so they had to wait 24 hours to scan that part of the sky again and 24 hours later, and every day since, and no similar signals have ever been found, so yeah, it's a mystery.

The way they had it set up, they couldn't tell which dish either, so there's 2 tiny slices of the sky that it might have come from, or it could have (theoretically) come from a ship of some sort transmitting it's location back to it's home planet, and the earth just passed through (or it could have been a mouse chewing on a wire, or a Russian prank) . . . it could have been lots of things.

In 1980 they set up the ability to have the dishes track a specific part of space, which would have come in handy in 1977, but new new signals of that strength were ever heard.

And, you're right, the big ear wasn't designed to record.  As I understand it, the system was designed to eliminate background signals, so if both dish's caught similar wavelengths while pointing at slightly different parts of the sky the duplication could be eliminated.  It was what one received that the other didn't that was printed on paper, but no way to tell which dish did the receiving.

Some cool pictures and write-ups about the Big ear here and here.  It was a monster of engineering for the 1950s when it was designed but it's primitive by today's standards.   The Wow signal has no triangulation to verify if it was from outer-space vs a closer event, and they couldn't turn their equipment to track it.   It was initially designed to look deep into space, by the way, not to record alien signals.   Seti began using it in 1973 after it was somewhat obsolete for more traditional astronomical pursuits, but throughout the 60s it was one of the better observatories.  Another problem, it didn't detect a wide range of radio waves, just a tight band, which is good for looking for a signal, but less good for eliminating rare stellar events that might have spanned a broader range of radio-frequencies.   Some possibilities here for stellar explanations and without triangulation, the possibility of rare atmospheric reflection of some kind seems possible too.

The 6 data points looks like a bell curve and that's exactly what we should expect if we pass through a tight signal.  It's not what I'd expect if a signal was directed at us, but passing through a tight beam would explain the bell curve of the spike.  A black hole in that part of space might explain it too.

  Is there a possibility that the burst could have carried information,
  however unlikely? It could have carried a one-time "message"?

The "one time message" idea strikes me as very unlikely because I don't see any point to it.   I also don't think a lot of information could be contained in 72 seconds, but some could, but there's no point in sending information for just 72 seconds and just once.  The point of sending information is so the person on the other end receives it.

But if it was Technology based and not natural in origin, Radio-waves can carry a lot of information, otherwise radios and antenna televisions wouldn't work, but the information is contained in signal variation usually in either amplitude or frequency, like these images below

or this:

Source

The big ear measured total signal variation between the disks every 12 seconds, so it wouldn't detect Amplitude or Frequency variations that would happen in much shorter intervals than 12 seconds.  We simply have no way of knowing.   Another possible method to transmit a information could perhaps be polarization variation in a consistent signal, but big ear couldn't pick that up either.

The problem is, we simply don't have enough information to know.  It's like somebody snapped a terrible picture of something that might have happened.   You can look at the picture forever, but it will never give you any new information.

More on it here, and I liked this article here.  Hope that wasn't too wordy or too much already known information.


Can equatorial mounts be used at lower latitudes? I wanted to order a Celestron 127EQ and I was going through the manual online and just noticed that the manual says that latitude is adjustable in about a 20-60° range.

The PowerSeeker equatorial mount can be adjusted from about 20 to 60 degrees (see figure 4-3)

I'm in Costa Rica which means that it's less than 10° latitude. Is the 20-60 limit a hard limit or it can be lower?
By looking at various telescopes equatorial mounts pictures in internet, I think all will have the problem that the counterweight might hit the legs of the tripod if lowering the angle a lot, so it's not like a problem unique to 127EQ.
I have seen that a cellphone can be used to measure this angle so that would not be a problem as long as the mount can physically rotate down to about 10° but I wonder if this setup would not trip the telescope because the center of mass would be too far to the front..... at least while setting up its axis.

Let me answer myself: Yes, they can. I got a 150mm x 1400mm Newtonian with a EQ mount and I am able to align its axis... even if it had no scale, it can be done with a cellphone (I have already done it... both for pointing north and adjusting latitude).
WARNING: Need to be careful that the weight doesn't go off-balance.


What is the level of tidal heating between bodies that are already in mutual tidal lock? As I understood tidal heating, it comes from tidal force acting upon a body as it spins, distorting it; the wave of distortion travels along the surface (along with apparent travel of the other body on the sky) and continued friction as the matter is strained into the traveling wave is the source of tidal heating.

Now, if the two bodies are in a tidal lock, the distortion remains constant - it doesn't move. There's no new work done as the bodies remain immobile relative to each other. The tidal heating should be a flat zero.

Meanwhile, Io has immense volcanic activity attributed to tidal heating - despite it being tidally locked to Jupiter. While it still heats Jupiter, dragging its own tidal wave around it, Jupiter shouldn't contribute any heat to Io as its distortion remains constant over time, a stable equilibrium.

Is that just residual heat from times when Io was spinning or am I missing something?

There are actually several components of tidal forces which serve to distort a planet or moon - diurnal, nonsynchronous rotation, ice-shell thickening, orbit obliquity, and polar wander.  A moon could be stressed by any combination of these mechanisms, leading to frictional forces and heating.

