Can I leave a planet without achieving escape velocity? I know that if you exceed orbital velocity, you will never fall-back to the planet.  My question is not about orbits.  It's about brute-force propulsion to achieve altitude.  I'm using an intentionally slow velocity to help illustrate my point.

Imagine I have a rocket with very efficient fuel storage.  My rocket can store enough energy to accelerate to 100kph shortly after leaving the ground, and continue to maintain that speed (100kph) for a very long period of time.

My rocket just goes straight up.  It doesn't try to enter an orbit.  As it leaves the atmosphere, it can throttle-back because there's no air resistance.  As it continues to gain altitude in inter-planetary space, it can throttle-back even more because Earth's gravitational influence diminishes with distance.  It just maintains enough throttle to continue moving away from Earth at 100kph.

At some point, Earth's gravitational influence would be moot, as other bodies (Jupiter, Sun), would gain relative influence.  Eventually, far outside the solar system, even the Sun's influence would be insignificant.

My rocket never achieved escape velocity, but it sure did escape.

Assuming that my fuel supply could last long enough, and I wasn't concerned about travel time, could this method allow my rocket to "leave" without achieving escape velocity?

The basic concept here, that you instead of relying on having a large enough velocity that the Earth can not pull you back in time just have a low and constant velocity, and keeps thrusting to counteract gravity instead, is not flawed by default, given your assumptions

  Assuming that my fuel supply could last long enough, and I wasn't
  concerned about travel time, could this method allow my rocket to
  "leave" without achieving escape velocity?

However escape velocity decreases by distance, so even though you are travelling at only 100kph, that is eventually going to be above escape velocity, as the escape velocity goes towards zero when the distance goes towards infinity. You can not really escape a gravitational field, so the only meaningful metric is if you have enough kinetic energy to not be pulled back. That is not possible by your constraint of not having escape velocity.

In a sense though, you can escape the Earth's gravitational field, using the common approximation of sphere of influence. It defines a region for Earth's gravitational influence, and ignores it outside of it. The border of the SOI can be reached without ever going at escape velocity.


Is there evidence for the speed of gravity? Is it an assumption that the speed of propagation of gravitational waves is exactly the same as the speed of light? If the speed of gravity is only slightly less than the speed of light, the effect might not show up over solar or intra galactic distances, but over extragalactic distances could it give the appearance that there is dark matter influencing motion?

It has long been one possibility that gravity does not work as in general relativity, or as in the Newtonian limit that is relevant to the gravity in a galaxy.  Thus, many different modifications to gravity have been tried.  But the reason they are generally not very successful is that dark matter seems to alter gravity on many different scales, from within a galaxy, to over galaxy clusters, to the expansion of the universe as a whole.  Thus it generally requires multiple different modifications to fit them all, and that makes for a very clunky model of gravity-- one can fit anything if given enough free parameters.  That sounds like it would be the problem for trying something based on the speed of gravity-- the timescale for propagation of changes would be very different over the three scales I just mentioned, yet dark matter has a similar influence on all three scales.


Variables in the Instability Strip I was thinking about the possibility of determining what phase of evolution a variable star is in when it is located in the instability strip (say, a Cepheid object). Can you differentiate whether it is evolving toward the blue (in HR diagram) or back toward the giant branch?

I was looking at a figure recently in Carroll and Ostlie's Intro to Astrophysics text, which brought this to mind. It shows on the HR diagram where Cepheids lie, which is right in this strip.

You can, in fact, differentiate between the crossings of the instability strip in certain types of variable stars. There have been quite a few evolutionary models devoted to this, and many, as well as scores of observations, show that the pulsational period $P$ of a Cepheid variable changes during each crossing. Look at, for instance, the period changes of 60 galactic Cepheids, shown in Figure 2 of Pietrukowicz:

The first and third crossings occur when the star is evolving redwards (towards the right of the HR diagram), and show positive $\dot{P}$. The second crossing occurs when the stars are evolving bluewards (towards the left of the HR diagram), and show negative $\dot{P}$.

Models and observations by Neilson et al. (2014) show the same thing (see Figure 1):

The initial large period change indicates the first crossing, the smaller negative period change indicates the second, and the third less drastic increase indicates the third.

Similar indicators should be present in further crossings (fourth and fifth crossings in the later stages of core and shell helium burnings), and while the amplitude of $\dot{P}/P^2$ would likely be different, the same general positive/negative trends should be present. For more information, see Turner & Berdnikov (2004).

There doesn't seem to be as much information on period changes in non-Cepheid variable stars, although Smith (2013) states that most Mira variables show only minute period changes, if any, and RR Lyrae variables may show period increases as they evolve redwards across the instability strip toward the end of their lives, as is the case with Cepheids. This, by the way, does not fully agree with evolutionary theory.


What is the apparent rotation time for a planet relative to an other planet? Just my curiosity. This is not my assignment etc.

Let planet A take m Earth days to complete one full round around the Sun, and planet B take n Earth days for one full round around the Sun. Assuming that both are orbiting in the same direction, what is the duration recorded by an observer on the planet A, as the number of Earth days the planet B took to complete one full round around planet A?

Note, an observer on Planet A can't see a rotation of Planet B. He can see a the Sun orbiting around him, and Planet B orbiting around the Sun. Thus

If Planet B has an orbit of smaller radius than Planet A, then the observer can see Planet B sometimes before, sometimes after its Sun.
If Planet B has an orbit of larger radius than Planet A, then the observer can see Planet B to cycle on the sky, roughly on the plane of the Ecliptics,  but a little bit faster as the Sun.

The calculation is this:

Planet A takes $\frac{1}{m}$ full circle in a day.
Planet B takes $\frac{1}{n}$ full circle in a day.
Thus, the observer on Planet A can see an $|\frac{1}{m}-\frac{1}{n}|$ full circle of rotation of Planet B in a day.
Thus, he can see a period of Planet B of $\frac{1}{|\frac{1}{m}-\frac{1}{n}|}$ Earth days, which is in simplified form

$$\underline{\underline{|\frac{mn}{m-n}|}}.$$


Could Mercury be the creator of Earth's Moon? Earth's Moon is thought to have been created in a glancing collision with a Mars sized planet.

I've just seen an expose on Mercury which claimed Mercury: formed in the region between Earth and Jupiter, migrated to it's current orbit through an unknown interaction,  consist of 60% core material which was explained as the result of a glancing blow with a planet sized object.

Has anyone looked in to whether a Mercury/Earth collision could be the creator of our Moon?

There's a principal in science that a strong hypothesis doesn't depend on numerous improbable factors.  I read that somewhere, I think it was Phil Plait who created the bad astronomy website, but I can't find it right now.

For Mercury to have done what you claim, a 2nd planet would have needed to form between Earth and Jupiter, OK, maybe that happened, in fact, the young solar system was probably full of protoplanets.

This planet would then have needed to change it's orbit.  This usually takes a more massive planet to toss the smaller one, but there's two problems here.  Could Mars push Mercury all the way into Earth's orbit?    Probably not.   Could Jupiter?   Sure, but then, where did Mercury get all it's Iron if it formed between Mars and Jupiter.  It's becoming problematic.

Planet on planet collisions are rare, obviously not impossible.

Then there's the problem of how did the collison result in Mercury flying off, losing some material, the moon forming and Earth rapidly rotating.  Planet on planet collisions tend not to be glancing blows because planets are too big, and gravitational wells are pretty deep.   I don't want to say it's impossible but that's an odd scenario which seems highly unlikely.

And then, Mercury changes it's orbit again, perhaps gravitational assist from Venus and it ends up where it is.

A hypothesis like that depends on that many rare events all happening isn't not going to be a popular one.   Building a hypothesis on a single improbable event is fine.   The giant impact theory began like that.

A good hypothesis also has evidence behind it.   Mercury looks like it formed about where it is now, clearing out it's orbital neighborhood and being made up of denser material that was more likely to orbit closer to the sun.    There's very little reason to stand behind the Mercury impacted Earth idea and there's no evidence to support it.


Isn't it absurdly unlikely that our Moon would constantly be "facing" us? I never thought about this until recently, but our Moon is constantly rotating in such a way that it happens to perfectly be "looking at us" at all times. That sounds incredibly unlikely to me.
Couldn't we just as well have had a moon which constantly shifts patterns by rotating in a different manner, rather than us always seeing the same craters on the "moon's front side"?
I can very much understand that this mislead people "in olden times" to assume that this universe is constructed in a very different way from what it actually is. If the moon had been "shifting in pattern" from day to day, or even hour to hour, or even minute to minute, it would be much easier to conclude that it's a spinning orb far away, orbiting our orb. Or at least that's what it feels like to me.
The fact that the moon is always "looking at" us seems highly unlikely. It's rotating around us, but also rotating itself in exactly such a way that it appears to be "fixed" from our surface as we look at it. It's very strange to me.
How did we end up with such a moon? Wouldn't the far, far more likely case be a "spinning" moon rather than "fixed" one? Maybe the moon would orbit us but not rotate at all? (I'm unsure what that would look like from here.)
Or is there something I'm missing here?

The key concept is called Tidal locking. Earth's gravity forced the Moon to rotate in that way during the first few tens of millions of years after the Moon formed and then kept it that way.
The wikipedia page describes the mechanism:

Consider a pair of co-orbiting objects, A and B. The change in rotation rate necessary to tidally lock body B to the larger body A is caused by the torque applied by A's gravity on bulges it has induced on B by tidal forces.[5]

The gravitational force from object A upon B will vary with distance, being greatest at the nearest surface to A and least at the most distant. This creates a gravitational gradient across object B that will distort its equilibrium shape slightly. The body of object B will become elongated along the axis oriented toward A, and conversely, slightly reduced in dimension in directions orthogonal to this axis. The elongated distortions are known as tidal bulges. (For the solid Earth, these bulges can reach displacements of up to around 0.4 metres (1.3 ft).[6]) When B is not yet tidally locked, the bulges travel over its surface due to orbital motions, with one of the two "high" tidal bulges traveling close to the point where body A is overhead. For large astronomical bodies that are nearly spherical due to self-gravitation, the tidal distortion produces a slightly prolate spheroid, i.e. an axially symmetric ellipsoid that is elongated along its major axis. Smaller bodies also experience distortion, but this distortion is less regular.

