The apparent brightness of a celestial object is the amount of light that we observe here on earth. If there is no absorption of light along the way, then it is related to the intrinsic luminosity of that celestial object through the inverse square law: The apparent brightness of an object goes down with the square of the distance from which that object is observed. As we are bound to earth (and its immediate neighborhood), we can only measure the apparent brightness of a celestial object, not its intrinsic brightness. IF we know our distance from that celestial object, we can then infer its true or intrinsic brightness.
Magnitude is a measure of brightness, both apparent and intrinsic. It was originally introduced around 200BC to organize stars according to visibility: 1 was most visible, and 5 was barely visible. We kept the magnitude ~1 for the stars which appeared brightest, and put the scale on a scientific (i.e. suitable for precise measurement) footing by defining that a difference of 5 in magnitudes correspond to a factor of 100 in the amount of light seen. Thus magnitude is a logarithmic scale: factors are translated into differences. This is useful as the range of magnitudes reaches from -26 for the Sun to +30 for the faintest objects visible with our largest telescopes; this span of 56 magnitudes would correspond to a linear scale spanning 22 powers of ten.
Parallax is our key to distances: the apparent motion of a nearby star in front of the distant background of fixed stars as the earth moves around the Sun in the course of one year. To maximize the baseline for this measurement we measure the apparent position of that nearby star in January and then again in July, getting a slight advantage from the eccentricity of the Earth orbit. This method of determining distances to stars gives rise to one of the most prominent distance units in astronomy: the parsec [pc]. One parsec is the distance at which a star would have a parallax of exactly one second of arc. The absolute luminosity of a star is defined as the apparent luminosity that star presents at a distance of 10pc. Absolute luminosity is typically measured in magnitudes: What magnitude would that star have if it were 10pc away from us rather than the XXpc that it really is. The answer to this question is what astronomers call a stars absolute magnitude, and it measures the star's absolute luminosity (distance does no longer play a role as per definition it now is the same for all stars: ten parsec...). Ground based parallaxes can be determined out to ~100pc (=330ly). Satellites are used to better apply this method to more stars. The method is limited to stars in our cosmic neighborhood; the next generation satellite missions are expected to obtain parallaxes for essentially all the stars that are visible and within our own galaxy.
"Proper Motion" is the motion of a star across the night sky that is not resulting from the apparent motion induced by the motion of the Earth around the sun, which is exploited as parallax in the distance measurements described above. Both types of motion, parallax induced apparent motion and proper motion, may exist simultaneously for any given star. Like the parallax induced apparent motion proper motion is difficult to detect for distant stars. Stellar motion visible as proper motion is perpendicular (orthogonal) to the stellar motion that is detected through blue- or redshift! The "True Motion" of a star is vector sum of its proper motion (which is perpendicular to our line-of-sight) and the motion expressed in the star's blue- or redshift (which is along our line-of-sight).
A star's surface(!) temperature is measured with color filters. Well defined astronomical color filters exist for the ultraviolet (U), blue (B), visible (V), red (R), and infrared (I) region of the spectrum. The difference of intensity measured through two appropriately chosen filters (most commonly U-B or B-V) reflects the temperature as this difference changes with where on the blackbody radiation curve the specific filters happen to measure. As the surface temperature changes, the characteristic blackbody curve moves its maximum around, and the relative emission in the various bands changes accordingly. Resulting from this method of temperature measurement through the difference of magnitudes as measured through the appropriate filters, the temperature axis in the Hertzsprung-Russell diagram is going in the "wrong" direction, i.e. temperature increases to the left rather than toward the right.
Only the nearest and biggest stars can be resolved in optical telescopes. Normally the size (diameter) of a star is estimated from a combination of its (absolute) luminosity and its (surface) temperature: the energy output per unit area increases with the fourth power of the temperature (Stefan's law), and the (surface) area is proportional to the square of the diameter, so that the luminosity (total energy radiated) must be proportional to the square of the radius and the fourth power of the temperature. Thus if we can measure its luminosity and its surface temperature as described above, we can deduce the radius or diameter of a star.