Diurnal Stress - Since orbits are ellipses, not circles, the moon will experience a differential gravitational field.  When the moon is closer to the planet, the tidal stresses will be slightly larger than when it is closer away, since the gradient will be steeper. In addition, since Kepler's 2nd Law informs us that a body moves faster when it is closer to its primary, this means that the tidal locking is imperfect.  When the moon is close to its primary, it is moving slightly faster than it is rotating.  Likewise, when it is further away it is moving slightly slower than it is rotating.  This leads to the moon seeing its primary as oscillating slightly in the sky.  Source
Nonsynchronous Rotation Stress - If the moon's crust is decoupled from its core by a liquid layer (either liquid rock or liquid water), the crust can rotate freely over the core.  The core will stay tidally locked to the primary, but the shell can move around.  Since the core will have a tidal bulge, when the crust moves around it will become stressed.  The crust feels a torque because its thickness varies across the moon's surface.  Source
Ice-Shell Thickening - Icy moons, such as Europa, can experience stresses caused by their icy outer shells freezing and thickening.  As the moon loses heat, the water at the bottom of the shell will freeze.  This will increase its volume, creating extensional stresses.  At the surface, the cooling of the ice will contract it, causing compressional stresses.  While this is not a tidal stress, it is still a source of stressing which can act on these moons, so I thought I'd throw it in.  Source
Orbit Obliquity - Most moons do not rotate exactly perpendicular to their orbital plane.  Rather, their rotational axis has some amount of obliquity.  This obliquity changes the latitudinal orientation of the tidal bulge as the moon orbits the planet.  This creates additional stresses as the tidal bulge gets pulled on.  Source
Polar Wander - Large impacts can cause a moon's lithosphere to rotate and reorient itself relative to its rotational axis.  When this rotation changes the apparent location of the rotational poles, it is called "polar wander".  Polar wander causes stress in a similar manner to nonsynchronous rotation.  The lithosphere rotates over the core's bulge, pushing on the crust.  Source

There may be other stress mechanisms that I am not aware of, but these are the main ones.  If you want more information about the topic, or want to see some visualizations of what these various stresses look like, check out SatStressGUI, a program I helped developed which models stresses on icy moons.


What is a debris field called before it becomes a planetary ring? As a body of a suitable size and mass moves within the Roche limit of a planet, it will disintegrate. This will form a debris field that will eventually form a planetary ring. Is there a name given to this debris field before it forms a ring? Something like a Roche train or Roche field seem appropriate but I can't find any reference to such

I don't think such an intermediate state exists for very long. Once you shred the moon, it's shredded and the particles orbit on locally Keplerian orbits.
This would look like Saturns rings, which are also well inside the planet's Roche limit.

The intermediate state you propose to name would have to form either
a) a steaty state of in some dynamic variable or
b) have a sufficiently long lifetime (which would be your job to prove)
in order to earn its own name.

Edit: As OP has asked for material on ring spreading, I can point to A. Crida's work who had a bit of a controversial idea about the origin of the gas giant moons. However his calculations involve also classical calculations of collisional disc spreading.


Strong Gravitational Lensing Dataset I'm a Machine Learning Student and I'm working on Strong Lensing Image Classification. this is a hot topic nowadays but strangely after days of searching and communicating with different people I still can't find an accessible Dataset of these objects!

Do you know any place I can find a simple(or not) Dataset containing images, labels and some extended features of Strong Gravitational Lensings?

The most comprehensive catalogue of strong gravitational lenses that I am aware of is the CASTLeS database with about 120 lens systems. Several of them have extended features, arcs, etc., while others are point sources. They provide the redshifts of lens and source where available, as well as time delays between multiple images and the velocity dispersion (a proxy for the mass) of the lens for a few.

You would have to create a comparison dataset without strong lensing yourself, but that should not be difficult: Go to Hubble's Ultra Deep Field web page, download the images with the same filters as used in CASTLeS and cut out likely looking galaxies. The UDF contains mostly very high-redshift galaxies, so the chance of accidentally picking a strong-lens system is very slim.

(Disclaimer: I've only ever used ACS UDF images, not the NICMOS ones; if it's too hard, just skip the IR data.)


What is the highest granularity focal-plane array on a dish radio telescope? Or is this the ONLY ONE? There is a short Wikipedia article Focal Plane Arrays that enumerates some projects, but my question is more along the lines of what is (at least) nearly complete or in "first light" phase, even if not commissioned yet.

I'd like to differentiate between focal plane arrays (an array of multiple feeds and amplifiers) used in single dish telescopes, and those integrated into multiple dish arrays, because I'm particularly curious about single dishes being used for spatial information, or even true imaging. Ideally, the answer will give some information for each case.

In the case of single dish instruments, are the elements - roughly speaking - used as pixels? Despite the longer wavelength, it's still optics and it is a telescope. If there are N individual, uncoupled feeds, does one build up an image roughly N times faster? Is the relative phase between the feeds ever used (for single dish instruments)?

For those like me who aren't already familiar with focal plane arrays, here is a random picture from one of the links I found in a quick internet search. It's from the Parkes 21cm Multibeam Receiver, has (had?) 13 receivers and sat at the focus of the 64m dish. The photo is dated 1997 - I have a hunch there's been some development in this technique in the intervening 20 years.

Is this actually the only one?

Edit: The Parkes array is still in use as shown below:

Above: Superposition of the half-power beam widths of the 13-element array of the Parkes 21-cm Mutlibeam Receiver, as used in a study of a Fast Radio Bursters.

It's likely the image is a screen shot from from The host galaxy of a fast radio burst Nature volume 530, pages 453–456 (25 February 2016), Keane et al. I can't find my archived copy now, but instead see Phys.org's New fast radio burst discovery finds 'missing matter' in the universe

where the caption there reads:

  This image shows the field of view of the Parkes radio telescope on the left. On the right are successive zoom-ins in on the area where the signal came from (cyan circular region). The image at the bottom right shows the Subaru image of the FRB galaxy, with the superimposed elliptical regions showing the location of the fading 6-day afterglow seen with ATCA. Image Credit: D. Kaplan (UWM), E. F. Keane (SKAO).