The material of B exerts resistance to this periodic reshaping caused by the tidal force. In effect, some time is required to reshape B to the gravitational equilibrium shape, by which time the forming bulges have already been carried some distance away from the A–B axis by B's rotation. Seen from a vantage point in space, the points of maximum bulge extension are displaced from the axis oriented toward A. If B's rotation period is shorter than its orbital period, the bulges are carried forward of the axis oriented toward A in the direction of rotation, whereas if B's rotation period is longer, the bulges instead lag behind.

Because the bulges are now displaced from the A–B axis, A's gravitational pull on the mass in them exerts a torque on B. The torque on the A-facing bulge acts to bring B's rotation in line with its orbital period, whereas the "back" bulge, which faces away from A, acts in the opposite sense. However, the bulge on the A-facing side is closer to A than the back bulge by a distance of approximately B's diameter, and so experiences a slightly stronger gravitational force and torque. The net resulting torque from both bulges, then, is always in the direction that acts to synchronize B's rotation with its orbital period, leading eventually to tidal locking.

The same process is, very slowly, slowing the Earth's rotation to synchronize with the Moon, although I'm not sure that process will ever complete.


What will happen if a black hole appears near our solar system? We know that black hole possesses immense power and can destroy anything coming in front of it's path. If such a black hole appears near our solar system, are we going to survive? Is there any way not to fall in a black hole?

The way not to fall into a black hole is essentially the same as the way not to fall into the Sun.  It's no more dangerous than any other body with the same mass, which is typically a few times the Sun's mass.

The danger of a passing black hole (or other star) is not that it could swallow us but that it could perturb our orbit, with unpredictable effects on climate; as well as stir up the comets of the (hypothetical) Oort Cloud, sending some of them toward the inner planets, with the possibility of an impact that could ruin your whole day.


What is "emission measure"? Many papers mention emission measure, and some of them give the expression of EM, BUT there is no clear definition of EM.

My question is what EM is on earth? Its unit is supposed to be cm-3.

How to calculate EM? here(4B5) gives an expression, but it can not be derived from a light curve directly.

Given an X-ray light curve(not from the Sun), probably I need to devide the light curve into many bins and do spectral fitting.

Then what should I do? Rererence papers with clear definition and expression are welcome.

Emission measure is (usually) used in X-ray and EUV astronomy, though I suppose also in cases of optically thin radio emission. It is defined as the square of the number density of free electrons integrated over the volume of the plasma.

$${\rm EM} = \int n_e^2 \ dV$$

The flux of optically thin emission from a plasma (e.g. thermal bremsstrahlung) is then directly proportional to the emission measure of the plasma multiplied by a temperature dependent cooling loss law.

In other words, when you measure the flux of X-rays from an unresolved optically thin emitter, there is a degeneracy between the electron number density (squared) and the overall plasma volume.

When you fit an X-ray spectrum with an optically thin model, the emission measure (divided by $4\pi d^2$, where $d$ is the distance to the object), is a multiplicative free parameter.

Your question about calculation is extremely difficult to answer. Suppose I measure a count-rate of $N$ X-ray counts per second using some X-ray telescope (I can only assume that's what you mean by a "X-ray light curve".).

The count-rate received at the telescope depends on: the emission measure (as defined above) multiplied by a term that depends on the temperature (or temperatures) of the source, the chemical composition of the source and the adopted emission process (is it free-free thermal bremsstrahlung, a thermal plasma or something else). It is then attenuated by any intrinsic absorption in the source and any absorption between us and the source and by the distance to the source (assuming the radiation is isotropic). Finally what is detected is determined by the response of the X-ray detector to X-ray photons as a function of energy.


Why does a particle feel no force at radii greater than itself? I am working through Liddle's "An Introduction to Modern Cosmology", and in a newtonian derivation of the Friedman equation he states:

  in a spherically symmetric distribution of matter, a particle feels no force at all from the material at greater radii, and the material at smaller radii gives exactly the force which one would get if all the material was concentrated at the central point.

He attributes this result to Newton, Can anyone give a mathematical or physical explanation as to why this is true?

Thanks

"in a spherically symmetric distribution of matter, a particle feels no force at all from the material at greater radii, and the material at smaller radii gives exactly the force which one would get if all the material was concentrated at the central point".

This may be poorly worded. The idea is not that the distance from the center of the body must be smaller than the radius of the particle. Rather, those parts of the (uniform, spherical) body which are further away from the center than is the particle have all their respective gravitational attractions cancelled out.


Changes of the velocity of the Moon due to impacts from other objects? Has the moon’s velocity changed since ‘shortly’ after it’s formation due to impacts?

Wouldn't it be likely that a significant collision would have occurred over such a long period of time?

It would seem an object with enough momentum to change the velocity of the Moon would have made an impact on its velocity at some point over the millennia.

Our best current understanding of the formation of the Moon has it forming quite close to Earth.  Because even then its orbit around the Earth took more than one day (Earth was spinning faster, also), the effect of tides has always been for the Earth to transfer some of its rotational energy to the Moon. This slows the Earth's rotation  and makes the Moon move outwards into a higher (and slower) orbit.

It's been around four billion years since the Moon last suffered a major impact, so collisions have not played a significant role since then.

Expanding (a lot) on this:

How do we measure Lunar ages? Two ways: Cratering and isotope dating. The cratering technique was developed by William K. Hartmann of the University of Arizona and makes use of the amazingly simple fact that when two craters overlap, you can tell which is older. Combine that with the observed fact that while the cratering rate varies with time, it changes smoothly and the distribution of craters is essentially random. Using that and a lot of painstaking analysis of photography, the order of events on the Moon can be determined quite closely (E.g., this crater, then this mare, then those craters, etc.) as well as getting rough estimates of the actual time.

Isotope dating of Lunar meteors (bits of Lunar rock blasted into space by impacts) and Apollo samples lets us put precise times to various bits of the sequence resulting in a reliable chronology going back nearly to the Moon's formation.

Arguably there were four stages in the Moon's history:

Stage 1: Formation of the Moon by Giant Impact

The Moon forms out of the debris in orbit after a Mars-sized impactor strikes the Earth about 4.5 billion years ago during the formation of the Solar System.  This was the final giant impact involving either the Earth or the Moon.

Stage 2: The Early Years

When formed, the early Moon was around 25,000 miles from Earth and the huge tides raised by the Moon in the Earth and by the Earth in the moon rapidly (by astronomical standards, anyway) further out.  The Moon (and Earth) continued to be hit by big rocks (dinosaur-killer-sized and bigger) but not by things in the many hundreds of miles in diameter.

Stage 3: The Late Heavy Bombardment

We can tell from cratering patterns and from dating Lunar samples, that 4.1 to 3.8 billion years ago there was an upsurge in big rocks (dinosaur-killer-sized) hitting the Moon. We also see evidence of this on other planets, though not on Earth since so little remains of the Earth's crust from that era.) We don't know why the LHB happened, though there are lots of theories.  (There are also scientists who think the LHB was not as heavy and more spread out, but they are currently in a minority.)

Stage 4: Retirement

Since the end of the LHB, the Moon has had few new craters created and has simply slowly receded from Earth (enjoying a serene and fairly uneventful old age.)  There are still craters being formed now and then -- the freshest big one, Tycho, is only 108 million years old!  Another recent one, Aristarchus, is around 400 million years old.

These are typical of the last few billion years: Very occasional craters caused by rocks in the 10-50 mile rage. They make a big hole, but happen rarely (ca. even hundred million years) and have had no significant impact (pun intended) on the Moon's orbit.


Why Venus evolved so differently from Earth? I read somewhere that the reason why Venus has a runaway greenhouse effect and Earth doesn't is the former's inability to form separate tectonic plates due to slightly higher temperature softening the then-cooling tectonic plate, preventing its fracture and thus disabling the $\mathrm{CO}_2$ scrubbing function of tectonic activity, which absorbs $\mathrm{CO}_2$ from air by rock weathering and then puts it back into the mantle. And once $\mathrm{CO}_2$ is a runaway in the atmosphere $\text{H}_2\text{O}$ and $\text{SO}_2$ will join forces with it multiplying the greenhouse effect. Therefore, the surface temperature of Venus rose to astonishing values long before life formed on Earth. Is this theory at least possibly correct?

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

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

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


Distance and orbital period for terrestrial binary planets This article, http://phys.org/news/2014-12-binary-terrestrial-planets.html, suggests binary planets could orbit each other at a distance of only three planet radii. For two earth-like planets, that is approximately 12,000 miles. Does this distance seem accurate? How long would it take for these planets to orbit their barycenter?

The formula for orbital period is $$T= 2\pi\sqrt{\frac{a^3}{G \left(M_1 + M_2\right)}}$$

For your example, $a=19000000$ metres, and $M_1 = M_2 = 6e24$

Which gives an orbital period of just over 5 hours.

Whether such an arrangement is likely depends on the nature of the the formation of the binary planet, and how the orbital period and rotation rates are affected by tides.


What is the average temperature of all planets at 1 bar? Jupiter's (and Saturn's) "surface" temperature is defined as that at 1 bar. It is inconsistent to say the other planet's surface is their hard surface. Applying the same measure what is the equivalent standard temperature of all the IAU defined planets?

At 1 bar (0.9869 atm / 14.5 psi) pressure level the planets have the following average temperatures:

Venus: 30 degrees Celsius (1 bar level is about 33 mi or 53 km high)
The Earth: 15 deg C
Jupiter: minus 145 deg C
Saturn: minus 178 deg C
Uranus: minus 197 deg C
Neptune: minus 200 deg C
Titan's temperature ranges from minus 180 deg Celsius on the surface to minus 93 deg Celsius in the upper stratosphere. Titan's surface air pressure is 1.45 atm (1.47 bar).