The Hertzsprung-Russell (H-R) diagram shows (intrinsic) luminosity on the vertical axis, and surface temperature on the horizontal axis (with the quirk that the temperature increases to the left rather than to the right; see above for a hint at the explanation...). Each star for which the two quantities are measured can be included as a dot in the Hertzsprung-Russell diagram, which is also called a color-magnitude diagram. Color relates to the surface temperature of a star, and magnitude to its energy output. Remember that to know a star's magnitude (absolute), we need to know its distance from us.
The prominent feature in the H-R diagram is the main sequence, where 90 percent of all stars can be found. It presents a unique relation between temperature and luminosity for the in the stable hydrogen burning phase of their lives. This relationship can be exploited to get distances for stars that are beyond the reach of parallax: this is called spectroscopic parallax. It has nothing to do with the traditional parallax and its apparent motion of a star as the Earth moves around the Sun, which is entirely based on simple geometry. But it extends our ability to measure distance as we can guess a star's intrinsic luminosity from its temperature using the H-R diagram, and then compare it to the measured apparent luminosity to figure out its distance...
Binary star systems are our only access to mass - through Kepler's third law. Most stars are not alone out there in space like our sun is - this is the only aspect in which our Sun is not just an average star. Most stars are gravitationally bound to other stars in binary systems or even systems with more than two stars orbiting each other. Visual binaries can be directly observed to orbit each other - so they must be close enough that our telescopes can resolve them as two stars. At larger distance we cannot separate the two stars in our telescopes, and we identify them as binary through changes as the spectral lines oscillate between blue- and redshift, and/or the two stars align to momentarily reduce the total amount of radiation (luminosity) we see as one obscures the other in our line-of-sight.
Using the mass information gleaned from the binaries, we can figure out what the mass hierarchy along the main sequence is. The high mass stars are also the high luminosity, high temperature ones. Lower luminosity and temperature on the main sequence also mean lower mass. Who lives longer? The high mass ones certainly have more fuel available, but they also spend it at a much faster rate as witnessed by their (surface) temperature and luminosity. So their lifetime in fact is much shorter. How can we verify this experimentally (actually: observationally...)? By looking at clusters of stars that were born from the same cloud, i.e. at the same time. Depending on the age of the cluster we can clearly see how the highest luminosities disappear first from the main sequence population in a cluster's H-R diagram. Calculation shows that the lifetime goes roughly as the mass to the power of -2.5. Mind the minus sign in that exponent.
The interstellar medium contains the primordial elements Hydrogen (H) and Helium (He) as well as dust particles made mostly from Carbon (C), Oxygen(O), and Silicon (Si). They are up to 1 micrometer "large" and scatter light of comparable wavelength quite efficiently. Larger wavelength (infrared and radio) can penetrate these dust clouds. Passing through this interstellar dust changes the spectrum of starlight: shorter wavelength are more readily scattered, reducing the amount of blue light in the spectrum more than the amount of red light. So the blackbody radiation spectrum is changed by the passage through the interstellar medium, an effect that has to be accounted for if we want to measure the right surface temperature of a star.
Gravitation slowly pulls larger dust clouds together to form denser clouds that for example obscure much of our view of the center of our own Milky Way galaxy. If the cloud is dense enough, light can simply not pass through any more. Infrared light and radio signals may still be able to penetrate the cloud though, also IR will also be attenuated to some degree. This preferred scattering of shorter wavelength (i.e. "blue" light) over longer wavelength (i.e. red light) is responsible for the fact that reflection nebulae appear blue. The fact that emission nebulae are red comes from the red Hydrogen alpha emission line at 656nm. It is indicative of a "hot" environment where a lot of Hydrogen is in an excited state. This excitation of Hydrogen typically is produced by the intense radiation of a nearby bright star. Radiation from stars born at the center of such dust clouds clears out the dust in a bubble around the radiating stars.