I am not a professional astronomer, so take this answer with a grain of salt, but just from visiting a few facilities, I know some.
For single dish applications, the 100m telescope in Effelsberg, Germany, uses a 7-beam receiver -- fewer than the Parkes array you mention, and I dont't know of any single dish setup with a larger number of beams.
Regarding arrays, the Westerbork Synthesis Array in the Netherlands uses the APERTIF arrays in most of its individual 25m dishes. With 121 elements per array, this seems to be on the higher-end side of granularity.

update:
From astron.nl/dailyimage for 31-01-2017 First image with Apertif: a new life for the Westerbork radio telescope

The first images made with the upgraded telescope that demonstrate this new 'wide-angle' capability is shown here. The first image shows the dwarf galaxy Leo T. The image is colour-coded and shows the gas (in blue) in this galaxy together with many distant radio galaxies in the background shown in orange. For comparison, the field of view of the previous Westerbork system and the size of the full moon are also indicated.
To make this new capability possible, ASTRON developed and built the hardware in-house. 121 small receivers are used in each telescope whose signals are combined electronically to produce the large field of view.
The upgraded telescope will also be used to search for and study new variable sources in the radio sky. With the new Apertif receivers, observations of large parts of the sky can be done much faster, and projects that used to be impossible as they would take tens of years can now be done in a much shorter time. Westerbork is therefore poised to make many new discoveries in the radio sky.


Expanding our observable universe suggestion Firstly, regarding the observable universe there is a certain radius of a circle, and we cannot see further than that. It is understandable, that due to photons travelling at a certain speed we can only see that far. But question is: "What happens with photons that might arrive later to our vision?", meaning outside our "ring" of the observable universe, to my understanding we could detect also photons or radiation coming from there, meaning that our radius of the visible universe is expanding, in the sense of areas now, where we can have layers of certain visibility and with a percentage of "accuracy" and not a static radius, i.e. "x light years beyond the radius the existence of one y object has been confirmed due to late photons coming to our vision. There might or might not be more around it but we have no certainty". Understanding human life is too short to experience that change, these humans might be living some thousands of years from now. Until here I have considered a static model, I haven't taken into account universe expansion yet.
When the universe expansion speed is put into the equation, things change. If the universe expansion is faster than the speed of light, we would be "losing ground" in visibility, meaning more objects would be leaving our radius than the ones coming, however nothing is faster than the speed of light, meaning that the universe expansion is smaller than speed of light (unconfirmed quotation), thus in reality to me, our vision should be expanding, but slower than the static model I described a little earlier.
On the other hand, at the edge of our imaginary ring, we can also have objects popping in and out from time to time. In this manner we can also create an uncertain map of what was and what will be in the outer ring of our observable universe, thus expanding our vision, and over time this map will be more certain.
Now, I haven't done the math, nor am I certain of many of my quotations, so please don't be angry at me, I am just a curious human being trying to learn more about our universe :)

Unfortunately, the limit of the observable universe isn't just "photons from the big bang arriving to us now" - so we won't see anything past it because nothing existed before; anything outside that sphere didn't arrive yet. This limit is called the Hubble Sphere, and is only the "first stage" of unobservability - and indeed, as time passes what we can see will expand and we'll be able to see further.
But there's another limit - called the cosmic event horizon - which, to current knowledge, is quite unconquerable. It's connected with space expansion. Space expansion is a phenomenon of far objects "moving" further apart, with the nasty property of bypassing special relativity with its property of light speed being the same in all frames of reference (which would normally imply no matter how far a light source, the light would reach us sooner or later). Instead of classic motion - change of position in space over time - the "motion" due to space expansion is "production of more space" between the objects - they aren't actually in motion relative to each other (or that motion is irrelevant), it's the very definition of distance between them that is changed.
And as result of that, there is a region distant enough, that light from there - traveling at speed of light towards us - will never reach us as there will be always more space inserted between it and us, per unit of time, than it can cover. And so, the cosmic event horizon describes the actual ultimate sphere of the observable universe, not just "for now" but "forever, unless we discover some way to bypass it, but currently not even a hint of such bypass exists."


If a giant object would approach the Earth, would Earth start to turn faster or slower while the object starts to become nearer and nearer? I am asking myself this question: if a giant celestial object would approach the Earth, would Earth start to turn faster (then day duration would be diminished) or slower (day duration increased) while the object starts to become nearer and nearer?
Thank you!

Could be, you'd be better to ask this on the Physics site, by the way.

No, the spinning of a perfectly round object is not affected at all, it just keeps spinning identically.

Let's say ..

say another Sun appeared, just like that.

Yes, the path that Earth is taking would go crazy.  We might fly off, head towards this new Sun, starting making a figure of 8, or some crazy path!

But the spinning as such of the Earth would be totally unchanged.

The "way we point" ("North up" so to speak) would be completely unchanged, and the speed we turn at (ie, 24 hours a turn) would be unchanged.

This might seem surprising but, well, you can learn all about it in Newtonian physics!

Once again, no change to the spin as such of the Earth.  the "direction we point" (ie the axis we spin around) and the time taken is totally unchanged if (for some bizarre reason!) we are "moved around in the heavens".

Note that there would be some absolutely tiny effects due to things like relativity (have a read!), the water on our planet sloshing around, and after all our Moon could be moved, bashed, etc.  These effects on the speed we are spinning would be extremely small over the short period of time you're considering, in the "sci fi" scenario you outline, as seen in the diagram.


Can angular information be known more precisely than the diffraction limit? The diffraction limit deals with the ability to determine if two things are separate.  I am interested in the ability to find the the centroid of a single object.

Imagine a star with no near neighbors would it be possible to determine the centroid of that star at a resolution higher than the diffraction limit?

The star is essentially a point source and I wonder if looking at the edges of the airy disk would allow a centroid to be determined if I had a magical camera with infinite pixels.