Mercury has no atmosphere while Mars' atmosphere doesn't reach 1 bar anywhere on the planet.


How far away is the light that would reveal the Big Bang? I guess theoretically if we could go faster than light, which we clearly cannot, at some point we'd be able to see the big bang itself. Just curious - how far away is that light?

The particle horizon is the point where light arriving at us now would've had to have been emitted at the big bang (though note this is hypothetical as the Universe did not become transparent to light until it was several 100,000 years old).. How far away it is depend on the model you use (not to mention there are several different distance measures in cosmology), but it's proper distance is reckoned to be about 46.9 billion light years.

A note of caution though, when calculating the particle horizon cosmic inflation is usually ignored. What is light arriving (again hypothetically) from usually called the particle horizon therefore would be actually light that was emitted from just the end of the inflationary epoch. If you were to take a model with a big bang singularity and inflation the "true particle horizon" would in fact be much further away.


Aphelion and the solstice We have just passed one of the solstices and are approaching aphelion.  The two events are close but not simultaneous.  There is no very obvious (to me) reason why the two should coincide or be close.  Is there a less obvious reason why they are close?  If so, is the gap closing, growing, or remaining approximately constant?

The two coincided about 800 years ago. The December solstice and perihelion date coincided in 1246, 773 years ago.

There are many different concepts of what qualifies as a "year". Three of them are the sidereal, tropical, and anomalistic years. The sidereal year measures how long it takes for the Earth to complete one orbit about the Sun with respect to the remote stars.

The tropical year measures how long it takes for the Earth to pass from one March equinox to the next. The tropical year is about 20 minutes shorter than the sidereal year due to the slow changes in the Earth's rotation axis caused by the Moon and the Sun.

The anomalistic year measures how long it takes for the Earth to pass from one perihelion to the next (or one aphelion to the next). The anomalistic year is about 7.8 minutes longer than the sidereal year due to gravitational perturbations from the other planets, with Jupiter and Venus dominating over all the others. The tropical year is thus shorter than the anomalistic year by about 28 minutes. That the tropical year and anomalistic year have slightly different frequencies means that there is a very slow beat frequency with a beat period of about 26000 years.

Human civilization arose about half of this period ago (about 12000 years ago) when the last glaciation ("ice age") ended. At that time, the anomalistic and tropical years were almost 180 degrees out of phase with respect to where they are now. The end of a glaciation typically occurs when northern hemisphere summers coincide with perihelion. That makes summers in far northern latitudes short in duration but rather hot.

One of the key proxies regarding whether glaciations start or end is the mean July temperature at far northern latitudes. When perihelion occurs a few weeks after the June solstice, the short but hot summers that result makes snow from the previous winter melt rather than build up. The melting that results from such short but hot summers triggers a positive feedback that makes the end of a glaciation very rapid, in a geological context of "rapid".

To clarify things,

  So, there is no particular reason in orbital mechanics; it just happens that we live in an era when they are close together?

Exactly.

Or maybe not. The end of the last glaciation, which occurred about 12000 years ago, massively spurred human development. The cycle then was almost exactly opposite what it is now. It took about 10000 years for humanity to develop to a somewhat advanced level (e.g., Roman Empire level), and then only another couple of thousand to develop to the stage where we are now. Whether humanity can survive for another 13000 years without wiping itself out will determine whether this was happenstance.


How Do we know about redshift? Okay so scientists use the red shift to know that galaxies are moving away from us faster than light. How do we know that its not just red light from red stars. i mean most stars are in the infrared spectrum.

Let's start with a quick clarification: Red Shift is not the same as red light.

Red light is just electromagnetic radiation with a 400–484 THz frequency range, the lowest our eyes can see - highest being violet light, with a 668–789 THz frequency.

Red Shift is an observable effect when you analyze the spectroscopic signature of an electromagnetic radiation source that is so far from us that the expansion of the universe (whose unit of measurement is called the Hubble Constant) is noticeable because of the frequency drop, called Doppler effect.

In short, it works like this:

Pick one of the elements the universe is made of - hydrogen (75%), helium (24%) and trace amounts of everything else (1%). For this example, let's stick with hydrogen.
Determine the spectrum signature of hydrogen by running the light generated via electric discharge on the gas through a spectrometer:

source: flatworldknowledge.com

You'll notice some markings as you examine the light dispersed by the prism. Those markings are called narrow bands, and they appear as a consequence of quantized energy states. Hydrogen narrow bands are well known : 656.2, 486.1, 434.0 and 410.1 nanometers.
Now look at the sky. Pick any bright dot, run its light through a prism, and knowing that it is also made of roughly 75% hydrogen and 24% helium compare its spectroscopic signature to the bands you obtained earlier.

Because of the Doppler effect caused by the expansion of the universe the light waves are stretched, making the frequency drop - shifting the whole spectroscopic signature towards lower frequencies. The further the object is, more noticeable the shift towards red is.

Thus, the term Red Shift.

Now, answering your question:

  How do we know that its not just red light from red stars?

Because even regular light from red stars will shift towards the infrared side of the spectrum as a consequence of the Doppler effect.

Image Sources:
General Chemistry: Principles, Patterns, and Applications, v. 1.0
Starts With A Bang!

More information:
Spectral Lines of chemical elements


What is the antonym of "closest approach"? The distance from Earth to Mars, during their closest approach, is about 55 million kilometers.
At their furthest apart, that distance would be about 401 million kilometers.

Distance
at closest approach
at ___________

Earth–Mars
55 million km
401 million km

Earth–Jupiter
629 million km
928 million km

Is there a standard astronomical name for "their furthest apart"?
My vague understanding is that if one body orbits around another, then we have words like "at perigee" (closest approach to Earth) "at apogee" (furthest apart from Earth); "perihelion" and "aphelion"; and in general "periapsis" and "apsis." However, it doesn't seem correct to talk about the distance from Mars to Earth "at apsis" because neither body orbits the other.
IIUC, in the specific case of Earth and Mars, we might also say that their closest approach happens (more or less) when Mars is in opposition (relative to the Sun, as seen from Earth); but I don't think I can infer from that any useful terminology for their maximum distance — which would happen more or less when Mars is in opposition relative to the Earth, as seen from the Sun. [eshaya's answer indicates that the phrase I'm looking for here is "...when Mars is in conjunction (relative to the Sun, as seen from Earth)." Contrariwise, for Venus, both extrema occur during conjunctions.]
(I'm interested in names for the distance and/or the position. "Closest approach" applies to both distance and position; "apsis" applies only to the position AFAIK. You wouldn't say "the apsis of the Earth and the Moon is 406,000 km.")

The term "maximum separation" is often used, though maximum separation can also refer to the maximum angle between two bodies in the celestial sphere.
Here is an example from Quintana and Lissaur of the usage referring to distance:

close binary stars with maximum separations $Q_B≤0.2 AU$

and here is an example from Nouh of the usage referring to angle:

In this paper, an efficient algorithm is established for computing the
maximum (minimum) angular separation ρ max(ρ min) [...] of visual
binary stars


Estimating a star's radius, temperature, and luminosity based on its mass (See updated figure and description below.)

I've been trying to generate ballpark estimates for the radius, temperature and luminosity of stars in the main sequence based solely on their masses (assuming the same composition for all stars).  The idea is to iterate through masses in steps of, say, 0.1 solar masses from 0.1 to 100, and roughly trace out the curve of the main sequence on an HR diagram.

For luminosity, I'm using something like this:

  if ( mass < 0.43 ) {
    L = 0.23*L_sun*pow(mass/M_sun,2.3);
  } else if ( mass < 2 ) {
    L = L_sun*pow(mass/M_sun,4);
  } else if ( mass < 55 ) {
    L = 1.4*L_sun*pow(mass/M_sun,3.5);
  } else {
    L = 32000*L_sun*mass/M_sun;
  }

For radius, I'm using something like this:

  if ( mass < 1 ) {
    R = R_sun*pow(mass/M_sun,0.57);
  } else {
    R = R_sun*pow(mass/M_sun,0.78);
  }

And, using these, I'm calculating the temperature from the luminosity-radius-temperature relation like this:

T = pow(L/(4.0*M_PI*R*R*sigma), 0.25);

where sigma is just the Stefan-Boltzmann constant.

All of this works about as well as I'd expect for stars of 1 solar mass or above, but breaks down completely for M dwarfs, as you can see from the attached image which shows my meager effort superimposed on HYG data.

Note that I'm plotting B-V here, calculated from temperature like this:

BV = pow(5601.0/T,1.5) - 0.4;

and absolute magnitude, calculated from luminosity like this:

magnitude = -2.5*log(L)/log(10) + 71.1974;

What can I do to improve this a little?  I'll add that this is aimed at being part of an exercise for undergraduates who are beginning programmers, so I'm looking for simple ballpark approximations, not anything fancy.

Update:
Based on Rob Jeffries' recommendation below, I took a look at the Mamajek data.  Here's a plot of B-V versus temperature from that data:

I've superimposed a fitting function of the form:

bv(t) = a/(b*t) - c

where the best-fit parameters are:

a = 4.2413
b = 0.000576479
c = 0.607144

Using this function to calculate B-V moves my numbers in the right direction, but it still doesn't quite do the trick, as shown in the figure below (the new B-V values are the curve that's lower on the right-hand side):

You could plot $L$ vs $T$ and compare your model result to the measured/estimated $L$ and $T$ of main sequence stars?

You have not explained or labelled what your diagram is, but it isn't $L$ versus $T$; I suspect that it absolute $V$ magnitude versus $B-V$ colour.

The transformations between $L,T$ and $V$, $B-V$ are highly non-linear, especially at low temperatures (which is why your plot fails to reproduce the M-dwarfs) and involve folding a model atmosphere through some standard filter responses or using relationships between $T$ and $B-V$ and bolometric corrections and $T$. (e.g. Mamajek 2019)

EDIT: For the clarified question.