In the regime of radio wavelength cold hydrogen can be measured with ground based instruments (i.e. through the atmosphere) by its 21cm emission that is associated with an electron spin flip in the isolated Hydrogen atom. It does not happen in molecular Hydrogen. This so-called 21cm-radiation produced the first insight into the large scale structure of our galaxy: it revealed that our Milky Way galaxy is a spiral galaxy by mapping the cold atomic Hydrogen in the galaxy's interstellar medium.
Star formation happens when a cold dust cloud finally contracts (collapses) under the influence of its own gravity. This is thought to be triggered by an external event, for example a shock wave from a stellar explosion moving through and around the dust cloud.
Small imbalances in the dust concentration throughout the cloud lead to fragmentation: as it contracts, the cloud does not do so uniformly all moving toward one center, but local overdensities have stronger gravitational pull on the surrounding material than another overdensity further away. So the cloud breaks up into fragments as local overdensities collect the material in their respective neighborhood. These cloud fragments then collapse to form individual protostars. Stars are therefore born in clusters.
Gravitational infall releases energy; this energy gets transfered into heat. The gas/dust heats up as it contracts. Heat means internal random motion, and the bouncing off each other of the randomly moving particles in the gas that comes with heat produces the pressure that will ultimately counterbalance the relentless inward pull of gravity.
Initially the heat from gravitational infall can very easily be radiated off through the still not very dense material that is collecting to form a protostar. But as the density increases, the radiation gets trapped on the inside and can only escape from a surface at a certain distance from the center: a photosphere has developed. Now the temperature really starts to rise on the inside.
To conserve angular momentum, the dense region on the inside is shaped like a disk. As the protostar heats up at the center of that disk from gravitational infall the radiation pressure from that center increases, and jets may form where radiation pushes material out along an axis perpendicular to the disk.
The protostar enters the Hayashi-track or T-Tauri phase, where violent surface activity goes hand in hand with further contraction and only a slight increase in surface temperature as more and more of the heat generated from gravitational infall gets trapped further inside. As the surface area shrinks at almost constant surface temperature, the luminosity declines. The Hayashi-track is almost vertical in the H-R diagram. There is NO fusion going on yet! All energy heating the inside of our protostar is generated from the gravitational energy of the infalling matter.
At the bottom of the Hayashi-track contraction of the star finally the core is hot enough to start fusing Hydrogen into Helium. Finally it has an internal heat source and is a real star. It takes some time still to adjust and bring the internal pressure from heat and its repercussions on the fusion reaction rate into equilibrium with the steady inward pull of gravity. Once this is achieved, the star has settled into the main sequence, where it will now burn Hydrogen to Helium at its core for the vast majority of its lifetime. As with this time spent on the main sequence, the time it takes to settle onto the main sequence is much longer for small masses than it is for large mass stars.
The evolutionary track is the path a star follows in the H-R diagram over its whole life. It is completely determined by mass. During its time on the main sequence a star stays largely where it started. It will move up in both surface temperature and luminosity only very slightly over that long time on the main sequence.
Brown Dwarfs are failed stars, meaning that they never get hot enough to start fusing Hydrogen into Helium. Their mass is too small. If their mass is at least twelve times that of Jupiter, they will experience a very brief period in which they fuse the small amount of primordial Deuterium (Hydrogen with and extra neutron in the nucleus) that they have into Helium. But again: they are not massive enough to start fusing normal Hydrogen into Helium and run out of Deuterium fuel very, very shortly. Below 12 Jupiter masses not even Deuterium fusion is possible.
After burning its Hydrogen near the stellar core a typical solar mass star looses its long maintained equilibrium between outward pressure from the heat generated by the fusion at its core and the inward gravitational pull on its mass. As the core fills up with the Helium ash from the Hydrogen fusion process it does no longer produce the energy needed to maintain the pressure. The core shrinks and heats by gravitational infall. Helium cannot yet be fused into Carbon, and so no heat from fusion opposes the infall under gravity, which greatly heats the core and thus increases the fusion rate in the shell of Hydrogen fusion around this non-fusing Helium core. The extra heat from the gravitational infall and the increased Hydrogen fusion make the outer layers of the star expand and cool down in the expansion process. Expansion means larger surface area to radiate from, cooling means less energy radiated per unit area, and together the two effects keep the luminosity constant while the surface temperature drops: The star entered the subgiant branch in the H-R diagram.