It seems to me that I should be able to see the disk and then calculate a centroid that is smaller than the airy disk.

Thanks for any help you can provide =)

Yes, Gaia makes extensive use of this technique.

The measurement accuracy of star positions for the brightest stars is 10 to 20 micro- arcseconds – up to 100 times better than that the precursor mission, Hipparcos; this angular resolution corresponds to the observed diameter of a Euro coin on the Moon, viewed from Earth.

The underlying principle is the standard deviation of the mean. It depends on the sample size, more precisely on the square root of the sample size. In this case the sample size is the number of the measured photons. Therefore it works better for bright stars.


What are the concrete technical arguments supporting the idea that the wave function of the universe can be written as partition function? In this talk, the not yet settled down idea that the wave function of the universe could potentially be written as  the partition function of a scale invariant statistical field theory is mentioned:

$$
\Psi[g] = Z[g] = \int D(\text{Fields}) e^{-S[g,\text{Fields}]}
$$

If our universe were AdS, this relation could already be well enough explained by the AdS/CFT correspondence, but as our expanding universe correspond to a dS geometry, things are less clear.

What are the concrete technical arguments, ideas, or hints that this relationship should hold for our dS universe too? What work has already been done on this?

Similar approaches have been generalized to more general spacetimes.

Technical details for dS space and further references in this paper about the holographic principle, p.43 ff., may provide some idea.


Why can't we use the semi-minor axis in Kepler's third law? Why can't we use the semi-minor axis in Kepler's third law? The formula for the third law is:
$$\dfrac{a^3}{T^2} = \text{const.} $$
Instead of semi-major axis (a) why can't we use the semi-minor axis (b)?
$$\dfrac{b^3}{T^2} \neq \text{const.} $$
I just checked it (I calculated the results for some planets using the semi-minor axis) and I really don't know why it's not constant.

The relationship between $a$, $b$ and the eccentricity of an orbit $e$ is
$$ a = b\left(\frac{1+e}{1-e}\right)\ .$$
It is clear therefore from a mathematical point of view that you cannot get a similar constant relationship by replacing $a^3$ by $b^3$ in Kepler's third law. You would have to use an expression involving both $b$ and $e$.
Another way of saying this is that you can have (infinitely) many orbits that have the same $a$ and $T$, but which have different $b, e$ combinations.
I guess from a fundamental point of view it boils down to being able to express the total energy of an orbit as
$$ E = -\frac{GMm}{2a}$$
which is independent of the shape of the ellipse.


Need Simple equation for Rise, Transit, and Set time I've been looking, unsuccessfully, for hours for a simple set of equations:

Input:
RA and Dec of an object
Observers Lat and Lng
Current Time

Output:
Rise Time
Transit Time
Set Time
(and, hopefully) max elevation at transit
(and, hopefully) current alt and az of object

It seems like this should be a simple relationship, but I haven't been able to find it.
If an equation simply gives the current alt/az for an object, I guess I could solve for az and then find the maximum for a given time range.

Questions:
1. Anybody know of such equations? Or, even better, some sample code.
2. If not, can anybody point me in the right direction?

Thanks

I'm not sure it qualifies as "simple", but, using
http://idlastro.gsfc.nasa.gov/ftp/pro/astro/hadec2altaz.pro (and some
additional calculations/simplifications):

$
   \begin{array}{|c|c|c|c|}
\hline
    \text{Event} & \text{Time} & \phi  & Z \\
\hline
    \text{Any} & \text{t} & \tan ^{-1}(\cos (\lambda ) \sin (\delta )-\cos
      (\delta ) \cos (\alpha -t) \sin (\lambda ),\cos (\delta ) \sin (\alpha
      -t)) & \tan ^{-1}\left(\sqrt{(\cos (\lambda ) \sin (\delta )-\cos (\delta
      ) \cos (\alpha -t) \sin (\lambda ))^2+\cos ^2(\delta ) \sin ^2(\alpha
      -t)},\cos (\delta ) \cos (\lambda ) \cos (\alpha -t)+\sin (\delta ) \sin
      (\lambda )\right) \\
\hline
    \text{Rise} & \alpha -\cos ^{-1}(-\tan (\delta ) \tan (\lambda )) & \tan
      ^{-1}\left(\sec (\lambda ) \sin (\delta ),\cos (\delta ) \sqrt{1-\tan
      ^2(\delta ) \tan ^2(\lambda )}\right) & 0 \\
\hline
    \text{Transit} & \alpha  &
\begin{cases}
                   \delta >\lambda  & 0 \\
                   \delta =\lambda  & \text{Zenith} \\
                   \delta <\lambda  & \pi
                  \end{cases}
    & \frac{\pi }{2}-\left| \delta -\lambda  \right| \\
\hline
    \text{Set} & \alpha +\cos ^{-1}(-\tan (\delta ) \tan (\lambda )) & \tan
      ^{-1}\left(\sec (\lambda ) \sin (\delta ),-\cos (\delta ) \sqrt{1-\tan
      ^2(\delta ) \tan ^2(\lambda )}\right) & 0 \\
\hline
    \text{Lowest Point} & \alpha +\pi  &
\begin{cases}
                   \delta >-\lambda  & 0 \\
                   \delta =-\lambda  & \text{Nadir} \\
                   \delta <-\lambda  & \pi
                  \end{cases}
    & \left| \delta +\lambda  \right|-\frac{\pi }{2} \\
\hline
   \end{array}
$

where:

$\phi$ is the azimuth of the object
$Z$ is the altitude of the object above the horizon
$\alpha$ is the right ascension of the object
$\delta$ is the declination of the object
$\lambda$ is the latitude of the observer
$t$ is the current local sidereal time