Your relationship for $T$ vs $B-V$ is not applicable at lower temperatures.

Your relationship between absolute magnitude and luminosity takes no account of the spectral energy distribution of the light from the star (i.e. it assumes the same bolometric correction for all temperatures, which isn't correct). You must calculate the bolometric magnitude from the luminosity and then apply a temperature-dependent bolometric correction to calculate a V-band magnitude.

The solution is to construct relationships that are valid over the full range of temperatures using the resource I referenced above.


How do we know the big bang didn't happen in an existing universe? I understand the evidence for the big bang (expansion, background radiation, etc), but how do we know that it was the start of the universe? Why couldn't it have occurred in an existing, but very sparsely populated universe?

We don't know what's on the other side of the Big Bang, if anything at all. The Big Bang theory as well as the rest of physics is agnostic on this possibility. During the time period after the Big Bang, the entire universe was basically in a hot, dense plasma state; any physical signals propagating from a period before the Big Bang that could potentially influence the present or provide us with information of a pre-Bang past would be eradicated by the extreme conditions.

So the singularity of the  Big Bang is effectively the start of our universe because it is effectively an impenetrable cosmic firewall disconnecting us from anything that came before. Maybe there's another room on the other side of the wall, or maybe it's just more wall. Until someone comes up with a more refined theory or an experiment finds a way to detect a tiny signal that somehow made it through, it's pointless to speculate.


To several decimal places, how many days are in one year? This morning I was interested in finding out how many days are in one year on average, to several decimal places (because I am a mathematician, so I like to be accurate). Unfortunately, Googling the phrase "days in a year" results in several answers that differ as soon as the third decimal place (taking into account rounding).

A second question in addition to the title question: Does the duration of earth's orbit increase and decrease, or change only monotonically over time?

A mathematician would probably find the imprecision of precise orbital time a little frustrating.
A second can be defined as

"the duration of 9 192 631 770 periods of the radiation corresponding
to the transition between the two hyperfine levels of the ground state
of the caesium 133 atom"

A 24 hour day is 86 400 of those, well, more or less.
A problem that early time-keepers ran into was that the length of the day varied based on the time of year.   Earth does a complete rotation in about 23 hours, 56 minutes, so, the rotation really doesn't factor into it directly, but because Earth orbits the sun about 1 degree of arc every 24 hours, an apparent solar day is has been defined as 24 hours long for centuries.    Days are shorter than that when the earth is closer to the sun, at perihelion, (by about 20 seconds) and days are longer than the 24 hour average by about 30 seconds at aphelion.   Not enough for anyone to notice unless they were measuring solar time very precisely.
So a day has to be averaged out, called a mean solar day, and that's what 24 hours was based on.   The problem is, Earth's rotation is slowing down.  The Moon is very gradually slowing down the rotation of the Earth by tidal interaction, so an average day is currently about 86 400.002 seconds.   Leap seconds are added almost every year and in the future, leap seconds will be added more often.   Officially a day is still 86 400 seconds, and when a leap-second is added, the "leap-day" is 86 401 seconds.   But you could use either number for number of days in a year, average day (currently 86 400.002) or SI day* (86 400).  As I understand it, a year is usually measured in 86 400 second days or SI days, not "average" days, but it's important to be specific about which one you're using.  I've seen the occasional article that says that 200 million years ago a year was about 400 days long.  That's clearly using average says, not SI days.
And if you live in high altitude, your clock will run even faster and a day will run a tiny bit longer thanks to relativity but . . . lets not even go there for the calculations).
A solar year, based on the position of the sun and the precise time of the solstice of your choice, which is a precise and momentary alignment.   The length of a solar year changes a bit on a year-to-year basis.   The position of the moon and gravitational perturbations from other planets, affect Earth's period of rotation a little bit, so like the day, the year should be averaged out also.   Modern leap-year calendars adjust days to years quote well, but not to the accuracy you're asking about.
Looking at the time of recent summer solstices (I've copied them down below for the June solstice).
2010-06-21 11:28
2011-06-21 17:16
2012-06-20 23:09
2013-06-21 05:04
2014-06-21 10:51
2015-06-21 16:38
2016-06-20 22:34
2017-06-21 04:24
2018-06-21 10:07
2019-06-21 15:54
2020-06-20 21:44
There's as much as 15 minutes variation in the length of a year over this 11 year sample.  That's not because Earth's orbit is permanently changing.  These are mostly fluctuations due to the Moon and nearby planets.
An "average year", or mean tropical year, is currently about 365.2422 86 400 second days and if you want to get more accurate, 365.2421897 days on 1 January 2000.  Formula in the link for adjusting to different years.
I suspect, but I'm not 100% certain that the Earth's mean tropical year moves back and forth with some of Earth's Milankovich cycles.   It's not slowing consistently in the sense that a day is slowing due to the Moon's gravity and inverse tidal force.
According to Kepler's laws, the orbital period is based mostly on Earth's semi-major axis.  The problem is that Kepler's laws aren't 100% accurate.   Relativity is a factor when you try to pinpoint a year precisely to the second and variation in eccentricity, which affects Earth's orbital speed and the precise timing of an average year.
What I know about orbital stability, is that Earth's semi-major axis changes very little, even during the Milankovich cycles, and orbital projections say, that it will probably change little even over time-frames of millions of years.
The Sun is also losing mass, which should slow the Earth's orbit over time, but that effect is also very tiny.   Over millions of years, it's thought that the length of a year doesn't change very much.  The length of a day, over time-periods that long, changes quite a bit more rapidly.  An unanticipated orbital change is always possible, but not thought to be likely.   Orbits don't leave much in the way of track-able footprints, so they're hard to pinpoint over long periods of time and are mostly estimated by mathematical models so, take with a grain of salt, but models do suggest that a year is pretty much constant even over periods of millions of years.
For further reading, maybe start here:   https://en.wikipedia.org/wiki/Equation_of_time
Hope that helps.  Some of the math gets kinda tricky.

Thanks zephyr.  I wasn't familiar with that term.


Can weak gravitational lensing or microlensing-induced wavefront distortion limit resolution of absurdly large aperture telescopes? This is a theoretical question.

This answer to the question If we had the right technology could we see a distant star in detail? (presumably space-based) primarily addresses the scaling of diffraction-limited resolution with aperture $D$. For existing and near-future telescopes this is assumed to follow $\lambda/D$.

Imagine a future optical or radio interferometric telescope of absurdly large effective aperture or baseline, say AU or even larger, operated similarly to the Event Horizon Telescope (local downconversion and timestamping, interferometry performed offline)

Could weak gravitational lensing effects or gravitational microlensing result in wavefront distortion that if not corrected, would result in degraded resolution? Note I'm not asking about distortion of the field, but instead a limitation to the resolution; degraded point spread function.

If so, would this problem be amenable to correction in the same way that atmospheric and ionospheric effects (visible and radio) are dealt with? Would it generally be static on the timescale of the observation?

As far as microlensing is concerned:

Not likely

This is a bit of a dodge on the question, but it is hard to come up with a scenario where microlensing becomes the dominant PSF term.

Microlensing occurs during conjunction with a stellar or planetary object, and results in a momentary (seconds-to-days timescale) increase in the brightness of the observed star. It is an extremely rare event, with rates around 100 in 17 million stars in the most recent surveys.

Stellar densities ($\rho$) around our solar system are on the order of 0.14 stars per cubic parsec. With such a low number density, microlensing does not pose any practical limitation on the resolution of radio telescopes. Lets do some scratch-pad estimates to see:

If we assume a synthetic aperture ($D$) of 1AU ($4.84\times10^{-6}$parsecs) , a wavelength ($\lambda$) of 1mm (similar to the view of Betelguese by the ALMA observatory) , and a desired spatial resolution ($s$) of 10,000 km at a distance $L$:

$$
\begin{align}
\frac{s}{L}  &= 1.22 \frac{\lambda }{D} \\
L &= \frac{sD}{1.22 \lambda } = 40\;\text{kilo-parsec}
\end{align}
$$

This is roughly the diameter of the milky way galaxy. So an interferometer operating in the 1mm wavelength with a 1AU baseline has a 10,000 km resolution at a distance of 40 kpc.

How many stars are between the 1AU-wide observatory and the target star? Using a cone of base diameter $D$ and height $L$ as a volume and the above stellar density, we get the average number of stars ($N$) between our aperture and the observed star:
$$ N = \rho \frac{\pi}{12}D^2 L = 3 \times 10^{-8} \; \text{stars}$$

You would have to ramp $D$ up by a factor of $10^4$ before $N \approx 1$ and 'microlensing aberrations' were more than just an extremely rare event.


What is Earth believed to have looked like before the collision with Theia? Let's assume the moon was formed because Earth collided into Theia. (There are numerous other theories, but this one is the most widely accepted.)

In that case, are there theories that address what the Earth surface looked like? E.g. would the Earth have had bodies of water at the time?

I've read some websites and a book!, and some say it did, some say it did not.

Greenwood et al suggest that Earth had a lot of its water (maybe upto 70%) before Theia, but I can't find anything definite saying that it was liquid, although several sites reporting on the Greenwood paper assume that Earth was cool enough to have an ocean. It almost certainly had a crust, though. And immediately after the impact it had a magma ocean for quite a while. ;)


Is the M81 group bound to the Local group? Andromeda and the Milky Way are set to collide in 4bn years. Will the Local Group collide with M81, and what about further away groups? if so, when?

Interestingly enough, although we tend to group galaxy clusters together into superclusters, superclusters are actually not gravitationally bound. This suggests that, due to Hubble flow, galaxies in separate clusters will never "collide". In fact, it may be impossible to ever leave one's galaxy cluster.