At some point the energy transport in the outer layers becomes convective, at which point the temperature of the surface stops dropping further. As the expansion is still driven by increased Hydrogen shell burning, the luminosity now goes up. At this stage the surface activity is very turbulent, and a significant fraction (up to 30 percent) of the star's mass can be ejected into the surrounding space.
As the non-fusing Helium core keeps collapsing, it hits the limit where it is supported against further contraction by electron degeneracy. Now the pressure is independent of the temperature, and maintained by the quantum nature of the fermionic electrons rather than the internal motion of the plasma particles. At some point the core does get hot enough to start fusing three Helium nuclei into one Carbon nucleus. Energy is generated again and the normal reaction would be to react by expansion and thus cool down and control the fusion rate a little. But the core is supported by electron degeneracy and its size independent of the temperature and pressure. So the Helium fusion dramatically increases the temperature until pressure takes over from the electrons again and the subsequent expansion of the core cools the Helium burning to a more manageable rate. As the outer layers contract and heat in response to this, the star actually looses luminosity and enters the horizontal branch in the H-R diagram. Where exactly on the horizontal branch it settles depends on the amount of mass lost during the red giant phase. Stars of more than two solar masses reach the temperature necessary to fuse Helium into Carbon before the core gets to be supported by electron degeneracy and thus can use the normal regulation mechanism of temperature and pressure to avoid the helium flash phenomenon. Lower mass stars never reach the temperature to start fusing Helium.
As the Helium is burned at the core, Carbon ash builds up just like before the Helium ash from the Hydrogen burning. Again it cannot fuse and the carbon core contracts, heating up the surrounding shell of Carbon fusing Helium, which itself is surrounded by a Hydrogen burning shell. Again the outer layers are expanding, and the star ascends the asymptotic giant branch.
Stars with less than about eight solar masses cannot reach the temperatures necessary to fuse Carbon. The Carbon core shrinks under its own gravity until it is supported by electron degeneracy again. The outer layers are still fusing Helium and Hydrogen in their respective layers, until they become oscillatory and unstable and are blown off the Carbon core to form a planetary nebula. The name "planetary nebula" is a misnomer surviving from times when we did not understand what is going on that well.
The non-fusing Carbon core is now naked and glowing white from the heat stored in it: it is a white dwarf (WD). As it cools by radiating of this stored energy it slowly turns into a black dwarf. As stated before it is supported by electron degeneracy and does therefore not change its size any more, even as it cools.
Most stars are in (at least) binary systems. The concept of a Roche Lobe describes the area where each of the stars' gravitational influence dominates. The Roche Lobes of the two stars in a binary system touch in the point where the gravitational pull of the two stars is identical. If e.g. during the giant phase of stellar development one star's outer layers leave the star's Roche Lobe and enter into the companion star's Roche Lobe, mass gets transfered from these outer layers to the other star. Such effects would certainly alter the development of both stars, the one that looses mass and the one that gains mass. If the two stars in the binary system are sufficiently separated, no mass transfer will ever happen and the two stars will evolve just as if they were isolated stars rather than partners in a binary system.