Note the two-argument form of arctangent is required so that the
results are in the correct quadrant:
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Two-argument_variant_of_arctangent

Additional caveats:

If $\left| \delta -\lambda \right|>\frac{\pi }{2}$, the object is
always below the horizon, and the equations for rising time and
setting time will not work.
If $\left| \delta +\lambda \right|>\frac{\pi }{2}$, the object is
always above the horizon (circumpolar), and the equations for rising
and setting time will also not work.
The measurements above are in radians. You can convert $\pi \to
  180 {}^{\circ}$ for degrees.
Because we use the local sidereal time, the longitude doesn't
appear in any of the formulas above. However, we do need it to find
the local sidereal time, as below.
To find the local sideral time $t$ in radians, we use
http://aa.usno.navy.mil/faq/docs/GAST.php and make some substitions
to get:

$t = 4.894961212735792 + 6.30038809898489 d + \psi$

where $\psi$ is your longitude in radians, and $d$ is the number of
days (including fractional days) since "2000-01-01 12:00:00 UTC". Traditionally, we use $\phi$ for longitude, but I'm already using it in the formulas above for azimuth.

If you combine the formula for local sidereal time and
azimuth/altitude and assume excessive precision, you get my answer to
https://astronomy.stackexchange.com/a/8415/21

Additional computations for these results at:
https://github.com/barrycarter/bcapps/blob/master/STACK/bc-rst.m

I was going to add some graphs to show how the altitude is NOT a sine wave and how the azimuth is NOT a straight line (although you might expect them to be), but they turned out not to be terribly instructive/helpful.

You might also be able to get simpler formulas if you set $t$ to be the "hour angle" (which is $\alpha-t$ in the current setup).


Has stellar evolution ever been modeled analytically? I'll still remember the handout my astronomy professor gave us more than several decades ago; Our Friends the Polytropes. We spent a lot of time learning how polytropes, simple analytical power-law models could reproduce some basic thermodynamic properties of stars, and how those properties would tend to vary with mass (and perhaps other parameters, I don't remember now).

Have polytrope models of stars (or other simple analytical models of stars) ever been "evolved" analytically in time? In other words, in addition to variations on radius, have a set of equations ever been written which also included a $\frac{\partial}{\partial t}$ in such a way that they were amenable to an analytic solution?

note: I've added the history tag since I'm asking about the "gold 'ole days" rather than about modern techniques.

Analytic models have been applied to various phases of the evolution, though it would be impossible to apply a single model to all phases because the physics changes so much.   Also, a star will often have very different physics going on in various different parts of the star, so analytic models sometimes have to treat different parts separately and then unify the results across boundaries (just as numerical simulations do).  For example, I've seen different polytrope indices used in the convection, radiative, and core regions of the Sun.

I realize that your interest is when there is not a steady-state assumption, but rather time is a dynamical variable.  But it is not always necessary to use time as a dynamical variable to do evolution.  You can simply do a steady-state analytical model (like a polytrope, perhaps supported by a numerical piece that solves for some parameter like core pressure), and allow the parameters in your solution to be time varying in some simple analytical way.   For example, you could study how mass loss affects a star by doing steady-state analytical models and just let the mass change with time according to some analytical mass-loss prescription (such as "Reimers' Law", something simple).  So steady-state analytical models can be elevated into evolutionary models if you simply have an analytical way to change the parameters with time.

Another simple example is the evolution of a fully convective protostar.  You simply fix the mass and initial radius, and you solve the interior by assuming it's all at the same entropy (a reasonable approximation for fully convective stars).  Then you fix the surface temperature to lie on the Hayashi track, which for a very simple model could mean just pegging the surface T to 3000 K.  That and the initial radius determines the luminosity, and the constant entropy assumption fixes the internal structure, so then you simply let it lose energy at the rate of the luminosity, and use the internal energy as the variable that changes with time, always updating the radius to be consistent with the new internal energy (a la the virial theorem), and that gives the new luminosity and so forth.  A simple fully analytic model until the star is no longer fully convective (and we then might call it a pre-main-sequence star).

I've also seen white dwarf cooling done analytically.  It's a similar idea-- take the surface temperature as an initial condition and just let it lose heat via its luminosity.  The radius doesn't really change, so all you need is to keep updating the surface temperature as a function of the internal energy, which can be handled via some analytic internal heat transport treatment.  If you are willing to make approximations that address the key physics, you can do almost anything analytically.


Do you know about any Image Processing Challenges known in Space Telescope like Hubble? For a computer science project  in University;

I need to find image processing problems on space exploration and Space Telescopes, only related to the image itself.

I have wrote a computer program that is able to separate every pixel of an image according to its pixel intensity (the whitest to the darkest) in 3 (or even more) big groups.

I was thinking that this might be somehow useful in a space image processing situation in a well known challenge of the technology in the Telescopes or machine vision devices around space.

Situation can be very specific, just as an example calculating the mass of a star by the look of the pixels which compose it, or something like this.

First, I'll talk a little bit about the state of what is being done: (according to my knowledge)

In Astronomy on-board processing is usually avoided if possible.

The reason for this is to have the pure data as unspoiled as possible, because instrument effects, cosmic rays, the light from Earth's corona, etc. can introduce many systematic errors into the data. If that would be removed on-board and then sent to Earth, then there would be no control later on what the actual, original data was.
This became relevant e.e. for the exoplanet-finding Kepler(see also the technical data there) space telescope where the photometric data was showing very unexpected behaviour at first, before the reasons for it were realized, obviously long after the spacecraft was designed and built.

Thus, the image processing challenges are usually resolved on the ground stations and with the astronomers on their big computers. Also the interpretation of the data, after being cleaned, debiased, etc. is non-trivial.