EDIT:

In a previous version of this post, I stated the the Great Attractor may be reason to suspect a gravitational binding of Laniakea. In reality, the Great Attractor only reduces the expected relative velocities of these galaxies away from one another by between $200-400$ $\mathrm{km}\cdot\mathrm{s}^{-1}$—I would like to thank eshaya for pointing out my error here. Due to the current expansion rate of the universe of $67.6^{+0.7}_{-0.6}$ $\mathrm{km}\cdot\mathrm{s}^{-1}\cdot\mathrm{Mpc}^{-1}$, some galaxies in Laniakea have recession velocities of up to $30000$ $\mathrm{km}\cdot\mathrm{s}^{-1}$, as given by Tully et al. (2014)—far greater than could ever hope to be counteracted by the Great Attractor.

The question now becomes one of whether the M81 Group is gravitationally bound to the Local Group. Given the M81 Group's status as being one of the closest to the Local Group, at only $3.6$ $\mathrm{Mpc}$ distant, I would not be entirely surprised if through some chance it managed to be. Given the current estimate for Hubble's constant, due to Hubble flow the M81 Group would be expected to recede at $\sim243.4$ $\mathrm{km}\cdot\mathrm{s}^{-1}$. This is what would need to be countered through gravitational effects for the two groups to be bound, and thus eventually coalesce.


Eccentric Earth Orbit and Day/Night Cycles If the Earth's rotational axis was perpendicular to the orbital plane, but the orbit also highly eccentric (sun at center), how would that affect the day/night cycles?

What I think is that during the time the Earth's closer to the sun, because the gravitational pull is stronger, it would make the Earth spin faster than thereby creating much shorter days than if the Earth was near the focii of the orbit (on which the days/nights would be longer). Is this line of thinking correct?

Your thinking is incorrect.    High gravity, or, more specifically, strong tidal forces lead to tidal locking not faster spinning.   Moons are a good example of this.   The Earth's gravity keeps the same side of the Mon facing the Earth at all times.   The Moon still rotates but it's rotation is in sync with it's orbit around the Earth, 1:1, so the same size of the Moon (the heavier/denser side) always points towards Earth.   Many moons are tidally locked to their planets as a result of the strong tidal force.  Pluto and Charon are tidally locked to each other.

That said, tidal locking takes a long time.  Just passing close to a massive object wouldn't stop a planet from rotating.   It takes many orbits to create the tidal locking, usually millions or billions.  Because planets tend to be pretty uniform in density, gravity doesn't affect their rotation very much.

A highly eccentric orbit, the sun isn't close to the center, either.   A low eccentricity orbit, the sun is close to the center.

A highly eccentric orbit would affect seasons.  With no Axial tilt, the seasons would be entirely governed by the distance to the sun, not the position of the sun in the sky which governs seasons on Earth as it is.   To clarify, it's the Axial tilt that causes the seasons, not the distance to the sun.  But as you specified with zero axial tilt, then the distance would govern the seasons, but at Earth's current eccentricity, that effect would be small.   The greater the eccentricity the greater the seasonal variation in this hypothetical and the seasons would be planet wide, not alternating between the two hemispheres.

Summer would also be shorter than winter due to the Earth moving faster around the sun at the closest point,

The days themselves wouldn't be affected much.   The length of a typical day (Synodic) would vary by a minute or two based on the Earth's orbital speed.  It wouldn't be very noticeable but with a perfect 0 degree axial tilt, the 12 hours of sunshine would vary slightly based on orbital speed and position in the eccentric orbit. . . and the size of the sun in the sky and atmospheric refraction. . . . but only a couple minutes here or there with the Earth's current rotation speed.


How exactly is the Initial Mass Function (IMF) calculated? The Initial Mass Function (IMF) is the empirical function which describes the initial masses of a population of stars. My questions are,

1) What are the various IMF's which are used?

2) For each, what type of population do they describe? (e.g. - galaxy, dwarf galaxy, globular cluster, etc..)

3) And how are they actually calculated? (meaning, do they come from simulations/observations, and what assumptions are made about each?)

Entire answers and pieces of answers are all welcome. Formulae (in latex, please) are encouraged.

What is it?
An IMF, $\Phi(m)$, is defined such as $\Phi(m){\rm d}m$ gives the fractions of stars with a mass between $m - {\rm d}m/2$ and $m + {\rm d}m/2$, and with a normalized distribution
$$\int_{m_{\rm min}}^{m_{\rm max}}m\Phi(m){\rm d}m = 1\ M_{\odot}.$$
Note that these boundaries ($m_{\rm min}$ and $m_{\rm max}$) are ill-defined, but typically of the order of 0.1 $M_{\odot}$ and 100 $M_{\odot}$, respectively.
IMFs
The various IMF used are the following, with their main characteristics:

the Salpeter's IMF, that is a parametrization of the IMF by a simple power-law, of the form
$$\Phi(m){\rm d}m \propto m^{-\alpha}{\rm d}m;$$
the Miller & Scalo's IMF, that is a parametrization of the IMF by a log-normal distribution of the form
$$\xi\left(\log(m)\right) = A_0 + A_1 \log(m) + A_2\left(\log(m)\right)^2;$$
the Kroupa's IMF, that is a parametrization of the IMF by a broken power-law;
the Chabrier's IMF and Chabrier's system IMF, that is a combination of log-normal distribution (for low mass stars with masses less than 1 $M_{\odot}$) and and a power-law distribution (for larger masses). The difference between the IMF and the system IMF is to merge resolved objects into multiple systems to compute the magnitude of systems instead of individual stars.

Determination
As you see, all these IMF are parametrization, deduced from observations. In general, the observations used to infer these mass functions come from star clusters in our galaxy. All you need to do it is to find a mass-magnitude relationship to deduce, from an observed luminosity, a mass function. In general, the number density distribution per mess interval, ${\rm d}n/{\rm d}m$, takes the following form
$$\frac{{\rm d}n}{{\rm d}m}\left(m\right)_\tau = \left(\frac{{\rm d}n}{{\rm d}M_\lambda(m)}\right) \times \left(\frac{{\rm d}m}{{\rm d}M_\lambda(m)}\right)^{-1}_\tau,$$
for a given age $\tau$ and an observed magnitude $M_\lambda$. Then, it is just a matter of parametrization, but also of how well it can arise from a proper theory.
For this matter, the Chabrier's IMF is probably the one that is best back up by theoretical arguments. It relies on a gravo-turbulent theory, taking into account all the possible supports (thermal support, turbulent support and magnetic support) plus the dual nature of turbulence, that both favors star formation by compressing the gas, and impedes star formation, by dispersing the fluid. All the dirty details are given in Hennebelle & Chabrier (2008) and Hennebelle & Chabrier (2009), showing how you can analytically deduce an IMF from these theoretical considerations.
Applications
As far as I know, these IMF are more or less used for every type of population. However, you won't favor Salpeter's IMF if you have enough resolution to resolve low-mass objects, that are not at all well-taken into account with this IMF. You should also favor the Chabrier's system IMF in case of unresolved objects.
To know if all these IMFs are really well-suited to any kind of population is an open and difficult question (the so-called question of the universality of the IMF), in particular because you need to resolve individual stars in clearly identified clusters to deduce an IMF. There are some papers investigating the question (for example, you could have a look at Cappellari et al. (2012) for a recent discussion of the problem).


Climate modeling of exoplanets These notes from an exoplanet talk discuss modeling the climate of exoplanets. How would we do that with our current technology? Would data be primarily gathered by spectrum analysis?

The presentation linked to seems to primarily  discuss General Circulation Models, solutions to the equations of fluid dynamics which predict the behavior of a planet's atmosphere, a component of more complete Global Climate Models (confusingly, both are GCMs).

Spectral data on exoplanets is extremely limited to date and poorly constrains the composition, but a lot can be inferred from a planet's orbit and eccentricity (how far from circular the orbit is). At the simplest level, planet temperature depends on how much light it receives from its parent star, and this will vary as a planet goes around a non-circular orbit. Starting from there and making reasonable assumptions about the composition of the planet (usually starting with a solar system analogy) you can test out a GCM model in the various extreme cases we observe (hot Jupiters etc.), to be subjected to future observational constraints.

One exciting new development is looking at a planet's phase curve (the observed brightness of the planet as it orbits its parent star) and using that to infer properties of the atmosphere. One team recently found that Kepler 7b's phase curve  may best be explained by the presence of clouds:

Demory, et al. 2013. “Inference of Inhomogeneous Clouds in an Exoplanet Atmosphere.” arXiv:1309.7894 [astro-Ph] (September 30). http://arxiv.org/abs/1309.7894.


Black Holes emitting Hawking radiation I've read that even light cannot escape from black holes.
The speed of light is theoretically predicted to be the escape velocity of a black hole.
I've also read that the general notion that nothing can escape from a black hole is wrong.  Even black holes emit particles and radiation known as the Hawking radiation.
My question is: How can black holes emit any kind of radiation?
If they even do emit, wouldn't the speed of the radiation be less than or equal to velocity of light?
If the escape velocity of a black hole is c, then how can the Hawking radiation even escape the black hole? Wouldn't it be pulled back into the black hole? Then, how is it even
emitting anything?

The escape velocity is c at the event horizon of a black hole. Above the event horizon the escape velocity is below c. Escaping particles of the Hawking radiation form above the event horizon; that's why they can escape, if they are pointing towards a sufficiently narrow angle to vertical upward, and if they are sufficiently energetic.

Escaping particles form as virtual particle-antiparticle pairs in the "infalling" coordinate system: One of the two particles forms outside the event horizon; the counterpart forms below the event horizon. Thus the originally virtual particles cannot annihilate, and therefore become real particles; one particle can escape; the counterpart falls toward the singularity.

The energy needed to form the escaping particle, and its remaining kinetic energy after escape, is subtracted from the mass of the black hole.

The described mechanism probably works also for both particles forming very close above the event horizon if the tidal forces are high enough to separate the virtual particle pair, before it can annihilate.