An isolated WD is pretty much the end of that star's evolution - if one does not count the slow cooling to Black Dwarf status, which is rather booring. If a WD can wrestle some matter from a binary companion that overflows its Roche Lobe as it ascends the giant branches, it may give rise to one or more nova explosions. Please always remember that there is a fundamental difference between novae and supernovae: The latter destroy either an Iron core or a WD while the nova leaves the WD intact. As material from the companion falls under the WD's gravitational spell, it necessarily has angular momentum, as living in a close binary system where overflow can happen means rotating around each other rather rapidly. So the infalling matter accretes in a disc around the WD, and only falls onto its hot surface from the innermost edge of that disc. As Hydrogen slowly builds up in the strong gravitational field at the surface of our WD, it reaches a point where H-fusion ignites and proceeds explosively at the WD surface, blowing the inner parts of the disk and most of the material at the surface into the surrounding space: A nova has occured. As the WD does survive this explosion at its surface, it can continue to accrete after the spectacle is over and do the whole sequence again. As with every explosion its mass grows a little bit (not all material is blown off into space), it may ultimately be pushed towards the Chandrasekhar limit of 1.4 solar masses, at which a WD would become unstable and collapse into a neutron star (see below). Before the WD has grown to the Chandrasekhar limit though it will explosively start C-fusion and blow itself to pieces in a type one supernova which leaves no remnant at its core. Remember that WD are supported by electron degeneracy, so the heat released in the fusion process does not lead to an expansion of the WD, leading to the catastrophic nature of this particular runaway fusion reaction. No outer shell is there to stop the exploding material from forever expanding into space, as is the case with the Helium flash in solar mass stars. The accretion disk does not matter much against the violence of the explosion as it does not cover much of the sky surrounding the WD.
Stars with more than eight solar masses go on to fuse the Carbon at their cores and continue to fuse ever heavier elements as the ashes of each earlier fusion process build up in the core. Each new fusion process gets its own layer in the onion type structure, and it takes ever shorter times before a new layer is added. Iron is the most stable element, and no more energy can be won by adding Helium nuclei to the Iron nucleus. Here the fusion stops and the Iron core is being compressed and heated by gravity. The heat gets so large that its associated radiation starts splitting up the nuclei: photodisintegration. All the nuclei are destroyed, and we have just electrons, protons, and neutrons again. Instead of restarting the fusion of protons (Hydrogen) into Helium again the pressure is now so great that the protons combine with the electrons to form neutrons and thus circumvent the barrier being erected by electron degeneracy in the core contractions discussed earlier. Both processes, the splitting of the most stable Iron nuclei and the combining of electrons and protons to form neutrons (and a neutrino that will escape and take even more energy out of that collapsing core) use energy to proceed, thereby cooling the core and further undermining its ability to counter the relentless force of gravity with thermal motion. The quantum mechanical neutron degeneracy is the analog to the electron degeneracy we encountered earlier, but allows matter to be much further compressed than electron degeneracy does before it puts an absolute stop to further contraction. The stop is so abrupt and absolute that it leads to a tremendous "bounce" as all the infalling matter behind the neutrons hits that brick wall and literally bounces back from it. The result is a tremendous explosion that blows away the outer layers of the star, ejecting them into a rapidly expanding bubble called a supernova remnant. So the supernova remnant is not the neutron star left at the center of this supernova type two explosion, but the expanding bubble of hot gas that consists of the blown off outer layers of the original star. As it turns out from measurements on binary systems where one star exploded into a supernova of type 2, most neutron stars have a mass of about 1.4 solar masses: the Chandrasekhar limit for contracting into a neutron star, where gravity becomes so strong that it forces neutronization to have its way and contract matter to the next quantum mechanical limit, that of neutron degeneracy.
Where are the heavy elements made? They are bred in the cores of heavy stars, but then photodisintegrate again as their Iron cores collapse... So they are not made by the alpha-process (the capture of He nuclei on other nuclei) at the cores of heavy stars. In fact they are made in the hot outer shells of red giants, through the "slow" process, the s-process: neutrons are added one by one to existing nuclei. The process is so slow (order one capture per year per nucleus) that radioactive decay can proceed in between, changing neutrons in those nuclei back to protons and thus moving up all the way to the chemical element of Bismuth. Beyond Bismuth the slow process cannot make any heavier elements as they would radioactively decay on much shorter timescales than the year it takes to add another neutron back on. The elements up to Uranium can only be made in a supernova explosion: There are so many neutrons around in that explosion, that more neutrons can be added to even the heavier elements before they have a chance to decay again. This is called the r-process, for "rapid" neutron capture. As they are made in the outer shells of the dying stars, the heavy elements are then blown out into the surrounding space as either a planetary nebula for lower mass stars or a supernova remnant for massive stars. Still the vast majority of the matter blown into space there is Hydrogen and Helium. But the heavier elements do exist now, and a next generation of stars born from a cloud of material that is gathered together from these nebulae and remnants will have heavy elements in them from the start. And it will have the heavy elements available to build rocky planets like Earth.