As however the downlink speed depends on the distance $d$ to Earth like $\sim \frac{1}{d^3}$ so satellites that orbit far away do have to apply some clever ways to reduce data sometimes. The Hubble is not a candidate for this, as it orbits Earth at only some 500 km a.s.l, its data can be transferred to Earth without problems.

The Gaia satellite however, orbits the second Earth-Sun Lagrange point and is thus farther away from Earth, 1.5 million km. This mission is an astrometric mission, meaning it's there to measure positions and velocities of stars to extreme precision. If all the pixels of all onboard systems would continuously be read out and produce data, that would be too much to handle and some would be lost. So Gaia has a pre-detector that tells the main detector "Hey a star is coming". The main CCD then creates a moving window of 12x12 pixels from which all electrons are read-out and accepted as data. So that's some limited processing there.

In terms of data transfer, missions like New Horizons (Pluto encounter last year) or the Voyagers, are too far out to simply send their data over. Therefore the Deep Space Network exists, to look out for radio signals from those probes and actively amplify them.

What you could do:
So now that I've spent time explaining that on real spacecraft it's not being done what you want to do, we can bend the rules and just try to think about what's being done on the ground (if you want to):

Image calibration: A detector taking an image wants to be very well understood, before claiming any interesting discoveries or measurments. Part of this process is taking a dark image to characterize what your CCD is doing when nothing is measured. Then a flatfield image is done, an image of an ideally uniformly white surface, in order to characterize pixel-to-pixel variations. More. For the measured image then, the dark image is substracted and divided through the flatfield image.
Cosmic ray removal: Cosmic rays fly through your camera in a random direction at random times (in space as well as on the ground). They produce streaks in CCD cameras, and can be automatically removed by taking the same image three times, and discarding one of them or only a pixel-region if a dramatic intensity streak somewhere is measured.
Measuring a point-spread function:  Stars are so far away, that they appear as point sources. However through instrumental effects this ideal point will always be spread out into a finite shape, called the PSF. Because this function can be easily a few pixels wide, it is very important to know the shape of the PSF to be able to assign photons to a star that have been measured in off-center pixels. Knowing then a star's photon output and it's distance one can calculate the luminosity. But even that can't be done on an on-board chip. What you could to is to try to automatically fit an Airy-function to the PSF.

Ok so that was just a few ideas, but I'm not an observer, there may be better ones out there. Also you got keywords not to google further.


Are we center of universe? So I just learned that CMB redshift is 1100 regardless where we look (up down left right). According to Hubble's Law that makes it around 46 billion light years away, making it the farthest matter away from us. If CMB redshift is the same regardless which direction we look, then how could that be unless we're in the center of the universe?

At the time when the universe was ~380,000 years old it was completely filled with a plasma and light. Light could only travel very short distances (I forget the specifics, but think a couple meters at the most) before being absorbed. So the entire universe was a fog, if you were there and there was an object 10 meters in front of you, you couldn't see it because all the light it emitted or reflected would be reabsorbed before it reaches your eyes. There are two things to keep in mind: 1) This fog disappeared across the entire universe simultaneously (plus or minus some time, but effectively simultaneously); at the same point in time after the Big Bang, 380,000 years after the Big Bang, the fog dissipated due to recombination. And 2) Light takes time to travel.

So let's say you are sitting in the fog-filled universe a few seconds before recombination happens. Light travels about 1 foot per nanosecond. So if the fog disappeared and there was a street lamp 10 feet in front of you, you wouldn't receive the first light from that lamp for 10 nano seconds. All of the light the lamp was emitting before recombination was just reabsorbed, but after recombination the first photons to leave the lamp no longer have anything to run into, so they continue on until hitting your eyes. A street lamp 20 feet away wouldn't be visible for 20 nanoseconds after recombination. The light of the plasma itself is effectively a street lamp at every location in space, so due to the 2 facts above, you would see successively farther street lamps in a shell centered on you.

After recombination happened and the entire universe became transparent. The fog is gone, but the farther you look the farther back in time you look, so you can only see back as far as when the fog existed, and the farthest you can see is the fog itself. After 13.8 billion years you can see all the way out to a shell of radius 13.8 billion light years around you (ignore expansion of the universe for now), and what you see is the light of the plasma (the fog) that was filling the universe 13.8 billion years ago, the CMB.

This scenario is the same if the universe is infinite in extent. We are not the center of the universe, we are simply the center of the observable universe, as is anyone else. Aliens looking towards Earth but are at the current location of edge of the CMB in some direction will not see Earth, they will see the last light of the CMB itself at our location when it was emitted 13.8 billion years ago; we are the edge of their observable universe.


What experiment does this "Tracing the History of Starlight" video try to explain? The YouTube video Tracing the History of Starlight with NASA's Fermi Mission describes observation by the Fermi gamma ray telescope  of EBL (extragalactic background light) gamma rays from 2,000 Blazars. It seems to suggest that they used an attenuation measurement of the gamma rays to determine... I'm not exactly sure exactly what was determined by the investigation discussed in the video.

The attenuation is due to reverse pair-production, two photons in, and an electron-positron pair out. I am not sure if they measured 511 keV gamma rays from subsequent annihilation of the positrons, or attenuation of the primary gammas, and I am also not sure if the experiment was used to measure the primary gamma ray rates, or the total density of UV-Vis-IR light photons with which the gamma rays annihilated.

Question: Is there a simple but physically clear way to explain all of this; the experiment the video is supposed to be explaining?

How do they differentiate attenuation due to revere pair production from attenuation by say Compton scattering considering dust and gas?