Formation of virtual particles is due to the Heisenberg uncertainty applied to time and energy: Very short time intervals require energy uncertainty, leading to short-lived particle-antiparticle pairs.


The Solar System formation I've recently watched one conversation (I won't post a link since the conversation is not in english) involving a physicist (at least he claims so), who spoke about probabilistic estimates of the Solar System formation and the birth of life. Which are:

There are approx. 2000 planetary systems discovered, and all of them, except the Solar System, have angular momentum mostly in a star, which agrees to Kant-Laplace theory (probably it's the Nebular hypothesis). But the Solar System is the only one amongst them with angular momentum mostly in planets. This was sort of explained by the Schmidt's hypothesis (it's about the Accretion as I understand), that the Sun was flying through some debris and "picked" some. But the probability of such an event is very low, given the speeds of participating celestial bodies, read quantity of moving stars with requisite speed (why those stars have that speed is also unknown), and also there should have been supernova explosion at the same time. So the Solar System formation explanation is based on the chain of unlikely events.
Earth is the only planet in the Solar System that has continental crust and oceanic crust (I guess it is important for the water cycle and life formation on Earth in general). The only explanation of that fact also relies on the chain of low probable events:

once Earth had an Lagrangian satellite (near Earth), comparable to Mars in size, which is itself unlikely, since Lagrangian satellites are typically much smaller;
each of them were each other's Lagrangian satellites, which is also unlikely, nothing like that was discovered up to date;
then some unknown power brought out that satellite from Lagrangian point. Such power could have resulted from an event of some wandering star or another massive object moving through the plane of the Solar System, which is nearly impossible since even at the event of galaxies collision, the probability of collision of stars is vanishingly small;
as a result, that satellite "rubbed" Earth in an event of collision, tore off some amount of crust, that was how place for oceans was formed, also the Moon was formed from the resulting debris.

Without the chain of events above the birth of life as ours is impossible, and since the aforementioned planetary systems have some consequences of formation according to Kant-Laplace theory, there is no intelligent life possible there.

Are those statements true?

I'll touch on a few of these because I disagree with some of those ideas.

  There are approx. 2000 planetary systems discovered, and all of them,
  except the Solar System, have angular momentum mostly in a star

There's a few problems with that argument because planets close to their star are easier to identify.  We don't have a complete picture of the solar systems around other stars, so we don't know the full answer to this.

  Earth is the only planet in the Solar System that has continental
  crust and oceanic crust

Again, we don't know.   I think it's interesting that Earth has entire continents of mostly granite and Mars/Venus have almost no granite and maybe that is a byproduct of oceans and water mixing in the crust, or, maybe something else, like the giant impact . . . but we don't know enough about exoplanets to say how common continents of granite might be.    As for plate tectonics, there's some models about that which suggest the size of the rocky planet is a factor.   I think plate tectonics might be fairly common, at least based on models.

  Earth had an Lagrangian satellite (near Earth), comparable to Mars in
  size, which is itself unlikely, since Lagrangian satellites are
  typically much smaller

I have a big problem with this statement because during planetary formation there were perhaps a few dozen planetesimals.    That a planet might have a lagrange point minor planet when planet formation is very young doesn't seem that strange to me, and we also don't actually know if Theia was in a Lagrange orbit, it's often speculated that it was, but nobody really knows.    If Theia was in a Lagrange point, it was only there for about 1/50th of the age of our solar system.   Not all together that strange.

  then some unknown power brought out that satellite from Lagrangian
  point

Orbital perterbations could have done it.   No need for anything so exotic as a wandering or, as I like to call it "just passing through" star.

There are reasons to think that life friendly Earth-like planets might be rare.   Earth's relatively stable orbit, though Earth did spend significant periods frozen, it's still mostly been in the right distance from the Sun for a few billion years.   That requires a very stable orbit and the right size star - too big, the star would age too fast.  Too small, red dwarfs get colder over time, not warmer.   I think a star that grows larger and more luminous works well with a photosynthesis planet which removes CO2 over time.

The perfect combination of events to allow life to grow (it took a billion years just to make the first nucleus), might be rare, but I think events like the giant impact would still happen from time to time.

The James Webb space telescope might provide more information on this subject.  I think for now, it's more speculative and the information that can be observed is very limited.

I disagree with some of the arguments but I agree with the overall premise that planets with Earth's combination of seemingly just right circumstances - oceans of water but not covered by water, continents of granite that basically float over the basalt, an atmosphere that's thick enough to protect the planet but thin enough that it can be changed by life on the planet over time, making a partially Oxygen atmosphere possible.  A magnetic field, plate tectonics and pretty impressive long term orbital and axial tilt stability and relatively few globally catastrophic meteor impacts.     Earth got a lot right and that combination of stuff that went right might be pretty rare.    But conclusions based on assumptions are limited.   We need better telescopes to get real answers and some bigger and better telescopes are on the way.


Could a "burping" supermassive black hole be responsible for a spiral galaxy's look? I read an article which suggests black holes could influence the production of stars in the galaxy as the black hole can burst out powerful X-rays to "snowplow" the surrounding dust and gas.

Although the article didn't suggest that spiral galaxies are made this way, I like to think that a supermassive black hole that "burps" lies in the core of every spiral galaxies. Am I mistaken?

Complementary question: does the supermassive black holes inside the Milky Way still "burp" or it is already cured?

Does the supermassive black holes inside the Milky Way still "burp" or
  it is already cured

Technically, black holes don't burp it's the material that falls into a black hole, as it gets tightly squeezed and under enormous spiral and magnetic forces it shoots out gamma rays at the poles perpendicular to the accretion disk.   More details here.

The super-massive black hole (not holes as there's just one) in the center of the milky way isn't currently "burping" but the next time it eats a star it's expected to "burp" again.   I remember reading that this may happen sometime in the next 10,000 years or so, but I couldn't find the article just now.

"Burping" or Jets of material are more active when galaxies are young.  The quasars observed by large telescope operate in that way.

  I like to think that a supermassive black hole that "burps" lies in
  the core of every spiral galaxies. Am I mistaken?

This is essentially correct.   Quoting this article

  Astronomers believe that supermassive black holes lie at the center of
  virtually all large galaxies

Black holes are surprisingly good at shooting material back into space, strange as that may sound.  Source.


Is there a limit to how hot a star can be? I think that size and mass do not correlate to temperature, but then again these factors contribute to the internal pressure.

I would like to know if there is a limit to how hot a star can get and what mechanism(s) could drive a star to get unusually hot.

I also know that negative temperature occurs in laser is hotter than a positive temperature, and can a star produce negative temperature?

Yes, there is a limit. If the radiation pressure gradient exceeds the local density multiplied by the local gravity, then no equilibrium is possible.

Radiation pressure depends on the fourth power of temperature. Radiation pressure gradient therefore depends on the third power of temperature multiplied by the temperature gradient.

Hence for stability
$$ T^3 \frac{dT}{dr} \leq \alpha \rho g,$$
where $\rho$ is the density, $g$ is the local gravity and $\alpha$ is some collection of physical constants, including how opaque the material is to radiation. Because there must be a temperature gradient in stars (they are hotter on the inside than the outside) this effectively puts an upper limit to the temperature. It is this that sets an upper limit of around 60,000-70,000 K to the surface temperature of the most massive stars, which are dominated by radiation pressure.

In regions of higher density or higher gravity, radiation pressure is not such an issue and temperatures can be much higher. The surface temperatures of White dwarf stars (high density and gravity) can be 100,000 K, the surfaces of neutron stars may exceed a million K.

Of course stellar interiors are much denser and consequently can be much hotter. The maximum temperatures there are controlled by how quickly heat can be moved outwards by radiation or convection. The very highest temperatures of $\sim 10^{11}$ K are reached at the centres of core-collapse supernovae. Ordinarily, these temperatures are unattainable in a star because cooling by neutrinos can carry energy away highly effectively. In the final seconds of a CCSn, the density gets high enough that neutrinos become trapped and so the gravitational potential energy released by the collapse cannot escape freely - hence the high temperatures.

As to the last part of your question, yes there are astrophysical masers found in the envelopes of some evolved stars. The pumping mechanism is still debated. The brightness temperatures of such masers can be much higher than anything discussed above.


Is the expansion of the universe proof of the big bang? Is the expansion of the universe proof of the Big Bang theory?

Or are there other proofs?

In and of itself, no the expansion of the universe is not proof of the big bang. Other theories could be constructed that would also be explained by the expansion. For example Fred Hoyle and others proposed the Steady State Theory which proposed that new matter was constantly being created thus causing the expansion. A the time there was no other evidence for either theory so either could have been true.

However, when you couple the expansion with other evidence such as the cosmic microwave background radiation discovered in 1964 then it does strengthen the case for the big bang being the cause.


Since when are we able to discover interstellar objects similar to 1I/ʻOumuamua? Since when are our observation capabilities good enough to discover objects similar to the first discovered interstellar object 1I/ʻOumuamua (October 2017) with high probability?

Could we make rough estimate how frequent are passes of similar objects through the inner Solar System?

What could be the probability that during this century we will be able to gain more observational knowledge about interstellar objects?

In general terms we have had the observational capability to discover interstellar objects like this for quite some time - before digital imaging, photographic plates were used, which allowed accurate and verifiable measurements to be made. The issue is not so much in the observational and recording capability, but in collecting sufficient data to be able to compute an orbit.

With a few hours of observation you can compute a path that allows further study on subsequent days. This usually assumes a circular orbit as a first approximation, but with increasing elapsed time after the observations the computed path increasingly diverges from the actual orbit.

With observations over a span of days you can compute a better orbit, but again with increasing elapsed time after the observations the computed path increasingly diverges from the actual orbit.

With observations over a span of weeks you can compute an orbit that allows you to find the object again in subsequent years... and so on. Increasing elapsed time correlates to increasing divergence from computed orbits.