Neutron stars are very compact indeed: they pack 1.4 solar masses into a sphere that has a diameter of only about 20 miles. The conservation of angular momentum means that they must be spinning very fast. Also at least part of the magnetic field of the original star must have shrunk with the collapsing core, and if magnetic field lines are concentrated to pass through a smaller area, that means that the field strength is greatly enhanced. So newly born neutron stars are spinning very rapidly and have very strong magnetic fields. As charged particles from the surrounding space are pulled into the neutron star by gravity, they will have to follow the magnetic field lines and therefore are funnelled towards the magnetic poles. As they strike the surface there at the magnetic poles, electromagnetic radiation up to gamma ray energies is generated and leaves the neutron star in narrow beams emanating from the magnetic poles. As the magnetic axis is not normally aligned with the rotation axis of the neutron star, these beams sweep out a cone in space. If Earth happens to sit in that cone, we will see the radiation signal sweeping across our detectors once every rotation of the neutron star. We call this phenomenon a pulsar, as we see its beamed radiation as a pulse of radiation every time the beam sweeps across our detectors. Typical periods for the rotation of these newborn pulsars are second to tenth of seconds. As energy is radiated off in this process, the pulsar is slowly loosing energy and angular momentum, and will ultimately stop to be a pulsar as it retires to neutron star status. At birth all neutron stars are pulsars though; we may not be in the path (cone) of their beam though.
Microsecond pulsars are not as easily explained. If a 20 mile object turns in microseconds, its outer layers must move around at a significant fraction of the speed of light. How could they possible be spun up that fast without the centrifugal forces preventing the infall to start with? The answer lies in binary systems again: The slow influx of matter syphoned off a giant companion growing beyond its Roche Lobe comes with angular momentum, which is thus continually added after the neutron star was born, maybe even after it spun down from its original pulsar activity already.
Accreting matter on a WD led to the phenomenon of novae. On neutron stars, it leads to x-ray bursts. As H-fusion ignites briefly, x-rays are emitted from the whole surface of the neutron star, now being visible irrespective of the particular orientation of its beams. As in the case of novae, this process can be repeated. The repetition rate here is much faster though; for novae it is on the scale of years, for x-ray bursters on the scale of hours.
The most massive stars pack much more mass in their collapsing cores and subsequently emerging neutron stars than is required by the Chandrasekhar limit. For every mass we can calculate a theoretical radius where if the mass was concentrated in a sphere smaller than that radius gravity on that radius would be strong enough to keep light from escaping. Gravity would pull back even light trying to emerge from anywhere inside this radius, pull it back towards the center of that sphere. Why is that significant? Light particles (photons) are the only particles that have no rest mass, which means they cannot be at rest and always have to race around at the speed of light. As such they are the fastest particles (all particles with rest mass have to be slower), and the particles that gravity has the least hold on, as gravities ability to influence a particle is dictated by its mass. Let me remind you: Photons are subject to gravity, as they have energy and thus acquire a so-called dynamical mass. But as stated before: Of all particles photons - light particles - are the ones that gravity has the least sway over. So if they can no longer escape the pull of gravity, nobody else can. So if we can pack the requisite amount of mass inside the radius where its gravity becomes strong enough to hold back even light, no signal can reach us from that mass. We cannot figure out anything about its state or doings any more. As neutron stars become more and more massive, their radius grows slower than this theoretical radius, which we call either Schwarzschild radius after a scientist who first studied the phenomenon, or "event horizon" to allude to the fact that nobody can see beyond that horizon. The most massive neutron stars will be hidden behind this event horizon. Are they still neutron stars or does matter take on other forms under those extreme conditions? We cannot know: they are beyond a horizon from which we could receive any information about them. They are in a Black Hole (BH). Things (and radiation) can fall into a BH, but not get out any more. And no receipt can be sent for their reception from beyond the event horizon, from inside the Black Hole.