Blazar sequence
Let's first look at which kind of sources are studied.
Blazars are special type of radio-loud AGN. One of their special properties is that their spectral energy distributions (SED) form the so-called Blazar sequence. They all have a characteristic double peak SED and the position of the peaks mainly depend on their luminosity. The higher the luminosity the lower the peak frequencies (see Fossati et al., 1998 for details)

(image source: Donato et al., 2001)
Leaving details like sub-classes aside, this means that SED of Blazars can be modelled very well, especially if looking at a huge ensemble of Blazars.
Extra-galactic background light (EBL) absorption
Now the high energetic light has to travel from the Blazar to us. The Blazars being far away, much can happen during this travel, notably it can interact with the extra-galactic background light (EBL absorption). If a high energetic photon from the Blazar encounters an EBL photon in the optical or ultraviolet range they can pair produce an electron/positron pair (this is the dominant process for the involved energies).

These charged particles will most likely be subject to magnetic fields or other interactions diverting them from their direct way towards earth.
This means not all of the high energy photons from the Blazar reach Earth and how large the absorption is depends on:

the distance between the Blazar and Earth
the energy of the high energy photons from the Blazar
the energy of the interaction partners from the EBL

Bringing both together
Fermi looked at a large sample of SEDs from Blazars of various luminosities and grouped them based on their place in the Blazar sequence. For each group they could compare the expected SED from the Blazar model with their measured SED and thus determine how large the EBL absorption is depending on the energy. This allows to calculate the density and energy distribution of the EBL.
Knowing the spectral energy distribution and density of the EBL allows to go even one step further and make assumptions on where the photos in the EBL came from. How many stars it took, how long they had to emit photons etc.
As if this alone would not already be interesting enough, Fermi even repeated this for samples of Blazars at different distances, thus probing the EBL at different ages of the universe (remember that we observe the light that reaches the Earth now, thus the light from Blazars further away was emitted earlier than the light from "nearby" Blazars)

(image source: https://fermi.gsfc.nasa.gov/science/eteu/ebl/)


Right Ascension to Zodiac House At the moment (01-04-2020 09:36) the Moon is situated in Gemini:

"The Moon is currently in the constellation of Gemini. The current Right Ascension is 06h 37m 56s and the Declination is +23° 42' 09”." theskylive.com, 2020.

If there are 12 houses (zodiac) and Right Ascension is in a 24hr format how is the moon calculated to be in Gemini?

The calculation i'm using has it in Cancer. This is obviously a mistake but here's how i'm getting it:

Aries 0h 0m 0s
Taurus 1h 51m 39s
Gemini 3h 51m 16s; <- (Moon is  actually in Gemini)
Cancer 6h 0m 0s; <- (here's where ive got the    moon at 06h 37m 56s)
Leo 8h 8m 44s;
Virgo 10h 8m 21s;
Libra 12h 0m 0s;
Scorpio 13h 51m 39s;
Sagittarius 15h 51m 16s;
Capricorn 18h 0m 0s;
Aquarius 20h 8m 44s;
Pisces 22h 8m 21s.

(Courtesy of https://astronomy.stackexchange.com/a/27488)

How have I misunderstood right ascension and how would I derive Gemini from RA 06h 37m 56s?

Zodiac signs are equal 30° divisions of the ecliptic.
In the Western tropical zodiac, these are linked to the equinoxes and solstices, and named after the nearest constellations at the time Ptolemy wrote about them.
The equinoxes precess westward 1.4° per century, so the signs currently lie 25° to 30° west of their namesake constellations.
There are also sidereal zodiacs, which account for precession and keep signs aligned with stars.

A celestial object's ecliptic longitude of date determines its zodiac sign.
If the lunar RA and Dec in question are for epoch J2000, then the Moon's epoch 2020.25 ecliptic longitude is 99.0°, in the tropical sign of Cancer (longitudes 90° to 120°).
Relative to the IAU boundaries, drawn along lines of RA and Dec for epoch B1875, this is in the constellation Gemini.

Approximating ecliptic longitude by right ascension alone can err as much as 2° in the case of the Moon, whose orbit is inclined 5° to the ecliptic.
For example, at 2020-04-06 19:40 UT, the Moon is at equatorial RA 12h4m (1° east of 12h0m) but ecliptic longitude 179° (1° west of 180°).
Since the ecliptic is oblique to the celestial equator, lines of ecliptic longitude are oblique to lines of right ascension except at the solstices.

Houses are completely different, dividing the sky in reference to the local horizon.
The precise definition depends on the particular astrological tradition; roughly, a celestial object enters the 12th house when it rises and the 6th house when it sets, decrementing house number every 2 hours or so.
The most closely related astronomical coordinate is hour angle.


Has the Big Crunch been ruled out? If the dark energy equation of state $\omega$ evolves with time, and becomes larger, this would result in decelerated expansion and eventually a Big Crunch. If $\omega$ decreases with time, becoming more negative, then a Big Rip would occur. I understand that the Big Rip has not yet been ruled out by current observations, but I often hear that the Big Crunch has been ruled out since we know the universe is expanding (and accelerating).

If the Big Rip can't currently be ruled out, then how can the Big Crunch be ruled out? Shouldn't they both still be on the table, unless we can somehow infer which direction $\omega$ evolves in under a quintessence model?

The Big Rip happens if the equation of state for the dark energy has $p/\rho = w<-1$, and all empirical data give us $w\approx -1$. A Big Crunch requires a pretty high value of $w$ (it must go above -1/3 to just stop the acceleration), but of course if dark energy is changing over time it might do that.

So tentatively ruling out Big Crunches is the reasonable thing to do given observational evidence and the assumption that the future will not be wildly different. If one thinks we should be more open to dark energy changing (which is going to be rather unconstrained by observation for the foreseeable future) then we should also be more open to $w$ moving across the Big Rip boundary.