So, with a moderate span of observations, we would find that an interstellar object is diverging from its predicted path (orbit around the Sun) and that it is passing through the Solar System instead of being part of it.

But with the greater number of automated surveys (especially compared to pre-digital imaging) that routinely flag asteroid and comet detections, the likelihood of interstellar objects being detected is greater now than in the past.

But someone has to be looking for it. The Minor Planet Center likely has a large number of objects in its database which are actually interstellar objects. But it's set up to correlate detections based on Solar System orbits and the interstellar objects will exist as unconnected short-path observations that don't fit to an orbit. Maybe someone could find more by mining the data, but unless they are still near enough to follow up with further observations these will be unverifiable. But it would be an interesting project!


What effects does the motion of the Sun have on the perihelion precession of Mercury? According to the wikipedia page, there are multiple effects contributing to the perihelion precession of Mercury.
Of course, according to this link and this link we know that the Sun is itself orbiting around a barycenter that sometimes crosses its own surface. Are there any calculations that tell us what effects this has on the perihelion precession of mercury?
Edit: I see now that you can do calculations in a barycentric reference frame or in a heliocentric reference frame. If one chooses to use the heliocentric frame (which is an accelerating frame), one would have to take into account "third body effects" on Mercury.
My question now comes down to the following: do we know whether or not these third body effects were taken into account in the calculation for the 532 arcsec/century value listed in the wiki link?

What effects does the motion of the Sun have on the perihelion precession of Mercury?

A better way to phrase that question is "What effects do the planets have on the perihelion precession of Mercury?"
When calculating the perihelion precession of a planet, one is implicitly working in a heliocentric frame, one in which the Sun is viewed as fixed. Perihelion precession is defined as the precession of the motion of a planet relative to the Sun. The motion of the Sun is irrelevant. The motion of a planet with respect to the barycenter is significantly more complex than is the motion of a planet with respect to the Sun. To illustrate, I'll quote from an answer I gave on physics.SE five years ago:

The following plot shows the distances between Venus and the Sun (red) and Venus and the solar system barycenter (black) from January 1970 to December 2014. The horizontal (time) axis is in days from 12 Noon TT, 1 January 2000.

Note that the red curve, the distance between the Sun and Venus, exhibits a key characteristic of an elliptical orbit, which is a repetitive, nearly sinusoidal distance curve. The black curve, the distance between the solar system barycenter and Venus, does not. It exhibits beats and other nastiness.
One way to model the behavior of a planet orbiting the Sun while acknowledging the presence of other planets is to treat the center of the Sun as the center of an accelerating frame of reference. This results in what aerospace engineers and solar system modelers call "third body effects". The effective acceleration of Mercury toward Jupiter in a heliocentric frame is the gravitational acceleration of Mercury toward Jupiter less the gravitational acceleration of the Sun toward Jupiter.
This approach combined with numerical integration could be used to model the entire solar system. Doing so would have the advantage of not having to worry about where the barycenter is. It has the disadvantage of making an already highly coupled set of differential equations even more highly coupled. That disadvantage outweighs the advantage, making solar system modelers use a barycentric approach when modeling the entire solar system.
Neither approach (numerically integrating the solar system from a heliocentric vs barycentric approach) was used to discover the problem with Mercury's orbit by  Urbain Le Verrier in the 19th century. The numerical integration techniques currently used to model the solar system very much depend on digital computers, something that didn't exist in the 19th century. The number of calculations needed far exceeded the capabilities of the human computers available in the 19th century.
Instead, Le Verrier and others who followed used Lagrange's planetary equations, or variations of those equations to model Mercury's behavior. These equations yield the contributions of perturbing forces (or perturbing potentials) to the time derivatives of various orbital elements. In particular, what is $\dot\omega$, the time derivative of the argument of perihelion, for Mercury?
Le Verrier calculated that the planets would cause Mercury's orbit to precess by 526.7 arc seconds per century. By 1912, Doolittle (and others) had found some issues with Le Verrier's calculations and had refined the Newtonian effects of other planets on Mercury's orbit to a precession of 532.36 arc seconds per century.
Neither Le Verrier's value nor Doolittle's refinement agreed with observation. There was a 43 arc second per century discrepancy between Mercury's observed perihelion precession and the calculated values, which Einstein showed was very nicely explained by general relativity. Note that the relativistic effect is small, less than 10% of the combined planetary effects.

References:
Doolittle, Eric. "The secular variations of the elements of the orbits of the four inner planets computed for the epoch 1850.0 GMT." Transactions of the American Philosophical Society 22.2 (1912): 37-189.


Are there any examples where the transverse doppler effect is applied in astronomy? Are there any astronomical examples where the transverse doppler effect(Horizontal doppler effect) is applied (Derives a meaningful result)?

Accounting for the transverse Doppler effect (and other relativistic effects) is essential in modelling the X-ray spectral emission lines from the accretion discs around black holes (e.g. Cadaz & Calvani 2005). In this case the transverse Doppler effect is "mixed up" with gravitational redshift and it is treated holistically in the Schwarzschild or Kerr metrics.

The transverse Doppler effect is also essential in interpreting the signals from binary pulsars and even in single pulsars because of the rotation of the Earth and its motion around the Sun.


How to cool down a moon? I'm doing an Astronomy project and need to terraform a moon. It is about the size of Mars and orbits a gas giant which is at a distance of 1.28 AU of its star. I though about putting a sunshade at the Lagrange point L1 between the planet and the star, but I don't think that would be consistently cooling the moon (by about 30K). For the purpose of this project, it's surface is mostly water, and the atmosphere is mostly water vapor. Any ideas how to tackle this problem? Any way to raise the albedo, or maybe getting a sunshade to work? Be it at a Lagrange point or in orbit around the moon? I still want to do this project by myself, I could just use an idea or two, maybe some initial help with the calculations. Thanks in advance :)

Assuming the major heat input is from the star (the gas giant may also radiate significant IR) then you could seed clouds or freeze the water in order to increase the albedo.

Alternatively, you could increase the radiative efficiency by somehow painting the side of the moon facing away from the star black. Tricky, as no doubt this hemisphere changes with time quite rapidly.

If tidal heating from the gas giant is significant then you need to get the moon into a wider orbit!

The sunshade at the L1 point idea is widely discussed. Usually it is lenses that are considered, to avoid big problems with radiation pressure. I'm sure you could calculate some appropriate numbers to block a certain percentage of light in your system.


How does the ITRS frame related to ECEF frame? How does the International Terrestrial Reference frame related to the Earth-centered, Earth-fixed frame, as both of them rotate with the Earth?

The simple explanation:
The International Terrestrial Reference (ITR) Frame is an Earth-centered, Earth-fixed (ECEF) frame, but with the axes of the ITR frame precisely specified taking into consideration the movements of the Earth's crust due to plate tectonics.

This website may be useful.


How to construct a sky map First of all, I have very basic knowledge of astronomy so sorry if I make some mistakes in my terminology.

I'd like to plot the Southern Cross. I can find the name of its 5 stars. Also, I have some knowledge of Python and I've tried to use PyEphem and AstroPy.

What I don't know is: what kind of information do I need to plot the cross on the circle? I mean, I'd like to plot it exactly at the right position.

The most difficult part is: the observer should be in Rio de Janeiro and the date should be 1889-11-15 08:30:00 UTC-03.

I tried to collect some info with the Astropy function FixedTarget.from_name() but, if I understood correctly, the date is too far in the past.

If anybody could give me some guidelines (maybe an online tool) to create a sky map with the cross in the right position, I'd really appreciate it.

There are a few details here that make this non-trivial. The year doesn't matter much, since the same stars are visible each year. Further, in Stellarium, you can set the date to 1889 so this is not an issue but we have to turn off atmospheric effects to make the stars visible.

More tricky is that the Brazilian flag is designed as if viewed "from above". We can flip the image from Stellarium to simulate this.

Now, the stars are on a globe and the flag is flat, so a projection is needed to represent it. It's not clear what projection the flag designers used, but perhaps it is "orthographic". Stellarium can do that projection.

The projection also seems to be slanted, so "north-south" isn't vertical, although it's not clear by how much. We can estimate this slant from the position of the equator.

So here is an image of the sky from Rio. The Southern Cross is visible towards the bottom of the image.

If you want to do this yourself in Pyephem you can create a "fixed body" object for each star (Pyephem has a catalogue of brighter stars), create an observer in Rio, and call "body.compute(observer)" which will calculate the azimuth and altitude of the star. Their orthographic projection is then the application of the formulas found on Wikipedia, and you can plot that using any plotting package.


Is there a standardized "Astronomical Alert" system? “Who saw” the binary neutron star merger first? What was the sequence of events? (GRB/GW170817) highlights a particularly notable astronomical alert, but I am sure that there are alerts triggered by various systems sent to various other systems.
Question: Is there a standardized "Astronomical Alert" system? Or are there many decentralized and specialized systems for each of various specific scenarios, such as the one described in Does the Zwicky Transient Facility only run when triggered by a cell phone?
The question occurred to me after posting Will LSST make a significant increase in the rate of astronomical event alerts? and reading the excelent answer(s) there!

In general, no there is not a standardized alert system (unless you count email...). Most subareas of time-domain astronomy that deal with alerts tend to have their own central clearing houses and alert distribution mechanisms. Most of the high energy/explosive (supernovae, novae, blackhole transients (stellar or massive)) transient community tends to announce things via Astronomers Telegram which have basically replaced the older IAU Circulars and the Central Bureau for Astronomical Telegrams for announcing things. More recently, those working on comet outbursts have also sent notifications via ATel e.g. the recent Comet 29P outburst.

The NEO community tends to coordinate through the Minor Planet Center NEO Confirmation Page (NEOCP) for newly discovered NEO candidates found by the surveys. In recent years, this has been supplemented by the JPL/CNEOS Scout system which performs risk analysis on the NEOCP candidates and sends out email alerts for close passing objects or where they may be issues with the measurements - follow-up and discussion on these is done via email.