Effect of the obliquity of the ecliptic / tilt of the Earth on the equation of time In an attempting to answer the question “why aren’t changes in sunset and sunrise times symmetrical  around the solstices ?” I’ve got stuck on being able to explain the effect of the obliquity of the ecliptic.

I understand the general idea of the equation of time and how the difference in clock time and solar time answer my original question, but I’m struggling to grasp how the tilt of the Earth contributes to the equation of time.

I’m trying to visualise the effect, and assuming it is to do with the point at which the sun is highest in the sky not being in the middle of sunset and sunrise (except at solsitces and equinoxes?). So, is this correct? Is there any good way of trying to visualise it ?

I’m a bit confused by the wikipedia entry
https://en.wikipedia.org/wiki/Equation_of_time#Obliquity_of_the_ecliptic saying

“If this effect operated alone, then days would be up to 24 hours and 20.3 seconds long (measured solar noon to solar noon) near the
solstices, and as much as 20.3 seconds shorter than 24 hours near
the equinoxes”
“the inclination of the ecliptic results in the contribution of a sine wave variation with an amplitude of 9.87 minutes and a period
of a half year to the equation of time. The zero points of this sine
wave are reached at the equinoxes and solstices”.
Is point 1 correct ?

Also found these links helpful to some extent, but haven’t fully explained it for me.
https://web.archive.org/web/20150910174438/http://www.rmg.co.uk/explore/astronomy-and-time/time-facts/the-equation-of-time
http://totaleclipse.eu/Astronomy/EOT.html

As the solar time
Wikipedia page illustrates, we can split a solar day into a constant 23h56m to complete a rotation relative to the stars
(a sidereal day)
and a variable ~4m to compensate for one day's orbital motion around the Sun.
The Sun's apparent motion around the equator determines how much more than 360$^\circ$ the Earth must rotate from one solar noon to the next.

But the Earth orbits the Sun in the ecliptic plane, which is oblique to the equator.
If that orbit were circular, the Sun's
ecliptic longitude
would change at a constant rate (see Note below).
One degree along the equator spans exactly 4m00s of
right ascension;
one degree along the ecliptic, measured around the equator, may span ~8% more or less than that.

At an equinox, the ecliptic crosses the equator at a 23.4$^\circ$ angle,
and one degree of ecliptic longitude spans 3m40s of right ascension.
The Sun appears to move slower than average around the equator,
so the apparent solar day is slightly shorter.

At a solstice, the ecliptic is tangent to a $\pm$23.4$^\circ$ declination parallel,
and one degree of ecliptic longitude spans 4m22s of right ascension.
The Sun appears to move faster than average around the equator,
so the apparent solar day is slightly longer.

Stellarium images around the March equinox and the June solstice,
showing 30 degrees of ecliptic longitude and 2 hours of right ascension for comparison.
Equatorial grid is blue, ecliptic grid is orange, ecliptic is yellow.

The equation of time is the cumulative sum of differences between mean and apparent solar day length.
Like a mathematical function and its derivative, zero crossings of one should roughly correspond to maxima or minima of the other.
Differences of a few seconds per day, several weeks in a row, add up to a few minutes.

The earliest/latest sunrise and latest/earliest sunset of the year do occur several days on either side of the summer/winter solstice.
However, on any given day, sunrise, noon, and sunset are all behind or ahead of mean solar time by approximately the same amount.

Note: the Earth's orbit is not quite circular.
Following Kepler's second law,
the Sun's ecliptic longitude changes ~3% slower at aphelion and faster at perihelion,
making the apparent solar day slightly shorter or longer respectively.
This Wikipedia entry
discusses it in more detail.


Black hole without singularity? My question is about the equivalence about having an event horizon and having a singularity.

In one side the implication looks pretty obvious:

A singularity implies having an event horizon and therefore a black hole. Since the mass is compressed in a zero volume space, if you get close enough there will be a point where the escape velocity gets bigger than the speed of light so you will get a black hole by definition.

But what about the opposite? Does having an event horizon imply the existence of a singularity?

Could it be that you have a neutron star massive enough to reach a escape velocity equal to the speed of light but not strong enough to make the matter collapse?

Even if such star can not exist because the strong force collapses before reaching an event horizon, this doesn't mean an equivalence.

It just means that for some specific value of the maximum strong force this is not possible, but image now an imaginary exotic matter that has a way bigger strong force.

For such "science fiction" matter, it would be possible to reach an event horizon without collapsing to a singularity, right?

Or is it really an equivalence between this two concepts, such that no matter how resistant matter is to collapse it will never reach an event horizon?

Does having an event horizon imply the existence of a singularity?

An event horizon is not an inherent component of any given object. It's not like once a star turns into a black hole, it suddenly gets an event horizon. The event horizon is merely a mathematical boundary which defines the distance from a mass $M$ where the escape velocity equals the speed of light. I can calculate such a boundary for a black hole, for the Sun, the Earth, or even you. So I guess the answer here is no, having an event horizon does not imply the existence of a singularity.

  Could it be that you have a neutron star massive enough to reach a escape velocity equal to the speed of light but not strong enough to make the matter collapse?

The answer here, technically, is no. The reason being that once it requires a speed greater than or equal to the speed of light to escape your object, it is necessarily a black hole. That is the definition of a black hole. So that means this neutron star you propose is actually a black hole. Another equivalent definition of a black hole is any object whose mass is concentrated inside that object's event horizon.

But you might still ask, could you have a black hole where the mass inside the event horizon is not a singularity. This would require some sort of support to prevent the matter from collapsing down to the singularity. The answer to this is that it is currently unknown. The problem is that inside event horizons, suddenly you need to work with both GR and quantum field theory but those two theories don't play nice. Instead you should be using a Quantum Gravity theory but this theory hasn't been developed. So ultimately any answer to this would be a guess until this theory gets fully fleshed out.