The Gamma Ray Burst (GRB) community has had the GCN Circulars as previously mentioned which started around the end of Compton Gamma Ray Observatory and the start of BeppoSAX when better location of GRBs become possible. An issue with both ATels and GCN Circulars is that they are written in English text and are sent via email and are mainly designed for human consumption and reaction. This has been recognized as an issue for many years and lead to the development of the VOEvent 2.0 which became an IVOA Recommendation in 2011. Quoting from the Abstract of the document:

  VOEvent [20] defines the content and meaning of a standard information packet for representing, transmitting, publishing and archiving information about a transient celestial event, with the implication that timely follow-up is of interest. The objective is to motivate the observation of targets-of-opportunity, to drive robotic telescopes, to trigger archive searches, and to alert the community. VOEvent is focused on the reporting of photon events, but events mediated by disparate phenomena such as neutrinos, gravitational waves, and solar or atmospheric particle bursts may also be reported. Structured data is used, rather than natural language, so that automated systems can effectively interpret VOEvent packets. Each packet may contain zero or more of the "who, what, where, when & how" of a detected event, but in addition, may contain a hypothesis (a "why") regarding the nature of the underlying physical cause of the event.

It's fair to say that adoption of VOEvent has been slow and limited, driven in part by a lack of easy-to-use libraries or toolkits, but usage is picking up with the LIGO/VIRGO gravitational wave alerts going out in VOEvent XML format (Alerts User Guide) and a standard for reporting Fast Radio Bursts. Sending of VOEvents from the LSST brokers is also planned and so the standard is likely to see greater adoption.


What is the typical resolving power of the largest optical telescopes? I'd like to estimate the typical resolving power of the largest optical telescopes.

I've calculated the theoretical resolving power of the VLA for 21 cm light, $$R=\frac{\lambda}{B}=\frac{2.1 \times 10^{-1} m}{2.7 \times 10^4 m}=1.6 ''$$

And I know that the largest optical telescopes on Earth are GTC, Keck 1 & 2, SALT, VLT, etc.

But I don't know how to calculate a typical resolving power in practice. How can I make this estimate taking into account astronomical seeing and other effects?

The angular resolving power of a telescope is approximately $1.22 \lambda/D \times 648000/\pi$, where this is the Airy disc diameter in arcseconds, $\lambda$ is the working wavelength in metres and $D$ the diameter of the telescope primary mirror in metres. For an 8-m telescope working at 500 nm, this gives an angular resolution of 0.05 arcseconds.

There is then a separate question about whether this theoretical limit is ever achieved because of detector limitations or degradation by blurring due to atmospheric turbulence (seeing).

In practice, at visible wavelengths, seeing always trumps the intrinsic angular resolution and has a typical value of 0.5-1 arcsecond at the world's best astronomical sites. There is no set number for this. It varies with atmospheric conditions from night-to-night or even hour-to-hour. Most observatories publish statistics on this so that you can see what the mean (or median) seeing is. For instance here are some statistics for the Paranal (VLT) site, where the median is about 0.8 arcsec.

Special techniques can mitigate the seeing degradation. You may have heard of adaptive optics - you can analyse the incoming wavefront and send a correcting signal to a tip-tilt or fully deformable mirror that removes the distortion, based either on an adjacent star or a laser guide star. However, these techniques are most effective in the near infrared and are not usually attempted at visible wavelengths. See http://en.wikipedia.org/wiki/Adaptive_optics

Another method is Lucky Imaging - which can be used by amateur astronomers too. Take many pictures with short exposures. If there is a bright point source in the image somewhere, you can use this to select the best images and to shift and stack the images to produce a cleaner final product. The shorter the exposures the better this works. Very close-to the theoretical maximum resolution can be obtained in this way - but it needs a special high speed, low noise CCD and you end up throwing away more than 90% of the images.


UTC to UT1 time corrections Astronomic observations in geodesy.

It is well understood that sun and star positions may be expressed in Greenwich hour angle and declination. Or Right Ascension Declination. And may be published reckoned to Universal Time (UT), or TT.

An approximation of UT can be obtained by apply a correction (DUT) to UTC. That results in an expression of time described as UT1. DUT time offsets are published and broadcast (WWV shortwave radio) in 0.1" increments.

I'm looking for daily precise (better than 0.1") DUT corrections.

Current broadcast DUT correction is -0.2".

I have retrieved IERS Bulletin B, which has daily values of UT1-UTC in ms. Current value is -21.996 ms. Is it just me? Is -21.996 ms = -0.022"?

Or is there something in the IERS Bulletin B that I'm missing?

I think you may be looking at the wrong date? -21.9964 is the first value in the current bulletin B but that is for 3/2/2016. The current bulletin B only seems to go up to 5/1/2016, with -134.3179 as the value. I think you may instead want bulletin A, which shows -0.154825s for today (5/12/2016)?

http://maia.usno.navy.mil/ser7/ser7.dat


Do other planets in our solar system experience eclipses or is this unique to Earth? Here on Earth we get to experience lunar and solar eclipses. Is this pure coincidence that everything happens to line up?

Do other planets in the solar system experience this phenomenon or is this something unique on Earth?

Eclipses are common, but solar eclipses like the ones our moon causes would be unusual since it takes a combination of factors, the angular size of the moon and sun are nearly the same from the viewpoint of Earth so that the moon (depending on where it is in its slightly ellipitcal path around the Earth) can cover the sun nearly exactly.

NASA has a nice picture of an eclipse as seen from the Mars Curiosity rover showing an eclipse by a moon with an angular size less than that of the Sun.

Phobos eclipsing the Sun


Black Hole / Hawking Radiation: Why only capture anti-particle? I may have some specifics wrong here.  If so, don't focus on those.  Just focus on the general thrust of my question.

I "understand" (cough) that particle/anti-particle pairs form spontaneously in space.  I understand that they can form near the event horizon of a black hole, and that one particle can fall in, where as the other particle can barely escape.  I understand that an anti-particle will annihilate with a particle.  What I don't understand is why only the anti-particles of these virtual particle pairs fall into the black hole, while the other ones just manage to escape.  Shouldn't both particle and anti-particle have equal chance to be the one to fall in, or just manage to escape?

It seems that there should be an equal chance of either the particle, or the anti-particle, would be captured while the other "ejected."  So it seems that the black hole should be somewhat steady-state as far as mass change with respect to virtual particles.

Explain?

I thought that the anti-particle was annihilating with "normal" mass inside the black hole? No?

No. First, both particles and anti-particles have "normal" mass (should they have mass in the first place) and "normal" (positive) energy. The distinction between them is either a matter of convention or a question of which type is more common in the universe. Furthermore, for typical-massed blacked holes, the bulk of Hawking radiation would be made of photons, which properly speaking do not even have anti-particles, though one could also say that they are their own anti-particle.

  Shouldn't both particle and anti-particle have equal chance to be the one to fall in, or just manage to escape?

Yes, and uncharged ones do. A smaller black hole would radiate both neutrinos and anti-neutrinos, assuming all neutrinos are massive (otherwise, all black holes would do it already), and a sufficiently small (and thus sufficiently hot) one would radiate both electrons and positrons. Very roughly, a black hole will radiate non-negligible amounts of massive particles when the temperature of the black hole is on the order of the particle mass or greater, in natural units.

  It seems that there should be an equal chance of either the particle, or the anti-particle, would be captured while the other "ejected."

Correct, with a minor exception that if a hot black hole has electric charge, it is more likely to radiate particles of the same sign of charge.

  So it seems that the black hole should be somewhat steady-state as far as mass change with respect to virtual particles.

If either a particle or an anti-particle falls into a black hole, its mass will go up. It doesn't matter. Fundamentally, the "reason" for Hawking radiation is that the vacuum state in quantum field theory is a state of lowest energy, but different observers can disagree about which state is the vacuum. Thus, since particles are fluctuation on top of the vacuum, they can disagree about whether or not there are particles.

I don't think there is a good way to repair the "antiparticle falls in" story except some roundabout appeal to energy conservation: if the escaping particle is real and have positive energy, the one that fell in must have negative energy, and would therefore decrease the mass of the black hole. Unfortunately, that only shows what must happen for the situation to be consistent, not that it does actually happen.

Although with some knowledge of general relativity, one can motivate this slightly further--e.g., for the Schwarzschild black hole, there is energy conservation given by a Killing vector field, which goes from timelike to spacelike at the horizon--so what an external observer considers time/energy would be space/momentum inside the black hole, and momentum is allowed to be negative.


Stellar electromagnetic signatures By using only the electromagnetic signature of a star, could a star be distinguished with reliable accuracy from any other star?
To elaborate a little, say we have a collection of about 200,000 stars. By using EM spectrum alone (and any means of analysis, such as spectral analysis), and by tagging each star with its corresponding EM signature, could there be similarities too close to fail a distinction between any two or more stars?

Now, of course, I know that such a thing is possible. I'm not asking whether it is possible. I just want to know the odds of such a lack of distinction, considering all the factors of stellar evolution. It's either in the ballpark or not.

The spectrum of a star is almost certainly a unique fingerprint. Even though stars are born in clusters, formed from the reasonably chemically homogeneous environment of a giant molecular cloud, there are likely to be small differences in their local environment. Furthermore their formation environments, and the stars themselves at later times, can be polluted by external events (e.g. nearby supernovae). Even if their chemical abundances were identical, their spectra would not be because of differences in their mass, rotation and hence their temperatures.

But, the differences between stars may be smaller than the measurement precision, rendering their spectra effectively indistinguishable.

To give any sort of definitive answer requires some definition of what types of stars are being observed, their temperature, gravity and chemical abundance dispersion, at what spectral resolution and with what signal to noise ratio in what waveband.

At the moment I would say this is impossible for any star in our Galaxy. The differences in the spectra between one star and its most similar "twin" (the word might be appropriate here, as they were likely born in the same stellar nursery) are far smaller than we can distinguish with current data and techniques.